INTRODUCTION
Some problems of helical cutter with constant pitch are discussed
in references (Liu, 1998; Tai and Fuh, 1995). The researches are not so
deeply and versatile as that on the cutter with constant helical angle,
such as references (Liu, 1998; Liu and Liu, 1997, 1998a; Aoyama, 1986;
Kataev, 1989; Kaldor et al., 1984, 1985; Kang et al., 1996;
Liu and Liu, 1998b). The researches on the revolving cutter have problems
existing at four aspects.
• 
Only single problem is discussed in most of the references. For
example, reference (Liu and Liu, 1998a) only discusses the cutting
edge. Reference (Liu and Liu, 1997) only discusses the design of groove.
References (Aoyama, 1986; Kataev, 1989; Kaldor et al., 1984,
1985; Kang et al., 1996) discuss the design of this type of
cutter. References (Kang et al., 1996; Liu and Liu, 1998a)
only discuss the NC machining. Such references are far away from the
engineering application. There are still many works need the reader
to finish. 
• 
The manufacture cost is high. 3axis or 4axis NC machining is adopted
in most cases. The rake face of the cutter is machined first; other
surfaces are machined gradually (Liu and Liu, 1998a). In this way,
one groove is machined in multiple processes, so that the manufacture
cost is much higher. 
• 
Planar cutting edge is used in most of the ballend cutters (Liu
and Liu, 1998a; Zhou et al., 1991). This is not benefit to
the chip removal. 
• 
Continuities of the cutting edge and the feeding speed of grinding
wheel at the connecting part are not verified. 
Based on the above cases, all the problems (including the design, the
NC machining, remedy and postprocess) of ballend taper cutter with constant
pitch will be discussed in the study. This is benefit to extension of
the equation. These problems are not discussed deeply in references (Liu,
1998) and (Tai and Fuh, 1995). The basic groove is finished mainly in
once by 2axis NC machining so as to decrease the manufacturing cost of
cutter. Only about oneseventh of the cutting edge is planar curve at
the tip part of the ballend in the study, this is benefit to the chip
removal. The continuities of cutting edge and the feeding speed of grinding
wheel along the axial and radial direction will be verified.
The fifth problem in the most references is lack of generality. It is
difficult for the research work on certain type of cutter to be extended
to the similar cutters. Therefore, this paper will present the general
equations first; the detailed equation of the ballend taper cutter can
be deduced from the general equation. The related contents will be discussed
as follows.
THE CONTINUITY OF THE CUTTING EDGE
The revolving surface of special revolving cutter can be expressed
by the following general equation:
For the cutting edge of cutter with a constant pitch, the helical lead
is also constant, i.e.,
is constant. Since f(u) varies with the variation of parameter u, so
the angle φ between cutting edge and the generator of revolving surface
is also the function of u, it is defined by the follows:
where, b is the spiral parameter and
In order to obtain the cutting edge on the surface of Eq.
1, calculate the partial differential of the surface as following:
The coefficients of first fundamental form is as follows:
By use of the definition of angle φ between the tangent line dr
of cutting edge and the generator δr (δφ = 0) of revolving
surface, it is known that:
Equation 5 can be obtained:

