A novel mechanics model of parametric helical-end mills for 3D cutting force prediction

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W Yongqing and L HaiboA novel mechanics model of parametric helical-end mills for 3D cutting force prediction

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A novel mechanics model of parametric helical-endmills for 3D cutting force predictionW Yongqing and L Haibo*

Key Laboratory for Precision and Non-traditional Machining Technology of Ministry of Education, Dalian University of

Technology, Dalian, People’s Republic of China

The manuscript was received on 2 October 2010 and was accepted after revision for publication on 14 February 2011.

DOI: 10.1177/0954406211402757

Abstract: Cutting force prediction plays an important role in modern manufacturing systems toeffectively design cutters, fixtures, and machine tools. A novel mechanics model of parametrichelical-end mills is systematically presented for three-dimensional (3D) cutting force predictionin the article, which is different from mechanistic approach and Oxley’s predictive machiningtheory in model formulation and shear stress identification process. The single-flute cutting edgeand multiflute cutting edge of helical-end mills are modelled according to kinematic analysiswith vector algebra. Based on Merchant’s oblique cutting theory, a new mechanics model of 3Dcutting force with runout has been developed. Meanwhile, the asynchronous problem betweenpredicted and measured curves is solved by adjusting phase angle to minimize the average devi-ation. After minimizing the asynchronous phase angle deviation, shear stress can be estimateddirectly using corresponding peak-to-peak ratio or valley-to-valley ratio of the predicted curvesand the measured curves in X- and Y-directions of an arbitrary selected milling test. To assess thefeasibility of the general model, over 100 milling experiments of aluminium alloy (7075) usingflat-end mills and ball-end mills were conducted, respectively, and numerical tests implementedin time domain on MathWorks platform. The comparative results indicated that the predictedand the measured waveforms were quite satisfied in both pulsation pattern and period.

Keywords: mechanics model, parametric helical-end mill, 3D cutting force, runout effect

1 INTRODUCTION

In modern manufacturing systems, a variety of heli-

cal-end mills are adopted to satisfy complex shape,

tight tolerance, and high surface accuracy in aero-

space, automobile, and die/mold industries. The

understanding of kinematic and mechanics relation-

ships between cutters and workpiece plays an essen-

tial role in design of mills, fixtures, and machine tools

as well as in the optimization of milling process [1, 2].

Therefore, three-dimensional (3D) cutting force of

helical-end mills must be analysed clearly. The

classical orthogonal and oblique cutting processes

have been studied based on maximum shear stress

and minimum energy principles by a great number

of research studies for more than half a century [3, 4].

These achievements can be used as the fundamental

theory in cutting force prediction.

Literature review shows that two technologies have

been developed to predict 3D cutting force, mecha-

nistic approach [5, 6], and predictive machining

theory [7]. The mechanistic method views the

machining process as a combination of the following

components: cutter geometry, cutting process geom-

etry, workpiece characteristics, and machining

conditions. From a mathematical formulation point

of view, two types of mechanistic models were pre-

sented, lumped-model [8] and dual-model [9, 10].

The dual-model takes shearing and ploughing

*Corresponding author: Key Laboratory for Precision and Non-

traditional Machining Technology of Ministry of Education,

Dalian University of Technology, Dalian 116024, People’s

Republic of China. email: [email protected]

1693

Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science

at UNIV OF WISCONSIN on September 8, 2014pic.sagepub.comDownloaded from


forces into consideration, respectively, with smaller

model residual errors but more coefficients than the

simplified lumped-model. A unified method for dual-

model was developed to identify all force coefficients

by bridging a relation between oblique cutting and

orthogonal cutting [11, 12]. However, it was essen-

tially a choice to get rid of the constraint from

traditional cumbersome identification process. As

an alternative model, a comprehensive predictive

machining theory, i.e. Oxley’s theory, has been devel-

oped [7, 13–15], which takes two steps to calculate

cutting forces: (1) radial and tangential cutting

forces are calculated according to orthogonal cutting

operations and (2) axial cutting force is obtained from

oblique cutting process. As further achievements,

cutting edge effect and tool nose radius effect have

been taken into consideration for the first slice of

each tooth in references [14] and [15]. From the

research studies mentioned above, two points

should be concerned. First, those research studies

pay close attention to some special helical-end mills

with different envelops and edge geometries. Second,

the determination of shear stress is an indirect pro-

cess due to its dependence on orthogonal cutting

data. Therefore, it is worthwhile to develop a general

cutting force model, which can be applied to any heli-

cal-end mills with different geometric configurations,

and can use experimental results of milling to esti-

mate shear stress directly.

