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A novel wheel path generationapproach for
grinding taper end-mills subject to adaptive cross-
section
Liming Wang ( liming_wang@sdu.edu.cn )
Shandong University https://orcid.org/0000-0001-5716-7010
Jianping Yang
Yang Fang
Jiang Zhu
Jianfeng Li
Research Article
Keywords: Flute-grinding, 5-axis CNC grinding, Taper end-mill
Posted Date: January 20th, 2023
DOI: https://doi.org/10.21203/rs.3.rs-2485917/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License.
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A novel wheel path generation approach for grinding taper end-mills subject to
adaptive cross-section
Liming Wang1,2,3,*, Jianping Yang1,2,3, Yang Fang1,2,3, Jiang Zhu 4*, Jianfeng Li1,2,3
1 School of Mechanical Engineering, Shandong University, Jinan, China
2 Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Ministry of Education,
Shandong University, Jinan, China
3 Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Ministry of Education,
Shandong University, 17923 Jingshi Road, Jinan 250061, China
4 School of Electrical and Information Engineering, Hubei University of Science and Technology,
Xianning 437100, China
*Corresponding Author: liming_wang@sdu.edu.cn
Abstract: Taper end-mills are widely used in CNC machining and the flute are crucial to the cutting
performance of the end-mill. However, it is very difficult to form the wheel path of grinding taper end-
mill due to its complex structure and variable cross-section. In the traditional methods, the wheel path is
generated by experience, and the accuracy of flute parameters cannot be guaranteed. To overcome those
problems, this paper proposed a wheel path generation approach for taper end-mills flute-grinding subject
to adaptive cross-section, which is capable to ensure the accuracy of parameters of variable cross-section.
In the wheel path generation, the taper end-mill is discretized into a series of finite slices, which can be
transformed into solving a series sub-problem of grinding cylindrical end-mills. To eliminate the
interference and discontinuity caused by the discretization, the enhancement based on sensitivity analysis
and smoothing method are used to optimize the grinding wheel path. Finally, the approach is validated
by both simulation and experiments. The results indicated that this approach is capable to generate the
wheel path for grinding the taper end-mill flute accurately and efficiently subject to the adaptive cross-
section parameters.
Keywords: Flute-grinding, 5-axis CNC grinding, Taper end-mill
1 Introduction
Taper end mill is applied widely in manufacturing industries due to its high stiffness and efficiency
for complex surface machining. With the increasing requirements of machining free-form surface, the
cutting performance of taper end-mills become more significant, which is mainly determined by the flute
in cross-section [1-4]. In engineering, the flute is defined by three parameters: rake angle, flute angle and
core radius. In the CNC grinding, it is ground by a grinding wheel with a series of the positions and
orientations, which can be called the wheel path [5-7]. However, the variation of flute cross-section of
taper end-mills make it difficult to be ground. At present, there is no comprehensive and definite methods
of the taper end-mill flute-grinding. To formulate the method, the literature on three methods of the flute-
grinding is reviewed in the following [8].
In the process of flute machining, grinding wheel shape, grinding wheel path and flute profile are
closely related. The main stream research could be classified into three groups, including: 1) Given
grinding wheel shape and grinding wheel path to generate flute profile, usually called direct method; 2)
Given flute profile and grinding wheel path to find grinding wheel shape, usually called indirect method;
3) Given grinding wheel shape and flute profile to calculate grinding wheel path [9-11].
At present, the research on the direct method mainly includes three methods: graphical method,
analytical method and Boolean operation method [12]. The graphic method makes uses of the
discretization and spatial geometry theory to calculate flute profile. By discretizing the end-mill and the
grinding wheel into a series of elements, the method can get the intersection points of these elements to
form the flute profile [13,14]. The analytical method makes use of envelope theory to calculate the flute
profile. By taking the common normal of the contact points between the grinding wheel and end-mill
through the grinding wheel axis as the condition, the method can obtain the instantaneous contact curve
to solve the flute equation [15-17]. The Boolean operation method makes use of Boolean subtraction
operations to simulation the flute surface. By setting the grinding wheel to move spirally relative to the
blank step by step, and each step can be regarded as subtraction calculation, the method can get the 3D
model of the flute. In sum, the direct method can get the exact geometry of the flute, but the geometric
characteristics of the flute can not be controlled and it is greatly limited by standard grinding wheel [18].
