The international journal of advanced manufacturing technology, tập 59, số 1 4, 2012

Further, some work has also been donetowards cutting force modeling in circular end milling [5].The undeformed chip thickness in circular end milling wasapproximated by the same analytic

ORIGINAL ARTICLE

Dynamics of the guideway system founded

on casting compound

Bartosz Powałka&Tomasz Okulik

Received: 31 December 2010 / Accepted: 13 June 2011 / Published online: 30 June 2011

# The Author(s) 2011 This article is published with open access at Springerlink.com

Abstract The work presents a new technology for the

assembly of ball guideway systems which involves the use

of a thin layer of a casting compound The experimentally

verified simulation research presented in the work indicates

that the use of the casting compound between the guide rail

and the bed of the machine tool positively influences the

dynamics of the system The paper is concerned with the

comparison between the new solution with the guide rail

assembly technology presently in use on the basis of a

guideway system consisting of a body and a milling table

The dynamics was compared with the use of a frequency

response function which had been determined in an impulse

test The proposed solution is characterised by a higher

dynamic stiffness, which may directly influence the

precision of the machined surfaces

Keywords Casting compound Ball rail system

Guideways Dynamics

1 Introduction

Linear ball guideways, which are now being used more

often in modern machine tools, have replaced the

previ-ously used slide guideways A considerable disadvantage of

the slide guideway system was the stick–slip phenomenon

which occurred while machining at a low feed rate [1,2].This would contribute to the deterioration in the accuracy ofthe machine tool positioning Fortunately, the use of ballguideways eliminated this phenomenon The introduction

of ball guideway systems improved the operating properties

of the machine tool frame system by reducing the resistance

to motion and increasing the permissible feed speed It alsoassisted in simplifying the assembling technology com-pared to the slide guideway system However, the maindisadvantage of the ball guideway system is its lowdamping Low damping might lead to vibrations, whichmay, in turn, lead to the appearance of chatter marks on themachined surface

Machine tool constructors have tried to improve thedissipation parameters of the machine tool body system invarious ways One of the applied solutions is the use ofcomposite materials which have high damping qualitiescoupled with the high specific stiffness for the construc-tion of the machine tool body system Such properties ofthe composite are achieved by using a material with highYoung module and a material with high damping Choiand Lee [3] proposed a spindle construction of carbonfibre–epoxy, which resulted in an increased naturalfrequency and damping than that of the steel spindle.Suh et al [4] proposed the use of a carbon fibre compositelaminate for the construction of the spindle cover Otherexamples of the use of composite materials for theimprovement of dynamic stiffness regarded headstock [5]and machine tool columns [6]

Kim et al [7] designed a three-axis ultra-precision CNCgrinding machine whose bed was made of resin concretewhich contributed to the increase in the damping capacity.The effectiveness of the use of resin concrete for theconstruction of the bed was verified in the impulse test aswell as while machining hard and brittle materials

B Powa łka (*):T Okulik

Institute of Manufacturing Engineering,

Faculty of Mechanical Engineering and Mechatronics,

West Pomeranian University of Technology, Szczecin,

The article by Kim et al [8] presents a research on

sandwich structures composed of fibre-reinforced

compos-ite materials, polymer foams and resin concrete in regard to

their use for the construction of a micro-EDM machine

structure The constructed prototype was characterised by

good stiffness and dissipation properties

Another interesting way of increasing the damping in

machine tools is the use of viscoelastic materials to

dissipate energy [9, 10] The concept of viscoelastic

materials is based on the use of constrained layer damping

CLD [11, 12] The vibration damping mechanism in the

structures consisting of a viscoelastic layer bounded by

steel sheets on both sides was examined by Chen et al [13],

who used the theory devised by Ungar [14]

Wakasawa et al [15] examined structures packed with

balls The use of such structures allowed for a considerable

increase in the damping capacity The research took into

consideration the influence of the ball size, ball

arrange-ment as well as the degree in which they were packed

together and the direction of excitation on the increase and

stability of the damping capacity

An increase in the damping capacity might also be

achieved thanks to the use of cementitious materials [16]

Rahman et al [17] investigated the influence of machining

on two lathes, a ferrocement bed lathe and cast iron bed

lathe, on the tool life The improvement in tool life for the

ferrocement bed lathe was attributed to its higher damping

capacity

An increase in the dissipation properties might also beachieved using polymer inserts in the construction of theguide carriage [18] The use of a polymer impregnatedconcrete damping carriage was compared with a steeldamping carriage The polymer impregnated concretedamping carriage appeared to be a better solution than thesteel damping carriage due to the increase in the dampingcapacity within the frequency range of up to 650 Hz.The application of the casting compound, presented inthis paper, was motivated by the need to eliminate machinetool bed grinding required before guide rail assembly Thegrinding operation is expensive, especially in the case oflarge-size machine tools If the machine tool bed is finished

by milling instead of grinding, it will increase productivityand cut production costs considerably Milled surfaces areexpected to have a lower contact stiffness than groundsurfaces which is due to the lower real contact area In thispaper, a layer of EPY (tradename) casting compound [19] isapplied between a guide rail and the machine tool bed asthe damping material to compensate for the decrease in

Fig 1 a The view of the samples used in the investigation b The

characteristics of the load used in the investigation

Fig 2 The schema of the test stand for determining the deformations

of the contact layer

Fig 3 The graph of the deformations registered for sample C by the sensor located inside the sample for the examined thicknesses of the EPY resin layer in comparison with the reference sample A

contact stiffness EPY material is used in the seating of

main engines, gears, power generators, compressors,

bearings, stern tubes, tanks and many other naval

machinery First, we examined the effect of the thickness

of the EPY layer and machining method on the contact

stiffness and damping capacity The obtained results were

used to build a simulation model of a machine tool

guideway system presented in chapter three Model

research presented in the work indicated that the use of

a thin layer of EPY improves the dynamic characteristic

of the machine tool The positive influence of the use of

the EPY layer obtained as a result of numerical

simulations was confirmed experimentally

2 Static tests of the samples

In order to verify the new method of assembling the ball

guide rails with the use of a layer of EPY, it was checked

how its usage influences contact stiffness and damping of

the joint of the guide rail and the bed sample Therefore, the

experimental research was conducted by means of the use

of three bed samples of various surface quality and

roughness of the assembling surface The surface of sample

B (Ra=4.095μm, Rz=21.80μm) was precisely milled The

surface of sample C (Ra=8.152 μm, Rz=39.50 μm) wasmilled with a worn cutter, while the surface quality ofsample D (Ra=7.898 μm, Rz=39.28 μm) was like thesurface of a billet Figure 1a presents the samples usedduring the research Additionally, sample A (Ra=0.126μm,

Rz=1.27 μm), whose surface was ground in accordancewith the current assembling technology, was used in theresearch as a reference point for comparison

The guide rail was fixed to the bed sample using anintermediate layer of EPY resin with the thickness of 0 to

5 mm The thickness of 0 mm was assumed to be a state inwhich a surplus of the thin EPY layer on the sample wassqueezed out by the rail The intermediate layer filledonly the irregularities of the surface resulting from themachining

The samples were subject to quasi static compression

on the INSTRON testing machine and their force–displacement responses were measured Figure 2presents

a schema of the research stand for determining thedeformations in the contact layer on the joint of the rail,the EPY layer and the bed sample Figure 1b presents thecharacteristics of the quasi static load used during theresearch The loading force increased sinusoidally to thevalue of Fmax= 80 kN, in time t2 = 80 s Time t1 and t3were equal to 2 s

Figure3presents the displacements registered for sample

C by the sensor situated inside the sample for all theexamined thicknesses of the EPY resin layer in comparison

to the reference sample (A), which corresponded to theguide rail current assembling technology It might benoticed on the graph that the use of the thin (0 mm) layer

of EPY resin slightly increases the deformations of the

Fig 4 Comparison of the contact stiffness for various thicknesses of

the EPY resin layer

Fig 6 Schema of the physical model of the milling table

Fig 7 Difference in the investigated models

Fig 5 Comparison of the damping capacity for various thicknesses of

the EPY resin layer

contact layer, i.e there is a slight decrease in the static

stiffness It was found that an increased thickness, over

2 mm of the EPY resin layer significantly reduces the

contact stiffness of the joint

The contact stiffness and the damping capacity were

determined as a result of the conducted quasi static

compres-sion tests on the bed samples Figures4 and 5 present the

experimentally obtained coefficient values The first bar in

each graph corresponds to reference sample A machined in

accordance with the technology used so far The second bar

corresponds to the samples which were not ground (B, C, D)

and where the intermediate contact layer was not used The

subsequent bars correspond to the samples with an

increas-ing thickness of the intermediate layer It might be noticed on

graph 4 that the contact stiffness of the samples without the

EPY layer (13,085 N/μm—sample B) is close to the stiffness

with the‘0’ layer (11,789 N/μm—sample B) In the case

of sample B, for which the value of the contact stiffness

for the ‘0’ thickness is the highest, a 25% decrease in

stiffness was observed in comparison to sample A Based

on the comparison between Fig 5it might be concluded

that the highest damping capacity of 0.412 appears for the

‘0’ thickness of EPY layer The damping capacity for the

‘0’ thickness of the resin layer is the highest for sample Band it is ten times higher than for sample A (0.038) Thus,the use of the thin layer of EPY on the milled surfacemight have a positive influence on the dynamic stiffness

of the system: a slight decrease in static stiffness will becompensated by a significant increase in damping capacity.Since the most promising results were obtained for theassembly of guide rails on the milled surface (sample B)with the use of the‘0’ thickness layer of EPY (0 mm), onlythis solution is compared to the traditional solution inregard to its dynamic stiffness in the simulation research

3 Dynamic response of the milling table modelThe contact stiffness and damping capacity obtained fromthe static experimental research were used to build asimulation model of the milling table mounted to the bedusing guide rails The goal of the analysis was toinvestigate the influence of a decrease in contact stiffnesswith the simultaneous increase in damping capacity on thedynamic stiffness observed for the guide rail mounted to thebed via intermediate layer of EPY resin Figure6presents aschema of the physical model of the milling table togetherwith the assumed location of the machining force Thesimulation research was conducted for two variants of theguideway system assembly

In the first simulation model, the stiffness and dampingparameters corresponded to the current technology forassembling the guide rails (steel–steel contact, parameters

Fig 8 Frequency response function for direction Y for the analysed

models

Fig 11 Schema of the test stand used for the dynamics tests

Fig 10 The view of the stand for the milling table examination

Fig 9 Frequency response function for direction Z for the analysed

models

of sample A) The stiffness and damping parameters

implemented in the second model tested in the simulation

were those for assembling the guide rails on a thin layer of

EPY (steel–EPY layer–steel contact, parameters for sample

B for 0 mm EPY) Figure 7 schematically presents the

models used in the numerical simulation

It follows that since the machining force has a dynamic

character, its dynamic characteristics play a very important

role in the evaluation of machine tool performance The

dynamics of the machine tool are frequently represented in

terms of frequency response functions (FRFs) The

ampli-tude of FRFs and, in turn, the level of vibrations depends

on the stiffness and damping parameters of the machine

tool Thus, the frequency response function can be used to

evaluate the impact of a simultaneous decrease of the

contact stiffness and an increase of damping capacity on the

dynamic performance of the guide rail mounted via the

intermediate layer of the EPY

Figures8and9present the frequency response functions

obtained for the simulation model of the milling table, for

directionY (Fig 8) and direction Z (Fig.9), respectively

Direction X was disregarded due to the fact that for theprepared model the stiffness in this direction dependedmainly on the stiffness of the lead screw and, thus, thedifferences in FRF for the two considered models werenegligibly small It might be noticed on the graphs that theuse of the thin layer of EPY reduces the amplitude of thesystem’s response to the dynamic excitation For direction Y,there was a decrease in the amplitude of about 58%compared to the technology for assembling guide railsused so far as well as a slight decrease in the naturalfrequency from the value of 553 Hz to the level of 547 Hz.For direction Z, a decrease of about 54% in the amplitudewas observed, while maintaining the same resonancefrequency of the simulated system Numerical simulation

on a simple model indicated that the use of the thin layer ofEPY positively influenced the dynamic stiffness of theexamined guideway system