A novel mechanics model of parametric helical-

end mills for 3D cutting forces is systematically pre-

sented in this article. The helical cutting edge geom-

etry is modelled through forming transformation

from single-flute edge to multiflute edges in Section

2. Based on Merchant’s oblique cutting theory, a new

and accurate formulation of 3D cutting force is

derived through numerical integration of elemental

cutting force in Section 3. Also, in Section 4, the

instantaneous undeformed chip thickness is calcu-

lated considering runout effect. Finally, utilizing a

series of helical-end milling tests of aluminium alloy

(7075), both shear stress identification and model val-

idation are implemented with the aid of experimental

and numerical methods in Section 5.

2 FORMING PROCESS OF HELICALCUTTING EDGE

At present, there are two typical envelops to describe

the outer geometry of all helical-end mills: tapered

ball-end envelop [16] and parametric envelop [17–

19]. The latter can provide a convenient interface to

CAD/CAM software systems. In order to detail

the cutting edge geometry, a global Cartesian

coordinate system is defined: tool coordinate

system {Ct:OtXtYtZt}. The origin of Ct is located at

the tip of cutter with Xt pointing to the feed direction,

Zt aligning with the tool axis, and Yt following the

right-hand convolution, as shown in Fig. 1.

2.1 Single-flute cutting edge

The envelop of cutter is represented using vector r

drawn from tool tip Ot,

r ¼ �ðcos ’iþ sin ’jÞ þ zk ð1Þ

where ’ is counterclockwise position angle measured

from Xt-axis on XtYt-plane, � the sweeping radius,

and z the elevation along Zt-axis, i,j,k the unit vectors

corresponding to Xt-, Yt-, and Zt-directions,

respectively.

From a kinematic point of view, helical cutting

edge is generated through a motion, which is com-

posed of rotation around Zt-axis and movement

along the generatrix. Also, the inclination angle i is

defined as

tan i ¼tq

ts¼�d’

dsð2Þ

where tq is the rotational velocity around Zt-axis and

ts the moving velocity along generatrix. Therefore, the

position angle ’ for an arbitrary point on helical cut-

ting edge is expressed as

’ ¼ ’0 þ

Z f ðzÞ

g ðzÞ

tan i

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ d�=dzð Þ

2q

dz ð3Þ

where ’0 is the initial position angle and g(z) and f(z)

the integrating down boundary and up boundary

along Zt-axis.

The sweeping radius of each segment is calculated

as

�OM ¼ z tan�1 �

�MN ¼ Rr þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � Rz � zð Þ

2q

�NS ¼ Rr þ R cos�þ z þ R sin �� Rzð Þ tan�

9>>=>>; ð4Þ

tXtY

tZ

z

ϕ

Pφ

ρ

r

ρ

rτυ

sυ i

Fig. 1 Forming process of helical cutting edge

1694 W Yongqing and L Haibo




Then, for helical-end mills with constant helix

angle, combining equations (4) and (3), it yields

where

Further, the lead L is defined as a distance of a

moving point on helical-edge long axial direction

when mill rotates one unit (rad), which is expressed

as follows

L ¼dz

d’¼

�

tan i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ d�=dzð Þ

2q ð6Þ

Submitting equations (6) into (3), the helical-cutting

edge geometry with constant lead can be obtained.

2.2 Multiflute cutting edge

The single-flute cutting edge in Section 2.1 is assigned

to be No.1 flute, and multiflute cutting edge geometry

can then be formed through a forming

transformation

rj ¼ T �p,j

� r

T �p,j

¼

cos�p,j � sin�p,j 0sin�p,j cos�p,j 0

0 0 1

24

359>>=>>; ð7Þ

where Tð�pÞ is forming transformation matrix,

�p,j ¼ 2j�=Ne the cutter pitch angle for jth flute

(index j¼ 2 Ne), and Ne the number of helical

edges.

According to kinematic analysis above, once

sweeping radius, position angle, and pitch angle are

determined; the general cutting edge formulation of

N-flute milling cutter can be applied to any helical-

end mills, such as cylindrical flat-end mill, ball-end

mill, tapered ball-end mill, and radius-end mill, as

depicted in Fig. 2.

