The international journal of advanced manufacturing technology, tập 61, số 1 4, 2012

When the cutting speed adopted is below 1,400 m/min, the contribution order of the cutting parameters for surface roughness Ra is axial depth of cut, cutting speed, and feed rate.. It is

ORIGINAL ARTICLE

Surface roughness and chip formation in high-speed face

milling AISI H13 steel

Xiaobin Cui&Jun Zhao&Chao Jia&Yonghui Zhou

Received: 6 July 2011 / Accepted: 3 October 2011 / Published online: 22 October 2011

# Springer-Verlag London Limited 2011

Abstract Many previous researches on high-speed

ma-chining have been conducted to pursue high mama-chining

efficiency and accuracy In the present study, the

character-istics of cutting forces, surface roughness, and chip

formation obtained in high and ultra high-speed face

milling of AISI H13 steel (46–47 HRC) are experimentally

investigated It is found that the ultra high cutting speed of

1,400 m/min can be considered as a critical value, at which

relatively low mechanical load, good surface finish, and

high machining efficiency are expected to arise at the same

time When the cutting speed adopted is below

1,400 m/min, the contribution order of the cutting

parameters for surface roughness Ra is axial depth of

cut, cutting speed, and feed rate As the cutting speed

surpasses 1,400 m/min, the order is cutting speed, feed

rate, and axial depth of cut The developing trend of the

surface roughness obtained at different cutting speeds

can be estimated by means of observing the variation of

the chip shape and chip color It is concluded that when

low feed rate, low axial depth of cut, and cutting speed

below 1,400 m/min are adopted, surface roughness Ra

of the whole machined surface remains below 0.3μm, while

cutting speed above 1,400 m/min should be avoided even if

the feed rate and axial depth of cut are low

Keywords Cutting forces Surface roughness Chip

formation High-speed face milling AISI H13 steel

1 IntroductionThe primary objective of manufacturing operation is toefficiently produce parts with high quality The high-speedmachining processes can produce more accurate parts aswell as reduce the costs associated with assembly andfixture storage by allowing several process procedures to becombined into a monolithic one [1] For the purpose ofenhancing machining efficiency and accuracy at the sametime, many significant researches on high-speed machininghave been conducted

High-speed milling has been widely used in themanufacturing of aluminum aeronautical and automotivecomponents so as to generate surfaces with high geometricaccuracy The tool materials and rigid machine tools haveadvanced to be applied in hard milling, which can even be

an alternative for the grinding process to some extent [2,3]

In order to reveal the effects of cutting conditions especiallycutting speed on the machining efficiency and productquality in high-speed hard milling, comprehensive andthorough researches on surface roughness and chip forma-tion should be conducted

There are relatively few researches relating to surfaceroughness in the field of high-speed milling of hardenedsteels, and studies on chip formation are scant As is stated

by Ghani et al [4], when high cutting speed, low feed rate,and low depth of cut were adopted, good surface finish can

be obtained in semifinish and finish machining hardenedAISI H13 steel using TiN-coated carbide insert tools Theeffects of cutting parameters on surface roughness in high-speed side milling of hardened die steels were investigated

by Vivancos et al [5, 6], and mathematical models ofsurface roughness were established by means of the design

of experiment (DOE) method Toh [7] investigated andevaluated the different cutter path orientations when high-

X Cui:J Zhao ( *):C Jia:Y Zhou

Key Laboratory of High Efficiency and Clean Mechanical

Manufacture of MOE, School of Mechanical Engineering,

Shandong University,

Jinan 250061, People ’s Republic of China

e-mail: zhaojun@sdu.edu.cn

DOI 10.1007/s00170-011-3684-9

speed finish milling hardened steel, and the results

demonstrated that vertical upward orientation is generally

preferred in terms of workpiece surface roughness Ding et

al [8] experimentally investigated the effects of cutting

parameters on cutting forces and surface roughness in hard

milling of AISI H13 steel with coated carbide tools And

empirical models for cutting forces and surface roughness

were established The analysis results showed that finish

hard milling can be an alternative to grinding process in the

die and mold industry Siller et al [9] studied the impact of

a special carbide tool design on the process viability of the

face milling of hardened AISI D3 steel in terms of surface

quality and tool life It was found that surface roughness Ra

values from 0.1 to 0.3μm can be obtained in the workpiece

with an acceptable level of tool life

Previous studies provide much valuable information for

the understanding of surface roughness in high-speed hard

milling But very few researches were conducted to

investigate the surface roughness in high-speed face milling

of hardened steel And probably due to the relatively small

tool diameter and the high hardness of the workpiece, the

upper limits of the cutting speed in these studies mentioned

above are much lower than those (1,100 m/min) in the

researches on tool wear in high-speed face milling of

hardened AISI 1045 steel [1]

Because of the great high-temperature strength and

wear resistance, AISI H13 tool steel is widely applied in

extrusion, hot forging, and pressure die casting In the

present study, characteristics of cutting forces, surface

roughness, and chip formation obtained under different

cutting speeds in high and ultra high-speed face milling

of AISI H13 steel (46–47 HRC) are identified and

compared For the purpose of experimental investigating

the effects of cutting parameters especially cutting speed

on surface roughness, Taguchi method was used for the

DOE Because of the dynamic effects, runout, vagaries

of the table feed, and back cutting in the milling process,

the profile of the milled surface can vary substantially in

either the feed or perpendicular directions Wilkinson

[10] pointed out that, although some profiles were

measured in nonback cutting regions, it still seems that

such variations were realistic In the present study, for the

purpose of reducing such variation, the milled surface is

divided into four regions, and those regions are

investi-gated separately and integratedly

2 Experimental procedures

2.1 Workpiece material

A block of AISI H13 steel hardened to 46 to 47 HRC was

used in the present study The nominal chemical

composi-tion of the H13 tool steel under consideracomposi-tion is shown inTable 1 Dimensions of the block were designed so as toavoid back cutting as shown in Fig.1

2.2 Cutting tool and machining center

A Seco R220.53-0125-09-8C tool holder with a tooldiameter of 125 mm, major cutting edge angle of 45°,cutting rake angle of 10°, axial rake angle of 20°, and radialrake angle of −5° was used in the milling tests The toolholder is capable of carrying eight inserts The tungstencarbide insert SEEX 09T3AFTN-D09, which is coated withTi(C, N)–Al2O3, was used in the experiments In order tosimplify the analysis, only one of the teeth was used in allthe milling tests All of the surfaces were milled using freshcutting edges The milling tests were conducted on avertical CNC machining center DAEWOO ACE-V500 with

a maximum spindle rotational speed of 10,000 rpm and a15-kW drive motor without cutting fluid

2.3 Cutting tests

As has been mentioned, it has been found that the use ofhigh cutting speed, low feed rate, and low depth of cutleads to a good surface finish in semifinish and finishmachining hardened AISI H13 steel [4] Therefore, for thepurpose of acquiring better surface finish at high cuttingspeed (upper limit 2,400 m/min), low feed rate (0.02–0.06 mm/tooth) and low axial depth of cut (0.1–0.3 mm)were adopted in the milling tests Symmetric milling wasapplied, and the radial depth of cut was fixed as 75 mm asshown in Fig.1 In all the milling tests, the feed length wasset to be invariable 112.5 mm so that back cutting can beavoided

The effects of cutting speed on cutting forces, surfaceroughness Ra, and chip formation are focused on in thepresent study Firstly, experiments with all the cuttingparameters fixed except for the cutting speed v rangingfrom 200 to 2,400 m/min with 200 m/min as an intervalwere performed Axial depth of cut ap and feed rate fzwere set to be invariable 0.2 mm and 0.04 mm/tooth,respectively

The Taguchi method uses a special design oforthogonal arrays to study the entire parameters spacewith only a small number of experiments [11] After theexperiments with cutting speed in the range from 200 to2,400 m/min, in order to distinguish the differences of theeffects of cutting parameters on surface roughnessobtained within different cutting speed ranges, two L9orthogonal arrays, each of which has four columns andnine rows, were used in the present study For both of theorthogonal arrays, the three influencing factors werecutting speed, feed rate, and axial depth of cut, and one

column of array was left empty for the error of

experi-ments Table2 shows the three levels of the factors in the

two arrays The experimental layouts ME1 and ME2 are

shown in Tables3 and4

The machined surface of the workpiece material was divided

into four regions as shown in Fig.2 And the total machined

surface is represented by R5 In region R2 the entrance and

exit angles stay the same, while in the other regions those

angles keep changing Moreover, for any small time period,

the milling conditions in regions R1 and R2 can still be

considered as symmetric milling, but in regions R3 and R4

they seemed to be two different kinds of asymmetric milling

It is inferred that these differences will lead to varying

characteristics of the mechanical and thermal loads when

machining different regions, and finally affect the way how

the surfaces generate Taking these into consideration, in each

test for each region denoted in Fig.2, surface roughness Ra

was measured three times along the feed direction

Under given milling conditions, each test was replicated

three times The surface roughness Ra in different regions

was measured along the feed direction using a portable

surface roughness tester (Model TR200, China) The

sampling length and number of spans were set to be

0.8 mm and five, respectively As shown in Fig 3, the

cutting forces were measured using Kistler piezoelectric

dynamometer (type 9257B) mounted on the machine table

And the charge generated at the dynamometer was

amplified by means of a multichannel charge amplifier

(type 5070A) The sampling frequency of data was set as

7,000 Hz After the experiments the tool wear was

examined with an optical microscope and the chips were

observed using a Keyence VHX-600E 3D digital

micro-scope with a large depth of field

3 Results and discussion3.1 Cutting force

The effects of cutting speed on cutting forces are focused

on in the present study In the milling tests with cuttingspeed ranging from 200 to 2,400 m/min, the cutting forcesignatures were picked at the time when the milling cutterreached the midpoint of region R2 For per cutting forcecomponent, there were 7,000 data points in each recordedsignature The data point Fmof the resultant cutting force iscalculated from the cutting force components as shown in

Fsta ¼ 1N

XN m¼1

where Fmaxis the maximum data value of all the data points

of the resultant cutting force

Since each test was replicated three times, for eachcutting speed, there exist three values for Fsta and Fdyn,respectively Figures4and5show the developing trends ofthe average values of the static cutting force and thedynamic cutting force with the cutting speed, respectively

It can be seen from Fig 4 that as the cutting speedincreases, the static cutting force firstly increases approach-ing a peak value at a cutting speed of 1,000 m/min and thenbegin to decrease At the cutting speed of 1,400 m/min, thestatic cutting force reaches a valley value When the cutting

Fig 1 The setup of face milling

Table 1 Nominal chemical

composition of AISI H13 tool

steel (in weight percent)

