Abstract: This study presents the design models of the cutting edge and groove of the ball-end taper cutter with constant pitch. The section profile and relative feeding speeds of the grinding wheel in the NC machining of the cutter are deduced. The problems of remaining revolving surface and no cutting edge strip on the tip part of the ball-end of cutter are post-processed according to the computer simulation result of the actual groove surface. A ball-end taper cutter with ideal grooves may be obtained.
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. 3-axis or 4-axis 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 ball-end 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 post-process) of ball-end 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 2-axis NC machining so as to decrease the manufacturing cost of cutter. Only about one-seventh of the cutting edge is planar curve at the tip part of the ball-end 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 ball-end 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:
| (1) |
For the cutting edge of cutter with a constant pitch, the helical lead
is also constant, i.e.,
| (2) |
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:
| (3) |
where, b is the spiral parameter and
| (4) |
In order to obtain the cutting edge on the surface of Eq.
1, calculate the partial differential of the surface as following:
| (5) |
| (6) |
The coefficients of first fundamental form is as follows:
| (7) |
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:
| (8) |
| (9) |
Equation 5 can be obtained:
| (10) |
| Fig. 1: | Ball-end taper cutter |
In integration form, Eq. 10 becomes:
| (11) |
where, θ0 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 ball-end 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:
| (12) |
| Where: | ||
| R | = | Radius of the sphere |
| θ1 | = | Angular parameter and θ1 ∈[0,2π] |
| z1 | = | Parametric variable and z1 ∈[–R, –Rsin α] |
Corresponding to Eq. 12 and 1, we
have:
| (13) |
Corresponding to Eq. 10, we can obtain that:
| (14) |
After integration, Eq. 12 becomes as follows:
| (15) |
This is because that the initial θ1 = 0 when z1
= 0, so θ0 = 0, Eq. 15 may be obtained.
At the conjunction position, substitute z1 = R sin α into
Eq. 15, we have:
| (16) |
By referring to Fig. 1, the equation of cone may be
expressed as:
| (17) |
where z2 ∈[0,h]
| (18) |
Corresponding to Eq. 10, it is known that
| (19) |
| (20) |
Substitute the boundary condition that z2 = 0 and θ2
= θ20 = θ10 into the above equation, we
can obtain that:
| (21) |
By substituting Eq. 21 into Eq. 17 and 16 into Eq. 12, the helical angle of the ball-end 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 ball-end 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 ball-end decreases with increasing z-coordinate, the depth of the groove should also gradually decrease accordingly. Therefore, on the ball-end 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 ball-end part is formed by the enveloping surface of the disc-shaped 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:
| (22) |
where,
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 r1, then Q is on the line paralleling to the straight
line AB and with a distance r1. Q is also on the circle centered
at point O with radius r + r1. In this way, the coordinate
of point Q may be defined by the following equations:
| (23) |
where,
| (24) |
Since the arc BC is tangent to straight line AB at point B, the coordinate
of point B may be defined as:
| (25) |
By substituting Eq. 22 and 24 into
Eq. 25, λ1 and φ2 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,
| (26) |
where, λ2 is the length parameter for describing any
arbitrary point on segment EF and αe is the angle between
X-axis and segment EF at point F. The coordinate of point E may be defined
by the length l of line EF as:
| (27) |
Let straight line DE is at an angle αE to x-axis, the
equation of straight line DE can be obtained as:
|
|
(28) |
where, μ1 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 r2.
The center P of arc CD is on the line paralleling to the straight line
DE at a distance r2. P is also on the circle, which is centered
at point Q and with a radius of r + r2. 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.,
| (29) |
where, φ1 and μ2 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:
| (30) |
The coordinate of the tangent point C between arcs BC and CD are given
as:
| (31) |
After solving for ψ and φ2 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:
| (32) |
where, μ1 and φ2 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:
| (33) |
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:
| (34) |
where, b is the pitch of the spatial helical curve and can be expressed
as:
b = Rcotφ |
(35) |
DESIGN FOR THE SECTION PROFILE OF THE GRINDING WHEEL
A new coordinate system σ1 = [o1;x1,y1,z1]
is attached to the grinding wheel, as shown in Fig. 3.
