Force Prediction Models for Helical End Milling... 2015

8/18/2019 Force Prediction Models for Helical End Milling... 2015

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ORIGINAL ARTICLE

Force prediction models for helical end millingof nickel-aluminium bronze

Ruihu Zhou1

& Wenyu Yang1

& Kun Yang1

Received: 10 September 2015 /Accepted: 17 December 2015# Springer-Verlag London 2016

Abstract Nickel-aluminium bronze is used widely in seawa-

ter environment. The mechanistic method and analyticalmethod to predict helical end milling force are briefly de-scribed and compared. The mechanistic approach is shownto depend on milling force coefficients determined from mill-ing tests. By contrast, the analytical method is based on a

predictive machining theory, which regards the workpiecematerial properties, tool geometry, cutting conditions andtypes of milling as the input data. Each cutting edge of thehelical end cutter is discretized into a series of infinitesimalelements along the cutter axis and the cutting action of whichis equivalent to the classical oblique cutting process. Thus, thecutting force components applied on each element can be

calculated using a predictive oblique cutting model and thetotal instantaneous cutting forces are obtained by summingup the forces contributed by all cutting edges. The equationof equivalent plane angle is derivation through coordinatetransformation. Experiments on machining nickel-aluminium

bronze under different cutting conditions were conducted tovalidate the proposed model.

Keywords Nickel-aluminium bronze . Helical end milling .

Analytical model . Mechanistic model

1 Introduction

Nickel-aluminium bronze (NAB) is a copper-based alloywhich is widely used for propulsion and seawater handlingsystems in naval platforms, such as marine propellers, bear-ings [1]. However, the study of machinability of NAB in the

published literature is little. Maybe, the researchers consider the machinability of NAB is well, compared with difficult-to-machine alloys.

The helical end milling is used extensively in manufactur-ing of surfaces, such as propellers, turbines, and dies/molds.Modelling of cutting forces is the basis for prediction of ma-chine tool chatter, tool wear and breakage, cutting parameters

optimization and surface quality [2]. There mainly exist threemethods to model milling forces: mechanistic, numerical andanalytical methods. The mechanistic models are commonlyavailable to predict cutting forces especially for new materialsand tools. Typical approach for numerical modelling is finiteelement method (FEM). Currently available FE models can beused to predict cutting forces, stress and temperature distribu-tions. Simulations of FEM are time consuming and the phys-ical meaning is not clear. Due to the fact that analyticalmethods are more physics based rather than experimentalmodelling, it can reflect the actual physical phenomenon bet-ter than the other methods. Arrazola et al. [3] summarized the

capabilities and limitations of modelling approaches.Two types of mechanistic cutting force models are found in

the machining literature. In the first model, the effects of shearing mechanism and effects of rubbing and ploughingmechanisms are lumped into one specific force coefficient for each cutting force component. In the second model, theshearing and ploughing effects are characterized separately bythe respective specific cutting and edge force coefficients.Budak et al. [4] described and compared the mechanistic andunified mechanics of cutting approaches to the prediction of

* Wenyu [email protected]

1 State Key Laboratory of Digital Manufacturing Equipment andTechnology, School of Mechanical Science and Technology,Huazhong University of Science and Technology, Wuhan 430074,China

Int J Adv Manuf Technol

DOI 10.1007/s00170-015-8261-1


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milling forces. Gradišek et al. [5] developed expressions for semi-empirical mechanistic identification of specific cuttingand edge force coefficients for a general end mill from millingtests at an arbitrary radial immersion. Wan et al. [6] develop anew approach able to identify the cutter radial run-out andcutting force coefficients in the flat end milling. Recently,Wan et al. [7] developed a new ternary-mechanism cutting

force model including chipremoval, flank rubbing and bottomcutting effects to predict cutting forces in flat end milling.Wang et al. [8] used experiments to investigate the cuttingforce coefficients in the average cutting force model.