Moreover, the grinding process is inefficient and time-consuming. As a result, this method cannot be
used for end-mills with strict requirements of the flute parameters such as taper end-mills.
In the development of the cutting tool industry, the indirect method is developed when the standard
grinding wheel and the grinder coupling axis numbers in the direct method cannot grinding the end-mills.
The indirect method is generally used to calculate forming grinding wheel shape based on the
prespecified wheel path and flute profile to obtain the accurate flute parameters [19]. Liaw et al.
developed an indirect method to design and machine the concave-arc ball-end milling cutter. In this
method, based on the maximum sectional radius of the end-mill, the sectional profile of the grinding
wheel was established by inverse envelope theory [20]. Chen et al. used the similar approach to
established a comprehensive manufacturing model for the concave conical end-mill [21]. However, this
method would result in some complicated and impractical wheel shape. In general, the indirect method
can ensure the precision of the flute, but can generate complex shape of the forming grinding wheel
which is expensive to manufacture and difficult to repair.
In light of the above factors, it is necessary to use standard grinding wheel and multi-axis grinder
to generate the wheel path. Lai et al. proposed a high-precision grinding model for the ball end-mill to
calculate the grinding wheel path, which used the radial equidistant lines and the oblique equidistant
surfaces to establish the structure model of the GBEM cutter [22]. Based on the envelop theory, Zhu et
al. derived a closed-form solution of the swept envelope for a general rotary tool moving along a general
tool path defined in the NC programs, which could greatly simplify derivation and computation in the
end-mill process [23]. Wang et al. proposed a new projection method to generate grinding wheel path of
the end-mill. The projection method greatly simplifies the solution of the transcendental equation in the
enveloping method through the geometric relationship between the cutter and the grinding wheel on
projection section, and can quickly solve the grinding wheel path by optimization algorithms [24].
However, most of the above researches are applied to the cylindrical end-mill rather than the taper end-
mill. For the taper end-mill, the flute parameters are changing in different cross-section, which will lead
to the variations of flute parameters [25]. In the traditional grinding of taper end-mills, the wheel is first
set with a fixed orientation and position, then moving in axial direction with a fixed feeding speed while
keep a slight lifting in the radial direction. In this processes, trial and error is generally applied to adjust
the wheel path for the variable flute parameters in cross-section. What’s more, the interference would
happen in the adjusting processes.
In order to form accurate flute of taper end-mills, this paper proposes a grinding wheel path
generation approach subject to adaptive cross-section. In this method, by discretizing the taper end-mill
into a finite of cylindrical end-mills, the solution problem of the wheel path is transformed into solving
the problems of grinding cylindrical end-mills. Additionally, the geometrical conditions are developed to
avoid interference. Subsequently, the grinding wheel path can be further adjusted by the enhancement
and smoothing to improve the machining accuracy of taper end-mills. The outline of the paper is as
follows: Section 2 develops the machining process of the taper end-mill in 5-axis CNC flute-grinding.
Section 3 introduce the discrete method, enhancement and smoothing in detail. Following this, section 4
establishes an integral procedure of the wheel path generation. In Section 5, simulations and experiments
are used to verify the proposed approach. The conclusion is shown in Section 6.
2 Machining process of the taper end-mill
2.1 Basic of the taper end-mills
Flute is the main structure for the end-mills, which is the decisive factor of the overall performance
for the end-mills [26]. The flute could be defined by these parameters: the rake angle
γ
; the core radius
c
r
; the flute angle
φ
and the tool radius
T
r
. which can be seen in Fig. 1.
The tool radius
T
r
can be described as follows:
）（ T0eTT0Tr-r
L
r+= l
r
,
L] 0, [l
. (1)
where
T0
r
is the initial tool radius,
l
is the length from the section which the end-mill radius is
T
r
to the
initial end face.
Te
r
is the end of tool radius,
L
is the total length of the cutter.
Therefore, the design flute can be described by giving the initial values of the parameters
T
0
c00 rrφγ
.
Fig. 1 Flute parameters of the taper end-mill
2.2 Mathematical model of flute-grinding for the taper end-mill
In 5-axis CNC grinding, the flute is generated by the helix motion between the grinding wheel and
the end-mill. The flute-grinding processes can be divided into two steps: 1) grinding wheel set-up and 2)
grinding wheel moves along a 5-axis helical line.
