Abstract In the milling process, the major flank wear land area (two-dimensional measurement for the wear) of a small-diameter milling cutter, as wear standard, can reflect actual changes of the wear land of the cutter. By analyzing the wearing characteristics of the cutter, a cutting force model based on the major flank wear land area is established. Characteristic parameters such as pressure parameter and friction parameter are calculated by substituting tested data into their corresponding equations. The cutting force model for the helical milling cutter is validated by experiments. The computational and experimental results show that the cutting force model is almost consistent with the actual cutting conditions. Thus, the cutting force model established in the research can provide a theoretical foundation for monitoring the condition of a milling process that uses a small-diameter helical milling cutter.

Keywords helical milling cutter, major flank wear land area, cutting force model, characteristic parameter

1 Introduction

In the milling process, small-diameter helical cutters with multiple spiral cutting edges have such advantages as good milling performance, a wide process range, and high efficiency. Thus, they are widely used in operations such as planar milling, three-dimensional milling and multi-axis milling [1−2]. However, the milling process, which is affected by factors such as materials of the work pieces, cutting parameters, dynamic performance of the millers, and

conditions of the cutters, is a typical multi-input nonlinear system. Changes of conditions to the system can be sensed by physical parameters, such as cutting force, torque, acoustic emission signal, radial displacement and vibration accelera-tion of the spindle, and output power of the spindle motor. Each parameter reflects the cutting conditions from a certain aspect and is affected by its corresponding sensing mecha-nism. The wear process of a milling cutter is highly related to the cutting force, cutting heat, vibration of the machining system and other factors; it directly affects the cutter’s life span, cutting cost, work efficiency, and milling quality [3−4]. Therefore, modeling the wear process of a milling cutter has always been an important research subject. In recent years, various approaches have been put forward, most of which are cutting force models [5−9]. In the paper, by analyzing wearing characteristics of small-diameter helical cutters, a cutting force model based on the major flank wear land area is established numerically and validated by experiments.

2 Cutting force model for a small-diameter helical cutter

In the milling process, the cutting force acting on a cutter contains much information, including wear conditions of the cutter, technological system of the miller, machining accuracy and surface roughness of the work piece. Because wear conditions of a milling cutter are affected by many factors, the following presumptions are made to simplify the modeling process: the cutting process is stable without chatter; there is no built-up edge formation on the cutter; crater wear on the rake surface is negligible compared with flank wear; the width of the flank wear land of a sharp cutter (or a new cutter) is zero; when a particular edge point engages, the flank wear land next to that point is in contact with the work piece, and conversely, when a particular edge point disengages, the adjacent flank separates from the work piece [5].

2.1 Wear standard of a helical cutter

The wear of a cutter consists of rake wear, flank wear and grooving wear; of these three, flank wear affects cutting force

Front. Mech. Eng. China 2007, 2(3): 272–277DOI 10.1007/s11465-007-0047-1

RESEARCH ARTICLE

LI Xiwen, YANG Mingjin, XIE Shouyong, YANG Shuzi

Cutting Force Model for a Small-diameter Helical Milling Cutter

© Higher Education Press and Springer-Verlag 2007

Received December 12, 2006; accepted March 20, 2007

LI Xiwen ( ), YANG Mingjin, XIE Shouyong, YANG ShuziSchool of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, ChinaE-mail: xiwenli@vip.sina.com

YANG Mingjin, XIE ShouyongCollege of Engineering and Technology, Southwest University, Chong qing 400716, China

273

the most and is easy to measure. Therefore, in practice and research, the wear standard of a cutter usually refers to the allowable maximum mean width of flank wear land at the middle of the cutter. However, both the machining process of the machine tool and the wearing process of the cutter are complex for milling. The major flank wear land area AV of a milling cutter, as wear standard, can reflect real changes of the wear land. The width of flank wear land VB is a one-dimensional measurement for the flank wear, while flank wear land area AV is a two-dimensional measurement [2−4].

The major flank wear land of a milling cutter usually consists of three parts: the triangle wear land, trapezium wear land, and polygon wear land. For simplicity, the wear land area can be further considered as the parts of the triangle wear land and trapezium wear land as shown in Fig. 1. Thus, the major flank wear land area of a milling cutter is given as follows

A ab b f c deiV, ( )= + + +12

12

(1)

where i is the flute number of the cutter, and i = 1,2,..., Nf.

development of flank wear land [5−7]. Force Ff on the cutter flank surface then reflects the change in major flank wear land of a milling cutter in operation. By modeling cutting force acting on the cutter flank surface, based on the major flank wear land area, wear conditions on the flank surface can be monitored by means of information extraction technology.