For the analytical methods, the researchers try to establishmathematical relations between the milling forces and severalmechanical aspects likefriction, geometry and material behav-iour. Oxley [9] developed a predictive machining theory that allows for the high strain rate, high temperature and thermal

properties of the work material. Moufki et al. [10] developedm o d e l o f o b l i q u e c u t t i n g b y i n t r o d u c i n g t h ethermomechanical behaviour of the workpiece material and

the tool-chip interface friction characteristics. Li et al. [11] presents an analytical method based on the unequal divisionshear-zone model to study the machining predictive theory.Through introducing the thermomechanical behaviour of theworkpiece material and tool-chip interface friction character-istics, Fontaine et al. [12], Moufki et al. [13], Li et al. [14] andFu et al. [15, 16] presented a analytical force model for endmilling process using the oblique cutting approach. Fontaineet al. [12] present a predictive force model for ball end milling

based on thermomechanical modelling of oblique cutting.Moufki et al. [13] developed an analytical thermomechanicalmodelling of peripheral milling process using a predictive

machining theory. Li et al. [14] presented a theoretical model-ling for cutting forces in helical end milling based on Oxley’stheory, where the action of a milling cutter was modelled asthe simultaneous actions of single-point cutting tools. Recent-ly, Fu etal. [15, 16] also presented a predictive force model for helical end milling and ball end milling process based on ananalytical thermomechanical modelling of oblique cutting.Pang et al. [17] developed a modified Oxley’s predictive ma-chining theory to analyse the mechanics of cutting in endmilling using helical end mill tools.

However, some input parameters are based on assump-tions which may induce error in analytical model. The

shear angle and friction coefficient are the important pa-rameters which affect the accuracy of prediction of cuttingmodel. The main problem of finding a completely analyt-ical way to determine shear angle and friction law has not

been solved. Hu [18] summarized the shear angle equa-tions. Toropov et al. [19] developed a new equation for the determination of the shear angle for continuous cut-ting assuming the curvilinear form of the shear line. Ozel[20] comprised five friction models using FE simulationsof machining. Bahi et al . [21 ] developed a hybrid

analytical-numerical approach to determine frictioncoefficient.

Generally, two approaches have been used to modeloblique cutting process. One of them is the normal planemethod [10]. Many researchers made the assumption that analysis of oblique cutting in the normal plane is equivalent to that of orthogonal cutting, and then all the velocity and

force vectors are projected on the normal plane. The other isthe equivalent plane method [14]. The equivalent plane isdetermined by the cutting velocity and chip velocity; themechanism of oblique cutting is considered as the accumula-tion of a series of two-dimensional cutting processes. Li et al.[14] developed the equivalent plane in oblique cutting; thederivation of equivalent plane angle is brief which may bedifficult to understand. In this paper, the equivalent plane an-gle is obtained using another approach through coordinatetransformation.

The published papers use only one method to predict thecutting force. It is the first time using both mechanistic model

and analytical model to obtain the milling force of NAB. For simplicity, tool wear, tool run-out and deflection are not con-sidered in this presented model. In order to evaluate the mill-ing force coefficients, a set of milling experiments at different feeds per tooth, spindle speed and radial cutting depth have to

be run. According to the fact that along an engaged cuttingedge of an end mill, the present paper proposes to apply thethermomechanical model of oblique cutting developed by Liet al. [14] to calculate the end milling forces. Orthogonal cut-ting theory based on unequal division shear zone is extendedto oblique cutting using equivalent plane approach.

The organization of this paper is as follows: After introduc-

ing the geometry of the cutter and the related parameters inhelical end milling process in Section 2, mechanistic modeland analytical model are developed in Sections 3 and 4. Themodel validation is implemented and discussed in Section 5

by the end milling experimental data. Finally, conclusions of this study are drawn in Section 6.

2 Geometry of the helix end milling

The geometric model of end milling and oblique cutting mod-

el are illustrated in Fig. 1. The related parameters are definedin the global coordinates {o, X , Y , Z }. D is the cutter diameter,i0 is the helix angle, a p is axial depth, d r is radial depth of cut and N is the number of teeth.

For the point P on the j th cutting edge with elevation z , it islocated by the immersion angle ϕ j ( z ), which is determined by:

ϕ j z ð Þ ¼ ϕ þ j −1ð Þ 2π

N −

2 z

D tani0 ð1Þ

where ϕ is the rotation angle of the reference cutting edge.