4 Experimental dynamic tests

As numerical simulations presented in section3show there

is an improvement of dynamic properties of the millingtable model with its guideway when a thin layer of EPY isused The authors were encouraged to perform an experi-mental verification of the obtained results The test standFig 12 The location of the measurement points in the dynamics tests

Fig 15 FRF at point 12 due to excitation at point 23 in direction + Y

Fig 14 The location of the points used in the comparative analysis

Fig 13 A fragment of the guideway connection with a thin layer of

EPY resin —the second stage of the investigation

used for this purpose, presented in Fig.10, is geometrically

similar to the simulation model

The stand consisted of a body element with a table

supported by the linear ball bearings The body was

made of grey cast iron The mass of the body element

was ca 314 kg The guideway connection was made

with the use of ball guideway elements of Bosch-Rexroth

which consisted of two guide rails of 25 and 1,410 mm

in length, on which four guide carriages of 25 in length

and catalogue number 1605-213-10 were moving (two

carriages per each rail) The carriages of the guideway

system had the preload equal to 2% of their dynamic

load capacity The dynamic load capacity of each

carriage was 22,800 N A table, also made of grey cast

iron, with a mass of 69.4 kg was mounted on top of the

carriages The guideway elements were founded with the

use of side fixing slits in accordance to the

recommen-dations of the guideway system producer The screw

connections of the guideway system were tightened up

with a torque recommended by the producer In addition,

a turned lead screw with an external diameter of 24 mm

and the lead of 6 mm was used to position the table

Front-end Scadas III was used during the investigation

of dynamics for the data acquisition The excitation was

performed by means of a Kistler modal hammer Kistler and

PCB accelerometers were used for measuring system

response The measurement data were processed with the

use of LMS Test Lab software Figure 11 presents the

experimental set-up used for investigation of dynamics

Thirty-three measurement points were located on thetested object including: eight points on the guide rails, eightpoints on each body element, four points on guide carriages(one on each carriage) and 13 measurement points on thetable Triaxial accelerometers were used to measure thevibration signal in each measurement point The location ofthe measurement points is presented in Fig.12

During the investigation, the tested system wasexcited successively at two points One of the excita-tion points was located in the central point of the tableand the direction of excitation for this point corre-sponded to −Z The second excitation point was located

on the side surface of the table near the guide carriage Inthis case the direction of excitation corresponded to +Y.The frequency response functions were determined foreach of the tested directions based on 30 realizations ofthe excitation signal

Investigations were conducted for the guideway with andwithout a thin layer of EPY (Fig 7) First, the test standwas assembled in accordance with the assembling technol-ogy recommended by the producer of the guidewaysystems Between the guide rail and the body element therewas a steel–steel contact Then the guideway system wasdisassembled and a thin layer of EPY was inserted betweenthe guide rail and the body element During the assembly,the surplus of the intermediate layer was squeezed out only

to fill the irregularities on the contact surfaces of theguideway system Figure 13 presents a fragment of theguideway connection with a thin layer of EPY used duringthe second stage of the investigation Excitations andmeasurements points are shown in (Fig 14)

Figure 15shows FRF at point p12 due to excitation atpoint p23 (Figs 12 and 14) within the frequency rangefrom 30 to 1,000 Hz Application of the EPY results in adecrease of amplitudes in the vicinity of dominantresonances The observed amplitude reduction varies from18% for the dominating mode around 309 Hz to 40%, forFig 16 FRF at point 12 with excitation at point 22 in direction −Z

Table 1 Modal parameters of the investigated object

f n [Hz] ζ [%] f n [Hz] ζ [%] f n [Hz] ζ [%] f n [Hz] ζ [%] f n [Hz] ζ [%] f n [Hz] ζ [%] f n [Hz] ζ [%] f n [Hz] ζ [%] With EPY 62.0 3.41 77.6 1.93 86.9 1.49 305.9 0.64 329.1 0.66 452.0 0.94 488.2 0.60 606.3 0.54 Without EPY 62.8 3.23 80.3 1.71 88.1 1.71 308.43 0.57 332.2 0.55 455.4 0.47 495.5 0.31 621.6 0.47

Fig 17 Vibration mode at 455.4 Hz

the 605 Hz resonance Figure 16 presents the FRF

measured at point p13 due to excitation at point p22

Similarly, an improvement of the dynamic performance of

the system is observed A significant drop of FRF

amplitude has been observed in the vicinity of 500 Hz

(about 48%) The amplitude reduction and also a

decrease of resonance frequencies can be attributed to

the increased damping introduced by the application of

the EPY layer Table1summarizes the modal parameters

of the compared configurations

An improvement of damping ratios (ζ) is more

signifi-cant for modes that exhibit relative table–machine tool bed

motion For instance, the damping ratio of Mode 6 has

improved by 100% This mode is visualized in Fig.17

5 Discussion and conclusions

The impulse tests conducted on the described test stand

validate the statement that the use of a thin layer of EPY

improves the dynamic performance of the object An

increase in the dynamic stiffness of the system was

obtained for two tested perpendicular directions Z and Y

An increase in the dynamic stiffness resulting from the

increased damping capacity in the EPY layer occurred in the

areas of dominant resonances These resonances are

respon-sible for the dynamics of the tested system This means that

the decrease in stiffness resulting from the use of the thin layer

of EPY is compensated with a significant increase in damping

capacity The increase might be caused by the convection of

the mechanical energy of vibrations into the thermal energy on

the resin–steel contact [12, 13] or by an increase of the

effective contact surface as the resin fills all the irregularities

of the connected surfaces [19] The explanation for the

complex mechanism of the damping of vibrations in the thin

layer of EPY resin is beyond the range of this work

The presented proposal for the assembly of guideway

systems that make use of a layer of EPY resin might be

very attractive from the practical point of view The

attractiveness results from the reduction in the costs of

preparing the assembled surfaces for the guide rails by

eliminating the expensive grinding operation An

unde-niable advantage of the method is also its simplicity as it

does not require the construction of any special

equip-ment The solution presented in the work is the subject

of patent application no P388153 in the Patent Office of

the Republic of Poland

Acknowledgement The work was financed from the resources for

science in the years 2009 –2010 as a research project no N503

174637.

Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which per- mits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

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2 Marui E, Endo H, Hashimoto M, Kato S (1996) Some considerations

of slideway friction characteristics by observing stick –slip vibration Tribol Int 29:251 –262

3 Choi JK, Lee DG (1997) Manufacture of a carbon fibre –epoxy composite spindle bearing system for a machine tool Compos Struct 37:241 –251

4 Suh JD, Chang SH, Lee DG, Choi JK, Park BS (2001) Damping characteristics of composite hybrid spindle covers for high speed machine tools J Mater Process Technol 113:178 –183

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6 Lee DG, Chang SH, Kim HS (1998) Damping improvement of machine tool columns with polymer matrix fiber composite material Compos Struct 43:155 –163

7 Kim HS, Jeong KS, Lee DG (1997) Design and manufacture of a three-axis ultra-precision CNC grinding machine J Mater Process Technol 71:258 –266

8 Kim DI, Jung SC, Lee SH, Chang SH (2006) Parametric study on design of composite foam resin concrete sandwich structures of precision machine tool structures Compos Struct 75:408 –414

9 Marsh ER, Slocum AH (1996) An integrated approach to structural damping Precis Eng 18:103 –109

10 Bamberg E, Slocum AH (2002) Concrete-based constrained layer damping Precis Eng 26:430 –441

11 Plass HJ (1957) Damping vibrations in elastic rods and sandwich structures by incorporation of additional viscoelastic material In: Proceedings of Third Midwestern Conference on Solid Mechanics,

pp 388–392

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14 Ungar E (1979) In: L.L Beranek (ed.) Noise and vibrations control McGraw-Hill, Inc., New York

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of polymer impregnated concrete damping carriage for linear guideways for machine tools Intern J Machine Tools and Manufacture 41:431–441

19 Grudziński K, Jaroszewicz W (2004) Seating of machines and devices on foundation chocks cast of EPY resin compound Zapol, Szczecin

ORIGINAL ARTICLE

Drilling performance of green austempered ductile iron

(ADI) grade produced by novel manufacturing technology

Anil Meena&M El Mansori

Received: 17 January 2011 / Accepted: 13 June 2011 / Published online: 7 July 2011

# Springer-Verlag London Limited 2011

Abstract Machinability study on drilling of green

austem-pered ductile iron (ADI) grade was conducted using a

TiAlN-coated tungsten carbide drill The green ADI grade

was produced by a novel manufacturing technology known

as continuous casting-heat treatment technology to save

energy and time in foundry However, in spite of good

combination of strength, toughness and enhanced wear

resistance, the microstructural properties of ADI sometimes

lead to machinability issues The effect of cutting

param-eters on cutting force coefficients, chip morphology, and

surface integrity of the drilled surface were discussed

Results showed that the strength properties of novel ADI

are comparable to that of ASTM grade 1 ADI, whereas

percent elongation is comparable to that of ASTM grade 2

ADI Results obtained also showed that the combined effect

of cutting speed at its higher values and feed rate at its

lower values can result in increasing cutting force

coef-ficients and specific cutting energy At higher cutting speed,

hardness values increases at the subsurface layer of the

drilled surface due to plastic deformation

Keywords Austempered ductile iron Novel manufacturing

technology Drilling Cutting force coefficients Surface

integrity

1 Introduction

Austempered ductile iron (ADI) is an alloyed heat-treated

ductile cast iron [1] In recent years, ADI has emerged as a

major engineering material due to its high strength andhardness, coupled with substantial ductility and toughness[2] The attractive properties offered by ADI are attributed

to a unique “ausferrite” microstructure that is induced byaustempering heat treatment process [3] Ausferrite consists

of graphite nodules embedded in a matrix of acicular ferriteand carbon enriched austenite [4] However, this uniquemicrostructure significantly affects mechanical and thermalmachining properties due to its high strength, hardness, andthe inclination of its retained austenite to strain hardening,which leads to short contact length and higher mechanicalloads on the cutting tool’s edge [5] From the machinabilitypoint of view, the characteristics of ADI derived from theaustenite are a low thermal conductivity and high work-hardening coefficient [6] The austenite lattice has a highertendency to deform due to the greater number of slidingplanes, while the increase in strength and hardness during thedeformation also results as a transformation of retainedaustenite to martensite During machining, the chips areformed on the basis of catastrophic failure in narrow shearsurface due to low thermal conductivity of austenite In such away, unfavorable, segmental chips are formed [7] This limitsthe use of the material in the various industrial sectors andcauses: the formation of built up edges (BUEs) when carbidetools are used, low tool life, increased cutting forces, and theappearance of unfavorable tough chips during machining.From the open literature [8–12], it was found that veryfew studies have dealt with the drilling of ADI specifyingthe effects of cutting parameters on chip morphology andthe resultant chip formation process Drilling is, however, amajor machining process for many applications IndeedADI material, thanks to its high strength to weight ratio andenhanced mechanical properties, predestine this material toact as a substitute for forged steel and cast iron components

