Modeling the material removal rate in ultrasonic machining of titanium using dimensional analysis

ORIGINAL ARTICLE

Modeling the material removal rate in ultrasonic machiningof titanium using dimensional analysis

Jatinder Kumar & J. S. Khamba

Received: 21 June 2008 /Accepted: 27 August 2009 /Published online: 13 September 2009# Springer-Verlag London Limited 2009

Abstract Titanium is known as the metal of the futurebecause of its excellent combination of properties such ashigh strength-to-weight ratio, low thermal conductivity, andhigh corrosion resistance. Machining of titanium, however,is considered as cumbersome with the conventionalmanufacturing practices, and there is a critical need ofdeveloping and establishing cost-effective methods ofmachining. This investigation is focused on exploring theuse of ultrasonic machining, a nontraditional machiningprocess for commercial machining of pure titanium (Amer-ican Society for Testing and Materials grade-I) andevaluation of material removal rate under controlledexperimental conditions. The optimal settings of parametersare determined through experiments planned, conducted,and analyzed using Taguchi method. An attempt has beenmade to construct a micro-model for prediction of materialremoval rate in ultrasonic machining of titanium usingdimensional analysis. The predictions from this model havebeen validated by conducting experiments. The microstruc-ture of the machined surface under different experimentalconditions has been studied using scanning electronmicroscopy. A relation was established between the modeof material removal and the energy input rate correspondingto the different process conditions.

Keywords Titanium . Ultrasonic machining .

Material removal rate . Taguchi method .Micro-model .

Dimensional analysis

1 Introduction

Titanium and its alloys are alternative for many engineeringapplications due to their superior properties such aschemical inertness, high strength, stiffness at elevatedtemperatures, high specific strength, excellent corrosionresistance, and oxidation resistance. However, these prop-erties also make titanium and its alloys difficult to shapeand machine into a precise size and shape. As a result, theirwidespread applications have been hindered by the highcost of machining with current technology [13, 27]. Themachining characteristics for titanium and its alloys usingconventional machining processes are summarized below[10]:

– Titanium and its alloys are poor thermal conductors. Asa result, the heat generated when machining titaniumcannot dissipate quickly; rather, most of the heat isconcentrated on the cutting edge and tool face. About50% of the heat generated is absorbed by the tool whilemachining titanium alloy (Ti-6Al-4V) [1].

– During machining, titanium alloys exhibit thermalplastic instability that leads to unique characteristicsof chip formation. The shear strains in the chip are notuniform; rather, they are localized in a narrow band thatforms serrated chips [28].

– The contact length between the chip and the tool isextremely short (less than one third the contact lengthof steel with the same feed rate and depth of cut). Thisimplies that the high cutting temperature and the high

J. Kumar (*)Department of Industrial Engineering,National Institute of Technology,Kurukshetra, Indiae-mail: [email protected]

J. S. KhambaDepartment of Mechanical Engineering,University College of Engineering, Punjabi University,Patiala, India

Int J Adv Manuf Technol (2010) 48:103–119DOI 10.1007/s00170-009-2287-1

stress are simultaneously concentrated near the cuttingedge [6].

– Serrated chips create fluctuations in the cutting force;this situation is further promoted when alpha-betaalloys are machined. The vibrational force, togetherwith the high temperature, exerts a micro-fatigueloading on the cutting tool, which is believed to bepartially responsible for severe flank wear [30].

– The surface finish achieved by a single machiningprocess is poor.

Therefore, there is a crucial need for reliable and cost-effective machining processes for titanium and its alloys.Over the last few decades, there have been great advance-ments in the development of cutting tools, including coatedcarbides, ceramics, cubic boron nitride, and polycrystallinediamond. These have found applications in the machiningof cast iron, steels, and high temperature alloys such asnickel-based alloys and super alloys. However, none ofthese newer developments in cutting tool materials has hadsuccessful application in improving the machinability oftitanium alloys [6]. Most cryogenic machining studies ontitanium and its alloys have documented improved machin-ability when freezing the workpiece or cooling the toolusing a cryogenic coolant. However, inherent weaknessesexist in these approaches as well [10].

Machinists have developed a few methods for commer-cial machining of pure titanium in manufacturing industriesall over the world. Most of the machining work for titaniumis related to drilling, so twist drilling and vibration-assisteddrilling are two conventional machining methods beingused in recent times. For other machining operations,special considerations are to be taken care of whilemachining titanium parts on a machine tool. Titanium andits alloys are very sensitive to changes in cutting speed.Industry generally operates at cutting speeds providinglonger tool life. Moreover, because of the bouncy actiongenerated due to low modulus of elasticity of titanium, therigidity of a machine tool becomes an important consider-ation [1, 6]. The average unit power requirements forturning or milling of titanium have been found to be muchlower than that for high temperature Ni/Co-based alloys ortool steel grades. As far as the cutting tools are concerned,the straight tungsten carbide (WC) cutting tools, typicallyC-2 grades, perform best in operations such as turning andface milling, while the high-cobalt, high-speed steels weremost applicable in drilling, tapping, and end milling [30].Economic production techniques developed for titaniumand its alloys are based on a few general rules which havebeen summarized as:

& Use low cutting speeds. Tool tip temperatures areaffected more by cutting speed than by any other singlevariable. A change from 6 to 46 m per minute with

carbide tools results in a temperature change from 427°Cto 927°C.

& Maintain high feed rates. Temperature is not affected byfeed rate so much as by speed, and the highest feedrates consistent with good machining practice should beused. A change from 0.05 to 0.51 mm per revolutionresults in a temperature increase of only 149°C.

& Use generous amounts of cutting fluid. Coolant carriesaway heat, washes away chips, and reduces cuttingforces.

& Use sharp tools and replace them at the first sign ofwear, or as determined by production/cost considerations.Tool wear is not linear when cutting titanium. Completetool failure occurs rather quickly after small initial amountof wear takes place.

& Never stop feeding while a tool and a workpiece are inmoving contact. Permitting a tool to dwell in movingcontact causes work hardening and promotes smearing,galling, seizing, and total tool breakdown.

Despite the establishment of effective machining meth-ods using conventional technology, lower tool life, andpoor surface quality are two major concerns that continue tobe associated with the machining of titanium components.Besides this, poor surface integrity of conventionallymachined titanium parts is another area where moreconcentration is required.

