Applied Mechanics and Materials Vol. 163 (2012) pp 95-99
© (2012) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMM.163.95
Online: 2012-04-12
Investigation of Tool Deflection of Solid Carbide End Mill in Cutting
Process
Qiong Wu 1,a, Yidu Zhang 1,b, Xiangsheng Gao1,c, Lin Fang2,d
1.
State Key Laboratory of Virtual Reality Technology and Systems, School of Mechanical
Engineering and Automation, Beihang University, Beijing 100191, China
2.
Army Aviation Institute, School of Mechanical Engineering,Beijing,101123, China
a
c
wuqlc@126.com, bydzhang@buaa.edu.cn,
gaoxsh@ me.buaa.edu.cn, dfanglin@ me.buaa.edu.cn
Keyword: Deflection, FEM, Cantilever Beam, Mill
Abstract. Tool deflection is one of the important influencing factors for surface roughness and
surface integrity of work piece in cutting process. The excessive deflection even causes seriously
defects of work piece or failures of tool. This paper gives theory analysis and mathematic method to
predict the tool deflection by means of the cantilever beam deflection theory. Based on modeling of
3-dimmsion milling force model, the finite element analysis has been performed for calculation of
tool deflection. The experiment of tool deflection of solid end mill is performed to compare
simulation and theory. Results show the correction and reliability of research method. It lays a
foundation for fast calculation of tool deflection and optimization of milling parameters.
Introduction
In metal cutting operation, end mills are used for slot or peripheral milling. The cutting force acting
on the end mill generally results in tool deflection. The main factors are the tool geometry, the
cutting condition and the work piece properties. Tool deflection can affect the acceptable dimension
accuracy of finishing. To achieve the time reducing and low price with good quality, many
researchers have studied the form error caused by tool and work piece deflection and surface
roughness [1-3]. Wang studied the tool deflections due to the bending and the twist in end milling
based on the X and Y cutting force model [1]. The Z cutting force model was neglected because of
little effect. The cutting force model was built based on the chip load and cut geometry in end
milling. R. Jalili Saffar[4] investigated the tool deflection minimization in order to optimize the
cutting parameters using genetic algorithms. Wang calculated the moment of inertia of the 2-fluted
end mill cross-section by assuming that the cross-section area varies axially from the bottom to the
neck and to the shank. The area moment of the tool can also be calculated using an equivalent tool
radius. J. A. Nemes and E.B. Kivanc approximated the equivalent tool radius by cosine law [5].
In this paper, the deflection caused by x-, y- and z-directional forces has been estimated by
means of the cantilever beam theory and finite element analysis. These results are compared with
the results of experimental test using dial gage and the percent of difference is also described. The
relationship between cutting parameters and tool defection is investigated.
Cutting Force Model
The cutting forces based on the analytical theory [6] are used to investigate the deflection and
stresses in the end mill. Tangential ( dFt , j ), radial ( dFr , j ), and axial ( dFa , j ) forces acting on a
differential flute element with height dz are expressed in Eq.1.
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History of Mechanical Technology and Mechanical Design 2012
dFt , j (φ, z ) =  K tc h j (φ j ( z )) + K te  dz,
(1)
dFr , j (φ, z ) =  K rc h j (φ j ( z )) + K re  dz,
dFa , j (φ, z ) =  K ac h j (φ j ( z )) + K ae  dz.
The material of high-speed milling motorized spindle is 20Cr. The rotor of the spindle is
supported by two pairs of angular contact ball bearings. The milling cutter is made of hard alloy.
The chip thickness variation can be approximated as
h j (φ, z ) = c sin φ j ( z )
(2)
Where c is the feed rate and φ j ( z ) is the immersion angle for flute j at axial depth of cut z.
The elemental forces are resolved into feed (x), normal (y), axial (z) directions using the
transformation
dFx , j (φ j ( z )) = −dFt , j cos φ j ( z ) − dFr , j sin φ j ( z ),
dFy , j (φ j ( z )) = +dFt , j sin φ j ( z ) − dFr , j cos φ j ( z ),
(3)
dFz , j (φ j ( z )) = +dFa , j .
The resultant of the elemental force
dF = dFx2, j + dFy2, j + dFz2, j
(4)
doc = axial depth of cut ; woc = radial depth of cut or width of cut; The edge force coefficients
( K re , Kte and K ae ) are constant and have little effect on the resultant forces[7]. The values of cutting
force coefficients for aluminum workpiece are described in below [8]:
K rc
=1880.0N/mm2,
Ktc
=893.3N/mm2, K ac =157.1 N/mm2
Methods of Deflection Calculation
In this part, the cantilever beam theory will be used for analytical calculation to find tool deflection
and the finite element analysis (FEA) will be used for numerical calculation.
