ball end mill

1. Theoretical Estimation of Machined Surface Profile
Based on Cutting Edge Movement and Tool Orientation in Ball-nosed End Milling
Y. Mizugaki 1
, K. Kikkawa 1
, H. Terai 2
and M. Hao 3
1
Dept. of Mechanical and Control Engineering, Kyushu Institute of Technology, Kitakyushu, Japan
2
Dept. of Control and System Engineering, Kitakyushu National College of Technology, Kitakyushu, Japan
3
Dept. of Mechanical Engineering, Kyushu Kyoritu University, Kitakyushu, Japan
Submitted by T. Sata (1), Tokyo, Japan
Abstract
This paper presents the theoretical estimation method of machined surface profile without actual machining in ball-
nosed end milling. The fundamental simultaneous equations of identifying the cusp height at any point of a workpiece
in the simulated surface have been successfully derived from the geometric relationship between the cutting edge
movement and the normal line at the point. By the numerical calculation, the machined surface profile can be
estimated and illustrated graphically. It was found that the maximum and minimum cusp heights exist in a narrow
range of tool orientation less than 3 degrees near the normal direction.
Keywords:
Ball end milling, Machined surface geometry, Geometric simulation
1 INTRODUCTION
The ball-nosed end milling is a significant key technology of
machining for the generation of precise sculptured surface.
In order to achieve high qualified surfaces, there have been
reported some important studies on end milling. Kline [1] tried
to predict the surface accuracy from the viewpoint of the cut-
ting force and the tool deflection, but it was not a ball-nosed
end mill. In ball-nosed end milling, Zhu [2] presented the
mechanistic model of ball end milling for the prediction of
cutting force and discussed the cutting point geometry. Imani
[3] applied the solid modelling to the ball-end milling opera-
tion. Altintas [4] discussed the cutter geometry more precisely.
Nevertheless, there are few studies on the estimation of ma-
chined surface profile and cusp height based on the cutter
geometry. Although Mizugaki [5,6] revealed the generating
mechanism of machined surface from the viewpoint of spa-
tial analysis of moving cutting edge and tool orientation, it
was limited to the tool orientation with variable tilting angle
and fixed lead angle.
In this study, the general form to estimate a machined sur-
face has been newly formulated as the fundamental geo-
metric equations applicable to any tool orientations. The re-
sults of numerical calculation are illustrated graphically in
the machined surface profiles and the change of cusp heights
due to the tool orientations.
2 FUNDAMENTAL GEOMETRIC EQUATIONS
2.1 Ball-nosed end mill geometry
Figure 1 illustrates the schematic geometry among the ball-
nosed end mill, the offset surface to be machined and the
contouring tool path in spherical milling. This model is appli-
cable to any cases in ball-nosed end milling, for example a
reciprocating tool path, without losing the generality. Because
the target area of geometric estimation is enough small to
the order of the radius of the ball-nose or the radius of the
contouring tool path.
OT-XTYTZT in figure 2 is the local coordinate system on the
ball-nosed end mill. OT is the center of the ball-nose and ZT
is the revolving axis of the end mill tool. The initial direction
of the helical cutting edge is assumed to be on the OT-YT
plane. In this study, two teeth of cutting edges are assumed
tentatively as the number of teeth of the ball-nosed end-mill,
and this assumption does not lose the generality in the fol-
lowing discussion. The end mill is fed along the contouring
tool path with keeping constant the tool orientation to the
directions of the feed and the cross feed. The tool orienta-
tion is identified in the terms of the lead angle θlead and the
tilting angle θtilt in figure 3.
The fundamental approach to get the machined surface pro-
file is to detect the intersection between a normal line at a
point on the offset surface and the moving helical cutting
edge of the ball-nosed end mill. It is shown in figure 4. The
intersection gives the cusp height at the point on the ma-
chined surface. The normal line can be identified by a set of
direction angles ω and γ. The position and orientation of he-
lical cutting edge of the ball-nosed end mill can be given by
the revolving angle ψ. The cutting edge point on the flute j is
identified by directional angle ζ measured from the Zt axis.