0.32 –0.45 0.20 –0.50 0.80 –1.2 4.75 –5.50 1.10 –1.75 0.80 –1.20 0 –0.30 Bal

speed increases over 1,400 m/min, the static cutting force

keeps increasing

When the cutting speed is relatively low, the cutting

temperature is low and adhesion is less likely to happen

between the tool and the workpiece material Adhesion peaks

at some intermediate temperature [14] When the cutting

speed is below 1,000 m/min, the cutting temperature

increases with the cutting speed, leading to the more serious

adhesion It is inferred that, mainly due to the increase of the

friction coefficient induced by serious adhesion, the static

cutting force increases As the cutting speed increases over

1,000 m/min, higher cutting temperature occurred At high

cutting temperature, adhesion is reduced as thermal softening

has greater effect on the interface or on the workpiece

material [14] Higher cutting temperature arises in the shear

zone, leading to the reduction of the yield strength of the

workpiece material, chip thickness, and tool chip contact

area Moreover, the increase of cutting temperature results in

the decrease of the friction coefficient between the tool rake

face and the chip And the shear angle will increase Finally

the static cutting force will decrease When the cutting speed

surpasses 1,400 m/min, the tool wear increases greatly with

the cutting speed as shown in Fig.7 Because of the high

plowing forces induced by the increased contact area of the

larger flank wear face of the cutter acting on the workpiece,

the static cutting force increases with the cutting speed when

the cutting speed is above 1,400 m/min

Figure5shows that when the cutting speed increases, the

dynamic cutting force keeps increasing until it reaches a peak

value at about 1,000 m/min Then it decreases until the cutting

speed is 1,400 m/min As the cutting speed surpasses

1,400 m/min, the dynamic cutting force will increase Itseems that the developing trends of the static and dynamiccutting forces are similar This can be attributed to theprofound effect of the static cutting force on the occurrence ofcutter vibration Since the tool wear increases rapidly with thecutting speed when the cutting speed is above 1,400 m/min asshown in Fig.7, it is inferred that besides the effects of thefixturing elements and the machine tool system, the highertool wear also has great contribution to the increasing trend ofthe dynamic cutting force when the cutting speed increasesover 1,400 m/min The evolution of the dynamic cutting forcewith the cutting speed indicate that for the cutting parametersunder consideration, relatively stable cutting condition canstill be obtained at a high cutting speed of 1,400 m/min Therelatively stable cutting condition is beneficial to the surfacefinish of the workpiece It is concluded that the cutting speed

of 1,400 m/min can be considered as a critical value for both

of the static and dynamic cutting forces

3.2 Surface roughnessFigure 6 shows the surface roughness in different regions

vs cutting speed v The surface roughness yiin region Riiscalculated by means of the following equation:

yi¼1n

Xn j¼1

Table 2 Factors and selected

levels in the face milling

experiments

Table 3 Experimental layout ME1using an L 9 orthogonal array

Table 4 Experimental layout ME2using an L 9 orthogonal array

3, 4, 5; j=1, 2, 3) The average surface roughness y5jof the

total machined surface is determined by Eq.5:

y5j¼ y1jS1=S5þ y2jS2=S5þ y3jS3=S5þ y4jS4=S5 ð5Þ

where Skis the area of the region Rk(k=1, 2, 3, 4, 5)

It can be seen from Fig.6that the curves (solid line) of

the surface roughness in regions R1 and R2 with cutting

velocity are similar, while those in region R3 and R4 are

similar For surface roughness in all the regions, cutting

speed v=800 m/min is the optimum one, and v=1,400 m/

min can be considered as a transition value above which the

surface roughness in the five regions increase rapidly It

must be pointed that when the cutting speed v is at a rather

high value of 1,400 m/min, as for the total machined

surface R5 good surface quality (0.068 μm) can still be

obtained

Though the machined surface has been divided into four

regions and each test was replicated three times, for the

surface roughness in each region, there still seems to be some

randomness In order to reveal the developing trends of

surface roughness in a more clear way, the curves of the

surface roughness with the cutting speed are fitted as shown in

Fig.6 (dotted line) It can be seen from these fitted curves

that as the cutting velocity increases, the surface roughness

in different regions all exhibit similar developing trend: they

all decrease firstly and then increase Equations 6,7,8,9,and 10 are the fitted formulas for the surface roughness inregions R1, R2, R3, R4, and R5, respectively

600 and 900 m/min The surface quality is expected to beoptimal when the cutting speed adopted is in this speedrange

Since both the cutting forces and the surface roughnessare low at an ultra high cutting speed of 1,400 m/min,

Fig 5 Dynamic cutting force Fdynvs cutting speed v (f z =0.04 mm/ tooth, a =0.2 mm)

Fig 4 Static cutting force Fstavs cutting speed v (f z =0.04 mm/tooth,

a p =0.2 mm)

Fig 3 Photos of the experimental setup

Fig 2 Division of the machined surface

relatively low mechanical load, good surface quality, and

high machining efficiency are expected to arise at the same

time for the cutting parameters under consideration

Though the machining efficiency is a little lower, cutting

speeds below 1,400 m/min can still be used to obtain good

surface finish, but the cutting speeds above 1,400 m/min

should be avoided

Figure7 shows the evolution of the average flank wear

VB after one pass of the workpiece surface with the cutting

speed It can be seen that when the cutting speed is below

1,400 m/min, the tool wear rate is relatively small As the

cutting speed surpasses 1,400 m/min, the tool wear rate

increases rapidly with the cutting speed Taking the

developing trend of the surface roughness with cutting

speed into consideration, it is inferred that when the cutting

speed is below 1,400 m/min, the effect of tool wear on

surface roughness is small But as the cutting speedsurpasses 1,400 m/min, the higher tool wear rate contrib-utes greatly to the increase of the surface roughness withthe cutting speed

(a) Ra in R1 vs cutting speed (b) Ra in R2 vs cutting speed

(c) Ra in R3 vs cutting speed (d) Ra in R4 vs cutting speed

(e) Ra in R5 vs cutting speed

Fig 6 Surface roughness Ra in

different regions vs cutting

As 1,400 m/min is a transition cutting speed for surface

roughness, two experimental layouts ME1 and ME2 are

designed to investigate the effects of cutting parameters on

surface roughness within two different cutting speed

ranges, namely <1,400 and ≥1,400 m/min, as shown in

Tables3and4 The results of surface roughness show that

for all the regions surface roughness Ra remains below

0.3 μm can be obtained using the cutting parameter

combinations listed in the experimental layout ME1

Surface roughness below 0.3 μm is an acceptable value

for the comparison with other finishing process like

grinding [15], while the surface roughness Ra obtained

under some cutting parameter combinations with relatively

higher cutting speeds in ME2is much larger than 0.3μm

The signal to noise (S/N) ratio used in the Taguchi

method reflects both the average and the variation of the

quality characteristics Therefore, in the present study,instead of the average value, the S/N ratio is used so as toconvert the trial result data into a value for the evaluationcharacteristics in the optimum setting analysis The S/Nratioηifor region Rican be expressed in decibel units, and

it is defined by a logarithmic function based on the meansquare deviation around the target:

hi¼ 10log 1

n

Xn j¼1

yij2

!

ð11Þ

where all the symbols have the same meaning as they did in

Eq.4 It can be seen from Eq.11that the larger is the S/Nratio, the smaller is the variance of surface roughness Raaround the desired value

(a) The mean S/N graph for Ra in R1 (b) The mean S/N graph for Ra in R2

(c) The mean S/N graph for Ra in R3 (d) The mean S/N graph for Ra in R4

(e) The mean S/N graph for Ra in R

Fig 8 The mean S/N graph for

surface roughness in different

regions (ME 1 ) a The mean S/N

graph for Ra in R 1 b The mean

S/N graph for Ra in R 2 c The

mean S/N graph for Ra in R 3 d

The mean S/N graph for Ra in

R 4 e The mean S/N graph for

Ra in R 5

(a) The mean S/N graph for Ra in R1 (b) The mean S/N graph for Ra in R2

(c) The mean S/N graph for Ra in R3 (d) The mean S/N graph for Ra in R4

(e) The mean S/N graph for Ra in R5

Fig 9 The mean S/N graph for

surface roughness in different

regions (ME 2 ) a The mean S/N

graph for Ra in R 1 b The mean

S/N graph for Ra in R 2 c The

mean S/N graph for Ra in R 3 d

The mean S/N graph for Ra in

R 4 e The mean S/N graph for

Figures8and9show the mean S/N response graphs for

surface roughness Ra in different regions It can be seen

that, generally all the mean S/N ratios in Fig 8 are much

higher than those in Fig 9, indicating that if the cutting

speed surpasses 1,400 m/min, the surface quality will

deteriorate badly From Fig.8 which shows the mean S/N

graph for ME1, it can be seen that the feed rate fzhas little

effect on the surface roughness in the five regions As for

region R1, the effect of cutting speed v on surface

roughness is little However, for all the other four regions,

as the cutting speed increases, the surface roughness

decrease firstly and then increase In region R1, as the

depth of cut ap increases, the surface roughness increase

firstly and then decrease, while in the other regions the

surface roughness increase with the depth of cut It can be

seen from the mean S/N graph for ME2in Fig 9 that the

surface roughness in the five regions all increases with the

cutting speed v As the feed rate fz increases, the surface

roughness in all the regions decreases firstly and then

increases The surface roughness in all the five regions

except for region R3decrease with the depth of cut ap In

region R3, as the depth of cut increases, the surface

roughness decreases firstly and then increases It can be

concluded that within different speed ranges, the effects of

the cutting parameters on surface roughness in the five

regions change greatly

For the experimental layout ME1, the optimum

combina-tions of the cutting parameter levels are A1C2D1, A2C3D1,

A2C1D1, A2C1D1, and A2C3D1 for surface roughness in

regions R1, R2, R3, R4, and R5, respectively As for the

experimental layout ME2, the optimum combinations of the

cutting parameter levels are B1C2D2, B1C2D3, B1C2D2,

B1C2D2, and B1C2D3 The optimum combinations of the

cutting parameter levels for surface roughness in regions R3and R4are the same And it seems that low feed rate and lowdepth of cut is beneficial especially for surface roughness inthese two regions It can be concluded that because of thevarying characteristics of the mechanical and thermal loads,for different regions, there definitely exist differences in theformation of surface profiles

Tables 5 and 6 show the cutting parameters andcorresponding S/N ratios for the total machined surface R5obtained by means of Eqs 4 and 5 Based on the resultslisted in Tables5 and 6, the results of analysis of variance(ANOVA) for surface roughness in the total machinedsurface can be obtained as shown in Tables7and8 For theexperimental layout ME1, the contribution order of thecutting parameters for surface roughness Ra is axial depth

of cut, cutting speed, and feed rate, and the contribution offeed rate is very small As for ME2, the order is cuttingspeed, feed rate and axial depth of cut, and the contribu-tions of feed rate and axial depth of cut are approximatelythe same It can be concluded that as the cutting speedsurpasses 1,400 m/min, the degree of influences of cuttingspeed and feed rate on surface roughness Ra increasessubstantially especially for cutting speed, while that of axialdepth of cut decreases substantially

The surface roughness Ra in the total machined surface

R5is focused on in the regression analysis The form of theTaylor’s tool life equation in metal cutting is used, and afunctional relationship between the average value of surfaceroughness in region R5and the cutting parameters could bepostulated by:

Ra¼ avb

Table 7 Results of the ANOVA

for Ra in the total machined

Table 8 Results of the ANOVA

for Ra in the total machined

By means of a logarithmic transformation, the nonlinear

form of Eq.12 can be converted into the following linear

form:

lnRa¼ lna þ b ln v þ c ln fzþ d ln ap ð13Þ

where a, b, c, and d are the corresponding parameters After

regression analysis, the regression equations for the surface

roughness are obtained as follows:

where Ra(ME1) and Ra(ME2) represent the surface

rough-ness in experimental layouts ME1and ME2, respectively

Figure10 compares the fitted values of surface

rough-ness and the observed values for ME1and ME2 Figure11

shows the relative percentage error between the fitted and

the observed values The average value of the relative error

for Ra(ME1) and Ra(ME2) are 4.6921% and 4.8906%,

respectively It can be concluded that Eqs.14 and 15 can

describe the behavior of the data well 3.3 Chip formation

Figure 12 shows the chip formation under different cuttingspeeds with axial depth of cut apand feed rate fz fixed as0.2 mm and 0.04 mm/tooth, respectively As the cutting speedincreases, both the shape and the color of the chip changegradually When the cutting speed increases, the shape of thechip changes in the following order: washer-shaped chip(v=200 m/min), wave-shaped chip (v=400 m/min), arc-shaped chip (v=600 m/min), long strip of chip (v=800–1,400 m/min), short strip of chip (v=1,600–2,200 m/min), andpowder-shaped chip (v=2,400 m/min) It is found that thecolor of the chip is blue when the cutting velocity is relativelylow (v=200–600 m/min) At higher cutting speed (v=800–1,200 m/min), the color turns into purple When the cuttingspeed is no less than 1,400 m/min, the chip color is yellow