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The undeformed chip thickness h(ϕ j ) varies periodically

with the immersion angle ϕ j ( z ) and is written as:h ϕ j ¼ f t sin ϕ j Γi j ð2Þ

where f t is the feed per tooth. Γ ij is a unit step function that determines whether the teeth is in or out of cut,

Γ i j ¼ 1 f or ϕ st < ϕ j < ϕex0 f or ϕ j < ϕ st or ϕ j > ϕex

ð3Þ

where the entrance angle ϕ st and the exit angle ϕex are definedas:

ϕ st ¼

0; ϕ

ex ¼ arccos 1−2d

r . D up milling

ϕ st ¼ π −arccos 1−2d r .

D

; ϕex ¼ π down milling 8


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The tangential, radial and axial forces can be expressed inglobal coordinate system {o, X , Y , Z } as

d F x; j ϕ j ; z

d F y; j ϕ j ; z

d F z ; j ϕ j ; z

24

35 ¼ −cosϕ j z ð Þ −sinϕ j z ð Þ 0sinϕ j z ð Þ −cosϕ j z ð Þ 0

0 0 1

24

35 d F t ; j ϕ j ; z

d F r ; j ϕ j ; z

d F a; j ϕ j ; z

24

35

ð7

ÞThe average cutting force calculated as follows:

F x

F y

F z

264

375 ¼ N a p f t

8π

K t K r K a

24

35 cos2ϕ − 2ϕ−sin2ϕð Þ 02ϕ−sin2ϕ cos2ϕ 0

0 0 −4cosϕ

24

35ϕexϕ st

ð8Þ

Given the cutter geometry and immersion conditions, onlythe specific cutting force coefficient remain unknown in theright-hand side of Eq. (8). The coefficients can therefore bedetermined by equating the measured cutting forces with the

corresponding expressions. The details can be found in Wanet al. [6].

4 Analytical cutting force model

In Figs. 1 and 2, geometric parameters associated to obliquemachining are illustrated. The normal shear angle, normalrake angle, the undeformed chip thickness and the width of cut are given respectively by ϕn, αn, t 1 and w. In the tool rakeface, the chip flow direction is defined by the chip flow angle

η c. η s is shear flow angle.

For application to cutting forces modelling of helicalend milling, the cutting action of each infinitesimal ele-ment can be represented as the oblique cutting force. Andthe relevant equations derived from the oblique cuttingmodel [14] are established and summarized. A set of co-ordinate systems are defined. These coordinate systemsare associated to the reference planes, and are summarized

as follows:Coordinate system (x I , y I , z I ); x I is parallel to the cutting

velocity direction, (y I , z I ) is the reference plane P r and (x I , y I )is the cutting plane P s.

Coordinate system (xc, yc, zc); zc is the rank cutting edgedirection, yc is parallel to yn, rank face plane Aγ is determined

by yc and zc.Coordinate system (x fl , y fl , z fl ); z fl is chip flow direction and

x fl is collinear to xc.Coordinate system (x, y, z); y is collinear to yn, (x, z) is the

normal plane P n and z is perpendicular to the shear plane P sh.Coordinate system (x s, y s, z s), x s is shear flow direction, z s

is collinear to z and shear plane P sh is determined by x s and y s.Coordinate system (xe, ye, ze); xe is collinear to x s, ze is

parallel to z fl and equivalent plane P e is determined by xeand ze.