Forces acting on the rake and flank surface of a helical cutter are schematically shown in Fig. 2, and elemental areas and forces are extracted to obtain analytical expressions of the flank forces from force equilibrium. Some main symbols in Fig. 2 are described as follows:Ar rake surface of the helical cutter;Af flank surface of the helical cutter;dAr elemental areas of the rake surface, mm2;dAf elemental areas of the flank surface, mm2;N clockwise revolution of the cutter, rpm;Ra cutting depth, mm;ar rake angle of the cutter in the X−Y plane;ara axial engagement angle of the cutting point;a position angle measured from the bottom of the tool;hc chip flow angle; hh helix angle; S major cutting edge of the cutter;dS elemental width along the major cutting edge, mm;lr, lf widths along the rake surface and along the flank

surface, respectively, mm;nr, nf unit normal vectors of rake surface and flank surface,

respectively;tr, tf unit tangent vectors of rake surface and flank surface,

respectively;br unit vector orthogonal to vectors nr and tr;Tc unit vector in the chip movement direction

Cutting force on the flank surface of the cutter can be resolved into a normal Fnf and a tangential Fff as follows

d dnf nf f fF ( ) ( , )h h a=K An (3)

d dff ff nf f fF r( ) ( , ) ( )h h a h=K K A (4)

d df B hA V S= (cos )h

where Knf is the pressure parameter on the flank surface; Kff is the friction parameter on the flank surface; and h is the rotation angle for a single-fluted cutter. Knf and Kff are the functions of h and a.

For single-fluted cutters, Eqs. (3) and (4) can be integrated to obtain analytical expressions for the instantaneous flank forces at a particular orientation as follows

F Kxf ff( ) sin ( ) cos( )

( )h l a h a

a h

a h= +−[ ]∆ ∆

1

2

F Kyf ff( ) cos ( )sin( )

( )h l a h a

a h

a h= +∆ ∆[ ]

1

2

F zf ( )h =0

where l h h=K V Rnf B a h( ) tan ; ∆ = a-h; and a1(h) and a2(h) are the integration limits and functions of the rotation angle h.

Fig. 1 Schematic diagram of major flank wear land of a milling cutter1) Triangle wear land; 2) trapezium wear land; 3) polygon wear land, S major cutting edge, worn part of the cutter, wear land

Therefore, mean flank wear land area ArV can be expressed as follows

AN

A ii

N

V

fV

f

==

1

1,∑ (2)

2.2 Modeling procedure

In the milling process, the cutting force acting on a cutter can be divided into two components: force Fr on the cutter rake surface and force Ff on the cutter flank surface. As the major flank wear land area AV increases, Ff increases owing to greater cutter flank-work piece interaction. Nevertheless, Fr remains practically unchanged before and after the

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For multi-fluted cutters, the cutting force is obtained by summing the corresponding X, Y, and Z components acting on each flute

F Fx xi

N

if f

f

( ) ( )h h==1∑

F Fy yi

N

if f

f

( ) ( )h h==1∑

F zf ( )h =0

where h hi i N= + -2 1p( ) f, and i = 1,2, ..., Nf; i is the flute number; hi is the rotation angle for a flute i measured from the start of the engagement [6–7].

The relationship between mean cutting forces, in the directions of X and Y acting on the flank surface, and width of flank wear land VB is expressed as

F Kxf en ff en= + -a l a a− ( )⎡⎣ ⎤⎦sin cos1 (5)

F Kyf en ff en= - +a l a a1 cos sin⎡⎣ ⎤⎦ (6)

where a a a a= +( )ra en ; l h=K V Rnf B a tan ; K nf and K ff are mean pressure parameter and mean friction parameter on the flank surface respectively; ara is the axial engagement angle of the cutting point as mentioned above; and aen is the radial engagement angle of the cutting point.

For multi-fluted cutters, changes of cutting width Rd lead to multiple cutting edges engaged simultaneously. Hence, it is difficult to calculate the mean cutting force on each flute. For simplicity, while calculating mean cutting forces per revolution, Eqs. (5) and (6) are employed, while F xf and F yf are mean cutting forces in the time period when there is flute contact with the work piece.

