Cutting force prediction for circular end milling process
Wu Baohai
a ,*
,Yan Xue b ,Luo Ming a ,Gao Ge
a
a
Key Laboratory of Contemporary Design and Integrated Manufacturing Technology,Ministry of Education,Northwestern Polytechnical University,Xi’an 710072,China b
AVIC Commercial Aircraft Engine Co.Ltd.,Shanghai 200241,China Received 12April 2012;revid 8June 2012;accepted 26July 2012Available online 28April 2013
KEYWORDS Chip thickness;
Circular end milling;Cutting force;Machining;Radial depth;
Tool path curvature
Abstract A deduced cutting force prediction model for circular end milling process is prented in this paper.Traditional rearches on cutting force model usually focus on linear milling process which does not meet other cutting conditions,especially for circular milling process.This paper pre-nts an improved cutting force model for circular end milling process bad on the typical linear milling force model.The curvature effects of tool path on chip thickness as well as entry and exit angles are analyzed,and the cutting force model of linear milling process is then corrected to fit cir-cular end milling process.Instantaneous cutting forces during circular end milling process are pre-dicted according to the propod model.The deduced cutting force model can be ud for both linear and circular end milling process.Finally,circular end milling experiments with constant and variable radial depth were carried out to verify the availability of the propod method.Exper-iment results show that measured results and simulated results corresponds well with each other.
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1.Introduction
End milling is widely ud in machining moulds and dies,as well as various aircraft components.To e
nsure cutting quality,tool life prolongation and the productivity,accurate milling process analysis is critically necessary for beforehand process planning and adaptive controlling.During the entire milling process,cutting force is one of the most important issues and an efficient and preci cutting force model is thus crucial
for the lection of machining parameters,such as feed rate,and spindle speed.
Traditional rearches 1,2on cutting force model usually fo-cus on linear milling process.Chip thickness calculation 3–5as well as cutting force coefficients identification 6–9is also ana-lyzed specially for linear milling force simulation.The ap-proaches do not always meet other cutting conditions,especially circular milling process.In Refs.[10,11],cutting con-dition independent force coefficients concept was introduced into milling force modeling,however;preci instantaneous uncut chip thickness and runout offt and angle factors are needed in this method to get desirable instantaneous cutting force coefficients.Size effect and other characteristics during the process may influence the simulation results eventually.Therefore a particular model for circular milling is necessary to satisfy practical request.
Regarding force simulation of circular milling,relative re-arches have been done recently.Zhang et al.12,13propod
*Corresponding author.Tel.:+862988493232414.
E-mail address:wubaohai@nwpu.edu (B.Wu), (X.Yan),luoming@nwpu.edu (M.Luo).
生字成语Peer review under responsibility of Editorial Committee of
CJA.
Chine Journal of Aeronautics,2013,26(4):1057–1063
1000-9361ª2013Production and hosting by behalf of CSAA &BUAA.dx.doi/10.1016/j.cja.2013.04.003
Open access under CC BY -NC-ND licen.
Open access under CC BY -NC-ND licen.
an approach to predict the cutting forces in the end milling of rectangular circular corner profiles by discretizing the corner into a ries of steady-state cutting process,each with different radial depth of cut.Wu et al.14investigated the rela-tionship between working parameters and the corner coordi-nates by way of combination of tool tracing and cutting geometrodynamics.Yang et al.15developed a method for cut-ting force modeling related to peripheral milling of curved sur-faces including the effect of cutter runout.For curved surface milling process,Rao et al.16–20prented a ries of accom-plishment concluding process geometry,curvature effect,cut-ting force,surface error and so on.
Although the above-mentioned literature has extended to a bunch of stuff for circular milling,it does not mention the var-iation of feed rate along cutting tool envelope during circular milling process.For the existence of workpiece curvature,the feed rate along the cutting tool envelope will not remain the same as that of the tool center,and this is an important differ-ence from the linear milling process.When establishing a cut-ting force model,one of the key issues is the calculation of instanta
neous chip thickness,which has dependent relation-ship with feed rate.Abundant rearches have been done on this aspect.3,4,21The effects of runout,tooth trajectory as well as tool deflection on chip thickness are discusd,respectively or simultaneously.It is noticed that studying the cutting force in milling with a circular tool path,the variation of uncut chip thickness is too important to be neglected.Actually the key ele-ments involved in force model,for instance,feed rate and chip thickness,will deviate from their normal values during circular milling process.