Directional angle ζ gives the helix lag angle α as the phase
delay angle. So the intersection can be derived as the solu-
tion of the simultaneous geometric equations with variable ψ
and ζ.
2.2 Fundamental geometric equations
The coordinates of position P on the helical cutting edge are
represented in equation (1).
Pt = (Xt, Yt, Zt). (1)
Xt = r sin[α(ζ) + 2jπ / N] sin ζ.
Yt = r cos[α(ζ) + 2jπ / N] sin ζ.
Zt = r cos ζ.
Here, 0 ≤ ζ ≤ π/2, 0 ≤ α(ζ) ≤ π/2 .
α(ζ) + 2j π/N ≡ a ζ+ b (in j =0), 0 ≤ a ≤ 1.4, a,b: given.
r : the radius of the ball-nosed end mill. [mm]
j : the ordinal number of cutting tooth. In the case of the
first tooth, j is 0. It moves from 0 to N-1.
N : the number of teeth of end mill: tentatively two.
ζ : the directional angle of the cutting point P measured from
the ZT axis of the tool coordinate system. [radian]
α : the helix lag angle as phase delay of cutting edge mea-
sured from the initial plane of YTZT. α is a function of ζ.
[radian]

2. R0
sin θ ≡ rk
.
rk
τ = N f ψ / 2π.
τ = τ(ψ) = N f ψ / 2π R0
sin θ ≡ tψ .
t ≡ N f / 2π R0
sin θ << 1.
Position P can be represented as the matrix product of the
coordinate transformation matrices described below.
P = Rz(τ) Ry(θ) T(C) Rx(η) Ry(ξ) Rz(ψ) Pt. (2)
P corresponds to the workpiece coordinate system and Pt to
the tool coordinate system. Rz indicates a 4 by 4 homoge-
neous transformation matrix of rotation on axis Z specified
as suffix. T is a transformation matrix of translation.
Rz(τ) means the operation to rotate the tool coordinate sys-
tem to the feed direction at the position corresponding to the
tool revolution angle ψ on the contouring tool path. Ry(θ) is
also to rotate it to the normal direction at the position, and
T(C) is to translate the origin of the tool coordinate system to
the position on the contouring tool path.
Rx(η) means the operation to rotate it so as to justify the tool
orientation of lead angle θlead, Ry(ξ) is also to do so in the
tool orientation of tilting angle θtilt, and Rz(ψ) is to revolve the
tool coordinate system. The relationship of η and θlead and
that of ξ and θtilt are given below.
η = θlead
tan ξ = tan θtilt cos η.
Equation (2) can be represented in the form of the matrix
product as shown in equation (3).
On the other hand, position P on the normal line at the point
on the offset surface can be represented below.
P = (k sin ω cos γ, k sin ω sin γ, k cos ω). (4)
X cos ψ cos ξ – sin ψ sin η sin ξ -sin ψ cos η cosψ sin ξ + sinψ sin η cos ξ rk
cos τ Xt
Y sin ψ cos ξ + cos ψ sin η sin ξ cos ψ cos η sinψ sin ξ - cosψ sin η cos ξ rk
sin τ Yt (3)
Z - cos η sin ξ sin η cos η cos ξ R0
cos θ Zt
1 0 0 0 1 1
Through equation (1) to (4), three simultaneous equations
are obtained in each x, y and z coordinate below.
k sin ω cos γ = cos ψ A1(ζ) - sin ψ A2(ζ) + rk
cos τ(ψ)
k sin ω sin γ = cos ψ A2(ζ) + sin ψ A1(ζ) + rk
sin τ(ψ) (5)
k cos ω = A3(ζ)
Here, A1(ζ), A2(ζ) and A3(ζ) are given below.
A1(ζ)
≡ r cos ξ sin α sin[α(ζ) + 2jπ/N] + r sin ξ cos ζ
= cos ξ Sa - sin ξ Da
Figure 1: Spherical milling with a contouring tool path.