At the cutting speed of 800 m/min, the chip color turnsfrom blue into purple and the shape of the chip changesfrom arc-shaped chip to long strip of chip As has beendiscussed, at this cutting speed, optimum surface qualitycan be obtained as shown in Fig.6 When the cutting speedsurpasses 1,400 m/min, short strip of chip is about to formand the chip color changes into yellow This cutting speedcan be seen as a transition cutting speed for surfaceroughness as has been mentioned It seems that thecorrespondence between the chip formation and the surfaceroughness is obvious, indicating that the evolution of the

(a) Deviation of the fitted surface

(a) The fitted and observed surface

roughness in ME 1

(b) The fitted and observed surface

roughness in ME2 Fig 10 The fitted values of surface roughness vs the observed

values a The fitted and observed surface roughness in ME 1 b The

fitted and observed surface roughness in ME

surface quality can be estimated by means of observing the

variation of the chip shape and color

It is also found that the serrated chip begins to arise when

the cutting speed is about 400 m/min as shown in Fig.13b As

the cutting speed increases, the serrated chip becomes more

and more obvious As shown in Fig.13e at the cutting speed

of 2,400 m/min, the serrated chip is about to be separated

4 Conclusions

The effects of cutting speed on cutting forces, surface

roughness, and chip formation in high and ultra high-speed

face milling of AISI H13 steel were focused on in the

present study Taking the critical cutting speed 1,400 m/min

into consideration, the effects of cutting parameters on

surface roughness within two cutting speed ranges (<1,400

and ≥1,400 m/min) were investigated experimentally by

means of Taguchi method In order to reduce the

undesir-able variations of surface roughness, back cutting wasavoided, and the milled surface was divided and investi-gated separately and integratedly The following conclu-sions can be obtained:

& When the cutting speed increases from 200 to 2,400 m/min with feed rate fz and axial depth of cut ap fixed,both the static and dynamic cutting forces reach a valleyvalue at a cutting speed of 1,400 m/min It can beconcluded that relatively stable cutting condition which

is beneficial to the surface finish of the workpiece canstill be obtained at a high cutting speed of 1,400 m/min

& As the cutting speed increases from 200 to 2,400 m/minwith feed rate fz and axial depth of cut ap fixed as0.04 mm/tooth and 0.2 mm, the surface roughness indifferent regions all decreases firstly and then increases.The cutting speed of 1,400 m/min is considered as acritical value above which the surface roughness willdeteriorate badly When the cutting speed is between 600

(a) v = 200 m/min (b) v = 400 m/min (c) v = 600 m/min

(d) v = 800 m/min (e) v = 1000 m/min (f) v = 1200 m/min

(g) v = 1400 m/min (h) v = 1600 m/min (i) v = 1800 m/min

(j) v = 2000 m/min (k) v = 2200 m/min (l) v = 2400 m/min

Fig 12 Chip formation under

different cutting speeds

and 900 m/min, low surface roughness is expected to be

obtained When the cutting speed surpasses 1,400 m/min,

the higher tool wear rate has great effect on the increase

of the surface roughness with the cutting speed One

thousand four hundred meters per minute is considered

to be a critical cutting speed for both the cutting forces

and surface roughness At the cutting speed of 1,400 m/

min good surface quality, relatively low mechanical load

and high machining efficiency are expected to arise at the

same time for the cutting parameters under consideration

& Due to the variations of the characteristics of the

mechanical and thermal loads, the surface roughness

Ra in different regions of the machined surface respond

in varying ways to the changes of cutting parameters

For the experimental layouts ME1 and ME2, the

optimum combinations of the cutting parameter levels

for surface roughness in the whole machined surface R5

are A2C3D1 (v=700 m/min, fz=0.06 mm/tooth, ap=

0.1 mm) and B1C2D3 (v=1,400 m/min, fz=0.04 mm/

tooth, a =0.3 mm) The results of ANOVA for surface

roughness of the total machined surface show that forthe experimental layout ME1, the contribution order ofthe cutting parameters is axial depth of cut, cuttingspeed, and feed rate, and the contribution of feed rate isvery little; while for ME2, the order is cutting speed,feed rate, and axial depth of cut, and the contributions

of feed rate and axial depth of cut are roughly the same.When the cutting speed surpasses 1,400 m/min, thecutting speed and feed rate become much moreinfluential to the surface roughness especially forcutting speed, while the effect of axial depth of cutdeclines greatly It is found that when the cutting speed

is below 1,400 m/min, low surface roughness Ra below0.3 μm can be obtained By means of regressionanalysis, two equations for the surface roughness ofthe total machined surface R5in experimental layouts

ME1and ME2are fitted It is found that those equationscan describe the behavior of the data well

& As the cutting speed changes from 200 to 2,400 m/minwith invariable feed rate f 0.04 mm/tooth and axial

(a) v = 200 m/min (b) v = 400 m/min

(c) v = 800 m/min (d) v = 1400 m/min

(e) v = 2400 m/min

Fig 13 Magnified chip

forma-tion under different cutting

depth of cut ap0.2 mm, both the shape and color of the

chip change gradually There exists obvious

correspon-dence between the shape, color of the chip, and the

surface roughness obtained at different cutting speeds

When the cutting speed surpasses 1,400 m/min which is

considered as a transition speed below which good

surface finish can still be obtained, short strip of chip is

about to form and the color of the chip turns into

yellow It seems that the evolution of the surface quality

with cutting speed can be estimated by means of

observing the variation of the chip formation

Acknowledgments This research is supported by the National Basic

Research Program of China (2009CB724402), the National Natural

Science Foundation of China (51175310), and the Graduate Independent

Innovation Foundation of Shandong University, GIIFSDU (yzc10119).

References

1 Liu ZQ, Ai X, Zhang H, Wang ZT, Wan Y (2002) Wear patterns

and mechanisms of cutting tools in high-speed face milling J

Mater Process Technol 129:222–226

2 Nelson S, Schueller JK, Tlusty J (1998) Tool wear in milling

hardened die steel J Manuf Sci Eng 120(4):669–673

3 Iqbal A, He N, Li L, Dar NU (2007) A fuzzy expert system for

optimizing parameters and predicting performance measures in

hard-milling process Expert Syst Appl 32(4):1020 –1027

4 Ghani JA, Choudhury IA, Hassan HH (2004) Application of Taguchi method in the optimization of end milling parameters J Mater Process Technol 145(1):84 –92

5 Vivancos J, Luis CJ, Costa L (2004) Optimal machining parameters selection in high speed milling of hardened steels for injection moulds J Mater Process Technol 155–156:1505–1512

6 Vivancos J, Luis CJ, Ortiz JA (2005) Analysis of factors affecting the high-speed side milling of hardened die steels J Mater Process Technol 162 –163:696–701

7 Toh CK (2006) Cutter path orientations when high-speed finish milling inclined hardened steel Int J Adv Manuf Technol 27:473 –480

8 Ding TC, Zhang S, Wang YW, Zhu XL (2010) Empirical models and optimal cutting parameters for cutting forces and surface roughness in hard milling of AISI H13 steel Int J Adv Manuf Technol 51:45 –55

9 Siller HR, Vila C, Rodríguez CA, Abellán JV (2009) Study of face milling of hardened AISI D3 steel with a special design of carbide tools Int J Adv Manuf Technol 40:12 –25

10 Wilkinson P, Reuben RL, Jones JDC, Barton JS, Hand DP, Carolan TA, Kidd SR (1997) Surface finish parameters as diagnostics of tool wear in face milling Wear 205:47–54

11 Yang WH, Tarng TS (1998) Design optimization of cutting parameters for turning operations based on the Taguchi method.

J Mater Process Technol 84:122–129

12 Dimla DE, Lister PM (2000) On-line metal cutting tool condition monitoring I: force and vibration analysis Int J Mach Tools Manuf 40(5):739 –768

13 Toh CK (2004) Static and dynamic cutting force analysis when high speed rough milling hardened steel Mater Des 25:41 –50

14 Childs T, Maekawa K, Obikawa T, Yamane Y (2000) Metal machining: theory and applications Wiley, New York

15 Boothroyd G, Knight WA (2005) Fundamentals of machining and machine tools, 3rd edn CRC, New York

ORIGINAL ARTICLE

A study on helical surface generated by the primary

peripheral surfaces of ring tool

S Berbinschi&V Teodor&N Oancea

Received: 1 June 2011 / Accepted: 3 October 2011 / Published online: 22 October 2011

# Springer-Verlag London Limited 2011

Abstract Often in the engineering practice, cutting tools

bounded by primary peripheral surfaces of revolution are used

because of their effectiveness Among these, ring and

tangential tools can be used for the generation of constant

pitch cylindrical helical surfaces In this paper, we present an

algorithm for the profiling of these types of tools The

algorithm is based on the topological representation of the

tool’s primary peripheral surface The main goal is to devise a

methodology for the profiling of tools whose surfaces are

reciprocally enveloping with cylindrical helical surfaces We

present a numerical example for the numerical determination

of the axial section form for this type of tools The application

method for this algorithm was developed in the CATIA

graphical design environment within which the procedure is

developed as a vertical application In addition, we present a

solution for the shape correction of the tool’s axial

cross-section by considering the existence of singular points on the

1 Introduction

The ring and tangential tools are tools bounded by a

revolution primary peripheral surface The ring tools are

tools designated for the generation of the cylindrical helicalsurfaces with constant pitch (threads) on specializedmachine tools or using specialized technological equip-ments for longitudinal turning machines

The ring tool is frequently made as an enwrappingmilling tool The advantage of this technological solution isthe increased productivity of this process Although thetools of this type generate in the cutting motion a revolvingsurface, the issue of profiling the primary peripheral surface

of this surfaces reciprocally enveloping with cylindricalhelical surfaces with constant pitch, is a problem differentfrom the profiling of the side mill [1–4]

The profiling method of this type of tool uses thefundamental theorems of the surfaces generation [1, 5] orthe complementary methods as “the minimum distancemethod” [1], “the in-plane generating trajectories method”[6] Also, the development of the graphical design environ-ment allows solving these problems using 3D designenvironment [7–10] or using solid modeling [11,12]

2 Ring tool’s profiling algorithmThe basic idea behind the proposed algorithm is that ahelical movement, with ~V axis and p helical parameter, can

be decomposed in two rotations conjugated with thisrelative motion as shown in Fig.1

The three movements, as shown in Fig.1, are:

I is the translation movement correlated with therotation movement II, in order to produce a helicalmotion with the same helical parameter p as of thesurface to be generated In most cases, this motion isexecuted by the workpiece;

S Berbinschi:V Teodor ( *):N Oancea

Manufacturing Science and Engineering Department,

“Dunărea de Jos” University of Galaţi,

Gala ţi, Romania

e-mail: virgil.teodor@ugal.ro

DOI 10.1007/s00170-011-3687-6

profile of the helical surface to be generated where multiple

normals to the surface exist

Keywords Ring tool Helical surfaces generation

Topological representation

II the rotation movement of blank around its own axis,

B In the most practical cases, this motion is uniform;

and

III the rotation motion of tool around its own axis, the

axis ~A This axis is positioned regarding the axis B at a

distance and inclined with angleα regarding the Z

p¼ a  tan bð Þ ¼ b  tan að Þ ð2Þ

with p is the helical parameter

In this way, it is possible to choose as revolution

surface’s axis, the B axis, which may be established as the

axis of the ring surface Also, it is possible to arbitrarily

choose the values b andβ according to the dimension of

the helical surface and the diameter of the ring tool (see

Eqs.1 and2)

The Nikolaev’s theorem [5] for the determination of the

characteristic curve between the Σ helical surface, Fig.2,

and the primary peripheral surface of ring tool allows to

determine the geometrical locus of points which belongs to

theΣ surface where the condition is accomplished:

NΣ is the normal to the helical surface; and ~r2 is the

position vector of the current point that belongs to the

helical surface, regarding the O2 origin of the reference

system joined with the axis X

The condition3has the geometrical significance, the factthat the three vectors are in the same plane Moreover, thecontact points (the tangency points) between theΣ helicalsurface and the primary peripheral surface of the ring tooldefines the characteristic of Σ surface, in the rotationmovement of this around the ~B axis The characteristiccurve represents the geometric locus of intersection pointsbetween the normal draws from the points belongs to the ~Baxis to the surface to be generated (so, the projection of the

~B axis to the Σ surface).

