Comprehensive Simulation of Surface Texture for End-milling Process
Behnam M. Imani, Ehsan Layegh
Assistant Professor, Graduate Student
Ferdowsi University, Mechanical engineering department, Mashhad, Iran
Email: imani@um.ac.ir, ehsan_layegh@yahoo.com
Abstract
Analysis and simulation of manufacturing process require extensive and complicated computations. Nowadays,
computer resources and computational algorithms reach to the state that can model and simulate the problem
efficiently. One of the important processes in manufacturing is machining. In this research end-milling process
which is one of the complex and wide-spread processes in machining is chosen. Most important parameters in
end-milling are surface roughness and surface location errors. A comprehensive simulation software is
developed to model end-milling process in order to anticipate finishing parameter such as surface roughness and
errors. The proposed algorithm takes into account cutting conditions, such as feed, doc, woc, tool run out, etc. In
addition, dynamic simulation module of the software can accurately model flexible end-mill tool, the milling
cutting forces and regeneration of waviness effects in order to construct realistic surface texture model. The
software can accurately determine the most commonly used index of surface roughness parameters such as Ra,
P.T.V. and surface errors.
Keywords: Machining Simulation, Regeneration, End-milling, Surface Texture, Surface roughness
geometry, cutting coefficients for oblique cutting are
determined.
In general, one of the important sources of vibrations
in milling process is self-excited vibration. In this kind
of vibrations, internal sources will provide energy for
the system. In some cases, the energy will exceed
damping thus vibrations sustain or even grow in
amplitude. One of the important principles observed in
machining dynamics is the regeneration of waviness
which affects chip thickness [3].
In current research, a dynamic simulation model for
machining dynamics is proposed. The surface is
modeled using a 3D algorithm which is an extension
to available 2D methods [4]. The developed
simulation model can anticipate surface roughness in
any direction of interest on the machined surface.
1. INTRODUCTION
End-milling operations are extensively used in the
production of precise parts in the industry. Milling
process dynamics is investigated in many research
works, however due to complexity of the process
different fields of research works are required. One
area in analyzing and simulation of milling process is
developing a comprehensive model in order to
anticipate surface quality, dimensional error and trend
of them in mass production. In various machining
process the following steps are required to build up
the model:
1- analyzing and modeling machining forces
2- modeling dynamic system of machine tools
3- modeling surface texture
A comprehensive model for accurate anticipation of
surface errors requires a "precise" force model. The
model must take account of phenomena such as, tool
deflections, tool run-out, tool wear, machining
dynamics, etc. In the most of previous research works,
machining forces are obtained by mechanistic force
model based on oblique cutting tests [1]. In this work,
orthogonal cutting theory and transforming method
proposed by E. Budak et al is used [2]. In contrast to
the previous methods, there is no need for oblique
cutting test for each tool shape. At first, a general
machining data base is provided by orthogonal cutting
tests with various work materials and cutting tool
materials. Using transforming relations and tool
2. SIMULATION OF CUTTING FORCES
Figures 1-a and 1-b illustrate coordinate system and
end-milling geometry required for the model.
Differential forces, dft tangential, df r radial and df a
axial are depicted which act on the jth cutting edge for
an ideal system (without any deflection). Values of the
differential forces are obtained by (for tooth j):

( , z )  K
( , z )  K

h ( , z )dz
h ( , z )dz
dftj ( , z )  K te  K tc h j ( , z ) dz
df rj
df aj
1
re
 K rc
ae
 K ac
j
j
(1)
In equations (1) there are two separate terms, the first
one is related to friction and ploughing forces and the
second term is considered for the cutting action which
takes place in the shear zone. Coefficients for the first
phenomena are represented by K te , K re , and K ae
showing amount of force per unit length of the cutting
edge. Coefficients represented by K tc , K rc , and K ac
are considered for cutting action showing amount of
force per unit area of cutting edge.