Figure 2: Helical cutting edge at the tool tip of ball-nosed
end mill.
Figure 3: Tool orientation specified by lead and tilting angles.
Figure 4: Schematic geometry to detect the machined sur-
face profile as the intersection of a normal line
and a cutting edge.
R : the radius of the sphere to be machined. [mm]
Ro
: the radius of the offset surface equal to R + r. [mm]
f : the feed per tooth per revolution. N f ψ / 2π indicates the
feeding distance of the tool center from the start. [mm]
ftrack : cross feed, i.e. feed per track. [mm]
θ : the directional angle of the normal line at the conturing
tool path measured from the Z axis of the workpiece co-
ordinate system. This specifies the position of the
conturing tool path and is dealt as a constant. [radian]
rk
: the radius of the contouring tool path. [mm]
τ : the central angle along the contouring tool path from the
start point in X-Z plane to the corresponding position of
the ball center of end mill with the revolution angle ψ.
[radian]
ψ : the revolving angle of the end mill around the ZT axis.
[radian]
θlead: the element angle of tool orientation toward the feed
direction. It is measured from the Z axis of the workpiece
coordinate system to the projected line of the ZT
axis onto
the Z-Feed plane. [radian]
θtilt: the element angle of tool orientation toward the feed
direction. It is measured from the Z axis of the workpiece
coordinate system to the projected line of the ZT
axis onto
the perpendicular plane to Z axis and the Feed direction.
[radian]
ω: the directional angle of the normal line at a point on the
shpere where the cusp height is calculated. It is
measeured from the Z axis of the workpiece coordinate
system. [radian]
γ: the directional angle of the normal line measeured from
the X axis of the workpiece coordinate system. [radian]
k : the distance from the origin of the workpiece coordinate
system to the cutting point P on the normal line. [mm]
αααα : Helix lag angle
XT
ZT
YT
ζζζζ : Z angle
of edge Helical cutting
edge
αααα
ζζζζ
ωωωω
Normal line of a
workpiece
(ωωωω, γγγγ)
X Y
ZTool
axis
γγγγ
Revolving
direction
Helical
cutting
edge
Boundary shape
of ball end mill
cutting point
Z
X
Y
O

3. A2(ζ)
≡ r sin η sin ξ sin ζ sin[α(ζ) + 2jπ/N] + r cos η sin ζ cos[α(ζ)
+ 2jπ/N] - r sin η cos ξ cos ζ
= sin η sin ξ Sa + cos η Ca + sin η cos ξ Da
A3(ζ)
≡ - r cos η sin ξ sin ζ sin[α(ζ) + 2jπ/N] + r sin η sin ζ cos[α(ζ)
+ 2jπ/N] + r cos η cos ξ cos ζ + R0
cos θ
= - cos η sin ξ Sa + sin η Ca - cos η cos ξ Da + rk
cot θ
Sa ≡ r sin ζ sin[α(ζ) + 2j π/N]
Ca ≡ r sin ζ cos[α(ζ) + 2j π/N]
Da ≡ - r cos ζ
By deleting variable k from equation (5), the following equa-
tion (6) is obtained with variables ζ and ψ.
A3(ζ) tanω cosγ = cosψ A1(ζ) - sinψ A2(ζ) + rk
cosτ(ψ)
A3(ζ) tanω sinγ = cosψ A2(ζ) + sinψ A1(ζ) + rk
sinτ(ψ) (6)
When the above simultaneous equations are rewritten in the
form of complex representation, the following equation (7) is
equivalent to equation (6). Namely the real part of equation
(7) corresponds to the former equation of (6), and the imag-
ine part of (7) to the latter equation of (6).
[A1(ζ) + i A2(ζ)] exp (iψ) - A3(ζ) tan ω exp (i γ)+ rk
exp [i τ(ψ)]
= 0 (7)
(a) the best case (b) the worst case
Figure 5: Simulated profiles of machined surface.
Real and broken curves of cutting edge positions generate
the highest and the lowest cusp height respectively.