In this way, for the Σ surface known in the X2Y2Z2reference system by equations form:

Σ ~r 2¼ X2ðu; vÞ~iþ Y2ðu; vÞ~jþ Z2ðu; vÞ~k; ð4Þwith u and v as independent variable parameters In the

X2Y2Z2 reference system, it is defined the normal to thesurface,

In this way, the characteristic curve on theΣ surface is

given by the4 and9 equations assembly:

CΣjX2¼ X2ðuÞ; Y2¼ Y2ðuÞ; Z2 ¼ Z2ðuÞ: ð9Þ

By revolving the characteristic curve 10 around the X2

axis, the axis overlapped to ~B is generated the primary

peripheral surface of the ring tool—SB The axial section of

the primary peripheral surface of the ring tool,ð ÞCΣ X2Y

2 Z 2isdetermined as:

2.1 Profiling of tool in CATIA design environment

An application of the presented algorithm for the profiling

of the ring tool for the generation of a trapezoidal thread

was presented (see Fig.2) The 3D method to profile the

ring tool—HSGT (helical surface generating tool)—is

grounded on the generative shape design environment

facilities The worked piece (in fact, the generated surface)

is 3D modeled, as it can be observed in Fig.3 The worked

piece reference system, XYZ, and the ring tool reference

system, X2Y2Z2, the last one as Euler system, are created

(see Fig.2)

By giving the “projection” command, the ring tool

axis projection onto the Σ surface is realized; thus, the

characteristic curve is determined By subsequently using

the “revolve” command, the tool primary peripheral

surface (S) results after rotating the characteristic curve

around Z2axis

The ring tool axial section is then obtained as an

intersection between the surface S and a plain which

includes the Z2 (~A) axis—by applying the “intersection”

command The coordinates of points that defined the

generating profile of the thread and the distance between

axes are given in Table1 In Fig.3, the 3D model of the

thread axial profile was shown In Fig.4, the HSGT–visualbasic application (VBA) was presented, where the profile’selements and the tool’s type are given Figure 5 presentedthe relative position of the primary peripheral surface forthe ring tool, its axis, the characteristic curve, and the axialsection of the surface Because the axial section of thehelical surface has singular points, the points C and D willresult in discontinuity points on the surfaces in enveloping(see Fig.5)

2.2 Numerical results—ring tool for trapezoidal thread

In Table 2 and Fig 6, we present the coordinates of thecharacteristic curves and of the axial cross-section of ringtool The existence of the singular points (see Fig 5, thepoints C′ and D′) leads to intersection points for thecharacteristic curves on adjacent flanks, which impose totake a decision regarding the form of the axial section Thesimplest solution is to eliminate the portions from thecharacteristic curve, in points C′ and D′, if there areacceptable modifications of the generated surfaces at thethread bottom In Fig 6, the intersection zone of theprofiles that forms the axial section and eliminates further

in the profile board construction for the ring tool wasshown

Fig 3 Generating profile of the trapezoidal thread

Table 1 Input parameters of the trapezoidal thread

b Distance between tool ’s axis

and thread axis

100

zA xA zD xD

zE xB

xC xE

Fig 4 The HGST-VBA application

2.3 Ring tool for ball thread

Figure7 showed the model of the ball thread and its axial

section (an assembly of circle’s arc, filleted), as so as the

following reference systems:

Xyz is the reference system associated

with the ball thread;

X1Y1Z1 reference system associated with

the ring tangential tool’s primaryperipheral surface; and

x0y0z0and x0′y0′z0′ additional reference systems

The generation movement assembly, the movements I,

II, and III, has the significances given in Fig.1 In Table3,

the input parameters correlated with the HGST-VBA

application Figures 7 and 8 present the forms and the

coordinates of the characteristic curve and the axial section

for the helical surfaces assembly that composes the ball

thread flute Obviously, in this case, singular points on the

profile do not exist

3 Tangential toolThe tangential tool is a tool bounded by a revolutionprimary peripheral surface The tangential tools are toolsdesignated for the generation of the cylindrical helicalsurfaces with constant pitch (threads) on specializedmachine tools or using specialized technological equip-ments for longitudinal lather machines The tangential toolmay used on a grinder machine

3.1 Ring tangential tool: algorithmThe problem of profiling the ring tangential tool, Fig.9, is, inprinciple, similar with the known problem of the side milltool’s profiling In principle, the Nikolaev [5] condition forthe determination of the characteristic curve—the tangencycurve between a cylindrical helical surface with constantpitch,Σ, and a revolution surface with ~A axis, with positionknown in the reference system of theΣ surface—is:

~A; ~NΣ;~r1

Fig 5 The primary peripheral surface for the ring tool, the axis, the

characteristic curve, and the axial section of the surface

Table 2 Coordinates of points on the characteristic curves and the axial section of ring tool

where ~A is the versor of the rotation axis of the tool bounded

by a revolution surface;

~

NΣ is the normal at the helical surface; and

~r1 is the vector which link the current point onto theΣ

surface with a point of the ~A axis (frequently, the

origin of the reference system joined with this axis,

here X1Y1Z1)

Condition 11 is equivalent with the statement: the

characteristic curve of a cylindrical helical surface with

constant pitch,Σ, in the rotation motion around a fixed axis,

~A, is composed by all the points belongs to the Σ surface,

which represent the projection of the ~A axis to the Σ surface

The specific problem is that the tool’s axis position is deferent

regarding the position of the side mill, regarding the blank

The generation process kinematics presumes the

follow-ing motions:

I is the rotation motion of the blank,

II translation motion of the blank correlated with the

They are defined the reference systems:

xyz is the reference system where is defined the

helical surface (the Z axis is the axis of the helicalsurface)

X1Y1Z1 reference system joined with the ring tangential

tool (the X1axis is the axis of the ring tangentialtool)

If, in the XYZ reference system, it is defined the Σhelical surface:

Σ : r! ¼ x u;vð Þ  i! þ y u; vð Þ  j! þ z u; vð Þ  k! ð12Þwith u and v variable parameters, then, by the coordinatestransformation, see Fig.9,

with a, b, and c technological constants

Ring Tool's Axial Section Characteristic

Curves

Fig 7 Characteristic curve and axial section of the ring tool

Table 3 Input parameters of the ball thread

e Half distance between the centers

of circles with radius r

0.155 mm

D j Diameter of centers cylinder of the

axial profile

49 mm

h Distance between the D j diameter

and the center of circle with radius r

0.17 mm

Fig 8 Axial section of ring tool

In the condition for the determination of the

character-istic curve11defining:

~A ¼~i the versor of the ring tool;

~

NΣ the normal to theΣ surface, in the reference system

X1Y1Z1

~r1¼ X1ð Þ ~iþ Yu;v 1ð Þ ~jþ Zu;v 1ð Þ ~k;u;v ð15Þ

the current vector on theΣ surface, in the reference system

X1Y1Z1, Eq.14

The Eqs 11 and 14 assembly represents the

character-istic curve, in principle, in form:

By revolving, the characteristic curve around the X1axis

is determined the primary peripheral surface of the ring

tangential tool The constants a, b, c, andβ are determined

from the condition that the trajectory of the S point,

corresponding to the external diameter of the Σ surface,

to be tangent at the helix (see Figs.9and 10)

Also, the projection of the helix corresponding to the Re

blank radius, in the same plane yz, is a curve with form:

In the following, an application of the proposed algorithmfor the determination of the primary peripheral surface ofthe ring tangential tool, for generation of a trapezoidalthread, with generatrix of helical surface presented in Fig.3

Fig 9 The generation process kinematics with the ring tangential tool

M

Rs

θ

Helix Projection (Re)

z x y

o V,p

Rs cos

T

Tool's frontal circle Projection

Helix (Re)

Rs Re

Fig 10 The ring tangential tool ’s axis position

is presented The method is the same with those described

in paragraph 2.1

The input data for the profile of the thread, the helix pitch,

and the distance between the tool’s axis and the thread’s axis

are inserted in the HSGT application, presented in Fig.11,

according to Table4 Figure12 represented the surfaces of

the trapezoidal thread’s flank, characteristic curves on the

thread’s flanks, primary peripheral surfaces of the ring

tangential tool, and the axial section

The form of the axial section of the ring tangential

tool (the plane X1Y1) is represented in Fig.13 We have

to notice that the axial tool’s profile is asymmetric

Obviously, in the points B and C (see Fig 12) on the

composed profile of the tool, emerged discontinuities that

may be solved by link this zones and accepting a

deformation of the thread bottom, according to a requiredtarget

3.2.2 Ring tangential tool for ball threadFigure 14 presented the model of the ball thread and itsaxial section, an assembly of circle’s arc, filleted, as so asthe reference systems:

Xyz is the reference system associated

with the ball thread;

X1Y1Z1 reference system associated with

the ring tangential tool’s primaryperipheral surface; and

x0y0z0and x0′y0′z0′ additional reference systems (see

Fig.15)

The generation movement assembly, the movements I,

II, and III, has the significances given in Fig.9

In this way, the helix belongs to the ball thread flute andsituated onto the cylinder with radius Re:

5 

00

Fig 11 HSGT application —ring tangential tool, trapezoidal thread

Table 4 Input parameters of the trapezoidal thread (straight lined segments)

a x coordinate of tool ’s origin −50 mm

b y coordinate of tool ’s origin −32 mm

c z coordinate of tool ’s origin 150 mm

U e Tool ’s rotation around Y 1 axis −18°

The helix, in the reference system X1Y1Z1, associated

with the ring tangential tool with the circle:

X1¼ RScosq;

Y1¼ 0;

Z1¼ RSsinq;

ð24Þ

of the ring tangential tool’s primary peripheral surface,

allow to determine the parameters:α, β, θ, and 8

Other solution may be obtained by knowing the angle of

helix for the cylinder with radius Re,

a ¼ arctg p

and the normal plane to the helix in the point O0 (see

Fig.15)

The plane of the circle RSis revolved around the axis x0

(x0′), with the angle β determined from constructive point

of view from the condition to avoid the interferencebetween the tool and the opposite flank, see Fig 14 The

x axis is symmetrical with the arcs with radius r In Table5,the input parameters correlated with the application HSGT