Figure 2 Simulated cutting forces for a four-flute HSS
tool and aluminium work-piece 7075-T6 with yield
stress of 542 MPa. Cutting conditions are d (Tool
diameter) =10mm, ft (Feed per tooth) = 0.05mm/tooth,
doc (Depth of cut) =10mm
3. REGENERATION OF WAVINESS
In general, the cutting edge of the tool is cutting a
surface which is generated in the previous passes. If
there exists any vibrations between tool and work
piece, the newly generated surface undulates which
results in the variation of chip thickness. Figure 3
illustrates the model used for simulation of milling
process. Dynamic characteristic of the system in x and
y direction is also depicted. Differential forces can be
obtained by:
dFtn 1  K tc da(h0  z n  z old1  z old 2  z old 3 )
(2)
dFrn 1  K rc da(h0  z n  z old1  z old 2  z old 3 )
Figure 1 Coordinate system and end-milling geometry
The simulation software which written in MATLAB
V7.0 first computes cutting and edge force
coefficients and then computes cutting forces for each
cutter angle [5]. Table 1 summarizes computed
coefficients for a four-flute HSS tool and aluminium
work-piece 7075-T6 with yield stress of 542 MPa.
The computed cutting forces are depicted in Figure 2.
Table1 The computed coefficients for a four-flute
HSS tool and aluminium work-piece 7075-T6 with
yield stress of 542 MPa, in this table  n is normal
Figure 3 Model used for simulation of dynamic
milling process
rake angle of tool
n
(deg)
Ktc
[ N/mm2]
Krc
[ N/mm2]
Kac
[ N/mm2]
0
1642.3
609.5
512.2
12
1396.2
519.8
258.1
15
1356.5
504.4
215.3
Due to the fact that axial stiffness of the tool and the
spindle assembly is high and also axial force is
smaller than the others, axial vibrations are ignored. In
the above relations, n, h0, zn, zold1 to zold3 and da are
respectively: index of time increment, nominal chip
thickness, tool deflection normal to the cut surface,
tool deflection in previous passes and axial increment.
In each instance, differential cutting forces on engaged
cutting edges are computed and projected on X and Y
axis. Then the total cutting forces are computed by
2
integration of differential forces over engaged edges.
Using Newton's second law, differential equations of
motion are written and then by double Euler
integration the dynamic deflections are obtained.
Fxn 1  m x xn  C x x n  K x x n
(3)
Fyn 1  m y yn  C y y n  K y y n
Where mx, my, Kx, Ky, Cx and Cy are mass, stiffness
and damping matrices for x and y directions,
respectively. Input data for a four flute end-mill with
helix angle of 30º are considered as follows [6, 7]:
f x  f y  660 HZ
K x  K y  2  10 7 N / m
M x  K x /( 2f x ), My  K y /( 2 f y )
 x   y  0.05
Figure 4 dynamic simulation of cutting forces for
aluminium 7075-T6 work piece. Cutting conditions
are: d = 10 mm, h = 30deg, ft = 0.1 mm/tooth,
woc=1.2
Cx   x  2 K x M x ,C y   y  2 K y M y
r  5mm, n  1200 rpm,
doc  10mm
In above relations f x , f y , x , y are natural frequency
and damping ratio for x and y directions, respectively.
The tool radius, spindle speed, and depth of cut are
represented by r, n, doc. Nominal chip thickness at
angular position  is:
h0  ft  Sin
(5)
Normal to cut deflection is:
Z n  xn Sin  yn Cos
(6)
Using the proposed model, it is possible to include
nonlinearities such as jump-of-cut and chatter
vibrations. Figure 4 shows the simulation results for
aluminium 7075-T6 work piece. Cutting conditions
are as follow:
d = 10 mm, h (helix angle) = 30deg, ft = 0.1
mm/tooth, woc (width of cut) = 1.2 mm, n = 1200
rpm, doc (Depth of cut) = 10 mm.
Dynamic milling forces for woc = 3 mm are shown in
Figure 5. Fourier transform of milling forces are
computed and depicted in Figures 6 and 7. Dominant
frequency in the stable cutting is tooth passing
frequency, however as inferred from Figure 6 and 7
the dominant frequency in the unstable cutting is close
to natural frequency of flexible mode shape, known as
chatter frequency [6]. When woc = 3 mm, milling is
unstable and vibration grows in amplitude gradually.
Figure 5 dynamic simulation of cutting forces for
aluminium 7075-T6 work piece. Cutting conditions
are: d = 10 mm, h = 30deg, ft = 0.1 mm/tooth,
woc=3mm
Figure 6 Fourier transform for Fx
3
Input Data
Cutter radius, woc, doc, Feed
per tooth, Specific cutting
force, Spindle Speed
Calculate mass, stiffness
and damping matrices
x0  x 0  y 0  y 0  zeros (n)
Fx 0  F y 0 0
x  M /( Fx  k x )
y  M /( Fy  k y )
For
Figure 7 Fourier transform for Fy
j  1 : a da
4. DYNAMIC VIBRATIONS OF TOOL AND
SPINDLE
Surface
roughness
prediction
requires
a
comprehensive model in time domain which can
simulate tool vibrations at locations of interest on the
work piece. In the current research milling vibrations
are modeled by FEM using CALFEM toolbox
implemented in MATLABV7.0 software [8].