Equivalent plane is determined by cutting velocity and chipvelocity, P e plane through x s and z fl . η e is the equivalent planeangle and calculated from coordinate system transformation.(xe, ye, ze) is obtained by the rotation of angle η e of the coor-dinate system (x s, y s, z s) around the xe axis.

xeye

ze8


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ze ¼ −sinη ey s þ cosη ez s ð10Þ

(x s, y s, z s) is obtained by the rotation of angle η s of the coordi-nate system (x, y, z) around the z axis.

x sy sz s

8><>>>: ð19Þ

γ :

m ¼ q

þ 1

ð ÞV s

h ¼ q

þ 1

ð ÞV cosλ scosϕn

hcosη scos ϕn−αnð Þ ð20Þ

ƛ ¼ cosαn−cos ϕn−αnð Þcosη s cosϕncosη s þ tanλ ssinη sð Þcosαn

ð21Þ

V c ¼ V cosλ ssinϕncosη ccos ϕn−αnð Þ

; V s ¼ V cosλ scosαncosη scos ϕn−αnð Þ

ð22Þ

where γ :; γ :m; γ and represent the shear strain rate, the maxi-mum shear strain rate and the shear strain. In this paper,h = 0.025 mm, q = 3.

The workpiece material is supposed to be isotropic andviscoplastic-rigid, and its behaviour is described by theJohnson-Cook constitutive model:

τ s ¼ 1 ffiffiffi3

p A þ B γ ffiffiffi3

p n

1 þ C ln γ :

γ :0

!" # 1−

T −T r

T m−T r

m

ð23Þwhere τ s; γ ; γ

: and T represent the shear stress, the shear strain,the shear strain rate and the absolute temperature, respectively.The material characteristics are defined by the parameters A,



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B, C , n, m, which stand for the yield strength, the strength co-efficient, the strain rate constant, the strain hardening expo-nent and the thermal softening coefficient, respectively.

In cutting process, the boundary of the primary shear zonecan be considered adiabatic; the thermal conductivity is thennegligible. Assuming that a fraction χ (Taylor-Quinneycoefficient) of the plastic work is converted into heat, since

the temperature depends only on the coordinate z e under thesteady flow condition, consequently, the heat transfer equation

becomes:

dT

dz e¼ cosη e

ρcV sinϕncosλ sχτγ : ð24Þ

where ρ, c and χ represent the material density, the specialheat and the Taylor-Quinney coefficient. In this paper,χ= 0.85 [16].

Normal shear angle ϕn is given by the Li et al. [14]:

ϕn ¼ π 4 −β 2 þ αn

2 ð25Þ

As suggest by Dudzinski et al. [22], the mean friction co-efficient may be a power function of the chip velocity:

f ¼ f 0V c p ð26Þ

In this paper, f 0 =0.704, p = − 0.248.The shear angle is an important parameter in analytical

cutting force model. Table 1 summarized the shear angleequations.

Furthermore, the shear force dF s, which is proportional tothe shear stress τ s, along with the normal force dN s can beexpressed as:

d F s ¼ τ s⋅t 1cosλ ssinϕn

dz ð27Þ

dN s ¼ − tan ϕn−αnð Þ þ tanβ cosη c½ cosη s1−tanβ cosη ctan ϕn−αnð Þ

d F s ð28Þ

Therefore, for the infinitesimal element of the j th cut-ting edge, the three differential cutting force componentsdF t , j , dF r , j , dF a, j (tangential, radial, axial), applied to

point P in Fig. 2a, are evaluated from the following

matrix form:

d F t ; j d F r ; j d F a; j

8<:

9=; ¼

cosλ scosϕncosη sh þ sinλ ssinη sh cosλ ssinϕn−sinϕncosη sh cosϕn

sinλ scosϕncosη sh−cosλ ssinη sh sinλ ssinϕn

24

35 d F s

dN s

ð29Þ

The cutting forces in x, y and z directions at the smalldifferential elements may also be transformed as the following:

d F x; j ϕ j ; z d F y; j ϕ j ; z

d F z ; j ϕ j ; z

24 35 ¼−cosϕ j z

ð Þ −sinϕ j z

ð Þ 0

sinϕ j z ð Þ −cosϕ j z ð Þ 00 0 1

24 35 d F t ; j ϕ j ; z d F r ; j ϕ j ; z

d F a; j ϕ j ; z

24 35ð30Þ

By integrating Eq. (30), respectively, the total cutting forcein each direction can be obtained. The algorithm for predictingthe cutting forces in end milling is shown in Fig. 3.

F m ¼X N j ¼1

Z z 2 z 1

d F m; j dz ; m ¼ x; y; z ð31Þ

where z 1, z 2 are the lower and upper axial engagementlimits of the in-cut portion.