3 MATHEMATICAL FORMULATION OF 3D

CUTTING FORCE

3D cutting force of parametric helical-end mill will

be derived according to the following assumptions:

(1) the workpiece–cutter–machine system is rigid

enough, so that the cutting process is static and the

effect of chatter is ignored and (2) the load on clear-

ance face has high-order effect on total cutting force

and is neglected. First of all, helical-end mill is

divided into a large number of elementary slices

along cutter axis. Then, based on Merchant’s oblique

cutting theory, differential cutting force components

’OM0 ¼ 0

’MN0 ¼

tan i sin �

cos2 �ln Rz � R cos �ð Þ

’NS0 ¼ ’

MN0 þ

Z Rz�R sin�

Rz�R cos�

R tan i

Rr þ


2q� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2 � Rz � zð Þ2

q dz

9>>>>>>=>>>>>>;

ð5bÞ

’OM ¼ ’OM0 þ

Z z

0

tan i tan�

z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ tan�2 �

pdz

’MN ¼ ’MN0 þ

Z z

Rz�R cos�

R tan i

Rr þ


2q� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2 � Rz � zð Þ2

q dz

’NS ¼ ’NS0 þ

Z z

Rz�R sin �

tan iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ tan2 �

pRr þ R cos �þ z þ R sin�� Rzð Þ tan�½ �

dz

9>>>>>>>>>=>>>>>>>>>;

ð5aÞ

(a) Flat-end mill (b) Ball-end mill (c) Radius-end mill (d) Tapered ball-end mill

Fig. 2 Different helical cutting edges

3D cutting force prediction 1695




are derived in each infinitesimal element. Finally,

total cutting forces are calculated with the aid of

numerical integration method.

3.1 Differential cutting force

A cutting edge coordinate system is established,

{Ce:OeXeYeZe}, of which the origin is located on the

cutting edge with Xe normal to cutting velocity and

Ye parallel to cutting velocity. In order to derive the

mechanics relation between cutting edge and work-

piece, six important planes should be defined: basic

plane perpendicular to cutting velocity, cutting plane

determined by cutting velocity and cutting edge,

orthogonal plane perpendicular to basic plane and

cutting plane, normal plane, chip-flow plane coinci-

dent with rake face, and shear plane, as shown in

Fig. 3. The other local curvilinear coordinate system

is defined as {Cc:n, �, b}. In Cc, the differential pressure

dFn is perpendicular to rake surface in normal plane,

and the differential friction force dFf is assumed to be

collinear with chip-flow direction; dFf is decomposed

into two components along the b and �. In addition,

dFn has an inherent relation with dFf according to

Coulomb’s friction law, expressed as follows

dFfe ¼ dFf sin C

dFn ¼ �dFf

dFfn ¼ dFf cos C

9>>=>>; ð8Þ

where dFfe and dFfn are the two components of dFf,

C the chip-flow angle, and the friction coefficient.

Then, tangential force dFt, radial force dFr, and axial

force dFa are calculated using kinematic transforma-

tion matrix T1 from Cc to Ce, expressed as follows

dFa

dFt

dFr

24

35 ¼ T1 �

dFfe

dFn

dFfn

24

35 ð9aÞ

where,

T1 ¼

cos i sin i cos�n � sin i sin �n

� sin i � cos i cos�n � cos i sin�n

0 sin�n cos �n

24

35ð9bÞ

The differential shear force dFS along shear-flow

direction in shear plane is given in reference [3]

dFS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�dFa cos i þ dFt sin ið Þ

2þ dFt cos�n cos i � dFr sin�nð Þ

2q

ð10Þ

h

tOα

β βκ

RrR

zRMN

S

D

ω

z

zΔ

tX

tZ

tX

tY

θ

Pφ

zOb

h

eXeY

eZ

τ

n

nα

b

nφi

Chip-flow PlaneBasic Plane

Orthogonal Plane

Cutting Plane

Normal Plane

Shear Plane

beZ

eX

τeO

eO

n

eY

tdF

rdFnα Cη

fdF

ndFiadF

(a) (b)

(c)

Fig. 3 Cutting geometry and mechanics of helical-end mill





where �n is normal rake angle and �n the normal

shear angle.