Nontraditional machining processes such as electricdischarge machining and laser beam machining have beenapplied for drilling holes in workpieces made from titaniumand its alloys, but even these processes have their ownlimitations; the most prominent are the surface finish anddimensional inaccuracies besides their undesirable effectson the machined surface such as heat affected zone, recastlayer, and thermal stresses [16]. These adverse effects canlower the working life of the components critically. Loss offatigue strength and hence surface integrity is anotherproblematic area in machining of titanium. The basicfatigue properties of many titanium alloys rely on afavorable compressive surface stress induced by tool actionduring machining [1, 6, 28, 30]. Ultrasonic machining(USM) could be another alternative machining process thatcan be applied commercially to the machining of titanium,as this process is known to be free from all these adverseeffects on the machined component, and the repeatedimpacts of abrasive grains on the work surface lead to afavorable compressive surface stress thereby improving thefatigue life of titanium components along with the surfaceintegrity [4, 20]. However, there is critical lack of evidencefor the application of USM for machining of titanium in theliterature available till now. Hence, in the present investi-gation, USM has been explored as an alternative machiningmethod for pure titanium (American Society for Testing and

104 Int J Adv Manuf Technol (2010) 48:103–119

Materials (ASTM) grade-I). The material removal rate(MRR) in USM of titanium has been evaluated underestablished experimental conditions. Taguchi’s method foroffline quality control has been used to plan and analyze theexperiments. The optimal process settings have beenidentified, and the macro-model for MRR has beenconstructed. The macro-model thus obtained has been usedto develop a micro-model for prediction of MRR over awide range of input parameters. A comparison of thepredictions from the micro-model with the experimentalresults has been made for its validation.

2 Literature review

To identify the potential factors affecting MRR in USM, acause-and-effect diagram was constructed (Fig. 1). As thediagram indicates, the MRR in USM is dependant on fourprimary factors: workpiece, tool, slurry, and machine-related factors.

Various investigators [3, 7, 8, 12, 18, 19, 21] havereported results indicating that the rate of material removalfor a certain abrasive is a function of its concentration,grain size, and hardness besides the feed system. Onincreasing the abrasive grit size or slurry concentration, anoptimum value of MRR is reached. Any further increase ineither aspect results in difficulty in the larger grainsreaching the cutting zone [3, 18] or a subsequent fall inMRR. Guzzo and Shinohara [8] reported a substantialincrease in MRR obtained while using abrasive of largergrain size on account of the increase in the stress caused bythe impact of abrasive particle over the workpiece surface.Neppiras [19] and Markov [18] reported that when grainsize is comparable to the amplitude of vibration, theoptimum level of MRR can be reached. Experimentally,

the ratio of the double amplitude to the mean size of theprincipal fraction of abrasive is 0.6 to 0.8. Goetze [7] hasreported the optimum value of slurry concentration to beclose to 12% (by volume) for all the abrasive grit sizes usedin the investigation. The optimum concentration is thoughtto be one providing a single layer of abrasive over the entirework surface [16].

The amplitude of vibration (ξ) has been found to affectthe machining performance of USM [4, 10]. Higheramplitude is obtained by using a tool with a largertransformation ratio, i.e., the ratio of transducer/tooldiameter [27]. Smith [26] showed that MRR is proportionalto ξ3/4, while other researchers [7, 21] have advocated thatMRR is linearly proportional to ξ, and yet others [12, 18,19] have suggested that MRR depends upon ξ2 for constantfrequency and static load conditions. Experiments con-ducted by Neppiras [19] have shown that in the range of 20to 50 kHz, the removal rate is proportional to square root ofthe vibration frequency. However, Kazantsev [12] reportedthat the abrasion rate is proportional to the frequency, whilethe non-linear frequency dependence of machining rate isdue to the variation in abrasive concentration in theworking zone.

It has been reported that the machining rate is directlyproportional to the tool form [13, 27] and shape factor (ratioof tool perimeter to tool area). The tool form defines theresistance to slurry circulation: a tool of narrow rectangularcross-section yielding a better machining rate than one witha square cross-section of the same area [4, 13, 27]. Use ofhollow tools has been reported to result in higher rates ofmaterial removal than ones with solid geometry for thesame area of the cross-section [27]. Komaraiah and Reddy[15] investigated the influence of tool material properties,i.e., hardness on the MRR in USM of glass. The differenttool materials were arranged in the increasing order of

Fig. 1 Cause-and-effectdiagram for material removalrate

Int J Adv Manuf Technol (2010) 48:103–119 105

superiority as mild steel<titanium<stainless steel<silversteel<niamonic-80 A<thoriated tungsten. Also, the MRRhas been found to vary in a linear proportion to thehardness of the tool being used [15, 21]. Tools withdiamond tips have been shown to have good materialremoval characteristics [21].

It has been concluded that productivity by USM (interms of machining rate) is primary determined by thebrittleness of the work material [7, 13]. The plasticity ofwork material is associated with low productivity. Theimpact hardness has been found to have an adverse effecton machining rate. However, while machining annealedsteel, the machining rates observed have been found to besignificantly better than normalized or quenched ones [16].Guzzo and Shinohara [8] outlined the ultrasonic abrasion ofdifferent hard and brittle materials using stationary USM.Results show that machining rate decreased with increase inhardness of the work material. Similar results were reportedby other investigators [17, 26].

The literature review reflected that most of the researchwork carried out by different researchers focused on theimprovement of process efficiency and efficacy whilemachining hard and brittle materials. The application ofUSM for machining of relatively tough materials (such astitanium) has been explored by a few researchers [5, 23–25]. However, most of this work has been concentrated onthe application of USM for machining of titanium alloy.Almost no effort has been put forward to investigate themachining characteristics of commercially pure titaniumgrades using USM. Singh and Khamba [24, 25] haveinvestigated and modeled the machining characteristics oftitanium alloy (ASTM grade-V) and pure titanium (ASTMgrade-II) using static USM apparatus. In USM, the propertiesof work material such as hardness, toughness, and impactstrength play a significant role in the variation of machiningcharacteristics (MRR, TWR, and surface roughness). Puretitanium (ASTM grade-I) differs from titanium (ASTMgrade-II) as well as titanium alloy (ASTM grade-V) to asignificant extent (Table 1). From the comparison of themechanical properties of the three grades, it is evident thatpure titanium (ASTM grade-I) possesses considerably lowervalues of tensile strength and hardness; hence, there is acritical need to assess its machining behavior with a processlike USM. Hence, pure titanium (ASTM grade-I) wasselected as work material for the present investigation.