Cantilever Beam Theory
The end mill is replaced by the cantilever beam and the forces acting on the cutter is assumed as
several concentrated loads. The deflection is formulated by means of the integration of the
bending-moment equation. The deflection formulas for different part are written in the following
equations. These equations can be used to find the deflection of the cutter with both y-directional
lateral loads and x-directional lateral loads. (As shown in Fig.1)
Fig. 1
The loading state of the cantilever beam
Fig. 2
4-flute end mill cross section
The maximum deflection is
vmax =
1
[ F1 (l13 − 3L2l1 + 2 L3 ) + F2 (l23 − 3L2l2 + 2 L3 ) + F3 (l33 − 3L2l3 + 2 L3 ) + ... + Fn (ln3 − 3L2ln−1 + 2 L3 )]
6 EI
(5)
n is the number of forces acting on the beam; L is the effective length of the cutter; E is Modulus of
elasticity; I is moment of inertia; The z-directional cutting force is assumed as the axial load of the
beam. Thus the deflection for z-direction is the change in length of the segments.
m
δ=∑
i =1
Ni li
EAi
(6)
Applied Mechanics and Materials Vol. 163
97
is the internal force, li is the length and Ai is the cross sectional area of the each segment
and m is the number of segment. The computation of the cutter’s inertia is difficult because the
cross section of the cutter is complex. Thus the equivalent radius Req for region 1 is shown in Fig.2.
Ni
Finite Element Analysis
In order to analyze deflection of the milling cutter, the cutter model is modeled using 3D software
according to Tab.1. The model of cutter is imported to FEA software. Due to complicated geometric
model, tool model is divided into 24547 elements which are only 4-noded tetrahedral elements. And
the fixed boundary condition and the concentrated loads are applied to the model as described in Fig.
3. The isotropic material properties are input according to different cutter material. The elemental
cutting forces used are at 37.2˚ immersion angle. The solution type is chosen as ‘statics’ from
structural analysis. After computation, Fig. 4 shows one of MSC solutions of 4-flute carbide end
mill with 20mm diameter. From these figures, the maximum deflection and stresses occur at the
cutter edge within the loading region. The location of maximum shank stress can decide from Fig.4.
Fig. 3
Model of milling tool FEM mesh
Fig. 4
Deflection and stress of tool by FEA
Results and Discussion
Cutter dimension and type of material used are described in Table 1. Modulus of elasticity and
poisson’s ratio of HSS (high speed steel), cemented carbide and cemented carbide with coating end
mill are 210GPa, 0.3, 560GPa, 0.22 and 580GPa, 0.22, respectively.
Table 1
Cutter parameters (Unit is mm)
No
Material of tool
D1
D2
L1
Overall L
Helix angle
α(deg)
Effective
length
1
HSS
12
12
26
83
30
57
12
12
30
75
30
56
12
12
26
100
50
67
2
3
Cemented
Carbide
Carbide With
Coating
Here, 5mm depth of cut, 3mm width of cut, 8000rpm spindle speed and 500mm/min feed rate are
used in calculation. By simulation for these machining parameters [10], the average cutting force is
50N at the immersion angle 33.6˚ for HSS and carbide 4-flute end mill, 49.2˚ for carbide 4-flute end
mill. Fig.4 is the deflections of HSS end mill caused by 50N force achieved from MSC software. To
verify the analytical results and compare with the experimental results, the deflection is measured
by means of dial gage. The experimental set up is shown in Fig.5. A force is loaded to end point of
the tool and displacements at the tow points of the tool are measured.
98
History of Mechanical Technology and Mechanical Design 2012
Fig.5
Experimental set-up of tool deflection
The comparison and percentage difference of experimental, analytical results are described in
Table 2.
Table 2
No.
Comparison of analysis, experiment and simulation (Unit is µm)
Errors (%)
Errors (%)
Material of
Exp.
Analysis
FEA
Compare
Compare
tool
force = 50 N force =50N
force =50N
with Exp.
with Exp.
1
2
3
HSS
Cemented
Carbide
Carbide with
coating
20
21.013
5.1
21
5
8.5
8.41
1.1
8.70
2.4
11
11.32
2.9
11.7
6.3
The highest value is the deflection of No.1 cutter among three kinds of material. Hence,
deflection depends on the modulus of elasticity of the material to some extent. The deflection of
cutter No.3 is higher than No.2 although modulus E of No.3 cutter is higher. That is reason that
effective length of No.3 cutter is longer than that of No.2. Thus, the longer the effective length of
the cutter is, the more deflection the cutter will be caused. From the comparison of these errors, it is
reliable to adopt cantilever beam analytical method to solve deflection of tool deflection of solid
carbide end mill.
Conclusion
The study on the fast prediction of tool deflection has been performed. The cutting forces have been
determined and the deflection of cutter has been obtained analytically. Three kinds of tool material
have been analysis and computed by means of the cantilever beam theory, MSC Patran/ Nastran
software and the test with dial gage. The experiment of tool deflection of solid end mill is
performed to compare simulation and theory. Results show the correction and reliability of research
method. This research lays a foundation for fast calculation of tool deflection and optimization of
milling parameters.
Acknowledgement
The authors would like to thank to National Natural Science Fund (No.51105025) and
Chinese National Science and Technology of Major Projects (No.2009ZX04014-051) for providing
financial support.
Applied Mechanics and Materials Vol. 163
99
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History of Mechanical Technology and Mechanical Design 2012
10.4028/www.scientific.net/AMM.163
Investigation of Tool Deflection of Solid Carbide End Mill in Cutting Process
10.4028/www.scientific.net/AMM.163.95