Straight dotted line is the normal line of the sphere.
Figure 6: The best case with θlead = -1º, θtilt = -3º.
Figure 7: The worst case with θlead = 0º, θtilt = 1º.
Figure 8: The trajectory of cutting edge and the stationaly
point of velocity.
Figure 9: The change of cusp height due to the tool orien-
tation. Conditions : f = 0.5 mm/tooth/rev., cross
feed = 0.5 mm, N = 2 teeth, r = 5 mm, R = 100
mm, a (measured coefficient of helix lag angle) =
0.533, ω = π/2 radian. Tool orientation unit: 5º.
As easily understanding, equation (7) or simultaneous equa-
tion (6) cannot be solved analytically. Then it has been solved
by the numerical calculation.
3 SIMULATION RESULTS
3.1 Simulated profile of machined surface
Figure 5 shows sample simulated profiles of machined sur-
face with the conditions below.
r = 5 mm, R = 100 mm, f = 0.5 mm/tooth/rev., cross feed =
0.5 mm, N = 2 teeth, a (measured coefficient of helix lag
angle) = 0.533, ω = π/2 radian.
Figure 5 indicates the best and the worst cases on the cusp
height in the numerical calculation by 1 degree of unit.
The best case: cusp height = 7.7 µm.
The tool orientation: θlead = -1º, θtilt = -3º.
The worst case: cusp height = 24.6 µm.
The tool orientation: θlead = 0º, θtilt = 1º.
The above cusp heights are the average in the range of co-
efficient of helix lag angle 0 ≤ a ≤ 1.4.
In figure 5, there are three lines illustrated. A straight line is
the normal line at a position of the spherical surface. Two
curves indicate the positions of the cutting edge generating
the maximum cusp height and the minimum. The feed direc-
tion is from the bottom to the top of the figure. As easily seen
in (b), the right side of the feed direction shows complicated
profile such as a combination of conical and alluvial fan. This
area corresponds to so-called the down-cut movement of
the cutting edge. The cutting edge position generating the
worst cusp height seems to be nearly parallel to the feed
direction.
Figure 6 shows the same best case with superimposing the
cutting edge movement. Figure 7 shows the same worst case
too. Considering the cutting edge movement illustrated in
figure 6 and 7, it is clear that there exists the stationary point
of velocity along the feed direction as shown in figure 8.
Generally speaking, the cutting speed around the stationary
point is relatively low and some damaged profile is likely to
be observed often.
3.2 Cusp height due to tool orientation
Figure 9 shows the distribution of cusp height due to the tool
orientation specified by the combination of θlead and θtilt. In
order to illustrate its general form, the domain of the tool
orientation is set wide with -45º ≤ θlead, θtilt ≤ 45º, and the
calculation is done by every 5 degree unit. Figure 10 is the
front views of the same result. As easily seen, the cusp height
changes very sharply and quickly in a narrow area of tool
orientation parallel to the normal direction. This indicates the
difficulty of keeping a low cusp height in the tool orientation
parallel to the normal direction. In order to survey it more

4. Figure 12: The change of boundary form of machined sur-
face unit due to the tool orientation.
Consequently 785 combinations of tool orientation have been
tested. Through the results of simulation, many cases of tool
orientation show the cusp height nearly two times of the ap-
proximated height formulated by ftrack2
/8r. In the conditions
of simulation, ftrack2
/8r is 6.25 µm, but the average of simu-
lated cusp height is 11.9 µm.
Even in the best case, the cusp height simulated is 7.64 µm.
It occurs in the condition with the coefficient of helix lag angle
a = 1.0 and 0.3, and with the tool orientation of lead angle
±1º, tilting angle -3º. This peak is in a very small area of 0.07
µm square. The reason why it is greater than 6.25 µm, i.e.
ftrack2
/8r, is not clarified nor identified at the present time.
3.3 Boundary form of machined surface texture
If it is able to synchronize the tool revolution in adjacent tool
paths, the boundary form of machined surface texture
changes regularly according to the tool orientation. Figure
12 indicates the boundary form based on the tool orienta-
tion. It shows a complicated shape around the narrow range
near the normal line with lead angle 0º and tilting angle 0º.