Fig. 1: 
Ballend taper cutter 
In integration form, Eq. 10 becomes:
where, θ_{0} is the initial value of parameter θ, it
is defined according to the detailed condition.
It is obvious that the equation of cutting edge with a variable helical
angle ψ and constant pitch T may be obtained by substituting Eq.
11 and 1.
For ballend taper cutter, there are helical cutting edges on the taper
part of the cutter. The cutting edge can also be defined by use of Eq.
1 and 11. As shown in Fig. 1,
the equation of the sphere is as follows:
Where: 
R 
= 
Radius of the sphere 
θ_{1} 
= 
Angular parameter and θ_{1} ∈[0,2π] 
z_{1} 
= 
Parametric variable and z_{1} ∈[–R, –Rsin
α] 
Corresponding to Eq. 12 and 1, we
have:
Corresponding to Eq. 10, we can obtain that:
After integration, Eq. 12 becomes as follows:
This is because that the initial θ_{1} = 0 when z_{1}
= 0, so θ_{0} = 0, Eq. 15 may be obtained.
At the conjunction position, substitute z_{1} = R sin α into
Eq. 15, we have:
By referring to Fig. 1, the equation of cone may be
expressed as:
where z_{2} ∈[0,h]
Accordingly, we can get
Corresponding to Eq. 10, it is known that
After integration, we have
Substitute the boundary condition that z_{2} = 0 and θ_{2}
= θ_{20} = θ_{10} into the above equation, we
can obtain that:
By substituting Eq. 21 into Eq. 17
and 16 into Eq. 12, the helical
angle of the ballend taper cutter may be obtained. The helical angle
is variable but the pitch is constant.
The actual cutting edge is determined according to the section and relative
speeds of grinding wheel in the NC machining. These problems will be introduced
as follows.
DEFINITION OF GROOVE SHAPE AND THE EQUATION OF SPIRAL GROOVE
Figure 2 shows the groove section of the cylindrical
part and the section plane of the ballend passing through the center
of the ball. The whole section consists of 5 segments. The first segment
is a straight line AB, which is the rake face with an rake angle γ.
The second segment is arc BC, which helps chip coiling. The third segment
is arc CD, which provides adequate strength for the cutter and a smooth
path for the chip flow. The fourth segment is straight line DE, which
links with the cutting edge land. The cutting edge land designated as
the fifth segment is also represented by a straight line EF. Since the
radius of the cutter at the ballend decreases with increasing zcoordinate,
the depth of the groove should also gradually decrease accordingly. Therefore,
on the ballend the groove shape needs to be modified. Only part of the
five segments will be included, possibly with only section BC remaining.
The groove on the cylindrical part is formed by the continuous spiral
motion of the above section. The groove on the ballend part is formed
by the enveloping surface of the discshaped grinding wheel, which moves
at variable radial speed and axial speed and constant rotational speed.
The profile of grinding wheel is defined by solving the reverse problem
of finding the profile envelope fitting the above groove section.

Fig. 2: 
Sectional shape of the groove 
The groove section as shown in Fig. 2 is for a cutter
of four flutes. If the cutter consists of n grooves, the dividing angle
of each groove will be 360°/n. In this study, a cutter with four grooves
is taken as an example, so the dividing angle of the groove section should
be 90°. Let the inner radius of the cylindrical part be r, with rake
angle γ, the equation of straight line AB may be defined as:
where,
is the outer radius of cross section.
In order to define the coordinate of point B, the coordinate of center
point Q of the arc BC should be calculated first. If the radius of arc
BC is r_{1}, then Q is on the line paralleling to the straight
line AB and with a distance r_{1}. Q is also on the circle centered
at point O with radius r + r_{1}. In this way, the coordinate
of point Q may be defined by the following equations:
where, is
the radius of the outer circle, r is the radius of the inner circle, r_{1}
is the radius of arc BC, γ is the angle between Xaxis and segment
AB and λ_{1} is the length parameter to describe any arbitrary
point P on segment AB. The coordinate (x_{Q}, y_{Q}) can
be calculated after λ_{1} and φ_{2} are
solved from (23). The equation of arc BC may be defined
as:
Since the arc BC is tangent to straight line AB at point B, the coordinate
of point B may be defined as:
By substituting Eq. 22 and 24 into
Eq. 25, λ_{1} and φ_{2} can
be computed. The coordinate of point B can be then be estimated by the
equation of 22 or 24. Since the
coordinate of point F is (0,)
and the clearance angle of cutting strip is α_{e}, the equation
of straight line EF can be expressed as:
where, λ_{2} is the length parameter for describing any
arbitrary point on segment EF and α_{e} is the angle between
Xaxis and segment EF at point F. The coordinate of point E may be defined
by the length l of line EF as:
Let straight line DE is at an angle α_{E} to xaxis, the
equation of straight line DE can be obtained as:
where, μ_{1} is the length parameter for describing any
arbitrary point on segment DE._{ }In order to satisfy the requirement
of cutter strength and smooth chip flow, let the radius of arc CD be r_{2}.
The center P of arc CD is on the line paralleling to the straight line
DE at a distance r_{2}. P is also on the circle, which is centered
at point Q and with a radius of r + r_{2}. This is because arc
BC is tangent to arc CD at point C and the tangent point C is on the line
PQ. In this way, the coordinate of point P may be defined by the above
two conditions, i.e.,
where, φ_{1} and μ_{2} can be computed by the
above equation. The coordinate of point P can also be defined accordingly.
Therefore, the equation of arc CD can be expressed as:
The coordinate of the tangent point C between arcs BC and CD are given
as:
After solving for ψ and φ_{2} from the above equations,
the coordinate of point C can be obtained from Eq. 31.
Since the straight line of DE is tangent to the arc CD at point D, the
coordinate of point D can be obtained as:
where, μ_{1} and φ_{2} can be computed from
the above equations. The coordinate of point D can then be estimated accordingly.
Since the equation for each segment of the groove section is defined,
the coordinates of the starting point, connecting points and end points
are also obtained, the curves of the groove can now be expressed as:
If the groove is moving along the helical curve on cylindrical surface
of the cutter, the equation of the helical groove can be obtained as:
where, b is the pitch of the spatial helical curve and can be expressed
as:
DESIGN FOR THE SECTION PROFILE OF THE GRINDING WHEEL
A new coordinate system σ_{1} = [o_{1};x_{1},y_{1},z_{1}]
is attached to the grinding wheel, as shown in Fig. 3.
Here, the x_{1} axis lies on the xoy plane having an angle of
30° counted from the axis x. O_{1} is defined by the rake
angle γ and arc bc. The centerline axis of the grinding wheel lies
on the z_{1} axis passing through o_{1}. Here, o_{1}z_{1}
and oz are straight lines on different planes and o_{1}o is the
common normal line. The equation of o_{1}z_{1} in coordinate
system σ can be expressed as:
where, φ is the helical angle and a is the distance between the
origins O and O_{1} of both σ = [O:X,Y,Z] and σ_{1}
= [O_{1};X_{1},Y_{1},Z_{1}] coordinate
systems. Since the profile of grinding wheel is a revolving surface, the
normal vector of any point on the surface passes through the revolving
axis z_{1}. Therefore, in order to satisfy the condition that
every point on the groove surface is a common tangent point of grinding
wheel profile and the groove surface, the normal vector of this point
passes through the centerline axis of the grinding wheel. That can be
expressed as:

Fig. 3: 
Relative coordinate systems of σ and σ_{1} 
By equating three components respectively to obtain
From Eq. 40, one can easily obtain the parameter μ_{2
}as:
By substituting the above equation into Eq. 39, the
parameter λ_{3} is given as:
After substituting the above two equations into Eq. 38,
the relation can be obtained as:
By using the property of spiral surface, y*N_{x}*–x*N_{y}*
= bN_{z}*, the above relation becomes:
By simultaneously solving Eq. 34 and 44,
the contact curve between the profile of the grinding wheel and the spiral
groove may be obtained as:
In order to obtain the profile of the grinding wheel, the above equation
of the contact curve is transformed to the coordinate system σ_{1},
which is attached on the grinding wheel. In order to do so, a new coordinate
system σ´ = └o_{1};x_{1},y´,z┘
is introduced. The transformation from coordinate system σ to coordinate
system σ´ is given as:
Since the origin o_{1 }and both of x_{1} and z_{1}
axes in coordinate system σ_{1} are already defined, the
direction of y_{1} axis can be easily obtained by using the righthand
rule. The transformation from coordinate system σ´ to coordinate
system σ_{1 }can then be expressed as:
Thus, the equation of the contact curve can now be expressed in terms
of σ_{1} coordinate system as:
As the contact curve is rotated around the z_{1 }axis, the profile
surface of the grinding wheel can be computed. The intersecting curve
between the profile surface and the plane y_{1} = 0 is the profile
curve of the grinding wheel. This curve can be expressed as:
The grinding wheel having the above profile curve automatically yields a desired
groove profile as designed in Eq. 33. However, this is not
the case for the ballend portion of the cutter. To achieve better dimensional
accuracy at the ballend, additional refinement operations should be given to
control the wheel motion in radial direction.
CONTINUOUS SPEED OF THE GRINDING WHEEL AXIS AT THE DIRECTION OF RADIAL
FEED AND AXIS FEED
In order to adopt the twoaxis NC machining, let angular speed ω
be constant, the speed v_{z} along the axial direction can be
calculated by:
At the ballend of the cutter, the speed of grinding wheel along the
axial direction can be defined as:
For the cone surface, the speed of grinding wheel along the axial direction
can be defined as:
At the conjunction place, z_{1} = Rsinα for surface (1)
and z_{2} = 0 for surface (17), substitute the above boundary
conditions into Eq. 51 and 52, respectively,
we have:
Equation 53 demonstrates that the axial speed in the
NC machining of the whole cutter is a continuous function.
The feeding speed of grinding wheel at the radial direction must be defined
according to radius function f(u). If the feeding speed varies linearly
with the radius, overcut will happen at the part of .
If the radial feeding speed is neglected, the groove will not exist at
the part of radius less than r. The proper feeding amount should think
about the general variation
of radius and the general variation r of feeding amount. Let the feeding
amount S_{g} vary proportionally to the variation
of radius, i.e., ,
we can get:
Accordingly, the general equation of feeding speed at the radial direction
is as follows:
At the ballend of cutter, the radial speed of grinding wheel is as follows:
For the cone surface, the speed of radial motion can be defined as:
Substitute the conjunction condition into Eq. 56 and
57, we can get:
Equation 58 explains that the radial speed of the
grinding wheel is a continuity function.
The actual obtained groove on the cutter in the NC machining is defined
by the enveloping surface of grinding wheel. The actual obtained groove
should be verified by computer simulation.
COMPUTER SIMULATION AND REMEDY
The cutting edge and the section of actual obtained groove are to
be simulated by computer. According to the simulation results, the correctness
of initial assumptions can be evaluated.
It is obvious that f(u) expresses the radius of section circle for a
certain value of u. Use the following equations:
We can get two intersect points of section circle at the position of
z* = g(u), let the point on the cutting edge be (x_{1}*,y_{1}*,z_{1}*),
the other point be (x_{2}*,y_{2}*,z_{2}*). The
actual obtained cutting edge is at the outside of designed cutting edge
except the tip of ballend, but the distance is very small and the two
cutting edges are approximate offset curve with each other. At the area
near the tip of ballend, overcut exists. So remedy is necessary.
The actual groove section at position of z = g(u_{0}) may be
obtained by different value of x_{c}. It is obvious that
the cutting edge strip decreases gradually with the decrement of radius.
At the tip part of the ballend, the strip does not exist any more. The
remaining revolving surface exists between the adjacent grooves since
the center angle corresponding to each groove is less than 90°. According
to the above simulation results, the machining methods at the tip part
of the ballend should be modified so that the overcut can be diminished.