2.2.1 Model of the grinding wheel
Instead of using a traditional taper wheel, we use a taper wheel with rounded corner, Fig. 2 shows
the taper wheel in detail. The geometric features are described by the grinding wheel coordinate system
; , ,
(G) (G) (G) (G)
O X Y Z
in Fig. 2, where
(G)
Z
denotes the wheel axis from the large to small end face.
(G)
X
and
(G)
Y
denote the two axes on the large face. The grinding wheel surface is composed of two parts:
the straight part AB and the rounded part BC. In the grinding wheel coordinate system, the grinding
wheel can be expressed in Eq. (2):
(h,θ) = f(h) cosθ, f(h) sinθ,hG
,
0, Hh
,
0, 2πθ
. (2)
where
H
is the wheel thickness, and the formula for
)(hf
is as follows:
l
22
s s s l l
cot α, 0, h
()
R R (R h ) , h , H
−
=− + − − +
R h h
fh
R h h
. (3)
where
R
is the wheel radius,
R s
is the radius of the rounded corner,
l
h
is the thickness of AB, and
)cosα(1RHh 0sl +−=
. the angle
0
α π α=−
.
Fig. 2 Grinding wheel and its coordinate system
The normal of grinding wheel surface is derived as follows:
n( , ) -cos , - sin ,=hθ θ θ f'(h)
. (4)
2.2.2 Kinematics of flute-grinding for taper end-mills
To describe the flute-grinding operation, the tool coordinate system of the taper end-mill
: ; , ,
(T) (T) (T) (T)
T O X Y Z
is established in Fig. 3, where
(T)
O
is the center of the tool machining face.
(T)
Z
is the tool axis.
(T)
Y
axis is in the vertical direction, and
(T)
X
axis is horizontal. The rigid body
motion of the grinding wheel is determined by the Euclidean transformation for the wheel surface model
in this coordinate system. The transformation process is as follows.
In the wheel set-up operation, the position of grinding wheel is determined by the point
(G)
O
. The
motion of the grinding wheel relative to the cutter consists of the wheel position and orientation, which
can be seen in Fig. 3. In the tool coordinate system, it can be decomposed into two parts: 1) moving along
the vector P in space; 2) rotation about
(T)
Y
, which can be express by the rotation matrix
Y
R
. In the
wheel helix operation, spiral motion with
(T)
Z
can be express by the rotation matrix
Z
R
.
Therefore, the kinematics matrix corresponding to five-axis CNC grinding can be represented as
Eq. (5) in the tool coordinate system.
Z Y XYZ
( , ; ) ( , )( ( ) ( , ) ( ))=+
t t t t
hθtωtβhθx y zW R R G P
. (5)
where:
=
t
t
t
ttt
z
y
x
z y x )(
XYZ
P
,
( ) ( )
( ) ( )
−
=
tt
tt
t
ββ
ββ
β
cos0sin
010
sin0cos
)(
Y
R
,
( ) ( )
( ) ( )
−
=
100
0cossin
0sincos
),(
Ztωtω
tωtω
tωR
.
t
x
,
t
y
,
t
z
are the distances in the
(T)
X
,
(T)
Y
and
(T)
Z
axis.
tvz t=
(
v
is the feeding speed of grinding
wheel along
(T)
Z
axis).
t
β
is the rotation about
(T)
Y
in Fig. 3, which is the angle between the wheel axis
(G)
Z
and the tool axis
(T)
Z
.
ω
is the rotation speed about
(T)
Z
.
Then, the wheel moving surface can be express by substituting them into Eq. (5) results in:
( )
++
++++
++
=
ttt
tttt
tttt
zβhβθosf
θβtωftωθfβtωhtωxtωy
θβtωosftωθfβtωhtωytωx
tθ,h,
cossinc-
)cos(cos)sin()cos(sinsin)sin()sin()cos(
)cos(cos)(c)sin(sin-sin)cos()sin(-)cos(
W
. (6)
where, the parameters
ttt zyx ,,
are defined as the position of the grinding wheel,
t
β
is defined as the
orientation of the grinding wheel. Therefore, the grinding wheel path consists of the grinding wheel
position and orientation, which can be defined by
tttt βzyx ,,,
.