In Figs 16 and 17, we present the forms and thecoordinates of the characteristic curve and the axial sectionfor the helical surfaces assembly that composes the ballthread flute Obviously, in this case, it is not possible todefine singular points on the profile

4 ConclusionsThis paper presents algorithms and numerical applicationsfor the profiling of ring tools for the generation ofcylindrical helical surfaces with constant pitch, based onthe topological representation of the tool’s primary periph-eral surface The proposed method uses the capabilities ofthe CATIA graphical design environment This method

Helix Projection in xy

Fig 15 Ball thread; axial profile, and reference systems

Fig 14 Ball thread; axial profile, and reference systems

Fig 13 Axial section of the ring tangential tool ’s primary peripheral

surface

Characteristic Curves

V,p

A B

Axial section

Fig 12 3D model of the helical surface; 3D model of the ring

tangential tool ’s primary peripheral surface

allows to determine the composed characteristic curves of

the helical surfaces (the case of the helical flutes of the

motion threads) as so as, the highlighting of the singular

points on the tool’s profile, including modalities for the

solving of the inherent discontinuities by the method of

virtual extending of the profiles

The results obtained in graphical and numerical form

confirm the method quality Based on this method, an

original software, in VBA, was created

The profiling of the ring tangential tool is similar to the

profiling of the side mill tool The particular position of

tool’s axis may limit the length of the machined thread

The specific application HSGT allows the determination

of the characteristic curve (in particular for composed

characteristic curves for complex surfaces) and allows the

solving of problems due of the singular points The profile,

in the tool’s axial section, is rigorously determined in the

specifically HSGT application

The proposed method, developed in the CATIA

graphical design environment, for the profiling of the

ring tangential tool’s primary peripheral surfaces allows

determining the characteristic curves and the axial

section The HSGT application is based on the position of the helical movement—the self-generatingmovement of the surface to be generated—a cylindricalhelical surface with constant pitch They are presentedwith analytical and graphical solutions for the determi-nation of the constructive parameters of the generatingtool Also, four numerical applications for cylindricalhelical surface with constant pitch used in machinepart’s construction were presented

decom-Acknowledgments The authors gratefully acknowledge the cial support of the Romanian Ministry of Education, Research and Innovation through grant PN_II_ID_791/2008.

finan-References

1 Oancea N (2004) Surface generation through winding Volume 2 complementary theorems “Dunărea de Jos” University Publishing House, Gala ţi, ISBN 973-627-106-4, ISBN 973-627-170-6

2 Veliko I, Gentcho N (1998) Profiling of rotation tools for forming

of helical surfaces Int J Mach Tool Manuf 38:1125–1148

3 Rodin RP (1990) Osnovy proektirovania rezhushchikh tov (Basics of design of Cutting Tools) Vishcha Shkola, Kiev

instrumen-4 Shalamanov VG, Smetanin SD (2007) Shaping of helical surfaces

by profiling circles, Russian engineering research, vol 27, No 7,

pp 470-473, ISSN 1068-798X

5 Litvin FL (1984) Theory of gearing NASA Scientific and Technical Information Division, Washington, reference publica- tions 1212

6 Teodor V (2011) Contributions to the elaboration of a method for profiling tools —tools which generate by enwrapping Lambert, Saarbrucken, ISBN 978-3-8433-8261-8

7 Berbinschi S, Teodor V, Oancea N (2010) Contributions to the elaboration of a graphical method for profiling of tools which generate by enveloping I Algorithms, Buletinul Institutului Politehnic din Iaşi, tomul LVI (LX), fasc 2, secţia Construcţii

de Maşini, ISSN 1011-2855, pp 41-48

8 Berbinschi S, Teodor V, Oancea N (2010) Contributions to the elaboration of a graphical method for profiling of tools which generate by enveloping II Applications for rack-gear profiling, Buletinul Institutului Politehnic din Ia şi, tomul LVI (LX), fasc 2, sec ţia Construcţii de Maşini, ISSN 1011-2855, pp 49-56

Fig 17 Axial section of ring tangential tool

Fig 16 Characteristic curve and axial section of ring tangential tool

Table 5 Input parameters of the ball thread

e Half distance between the centers

of circles with radius r

0.155 mm

D j Diameter of centers cylinder of the

axial profile

49 mm

h Distance between the D j diameter

and the center of circle with radius r

0.17 mm

9 Berbinschi S, Teodor V, Oancea N (2010) Kinematical method for

rack-gear tool ’s profiling in CATIA design environment

Proceed-ings of the 14 International Conference ModTech, ISSN

2066-3513, pp 119-122

10 Berbinschi S, Teodor V, Oancea N (2010) Comparisons

between CAD method and analytical method—rack-gear tool’s

profiling The annals of the “Dunărea de Jos” University of

Gala ţi, fasc V, Technologies in machine building, ISSN

mathemat-ORIGINAL ARTICLE

Tool wear and surface quality in milling

of a gamma-TiAl intermetallic

Paolo Claudio Priarone&Stefania Rizzuti&

Giovanna Rotella&Luca Settineri

Received: 24 July 2011 / Accepted: 6 October 2011 / Published online: 30 October 2011

# Springer-Verlag London Limited 2011

Abstract Advanced structural materials for high-temperature

applications are often required in aerospace and automotive

fields Gamma titanium aluminides, intermetallic alloys that

contain less than 60 wt.% of Ti, around 30–35 wt.% of

aluminum, and other alloy elements, can be used as an

alternative to more traditional materials for thermally and

mechanically stressed components in aerospace and

automo-tive engines, since they show an attracautomo-tive combination of

favorable strength-to-weight ratio, refractoriness, oxidation

resistance, high elastic modulus, and strength retention at

elevated temperatures, together with good creep resistance

properties Unfortunately such properties, along with high

hardness and brittleness at room temperature, surface damage,

and short and unpredictable tool life, undermine their

machinability, so that gamma-TiAl are regarded as difficult

to cut materials A deeper knowledge of their machinability is

therefore still required In this context the paper presents the

results of an experimental campaign aimed at investigating the

machinability of a gamma titanium aluminide, of aeronautic

interest, fabricated via electron beam melting and then

thermally treated Milling experiments have been conducted

with varying cutting speed, feed, and lubrication conditions

(dry, wet, and minimum quantity lubrication) The results are

presented in terms of correlation between cutting parameters

and lubrication condition with tool wear, surface hardness and

roughness, and chip morphology Tool life, surface roughness,

and chirp morphology showed dependence on the cutting

parameters Lubrication conditions were observed to heavilyaffect tool wear, and minimum quantity lubrication wasshown to be by far the method that allows to extend tool life.Keywords Intermetallic alloy Machinability Tool wear Lubrication

1 IntroductionGamma titanium aluminides (γ-TiAl) are intermetallicalloys that contain 44–48 atomic percent Al (32–35 inweight percent), with element additions of Cr, or Mn toincrease ductility, and Nb to improve strength and oxidationresistance; γ-TiAl alloys can be used as an alternative toNi-based superalloys for thermally and mechanicallystressed components in aerospace and automotive engines[1, 2] Gamma-TiAl alloys show approximately half thedensity of Ni superalloys, high strength/weight ratio, highstiffness, high refractoriness, and high temperature strength.Furthermore, they show fatigue resistance values close to100% of yield strength [2]

In spite of these advantages, γ-TiAl alloys show somedrawbacks: low ductility at room temperature, whichtypically ranges between 0.3% and 4% in terms ofelongation at rupture (depending on composition andmicrostructure), together with low fracture toughness.Furthermore, these characteristics, along with low thermalconductivity and chemical reactivity with many toolmaterials, make γ-TiAl difficult to cut materials Otherfeatures impairing machinability are the sensitivity to strainrate, with a strong tendency to hardening, the saw toothchip shape, the built-up edge and the presence of abrasives

in the alloy microstructure that contribute to acceleratedwear of the cutting edge, and the formation of large crater

P C Priarone:S Rizzuti ( *):G Rotella:L Settineri

Department of Production Systems and Business Economics,

wear on the rake face and evident chipping phenomena On

the workpiece side, we can observe that these unfavorable

characteristics translate into surface hardening, residual

stresses, poor finish, and presence of microcracks,

impair-ing fatigue strength of the finished component [3–5]

Nevertheless, the interest for gamma titanium aluminides

applications is high because of highly stressed components

in the automotive and aerospace sectors, such as engine

valves for high-performance engines, exhaust nozzles, and

turbine blades, and there still is a need for further studies,

particularly to fully understand the machinability of these

alloys with conventional or nonconventional processes and

to optimize process parameters

In this paper the results of an experimental campaign

aimed at investigating the machinability of aγ-TiAl alloy

produced via electron beam melting (hereafter EBM) are

reported Milling experiments have been conducted with

varying cutting speed, feed, and lubrication conditions

Machining results are presented and discussed in terms of

tool wear, surface hardness and roughness, surface integrity,

and chip morphology

2 Experimental setup

The milling tests were carried out on a γ-TiAl specimen

with rectangular shape (120×120×30 mm), obtained via

EBM and subsequently thermally treated The sample is

illustrated in Fig 1 EBM is an additive manufacturing

process for metal parts that starts from powders and melts

them layer after layer with an electron beam in a high

vacuum Unlike some metal-sintering techniques, the parts

are almost fully dense and void free The high vacuummakes it suited to manufacture parts in reactive materialswith high affinity for oxygen, e.g., titanium [6, 7] Theprocess is followed by a heat treatment to improve densityand release residual stresses The applied heat treatmentwas 5 h at 1,095°C, then hot isostatic pressing for 4 h at1,285°C, then 2 h at 1,305°C

The chemical composition of the alloy is listed inTable 1, while the main properties at room temperatureare reported in Table2 Furthermore, the specimen showed

an average initial hardness of 273 HV30 (with a standarddeviation of 5.2 HV30)

Specimens to perform microstructural analysis wererandomly cut from the workpiece; the samples were groundand polished with suspension of colloidal silica (SPM) up

to 0.1 μm abrasive particle size, then etched in a Kellersolution of 100 ml H20, 2.5 ml HNO3, 1.5 ml HCl, and

1 ml HF prior to inspection A microstructural observation

on the material reveals a typical lamellar structure, asshown in Fig 2, with a different orientation of lamellae.The microstructure is not always homogeneous, as shown

in the Fig 3, where some porosity can be observed Asexpected, the EBM fabrication process does not deliver amaterial with full density [8,9]: in the present case, density

is around 98%

The experimental tests were performed using a three-axisCortini M500/F1 vertical CNC milling machine Themachine has a continuously variable spindle that reaches

up to 8,000 rpm, with a peak power of 3.7 kW and amaximum torque of 24 Nm

Tools used in the experiments were 10-mm diameterVergnano F405 carbide ISO K30/K40 end mills, with four

Fig 1 Sample used for the

mill-ing tests: as provided by the

supplier (a) and after preparation

for the milling tests (b)

Table 1 Chemical composition of the machined γ-TiAl alloy (in weight percent)