Fig. 4: 
Relative position between grinding wheel and top of the cutter 

Fig. 5: 
Define the factor λ 
The radius
of groove bottom should be greater than i.e.,
i.e.:
Otherwise, the cutting edge adopts planar curve when f<r_{o}.
Examples verify that the planar curve of the cutting edge at the tip of
ballend only takes about oneseventh of the cutting edge at the ballend
of the cutter.
The problem of remaining surface and no strip may be remedied by the
following approach. A coneshaped grinding wheel with a bottom angle of
90°–α_{e} is used to eliminate the remains. The
relative position between the grinding wheel and the cutter is as shown
in Fig. 4. By means of the relationship between designed
point (x., y) and the actual point (x*,y*) on the cutting edge (Fig.
5):



Fig. 6: 
The grinding wheel and sectional profile. (a) the groove section
of the cutter (b) sectional profile of the grinding wheel and (c)
the grinding wheel 


Fig. 7: 
Feeding speed of the grinder in both the axial and the radial directions.
(a) Axial speed of grinding wheel (b) Radial speed of grinding wheel 
the factor λ may be defined. Then calculate the revised radial speed
by:


Fig. 8: 
The desired and actual (dark line) obtained cutting
edge curve (dimension: mm) (a) Projected cutting edge curve of the
cutter on XY plane (b) Spatial cutting edge curve of the cutter 
Keep axial speed and revolving speed unchanged, replace radial speed
v_{g} with ,
the remains revolving surface can be diminished A ideal special revolving
cutter may be obtained.
A ballend taper milling cutter is selected as one example to illustrate
the effectiveness of the proposed mathematical modeling and residual compensation
method. This cutter possesses a helical angle of π/6, a rake angle
of γ = 5°,_{ }the_{ }radius of groove arc BC
of r_{1} = 2 mm, the radius of groove arc CD_{ }of r_{2}
= 4 mm, the length of the cutting edge strip of l = 1 mm, the clearance
angles of cutting strip of α_{e} = 6° and α_{E}
= 60°, cylindrical radius of 8 mm and a taper angle of 10°, a
taper height of h = 11.860 mm and the remaining circular end radius of
2 mm, z_{1}∈[6,1.042] mm, z_{1}∈[0,11.860]
mm, the distance between the origins O and O_{1 }of_{ }a
= 20 mm, a spiral parameter of b = 32/π mm, a helical lead of T =
64 mm. The computed typical sectional profile of the helical groove on
the ballend taper milling cutter is given in Fig. 6a.
The grinding wheel and the sectional profile of the grinding wheel are
Fig. 6b and c.