The cutting force model, based on width of the flank wear land, reflects the relationship between VB and the cutting force acting on the flank surface. However, the width of flank wear land VB—one-dimensional measurement of the wear—can only illustrate the wear process on the flank surface to a limited extent. The major flank wear land area AV—two-dimensional measurement of the wear—can better express/present the state wear process on the flank surface. As shown in Fig. 2, the major flank wear land area AV is approximately

A V RV B a h= ( cos )2 h (7)

Substituting Eq. (7) into Ffx(h) and Ffy(h), an instantaneous cutting force model based on major flank wear land AV can be obtained as

F A Kxf A V ffA( ) sin ( ) cos( )

( )h l a h a

a h

a h= +−[ ]∆ ∆

1

2 (8)

F A Kyf A V ffA( ) cos ( )sin( )

( )h l a h a

a h

a h= +∆ ∆[ ]

1

2 (9)

where l h h hA nfA h h=2K ( ) cos tan

Fig. 2 Schematic diagram of cutting forces acting on a helical cutter

275

Integration limits of equations defining the instantaneous cutting force model are related to different machining conditions such as cutting depth and cutting width. Substitut-ing Eq. (7) into Eqs. (5) and (6), a cutting force model based on major flank wear land AV and of mean cutting force per flute can be expressed as

F Kxf A en ffA en= +a l a a− −( )⎡⎣ ⎤⎦sin cos1 (10)

F Kyf A en ffA en= - +a l a a1 cos sin⎡⎣ ⎤⎦ (11)

l h hA nfA V h h=2 K A cos tan

2.3 Integration limits of the cutting force model

Depending on the magnitude of engagements with respect to each other, there are two possible situations to be discussed [5].

First, if the radial engagement angle is greater than the axial engagement angle, the integration limits can be expressed as

a a

a h a h h h a

a h a h a aen ra’

1 2 ra

1 2 ra ra

0, for

0, for >

= = h h= =

( ) ( )

( ) ( )

0

hh h= - = h h += =

h a

a h h a a h a a h a a

a h a h

en

1 en 2 ra en en ra

1 2

, for

0

( ) ( )

( ) ( ) ootherwise

⎧

⎨⎪⎪

⎩⎪⎪

Second, if the axial engagement angle is greater than or equal to the radial engagement angle, the integration limits can be expressed as

a a

a h a h h h a

a h h a a h h ara en’

1 2 en

1 en 2

0, for

, for i

= = h h= - =

( ) ( )

( ) ( )

0

een ra

1 en 2 ra ra ra en

1 2

, for

h h= - = h h +=

h a

a h h a a h a a h a a

a h a h

( ) ( )

( ) ( )==0 otherwise

⎧

⎨⎪⎪

⎩⎪⎪

2.4 Modification of the cutting force model

The cutting force model stated above covers ideal machining conditions without considering effects of axial errors on the model. These errors include those caused by cutter installation and by axial movement of the spindle. Assuming that the angle of inclination between the spindle axis and normal vector of the work piece surface is g, then the real cutting depth of a flute at rotation angle h can be expressed as follows

R R R Ra a( ) sin sin sinh g g h= - +( )

3 Experiment work

3.1 Methodology

Experimental conditions to validate the cutting force model of the helical cutter are as follows: the machine tool is a XHK5140 vertical machining center; the helical cutter is made of high speed steel with a diameter of 10 mm, helix angle of 30° and flute number 3; material of the work piece is gray iron and size is 190 mmx90 mmx50 mm; cutting mode is climb milling without cutting fluid; clockwise revolution of the cutter is 360 rpm; cutting depth is 2.0 mm; feed per flute is 0.083 3 mm; major flank wear land area of a normally worn cutter is 0.503 mm2 and that of a worn cutter is 1.207 mm2; and other cutting parameters are shown in Table 1. The width and area of flank wear land of a sharp cutter or new cutter are assumed to be zero.

Measuring instruments employed in the experiment are as follows: a Kistler 9257A three-component dynamometer platform for testing X, Y and Z component forces acting on the flank surface of a helical cutter; a JC10 microscope for measuring dimensions of flank wear land; Yanhua PCL818 HG high-speed multi-channel data acquisition board for analog to digital conversion of input signals, and for pulse signal numbering and timing; and an industrial PC for data acquisition and signals processing, with 2 000 samples per revolution.