Fig. 3 Grinding process of the taper end-mill
In addition, the angular velocity
ω
can be expressed by the velocity
v
, helix angle
λ
and tool radius
T
r
in Eq. (7):
v
r
ωcotλ
T
=
. (7)
At any time during grinding, the taper end-mill and grinding wheel contact in the form of a curve,
which is referred to the contact curve in Fig. 4. When using traditional grinding wheels, the curve consists
of two parts. The first part is generated from the surface of the grinding wheel; The second part is formed
by the edge of the grinding wheel, and it is due to the discontinuity of the contact curve. Therefore, a
taper grinding wheel with rounded corner is used to simplify the solution of the contact curve. The contact
curve formed by this grinding wheel is continuous and the not need to calculate the second part.
According to envelope theory, the surface normal of the wheel at any point on the contact curve is
perpendicular to the corresponding wheel velocity to the tool, which is shown in Eq. (8).
0)()( = tθ;h,tθ;h, vn
. (8)
where
ZX
( ; ) ( , ) ( ) ( )=t
h,θtωtβh,θn R R n
. And the instantaneous speed of the grinding wheel
d
( ; ) ( ; )
dt
=h,θt h,θtvW
.
θ
can be represented by
h
and
t
by solving Eq. (8), the swept envelope of wheel can be obtained:
( , ) ( , ( ); )
=h t h θh,t tTW
. (9)
Then, the section profile of taper end-mills can be obtained by setting
0= Z
in Eq. (9), which can be
shown in Eq. (10).
* * * * * *
c
* * * * * *
c
cos - sint + sin cos - sin sin + cos cos cos
x
( ) = =
ycos + sint + sin sin + sin cos + cos cos sin
t t t t t
t t t t t
x t y h βtf βtf β θ t
h
y t x h βtf βtf β θ t
W
. (10)
where
*
*sin cos - cos
=tt
fβ θ hβ
tv
.
2.3 Calculation of flute parameters on the cross-section
In Fig. 4, the flute parameters
c
rφγ ,,
can be expressed by the cross-section of the end-mill [24].
In order to ensure the cutter performance, these parameters should be guaranteed to have sufficient
precision in processing.
The flute angle
φ
is the included angle between
1
(T)PO
and
2
(T)PO
. It is the important factor that
determines the chip removal performance. The excessively large flute angle will result to interference
between teeth, an excessively small flute angle will cause insufficient chip removal performance. The
formulas of the flute angle are as shown in Eq. (11).
1
P
and
2
P
are the intersection point between the
cutter radius and the cross-section.
( )
= acos( )
t t t t
φx , y , z , β(T) (T)
12
O P O P
. (11)
The core radius
c
r
has the significant influence on the rigidity and chip removal of the end-mill.
Too large core radius leads to the reduction of chip removal, too small core radius leads to the reduction
of tool rigidity. The
c
r
could be computed by the formula:
( )
( )
22
min x +=
c t t t t c c
r x , y , z , βy
. (12)
The rake angle
γ
will affect the deformation degree of the cutting area and the edge strength of end-
mills, which can be calculated by the measured distance
PD = 5% T
r
in the following formula:
( )
a cos( )
=
t t t t
x , y , z , β(T)
P O P P
2 2 3
. (13)
Fig. 4 Illustration of the contact curve and flute parameters
3 Determination of grinding wheel path
As aforementioned, the flute parameters are required to be as close to the design as possible to
ensure the performance of the end-mills. The machining grinding wheel path
tttt βzyx ,,,
calculated
by optimizing algorithm could generate the machining flute.
Generally, intelligent evolution algorithm, such as GA, can be used to solve the grinding wheel path.
And the optimization model is defined as:
M1
ξγ
ξ
++ 2
c0
2
0
2
0)r -()φ -() γ- (0
min
c
r φ s.t.
. (14)
where
ξ
is the grinding error,
0
γ
, ,
c0
r
are the machining of the given parameters.
However, the cross-sections of taper end-mill are changing. This factor makes it very difficult to
calculate the wheel path. Traditional flute-grinding method of the taper end-mill is usually used to process.
The method approximates the wheel path by setting that the wheel moves in a straight line to a certain
direction in the XY plane during machining, and the grinding wheel orientation is set to a fixed value.
However, in actual machining, the grinding wheel moves in a conical curve in space, and its orientation
is constantly changing. The traditional method cannot will lead to the machining errors. Therefore, a new
approach to determining wheel path for adaptive different cross section of taper end-mills was proposed.