As such, ADI with current applications in connecting rodand crankshaft is of increasing interest in automobile

A Meena ( *):M El Mansori

Arts et Métiers ParisTech,

LMPF-EA 4106, Rue Saint Dominique, BP 508,

51006 Châlons-en-Champagne, Cedex, France

e-mail: Anil.MEENA-0@etudiants.ensam.eu

DOI 10.1007/s00170-011-3469-1

industries for which drilling is one of the most critical

machining processes An insightful understanding of the

machinability of ADI material can lead to a better process

economics, increased process stability, improved tool life, and

reduced tooling cost Due to its high strength and hardness

properties, the cutting tools for machining ADI should

fundamentally yield at the same time; have high temperature

hardness and strength, show excellent hot chemical inertness

as well as high toughness at the higher temperatures [13] To

meet all these requirements, TiAlN-coated tungsten carbide

tools have been used for all experiments Titanium-based

coatings, especially TiAlN (titanium aluminum nitride), are

used in a broad range of machining operations TiAlN

coatings are well known for their excellent wear and

oxidation resistances, which enable improved machining

process at high material removal rates During high

temperature applications of this coating, a very dense and

strongly adhesive layer of aluminum oxide (Al2O3) is formed

by aluminum atoms diffusing to the surface preventing

further oxidation Because of its super saturated metastable

phase, the TiAlN-coatings also show age hardening effect,

which increases its hardness at the higher temperatures [14]

This paper thus focuses on the feasibility of novel

manufacturing technology to produce ADI and its impacts

on the material properties of ADI It also focuses on the

experimental studies of cutting force coefficients, chipmorphology, and surface integrity of drilled surface whiledrilling green grade of ADI with TiAlN-coated tungstencarbide tools under flooded conditions for different speed–feed rate combinations

2 Experimental procedure and sample preparation2.1 Workpiece material

Specimens were produced by a novel manufacturingtechnology known as continuous casting-heat treatmenttechnology developed by the integration of the casting(metallic mold) and heat treatment process in the foundry[15] In this process, spheroidization and inoculation wereperformed when the molten material temperature reached1,450°C Spheroidization was done in tundish ladle andsubsequently inoculation was done in a pouring ladle Forspheroidization and inoculation FeSi–Mg (ferrosiliconmagnesium) and FeSi (ferrosilicon) were used, respectively.The molten material was then poured in a metallic mold tomake the specimens of size 180 mm × 30 mm × 15 mmrectangular blocks In the temperature range of 1,000–1,100°C, the casting was shaken free of the metallic mold

Austempering Austenitizing

Austempering

Austenitization Pouring

Austenite

Ausferrite Pearlite

Novel process

Fig 1 Schematic representation

of conventional and novel heat

treatment process for ADI

and put in a muffle furnace for austenitization The

austenitization treatment was then carried out at 900°C

for 90 min After austenitization, the specimens were

quenched down to 500°C in a fluidized bed furnace (at

room temperature) and then austempered in another

fluidized bed for 90 min at 370°C and then air cooled A

schematic representation of the conventional heat

treat-ment process using a sand mold and novel heat treattreat-ment

process using a metallic mold are shown in Fig.1 The

chemical composition of the obtained specimens is as

follows: 3.54% carbon (C), 2.54% silicon (Si), 0.29%

manganese (Mn), 0.18% molybdenum (Mo), 1.19%

copper (Cu), 0.63% nickel (Ni), 0.0095% sulfur (S), and

0.043% phosphorus (P)

Alloying elements such as nickel, molybdenum, and

copper are usually added in ADI to alter the transformation

behavior of ADI [16] The alloying elements added in ADI

also increase the hardenability of the matrix sufficiently to

ensure that the formation of pearlite is avoided during the

quenching process [17] Carbon in the range of 3–4%

increases the tensile strength but has a negligible effect on

elongation and hardness [17], Si within the range of 2.4–

2.8% increases the impact strength of ADI and lowers the

ductile brittle transition temperature [17], Mn level in ADI

should be restricted to less than 0.3%, it strongly increases

hardenability [18], Mo is added as a hardenability agent in

ADI, it should be restricted below 0.2% [17], the higher Cu

content increases the austenite fraction in the final matrix

[18] and up to 2% Ni increases the hardenability of ADI,

and it also decreases the reaction rate of austempering [19]

Spheroidization and inoculation practices are important

for the production of spheroidal graphite and to control the

graphite nodule counts and its size in the matrix [17]

Spheroidizing, or magnesium treatment, of cast iron is ameans of modifying the solidification structure It promotesthe graphite phase precipitates to grow as spherical particlesinstead of flakes, thus resulting in a cast iron withsignificantly improved mechanical properties [20] Itprevents unwanted slag to store as a mass, and it promotesthe dispersion of microparticles, which act as potential sitesfor graphite nucleation during solidification Hence, aneffective spheroidizing process also gives a good basis forinoculation [21] Inoculation is the process of increasing thenumbers of nucleating sites from which eutectic graphitegrows during solidification Most inoculants are based onferrosilicon that contains 70–75% silicon or ferrosilicon–graphite mixtures [22]

Tensile testing of samples was performed according toASTM E-8 standard The test was carried out at a constant

engineering strain rate of 10 mm/min on an MTS (material

testing system) servo-hydraulic machine (MTS810) at room

temperature in an ambient atmosphere Hardness test was

performed using a Buehler MacroVickers 5112 tester with

an applied load of 20 kgf The hardness value is the average

of five different values taken in the different zones for each

sample

2.2 Drilling test and experimental design

The drilling experiments were carried out using the Deckel

Maho five axis machining center of model DMU 80 P

(Fig 2) The machining tests were conducted using the

TiAlN-coated tungsten carbide tools of diameter 8 mm The

tool was produced by Sandvik Coromant with tool

reference R840-0800-30-A0A 1220 (helix angle 30°, point

angle 140°) The tool tip of the drilling tool is shown in

Fig 3 Ecocool cat+ has been used as a coolant with a

viscosity of 35 mm2/s at 40°C A 5 × 4 matrix wasdeveloped (see Table 1) that covered four tool feed rates0.06, 0.08, 0.10, and 0.12 mm/rev and five cutting speeds

10, 30, 50, 70, and 90 m/min (corresponding to the spindlespeeds of 398, 1,194, 1,989, 2,785, and 3,581 rpm,respectively) The resulting feed rates in millimeters perminute are shown in Table 1 For example, case A5corresponds to cutting speed of 90 m/min (3,581 rpm)and feed of 0.06 mm/rev (215 mm/min), respectively Eachexperiment has four repetitions Chips were collected aftereach experiment Thrust and torque components acting onthe tool post were measured with a two-componentdynamometer of type 9271A made by Kistler Surfaceroughness measurements Ra(arithmetic surface roughness)were performed inside the holes using a stylus-basedinstrument (Hommel wave) having an accuracy of 0.5 μmwith a cut-off length of 0.8 mm

2.3 Sample preparationADI samples for metallography were polished, etched, andexamined using standard metallographic techniques Thenumber of graphite nodules per square millimeter wasdetermined on the unetched sample surface by taking theaverage of 10 different regions using an optical microscope.The morphology, microstructure, hardness, and averagethickness of chips were investigated after mounting in coldresin and metallographic preparation Chip morphology andmicrostructure were investigated with an optical and scan-ning electron microscopes Microhardness measurementswere conducted using a 1600-5101 MicroMet analog micro-indentation hardness tester A load of 50 gf and a dwell time

of 10 s was used for microhardness measurements

Table 1 The drilling test matrix used in this work utilizing the four

feeds marked A –D (0.06–0.12 mm/rev) and five speeds marked 1–5

(10 –90 m/min)

RPM (m/min) 398

(10)

1,194 (30)

1,989 (50)

2,785 (70)

3,581 (90)

Fig 4 Microstructure of green

ADI grade produced by novel

manufacturing technology

3 Results

3.1 Workpiece characterization

The obtained microstructure (Fig.4) consists of a matrix of

acicular ferrite and carbon enriched austenite with graphite

nodules embedded in it The gray needles and the white region

between the needles in the micrograph is known as ausferrite

matrix and the white bulky region is untransformed austenite

volume (residual austenite) [23] Image analysis of optical

micrographs of unetched samples gave a volume fraction of

graphite of 9%, with a mean nodular graphite density of

1,200 nodules/mm² and a mean graphite nodule size of 8μm

Such a microstructure provides higher strength and

elongation properties as compared to the as-cast ductile

iron The graphite morphology and mechanical properties

of novel ADI and conventionally produced sand mold ADI

is shown in Table 2 As shown in Table 2, the graphite

nodule counts of novel ADI is higher than that of

conventional ADI This can be explained by the increase

in heat transfer and solidification rates in a metallic mold as

compared to the sand mold which gives rise to higher

nodule counts It can be seen (Table 2) that the strength

properties of novel ADI are comparable to that of ASTM

grade 1 ADI while elongation percentage is comparable to

that of ASTM grade 2 ADI

The novel manufacturing technology using the die castingleads to the production of ADI with fine-grained microstructureand low porosity by improving the feed of molten metal intothe casting It acts as an attractive alternative for lightweightalloys where the strength to weight ratio becomes a key designvariable It increases the graphite nodule counts and decreasesthe grain size as compared to the conventional sand-cast ADI

As the permanent metallic mold is used, direct heat treatmentcan be made without cooling the casting This reduces theprocess time and energy requirement for heat treatment In thisway, the novel manufacturing technology is a high-speedproduction process which gives a good dimensional accuracyand stability to the final product without any casting defect.3.2 Cutting force coefficients and specific cutting energyFor a given chip section, any variation of cutting force withcutting speed can be attributed to variation in cutting forcecoefficient The cutting force coefficient is influencedlargely by the cutting temperature [24] The average torque(N-m) and thrust force (N) generated during drilling foreach individual hole was calculated within the time periodcorresponding to the drill’s first contact with the workpiecesurface and its complete retraction at the end of the drillcycle, as shown in Fig.5(cutting speed 90 m/min and feedrate 0.06 mm/rev) The average thrust force and torque

Table 2 Comparisons of the graphite morphology and mechanical properties of novel ADI and conventionally produced ADI

ADI Graphite (%) Tensile strength

(MPa)

Yield strength (MPa)

Nodule count (nodule/mm2)

Fig 5 Measured thrust and

torque for the case A5 (cutting

speed=90 m/min and feed=

0.06 mm/rev)

components are then used to calculate the cutting force

coefficients (N/mm2) and specific cutting energy (J/mm3)

The cutting force coefficients corresponding to thrust force

(Kcf) [25] and torque (Kcc) [25,26] and specific cutting

energy (U) [25] are respectively defined by as follows:

d Drill diameter (mm)

f Feed per revolution (mm/rev)