32.0–33.5 4.5–5.1 2.2–2.6 <0.08 <0.02 <0.015 <0.04 <0.001 <0.05 Balance (<60%)

uncoated edges Table3reports the geometrical parameters

of the tools, while a new cutting tool is shown in Fig.4

In order to investigate the alloy’s machinability, the

experimental plan listed in Table4 was carried out Axial

depth of cut daand radial depth of cut drwere kept constant

and both equal to 0.3 mm, while cutting speed V and feed

per tooth f were assumed as independent input variables

The cutting parameters were changed according to a 22+*

experimental plan, enriched by other experimental points

explored to better point out the dependence on the

lubrication conditions [10] The milling operations were

undertaken in down-milling direction, in dry, MQL, and

wet conditions MQL lubrication was performed with a

Novatea Accu-Lube Minibooster microlubrication system,

with a vegetal lubricant flow of 0.3 ml/min at an air

pressure of 5.5 bars, while wet cutting was carried out with

a 5% emulsion of mineral oil in water, with a flow of 10 l/

min Three repetitions for each experimental point have

been performed, and the tests were executed following a

random order to reduce the time-dependence effects

Tools were periodically examined in order to measure

wear at different cutting times, by means of a stereo

microscope Leica MS5 (with ×40 magnification), equipped

with a high-resolution camera Leica DFC280 for image

acquisition The tests were performed until the fixed limit

of 100μm for tool wear was reached: the parameter used to

evaluate tool wear as a function of processing time is the

maximum value between flank wear and corner wear

measured on the secondary flank surface, as shown in

Fig 5 Such restrictive condition, typical of a finishing

operation, has been chosen since the manufacturing of a

γ-TiAl component typically starts from a semimanufactured

part, obtained by Rapid Manufacturing processes or castingprocesses, in a shape that is close to the finished part.Therefore cutting is often limited to semifinishing orfinishing Furthermore, surface integrity and tolerances of

a component made ofγ-TiAl alloy are critical requirementsthat must meet the severe specifications of the automotiveand especially aerospace industry In fact excessively worntools generate more deformations, with surface hardening,residual stresses, and poor finish

The roughness of the milled surface was measured by aHommelwerke Tester T1000 according to DIN EN ISO

4287 In particular, the arithmetic mean roughness value Ra,the maximum roughness profile height Rt, the skewness Rskand the curtosis roughness Rkuwere measured in the feeddirection It is useful to remark that Rsk and Rku areimportant indexes when the machined surface load-bearingcapability needs to be investigated

The HV30 hardness of the generated surface wasmeasured by an Emcotest M4U 025 universal hardnesstester, according to the reference standard DIN EN ISO

6507 Microstructural analysis of the material and tion of the generated surface has been performed using aninverted optical microscope Leica MEF4U with magnifica-tion up to ×1,880, while chip morphology was investigated

observa-Table 2 Mechanical properties of γ-TiAl alloy

500 µm

A

100 µm Detail: A

Fig 2 Lamellar microstructure

of the sample

100 µm

Fig 3 Optical micrograph showing porosity

by a scanning electron microscope LEO 1450 VP, with

theoretical resolution equal to 10 nm

3 Results and discussion

The obtained experimental data on tool wear, surface

roughness, and hardness have been analyzed: Table 5

summarizes the results discussed below

3.1 Tool wear

Figure6 illustrates examples of worn tools, for the cases V=

50 m/min and f=0.08 mm/tooth, after 30 min of cutting time

and for each lubrication condition From the observation of

the worn tools, it can be assessed that the tool fails more

often for corner wear, especially in the case of higher feed

per tooth This is caused by chipping of the cutting edge tip

Figure 7 shows two SEM-backscattered images of the

tool rake face and of the tool tip The backscattered images

show some adhesion of workpiece material, besides evident

wear signs and microcracks

The results in terms of tool life listed in Table5highlight

that tool wear rate, in dry condition, increases as the feed

per tooth and the cutting speed increase, as expected These

trends resulted also in agreement with the results reported

by Branoagirre for other gamma-TiAl alloys, manufactured

as solidified ingot and extruded after solidification [11,12]

In dry milling, the experimental data for tool life estimationare well fitted by the following full quadratic model(correlation index R=0.99), whose behavior is shown inFig 8:

TL¼ 658:38  6:1748  V  9517:6  f þ 41:555  V  f

þ 0:014238  V2þ 36593  f2

In addition, variability of tool wear results for the repeatedtests remained lower than 10% in terms of relative range.Lubrication condition is also a factor strongly affectingwear mode and tool life Typical tool wear curves are

Table 3 End mills geometrical parameters

Fig 4 Uncoated Vergnano F405 carbide end mill

Table 4 Experimental plan

300 µm

Fig 5 Tool wear measurement criteria

shown in Fig 9, for the case f=0.10 mm/tooth, and in

Fig 10 for the three different lubrication conditions (wet,

dry, and MQL) Each experimental point plotted into the

graphs is the average value of the wear measured on the

four edges of the mill

In the case of wet cutting, the cutting edge abrupt

cooling causes severe thermal shocks, leading to the

breakage of the cutting edges This explains the very low

tool life achieved MQL appears to be, by far, the most

advantageous lubricating system in terms of tool wear

3.2 Surface hardnessFigures 11and 12show the trends of the surface hardness

as a function of processing time Each point in the graph isthe average value of five experimental measurements Inaddiction Table 5 reports the estimated hardnesscorresponding to the maximum tool life The effects ofcutting time are visible: hardness increases as a result ofstrain hardening due to tool wear This occurrence iscommon to all tests performed

In dry conditions, as far as the effects of the cuttingparameters are concerned, it can be assessed that thesurface hardness increases as the feed per tooth increase,and this trend is stronger in the range between 0.08 and0.1 mm/tooth On the other hand, the dependence on thecutting speed is more complex: this is due to a prevailinghardening effect at intermediate speed, compensated bythe temperature increasing at higher speed that softens thesurface [13]

The effects of the different lubrication conditions onhardness are more complex to explain: in wet conditions,the lubrorefrigerant has a hardening effect, as if the uppermaterial layers underwent a heat treatment In dry con-ditions, this effect is not present, but the high frictioncoefficient and therefore the high strain keep the hardnessrelatively high In MQL conditions, the low frictioncoefficient, combined with the gentle cooling effect, limitsstrain hardening with respect to the other cutting conditions[13]

3.3 Surface roughnessTable5 shows the results of the roughness measurements:the values are the average of all measurements taken until

Table 5 Experimental results

Lubrication condition V (m/min) f (mm/tooth) Tool life (min) Surface hardness (HV 30 ) Roughness indexes

Fig 6 Worn tools for the case of V=50 m/min and f=0.08 mm/tooth,

after 30 min of cutting time

the tool wear limit was reached As for the generatedsurface quality, the mean roughness Ra and the maximumroughness profile height Rt present a clear trend vs thecutting speed and the feed per tooth As expected, in drylubrication conditions, Ra and Rt indexes decrease as thecutting speed increases and the feed per tooth decreases.

In Fig 13a full quadratic model of Rtas a function ofcutting speed and feed per tooth in dry conditions isrepresented, according to the following equation (correla-tion index R=0.99):

Rt¼ 0:11522  0:03584  V þ 62:784  f  0:52924  V  f

þ 0:000587  V2 139  f2

Rahas a very similar behavior

On the other hand, Table5shows that Rkuand Rskhave amore complex dependence on the cutting speed and feed

Fig 8 Tool life T L (in minutes) as a function of V and f, for dry

lubrication condition The model used is reported in the full quadratic

model as shown in Section 3.1

0 20 40 60 80 100 120 140

Tool wear limit

Fig 9 Typical tool wear curves

for the case f=0.10 mm/tooth

Fig 7 Backscattered SEM

images of one cutter after

machining Tool rake face

(a) and tool tip (b), for the case

of V=50 m/min and

f=0.08 mm/tooth, dry

conditions after 30 min of

cutting time, respectively, show

microcracks and material

adhesion signs

per tooth: Rku shows a maximum for the intermediate

cutting speed and feed and decreases thereafter, while Rsk

behaves in the opposite way This effect is due to the

characteristics of the chip formation mechanism that, for

intermediate values of cutting speed and feed, generates

microcraters on the machined surface, being the material

obtained via a powder-melting process [8,9]

The effects of different lubrication systems on the

surface roughness do not follow strictly the indications of

the literature, being the results for dry cutting better than in

the other cases ([5] and ref therein) In Fig 14 the

generated surface is observed, in the case of V=50 m/min

and f=0.08 mm/tooth, in dry and MQL conditions The

presence of microcraters is evident, due to the

microstruc-ture of the sample, manufacmicrostruc-tured via a powder-compacting

process [8, 9] and to the chip formation mechanism This

phenomenon is less evident at different cutting speeds

Tool wear limit

Fig 10 Typical tool wear curves

for the case V=50 m/min,

f=0.08 mm/tooth

35 40 45 50 55 60 65 70

3.4 Chip morphology

Figures 15 and 16 obtained by SEM show, with different

magnifications, the chip morphology for f=0.06 mm/tooth

and for dry lubrication condition, at different cutting

speeds Chips are in general very small, almost powder

like, and with sharp edges, due to the low deformability of

the material

Looking at Fig 15, a clearly perceivable increase in the

average chip size at higher cutting speed can be observed: this

phenomenon is due to a prevailing effect of the temperature

increase (resulting in a greater deformability of the material)rather than of the higher strain rate A further effect of thehigher temperature is that the chip width is higher and moreregular The higher chip temperature is confirmed by thedifferent color of the chip that is darker at higher cutting speed,

an effect that cannot be seen from the SEM images

From the observation of Fig 16, obtained with highermagnification, we can clearly see the shear planes and thelower thickness of the lamellae as the cutting speedincreases The chip formation mechanism recalls the “carddeck” model of Piispanen [14]

4 Conclusion

In this paper, the results of milling experiments conducted

on a particularγ-TiAl, manufactured starting from powders

via electron beam melting, followed by a particular thermal

treatment are presented The experiments are conducted in

finishing or semifinishing cutting conditions, with three

different lubrication conditions

The results show that, in dry conditions, tool life is well

modeled by a full quadratic equation as a function of cutting

speed and feed per tooth Lubrication conditions heavily affect

tool wear: in wet conditions, the tool wear rate is higher than

in dry conditions, while minimum quantity lubrication is by

far the method that allows to extend tool life

Surface roughness parameters show dependence on the

cutting parameters, at least in the explored range Ra and Rt

increase with the tool wear and show classical dependence

on cutting speed and feed, while the nonmonotonic behavior

of Rsk and Rku on cutting speed and feed is due to the

mechanism of chip formation that generates microcraters on

the surface with intermediate values of speed and feed

Chip morphology shows a typical“card deck” chip formation

mechanism, with chip length and width dependent on the cutting

speed Further work is needed to optimize tool geometry in terms

of cutting angles and cutting edge pretreatment as well as to

explore the effects of advanced tool coatings

Acknowledgments The authors wish to acknowledge the

contribu-tion of the Region Piedmont and of the EU that funds this research

activity in the framework of a MANUNET Project, the company

Fratelli Vergnano S.r.l and in particular Mr Guido Vergnano for

supplying the tools, and the company AVIOPROP for supplying the

workpiece material.