The tool orientation with large and/or small tilting angle is
likely to bring the square boundary form of machined sur-
face texture as shown in figure 12. This result partially re-
flects and supports the experimental investigation and con-
trol of the boundary form in actual ball-nosed end milling by
Saito [7].
4 CONCLUSION
1. In ball-nosed end milling, there are many tool orienta-
tions that make the cusp height about two times of that
estimated by the approximated formula ftrack2
/8r.
2. The maximum cusp height in the worst case is nearly
equal to the value of f2
/2r in the conditions of this study.
3. The maximum and minimum cusp height exist in a nar-
row range of tool orientation within few degrees near the
normal direction of a workpiece.
4. The boundary form of machined surface texture changes
widely according to the tool orientation.
5 REFERENCES
[1] Kline, W. A., DeVor, R. E., Shareef, I. A., 1982, The Pre-
diction of Surface Accuracy in End Milling, Transactions
of the ASME, Jour. Engineering for Industry, 104: 273-
278.
[2] Zhu, R., Kapoor, S. G., DeVor, R. E., 2001, Mechanical
Modeling of the Ball End Milling Process for Multi-Axis
Machining of Free-Form Surface, Transactions of the
ASME, Jour. Manufacturing Science and Engineering,
123: 369-379.
[3] Imani, B. M., Elbestawi, M. A., 2001, Geometric Simula-
tion of Ball-End Milling Operations, Transactions of the
ASME, Jour. Manufacturing Science and Engineering,
123: 177-184.
[4] Altintas, Y., Lee, P., 1998, Mechanics and Dynamics of
Ball End Milling, Transactions of the ASME, Jour. Manu-
facturing Science and Engineering, 120: 684-692.
[5] Mizugaki, Y., Hao, M., Asao, T., Terai, H., 1999, Machin-
ing Error Estimation Based on Ball-nosed End Milling
Behavior, Proc. 32nd CIRP Int’l Seminar on Manufactur-
ing Systems, 283-288.
[6] Mizugaki, Y., Hao, M., Kikkawa, K., / Nakagawa, T., 2001,
Geometric Generating Mechanism of Machined Surface
by Ball-nosed End Milling, Annals of CIRP, 50/1: 69-72.
[7] Saito, A., Zhao, X., Tsutsumi, M., 2000, Control of Sur-
face Texture of Mold Generated by Ball-End Milling, Jour.
Japan Society for Precision Engineering, 66/3: 419-423
(in Japanese).
precisely in the narrow domain, it had been calculated by 1
degree unit. Figure 11 shows the result. The worst cusp height
was newly found and increased from 23.9 µm to 24.6 µm.
As easily seen in figure 9, the series of the cusp peak in the
worst case exist in the narrow lane along the tool path. This
phenomenon has a significant meaning in considering the
estimating formula of cusp height. Because the formula pre-
sented previously has been based on the geometry in the
direction of cross feed, or the perpendicular direction to the
feed direction. According to the geometry of moving the cut-
ting edge, the maximum cusp height of the worst case might
be approximated by an formula of (2f)2
/8r or f2
/2r, if the anal-
ogy of the estimating formula in turning, ftrack2
/8r, can be
applicable in ball-nosed end milling. Because the feed in the
formula of turning can be regarded as nearly two times of
the feed per tooth per revolution in this geometry of generat-
ing the maximum cusp height. The value of f2
/2r is about
0.025 mm in this study.
The maximum cusp height occurs in the condition with the
coefficient of helix lag angle a = 1.0, and is 2.485 µm with the
tool orientation of lead angle 0º, tilting angle 1º. This peak is
in a small area of 0.47 µm square.
Figure 10:The front and side view of the change of the cusp
height due to the tool orientation based on the
calculation with 5 degree of unit.
Figure 11:The front and side view of the change of the cusp
height due to the tool orientation based on the
calculation with 1 degree of unit.