The cutting force coefficients are normally the specificcutting pressures developed during the machining process.The variations of average thrust cutting force coefficients,

Kcf and Kcc, corresponding to different cutting speed andfeed rate are shown in Figs.6a and6b, respectively It can

be seen from Fig 6a, b that cutting force coefficients Kcf

and Kcc decreases with the feed rate It can also be seen(Fig 6a, b) that the combined effect of cutting speed at itshigher values and feed rate at its lower values can result inincreasing cutting force coefficients The variation ofspecific cutting energy, the ratio of total cutting energyinput rate to the material removal rate, is shown in Fig.7 Itcan be seen (Fig 7) that specific cutting energy (U)decreases with increasing the feed rate and increases withincreasing the cutting speed Results also showed that thecombined effect of cutting speed at its higher values andfeed rate at its lower values can result in increasing specificcutting energy From the obtained results, it can beconcluded that the strain hardening effect is more at thehigher cutting speed and lower feed rates, whereas thermalsoftening effect is more dominated on increasing feed rate

at lower cutting speed

Feed rate (mm/rev)

Fig 7 Specific cutting energy

Fig 6 Cutting force coefficients a thrust force and b torque

3.3 Chip analysis

Chips were collected after each experiment The main

purpose of collecting chips is to investigate the effect of

cutting parameters on chip morphology, chip

microhard-ness, and chip thickness From the literature review [27–

29], it can be concluded that size and shape of the generatedchips during drilling operation have a great influence on thesurface roughness of the machined holes and cutting forces.For drilling, small broken chips are desirable for theirability to efficiently move along the flute and get out of thehole The chips rotate with the drill and impact the wall of

h p a r g r i m p i h C y

o l o h r o m p i h C Cutting speed (50 m/min)

Feed rate (0.12 mm/rev)

Feed rate (0.12 mm/rev)

the drill or the interior of the flute This impact produces

bending moment in the chips Once the bending moment

causes chip’s maximum tensile strength to be exceeded, it

will fracture [30] The morphological analysis of chips was

carried out with the purpose of studying the influence of

cutting parameters in the formation mechanism and to

identify cutting parameters, which promote better chip

evacuation as in drilling process The average chip

thickness was measured to study the material deformation

involved in the chip formation process, and the chip

hardness was measured to assess the strain hardening and

thermal softening effect during machining

The variations of chip morphology and chip

micro-graph with respect to different cutting speed and feed

rate are shown in Fig.8 It is observed that on increasing

the cutting speed from 10 m/min to 90 m/min, the

needle-shaped chips transformed into the cone-needle-shaped chips and

then finally into amorphous chips The needle and small

cone-shaped chips are formed at low cutting speed as the

chips cannot curl sufficiently to follow the flute and

fracture prior to a complete revolution At high cutting

speed and feed rate, the material removal rate is very high,

which is more likely to cause jamming of flute and causing

chips shape as amorphous, and it did not have consistent

chip curl radius The chip is an arc-shaped type at the

lower feed rate (0.06 mm/rev) and high cutting speed

(50 m/min); it gets more helical with the cone diameter of

chip increases with feed rate and cutting speed Feed rate

is considered as the most significant variable effecting

chip size An increase in feed resulted in larger chips [31]

It is observed (Fig.8) that as the cutting speed increases

the extent of discontinuities on the sliding sides of chips

decreases Presence of streaks and micropores were more

evident as the cutting speed increases as a result of

softening and probably partial melting due to high rise of

temperature in the cutting zone

The percentage change in chip microhardness with

respect to bulk material (Fig 9a) presented here was

carried out in order to investigate the influence of cutting

parameters on the chip plastic deformation during drilling

Astakhov et al [32] stated that the microhardness of the

plastically deformed chips is uniquely related with the

preceding deformation and with the shear stress gained at

the last stage of deformation before the fracture The

percentage change in chip microhardness is calculated

according to Eq 4 [33] It can be seen (Fig 9a) that for

all feed rates, as the cutting speed increases, the percentage

change in chip microhardness first increases and then starts

to decrease This can be explained by the fact that as cutting

speed increases, the material thermal softening during the

process of plastic deformation becomes greater [34]

Plot of average chip thickness for feed of 0.12 mm/rev

against cutting speed is shown in Fig.9b It can be seen that

the average chip thickness decreases between cutting speed

of 10 and 50 m/min and then increases slightly betweencutting speed of 50 and 70 m/min and then furtherdecreases between cutting speed of 70 and 90 m/min Thecontinuous and asymptotic decrease in the average chipthickness (Fig 9b) was due to the continuous andasymptotic increase in the shear angle that occurs whenthe cutting speed increases due to the strain hardening effect[35]

ΔHvð%Þ ¼ HvðchipÞ  HvðmaterialÞ

HvðmaterialÞ  100 ð4Þ

0 10 20 30 40 50 60 70

Cutting speed (m/min)

20 40 60 80 100 120 140 160 180

Cutting speed (m/min)

3.4 Workpiece surface integrity

The term surface integrity is used to describe the quality

and condition of the final machined surface and, therefore,

encompasses both the surface analysis and subsurface

metallurgical alterations For this purpose, the surface

roughness of the drilled holes, microhardness of the

subsurface layer of the machined surface, and hole diameter

deviations were examined

Figure10shows the influence of cutting speed and feed

rate on the machined surface roughness (Ra) value It was

found that surface roughness values at various cutting

conditions varied within the range of 0.32 to 1.73μm It

can be (Fig.10) revealed that cutting speed has a significant

influence on the surface roughness produced It is seen

(Fig 10) that as the cutting speed increases from 10 to

50 m/min, the surface roughness value decreases for all

feed rates Further increase in cutting speed from 50 to

90 m/min increases the surface roughness values At higher

cutting speed, the low feed values give the better surface

finish This can be explained by the higher temperature

generated in the cutting at higher cutting speed Hence, the

shear strength of the material reduces and the material

behaves in a ductile fashion

Results on the microhardness of the subsurface layer of

the machined surface are presented in Fig 11 All values

were taken at a distance of 50 μm beneath the machined

surface The mean hardness value of the subsurface layer

was higher than the mean hardness of the workpiece

material It was due to the high cutting temperature

generated between tool–workpiece interfaces during the

drilling process It can be observed (Fig 11) that

micro-hardness values increases with cutting speed These resultsare expected due to high cutting temperature generated athigher cutting speed

Figure 12 shows the variation of hole diameter atvarious cutting conditions It can be observed (Fig 12)that at lower cutting speeds up to 50 m/min, there is nosignificant variation in hole diameter For cutting speedhigher than 50 m/min, variation of hole diameterincreases rapidly, which can be explained by the highertemperature generated during machining at higher cuttingspeed

8 8,01 8,02 8,03 8,04 8,05

Cutting speed (m/min)

Fig 12 Influence of the cutting parameters on hole dimensions

Cutting speed (m/min)

Fig 10 Workpiece surface roughness of drilled surface at various

cutting parameters

200 250 300 350 400 450 500

Cutting speed (m/min)

Fig 11 Microhardness variations at a distance of 50 μm beneath the machined surface for different cutting parameters

4 Conclusions

In this study, a novel manufacturing technology is developed to

produce green grade of ADI The novel technology is

developed by the integration of casting (metallic mold) and

heat treatment in the foundry to save energy and time

Conventionally, the ADI is produced by using sand mold and

heat treatment is performed after the cooling of casting at room

temperature The metallic mold casting leads to the higher

production rate of near net shape ADI castings and the in situ

heat treatment for austempering The green ADI grade has a

good combination of mechanical properties with strength

properties comparable to that of ASTM grade 1 ADI while

ductility is comparable to that of ASTM grade 2 ADI Also, the

graphite nodule count of green ADI grade is higher as

compared to conventional ADI This can be explained by the

increase in heat transfer and solidification rates in a metallic

mold as compared to the sand mold Hence, drilling study of

green ADI grade was conducted using a TiAlN-coated

tungsten carbide drill The effect of cutting parameters on

cutting force coefficients, chip morphology, and surface

integrity of the drilled surface were discussed The following

conclusions can be drawn from drilling of green ADI grade:

1 The combined effect of cutting speed at its higher values

and feed rate at its lower values can result in increasing

cutting force coefficients and specific cutting energy

2 The variation of cutting speed and feed rate has a

significant effect on chip morphology as the

combina-tion of higher cutting speed and feed values increases

the chip’s shape and size

3 The percentage change in chip microhardness first

increases and then starts to decrease This can be

explained by the fact that as cutting speed increases, the

material thermal softening during the process of plastic

deformation becomes greater

4 Average chip thickness decreases with cutting speed

5 Increase in cutting speed in the range of 50–90 m/min

deteriorates the surface finish and at higher cutting

speed the low feed value gives the better surface finish

6 At higher cutting speed, hardness values increases at

the subsurface layer of the drilled surface due to plastic

deformation

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ORIGINAL ARTICLE

Geometry of chip formation in circular end milling

Avisekh Banerjee&Hsi-Yung Feng&

Evgueni V Bordatchev

Received: 20 April 2010 / Accepted: 14 June 2011 / Published online: 28 June 2011

# Springer-Verlag London Limited 2011

Abstract Machining along continuous circular tool-path

trajectories avoids tool stoppage and even feed rate

variation This helps particularly in high-speed milling by

reducing the effect of the machine tool mechanical structure

and cutting process dynamics With the increase in

popularity of this machining concept, the need for detailed

study of a valid chip formation in circular end milling is

becoming necessary for accurate kinematic and dynamic

modeling of the cutting process In this paper, chip

formation during circular end milling is studied with a

major focus on feed per tooth and undeformed chip

thickness along with their analytical derivations and

numerical solutions At first, the difference in the feed per

tooth formulation for end milling along linear and circulartool-path trajectories is presented In the next step, validformulation of the undeformed chip thickness in circularend milling is derived by considering an epitrochoidal toothtrajectory with a wide range of the tool-path radius Thecomplex transcendental equations encountered in thederivation are dealt with, by a case-based approach toobtain closed-form analytical solutions The analyticalsolutions of undeformed chip thickness are validated withresults of numerical simulations of tool and toothtrajectories for circular end milling and also compared

to the linear end milling The close resemblance betweenanalytical and numerical calculations of the undeformedchip thickness in circular end milling suggests validity ofthe proposed analytical formulations As a case study, thecutting forces in circular end milling are calculated based

on the derived chip thickness formulations and anexisting mechanistic model The calculation resultsreiterate the need of taking into account adjusted feedper tooth and valid chip thickness formulations incircular end milling, especially for small tool-path radii,for more realistic process modeling

Keywords Circular end milling Epitrochoidal toothtrajectory Feed per tooth Undeformed chip thickness Closed-form formulation Cutting force

1 Introduction

In the past decade, aerospace and die/mold industries haveidentified the potential of tremendous time and cost savingthat can be achieved by high-speed machining (HSM)technology Hence, the popularity of HSM in end milling