5 Model validation

5.1 Experiment details

To validate the proposed modelling of end milling, a series of helical end milling tests have been performed on a vertical 5-axis machining center (Tuopu VMC-C50) with dry cuttingand experimental setup is shown as in Fig. 4. The workpiece

is mounted on a three-component Kistler table dynamometer (9257B) which is used to measure cutting forces. The sampling frequency is 25 kHz and sampling time is 10 s for eachtest. The carbide cutter is Sandvik Coromant R215.34C12040-DC26K 1640, which has four-flute cutting edge, 40°helix angle, 12° rake angle and 12 mm diameter. The compo-sition of nickel-aluminium bronze (NAB) has the followingchemical composition:

Cu ¼ 85%; Al ¼ 8:5‐10%; Fe ¼ 4:0‐5:0%; Ni ¼ 4:0‐5:0%;Others ≤ 0:1%:

Table 1 Summary of shear angle relationship

Researchers Relationship

Krystoff (1939) [18] ϕ=π /4 +α− λ

Mechant (1945) [18] ϕ=π /4 +α/2 − λ/2

Lee, Shaffer (1951) [18] ϕ=π /4 +α− λ+ (σ s/ τ s − 1)/2

Hucks (1951) [18] ϕ=α− λ+ (C M − arctan 2μ)/2+ arctanμ

Shaw (1953) [18] ϕ=α− λ+ arctan(σ s/ τ s)

Weisz (1957) [18] ϕ=α− λ+54.7∘

Wright (1982) [18]ϕ ¼ cos−1 τ s

τ μcos π 4 −

α2

sin π 4 −

α2

þ αA Toropov (2007) [19] ϕ=π /4 − γ f + 5γ /8



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In previous work [16], the flow behaviour of material NABis described by the modified Johnson-Cook constitutive lawand its involved parameters are given as:

A¼295MPa; B¼759:5MPa; C ¼0:011; n¼0:405; m¼1:09;γ :0 ¼0:001 1=s;T m ¼1311K ; T r ¼293K

The other material parameters are given by:

ρ¼7530kg=m3; c¼419:0 J= kg⋅K ð Þk ¼41:9 W= m⋅K ð Þ:

where ρ is density, c is specific heat and k is thermalconductivity.

Thirteen milling tests are conducted with different cuttingconditions, each test repeated three times. The instantaneousmilling forces for one revolution of the cutter can be obtained

by taking the average values of the cutting forces measures inthe neighboring ten periods. Fx, Fy, Fz represent forces in thefeed, normal to the feed, and axis directions, respectively. Theworkpiece size is 100× 100× 10 mm. The cutting conditionsfor the test are shown in Table 2. All the cutting is down

Input parameters

Work piece material:A B C n m

Helical end geometry:D Ni0Cuttingconditions: , , ,r p en a a f

Determine number of angular integration steps

L,number of axial integration steps K,entry

angle exit angle

Initialize force variation registers

Angular integration loop: i=1 to L

Initialize the instantaneous force integration

registers

Calculate the tool rotation angle

Tooth integration loop:j=1 to N

Calculate the immersion angle ofthe jth flute

bottom angle

Axial integration loop:s= 1 to K

Calculate the immersion angle of the sth discrete

point (height z)on the jth edge from Eq.(1)

Is the cutting

region or not?

Calculate the chip thickness ,Normal rake angle,

friction angle,shear angle,cutting velocity,widthof cut

Calculate the shear flow angle,chip flow angle

and equivalent plane

Calculation the force components(dFx,dFy,dFz)

From Eq.(30)

Oblique cutting model of

infinitesimal cutting edge

Calculate the temperature distribution in the

primary shear zone

Calculate the shear flow stress,shear and

normal forces

Calculate tangential,radial and axial force

K>s

Sum the force of aallengagededges

Output the cutting force variation with tool

rotation angle

N

s=s+1

N>j

L>i

N

Y

Y

j=j+1

Y

i=i+1

N

N

Fig. 3 The flow chart of cuttingforces calculation



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milling and axial depth of cut is 1.5 mm. The program of cutting force simulations was all executed with Matlab2010bon a computer.