Hu et al. [11] presented the modeling of MRR in rotaryUSM of alumina-based advanced ceramics. An approach tomodel the MRR during rotary USM of ceramics wasproposed and applied to predict the MRR for the case ofmagnesia-stabilized zirconia. In this investigation, a five-factor two-level factorial design was used to study therelationship between MRR and the controllable machiningparameters. The model developed had practical application T

able

1Chemical

compo

sitio

nandim

portantprop

ertiesof

titanium

andits

alloy

Chemical

compo

sitio

n(bywt.%

)of

titanium

(ASTM

grade-I)

Chemical

compo

sitio

n(bywt.%

)of

titanium

(ASTM

grade-II)

Chemical

compo

sitio

n(bywt.%

)of

titanium

(ASTM

grade-V)

ON

CH

Fe

Ti

Balance

ON

CH

Fe

Ti

ON

CFe

Al

VTi

0.18

0.03

0.08

0.01

0.2

99.1

0.4

0.14

0.01

40.00

60.00

070.05

99.78

0.15

0.05

0.06

0.25

5.8

4.1

89.3

Yield

streng

th22

0MPa

491

MPa

Ultimatestreng

th34

0MPa

650

MPa

910

MPa

Hardn

ess

115

HV

142

HV

187

HV

Density

4.51

g/cm

34.45

g/cm

34.4

g/cm

3

Mod

.of

elasticity

103

GPa

108

GPa

114

GPa

Fracturetoug

hness

68Mpam

1/2


to USM of extremely hard and brittle materials such asadvanced ceramics.

Wiercigroch et al. [29] presented a model for predictionof MRR in ultrasonic drilling of hard materials using impactoscillator approach. Micro-cracking of work material due toimpact of grains was assumed to be the material removalmechanism while constructing the model. However, themodel was applicable only to hard materials. Moreover, theassumption of uniform wear over all the surface of the toolalso proved to be false, as a non-uniform wear pattern wasobserved for the tool (over the length as well as the cross-section of the tool).

In the present investigation, an attempt has been made tomodel the MRR in stationary USM of commercially puretitanium (ASTM grade-I), a relatively tough and ductilematerial. Buckingham’s pi theorem has been used for thedimensional analysis and hence construction of the model.The model developed is mechanistic in the sense that theoutcome of the macro-model can be used to predict theMRR over a wide range of parameters. Also, the model isbased on realistic assumptions (refer section 7), and thechange in process conditions such as tool geometry andslurry concentration has been taken into account whileconstructing the model. Moreover, the predictions from themodel have been found to agree well with the experimentalresults (Figs. 6, 7, and 8).

3 Materials and methods

Commercially pure titanium (ASTM grade-I) has been usedas the work material in the present investigation. Fivedifferent tool materials were used: high carbon steel, highspeed steel, cemented carbide, titanium (ASTM grade-I),and titanium alloy (ASTM grade-V). The chemical com-position and other mechanical properties of titanium and itsalloy are shown in Table 1. All the tools except cementedcarbide were made as one-piece unit and attached to thehorn by tightening the threaded portion of the tool with thehorn. Tool of cemented carbide was prepared by silverbrazing the tip with replaceable threaded part at 1,200 F.

Three types of abrasive materials were used: siliconcarbide, aluminum oxide, and boron carbide. The abrasive

slurry was prepared with a concentration equal to 25% (bymass of the abrasive to water). Three different grit sizeswere selected for each abrasive material: 220, 320, and 500.The mean abrasive particle size corresponding to thesemesh numbers has been detailed in Tables 2 and 3. Theselevels were selected by means of pilot experimentationperformed to study the influence of the parameter grit sizeon the MRR in ultrasonic drilling of titanium. The pilotexperimentation is performed to study the effect of thechange in the levels of the factor of interest on the responsevariable. Experiments were conducted by varying the gritsize of the abrasive over a wide range (from 100 to 600)while keeping all the remaining factors as unchanged. Highcarbon steel was used as tool material along with two typesof abrasives-alumina and silicon carbide (with five differentgrit sizes ranging from 100 to 600), while the power ratingwas kept at two levels, 100 and 400 W, the ratio betweenthe excitation power and amplitude of vibration being 1:1.The 20 experimental runs were replicated twice to obtainthe results for pilot experimentation. Slurry grit sizes of220, 320, and 500 contribute most to variation in MRR(Fig. 2); hence, these three levels were selected for finalexperimentation. Power rating of the ultrasonic machinewas selected as another process parameter for this investi-gation as the effect of this parameter on MRR in USM hasnot been explored to a significant extent by any researcherby now. Three levels of power rating were finalized fromthe pilot experimentation: 100, 250, and 400 W. Theprocess parameters and their levels selected for the finalexperimentation has been depicted in Tables 2 and 3.

The experiments were conducted on an “AP-500 modelSonic-Mill” ultrasonic machine (Sonic-Mill, Albuquerque,NM, USA). The complete setup consisted of four sub-systems: power supply, module unit, slurry re-circulatingsystem, and workpiece holder. The USM equipment usedfor this research has been depicted in Fig. 3, with all itscomponents clearly marked. The ultrasonic drilling actiontakes place by means of excitation of the tool. The vibratingtool hammers the abrasive particles flowing in the cuttingzone, and machining takes place by microchipping of thework surface. The horn of the USM machine was made oftitanium alloy (ASTM grade-V), and the same horn wasused for conducting all the experiments for uniformity in

Table 2 Process parameters and their values at different levels

Symbol Parameter Level 1 Level 2 Level 3 Level 4 Level 5

A Tool material HCS HSS Titanium Ti alloy Cemented carbide

B Abrasive type Alumina SiC Boron carbide

C Grit size 220 (64µm) 320 (36µm) 500 (19µm)

D Power rating 100 250 400

E Slurry concentration 20% 25% 30%


results. In USM, the amplitude of vibration at the tool tipdepends on the mass of the tool to a large extent. As thetool materials involved in this investigation possessed awide spectrum of the value of density, the tool design wasgiven a lot of practical consideration. The dimensions of theeach tool were determined to keep the mass of the toolfixed at 50 gm. Using a tool of mass greater than this value(50 gm) results in overloading of the machine, and themachining is automatically stopped by the unit. To measurethe MRR, the time taken for drilling each hole was recordedusing a stopwatch. Each hole was drilled with a diameter of8 mm and straight cylindrical geometry. The workpiece wasweighed before and after drilling each hole using electronicbalance. The weight loss for drilling each hole was thusrecorded. The volumetric MRR (mm3/min) was calculatedby taking the ratio of weight loss of the workpiece per holeto the product of drilling time per hole and density of thetool material.