Combining equations (8) and (10), it yields

dFS ¼ dFf M ð11aÞ

where

M ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Aa cosiþAt sinið Þ

2þ At cos�n cosi�Ar sin�nð Þ

2

qAa¼cosi sin c�sini cos�n=�sini sin�n cos c

At ¼�sini sin cþcos�n cosi=�cosi sin�n cos c

Ar¼�sin�n=þcos�n cos c

9>>>=>>>;

ð11bÞ

Shear stress � is determined according to material

mechanics theory, as follows

� ¼dFS

bhsin�n cos i ð12Þ

where b is the axial depth and h the radial depth in

oblique cutting process.

Substituting equations (8), (11a, b), and (12) into

(9a, b), it gives

dFa ¼�bhAa

M sin �n cos i

dFt ¼�bhAt

M sin �n cos i

dFr ¼�bhAr

M sin �n cos i

9>>>>>>=>>>>>>;

ð13Þ

3.2 TOTAL CUTTING FORCE

In global Cartesian coordinate system Ct, 3D cutting

force of jth flute is formulated using a transformation

matrix T2 from Ce to Ct, given as follows

dFx,j

dFy,j

dFz,j

24

35 ¼ T2 �

dFr ,j

dFt :j

dFa:j

24

35 ð14aÞ

where

T2¼

�cos j sin� �sin j �cos j cos�sin j sin� �cos j sin j cos��cos� 0 sin�

24

35 ð14bÞ

j is the rotation position of the point on jth edge at

elevation of z and � the axial immersion angle; j is

calculated using the following equation

j ¼ �� þ ð j � 1Þ�p þ ’j ðzÞ ð15Þ

where � is the clockwise rotational angle from Xt-

direction at the time t,� ¼ !t ,! the spindle speed,

and’j ðzÞ the radial lag angle determined by the

geometry of cutting edge, in equation (5a).

Also, the axial immersion angle � is described as

follows

�OM ¼ �

�MN ¼ sin�1ð�� Rr

RÞ

�NS ¼ �=2� �

9>=>; ð16Þ

Finally, total cutting forces produced by Ne flutes

are integrated, shown as follows

Fx ¼XNe

n¼1

Z zup

zdown

dFxdz ¼XNz

m¼1

XNe

n¼1

�Fx

Fy ¼XNe

n¼1

Z zup

zdown

dFy dz ¼XNz

m¼1

XNe

n¼1

�Fy

Fz ¼XNe

n¼1

Z zup

zdown

dFzdz ¼XNz

m¼1

XNe

n¼1

�Fz

9>>>>>>>>>=>>>>>>>>>;

ð17Þ

where Nz is the number of differential intervals along

the contact length between cutter and workpiece and

Zup and Zdown the integration boundaries.

4 INSTANTANEOUS CHIP THICKNESSCONSIDERING RUNOUT EFFECT

Cutter runout is a common and unavoidable

phenomenon of multiflute milling process [20, 21].

The presence of radial runout causes un-uniform dis-

tribution of chip load on cutting points over a rota-

tion, which has adverse effect on finished surface,

cutter life, and stability of cutting process.

4.1 Formulation of instantaneous chip thickness

The uncut chip thickness can be described as sinu-

soidal curve, named Martelloti’s model. However, the

model must be modified due to axial immersion

angle �. Therefore, the instantaneous chip thickness

of a point on the jth edge at elevation of z is improved,

as follows

h j

¼ ft sin j

sin �w j

ð18Þ

where, hð j Þ is instantaneous chip thickness, ft the

feed rate per tooth, and wð j Þ a windows function,

defined as

w j

¼

1, en � j � ex

0, otherwise

�ð19aÞ

Uen or D

ex ¼ �=2þ sin�1ð ft=2R0Þ ð19bÞ

Uex or D

en¼

sin�1 1� rð Þ 05 r�1�sin�1 r�1ð Þ 15 r 52��=2�sin�1 ft=2R0

2� r or slot milling

8<:

ð19cÞ

where en and ex denote entry and exit angles, U�

and D� the rotation angles in up- and down millings,





R0 the designed rotation radius, r the radial immer-

sion ratio of radial depth magnitude to cutter radius,

r ¼ dr=R.

4.2 Runout effect

As shown in Fig. 4(b), runout is described by two

values: the shift �r and the angular �. The real cutting

radius of cutter teeth should be altered due to the

presence of runout, i.e. Kline’s model.