Figure 8 schematically shows FEM model for the endmill and spindle assembly, which is modeled as a step
beam with two simple supports.
For N  1 : Nr  1
T ( N )  ( N  1)  dt
X ( N , j)  X 0
Y ( N , j )  Y0
Calculate F , F , x, y from
x
y
regeneration principle
Fx 0  Fx
Fy 0  Fy
Calculate x , y , x, y from Newmark
method
No
N=Nr-1
Yes
j  a da
No
Yes
Figure 8 FEM model for the end-mill and spindle
assembly
End
Figure 9 Developed flowchart of the FEM analysis
Axial variation of the tool deflection is computed for
the engaged portion of tool which is used by the
surface location error algorithm. Therefore, finer
elements in this portion are introduced.
Inputs for FE analysis are given in the following: [9]
modulus of elasticity of steel = 207×109 (N/m2), inner
diameter of spindle axis = 0.02 m, outer diameter of
spindle axis = 0.1 m, density of steel = 7800(Kg/m3).
Figure 9 illustrates the developed flowchart of the FE
analysis.
In the developed algorithm Nr, a, da, F0, T (N), X (N,
j) and Y (N, j) are number of revolution, axial depth of
cut, axial increment, initial force, time, x-deflection
and y-deflection, respectively. Inputs to the algorithm
are tool radius, woc, doc, feed, cutting force constants,
and spindle speed. A function written in MATLAB
V7.0 is used to calculate mass, stiffness and damping
matrices. Using mesh and node positions, mass,
stiffness and damping of each element are computed
and then assembled together to get overall matrices.
In addition boundary conditions and external cutting
forces are applied. Differential equations of motion
are integrated numerically by the Newmark method.
M x X  C x X  K x X  F X

 M y Y  C y Y  K y Y  FY
(7)
For selected initial values of y  x  y  x  0 the
initial acceleration is expressed as
4
F  C y y  k y y
F  C x x  k x x
X 
, Y 
Mx
My
(8)
The iteration loop starts from 1 and end to a/da (where
a is DOC and da is axial increment), thus the variation
of axial deflection is obtained. Counter N varies from
1 to Nr-1 which is considered for the time increments.
After computing Fx, Fy, x, and y at current time
increment, these values are used as initial values for
the next step. Results of the milling tool in time
domain by FE analysis are shown in Figure 10.
Vibrations of the tool center point are plotted at
different axial levels.
In the next section amount of vibrations is used for
determining cutting edge position and surface
generation method.
Figure 11 Trochoidal path of a point on cutting edge
Where X l / c and Yl / c are coordinates of the edge with
respect to tool center point. Also i, r,  (t ) , (i ) and
 (Z ) are tooth index, tool radius, angular position of
the lowest point of the reference cutting edge, pitch
angle and phase lag of the point at z levels,
respectively.
 (Z ) is computed by:
z tanh
 (Z ) 
(10)
R
Where h is tool helix angle.
On the other hand, displacement of spindle center with
respect to work piece can be found by:
X s / w  f r   (t ) /( 2 )
(11)
Ys / w  0
Figure 10 Results of time domain FE analysis of the
milling tool for d= 12 mm, ft=0.3 mm/tooth,
Ktc=2000N/mm2, doc=5mm, woc=1.5 mm
5. TOOL PATH SIMULATION
Final machined surface is produced by subsequent
cutting actions of end-mill cutting edges. Ideally,
points on the cutting edge have trochoidal paths which
leave feed marks on the surface. Additionally, any tool
deflection, vibrations or geometric error results in
undesirable deformation and deteriorate required
surface roughness on the work piece. Ideal position of
the cutting edges with respect to tool cutter is shown
in Figure 11.