4 Experimentation and data collection

Before finalizing a particular orthogonal array for thepurpose of designing the experiments, the following twothings must be established [22]:

1. The number of parameters and interactions of interest2. The number of levels for the parameters of interest.

In the present investigation, four different processparameters have been selected as already discussed. Thetool material factor has five levels, whereas all otherparameters such as abrasive type, grit size, and powerrating of the machine have three levels each. Hence, L18array (in modified form) was selected for the presentinvestigation. L18 array has a special property that the two-way interactions between the parameters are partiallyconfounded with various columns. Hence, their impact onthe main effects of the parameters under consideration isminimized [2]. It is not possible to assess the possible twofactor interactions in L18 array but the main effects ofdifferent process parameters can be assessed with reason-

Table 3 Constant parameters

Frequency of vibration 21 kHz

Static load 1.63 kg

Amplitude of vibration 25.3–25.8µm

Depth of cut 2 mm

Thickness of workpiece 10 mm

Tool geometry Straight cylindrical (with diameter 8 mm)

Slurry temperature 28°C (ambient room temperature)

Slurry flow rate 36.4× 103mm3/min

Slurry media Water

Effect of grit size on MRR

0

0.3

0.6

0.9

1.2

1.5

1.8

100 220 320 500 600

Grit Size

MR

R (

cub

ic m

m/m

in) P=100 W, Alumina slurry,

HCS tool

P= 100 W, Silicon carbide slurry, HCS tool

P= 400 W, Alumina slurry, HCS tool

P= 400 W, Silicon carbide, HCS tool

Fig. 2 Effect of grit size onmaterial removal rate

Fig. 3 Ultrasonic machining apparatus used for experimentation


able accuracy. According to the scheme of the experimen-tation outlined in the L18 orthogonal array (Table 4), holeswere drilled in the work pieces which were prepared in theform of rectangular discs with thickness of 10 mm. Eachtrial was replicated twice; hence, three holes were drilledfor each of the 18 trial runs and, moreover, all the 54 trialruns were executed in completely randomized fashion toreduce the effect of experimental noise to the maximumpossible extent. Figure 4 shows the workpiece containingsome holes drilled with USM. The flow rate of the abrasiveslurry was maintained constant at a value of 36.4 × 103

mm3/min. To avoid any possibility of dullness of the edgesof the abrasive grains, a large volume of slurry wasprepared. The experimental results for MRR are summa-rized in Table 5. A wide variation in the MRR values wasobserved, with a mean of 0.57 mm3/min, the lowest valueat 0.11 mm3/min, and the highest value at 1.44 mm3/min.

5 Analysis of data

5.1 Evaluation of S/N ratios

The S/N ratio is obtained using Taguchi’s method. Here, theterm “signal” represents the desirable value (mean), and the“noise” represents the undesirable value (standard devia-tion). Thus, the S/N ratio represents the amount of variationpresent in the performance characteristic. Depending uponthe objective of the performance characteristic, there can bevarious types of S/N ratios. Here, the desirable objective is

larger values of MRR. Hence, the higher-the-better type S/N ratio was used for transforming the raw data [9].

h ¼ �10 log 101

n:Xn1

1

y2i

( )

where yi is the value of the characteristic in an observationi, and n is the number of observations or number ofrepetitions in a trial.

5.2 Assessment of the main effects

The main effects can be studied by the level averageresponse analysis of raw data or of S/N data. The analysis isdone by averaging the raw and/or S/N data at each level ofeach parameter and plotting the values in graphical form.The level average responses from the raw data help inanalyzing the trend of the performance characteristic withrespect to the variation of the factor under study. The level

Experiment number Tool material Abrasive Grit size Power rating

1 HCS Alumina 220 100

2 HCS SiC 320 250

3 HCS B4C 500 400

4 HSS Alumina 220 250

5 HSS SiC 320 400

6 HSS B4C 500 100

7 Titanium Alumina 320 100

8 Titanium SiC 500 250

9 Titanium B4C 220 400

10 Titanium alloy Alumina 500 400

11 Titanium alloy SiC 220 100

12 Titanium alloy B4C 320 250

13 Cemented carbide Alumina 320 400

14 Cemented carbide SiC 500 100

15 Cemented carbide B4C 220 250

16 HCS Alumina 500 250

17 HCS SiC 220 400

18 HCS B4C 320 100

Table 4 Experimental controllog based on L18 OA

Fig. 4 Holes drilled by ultrasonic machining in the titanium workpieces


average response plots based on the S/N data help inoptimizing the objective function under consideration. Thepeak points of these plots correspond to the optimumcondition. The main effects of raw data and those of theS/N ratio are shown in Fig. 5.

5.3 Analysis of variance (ANOVA)

The percentage contribution of various process parameterson the selected performance characteristic can be estimatedby performing analysis of variance (ANOVA). Thus,information about how significant the effect of eachcontrolled parameter is on the quality characteristic ofinterest can be obtained. The total variation in the result isthe sum of variation due to various controlled factors andtheir interactions and variation due to experimental error.

The ANOVA for raw data and S/N data have beenperformed to identify the significant parameters and toquantify their effect on the performance characteristic. TheANOVA based on the raw data signifies the factors, whichaffect the average response, but it misses the insight into theeffect of the parameters on the variability of the process[20]. However, ANOVA based on S/N ratio takes intoaccount both these aspects and, hence, it is used here. Thepooled ANOVA results for S/N data and raw data are givenin Tables 6 and 7, respectively. The most favorableconditions or optimal levels of process parameters havebeen established by analyzing response curves of S/N ratioassociated with the raw data.