References

1 Austin CM (1999) Current status of gamma titanium aluminides for aerospace applications Curr Opin Solid State Mater Sci 4:239–242

2 Loria EA (2000) Gamma titanium aluminides as a prospective structural materials Intermetallics 8:1339–1345

3 Aspinwall DK, Dewes RC, Mantle AR (2005) The machining

of γ-TiAl intermetallic alloys Annals of the CIRP 54(1):99– 104

4 Mantle AR, Aspinwall DK (2001) Surface integrity of a high speed milled gamma titanium aluminide J Mater Process Technol 118:143 –150

5 Sharman ARC, Aspinwall DK, Dewes RC, Bowen P (2001) Workpiece surface integrity considerations when finish turning gamma titanium aluminide Wear 249:473 –481

6 Cormier D, Harrysson O, Mahale T, West H (2007) Freeform fabrication of titanium aluminide via electron beam melting using prealloyed and blended powders Research Letters in Materials Science, Hindawi Publishing Corporation 1–4

7 Yu KO (2001) Modeling for casting and solidification processing Taylor & Francis, New York

8 Petropoulos GP, Pandazaras CN, Davim JC, West H (2010) Surface texture characterization and evaluation related to machin- ing Springer-Verlag, London

9 Cotterell M, Byrne G (2008) Dynamics of chip formation during orthogonal cutting of titanium alloy Ti –6Al–4V Annals CIRP 57 (1):93 –96

10 Montgomery DC, Runger GC (2007) Applied statistics and probability for engineers Wiley, London

11 Beranoagirre A, Lopez de Lacalle LN (2010) Optimizing the milling of titanium aluminide alloys Int J Mechatron Manuf Syst

ORIGINAL ARTICLE

NURBS approximation to the flank-milled surface swept

by a cylindrical NC tool

Chenggang Li&Sanjeev Bedi&Stephen Mann

Received: 1 May 2009 / Accepted: 18 October 2011 / Published online: 12 November 2011

# Springer-Verlag London Limited 2011

Abstract In this paper, a method to approximate the

flank-milled surface swept by a cylindrical cutter with a

non-uniform rational basis spline (NURBS) surface is presented

The swept surface produced by the moving tool can be

calculated as a collection of points organized as a series of

grazing curves along the surface The generated NURBS

surface closely matches the grazing surface The deviation

between this surface and the grazing surface is calculated

and is controlled by increasing the number of control points

used to represent the surface

Keywords Curved surface design Flank milling Surface

numerical analysis Surface error control

1 Flank milling

Flank milling is a widely used machining method in today’s

manufacturing industry In flank milling, the side of the

cutter machines the surface, removing the stock in front of

it Compared to other machining methods, flank milling can

offer higher machining efficiency, higher material removal

rate, and provide a better surface finish Flank milling isused in the machining of turbine blades, fan impellers, andother engineering objects Researchers working on improv-ing flank milling have developed various tool positioningtechniques in the past decade In general, these techniquescan be categorized into three classes, namely, direct toolpositioning methods, step-by-step tool positioning methodsand improved tool positioning methods

In the direct tool positioning method, the cylindrical cuttingtool is used to machine a ruled surface and the tool ispositioned to be tangential to the given surface at one point onthe ruled line either in the middle or end (or near the end)while the tool axis is parallel to the same ruled line.Alternatively, the tool is positioned to directly touch twopoints on the ruled line The methods that belong to this classinclude early methods [1,2], Rubio et al.’s method [1], Stute

et al.’s method [3], Liu’s method [4], etc The error defined

as the difference between the machined surface and thedesired surface in this class is higher than errors from theclasses described below, but the cutting tool is easy toposition and the computation time of tool positioning is low

A step-by-step tool positioning method is an ment over the direct tool-positioning method In this class,the cutting tool is first positioned on the given surface withone of the direct tool positioning methods and then the tool

improve-is lifted and/or twimprove-isted to reduce the error between themachined surface and the desired surface Methods likethose developed by Rehsteiner et al [5], Bohez et al [6],Tsay and Her [7], and Bedi et al [8–11] belong to this class

In comparison to the direct tool positioning methods, thestep-by-step tool positioning methods result in a machinedsurface that is close to the desired surface but thecomputation time of these methods is long

An improved tool positioning method is a combination ofthe techniques used in the two classes described above In this

method, the cutting tool is positioned on the given surface so

that it touches at three contact points A machined surface can

be generated with many tool positions, each of which has

three contact points (two on the guiding curves and one on the

rule) Three contact points at any tool position can be obtained

directly by solving seven transcendental equations based on

the given geometrical conditions The error between the

machined surface and the desired surface is small in this type

of tool positioning method Redonnet et al.’s method [12] and

Monies et al.’s method [13–15] belong to this class This

class of methods results in high accuracy machined surfaces

However, it requires the solution of seven transcendental

equations at each tool position, which makes it

computa-tionally cumbersome

All of the methods described above focus on ruled surfaces

and attempt to use different techniques to reduce the deviation

between the machined surface and the given surface In spite

of it being widely recognized that flank milling produces

curved surfaces, no one has attempted to design free-form

surfaces that can be flank milled with accuracy using no

approximations Some researchers, like Elber and Fish [16],

Bohez et al [6], tried to use flank milling techniques to

machine free-form surfaces In their methods, they first

divide the target surface into multiple ruled surfaces, and

then machine these ruled surfaces in pieces with one of the

above techniques Obviously, there are spatial limitations to

this method and it results in long tool paths

One of the key applications of flank milling is machining

of impellers Engineers design impeller surfaces to extract

power from fluid flowing over them Designers use

sophisticated aerodynamic analysis to improve the efficiency

and performance of impeller However, to machine the

impellers, these surfaces are approximated with ruled

surfaces or produced at large cost with point machining

techniques Manufacturing engineers simplify the curved

surface to a ruled surface to machine the part using flank

milling even though a curved surface is better for efficiency

and other requirements If a curved surface that can be flank

milled directly can be designed, it will be of great benefit not

only for manufacturing but also for application engineering

With this surface design technique, engineers would be able

to design impellers and optimize their performance without

worrying about compromises during machining How to

achieve this goal is a big challenge in surface design and

machining In this paper, this challenge is probed and a

solution is presented First, the surfaces that can be machined

with the flank milling method (by a cylindrical tool) are

identified and then a method to design such surfaces is

developed and tested

The focus of this study is to develop a method to

represent the machined surface with a non-uniform rational

basis spline (NURBS) or B-spline representation so that the

surface can be generated accurately by flank milling

technique Such a surface-fitting method can be used byengineers to design impellers and blades geometrically.The key idea behind this method is to approximate thegrazing surface with an NURBS definition based on theproperties of guiding curves and the cylindrical cutter Ateach tool position, the corresponding grazing curve lies onthe cylindrical tool surface When projected onto a planeperpendicular to the tool axis, this grazing curve becomes

an arc and can be represented by an NURBS Theinvestigation shows that this NURBS can be constructedwith three or four control points with their correspondingweights (as shown in Fig 1) A method of moving thesepoints off the plane along the tool axis direction isdeveloped to accurately model the 3D grazing curve Asthese control points (representing the grazing curve) aremoved along guiding curves in a manner that nearly retainstheir grazing curve character, a surface is generated ThisNURBS surface closely represents the grazing or sweptsurface and can be used in design

Our method is similar to that of Yang and Abel-Malek[17], who gave an algorithm for approximating the sweptvolume of an NURBS solid However, since we are focused

on flank milling with a cylinder, our method is significantlysimpler: we do not have to intersect a set of swept surfacesand the computation of points on the swept surface issimpler Further, while they compute a volume, we onlyneed compute a single swept surface to use for machining.Multiple tool passes with our method would need tointersect multiple swept surfaces, but Yang-Abel-Malek’smethod would have to intersect multiple swept volumes inaddition to intersecting surfaces for each pass

In the following sections, the proposed method isinvestigated and studied In Section 2, the tool positioningmethod used to test the proposed solution, Bedi et al.’s

Fig 1 The cylindrical tool and guiding curves: T(u) and B(u) are two guiding curves P T , P 0 , and P B are three control points of a grazing curve; w T , w 0 , and w B are their corresponding weights The tool moves along feed directions

method is given along with a method of computing the

grazing surface (the swept surface) and a surface error

measurement technique Section3 presents the strategy of

modeling the grazing surface with NURBS definition The

basic theory of this method is developed A flow chart to

describe the implementation procedure is given in Section

4 Accuracy control of the surface design for flank milling

with examples is discussed in Section 5 The proposed

method is compared to the developed least square surface

design method in Section6 The paper is closed with the

conclusion in Section7

2 Tool positioning method and swept surface

Different tool positioning methods result in different tool

locations, orientations, and directions of motion for

ma-chining the same surface Thus, a surface designed for flank

milling will apply to a specific method of tool positioning

In this work, a surface design method is developed for

designing a surface that can be flank milled The tool

positioning method developed by Bedi et al [8–10] is

applied to test the proposed method, although our method is

general enough that it should be possible to use it with any

tool positioning methods A cylindrical tool is used in this

study

In addition to the tool positioning method, the surface

design technique also depends on the actual shape of the

surface produced by a moving tool In previous work, the

surface that is produced by flank milling has been evaluated

Bedi et al [8] suggested a cross-product method to calculate

the envelop surface Mann et al [18] generalized this

method and applied it to tools with a general surface of

revolution The method evaluates the grazing surface (the

swept surface) as a collection of discrete grazing points

calculated using the cross-product method Li et al [10] and

Menzel et al [9] applied this method to conical and

cylindrical tools to simulate the machined surface and used

it to optimize each tool position Li et al [19] also used this

method to study the surface error

Lartigue et al [20] presented a similar method to

determine an envelop surface in their surface deformation

analysis Senatore et al [21] used a similar method to

define the grazing points and envelop surface They also

geometrically proved that, at each tool position, the contact

points (between cutter and guiding curves, cutter, and rule)

are on the envelop surface All these techniques use discrete

points to simulate the grazing surface (or the envelop

surface) Furthermore, they use the grazing surface to

approximate the machined surface The accuracy of this

approximation can be assessed with one of the error metrics

used by different researchers [19] The error metric used in

this study is also described below

2.1 Positioning a cylindrical tool on the surfaceBased on Bedi et al.’s technique [8], a cylindrical cutterwith radii R is positioned tangent to two curves at the sameparameter value as shown in Fig 2 The geometry of thetool and its relation to the guiding curves result in a set ofsimultaneous equations When these equations are solved,the cutter position (PTand PB) at parameter value u can beobtained [8] As the tool moves along the curves, itproduces a swept surface

2.2 The swept surface

A model of the swept surface is the basis of the proposedmethod The swept surface is composed of grazing curvescalculated at the various tool positions After a tool position

is defined, the grazing curve at each tool position can bederived using the cross-product method given in [8] andreviewed in the next paragraph

As shown in Fig.2if the velocity at point PTis VTand atpoint PB is VB, then the velocity between PB and PTalongtool axis direction can be linearly interpolated and is given by

V ¼ VBð1  vÞ þ VTv; 0 v  1 ð1ÞThe coordinate between PB and PT along the tool axisdirection can also be linearly interpolated and is given by

P¼ PBð1  vÞ þ PTv; 0 v  1 ð2ÞThe grazing curve between T and B is calculated as

G¼ P þ V Taxis

where, Taxisis the cutter axis direction

Fig 2 Cutter rolling on two rails

Using Eq.3, a continuous grazing curve can be obtained.