A Banerjee

Department of Mechanical and Materials Engineering,

The University of Western Ontario,

London, ON, Canada N6A 5B

H.-Y Feng

Department of Mechanical Engineering,

The University of British Columbia,

Vancouver, BC, Canada V6T 1Z4

e-mail: feng@mech.ubc.ca

E V Bordatchev

Centre for Automotive Materials and Manufacturing, Industrial

Materials Institute, National Research Council of Canada,

London, ON, Canada N6G 4X8

has been rapidly growing A major concern in HSM is the

higher risk of instability in the end milling system resulting

from the higher acceleration and deceleration of the

machine tool during cutting motions with high spindle

speeds This problem is aggravated especially when the tool

path has tangent or G1 discontinuities that lead to tool

stoppages Also, fast ramping feed rate leads to higher

jerks The discontinuous tool motions directly translate into

lower productivity and even damage through excessive

wear of the machine tool mechanical components Thus,

innovative machine tool and motion drive designs, and

motion control algorithms have become popular research

topics Moreover, researchers have also started looking into

tool-path optimization to obtain smooth and consistent

machining This effort has lead to the use of higher order

tool-path trajectories, such as splines [1], quintic splines

[2], NURBS [3], etc Such tool-path trajectories require

advanced interpolators, which are not commonly available

in commercial CNC controllers A simplistic alternative to

higher order tool paths are biarc and arc-spline tool-path

segments, which provide G1continuity along the entire tool

path End milling along circular trajectories are preferred in

finishing operations for lower fluctuation of its curvature

along the tool path and the ability to maintain a limiting

feed rate calculated based on dynamic characteristics of

machine tool [4] over the entire trajectory Moreover,

circular end milling also finds wide applications for

roughing operations like removing excess material, enlarge

holes and cavity openings [5] They are preferred to the

alternate boring operations due to higher productivity

achieved through multiple cutting edges

Physics-based and kinematic models of the cutting

process are commonly employed for modeling used in

process planning and optimization Therefore, accurate

analysis and modeling of the chip formation is very

important for optimal planning of circular end milling

operations The chip thickness formulation for circular end

milling has been addressed in some previous work with

numerical [6] and analytical [7] approaches In the

numerical approach [6], parametric equations representing

tool-path trajectory and workpiece surface to be machined

are numerically solved to determine the undeformed chip

thickness However, this approach does not provide any

insight into the geometry of chip formation and material

removal mechanism The analytical modeling approach [7]

proposes a single closed-form formulation of the

unde-formed chip thickness without considering greatly varying

geometric cases of tool-path radius arising in practical

circular end milling operations The assumption of the

defined delay angle being negligibly small is true for linear

end milling, but may not be applicable for all practical

circular end milling conditions Moreover, the presented

results suggest that the chip thickness increases as tool-path

radius decreases without determining its value at very lowtool-path radius Further, some work has also been donetowards cutting force modeling in circular end milling [5].The undeformed chip thickness in circular end milling wasapproximated by the same analytical formulation derivedfor a linear end milling with a feed per tooth compensationstrategy The paper also provides experimental measure-ment and analysis of the cutting force at low tool-pathradius, which is observed to be consistently smaller than thepredicted value Although the results presented do notaddress this observation from the chip formation point ofview, they clearly indicate the significant differencebetween linear and circular end milling operations

In this work, the mechanism of chip formation incircular end milling is studied in detail In Section 2, thedifference in feed per tooth for end milling along linearand circular tool-path trajectories is studied Based on thisdifference, in Section3, a case-based approach is adopted

to derive valid closed-form analytical formulations ofdelay angle and undeformed chip thickness in circular endmilling, for different ranges of tool-path radius InSection 4, a numerical procedure is also implemented tocalculate the undeformed chip thickness and results arecompared to the proposed analytical formulations forlinear and circular end milling Further, the proposedcase-based analytical formulation of undeformed chipthickness in circular end milling is integrated with acalibrated mechanistic cutting force model and cuttingforce simulation results for different geometric test casesare presented in Section 5 Finally, conclusions of thederivation, numerical validation, comparison of chipformation in circular and linear end milling, and theapplication of the proposed analytical formulation forimproved cutting force calculation are presented inSection 6

2 Feed per toothFeed per tooth is defined as the distance advanced by thecutting tool into the workpiece material per tooth revolution[8] In this section, feed per tooth for linear and circular endmilling operations are compared to study the materialremoval characteristics Figure 1a shows the comparison

of both linear and circular end milling operations with aconstant radial depth (say dr) while moving cutting toolfrom position P1to P2 During the linear end milling, areasmachined at different radial distances (da, db, dc) from toolcenter O are equal (Aabb′a′a= Abcc′b′b) However, for circularend milling with tool-path radius (R) and center C, it can beobserved that Aabb′a′a < Abcc′b′b Different amounts ofmaterial are removed at a, b, and c to accommodate thedifference between machined areas Hence, in circular end

milling, the value of feed per tooth (f), programmed for tool

center, O shown in Fig.1b does not remain constant along

the tool-path radial axis CO A simple formulation for

adjusting feed per tooth (fadj) for circular end milling

measured along the tangent of the circular tool-path

trajectory is given as [5]:

whereΔβ is the angle the tool center rotates about the

tool-path origin in one tooth period, as shown in Fig.1b The

above formula is applicable for cases where R is relatively

large and f is very small that allows the approximation of

the subtended arc length (OO′) by f as shown in Fig.1b To

obtain a more comprehensive and meaningful feed per

tooth adjustment formulation in circular end milling, the

unit machined area which is the area machined during one

tooth period and the tool-path radius (r), is also considered

into the adjusted feed per tooth formulation The detailed

derivation of the formulation of adjusted feed per tooth

(fadj) is provided in AppendixAand given as:

fadj¼ f  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ ðr=RÞ2þ 2  ðr=RÞ  cosðqÞ

q

ð2Þ

A detailed analysis of the above equation with respect to

the possible values of R results in a variety of fadj

represented by the gray-colored area in Fig 1b For

relatively large values of R (R >> r), the term (r/R)

approaches zero, and results in that Eq.2becomes fadj≈ f

In the range of 0≤R≤r, as R decreases, fadjincreases rapidly

Ultimately, for R→0, fadj→ ∞, implying that the amount of

material removal in one tooth period increases infinitely

Although this situation exists mathematically, it is difficult

to imagine it happening in actual machining One

explana-tion is that although fadjbecomes∞, engagement domain of

the tool also becomes zero as the tool simply rotates about

its own axis without tool movement and material removal

Moreover, the assumption that R >> f is no longer valid,

and the mathematical formulation in Eq 2 takes an

indeterminate form This implies the need for further

rigorous study in to the chip formation in circular end

milling For this purpose, the next section focuses on a

valid undeformed chip thickness analytical formulation to

understand the geometry of chip formation in circular end

milling operations

3 Analytical formulation of undeformed chip thickness

The difference in the feed per tooth between linear and

circular end milling clearly suggests that process models for

linear end milling will provide only an approximation for

the circular end milling The indeterminate form of the feed

per tooth in Eq.2 for low tool-path radius (R) implies thatonly variation in feed per tooth cannot provide acomprehensive understanding of chip formation in circu-lar end milling In this section, the geometry of chipformation is studied by considering the chip thicknesswhich is defined as the amount of material removed bytwo consecutive tooth trajectories For linear end milling,

a simplified formulation of the undeformed chip ness (tc) in terms of tool rotation angle (θ) and feed pertooth (f) is given as [9],

For circular end milling, the variation of f suggests thatthe simplified formulation of undeformed chip thicknesscannot be used [6,7] Two consecutive (j−1)th

and jthtoothtrajectories are shown in Fig 2 in gray and black,respectively The figure represents up-milling operation of

a circular slot cut with entry angle of 0° and exit angle of180° The tool radius (r) and different values of tool-pathradius (R) with a center at point C are selected Twodifferent coordinate systems are used and shown in Fig.2a.The first system is a global coordinate system represented

by X–Y axis with origin fixed at the tool-path center C Thesecond system is a local coordinate system represented byx–y axes about the instantaneous tool center O, whichrotates about center C Figure2a shows the case for R > r,where for any point N on the current jth tooth trajectorysubtending a tool rotation angle ofθN, there exists a point

θNcan be represented by the distance MN Figure2b and cshow the tooth trajectories for R = r and R < r, respectively

It can be observed through comparison of Fig.2a, b, and cthat the nature of the feed per tooth variation along the tool-path axis CO is quite different In Fig.2a and b, the entireengagement domain of the two fluted tool for a slot cut isbetween 0° and 180°, but the starting tc(atθN=0°) is higherfor R = r and it becomes zero at the exit point In Fig 2c,the tool engagement starts with even higher values of tc

than that in Fig 2b and becomes negative beforeapproaching the exit point As the tool-path center (C)moves inside the tool circumference, a negative feed pertooth in the reverse direction is also observed, as shownwith dotted arrows in Fig 2c This occurs due to the factthat near the exit point of the current tooth pass, no material

is left to be machined This can be called as backwardcutting, where the current tooth within the engagement

domain does not machine material as it has been already

removed by the previous tooth trajectories

3.1 Geometric representation of epitrochoidal tooth

trajectory

Figure2shows that undeformed chip thickness in circular

end milling is a function of several parameters related to

tool and tool path, such as tool-path radius (R), tool-path

angle (β), and tool rotational angle (θ) Hence, tooth

trajectory cannot simply be considered as a trochoid as inthe case of linear end milling [9] However, the epitrochoi-dal tooth trajectory, defined as the trajectory of a point onthe circumference of a circular disk rolling over anothercircular disk, will provide more accurate representation forthe circular case The parametric equation of an epitrochoid

b

c

f

P1 P2

C

R

O a

fadj

current tooth period

previous tooth period

dr

θ

Fig 1 Feed per tooth: a comparison between linear and circular end milling, and b variation in circular end milling in one tooth period

where P is the circumferential point on a disk with

radius r rolling over another disk with radius R The

anglesθ and β correspond to the subtended angle by the

point P to the tool and tool-path centers, respectively

Using the parametric representation of an epitrochoid

given in Eq.4, the point M for the (j−1)th

tooth trajectoryand the point N for the jthtooth trajectory (see Fig.2) can

whereΘ is the tooth position angle along the epicycloidal

trajectory and can be represented by,

where,θ0is the initial tooth orientation angle;θ is the tool

rotational angle and Δ=2π/N the lag angle between

adjacent cutting teeth, with N being the number of cutting

teeth Moreover, the proportional relationship between θ

andβ for a selected feed per tooth (f) at the tool center can

be formulated as:

where,

k¼ fN 2pR= ¼ M R= and M ¼ fN 2p= ð8bÞ

Hence, using Eqs 7 and 8, the angles ΘM,βMand ΘN,

βNat points M and N, respectively, and considering θ0=0,

in Eqs 5and 6, can be represented as:

MN

M N

X

Y

entryexit

(b)

MN

Fig 2 Adjustment of feed per

tooth and undeformed chip

thickness determination for

circular end milling based on

consecutive tooth trajectories:

a R > r; b R = r; and c R < r

Based on fact that points ON, M, and N are collinear to

represent the undeformed chip thickness as MN, the

following condition is implied:

XM XO N

ð Þ Y=ð M YO NÞ¼ Xð N XO NÞ Y=ð N YO NÞ

) Xð M XO NÞ  Yð N YO NÞ  Xð N XO NÞ  Yð M  YO NÞ ¼ 0

Substituting relations from Eqs.5–10 into the equation

above, the following equation is obtained:

rfR sin b½ M bð Nþ DNÞ  R sin b½ N bð Nþ DNÞ

r sin b½ð Mþ DMÞ  bð Nþ DNÞg ¼ 0

A delay angle (δ) defined as the difference between the

current (ΘM) and previous (ΘN) tooth positional angles is

introduced and given by,

3.2 Undeformed chip thickness

The undeformed chip thickness (tc) can be represented bythe distance MN (see Fig.2) and can be given as:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