Rj ðzÞ ¼ RðzÞ þ �r cos ’j ðzÞ � �þ ð j � 1Þ�p

� �ð20Þ

where Rj(z), R(z) denotes the actual and ideal cutting

radii of the corresponding disk element, respectively.

According to decentration ratio o of runout,

o ¼ �r=ft , the instantaneous chip thickness at any

point on cutting edge is affected by previous cutting

edges, expressed as

hð j Þ ¼ minm¼1 N

nhmð j Þ ¼

�mft sinð j Þ sin �

��þRj ðzÞ � Rj�mðzÞ

�wð j Þj

oð21Þ

where m is the weight factor, m¼ 1 Ne.

Generally, for common-end mills with two to four

flutes, m can be directly determined. When o is very

small, m¼ 1. As the cutter feeding, the runout effect is

mapped on the cutting face with overcut (slash area)

and undercut (grid area) relative to cutting trajectory

without runout, a case of four-flute end mill with

o ¼ 1=3 is shown in Fig. 4(c).

5 MODEL VALIDATION

Model validation was implemented through three

steps: (1) a great number of ball-end milling and

flat-end milling experiments were designed and cut-

ting forces were captured using dynamometer, (2)

shear stress was directly identified using a group of

identification tests and the 3D cutting forces were

then numerically predicted, and (3) comparisons of

predicted and measured cutting forces were con-

ducted under different cutting conditions.

5.1 Experiment design

Two-flute ball-end and four-flute cylindrical flat-end

mills with TiAlN-coated carbide were employed, listed

in Table 1. Over 100 cutting tests of 7075 aluminium

zO tX

tY

ωrρ λ

With runout

Without runout Overcut

Undercut

Rotation Cycle

tOtX

tY

Feed

A-A

A-A

(a) (b)

(c)

Fig. 4 Cutter runout and its effect on cutting surface

Table 1 Geometry of end mills and runout parameters

End mill D (mm) Le (mm) Ne i () �n () �r (mm) � ()

Flat 10 25 4 30 10 7 70Ball 10 15 2 30 10 18 60





workpiece were conducted with wind cooling on

Wahli-51 five-axis CNC horizontal machining centre.

The runout parameters were manually detected using

micrometer gauge. The cutting forces in feed, cross-

feed, and axial directions were measured using a

three-component dynamometer. The signals were

sampled at a frequency of 10 kHz. The set-up of the

cutting experiment is shown in Fig. 5.

5.1.1 Cylindrical flat-end mill

The envelop characteristics of four-flute cylindrical

flat-end mill are �¼ �¼ 0, R¼Rz¼ 0, Rr¼D/2. The

sweeping radius and lag angle are �(z)¼D/2,

’ ¼ 2z tan i=D. Cutting tests were performed with

two spindle speeds (1000 and 2000 r/min), two feed

rates (100 and 240 mm/min), three axial immersion

ratios (0.1, 0.2, and 0.3), and three radial immersion

ratios (0.2, 0.4, and 0.8), as listed in Table 2.

5.1.2 Ball-end mill

The envelop characteristics of two-flute ball-end mill

are �¼ �¼ 0, R¼Rz¼D/2, Rr¼ 0. The sweeping

radius and lag angle are given

�MN ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2

4�

D

2�z

� �2s

�NS ¼D=2

9>=>; �

’MN ¼tan i

2ln

Dz

D�z

’NS ¼tan i

2lnDþ

2tan i

Dz

9>=>;

ð22Þ

Cutting tests were performed with two spindle

speeds (3000 6000 r/min), two feed rates (1000 and

2000 mm/min), and four axial immersion ratios (0.1,

0.2, 0.4, and 0.8), as listed in Table 3.