XOY coordinate system is attached to the work piece
and original O is at the initial center position of the
spindle. Trochoidal motion of a point on the cutting
edge is computed by [9]:
X l / c (i, z, t )  R sin(  (t )   (i )   ( z ))
(9)
Yl / c (i, z, t )   R cos( (t )   (i )   ( z ))
If the runout error is not negligible, the location of
tool center with respect to spindle axis will be
computed by:
Yc / s (t , z )     cos( (t )   )  YFEM (t , z )
X c / s (t , z )    sin(  (t )   )  X FEM (t , z )
Where  is runout offset and
position, refer to Figure 12.
5

(12)
is its angular
Figure 12 Runout parameters
Figure 15 Trochoidal motions of the cutting edges
with considering vibrations (d=12 mm, ft=0.3
mm/tooth)
and YFEM (t , z) are dynamic tool
deflections at time t and z level, which are calculated
by FE analysis. Thus, instantaneous positions of a
point on the cutting edge can be founded by:
Xl / w  Xl /c  Xc/ s  X s / w
(13)
Yl / w  Yl / c  Yc / s  Ys / w
Figure 13 shows trochoidal motion of the cutting
edges without considering vibrations and runout. In
order to show the effects of the runout, its value is set
to 1mm, which is exaggerated intentionally. The other
cutting conditions are kept constant. The simulation
results are shown in Figure 14. Finally, Figure 15
shows path of cutting edge including tool vibrations.
X FEM (t , z)
6. SURFACE TEXTURE SIMULATION
Feed mark profiles can be produced by removing
point on the computed paths which do not make
contact with newly generated surface. The z-level
profiles are used in the mesh function to construct a 3d
surface. Figures 16, 17 and 18 illustrated 3d mesh
surfaces for different cutting conditions. Cutting
conditions used for simulation of figure 16 results in
negligible vibrations. In order to show the runout
effects an exaggerated runout offset (   1mm ) is
chosen and the result of simulation is shown in figure
17. Extreme cutting conditions which results in chatter
vibrations is used for simulation of figure 18.
Figure 13 Trochoidal motions of the cutting edges
without
considering
vibrations
and
runout
(ft=1mm/tooth, d=20mm)
Figure 16 Surface texture for d=12mm, woc=1mm,
doc=5mm,
ft=0.1mm/tooth,
Ktc=1500
N/mm2
(Ra=1.4  )
Figure 17Surface texture for d=10mm, woc=1mm,
doc=5mm, ft=0.1mm/tooth, Ktc=1500 N/mm2,
 =1mm (Ra=3.5  )
Figure 14 trochoidal motions of the cutting edges
with
considering
exaggerated
runout
(   1mm ,  =45)
6
[5]
[6]
[7]
[8]
Figure 18 Surface texture for d=12mm, woc=4.5mm,
doc=5mm, ft=0.6mm/tooth, Ktc=1500 N/mm2,
 =0.01mm (Ra=14  )
[9]
As shown in figures 10 and 18, amplitude of
vibrations at the tool tip is much higher. Table 2
summarizes Ra values of surface roughness computed
at tool tip.
Table 2 Surface roughness computed at tool tip.
d
(mm)
10
10
10
12
12
12
a
(mm)
5
5
5
5
5
5
WOC
(mm)
1.0
1.5
1.5
1.5
2
3
ft
(mm/tooth)
0.05
0.1
0.25
0.1
0.1
0.15
Ra

1.14
3.99
5.85
2.24
3.07
4.61
7. CONCLUTIONS
In this research work a comprehensive simulation
model is developed for anticipating dynamic cutting
forces and 3D surface texture for end-milling
operations. Variation of instantaneous tool deflections
are computed by FE analysis using CALFEM toolbox
implemented in MATLAB V7.0 environment.
Variations of dynamic tool deflections are also
computed along axial direction for constructing 3D
surface texture model. The final model contains all
information required to compute surface roughness
parameter and surface location errors.
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Milling Force Model for Surface Error
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pp. 89-94
[2] E. Budak, Y. Altintas, E. J. A. Armarego, (1996)
“Prediction of Milling Force Coefficient from
Orthogonal Cutting Data”, 1st ed., Cambridge
University Press, NY, USA.
[3] S. Smith, and J. Tlusty (1991), “An Overview of
Modelling and Simulation of Milling Process”,
ASME Journal of Engineering for Industry,
vol.113, No.2, pp.169-175.
[4] Won-Soo Yun, Jeong Hoon KO, Dong-Woo Cho
and Kornel Ehman, “Development of a Virtual
7
Machining System, Part2: Prediction and
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www.mathworks.com
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