The method can be divided into three parts: discretization, modification and smoothing.
3.1 Discretization
0
φ
In this section, to calculate the exact grinding wheel path, the taper end-mill is discretized into a
series of finite slices, as shown in Fig. 5(a). These slices can be projected as a series of cross-sections of
the taper end-mill in Fig. 5(b). As previously mentioned, the core radius is ground by the position of the
maximum diameter of the grinding wheel. Due to the constant change of flute parameters such as core
radius on the cross-sections, the grinding wheel path also changes. Therefore, by solving the grinding
wheel path corresponding to each cross-section and combining them, the overall grinding wheel path of
machining taper end-mill can be obtained. The problem is then transformed into how to adaptively solve
the wheel path with varying cross-sections. To achieve adaptive cross-sections, the grinding of each slice
after discretization can be considered as machining a cylindrical end-mill with the radius
i
rT
, the
coordinates of the center of each discrete end-mill on the tool axis is defined as
i
z
. Therefore, the wheel
path of taper end-mill flute-grinding is simplified as solving a series sub-problem of cylindrical end-mills
flute-grinding.
Fig. 5 Discrete process of taper end-mills: (a)Wheel path generation with the discretized method; (b)
Variable cross-section after projection
Fig. 6 Description of post-discrete tool grinding
For discrete cylindrical end-mills, a new tool and grinding wheel coordinate system can be
established in Fig. 6. The series of grinding wheel positions and orientations for cylindrical end-mills
iiii βzyx ,,,
can be called as the wheel path. In addition, by updating the Eq. (9), the flute surface of the
cylindrical end-mills can be expressed as Eq. (15):
( )
++
++++
++
=
iii
iiiiii
iiiiii
ii
zβhβθosf
θβtωftωθfβtωhtωxtωy
θβtωosftωθfβtωhtωytωx
t,θh,
cossinc
)cos(cos)sin()cos(sinsin)sin()sin()cos(
)cos(cos)(c)sin(sin-cos)cos()sin(-)cos(
W
. (15)
It is assumed that the forward speed
v
of the grinding wheel is fixed during machining, so
i
z
can be
solved by Eq. (16):
n
L
=i
zi
. (16)
where n is the discrete number, L is the length of the taper end-mill.
By updating M1, the optimization model is defined as:
M2
ξγ
ξ
++ 222 )r -()φ -() γ- (0
)( min
c0ici0ii0ii
iii
r φ s.t.
βyx
. (17)
where
0i
γ
,
0i
φ
,
c0i
r
are the corresponding given parameters of the discrete cylindrical end-mill.
Additionally, in practical application, the wheel path is also limited by some geometric conditions,
such as: over-cut and grinding wheel swinging violently. Therefore, some constraints are stipulated to
prevent the occurrence of these phenomenon.
Constraint 1: To avoid over-cut, a limited range of core radius is specified, which can be expressed
as follows:
ric0ci
r
. (18)
Constraint 2: To avoid the dramatic variation or zigzagging for the grinding wheel orientation, the
difference between adjacent
i
β
should be controlled in a certain range as
r
β
in Eq. (19):
r
ββ Δ
. (19)
3.2 Sensitivity analysis of the wheel position and orientation
The formula in above part individuals that there are complex geometric relationships between the
flute parameters and the wheel position and orientation, which makes the trend of flute parameters
changing with the wheel position and orientation can be predicted by sensitivity analysis for further study.
Therefore, the sensitivity coefficient can be defined as SFP, which can be shown in Eq. (20):
/
/
=FF
FP
PP
ΔSS
SΔSS
. (20)
where SF is the value of the flute parameters and SP is the value of the wheel position and orientation,
ΔSF and ΔSP is the variation of flute parameter and grinding wheel position and orientation.
To ensure the reliability and validity of the sensitivity analysis, ordinary cylindrical end-mill is used
to avoid the interference of the other cutter parameters. The cylindrical end-mill parameters can be seen
in Table 1. Whereafter, wheel position and orientation
iii βyx ,,
can be calculated and used as the
initial value for sensitivity analysis.
To obtain the sensitivity of the flute parameters to wheel position
i
x
,
( )
ii βy,
should be constant
when
i
x
changes. By putting the
i
x
values into the flute parameter solving formula, the variation can be
obtained in Fig. 7, (a). In the same way, the variation of the flute parameters to
i
y
and
i
β
c ould be
obtained in Fig. 7, (b) and (c). The sensitivity coefficient can be shown in Table 2, which reflects the
degree of influence to wheel position and orientation on flute parameters.
i
x
has little influence on the
flute parameters, and
i
y
has the significant influence on the core radius,
i
β
has the significant influence
on the rake angle.