This paper propos a cutting force prediction model for circular end milling process by improving the chip thickness model.And this paper is organized as follows:a review of the related literature is prented in Section1,followed by mechanics of circular milling process in Section2.In Section3, the method for identification of cutting force coefficients is propod,followed by experimental validation and discussion in Section4.Finally,conclusions are given in Section5.For the sake of simplicity,the following assumptions arefirstly made:end-milling process is assumed to be chatter-free.And the deflection of the cutter and workpiece are assumed to be negligible.
2.Mechanics of circular end milling
As shown in Fig.1,in the end milling process,the cutter gen-erally rotates on an axis vertical to the wo
rkpiece.Cutting teeth are located on both the end face of the cutter and the periphery of the cutter body.The cutter can be tilted to ma-chine tapered surfaces.While in peripheral milling,the axis of cutter rotation is parallel to the workpiece surface to be ma-chined,and the cutter has a number of teeth along its circumference.
2.1.Geometry of circular end milling process
To analyze geometry of circular end milling process,a ries of plane Cartesian coordinate systems is definedfirst.As shown in Fig.2,a plane Cartesian coordinate system O W X W Y W Z W isfixed on the workpiece.For circular milling process,coordi-nate origin O W coincides with the arc center offinished work-piece,and the directions of X W axis and Y W axis depend on the cutting force measuring device.Coordinate system O C X C Y C Z C isfixed with the cutter center and has a synchronous motion in X W Y W plane corresponding to the motion of cutter along tool path.Its X C axis is always oriented in the instantaneous feed direction,Y C axis locates on the line of current cutter center and O W of global coordinate,and Z C axis coincides with the cutter axis.
As shown in Fig.3,during linear milling process,if the tool path is straight line,then the feed rate of every cutting point along the cutter envelope keeps constant,and equals normal feed rate f,here f ref
ers to the feed per tooth.But during cir-cular milling process,the cutter center path is circular and the instant feed rate of the cutting point on the cutter is f when it is just located on the path,as shown in Fig.3.In thefigure,R refers to radius of tool path,r is the radius of the cutter,/is the tool instantaneous angle position.The feed rates of other cutting points along the cutter envelope are relative to the radius of toolpath and the distance with cutter center.
(a) End milling
(b) Peripheral milling
End milling and peripheral
(a) Linear
(b) Circular
Geometry of linear and circular end
1058 B.Wu et al.
2.2.Cutting force model
The commonly ud mechanistic model of cutting force model
for milling process deduced by Altintas and Lee 1is adopted here.To calculate the total force,the cutting flute is divided into finite number of small differential elements along the cut-ting flute curve,and the total cutting force components acting on a flute at a particular instant are obtained by numerically integrating the force components acting on an individual dif-ferential element.
The tangential force d F T is in the direction of cutting speed.The radial force d F R is in the radial direction of cutter feed,and the axial cutting force d F A acts along the cutter axial direction.d F j ;T ð/j ;z Þ¼K T h ð/j ;z Þd b d F j ;R ð/j ;z Þ¼K R h ð/j ;z Þd b d F j ;A ð/j ;z Þ¼K A h ð/j ;z Þd b 8><>:
ð1Þwhere K T ,K R and K A are the tangential,radial and axial cut-ting force coefficients,h (/j ,z )is the instantaneous chip thick-ness which is related to the cutting edge gment with a differential length of d z ,and varies with the position of ting point and cutter’s rotation.d b is the projected infinitesimal cutting flute in the direction along the velocity,/j (z )is the instantaneous angular position of tooth at axial elevation z ,and it can be expresd as /j ðz Þ¼/1ðz Þþðj À1Þ/p
where /1(z )is the rotation angle of reference flute from positive direction of Y -axis,/p =2p /N is the and N is the number of cutter teeth.
Three orthogonal force components in Cartesian nates system of i th cutter element for j th tooth can be by following transformation of the tangential,radial forces defined on the cutting flute element:
d F i ;j ;X ð/i ;j ;z Þd F i ;j ;Y ð/i ;j ;z Þd F i ;j ;Z ð/i ;j ;z Þ
2
6437
5¼M d F i ;j ;T
d F i ;j ;R d F i ;j ;A
264375where
M ¼Àcos /i ;j
Àsin /i ;j 0sin /i ;j
Àcos /i ;j
00
À126
43
75Then the total force acting on the j th cutting edge can be obtained by integrating along the axial depth of cut:
F Áj ¼Z z 2
z 1
d F Áj d z ð4Þ
And z 1,z 2depend on the immersion condition of j th tooth.Finally the total cutting forces on the milling cutter in feed (X ),normal (Y )and axial (Z )directions are obtained by summing the contribution of all cutting edges.F X ;Y ;Z
¼X N f j ¼1
F Áj
ð5Þ
2.3.Mechanics of circular end milling
Chip thickness for linear and circular end milling process are different.For linear milling process,chip thickness is deter-mined as follows Model 1:h ð/Þ¼f sin ð/Þ贺平安
ps发光效果where f is the normal feed per tooth, [0,p ].