5.2 Numerical test

Numerical tests were conducted for various paramet-

ric helical-end mills with different outer and helical

edge geometries, in which some approximation tech-

nologies were employed. The chip-flow angle c was

approximated by the inclination angle according to

Stabler’s rule. From the deformation geometry as well

as the continuity and incompressibility conditions,

the relationship between normal shear angle �n and

normal rake angle �n could be derived from orthogo-

nal cutting, as detailed in reference [3]. Furthermore,

allowing for collinearity condition, the friction coef-

ficient was determined by employing the well-

known equation in oblique cutting process

¼ tan tan�1 cos �n tan i

tan c � sin�n tan i� �n

� �ð23Þ

Generally, there is a phase angle deviation between

instantaneous predicted force and measured force

acquired by the dynamometer, named asynchronous

problem. Some research studies focused on that, such

as peak value method [22] and stable point method

[23]. However, both of the two methods cannot be

Fig. 5 Set-up of the cutting experiment

Table 2 Cutting conditions of flat-end mill

Test r/min U/D a r ft (mm/tooth)

1 2000 U 0.1 0.2 0.032 2000 D 0.2 0.4 0.033 2000 U 0.3 0.8 0.034 2000 D 0.1 0.2 0.012 55 2000 U 0.2 0.4 0.012 56 2000 D 0.3 0.8 0.012 57 1000 U 0.1 0.2 0.068 1000 D 0.3 0.4 0.069 1000 U 0.2 0.8 0.0610 1000 D 0.1 0.2 0.02511 1000 U 0.2 0.4 0.02512 1000 D 0.3 0.8 0.025

Table 3 Cutting conditions of ball-end mill

Test r/min a ft (mm/tooth)

13 6000 0.1 0.08314 6000 0.2 0.08315 6000 0.4 0.16716 6000 0.8 0.16717 3000 0.1 0.33318 3000 0.2 0.33319 3000 0.4 0.16720 3000 0.8 0.167





easily conducted since the wave shape is affected by

signal noise and cutting chatter. A simple method is

employed to add an adjustment angle & to rotation

angle j . The average deviation percentage E is min-

imized in a period, as follows

min E ¼1

Nf

XNf

n¼1

F P� � F M

�

�� max F P

�

�� , F M�

�� 100 %

j ¼ �� þ ð j � 1Þ�p þ ’j ðzÞ þ &

9>=>; ð24Þ

where Nf is the number of sample points or simula-

tion points in a period, F P� and F M

� the instantaneous

predicted and measured force, respectively, and � the

x, y, z.

Thereby, 3D cutting forces can be predicted if shear

stress �, instantaneous cut parameters b and h are

known. However, the identification of shear stress is

a complex and difficult problem. Following, a simple

and effective approach to determine the shear stress �

is introduced. A group of cutting experiments in

Section 5.1 was selected as an identification test,

and the value of � is estimated according to the fol-

lowing steps: (1) the shear stress is first assumed to be

a dimensionless unit, and the cutting forces along

X- and Y-directions are then predicted using cutting

edge and cutting force formulations in Sections 2, 3,

and 4, and the curves are then adjusted using equa-

tion (24); (2) the peak and the valley values of

predicted and measured curves are selected, respec-

tively; (3) the peak-to-peak and valley-to-valley ratios

are estimated as the shear stress; and (4) in order to

weaken the effect of cutting noise and chatter, the

shear stress � is averaged to be assessed as the final

value. However, the parameter � contains a wealth of

cutting process information in addition to shear

stress, like the coefficient of lumped mechanistic

model.

The numerical tests were implemented on

MathWorks platform. The simulation flow chart is

shown in Fig. 6. First, cutting condition should be pre-

pared: cutter parameters and cutting parameters. The

former includes seven independent parameters of the

parametric envelop (in Fig. 3(a)), and helical edge

parameters: the number of cutting edge Ne , normal

rake angle �n, constant helix angle i, or constant lead

L. The latter contains workpiece material, spindle

speed !, feed rate ft, axial immersion ratio a , radial

immersion ratio r, runout parameters(�r and �), and

U/D cutting mode (up- or down-milling). Second,

multiflute cutting edge geometry is derived according

to forming transformation process. Third, the

chip-flow angle, the normal shear angle, the friction

coefficient, and the axial immersion angle are approx-

imated and calculated. Then, there are two steps to

calculate the instantaneous chip thickness for a

Cut

ting

forc

es (

N)

Rotation angle (°) 0 100 200 300 400 500 600 700 800

200(a)

(b)

100

0

–100

–200

–300

–400

–500

Fy

Fz

Fx

Rotation angle (°) 0 100 200 300 400 500 600 700 800

200

100

0

–100

–200

–300

–400

Cut

ting

forc

e (N

)

–500

300

400

Fx

Fz

Fy

Fig. 7 Predicted and measured cutting forces withrunout

Fig. 6 Flow chart of numerical test





specific milling process, the determination of

windows function in equation (19a) and the optimi-

zation of positive possible chip thickness using

a direct numerical iteration procedure in equation

(23). Fourth, shear stress is estimated using the

new approach. Finally, the cutting forces are

predicted using equation (17) and improved using

equation (24).