Experiment number MRR (mm3/min) Average MRR (mm3/min) S/N ratio (dB)

R1 R2 R3

1 0.31 0.27 0.36 0.31 −10.262 0.64 0.58 0.75 0.66 −3.803 1.09 1.21 0.99 1.10 0.71

4 0.40 0.28 0.30 0.33 −10.025 0.74 0.63 0.80 0.72 −2.946 0.21 0.15 0.15 0.17 −15.707 0.13 0.12 0.07 0.11 −20.458 0.24 0.27 0.35 0.29 −11.179 1.33 1.09 1.24 1.22 1.64

10 0.27 0.26 0.38 0.30 −10.7211 0.18 0.20 0.15 0.18 −15.2412 0.47 0.55 0.36 0.46 −7.1513 0.59 0.50 0.72 0.60 −4.6714 0.32 0.27 0.36 0.32 −10.1715 1.25 1.40 1.16 1.27 2.00

16 0.14 0.17 0.18 0.16 −15.8917 1.30 1.45 1.56 1.44 3.07

18 0.30 0.45 0.36 0.37 −8.99

Table 5 Experimental results ofmaterial removal rate

Average Response Graph

0.00

0.20

0.40

0.60

0.80

1.00

A1 A2 A3 A4 A5 B1 B2 B3 C1 C2 C3 D1 D2 D3

Factor level

A1 A2 A3 A4 A5 B1 B2 B3 C1 C2 C3 D1 D2 D3

Factor level

MR

R (

cub

ic m

m/m

in)

S/N Ratio Response Graph

-14.00

-12.00

-10.00

-8.00

-6.00

-4.00

-2.00

S/N

val

ue

A1-HCS A2-HSS A3-Ti A4-Ti alloy A5-Carbide; B1-Alumina B2-Silicon carbide B3-Boron carbide; C1-220 C2-320 C3-500; D1-100W D2-250 W D3-400 W

Fig. 5 Effects of process parameters on material removal rate rawdata and S/N ratio. Main effects (A=tool, B=abrasive, C=grit size, D=power rating, E=slurry concentration)


5.4 Prediction of the mean

After determination of the optimum condition, the mean ofthe response at the optimum condition is computed. Thisvalue is calculated by considering only the significantfactors that are concluded by ANOVA. It may also happenthat the predicted combination of the parameters beidentical to one of the trial combinations executed alreadyduring the final experimentation stage. Under such situations,the most direct way to estimate the mean of that treatmentcombination is to average out all the results for the trials thatare set at that particular levels [22].

6 Results and discussions

It can be observed from Fig. 5 that tool material usedaffects the MRR very significantly. Moreover, the differenttool materials used in the experimentation can be ranked inthe order of increasing MRR as titanium alloy, high speedsteel, titanium, high carbon steel, and cemented carbide.The highest MRR has been recorded with cemented carbideas the tool material. This can be attributed to its very higherhardness (92 RC) as compared to the other tool materialsused in this investigation. In USM, the indentation ofabrasive grains in work and tool is inversely proportional tothe hardness ratio of tool and work materials. Hence, use ofa harder tool results in more indentation in the work piece

as compared to tool, increasing the MRR [12, 14, 21]. Useof a tougher and ductile tool such as titanium or titaniumalloy tends to lower the MRR as it encourages the plasticdeformation of the tool as well as work material, which wasconformed from the microstructure analysis of the machinedsamples. High carbon steel, being harder than high speed steeland titanium, has also been found to perform better than thesetools in terms of MRR. In fact, the difference in theperformance of high carbon steel and cemented carbide toolshas been marginal, the later being on the higher side. It canalso be concluded that the MRR increases with the increase inrelative hardness of the tool-work combination, which hasalso been suggested by many other researchers [13, 17, 26].

The type of abrasive used has been found to produce asignificant effect on MRR. It has been observed that use ofboron carbide as abrasive results inmoreMRR as compared tothat achieved with the use of alumina or silicon carbide. Boroncarbide has 50–80%more cutting power as compared to otherabrasive materials used in this investigation. Hence, the use ofboron carbide as abrasive results is faster erosion of worksurface thereby improving the MRR [3, 17].

The MRR obtained has been found to increase with theincrease in coarseness of the abrasive grains. This is again inaccordance with the findings of most investigators [7, 8, 18,19, 27]. As reported in the literature [27], two main wearmechanisms act on the USM process: hammering and freeimpact of the abrasive particles. The primary mechanism isthe hammering action of abrasive particles into the workpiece

Table 6 Analysis of variance results for material removal rate (S/N data)

Source DF Seq. SS Adj. SS Adj. MS F (% P)

Tool 4 114.812 114.812 28.703 5.77 14.2

Abrasive 2 175.197 175.197 87.598 17.61 21.7

Grit size 2 97.573 97.573 48.787 9.81 12.1

Power rating 2 384.277 384.277 192.139 38.63 47.6

Error 7 34.817 34.817 4.974 4.4

Total 17 806.677

Ftab value is obtained from statistical tables and represents probability of the event that the observed F value resulted just by chance. If F>Ftab, thefactor is significant at 95% confidence level

Ftab=F(4,43) = 2.60; F(2,43) = 3.21

DF degrees of freedom, Seq. SS sequential sum of squares, Adj. MS adjusted mean square error, % P percent contribution

Source DF Seq. SS Adj. SS Adj. MS F (% P)

Tool 4 1.25 1.25 0.31 18.5 13.7

Abrasive 2 1.97 1.97 0.98 58.1 21.3

Grit size 2 1.57 1.57 0.79 46.5 17.2

Power rating 2 3.88 3.88 1.94 114.5 42.0

Error 43 0.72 0.72 0.017 5.8

Total 53 9.39

Table 7 Analysis of varianceresults for material removal rate(raw data)

Ftab=F(4,43) = 2.60; F(2,43) = 3.21

Order of significance: (1) powerrating, (2) abrasive type, (3) gritsize, and (4) tool


material. Decreasing the surface density of abrasive particlesby increasing the grit size induces a greater augmentation ofthe effective stress due to each particle acting on the worksurface, thus increasing the MRR. The secondary mechanismis the impact of free abrasive particles (accelerated by thevibrating tool) over the work surface. In this case, an increasein grit size increases the weight. The impact force against theworkpiece surface also increases. Thus, under both mecha-nisms, the effective load acting on the work surface increaseswith grit size, which in turn increases the effectiveness ofmicro-cracking thereby increasing the MRR.