Fig. 7 Analysis of sensitivity: (a) The variation trend of flute parameters and xt; (b) The variation trend
of flute parameters and yt;(c) The variation trend of flute parameters and β
Table 1 Cylindrical end-mill parameters
Cutter
Cutter radius
rT0
(mm)
Helical
angle λ
(deg.)
Core radius
rc
(mm)
Rake angle
γ
(deg.)
Flute
angle φ
(deg.)
Cylindrical end-mill
15
30
9
6
75
Table 2 Sensitivity coefficient
Calculated values of the flute parameters
Wheel position and orientation
γ
φ
rc
xi
-0.60
-0.22
0.11
yi
-7.14
-4.37
17.03
βi
16.67
-0.95
0.08
3.3 Enhancement
By the discretization, the grinding wheel path consists of a series of cylindrical end -mill grinding
wheel positions and orientations can be obtained. Subsequently, the errors of flute parameters on the
section are calculated in Fig. 8. The errors of flute parameters on some cross-sections are higher than 4%,
which can be called “adjustable error”. The adjustable error is caused by the discontinuity of the
discretization and the limitation of the optimization algorithm, which will affect the machining accuracy
of the taper end-mill.
Therefore, enhancement is proposed to reduce the adjustable errors by sensitivity analysis. First,
the flute cross-section conforming to the adjustable error and corresponding wheel positions and
orientations are selected. For each cross-section, the small search domain is constructed in 3D space in
Fig. 9 (a). The boundary of small search domain is determined by sensitivity analysis. In this domain,
the more sensitive the wheel path parameter is to the improved flute parameter, the smaller the search
area is. The PSO optimization algorithm is used to calculate the grinding errors in the search domain.
The optimization model is shown as follows:
M3
++
maxim in
maximi n
maximin
c0ici0ii0ii
iii
βββ
yyy
xxx
r φ s.t.
βyx
)r -()φ -() γ- (0
)( min
222
γ
. (21)
Then, the optimal wheel position and orientation can be obtained and shown in Fig. 9 (b). The
optimal result is put into solving the errors of flute parameters to determine whether it meets the accuracy
requirements. If the accuracy requirement is not met, the search domain is expanded. Finally, the optimal
wheel positions and orientations are replaced by the original one to complete the optimization of the
wheel path. The entire enhancement process is described in Algorithm 1.
Fig. 8 Error of flute parameters on the cross-section of taper end-mills
(a) (b)
Fig. 9 Search process of the enhancement: (a) Search domain; (b) Search process
Algorithm 1 Enhancement
procedure Enhancement
initialize flute parameters and wheel position and orientation
construct search domain
for each iteration result calculated by PSO
Calculate grinding error ξ and flute parameter errors
if
<
min and flute parameter errors < 4% then
iteration result = optimal result
end if
end for
return optimal wheel position and orientation
end procedure
3.4 Smoothing
In order to further optimize the wheel path calculated by discretization and enhancement, the
smoothing based on cubic polynomial regression function is proposed to smooth the wheel path, the
fitting model functions of the cubic polynomial regression function for the wheel path can be written as
[27-29]:
=
=
1
z
z
z
dcba
dcba
dcba
)(
)(
)(
Y
X2
3
3333
2222
1111
zf
zf
zf
β
. (22)
The fitting problems actually can be described as the optimization problems, and the optimization
model can be shown in Eq. (23).
M4
MSEff s.t.
MSE
i
ii −
=
2
n
1
)(z)(z)(
n
1
0
min
(23)
where MSE is the mean squared error,
i
f(z)
is the optimized value of the function,
i
f
(z)
is the wheel path.
By smoothing the functions, the smoothed curve can be got and reconstructed to obtain the
smoothed wheel path, which is shown in Fig. 11 (d).
4 The procedure of wheel path generation approach
Combined with the new method, the calculation procedure of adaptive cross-section shown in Fig.
10 is presented. The calculation procedure is discrete to a series processes according to the discreted
cylindrical end-mill. In each process, the wheel path of grinding cylindrical end-mill can be calculated
by GA, and is optimized by enhancement. And the flute parameters can be solved by the golden section
method. Finally, the grinding wheel path is obtained by smoothing.