Then for circular end milling process,chip thickness could be expresd as
Model 2:h ð/Þ¼f ð/Þsin ð/Þ¼f sin ð/Þ½R þr cos ð/Þ =R Fig.4shows the chip thickness (h )of circular end milling process under chip thickness Model 1and Model 2.The results show that appearance of the maximum chip thickness corre-sponds to different tool rotation angles,and values of the max-imum chip thickness are almost the same.It reveals the difference between the two models.3.Identification of cutting force coefficients
Linear milling data are usually utilized for the identification of cutting force coefficients 1,and the same procedure is applied in this rearch.The cutting force coefficients are treated as con-Chip thickness of circular end milling with different
models.
(a) Linear
(b) Circular
Feedrate along cutter circum during linear Cutting force prediction for circular end milling process
1059
where/st and/ex denote the start and exit radial immersion angles,respectively.Especially for slot milling process,substi-tuting/st=0and/ex=p into Eq.(6),and following equations expressing the average cutting forces per tooth in the three directions can be obtained:
½F Xð/ÞF Yð/ÞF Zð/Þ T¼f z
2
W½A1A2A3 Tð7Þ
where
W¼
最苦与最乐ÀK T sinð2/ÞÀ2K R sin2/À2K A sin2/
2K T sin2/ÀK R sinð2/ÞÀK A sinð2/Þ0À2K A sin/2K R sin/
2
64
3
75;
A1¼
Z z2
z1d z;A2¼
Z z2
z1
sin jðzÞd z;A3¼
Z z2
z1
cos jðzÞd z
½F X F Y F Z T¼f z
/
p
D½K T K R K A Tð8Þ
where
D¼
C3A1ðC2ÀC1ÞA2ðC2ÀC1ÞA3ÀðC2ÀC1ÞA1C3A2C3A3
0ÀC4A3C4A2
2
64
3
75;
C1¼1
2
/j/ex
/st
;C2¼
1嘻唰唰歌词
4
sinð2/Þj/ex
/st
and edge wear.In this paper,tool deflection and edge wear are
not taken into consideration.Cutter runout is a common phe-
nomenon in milling operations,and it affects the chip load
distribution of every tooth.To remove the effects of cutter run-
out,theflute-average method22(FAM)is applied to the cut-
ting force in this paper.In this method,sample points
spaced by theflute-spacing angle are averaged over one revo-
lution to obtain an averageflute force profile.Fig.5shows
the cutting forces before and after FAM.
4.2.Identification of cutting force coefficients
瓜叶菊怎么养
Cutting forces were measured during full slot milling Titanium
alloy TC4at different feed rates.Tool geometrical parameters
are listed in Table1.
Fig.5Cutting force before and after FAM.
Table1Tool geometry.
Symbol Terminology Value英语音节表
D T Tool diameter(mm)10
H Overhang length(mm)38
L Tool length(mm)75
N Number of teeth4
b Helix angle(°)30
1Cutter; 2Workpiece; 3Kistler 9255 B
Fig.6Experimental tup.
7Cutting force coefficients under different cutting
conditions.
1060 B.Wu et al.
The axial depth of cut is kept constant at2mm,and the ra-dial depth changes from1mm to9mm.The feedrate is 320mm/min and the spindle speed is2000rpm.Cutting forces during the milling process were recorded with Kistler9255B cutting force measuring device shown in Fig.6.
Polynomialfitting method was then ud for different ra-dial depths of cut d e,as shown in Fig.7.The s
mall circle in thefigure reprents the identified cutting force coefficients un-der different redial depth of cut.
With this method,the effects of different radial depth can be taken into account.Finally,the expressions of the cutting force coefficients are as follows:
K T¼À2:92d3
北京钟楼
e
þ51:82d2
e
À309:3d eþ2727
K R¼À1:618d3
e
þ39:85d2
e
À318:7d eþ1947
K A¼À2:055d3
e
þ34:54d2
e
À183:0d eþ603:5
8
><
>:ð10Þ
(a) Circular end milling tool path
(b) Machined workpiece
Circular end milling tool path and machined workpiece.
(a) Simulated
(b) Measured
Simulated and measured cutting forces for circular with constant radial depth.
(a) X-direction
(b) Y-direction
(c) Z-direction
Fig.8Simulated and measured cutting force.
Cutting force prediction for circular end milling process1061