As shown in Fig. 5, MRR drops rapidly, while grit size ischanged from 220 to 320 and drops further from 320 to500, but at a diminishing rate. After crossing the grit size of320 (36µm), the mean particle size of the abrasivecontinues to come closer to the mean gap between the tooland work as well as the vibration amplitude (25.3–25.8µm),which promotes more efficient machining of work material. Ithas also been observed that for a given tool material, MRRobtained is highest at that particular grit size–power levelcombination which corresponds to the highest value for toolwear rate. In other words, MRR is maximum at the points ofmaximum TWR. This is an inherent characteristic of theprocess and has also been found to be applicable in USM oftitanium from this research.

As far as the effect of power rating of the ultrasonicmachine on MRR is concerned, any increase in power ratingproduced a substantial increment in the rate of machining(Fig. 5) as the momentum with which the abrasive particlesstrike with the work surface increased manifold with acorresponding increment in power rating [16]. The particleswith higher momentum remove larger chunks of materialfrom the work surface, promoting the increase in MRR. Astitanium is a tough material with good strain hardeningcapability, the relative tool-work hardening plays a large rolein the indifferent behavior of MRR with change in powerrating [16].

With regards to the S/N response, the values of S/N ratiohave been found to be highest for those factor levels thatcorrespond to highest average response. Hence, these factorlevels can be termed as optimum from the point of view ofaverage response as well as S/N response. Hence, it can beconcluded from this discussion that input parameterssettings of ultrasonic power rating at 400 W, with cementedcarbide tool and boron carbide slurry with a coarse grit sizeof 220, have given the optimum results for MRR whentitanium (ASTM grade-I) was machined.

6.1 Macro-model for MRR

MRR is “larger-the-better” type characteristic. Therefore,higher values of MRR are considered to be optimal. It isclear from Fig. 5 that MRR is highest at the fifth level of

tool material parameter (A5), third level of abrasivematerial parameter (B3), first level of the grit size (C1),and third level of power rating (D3). The main effects of theS/N ratio are also highest at these levels of the parameters.

To establish the relative significance of the individualfactors, ANOVA has been performed, both on raw data andS/N data (Tables 6 and 7). The F values for various factorsas estimated from the general linear model have beencompared with the tabulated F values (Ftab) against theparticular combination of factor degrees of freedom (DOF)and error DOF. If the estimated F value turns out to besignificantly greater than the tabulated F value, thecorresponding factor is termed as significant [2, 8]. Thefactors with larger F values (estimated) are deemed to behaving more significance from a statistical point of view. Thepercent contribution of each factor has also been tabulated.With regarding to the average response, power rating factorhas emerged as most significant with a percent contributionof 42% (Tables 5 and 7), followed by abrasive type (21.3%)and slurry grit size (17.2%). Tool material factor can betermed as the least significant for MRR with a percentcontribution of 13.7%. However, for S/N response, powerrating has been found to be having highest percentcontribution (47.6%) followed by abrasive type (21.7%).Remaining factors have been found to be almost equallysignificant. The percent contributions of various factors havebeen depicted in Fig. 6 for both raw data as well as S/Nresponse data.

6.2 Conformation experiments

The Taguchi approach for predicting the mean performancecharacteristics and determination of confidence intervals forthe predicted mean has been applied. Three confirmationexperiments for each performance characteristics have beenperformed at optimal settings of the process parameters,and the average value has been calculated. The averagevalues of the performance characteristics obtained throughthe confirmation experiments must be within the 95%confidence interval (α = 0.05), CICE (fixed number ofconfirmation experiments).

For MRR, the overall mean of the population isµ = 0.56.

The predicted optimum value of MRR is calculated as:

mMRR ¼ mA5þ mB3þ mC1þ mD3ð Þ � 3mð Þ ¼ 1:52

For calculation of CICE, the following equation [20] hasbeen used:

CICE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFa 1; feð Þ 1

neffþ 1

R

� �Ve

sð1Þ


Fa 1; feð Þ the F-ratio at a confidence level of (1-α) againstDOF1 and error DOF fe (for MRR, fe = 43, soFa is 4.05)

Ve error variance for MRRVe 0.017 (Table 7)

neff ¼ N

1þ Total DF involved in estimation of meanð2Þ

N total number of experimentsneff 54= 1þ 10ð Þ ¼ 4:90R sample size for confirmatory experiments=3

Hence, putting all the values in Eq. 1,

CICE MRRð Þ ¼ �0:18

The 95% confidence interval for µMRR is

CICE MRRð Þ ¼ 1:34 < mMRR < 1:70

From the confirmation experiments, the average value ofMRR obtained is 1.69, which is well contained by theconfidence interval. Hence, the validity of experimentaldesign is conformed, and the optimized process setting canbe used as a macro-model (Table 8) for further developmentof the micro-model for prediction of MRR under varyingconditions.

7 Micro-model for prediction of MRR

The macro-model developed through Taguchi design hasfurther been used to form a micro-model to predict the

values of MRR over a wide range of input parameters. Asper Taguchi design, MRR in USM is dependent on type oftool/abrasive, ultrasonic power, and slurry grit size.

The following assumptions have been made whiledeveloping the mathematical model for the above situation:

1. The frequency of ultrasonic vibration is fixed throughoutthe experimentation and it is of the order of 21 kHz±200 Hz.

2. The amplitude of vibration is constant throughout theexperimentation and is of the order of 25.3–25.8µm.The USM set up contains a provision for maintainingthe amplitude of vibration constant under variableexcitation power conditions.

3. The static feed force is maintained constant at a valueof 1.63 kg.

4. The factors that turned out to be insignificant from Taguchidesign have been omitted from the micro-modeling.

The Buckingham’s pi theorem proves that, in a physicalproblem including “n” quantities in which there are “m”dimensions, the quantities can be arranged into “n–m”independent dimensionless parameters. In this approach,dimensional analysis is used for developing the relations.