Fig. 9: 
The feeding amount S_{g} of grinding wheel 

Fig. 10: 
Te actual produced ballend taper milling cutter 
Figure 7 shows the feeding speed of the grinder in
both the axial and the radial directions. Figure 8 shows
the various views of the ballend taper milling cutter cutting edge curve.
Figure 9 shows the feeding amount S_{g} of grinding
wheel. Figure 10 shows the actual produced ballend
taper milling cutter.
CONCLUSIONS
This study presents a modeling method that can be used to design
and manufacture a ballend taper milling cutter by using a simple twoaxis
NC machine. From the related models of design and NC machining and the
simulated results, it is known that the general models for the ballend
taper cutter are presented in this study. The validity of machining this
kind of cutter by twoaxis NC machining is verified by example. The surface
shape around the interfacial crosssectional circle between the spherical
head and the taper cone is also derived to ensure the geometric continuity
of the model. The instantaneous feed rates in both the axial and the radial
directions are also derived based on the rotating speed of the cutter.
This study presents ideal method for the design and machining of helical
groove of ballend taper cutter with constant pitch, it is valuable for
reference.
ACKNOWLEDGMENT
The authors would like to thank the anonymous referees who kindly provide
the suggestions and comments to improve this study.
NOMENCLATURE
a 
= 
Distance between the origins O and O_{1} 
b 
= 
Spiral parameter 
E 
= 
Coefficients of first fundamental form (= r_{u}^{2}) 
F 
= 
Coefficients of first fundamental form (= r_{u}•r_{φ}) 
f(u) 
= 
Radius function 
G 
= 
Coefficients of first fundamental form (= r_{φ}^{2}) 
l 
= 
Length of line segment EF 
n 
= 
number of grooves 
R 
= 
Radius of the sphere 

= 
Outer radius of groove 
r 
= 
Equation of the revolving surface of special revolving cutter 
r 
= 
Inner radius of groove 
r_{1} 
= 
Equation of the sphere 
r_{1} 
= 
Radius of arc BC 
r_{2} 
= 
Equation of the cone 
r_{2} 
= 
Radius of arc CD 
r_{e} 
= 
Sectional profile of grinding wheel 
r_{j} 
= 
Contact curve between the profile of the grinding wheel and the
spiral groove 
r* 
= 
Equation of the helical groove 
S_{g} 
= 
Radial displacement of grinding wheel 
T 
= 
Pitch of spiral 
v_{g} 
= 
Radial feeding speed of grinding wheel 

= 
Actual modified radial feeding speed of grinding wheel 
v_{z} 
= 
Axial feeding speed of grinding wheel 
z_{1}, z_{2} 
= 
Parametric variable of z coordinate 
α 
= 
Angle of taper 
α_{e} 
= 
Clearance angle of cutting strip 
α_{E} 
= 
Angle of straight line EF to the Xaxis 
φ 
= 
Helical angle 
φ_{1} 
= 
Angular parameter of circle 
φ_{2} 
= 
Angular parameter of circle 
θ 
= 
Angular parameter of revolving surface 

= 
Angular parameter of revolving surface 
θ_{0} 
= 
Initial value of parameter φ 
θ_{1} 
= 
Angular parameter of the sphere 
θ_{2} 
= 
Angular parameter of the cone 
γ 
= 
Rrake angle of groove 
λ 
= 
Actual modified radial feeding speed parameter of grinding wheel 
λ_{1} 
= 
Length parameter 
λ_{2} 
= 
Length parameter 
λ_{3} 
= 
Length parameter 
μ_{1} 
= 
Length parameter 
μ_{2} 
= 
Length parameter 
μ_{3} 
= 
Length parameter 
ψ 
= 
Angular parameter 
σ 
= 
Coordinate system attached to the pinion cutter 
σ_{1} 
= 
Coordinate system attached to the grinding wheel 
σ´ 
= 
Temporary coordinate system; σ rotates 45° around zaxis

ω 
= 
Revolving angular velocity of the cutter 