5.2 Mechanistic model results

The cutting forces measured in tests 1 – 9 are used to determinethe cutting force coefficients. Based on the mechanistic mod-el, calibrated values of K t , K r , K a are shown in Table 3. Theaverage cutting force coefficients are:

K t ¼1494N=mm2; K r ¼1444N=mm2; K a ¼105N=mm2:

The results of comparison are shown in Fig. 5a – d. Althoughcutting parameters, such as spindle speed, feed per tooth andradial width of cut, are different, mechanistic forces matchwell experimental ones in both shape and magnitude for all

cutting tests, while slight errors exist and are tolerable. Due tonoisy signals such as uncertain factors of the piezoelectric

dynamometer or the tiny foundation vibrations of nearby ma-chines in the experimental environment, slight perturbationsof measured cutting forces exist when the magnitudes of cut-ting force components attain a relatively low level. Even inthis condition, the trends of simulation and experimental dataare in good agreement. In Fig. 5a, since the experimental data

has run-out, the magnitude of cutting force did not match well, but the shape of force profile during the cutter revolution iscaptured well.

5.3 Analytical model results

In the simulations, the cutter was discretized into 15 disks(calculation increments dz=0.1 mm); the force calculationswere performed every 1° of cutter rotation (calculation incre-ments d ϕ= 1°).

Figure 6 shows that under these cutting conditions, the

experimental values of milling force have a good agreement with the proposed analytical model on the waveform, ampli-tude and phase. Each of the plots captures the cutting forcesfor one revolution. There is a small amount of amplitude dis-crepancy between the predictions and the experiments. Themain reasons for this deviation may be attributed to the fol-lowing: (1) the cutter deflection is caused by the peak forcesdue to the stiffness in actual helical end milling process. Inaddition, the plough forces are introduced at the tool-workpiece interface zone due to the little flank wear of thecutter. (2) In the proposed model, the predicted results areinfluenced due to the errors caused by some measured or

identified parameters. (3) There is another factor which is

Table 2 Cutting conditions for helical end milling, down milling

Test no Spindle speed(rpm)

Feed per tooth(mm/tooth)

Radial depthof cut (mm)

1 1000 0.04 6

2 1000 0.05 6

3 1000 0.06 6

4 1600 0.04 6

5 1600 0.05 6

6 1600 0.06 6

7 2000 0.04 6

8 2000 0.05 6

9 2000 0.06 6

10 2000 0.04 4

11 2000 0.05 4

12 1800 0.05 6

13 1600 0.03 6

Fig. 4 Milling cutting test

Table 3 Cutting force coefficients (units K t K r K a = [N/mm2])

Test no. 1 2 3 4 5 6 7 8 9

K t 1321 1612 1743 1004 1339 1819 1648 1420 1543

K r 1010 1640 1566 1086 1496 1436 1610 1640 1519

K a 105 125 90 43 54 98 154 61 217



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difficult to avoid: the tool run-out. In milling, this geometricaldefault is mainly due to the offset between the positionof the tool rotation axis and the spindle rotation axis(Fontaine et al. [12]). The consequence is a tool rotation

around the spindle axis with an eccentricity. This eccen-tricity modifies the tool engagement and the local cut-ting conditions (cutting velocity and angles). Then, therun-out has a direct effect on the cutting forces leveland variation. Its effect is particularly significant whenthe undeformed chip thickness reaches small values, asshown in Fig. 6h.

Influence of cutting parameters on cutting force:

(1) Feed per teeth f t . It can be noted that the feed rate varia-tion affects directly the cutting forces; indeed, thesevalues depend proportionally on the undeformed chip

thickness. Figure 6h,d shows the three force componentsmaximum values double closely as the feed per teethincreases from 0.03 mm/tooth to 0.06 mm/tooth. In mill-ing parameters, f t has the greatest effect on cutting force.