For plotting, only a series of discrete grazing points are

generated to represent the grazing curve By connecting

consecutive grazing curves along the u direction into a

mesh, a swept surface (or a grazing surface) is generated

This surface represents the machined surface accurately

This surface is also composed of discrete points and does

not have an exact NURBS representation In engineering

applications, an NURBS equation would be more helpful

and acceptable especially when the target surface needs

to be connected to other NURBS surfaces around it

Thus, to define a surface (with NURBS) that can be flank

milled will be of significance in today’s engineering

applications

2.3 The surface error measurement

To effectively evaluate the difference between the

grazing surface and the approximate NURBS surface,

an error metric is required In the literature, different

error measurement methods are used In a previous

work, these methods [19] were analyzed and compared

Generally, there are four types of error metrics used in the

literature These are the radial method, the parametric

method, the tangent plane method and the closest point

method Even though these methods are designed for

comparing the machined surface and the designed surface,

they can also be used to measure the difference between

the grazing surface (or a grazing curve) and the

approx-imate NURBS surface (or an approxapprox-imate NURBS curve)

in surface design and are thus relevant to the context of

this work

If the grazing surface and the approximate surface are

close, the two surfaces will nearly coincide and the errors

from the different error measurement methods described

above should be close or the same In this work, the

approximating NURBS surface and the grazing surface are

usually close However, under some circumstances, the

parametric error method is not accurate enough for our

work Therefore, a modified parametric error measurement

method will be used in this research to better reflect the

surface error variation

Figure 3 illustrates the modified parametric error

measurement method At each specific tool position, a

grazing curve is calculated and plotted as the dash line An

NURBS curve is used to approximate the grazing curve A

plane perpendicular to the tool axis can be created The two

curves intersect the plane with two points, A and B The

distance between the A and the B is used as the

approximating error at the grazing point A We call this

method as the reparameterized parametric method and will

use it for error measurement in this paper

3 NURBS approximating a swept surfaceThe idea behind the proposed technique for approximating agrazing surface is to select a few representative grazing curvesand construct a surface that parametrically is close to thesegrazing curves If enough grazing curves are used and they areclose enough to one another, then the resulting surface should

be a good approximation to the swept surface

Each grazing curve is modeled using an NURBS tation (e.g., in Fig.9, the control points P0,0, P1,0, P2,0are theNURBS representation for one grazing curve) Since we areworking with a cylindrical tool, the projection of the grazingcurve into a plane perpendicular to the tool axis direction will

represen-be a circular arc Our approximation for the grazing curvestarts with this circular arc, and then the control points aremoved off the plane to form a good approximation to thegrazing curve A sequence of NURBS approximations to thegrazing curves are used for several tool positions as the tool ismoved along two guiding curves (T(u) and B(u) in Fig.9) Byincreasing the number of control points along both theguiding curve and tool axis directions, better representations

of grazing curves can be defined and a better NURBSapproximation of the grazing surface can be obtained Thedetails of this method are presented below

3.1 Approximating a grazing curveThe grazing curve is the contact between the grazingsurface and the cutting tool Thus, it lies on the cylindricaltool surface This grazing curve is illustrated in Fig.4 as adashed line This grazing curve can be approximated by aquadratic NURBS curve with control polygon P0P1P2shown as a solid line in Fig 4, where P0is at the bottom

of the guiding curve at B(u) and P2is on the top guidingcurve at T(u) For simplicity, the coordinate system is setup

at the bottom of the cylinder center with the Z axis lyingalong the cylinder axis P0and P2have the same parametervalue u along the guiding curves and are known;they alsolie on the grazing curve P1 needs be determined The

Fig 3 A modified parametric error measurement method

grazing curve with the end points P0 and P2 is projected

onto the x-y plane (P0 and P2 correspond to P0and P2)

This projected curve,Pp0\Pp2, is a two-dimension arc and can

be represented by a quadratic NURBS curve [22] with three

weighted control points P0, P1p and Pp2: The X and Y

coordinates ofPp0and Pp2 are known; however,Pp1 needs to

be calculated.Pp1 is calculated from the intersection of the

two tangent lines passing through Pp0 and P2p If α is the

angle of the arcPp0\P2p; then the weights w0, w1, and w2at

points P0, P1, and P2 are [22]:

w0 ¼ w2¼ 1; w1¼ cosða=2Þ

Figure5 shows this relationship graphically The arcPp0\P2p

can be represented exactly as a 2D NURBS curve Cp(u)

with control pointsPp0; P1p andPp2 as

CpðuÞ ¼ð1  uÞ2w0P0pþ 2uð1  uÞw1Pp1þ u2w2Pp2

ð1  uÞ2

w0þ 2uð1  uÞw1þ u2w2

: ð4Þ

3.1.1 Modeling of the grazing curve

Once the arc has been defined as a 2D NURBS curve, its

control points can be stretched along the tool axis direction

untilP0pmoves to P0,Pp2 moves to P2andPp1 moves to P1,

where P1 needs to be decided This changes the 2D

NURBS curve of (5) to the 3D NURBS curve

CðuÞ ¼ð1  uÞ2w0P0þ 2uð1  uÞw1P1þ u2w2P2

ð1  uÞ2

w0þ 2uð1  uÞw1þ u2w2

: ð5Þ

The X and Y coordinates of P1are the same asPp1: The Z

coordinate of P1must be properly selected to make the 3D

NURBS curve closely match the grazing curve at this toolposition

The grazing curve is a function of the magnitude anddirection of the velocities VTand VBas shown in Eqs.1,2,and3 Since the Z coordinate of P1must be determined bymeasuring the deviation from the grazing curve, it becomes

a function of VTand VB A convenient assumption would be

to assumejVTj ¼ jVBj: This assumption will put P1in themiddle of P0and P2 The impact of this assumption on theerror is studied in following section Figure 4 shows thisrelationship graphically The Z coordinates of P0and P2are

0 and h and P0and P2pass through the contact points B(u)and T(u), respectively

3.1.2 Error in grazing curveðjVTj ¼ jVBjÞ

If the Z coordinate of point P1 is set to half of theeffective contact length h (hx= h/2, with h being measuredalong the tool axis direction between points P0 and P2),then the curve generated by Eq 5 can be used to checkthe deviation of the grazing curve using the reparame-terized parametric error measurement method described inSection2.3 To check that the error in our approximationwas small enough, we ran a test For this test, theparameters of the cylindrical cutter and control pointswere

P0½R cosðp=6Þ; R sin p=6ð Þ:0; P1

R cos ð p=4 Þ cos ð p=12 Þ;R sin ð p=4 Þ

Fig 4 Grazing curve and its control points on the cylindrical surface

Fig 5 Arc with its control points

of the cylindrical cutter; h is the effective contact length

along the axis of the cylindrical cutter; VB and VT are

velocities at points P0and P2, their directions are along

tangent line directions of each circle and their magnitudes

are true velocities; w0, w1, and w2 are the weights of

points P0, P1, and P2

Using Eqs 1, 2, 3, and 5, the grazing curve and the

approximate NURBS curve can be obtained The deviation

between the two curves is calculated and is shown in Fig.6

The errors at v=0, v=0.5, and v=1 are zero The shape of

the error curve is symmetric and the maximum error is

smaller than 0.035 mm

A close study of this error shows that the deviation

between the NURBS curve and the grazing curve

depends on the angle α between VBand VTwhich lie in

the plane perpendicular to the tool axis, the radius of the

cylindrical cutter, etc The contact length (L) between the

cutter and machined surface, however, has little influence

on it Different parametric combinations were considered

and the resulting maximum deviations are listed in

Tables1,2, and 3 From these tables, it can be seen that

the influence of the angleα measured between VBand VT

and the radii of the cutter are significant The larger the

angleα, the larger is the deviation; the bigger the radius,

the bigger is the deviation (the deviation varies linearly to

the tool radius) To effectively control the deviation, more

control points can be used and curve tolerance

require-ment (the permitted error between the desired curve and

the grazing curve) can be satisfied In general, three

control points satisfy most engineering applications for

α<30° and R<30 mm

3.1.3 Modeling a grazing curve ðjVTj 6¼ jVBjÞ

In a general situation, the magnitudes of velocities VBand

VTare unlikely to be equal The magnitude of VB(or VT)depends on the geometry of the guiding curves and thecutting tool Different velocity magnitudes of VB and VTinfluence the velocity distribution along tool axis, and as aresult affect the shape of the grazing curve If the Z coordinate(hx) of the middle control point P1 is kept as hx=h/2, thedeviation between the given grazing curve and the approx-imate NURBS curve will increase This is shown byconsidering the same example as before but where themagnitude of VTis bigger than VB (jVTj jV= Bj ¼ 1:07) Themaximum deviation in this case is shown in Fig 7 and ismuch bigger than the maximum deviation in Fig.6

To reduce the maximum deviation along the grazing curve,

we moved the control point P1from the middle along the toolaxis direction toward the point with the smaller velocitymagnitude By moving P1along the tool axis direction, weensure that our approximation of the grazing curve willalways lie on the surface of the cylindrical tool The length

of movement depends on the difference of the twomagnitudes The bigger the difference, the longer themovement Our study shows that if the ratio between thetwo magnitudes is less than k=1.35 (k ¼ jVBj jV= Tj 1:35

ork ¼ jVTj jV= Bj 1:35Þ; then the movement is less than orequal to (k−1)h/2 For the above example, P1 was movedtoward P0 by 1.55 mm along the tool axis The resultingdeviation between the grazing curve and the approximationNURBS curve is significantly reduced as shown in Fig.8

If the ratio k is bigger than 1.35, moving the point P1does not reduce the maximum deviation enough to satisfyengineering requirement In this case, four or more controlpoints are needed to approximate the grazing curve Thesimplest way to increase the number of control points is touse knot insertion [22]

If four or more control points are used, their locationsalong the axis of the tool are uncertain These points aremoved along the tool axis in a direction that reduces thedeviation between the grazing curve and the approximatecurve Normally, four control points will satisfy require-

Fig 6 Deviation along the grazing curve ifjVBj ¼ jVTj

Table 1 Max error for varying L ( α=30°, R=10 mm)

ments of normal engineering applications and produce

surfaces that approximate the desired surface well

3.2 Modeling of a surface

In the NURBS representation of the grazing curve, the outer

control points move along two guiding curves T(u) and B(u) as

explained in Section 2 To build an NURBS surface

representation of the swept surface, we will make T(u) and

B(u) be two of the boundaries of our NURBS surface To do

so, the two guiding curves, T(u) and B(u), must be

constructed with the same number of control points If the

number of control points and the knot vectors are different for

T(u) and B(u), then the method requires that additional control

points be added to either or both T(u) and B(u) to ensure they

have the same number of control points and knots

Since T(u) and B(u) are the boundary curves of the

grazing surface, the number of control points in the

approximate NURBS surface along the generating lines

should be equal to or greater than the number of control

points in T(u) (or B(u)) As an example, let the number of

control points used to define T(u) and B(u) be three, and

consider the case when the ratio of velocity

magni-tudes (jVBj jV= TjorjVTj jV= Bj) is less than 1.35, the angle

α is less than 30° and tool radii is less than 30 mm In the

simplest case, we can choose the number of control

points in the approximating NURBS surface along theguiding curve direction (u) to be three Since eachgrazing curve is approximated by a three-control pointNURBS curve, we can use three control points to definethe approximating NURBS surface along the tool axisdirection (v) A 3 × 3 NURBS surface can be created toapproximate the grazing surface The control points ofthis surface along with their weights are calculated usingthe technique given below This new surface is com-prised of a 3 × 3 grid of control point as shown in Fig.9.Figure 9 shows the tool rolling along the quadraticguiding curves T(u) and B(u) The grazing curves at thestart (u=0), the end (u=1) and the interior position (u=u0)are shown as dashed lines Each curve is approximated by anNURBS curve with three control points The control points

at u=0 and u=1 form the boundary of the control polygon ofthe approximate NURBS surface along the v direction;control points of the guiding curves form the boundary controlpoints of the NURBS surface along the u direction Thisleaves only one control point, P1,1, undefined The weights

of the various control points also need to be determined.3.2.1 Definition of NURBS function

An NURBS surface is defined as [22]

Sðu; vÞ ¼

Pn i¼0

Pm j¼0Ni;pðvÞNj;qðuÞwi;jPi;j

where, m+1 and n+1 are the number of control points along

u and v directions; N and N are the basis functions; p and

Table 3 Max error for varying R (L=45 mm, α=30 0

)

Fig 7 Deviation along the grazing curve ifjVTj 6¼ jVBj

Fig 8 Deviation after P1position shifting