Characteristic parameters KnfA(h), KffA(h), K nfA and K ffA of the cutting force model, in Eqs. (8)–(11), are obtained by substituting tested data into the following equations

KF F

Ay x

nfA

h f f

V h

( )tan [ ( ) cos ( )sin ]

cos( )

( )

hh h a h a

ha h

a h

=-∆ ∆

21

2

(12)

KF F

F Fx y

x yffA

f f

f f

( )( ) cos ( )sin

( )sin ( ) cos(

hh a h a

h a h aa

=+

-−

∆ ∆∆ ∆

1 hh

a h

)

( )2

(13)

KF F

A

y xnfA

h f en f en

V h en

=- -

-a h a a

h a2

1

2 1

tan [ ( cos ) sin ]

cos ( cos ) (14)

KF F

F F

x y

y x

ffAf en f en

f en f en

=- +

- -

( cos ) sin

( cos ) sin

1

1

a a

a a (15)

3.2 Experimental result and analysis

Tested values Ffx(h) and Ffy(h) of three helical cutters under different wear conditions—the sharp cutter (or new cutter), normally worn cutter, and worn cutter—are shown in Fig. 3. It shows three wave crests and three wave troughs coinciding

Table 1 Cutting parameters employed in the experiment

Wear condition of a cutter Sharp cutter or new cutter Normally worn cutter Worn cutter Cutter machining from sharp to worn

Cutting width / mm 5.0 5.0 5.0 9.0

276

with the flute number three of the tested cutters, which is consistent with the actual machining conditions.

The calculated characteristic parameters KnfA(h) and KffA(h) of the normally worn cutter from Eqs. (12) and (13), which are based on the major flank wear land area AV as the evalua-tion index of cutter wear condition, are shown in Fig. 4. Tested characteristic parameters KnfA(h) and KffA(h) of the normally worn cutter and worn cutter are shown in Fig. 5. It can be concluded from the comparison of Figs. 4 and 5 that calculated characteristic parameters KnfA(h) and KffA(h) of the normally worn cutter are consistent with the tested parameters, which partially validates the cutting force model established in the research.

Figure 6 shows the relationship between the mean pressure parameter KrnfA and the major flank wear land area AV

and the relationship between the mean friction parameter KrffA and the wear land area AV of a cutter with progressive wear from sharp to worn. The parameters KrnfA and KrffA in Fig. 6 are obtained by substituting tested data into Eqs. (14) and (15). Such inclinations from Fig. 6 show that the mean pressure parameter KrnfA increases, while mean friction parameter KrffA decreases with an increase of the major flank wear land area AV.

The comparisons between calculated and tested values Ffx(h) and Ffy(h) of an instantaneous cutting force model of the normally worn cutter are shown in Fig. 7. A conclusion can be made that the instantaneous cutting force model is almost consistent with tested conditions, which also validates the cutting force model established based on the major flank wear land area.

Fig. 3 Tested values Ffx(h) and Ffy(h) of cutters of different wear conditions1) Sharp cutter; 2) normally worn cutter; 3) worn cutter

Fig. 4 Calculated characteristic parameters KnfA(h) and KffA(h) of a normally worn cutter

Fig. 5 Tested characteristic parameters KnfA(h) and KffA(h)1) Normally worn cutter; 2) worn cutter

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4 Discussion

The wear process of the major flank wear land of a helical cutter is an important phenomenon in the milling operation. The wear process can be sensed by physical parameters such as cutting force, output power of the spindle motor, radial displacement and vibration acceleration of the spindle.

For milling, the machining process of the machine tool and the wearing process of the cutter are complex. The major flank wear land area of a milling cutter, as wear standard, can reflect actual changes of the wear land. The width of the flank wear land is a one-dimensional measurement for the flank wear, while the flank wear land area is a two-dimensional measurement.

The cutting force model of a small-diameter helical cutter based on the major flank wear land area can reveal changes to cutter wear land. Computational and experimental results show that the cutting force model was almost consistent with the actual cutting conditions. The cutting force model established in the research can thus provide a theoretical foundation for condition monitoring of the milling process of a small-diameter helical milling cutter.

However, developing the cutting force model of a small-diameter helical cutter, which involves many factors, is very complicated. Only limited machining parameters have been validated in this research. Using the cutting force model established for more general applications needs further studies.

Acknowledgements This paper was supported by the National Basic Research Program of China (No. 2005CB724101).

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Fig. 6 Relationship between calculated parameters KrnfA, KrffA and major flank wear land area AV (progressive wear process)

Fig. 7 Comparisons of calculated and tested values Ffx and Ffy of a normally worn cutter1) Calculated value; 2) tested value