5.3 Results analysis

The predicted and the measured cutting forces in

three directions for flat-end mill and ball-end mill

were compared, respectively, in Fig. 7: (a) test 9 and

(b) test 19. It was clear that the predicted and mea-

sured waveforms are quite satisfied in both pulsation

pattern and period, except slight phase shifts due to

the asynchronous problem. The average deviation

percentages for both tests 9 and 19 were 20–

25 per cent for Fx, 25–40 per cent for Fy, but over

60 per cent for Fz due to its sensitivity to measure-

ment noise, machine vibration, and wind cooling

effect. The cutter runout has a bad effect on cutting

surface, especially at some instantaneous local peak

points.

6 CONCLUSIONS

A novel mechanics model of parametric helical-end

mills for 3D cutting forces prediction has been sys-

tematically presented in this article.

1. The single-cutting flute edge was generated by a

moving point on the parametric envelop and mul-

tiflute cutting edge was then modelled after form-

ing transformation process.

2. A general mechanics model with runout was

derived by the integration of elemental cutting

forces based on Merchant’s oblique cutting

theory. It was different from mechanistic approach

and Oxley’s predictive machining theory in model

formulation.

3. A simple and effective approach was employed to

estimate shear stress directly using a selected mill-

ing test. After minimizing the phase angle to elim-

inate asynchronous phenomenon, the shear stress

was actually the corresponding peak-to-peak ratio

or valley-to-valley ratio between the predicted and

the measured curves.

4. Numerical simulation procedure was performed in

time domain on MathWorks platform. An average

deviation percentage criterion was adopted to

solve the asynchronous problem between instan-

taneous predicted and measured cutting forces.

Comparative results indicated that the predicted

and the measured waveforms were quite satisfied

in both pulsation pattern and period.

ACKNOWLEDGEMENTS

This study is supported by National Natural Science

Foundation of People’s Republic of China (Key

Program grant no. 50835001) and Advanced research

Foundation (Key Program grant no.

9140A18020310JW0902).

� Authors 2011

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APPENDIX

Notation

b, h xial and radial cutting depths{Cc:n, �, b} curvilinear coordinate system{Ce:OeXeYeZe} cutting edge coordinate system{Ct:OtXtYtZt} tool coordinate systemdFfe, dFfn two components of friction forcedFn, dFf differential pressure and friction forcedFS differential shear force along shear-flow

directiondFt, dFr, dFa differential tangential, radial, and axial

forcesdFx, dFy, dFz differential cutting forces in Ct

D, R, Rr, Rz parametric radial dimensions of end millE average deviation percentageft feed rate per toothFx, Fy, Fz total cutting forcesF P� ,F M� instantaneous predicted and measured

forcesg(z), f(z) integrating down boundary and up

boundaryhð j Þ instantaneous chip thicknessi inclination anglei, j, k unit vectors corresponding to Xt-, Yt-, and

Zt-directionsL lead of helical cutting edgeLe effect length of cutting edgem weight factor of multifluteNe number of helical edgesNf number of sample points in a periodNz number of differential intervalsr, rj vectors of cutter envelop and jth cutting

edgeRj(z), R(z) actual and ideal cutting radiiR0 designed radius of cutting pointt machining timeT1 transformation matrix from Cc to Ce

T2 transformation matrix from Ce to Ct

Tð�pÞ forming transformation matrix in Ct

wð j Þ windows functionz elevation along Zt-axisZup, Zdown integration boundaries

�, � Parametric angles of end mill�n normal rake angle& adjustment angle� axial immersion angle C chip-flow angle o decentration ratio of runout r radial immersion ratio� clockwise rotational angle friction coefficient� sweeping radius�r , � parameters of cutter runout� shear stresstq rotational velocity around Zt-axists moving velocity along generatrix�n normal shear angle�p cutter pitch angle’, ’0 position angle and initial position angle j rotation position en, ex entry and exit angles, respectively U� , D

� rotation angles in up- and down-millings,respectively

! spindle speed





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A novel mechanics model of parametric helical-end mills for 3D cutting force prediction