Now, MRR “Z” depends upon four input parameters:ultrasonic power, tool hardness factor, slurry hardnessfactor, and grit size of slurry. The slurry hardness factorindicates the Knoop hardness of the slurry used. Again, byselecting:

M (mass)L (length)T (time)

as basic dimensions, the dimensions of the foregoingquantities would then be:

1. The MRR “Z” (mm3/min); L3 T−1

2. Power rating “P” (watt); M L2 T−3

3. Tool hardness factor “H” (GPa); M L−1 T−2

4. Slurry hardness factor “μs” (GPa); M L−1 T−2

5. Slurry grit size “ρ” (mm); L

Percent contribution in variation of MRR (Raw data)

A14%

B21%

C17%

D43%

Error5%

Percent contribution in the variation of MRR (S/N data)

A14%

B22%

C12%

D48%

Error4%

Fig. 6 Contributions ofsignificant parameters tomaterial removal rate (S/N dataand raw data)

Table 8 Macro-model for material removal rate

Optimization of MRR

Tool material Cemented carbide

Abrasive material Boron carbide

Grit size 220 (coarse)

Power rating 400 W (80%)


Now,

Z ¼ f P;H ;ms; rð Þ ð3Þ

In this case, n = 5 and m = 3; hence, we can have (n–m = 2)π1 and π2 two dimensionless groups.

Taking Z and H as the quantities (randomly) which willgo in π1 and π2 respectively, we obtain:

p1 ¼ Z msð Þa1ðPÞb1 rð Þg1 ð4Þ

p2 ¼ H msð Þa2ðPÞb2 rð Þg2 ð5Þ

Substituting the dimensions of each quantity and equatingto zero, the ultimate exponent of each basic dimension isachieved, since the “πis” are dimensionless groups.

Solving for π1, we get,

p1 ¼ L3T�1 ML�1T�2� �a1

ML2T�3� �b1ðLÞg1 ð6Þ

Here,

3� a1 þ 2b1 þ g1 ¼ 0

�1� 2a1 � 3b1 ¼ 0

a1 þ b1 ¼ 0

On solving,

a1 ¼ 1; b1 ¼ �1; g1 ¼ 0

Hence,

p1 ¼ Z msð ÞðPÞ�1 rð Þ0

p1 ¼ Zms=Pð7Þ

Similarly, we get

p2 ¼ ML�1T�2 ML�1T�2� �a2

ML2T�3� �b2ðLÞg2

1þ a2 þ b2 ¼ 0

�1� a2 þ 2b2 þ g2 ¼ 0

�2� 2a2 � 3b2 ¼ 0

Solving, we get,

a2 ¼ �1; b2 ¼ 0; g2 ¼ 0

Thus,

p2 ¼ H msð Þ�1ðPÞ0 rð Þ0

p2 ¼ H=ms

ð9Þ

The functional relationship is of the form,

p1 ¼ f p2ð Þ ð10ÞHence,

Zms=P ¼ f H=msð Þ ð11Þ

It has been found experimentally that Z directly goes withH [15, 21]. Hence,

Z:ms=P ¼ Hf 1=msð ÞZ ¼ C P:H=m2

s

� � ð12Þ

where C is a constant of proportionality.To calculate “C”, experiments were performed by

keeping P=m2s unchanged, varying H (different tool materials)

to find out Z.

7.1 Case I (400 W power rating, alumina slurry, and 220grit size)

To know the impact of H on MRR, experiments wereperformed by “changing one factor at a time” approach thuswith changing only the tool material and keeping all otherfactors constant. The actual experimental data has beenpresented in Table 9. The data collected has been furtherused for finding the best fitting curve as depicted in Fig. 7.Thus, the regression equation for MRR in this case is

Z ¼ 2:807� 3:681H þ 1:496H2 � 0:07040H3� �

:P=m2s

ð13Þ

Similarly, the experimentation was performed withsilicon carbide slurry and boron carbide slurry using allthe tool materials used in this investigation, keeping thepower rating (400 W) and grit size (220) unchanged. Thebest fitted curves for these two cases are given below(Figs. 8 and 9):

7.2 Case II (400 W power rating, silicon carbide slurry,and 220 grit size)

The regression equation for MRR is

Z ¼ 8:215� 10:91H þ 4:503H2 � 0:2125H3� �

:P=m2s

ð14Þ


Abrasive type Tool material Tool hardness (GPa) Mean MRR (mm3/min)

Alumina High carbon steel 1.9 0.85

High speed steel 1.68 0.70

Titanium 1.15 0.57

Titanium alloy 1.36 0.38

Cemented carbide 18.5 1.02

Boron carbide High carbon steel 1.9 1.55


Titanium 1.15 1.20



Silicon carbide High carbon steel 1.9 1.42


Titanium 1.15 0.92



Table 9 Effect of tool hardnessfactor on material removal rate

Power rating=400 W, gritsize=220

Best fitting curve vs. experimental results

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5 6

Tool

Mea

n M

RR

(cu

bic

mm

/min

)

Mean MRR (experimental data) Predicted value from model

Experimental conditions: Abrasive used: Alumina, Grit Size: 220, Power Input: 80% (400 W), Work material: Titanium

S = 0.11 R2 = 0.95

1- H.C.S 2- H.S.S 3- Ti 4- Ti alloy 5- WC

Fig. 7 Material removal rate vstool hardness factor (P=400 W,alumina slurry, 220 grit size)

0 1 2 3 4 5 6

Tool


Experimental conditions: Abrasive used: Silicon Carbide, Grit Size: 220, Power Input: 80% (400 W), Work material: Titanium


0

0.5

1

1.5

2

2.5

Mea

n M

RR

(cu

bic

mm

/min

) S = 0.38 R2 = 0.92

1- H.C.S 2- H.S.S 3- Ti 4- Ti alloy 5- WC

Fig. 8 Material removal rate vstool hardness factor (P=400 W,silica slurry, 220 grit size)


7.3 Case III (400 W power rating, boron carbide slurry,and 220 grit size)

The regression equation for MRR is

Z ¼ 11:30� 14:10H þ 5:614H2 � 0:2636 H3� �

:P=m2s

ð15ÞThe results of macro-model are based upon the concept

of robust design; Taguchi analysis highlights that MRR isaffected by ultrasonic power rating; tool hardness factor,slurry grit size, and optimized results for the said casecomes up with the use of cemented carbide tool at 400 W ofultrasonic power and boron carbide slurry with 220 gritsize. The three cases described here express MRR inmachining of pure titanium at a specific power rating forparticular slurry type by varying hardness of the tool, i.e., byselecting different tool materials. The same methodology can

be further enhanced for developing mathematical model interms of other input parameters such as static load, amplitudeof vibration, and frequency of vibration using optimum inputparametric settings based upon macro-model.