XM XON

ð Þ2þ Yð M YONÞ2q

Now, using Eqs.5–11,ρ can be derived as:

r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Substituting the above expression forρ into Eq 13, tc

can be expressed as:

In the above equation, tc can be estimated only after δ

has been obtained by solving Eq 12 for different

geometrical and physical conditions Hence, tc will also

have different case-based formulations

3.3 Case-based approach

The presence of tool-path radius in circular end milling

makes the formulation of undeformed chip thickness (t)

much more complex than that in linear end milling Thisnecessitates the use of different geometrical and machiningconditions for deriving valid closed-form analytical for-mulations of tc The input parameters involved in deter-mining the delay angle (δ) in Eq.12and tcin Eq.14can becategorized into three types The parameters of the first typeare geometric: tool radius (r) and tool-path radius (R) Thesecond type includes the process parameters: feed per tooth(f) and lag angle (Δ) The third type involves coefficients k

and M, which combine the geometric and process

param-eters The versatility of end milling operations causes large

variations in the involved process parameters and

subse-quently affects the output parametersδ, tcand type of the

end milling operation For example, linear end milling

implies R =∞, ∞>R>0 refers to circular end milling, and

for the extreme geometrical case of R=0 there is no

machining In order to systematically capture all the

possible geometric cases, five different geometric

relation-ships between R and r and their effect on the other

parameters are to be determined These geometric cases

(cases I to V) are provided in Table1that also includes the

corresponding assumptions and type of end milling The

assumptions have been made based on the mathematical

definitions of k and M (Eq.8b) and their possible values for

practical machining conditions

In order to evaluate the above cases, corresponding ranges

of values of R for a typical end milling tool with radius of

10 mm are also provided in Table1 The assumption for each

case is systematically applied to obtain closed-form solutions

of the transcendental Eq 12 for δ and subsequently

determine tc from Eq 14 As the practical situations for

these cases are different, the corresponding formulations ofδ

and tcare also different The derived case-based closed-form

formulations ofδ and tcare presented in Table 2

4 Comparative analysis

In this section the delay angle (δ) and undeformed chip

thickness (tc) are calculated and compared using three

different approaches given below as:

& Analytical solutions for linear end milling: The

trochoi-dal tooth trajectory is used as a special case of the

epitrochoidal tooth trajectory with R =∞ The analytical

formulations for Case I provided in Table2is applied to

calculate δ and tc for different ranges of R and tool

rotation angle (θ)

& Analytical solutions for circular end milling: Theanalytical solutions for circular end milling are calcu-lated based on the epitrochoidal tooth trajectory Thecase-based closed-form formulations of δ and tc areshown in Table 2 for different cases given in Table1.These formulations are also implemented usingdifferent ranges of R and tool rotation angle (θ)

& Numerical solutions for circular end milling: Anumerical approach is implemented to calculate δand tcfor circular end milling, based on the definition

of the uncut chip thickness and the tooth trajectoriesshown in Fig 2 As δ and tc cannot be measured, anaccurate numerical solution becomes necessary forvalidation of analytical solutions The flow chart forthe detailed numerical procedure is shown in Fig 3.The derived analytical formulations for circular endmilling for various cases listed in Table1are validatedwith the numerically computedδ and tc

The main purpose of this comparative analysis is tovalidate the similarities between derived analytical andnumerically computedδ and tcfor circular end milling withepitrochoidal tooth trajectory Table 1 gives the ranges oftool-path radius (R) that were selected to cover all possibletheoretical and practical application scenarios for conven-tional and micro end milling In addition, analyticalsolutions of δ and tc for linear end milling with trochoidaltooth trajectory are also presented for the entire engagementdomain This helps in showing its difference with circularend milling for different tool-path radius

All the calculations of analytical formulations andnumerical solutions were carried out with the same ranges

of tool-path radius (R) and tooth orientation angle (θN)within the engagement domain for slot cuts A high speedsteel (HSS) end milling tool was selected having a toolradius (r) of 10 mm with the number of cutting teeth (N)being four A feed per tooth (f) of 0.05 mm/tooth and acutting speed (V) of 26 m/min were selected based on

Case no Geometric condition Assumptions Numerical sub-cases

IV R < r k≪1, k≈1, k≫1 A: 10>R≥0.1 mm Circular

B: 0.1>R≥10 −4 mm C: 10−4>R≥10 −7 mm

Table 1 Different geometric

cases for deriving analytical

formulations and numerically

computing delay angles and

undeformed chip thicknesses for

circular end milling

recommendations for cutting mild steel workpieces listed

on the machining data handbook [10]

4.1 Case I: R→ ∞

When R→ ∞, the circular end milling is degenerated into

linear end milling Applying the limits δ→0 and k→0

results in significant simplification in the derivation of δ

and tcfrom Eqs.12and14, respectively The results of the

analytical linear and circular chip thickness solutions as

well as the numerical solutions are shown in Fig.4, from

where it can be seen that all the three approaches for

calculating δ and tc yield almost the same results This

confirms that the proposed closed-form analytical

formula-tion for circular end milling (Case I, Table 2) can be

simplified to represent linear end milling

4.2 Case II: R > r

This particular geometric case includes a wide range of

possible values of R To capture the changes in the range

of R > r, the case was divided into three sub-cases inorder to show different level of details—case IIA with

109≥R>104

mm, case IIB with 104≥R>100 mm, and caseIIC with 100≥R>10 mm as given in Table 1 for a toolradius of 10 mm Linear, circular, and numericalsolutions were studied and compared for each sub-caseand the results are presented in Fig 5 It can be seenfrom the figure that all the calculations are close forhigher R as in this sub-case, chip formation in circularend milling closely resembles to that of linear endmilling The linear and circular solutions have beencompared to show their similarity which suggests that thecircular solution can also handle the extreme cases ofhigher R and linear cuts As R reduces and becomescloser to r, the deviation inδ for linear and circular endmilling increases The deviation arises due to a highervalue of difference in the tool-path angle between twoconsecutive orientations of the tool-path axis Mathemat-ically, the value of k is no longer negligible wrt 1, andthis affects the calculations of δ and tc in circular endmilling

Table 2 Closed-form analytical formulations of delay angle and undeformed chip thickness for different geometric cases in circular end milling

I

d ¼MΔ cosðDN Þ rM cosðD N Þ

t c ¼

r 1  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ M r

  2

ðd þ ΔÞ 2  2 M

r ðd þ ΔÞ sin d þ D ð N Þ q

k ! 0

r 1  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ M r

  2

Δ 2  2 M

r Δ sin D ð N Þ q

 

r coskΔ 2

  coskΔ 2

 

cos kΔ2D N

cos ðkΔÞk cosðkΔD N Þ

IV

d ¼

 2 sin kΔ 2

 

r cos kΔ 2

  2 sin2 kðdþΔÞ2

4.3 Case III: R = r

For R = r, the comparison of the calculated δ for linear

and circular end milling shows deviation, but the circular

analytical and numerical solutions for circular end

milling are quite close, as shown in Fig.6 The calculated

tc for circular end milling (analytical and numerical)

solutions are slightly lower than those for linear end

milling for the tool rotation angle between 90°≤θN≤180°

due to the reduction in the feed per tooth, which

eventually becomes zero at the exit point, as shown inFig 2b

4.4 Case IV: R < rThis is the most critical case because the geometry of chipformation in circular end milling becomes completelydifferent from that in linear end milling as shown inFig 7 Ranges of tool-path radius were selected for threesub-cases in order to show different levels of details - Case

M 2 M M M

δ

Determine tci j in jth pass:

2 O M 2 O M

Intersection of (j-1)th tooth pass trajectory with jth tooth position vector (ONN)

to determine the point M , with an iterative numerical search routine

Fig 3 Numerical solution procedure for the undeformed chip thickness

IVA with 10>R≥0.1 mm, Case IVB with 0.1>R≥10−4mm

and Case IVC with 10−4>R≥10−7mm, for a tool radius of

10 mm It can be seen that higher deviation in δ existbetween linear and circular end milling calculations and δ

Circular Numerical Linear

Fig 6 Comparison of delay angles and undeformed chip thicknesses for R = r

Circular Numerical Linear

Fig 4 Comparison of delay angles and undeformed chip thicknesses

Fig 5 Comparison of delay

angles and undeformed chip

thicknesses for R > r: a case

IIA; b case IIB; and c case IIC

calculated for the circular end milling is close to the

numerical solutions The maximum value of tcdecreases in

circular end milling with decreasing R below r, where

backward cutting situation starts emerging near the exit

angle A sharp drop in bothδ and tcnear R=0.1 mm as the

assumption that R >> f is not valid anymore In Fig.7b, asδ

at the entry point in circular end milling becomes much less

than that in linear end milling, tc becomes higher at the

entry point and negative near the exit point due to backward

cutting near the exit In Fig.7c, the maximum tcstill decreases

with decreasing R, and, eventually,δ approaches −90°, and tc

approaches 0

4.5 Case V: R→ 0

This is the extreme case, where the cutting tool rotates

about its own axis and is not removing any material as the

engagement domain reduces to zero Figure8shows thatδ

is −90° and tcis equal 0 for all tool rotation angles Withlimiting case of R=0, the amount of material to be removed

Circular Numerical Linear

Fig 8 Comparison of delay angles and undeformed chip thicknesses for R →0

Fig 7 Comparison of delay

angles and undeformed chip

thicknesses for R < r: a case

IVA; b case IVB; and c case IVC

also decreases to zero even though the cutting elements at

the periphery of the cutter moves at infinite speed around

the tool-path center As it can be observed in Fig 8, the

analytical solution for linear end milling is completely

different from that for circular end milling On the contrary,

both analytical and numerical solutions for circular end

milling are similar and this ensures the applicability of the

case-based analytical formulations of δ and tc (case V,

Table2) to capture this extreme R→0 condition

5 Case studies: cutting force predictions

Undeformed chip thickness cannot be measured directly,

hence indirect methods like cutting force prediction

becomes important to show the changes in chip thickness

based on input geometrical and machining parameters

Cutting force calculations have been used to illustrate the

impact of the proposed chip thickness formulation on the

process states Cutting force prediction for end milling

operations is by itself a well-developed research area

Although there are various modeling approaches available,

the mechanistic modeling approach is one of the most

popular Cutting force prediction for circular end milling

generally approximates the geometry of chip formation the

same way as linear end milling [5] In a study by Kardes

and Altintas [5], the cutting force in circular end milling is

predicted and compared to the experimental result The feed

per tooth is adjusted based on the tool-path radius and the

feed angle per tooth period as given by Eq 1 without

considering the instantaneous tooth angular position angle

with respect to the tool center For a more realistic

prediction of the cutting force, the derived analytical

formulations of the undeformed chip thickness (Table 2)

need to be applied In this section, an existing mechanistic

cutting force model [11] is implemented to calculate the

instantaneous cutting forces Undeformed chip thicknesses

based on the proposed analytical formulations for circular

end milling (case II to V, Table 2) as well as linear end

milling (case I, Table 2) are used to compute the

instantaneous cutting forces and the results are compared

The principle of a mechanistic cutting force model is

based on dividing the cutting edge of a tool into elemental

cutting disks along the axial Z direction with an equivalent

height dz as shown in Fig.9 The elemental forces acting

on each disk are determined based on the angular

orientation (θ) of the engaged disk and its instantaneous

undeformed chip thickness tc(θ) These elemental forces are

in radial (R) and tangential (T) directions and their X and Y

components can be calculated The elemental forces in the

X and Y directions are summed for the entire axial depth of

cut (d) and engagement domain (θ ≤ θ ≤ θ ) The

basic formulations of the instantaneous forces in the X and

Y direction and their resultant components can be sented as [11]:

repre-F X inst ¼ q¼q X exit q¼q entry

X dz¼d a dz¼0

Finst, andθentry,andθexitare the entry and exit angles of theengagement domain KT, KR, mT, and mR are the empiricalcoefficients in the radial (R) and tangential (T) directionscalibrated from experimental result [11]