(2) Radial depth of cut d r which influences the cuttingforce by changing the immersion angles of entranceand exit. Figure 6e, f shows the cutting forces for the various immersion angles of exit. When cuttingwidth decreases from 6 to 4 mm, the immersionangles of entrance change from 90 to 138°. So,

cutting forces decrease with the cutter-workpieceengagement domain expansion.

(3) Spindle speed nr . Comparing the predicted force compo-nents in Fig. 6b, e, the global level of cutting forces

decreases theoretically with increasing values of spindlespeed, which is accounted by Eqs. (26). Actual observa-tions show obvious changes in measured force Fx but little changes in Fy (see Fig. 6b, e). In fact, the cuttingforce is greatly influenced by tool run-out when the spin-dle speed increase.

5.4 Comparison of two models

Figure 7a, b is a comparison of mechanistic and analyt-ical model result in cutting conditions for test 2 and test

8. It is shown that the proposed analytical methodclosely agrees with mechanistic method results. Themechanistic method requires a set of cutting tests for each milling cutter conditions, whereas the analyticalmethod requires input data and using oblique cuttingmodel extend to end milling. The mechanistic modelis used widely on new materials or the materials prop-erties not given. This method can be used in industrysince it calibrates cutting force coefficients efficiently.The analytic model has a clear physical meaning, but

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

200

250

Tool rotation angle (deg)

C

u t t i n g F o r c e s ( N )

(c) Test 11


C u t t i n g F o r c e s ( N )

(d) Test 12

Fy

Fz

Fx

Fz

Fx

Fy



Mechanistic Experiment

(b) Test 8


Fy

Fx

Fz

Fy

Fx Fz


(a)Test 2

Mechanistic Experiment

Mechanistic ExperimentMechanistic Experiment

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

200

250

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

200

250

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

200

250Fig. 5 Comparisons of mechanistic model andexperimental results



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(h) Test 13(g) Test12

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

200

250



Analytical Experiment

Fx

Fy

Fz

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

200

250


Toolrotation angle (deg)


(e) Test 8 (f) Test 11

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

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250


Fx

Fy

Fz

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

200

250


C u t t i n g F

o r c e s ( N )

Fx

Fy

Fz




(d) Test 6(c) Test 5

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

200

250



Analytical

Experiment

0 45 90 135 180 225 270 315 360-100

-50

0

50

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Analytical

Experiment

(a) Test 1

0 45 90 135 180 225 270 315 360-100

-50

0

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(b)Test 2

Fy

Fx

Fz

0 45 90 135 180 225 270 315 360-100

-50

0

50

100

150

200

250




Fx

Fy

Fz


Fig. 6 Comparison of proposed analytical model and experimental results



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some parameters are still uncertain such as shear angleand friction coefficient which attracts many researchersto study. The analytical force models present major ad-vantages and may dominate in future advances of metal

cutting modelling reviewed by Arrazola et al. [3].

6 Conclusion

In this paper, the mechanistic model and analytical model are proposed and compared with experiment result. The contribu-tions of proposed model are drawn as follows:

(1) Mechanistic modelling is an experimental approachwhere the cutting force coefficients are calibrated for agiven milling tool and workpiece pair at different radialdepth of cut, feed per tooth and spindle speed.

(2) An analytical model incorporates cutting force and ther-mal models derived for oblique cutting conditions con-sidering geometric transformations as well as locationeffects. A new approach is used to get the equivalent

plane angle through coordinate transformation.(3) The mechanistic model and analytical model results for

cutting forces are in good agreement with the experimen-tal approach. The combined use of analytical models andmechanical models can improve prediction accuracy, es-

pecially for new materials or the material property is littlein published papers.

However, the effect of tool run-out and deflection in mill-ing process need to be considered in future work for moreaccurate prediction of milling forces.

Acknowledgments This work is supported by the National Key BasicResearch Program (NKBRP) of China (No. 2014CB046704), Key Pro-

jects in the National Science & Technology Pillar Program of China (No.2014BAB13B01).

References

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(a) Test 2 (b) Test 8

Fig. 7 Comparison of analyticalmodel, mechanistic model andexperimental results



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Force Prediction Models for Helical End Milling... 2015

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