8 Microstructure analysis

The machined samples were etched with Kroll’s reagent(2% HF, 10% HNO3, and 88% distilled water), and themicrostructure was then studied using scanning electronmicroscope at a magnification of ×1,500. The micro-structures obtained have been shown in Figs. 10, 11, 12,13, 14, 15, 16, and 17.

In mechanical shaping processes for brittle materialssuch as USM, material has been found to be removed bythe propagation and intersection of cracks [4, 16, 18].

0 1 2 3 4 5 6

Tool


Experimental conditions: Abrasive used: Boron Carbide, Grit Size: 220, Power Input: 80% (400 W), Work material: Titanium

S = 0.38 R2 = 0.94

1- H.C.S 2- H.S.S 3- Ti 4- Ti alloy 5- WCµs= 27 GPa


0

0.5

1

1.5

2

2.5

Mea

n M

RR

(cu

bic

mm

/min

)

Fig. 9 Material removal rate vstool hardness factor (P=400 W,boron carbide slurry, 220 gritsize)

Fig. 11 Microstructure of titanium (experiment number 2)Fig. 10 Microstructure of titanium (experiment number 1)


Fig. 13 Microstructure of titanium (experiment number 4)







While the cutting of brittle materials is performed, ingeneral, by brittle fracture, the chip can be removedplastically at an extremely small depth of cut [11, 19, 21].In other words, the mechanism of material removal maychange from brittle fracture to plastic deformation atextremely small MRRs while machining with USM.

From the microstructures of the machined samples oftitanium (Figs. 10, 11, 12, 13, 14, 15, 16, and 17), the modeof material removal could be easily established. Further, theway the fracture propagates through the material could berelated to the rate of energy input (work surface) duringmachining which, in turn, depends upon the processconditions, i.e., the input parameter settings in thatsituation. Consequently, an attempt was made to describethe mechanism of material removal in USM of titanium interms of the process conditions. It was observed that theprocess settings that involve higher rate of energy input tothe work surface (experiment number 5 and 13) eventuallylead to purely cleavage type of fracture, where the materialis removed by propagation and intersection of the medianand lateral cracks. This could be very well recognized fromthe sharp edges and the elongated projected regions in themicrostructures (Figs. 14 and 17).

The process conditions that involve a moderate level ofenergy input to the work surface during machining lead torealization of a mixed mode of material removal where thebrittle fracture of the work surface is preceded byconsiderable plastic deformation and work hardening.Depending on the level of energy input (as the term“moderate” is used for a considerable range of energy), thedominance of one or the other of the two possible modescould be established. For experimental trials 2 and8 (Figs. 11 and 16), the dominance of brittle fracture orcleavage type of fracture can be easily recognized. Forexperimental trials 3 and 4, the dominance of ductile failurehas been evidenced, where a large number of equiaxedround dimples could be observed (Figs. 12 and 13).

Figures 10 and 15 (experiments 1 and 6) indicate thepresence of a network of large number of spherical dimplesor micro-cavities with uniform shape and even distribution,which could be related to the sheer ductile failure of themachined surface. Moreover, the fractured surface showswhite and dark contrast in which the brighter contrast is theprojected portion of the fractured surface which is compar-atively larger than that observed in case of high energyinput conditions. This indicates ductile type of failure, andslightly longer duration was required to fracture the surface.From the microstructure analysis, no evidence of micro-cracking or surface disintegrity could be observed. Hence,it can be presumed that USM does not produce any surfacedamage to the machined titanium component, which can bevery critical for enhancement of the service life of thecomponents.

9 Conclusions

This research is aimed at exploring the use of USM forcost-effective machining of titanium and establishingoptimized process settings for MRR through Taguchi’stechnique for designing the experiments. The followingconclusions have been drawn from this research:

1. Optimum MRR in USM of titanium can be achieved byusing a tool material of higher hardness (cementedcarbide) along with a higher power rating (400 W),coarse grit size (220), and a hard abrasive (boroncarbide) material.

2. With regards to the average response, power ratingfactor has emerged as most significant factor with apercent contribution of 42%, followed by abrasive type(21.3%) and slurry grit size (17.2%). Tool materialfactor can be termed as the least significant for MRRwith a percent contribution of 13.7%.

3. USM is a very random process. Minor fluctuations inthe process variables such as slurry concentration cancause significant alteration in the results; hence, it isvery important to maintain the fixed parameters asconstant with the highest possible accuracy.

4. The outcome of the macro-model has been used forfurther development of a micro-model for predictionof MRR over a wide range of input parameters.Buckingham’s π-theorem has been used for construct-ing the micro-model for prediction of MRR in termsof tool hardness, power rating of the ultrasonicmachine, and slurry hardness factor. The samemethodology can be extended for development ofmathematical model under different conditions ofinput parameters.

5. The material removal in USM of titanium could bedirectly related to the energy input rate for theparticular process settings used for the machining. Ahigher level of input energy rate promotes the brittlefracture or cleavage of the work surface (Figs. 14 and17). For extremely small values of energy input, purelyductile failure mode was observed (Figs. 10 and 15). Itis possible to obtain a mixed mode of removal bymanipulating the process settings.

Acknowledgements The authors will like to thank Dr. V.K. Jain(Professor, Mechanical Engineering Department, Indian Institute ofTechnology, Kanpur) and Dr. Simon Barnard (Consultant, BSIInternational, UK) for providing technical guidance for the experi-mental work. The authors are also thankful to Mr. Trilok Singh andMr. Sukhdev Chand (Lab Superintendents, MED, Thapar University,Patiala, India) for providing laboratory facilities. The authors arethankful to Mr. Charlie White (Sonic-Mill, Albuquerque, USA) forproviding the necessary materials needed for this research work.


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Modeling the material removal rate in ultrasonic machining of titanium using dimensional analysis