The case-based analytical formulations of undeformedchip thickness for circular end milling given in Table2wereapplied in a pre-calibrated mechanistic cutting force model[11] The instantaneous force profiles are calculated forcircular slot end milling of SAE 1018 cold rolled steel withHSS tool having a tool radius of 10 mm and an axial depth

of cut of 10 mm The feed per tooth of f=0.0381 mm/tooth,and a cutting speed of v=26 m/min were selected from themachining data handbook [10] The tool path consideredwas a complete circle implying that the tool-path rotationangle (β) varies between 0° and 360°

The undeformed chip thickness (tc) for the four teeth (1,

2, 3, and 4) separated by 90°, and the instantaneousresultant cutting force (Finst) at the tool rotation angle (θ),were calculated Different circular end milling situationswere simulated with R=1, 10, and 100 mm and their chip-thickness profile and the resulting instantaneous cuttingforce compared with the linear end milling approximations

as shown in Figs 10, 11, and 12, respectively It can beobserved that the difference in maximum tc and Finst

between circular and linear end milling decreases as Rincreases This indicates the need of using the proposedformulations of tc for valid prediction of cutting forces,especially with lower R At higher R, cutting geometry ofcircular end milling approaches that of linear end milling

In Fig 10, for R=1 mm, the profiles of tc near their exitpoints become slightly negative due to the backward cutting

or machining of a region already machined by a previoustooth trajectory This implies that the backward cutting

effectively reduces the engagement domain and

subse-quently the cutting force due to the lesser engaged cutting

disks Moreover, a skewness is observed in the profile of

tcespecially for lower R as observed in Fig.10a and very

minor in Fig.11a The profile starts at a non-zero value,

peaks before reaching the halfway into the cut and never

reaches the maximum programmed feed per tooth value

This is also observed for cases R < r in Fig.2c and case

IVb in Fig.7b

An experimental analysis of the cutting forces for

circular slot end milling of Aluminum Al7075-T6 is

available in the literature [5], where for the combination

of geometry, R=15 mm and r=10 mm, the predicted cutting

force utilizing the chip thickness formulation based on

linear end milling consistently yielded values higher than

the experimental values The stated reasons for this

deviation are changes in the feed rate, inaccuracies in

synchronization, and errors in the calibrated coefficients of

the cutting force model The proposed undeformed chip

thickness formulation for circular end milling with theconcept of change in engagement domain with the change

of tool-path radius and feed per tooth may also be helpful tobetter understand the above inconsistency It should benoted that the above comparison is only qualitative, and adirect comparison of the predicted cutting forces with themeasured cutting forces [5] would require the calibration ofthe cutting force coefficients for the particular cuttingtool-work piece material

6 Summary and conclusionsThe geometry of chip formation in circular end milling issignificantly different from that linear end milling, espe-cially for circular tool paths with small radii which may beoccurring as a result of G1continuous arc-spline and biarcfit of higher order curves In this paper, the mechanics ofchip formation geometry in circular end milling was studied

Circular Linear

(b) (a)

Fig 11 Cutting force calculation for circular end milling for R=

10 mm and r=10 mm: a undeformed chip thickness profile with tool rotation; and b instantaneous cutting force with unit tool rotation

(b) (a)

Circular Linear

Fig 10 Cutting force calculation for circular end milling for R=1 mm

and r=10 mm: a undeformed chip thickness profile with tool rotation;

and b instantaneous cutting force with unit tool rotation

Fig 9 Mechanistic cutting

force modeling: a cut geometry;

and b elemental forces

taking into account both feed per tooth and undeformed

chip thickness to reliably model the material removal

geometry A need for feed per tooth adjustment in circular

end milling was presented based on a comparison of that in

linear end milling A simplified formulation is derived

based on the tool-path radius, tool radius, and the tool

rotation angle The indeterminate form of the feed per tooth

formulation at extreme conditions suggests the need of

studying chip thickness as well

The derivation of undeformed chip thickness for circular

end milling based on an epitrochoidal tooth trajectory is

quite complex and leads to transcendental equations

Hence, case-based closed-form analytical formulations of

delay angles and undeformed chip thicknesses are

devel-oped For this purpose, the different geometric cases were

systematically considered Each case covered a range of

tool-path radius with the corresponding assumptions and

end milling type, which were selected based on practical

considerations A numerical solution procedure was also

implemented to calculate the delay angle and undeformed

chip thickness The analytical and numerical solutions for

both linear and circular end milling operations were

compared The close resemblance of delay angle and

undeformed chip thickness for circular and numerical

simulation solutions in Figs 4 to 8 provides numerical

validation of the derived formulations Case-based

formu-lations were used to consider all situations of circular end

milling, where a single formulation may not be possible

Further, comparison between linear and circular end milling

confirmed the need of deriving the undeformed chip

thickness formulation for circular end milling Finally, the

cutting force was calculated to illustrate the impact of the

proposed chip thickness formulation on the process states

The calculations reiterated the need of replacing circular

end milling models based on the linear end milling

approximation with the derived formulations, especially

for small tool-path radius

Attempts should also be made will be made to

experimentally verify the cutting forces for circular end

milling in micro milling applications, where the tool-path

radius can be very small Further experiments are needed

to confirm the predicted difference in cutting force profiles forcircular end milling with different tool-path radius

Acknowledgment This research was funded in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Appendix ADerivation of feed per tooth adjustment formulationMachined area (Am) during circular end milling with aconstant radial depth of (dr) shown by the hatched region inFig 1b can be expressed as,

Am¼ p  R þ rð Þ2 x2

1

ðA:1Þwhere r is the tool radius, R + r is the maximumcircumferential distance from center C, and x1 is thedistance to the point of exit P from C The number oftooth period required (nt) for a complete rotation about thetool-path center C with circular end milling, is given by,

nt¼2 p  R

where f is the programmed feed per tooth at the tool centerO.The area (A1) machined in a one tooth period, isrepresented by the section (AA′P′PA) in Fig.1b and givenby

Based on Fig.1b, feed per tooth for circular end milling

is considered to be varying along tool diameter, which can

be denoted as f(x), where x is the distance to the point ofentry P from center C, x (x∈[R + r, x1]) The area A1given

by Eq.A.3can also be expressed as the area under feed pertooth profile The profile of feed per tooth and the area A1

are shown by the gray-shaded region in Fig 1b The area

A1under the feed per tooth profile is given by,

Fig 12 Cutting force calculation for circular end milling for R=

100 mm and r=10 mm: a undeformed chip thickness profile with tool

rotation; and b instantaneous cutting force with unit tool rotation

Differentiating Eq.A.5wrt x1,

fðx1Þ ¼ f x1

Hence f(x1) linearly varies with x1for a fixed value of f,

and R as shown in Fig.1b The range of x1is [R + r, R− r],

which corresponds to extreme circumferential entry points

from C Using cosine rule forΔCOP in Fig.1b,

x1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2þ r2þ 2  R  r  cosðqÞ

p

ðA:7Þwhereθ is the angle subtended by entry point at tool center,

O Substituting Eq A.7 in Eq.A.6, the adjusted feed per

tooth value (fadj) for the entry point P is given by,

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Machin-11 Feng HY, Menq CH (1994) The prediction of cutting forces in the ball-end milling process —I Model formulation and model building procedure Int J Mach Tools Manuf 34(5):697 –710

ORIGINAL ARTICLE

An analytical expression for end milling forces and tool

deflection using Fourier series

Shi Hyoung Ryu

Received: 15 December 2010 / Accepted: 20 June 2011 / Published online: 1 July 2011

# Springer-Verlag London Limited 2011

Abstract In this research, a novel and generalized

analyt-ical expression of cutting force and tool deflection in end

milling is presented as a function of tool rotational angle

and other cutting parameters The discontinuous cutting

force function caused by periodic tool entry and exit is

changed to an integrable continuous function using Fourier

series expansion Tool deflection is also formulated explicitly

by the direct integration of the distributed loads along the

helical cutting edges Cutting conditions, tool geometry,

runout components, and the stiffness of tool clamping part

are considered in estimating the cutting force and tool

deflection Cumbersome computational procedures needed

to check whether segmented cutting edges are engaged in

cutting or not are eliminated by this proposed method The

presented analytical approach has advantages in flexibility,

prediction time, and accuracy as compared with other

numerical techniques In addition, the effects of cutting

conditions and run-outs, such as eccentricity and tilting on

the cutting force and tool deflection, can be analyzed

quantitatively in the time domain or frequency domain The

validity and effectiveness of the suggested method are verified

through a series of cutting tests The model presented in this

research can be used in real-time machining error estimation

and cutting condition selection for error minimization since

the form accuracy is easily estimated from the acquired tool

deflection curve

Keywords End milling Cutting force Tool deflection Fourier series Machining error

Nomenclatureδ(θ) Window function representing tool entry

and exit

dFi i directional cutting force fraction

kt, kr, ka Cutting force coefficients in tangential, radial,

and axial directions

θ Cutting edge position angle

ft Feed per tooth

R Cutter radius

α Cutter helix angle

β Cutting edge lag angle by cutter helix

a0, ak, bk Fourier coefficients

m Cutting edge number

N Flutes number of cutter

r Cutting edge radius

e Eccentricity between cutter geometric

center and rotational center

t Tool tilting angle

r Cutting edge radius

κ, l Angle parameters representing cutter initial

position

tc Undeformed chip thickness

wi i directional distributed load

vi i directional tool deflection

Mbi Bending moment about i axis

E Young’s modulus of tool material

I Area moment of inertia of cutter

fr Tool rotational frequency

ftp Tooth passing frequency

f Tilting frequency

S H Ryu ( *)