Fig. 10 Procedure of the wheel path generation
5 Case studies
The main advantages of this approach are high accuracy and wide applicability. In addition, the
interference is avoided in the processing. To demonstrate the advantage of the new approach, it is applied
to three different case. The first case study shows that this method has high accuracy compared to
traditional methods. In the second case, two end-mills with varying cross section parameters are
simulated to show the wide applicability of the method. Finally, an experimental verification is carried
out in the third case study.
5.1 Case Study Ⅰ
To verify the accuracy and practicability of the optimization model, the flute-grinding simulation
of taper end-mills is carried out by MATLAB software. The parameters of Wheel 1 can be shown in
Table 3, and the flute-grinding parameters are shown in Table 4. The wheel path and the flute surface
generated by simulation are shown in Fig.11. The flute parameters of the new method are compared with
the traditional method to verify the the machining accuracy. As shown in Table 5, the flute parameters of
two methods can be got by sections at different positions which are intercepted in the end-mills model.
The relative grinding errors of the two methods are compared in Fig. 12, which showed that the new
method proposed can meet the requirements of machining better than the traditional method, and could
guarantee the precision of the flute-grinding in CNC grinding.
Table 3 Parameters for grinding wheel
wheel parameters
Wheel 1
Wheel 2
wheel width
H
(mm)
15
30
wheel radius
R
(mm)
30
75
wheel angle
α
(deg.)
80
80
rounded corner radius Rs (mm)
1
2
Table 4 Flute-grinding machining parameters of Case 1
Case
Initial cutter
radius rT0
(mm)
End of cutter
radius rTe
(mm)
Cutter
length L
(mm)
Core radius
rc
(mm)
Rake angle
γ
(deg.)
Flute
angle φ
(deg.)
Case1
5
10
100
3-6
6
75
Fig. 11 The fitting image of the optimizing parameters
Table 5 Section parameter of Case 1
cutter parameters
traditional method
discrete method
cutter radius(mm)
γ(°)
φ(°)
rc(mm)
γ(°)
φ(°)
rc(mm)
5
6.1562
74.8941
3.060
6.0004
74.9388
3.010
5.5
6.0007
75.9376
3.311
6.0117
75.2388
3.302
6
6.2366
75.0576
3.641
5.9989
75.9388
3.614
6.5
6.2508
76.1758
3.903
5.9987
75.6653
3.924
7
6.2581
73.3146
4.220
6.0323
73.9139
4.212
7.5
6.3628
74.4723
4.510
5.9570
74.9332
4.510
8
6.2328
72.6464
4.804
6.0291
74.6643
4.821
8.5
6.4328
75.8366
5.130
5.9951
74.7659
5.110
9
6.3250
78.0406
5.409
5.9997
75.2564
5.400
9.5
6.4245
75.2576
5.710
6.0273
75.1640
5.704
Fig. 12 The relative grinding errors
5.2 Case Study Ⅱ
To verify the wide applicability of the adaptive cross-section, two kinds of the taper end-mills with
varying cross section parameters are proposed and verified by simulation. The flute parameters of taper
end-mills vary uniformly with the radius. The cutting force of these end-mills are more uniform and can
reduce the vibration during machining. Therefore ,this type of taper end-mills are used for verification in
this section. In addition, the experiment is further analyzed and verified by comparing them with the
normal taper end-mill in Simulation 1. The parameters of Wheel 2 can be shown in Table 3, and the flute-
grinding parameters are shown in Table 6. The changing parameter in Simulation 2 is the rank angle, and
in Simulation 3 is the flute angle. The wheel path obtained by simulation is shown in Fig.13, which
indicates that with the change of different cross-section parameters for the taper end-mill, the generated
wheel path is constantly changing. It is worth noting that when the rank angle changes, the orientation of
the grinding wheel β will be greatly affected. These phenomena prove that the new method can get the
corresponding wheel path according to the change of the flute parameters, and has the characteristic of
adaptive. Then the cross-section results are simulated and measured in Fig. 14. The results show that the
proposed method can realize the adaptive cross-section by adjusting the grinding wheel path, and has
high machining accuracy.