Department of Mechanical Engineering,

Chonbuk National University,

664-14 Duckjin-dong, Duckjin-ku,

Jeonju 561-756, South Korea

e-mail: ryu5449@jbnu.ac.kr

DOI 10.1007/s00170-011-3490-4

1 Introduction

End milling is one of the most important material removal

processes widely used for machining precise mechanical

parts in automotive, aerospace, and die and mold industries

Tool deflection resulted from the cutting force directly

affects form accuracy, dimensional accuracy, and surface

roughness, which are decisive factors in machining quality

Over the past several decades, many researchers have

developed metal cutting models theoretically and

experi-mentally However, in most cases, cutting forces and tool

deflection have been predicted by numerical methods

because milling is basically an intermittent process In

early research on metal cutting, Martellotti [1, 2]

investi-gated the principal mechanism of surface generation, chip

flow, and cutting power Merchant [3] estimated the cutting

forces in orthogonal cutting considering the shear angle,

friction angle, and tool rake angle Based on experimental

observation, Stabler [4] suggested the chip flow rule that

chip flow angle is equal to the inclination angle in oblique

cutting For calculating the cutting force in oblique

machining, Oxley [5] introduced shear flow stress

deter-mined by cutting temperature, shear strain rate, and cutting

speed at plastic shear zones based on the slip line theory

Usui [6] took into account the side cutting edge effect on

shear area formation and calculated chip flow direction

through the energy method Gygax [7] studied full cut

milling dynamics with a single tooth focusing on the

graphical pulse shape for force component evaluation For

predicting end milling force, Kline et al [8] divided the

helical cutting edge into disk elements and estimated the

cutting force by a numerical summation of fractional

cutting forces on the segmented elements Kline and DeVor

[9], and Li and Li [10] suggested an extended mechanistic

cutting force model including the cutter run-out effect Tool

deflection has also been calculated through a similar

numerical procedure which is used for surface texture

generation and machining error evaluation [11] Yellowley

[12] added the parasitic force effect in calculating the

peripheral milling force and showed that specific power is a

unique function of average chip thickness Armarego and

Whitfield [13] presented a computer-based model for

calculating force and power in peripheral milling Armarego

and Deshpande [14] suggested a numerical cutting force

estimation model in end milling considering eccentricity and

the tool deflection effect Budak et al [15] showed that the

milling force coefficients for all force components and cutter

geometrical designs can be predicted from orthogonal

cutting data Altintas and Engin [16] predicted end milling

cutting forces by transforming shear angle, shear stress, and

friction angle stored in the orthogonal cutting database to

local oblique edge segments along the helical flutes Altintas

and Lee [17] presented a generalized cutting mechanics

model considering milling cutter geometry Wan and Zhang[18] developed numerical algorithms for form error estima-tion based on finite element methods As mentioned above,

in predicting the cutting force and tool deflection during endmilling, all of the reviewed papers are based on the slicedsegments method of cutting edge and require numericalsummation procedures case by case But through thesenumerical approaches, it is difficult to understand therelationship between each cutting parameter and the cuttingforce or tool deflection Also, estimation accuracy isdependent on the number of sliced segments when usingthese numerical approaches To overcome these drawbacksfrom numerical approaches, Zheng et al [19] suggested anangle domain analytical method for cutting force evaluation

in end milling But they only addressed the ideal case whichassumes a runout-free condition To the author’s knowledge,the general cutting force and tool deflection functions asexplicit forms have not been reported yet in end milling

In this research, the analytical cutting force and tooldeflection functions in end milling are firstly derived whichconsider the cutting parameters, tool geometry, and toolrotational angle The discontinuous form of the cuttingforce function caused by intermittent tool entry and exit ischanged to a continuous one using Fourier series expan-sion Then, through the direct integration of infinitesimalcutting forces loaded on the cutting edges, the resultantcutting force is acquired as a function of tool rotationalangle Tool deflection function is also formulated by adirect integration of the distributed load on the helicalcutting edges The presented analytical approach hasadvantages over the numerical approaches in its predictionaccuracy and estimation time regardless of the number ofcutting edge partitions or flutes This research clarifies therelationships between the cutting force, tool deflection, andcutting parameters in the time or frequency domain andgives more basic information on machining error generationmechanisms in end milling

2 Cutting force formulation using Fourier series

In side wall machining with an end mill, the cuttingpressures loaded on the cutting edge have differentmagnitudes and directions according to the cutting edgelocation, as shown in Fig 1 The cutting edge whichengages in cutting varies as the tool rotates Based on theassumption that tangential and radial directional cuttingforces are proportional to the undeformed chip area,fractional cutting force components loaded on the infinitesi-mal cutting edge are represented as Eq.1using cutting forcecoefficients [20] X, Y, and Z directional cutting forcecomponents are acquired through a coordinate transforma-tion expressed in Eq 2, referring to Fig 1 The window

function,δ(θ) in Eq.2, is introduced to represent whether the

cutting edge engages in cutting or not When the cutting edge

locates betweenθ1andθ2, it participates in cutting Here,θ1

is the entry angle andθ2is the exit angle In Fig.2,δ(θ) is a

discontinuous function with an interval of 2π Hence, cutting

force calculation through a direct integration of Eq 2 is

impossible because a discontinuity function,δ(θ), is included

The Fourier series is a mathematical tool useful for

analyzing the periodic function by decomposing such

function into a weighted sum of much simpler sinusoidal

component functions Keeping in mind that end milling is a

periodic process by tool rotation, Fourier series can be

adapted to express the cutting force and tool deflection

formulation If the discontinuity ofδ(θ) is solved by Fourier

series expansion, each directional cutting force function of

Eq.2is directly integrable

9

=

;ð2Þwhere,

dz ¼ Rtan adbδ(θ) can be expressed like Eq 3 using Fourier series.Here, a0, ak, and bkrepresent the Fourier coefficients

R2p

0 d qð Þ cos kqdq ¼sin kq 2 sin kq 1

kp

bk¼1 p

Fig 1 Cutting mechanism and force distribution in end milling

Fig 2 Window function

repre-senting periodic tool entry and

exit

ð6ÞThe property of Eq 6 can be simply proven through

Eq.7 By a direct integration of Eq.5after substituting f(θ)

with 1, sinθ, sin2θ, or cos2θ, three directional cutting force

components are acquired as a function of the tool rotational

angle, so numerical procedures are no more necessary in

calculating the cutting force

In end milling, generally, cutter runout exists, bringing

cutting force deviations between cutting flutes [9] The main

runout component is a tool setting error which happens when

the tool is chucked on the machine tool The tool setting

error can be described using two parameters, eccentricity and

tilting, as shown in Fig.3 When a tool setting error exists,

the rotational radius of each cutting edge varies, as in Eq.8

The parametersκ and λ, introduced to specify tool runout

and coordinating system, are shown in Fig.4, where the

X′-axis is set to coincide with the tool tilting direction andλ is

the angle between a cutting flute end, which is assumed as

the first flute, and X′-axis κ is the angle between the X′-axis

and the Y-axis set as perpendicular to the tool feeding

direction In Eq.8, the second term represents a variation of

the cutting edge radius caused by eccentricity, and the third

term denotes the change of cutting edge radius by tool tilting

þ L  Rb tan a

sin t sin l þ b þ 2mp

N  p N

ð9ÞTherefore, the cutting force components in Eq 2 could

be modified as Eq.10considering tool runout effects

9

=

;ð10Þ

By substituting Eq.9 into Eq.10, the fractional cuttingforce components are expressed like Eq 11 Finally, threedirectional cutting forces are acquired as a function of thecutting conditions and tool rotational angles [21]

9

=

;

ð11Þwhere, x ¼ f  k þ l p

N; h ¼ f þ l  p

NFigure5shows three directional forces according to thetool rotational angle when the Fourier series order changes.Assumed are two fluted end mill with a 10-mm diameter,0.2-mm feed per tooth, 50-mm tool overhang, and fullimmersion without tool runout As the Fourier series orderincreases, the simulated graph profile converges to the idealcutting force shape

Figure 6a–c shows the contribution of the feed andrunout components to the cutting force, and Fig 6

represents the resultant cutting force acquired after posing feed and runout forces In simulation, we assumedthat the tool tilting angle is 0.01° and eccentricity between(a) Axis eccentricity (b) Tool tilting

super-Spindle rotation center

Tool center

Before tilt

After tilt

L sin e

Fig 3 Runout induced by tool setting error

θ

X

Y

λκ

the geometrical tool center and tool rotational center is

10 μm The assumed values are the most typical ones

measured in general end milling processes Other cutting

conditions are the same as in Fig.5 The geometrical runout

parameters, λ and κ, are set to 45° Figure 6a shows a

typical cutting force pulsation profile generated by pure

tool feeding in milling, and the cutting force frequency is

equal to the tooth passing frequency [7] Initial negative

value in the Y direction comes from tool pushing by

workpiece On the other hand, in Fig.6b, tool eccentricity

affects the chip load variation with tool rotational frequency

Tool tilting changes the cutting radius of the helixed cutting

edge depending on the Z directional position, resulting in an

undeformed chip thickness variation The cutting force shape

of Fig.6c is influenced by the tilting angle magnitude and its

direction Tool runout is widely observed and is an inevitable

phenomenon in end milling and causes an unequal cuttingload on each cutting flute By the superposition of Fig.6a–c,the resultant cutting force can be acquired as shown inFig.6d It is notable that cutting parameters such as feed andrunout have different effects on the resultant cutting force inmagnitude, shape, and frequency

3 Tool deflection formulation using Fourier series

Distributed cutting forces in the X and Y directions can beexpressed like Eq.12 Since the Z directional cutting force

is relatively small compared with the X and Y directionalones and its effects on tool deflection and machining errorare negligible, the Z directional force is not consideredhereafter When the distributed cutting force in the Xdirection is denoted as wx, shown in Fig 7, the Xdirectional tool deflection, vx, induced by wx can becalculated from Eq.13

After substituting the distributed loads, wx, wy, into

Eq.13, tool deflections, vx, vy, can be acquired as functions

of the tool rotational angle and axial position, Z, after someconsecutive integrations

Mby¼ zRz

0

wxdzRz 0

The constants of integration are calculated from continuityconditions at the boundaries between the cylindrical tool part,flute part, and the top position of the axial depth The areamoment of inertia of tool is calculated considering cylinderand flute parts’ geometries The area moment of inertia of theflute part is obtained from Kops’s equivalent diameter model[22]

Mby¼ zRz

0

wxdzRz 0

wxdbRb 0

Tool rotation angle [deg]

Tool rotation angle [deg]

Tool rotation angle [deg]

(a) X directional force

(c) Z directional force

Fig 5 Cutting forces with respect to Fourier series order, K

the bending moment by the distributed cutting load By

integrating the bending moment with respect toβ, the tool

deflection angle and tool deflection value are obtained

Cutting conditions, tool eccentricity, and tilting also affect

the tool deflection shape, magnitude, and frequency in a

similar way as the cutting force The detailed proceduresare given in Ryu [21]

wxdbdbRb

0

Rb 0

bRb 0

wxdbdbdb Rb

0

Rb 0

Rb 0

wxbdbdbdb

ð16ÞFigure 8 shows the simulation results of cutting forcesand tool deflection with 0.1-mm feed per tooth and 0.3-mmaxial depth of cut in slot end milling The cutting force andtool deflection diagrams show the same trajectory in shapedue to their linear relationships under static deformationassumption Figure 9 shows tool deflection trajectorieswhen tool runouts exist The differences of tool deflectiontrajectories come from not only magnitude differences oftool tilting and eccentricity but also their positionalrelationships defined by the parameters κ and λ If tooltilting direction and eccentric direction have a phasedifference of 180° to each other, the cutting force and tooldeflection differences between two cutting edges aremaximized If their directional phase differences approachzero, the cutting force and tool deflection between thecutting edges are minimized

900

(c) Tilting component (d) Combined resultant cutting forces

Fig 6 Cutting forces composed by feeding and runout