Table 6 Flute-grinding machining parameters of Simulation
Simulation
Initial cutter
radius rT0
(mm)
End of cutter
radius rTe
(mm)
Cutter
length L
(mm)
Core
radius rc
(mm)
Rake
angle γ
(deg.)
Flute
angle φ
(deg.)
Simulation1
15
25
100
15-25
9
75
Simulation2
15
25
100
15-25
6-9
75
Simulation3
15
25
100
15-25
9
75-80
Fig. 13 The grinding wheel path of the simulation: (a) x in Simulation; (b) y in Simulation; (c) beta in
Simulation
(a)
(b)
(c)
Fig. 14 The flute parameters within the various cross-section: (a) Simulation1; (b) Simulation2; (c)
Simulation3
5.3 Case Study Ⅲ
A machining experiment is carried out on WALTER HELITRONIC POWER CNC grinding
machine tool in this section. The 5-axis relative motion between the grinding wheel and the end-mill in
Fig. 15(a) is provided by this machine tool, and the end-mill is shown in Fig. 15(b). In this experiment,
Wheel 2 is selected as the machining tool, the grinding parameters are shown in Table 7. The diamond
wheel is used, and the end-mill material is tungsten steel alloy. Grinding wheel speed: 2500 RPM, Feed
rate: 15m/s.
Fig. 16 shows the finished taper end-mill. Then, the flute parameters are measured to verify the
machining accuracy. Because of the particularity of taper end-mill, it is necessary to ensure the
consistency of flute parameters in the machining process. Therefore, the taper end-mills not only needs
to be measured on the end face, but also on multiple sections to verify the machining accuracy. In Fig.
17, the flute parameters on different cross-sections are obtained by grinding the end face of taper end-
mill. The measurement results show that the taper end-mill meets the machining requirements.
Table 7 Flute-grinding machining parameters of Case 3
Case
Initial cutter
radius rT0
(mm)
End of cutter
radius rTe
(mm)
Cutter
length L
(mm)
Core radius
rc
(mm)
Rake angle
γ
(deg.)
Flute
angle φ
(deg.)
Case3
10
20
40
6-12
9
75
(a)
(b)
Fig. 15 (a) WALTER HELITRONIC POWER CNC grinding machine tool; (b)The end-mill
Fig. 16 Machining taper end-mill
(a)
(b)
(c)
Fig. 17 Flute parameters of the taper end-mill sections: (a) rt=10; (b) rt=10.5; (c) rt=11
6 Conclusions
In this work, the discretization was proposed to generated wheel path of the taper end-mill flute-
grinding. The enhancement and smoothing were used to second-optimize the wheel path. In addition, the
procedure of the wheel path generation was developed to optimize and simulate the flute-grinding.
Finally, the variety of end-mill cases were simulated and machined. The results show that:1) The wheel
path generation approach could achieve higher accuracy; 2) The approach can be applied to taper end-
mills of different sizes and changing parameters. It has a wide range of applications.
The main contribution of this paper is to propose a approach to calculate the wheel path of the taper
end-mill subject to adaptive cross-section. The accuracy and practicability of the approach is verified by
analyzing simulation and experimental results. There are still some shortcomings in this paper. This paper
only studies the taper end-mill which is a representative of the variable section end-mills. Therefore, this
work will be extended to more complex tool flute-grinding.
Acknowledgments
This research is supported by the National Key R&D Program of China (Grant No.
2020YFB1711603), and the National Natural Science Foundation of China (Grant No. 52175473), and
the Key Technology R&D Program of Shandong (Grant No. 2022CXGC010304). The authors would
like to thank all anonymous reviewers for their helpful suggestions to improve this paper.
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Statements & Declarations
Funding This research is supported by the National Key R&D Program of China (Grant No.
2020YFB1711603), and the National Natural Science Foundation of China (Grant No. 52175473), and
the Key Technology R&D Program of Shandong (Grant No. 2022CXGC010304).
Competing Interests The authors declare no competing interests.
Availability of data and material All data and materials used or analyzed during the current study are
included in this manuscript.
Code availability Not applicable
Ethics approval Not applicable
Consent to participate All authors agree to participate in the editing of the paper.
Consent for publication All authors agree to publish this manuscript in this journal.
Author’s contributions Liming Wang, Jianping Yang and Yang Fang conceived the approach. Liming
Wang performed the manuscript. Liming Wang and Jianping Yang discussed the idea and revised the
manuscript.