Here, the x1 axis lies on the xoy plane having an angle of
30° counted from the axis x. O1 is defined by the rake
angle γ and arc bc. The centerline axis of the grinding wheel lies
on the z1 axis passing through o1. Here, o1z1
and oz are straight lines on different planes and o1o is the
common normal line. The equation of o1z1 in coordinate
system σ can be expressed as:
| (36) |
where, φ is the helical angle and a is the distance between the
origins O and O1 of both σ = [O:X,Y,Z] and σ1
= [O1;X1,Y1,Z1] 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 z1. 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:
| (37) |
| Fig. 3: | Relative coordinate systems of σ and σ1 |
By equating three components respectively to obtain
| (38) |
| (39) |
| (40) |
From Eq. 40, one can easily obtain the parameter μ2
as:
| (41) |
By substituting the above equation into Eq. 39, the
parameter λ3 is given as:
| (42) |
After substituting the above two equations into Eq. 38,
the relation can be obtained as:
| (43) |
By using the property of spiral surface, y*Nx*–x*Ny*
= bNz*, the above relation becomes:
| (44) |
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:
| (45) |
In order to obtain the profile of the grinding wheel, the above equation
of the contact curve is transformed to the coordinate system σ1,
which is attached on the grinding wheel. In order to do so, a new coordinate
system σ´ = └o1;x1,y´,z┘
is introduced. The transformation from coordinate system σ to coordinate
system σ´ is given as:
| (46) |
Since the origin o1 and both of x1 and z1
axes in coordinate system σ1 are already defined, the
direction of y1 axis can be easily obtained by using the right-hand
rule. The transformation from coordinate system σ´ to coordinate
system σ1 can then be expressed as:
| (47) |
Thus, the equation of the contact curve can now be expressed in terms
of σ1 coordinate system as:
| (48) |
As the contact curve is rotated around the z1 axis, the profile
surface of the grinding wheel can be computed. The intersecting curve
between the profile surface and the plane y1 = 0 is the profile
curve of the grinding wheel. This curve can be expressed as:
| (49) |
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 ball-end portion of the cutter. To achieve better dimensional accuracy at the ball-end, 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 two-axis NC machining, let angular speed ω
be constant, the speed vz along the axial direction can be
calculated by:
| (50) |
At the ball-end of the cutter, the speed of grinding wheel along the
axial direction can be defined as:
| (51) |
For the cone surface, the speed of grinding wheel along the axial direction
can be defined as:
| (52) |
At the conjunction place, z1 = -Rsinα for surface (1)
and z2 = 0 for surface (17), substitute the above boundary
conditions into Eq. 51 and 52, respectively,
we have:
| (53) |
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, over-cut will happen at the part of
| (54) |
Accordingly, the general equation of feeding speed at the radial direction
is as follows:
| (55) |
At the ball-end of cutter, the radial speed of grinding wheel is as follows:
| (56) |
For the cone surface, the speed of radial motion can be defined as:
| (57) |
Substitute the conjunction condition into Eq. 56 and
57, we can get:
| (58) |
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:
| (59) |
We can get two intersect points of section circle at the position of z* = g(u), let the point on the cutting edge be (x1*,y1*,z1*), the other point be (x2*,y2*,z2*). The actual obtained cutting edge is at the outside of designed cutting edge except the tip of ball-end, 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 ball-end, over-cut exists. So remedy is necessary.
The actual groove section at position of z = g(u0) may be
obtained by different value of xc. It is obvious that
the cutting edge strip decreases gradually with the decrement of radius.
At the tip part of the ball-end, 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 ball-end should be modified so that the over-cut can be diminished.
| Fig. 4: | Relative position between grinding wheel and top of the cutter |
| Fig. 5: | Define the factor λ |
The radius
| (60) |
| (61) |
Otherwise, the cutting edge adopts planar curve when f<ro. Examples verify that the planar curve of the cutting edge at the tip of ball-end only takes about one-seventh of the cutting edge at the ball-end of the cutter.
The problem of remaining surface and no strip may be remedied by the
following approach. A cone-shaped grinding wheel with a bottom angle of
90°–αe 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 |
| (62) |
the factor λ may be defined. Then calculate the revised radial speed
by:
| (63) |
| Fig. 8: | The desired and actual (dark line) obtained cutting
edge curve (dimension: mm) (a) Projected cutting edge curve of the
cutter on X-Y plane (b) Spatial cutting edge curve of the cutter |
Keep axial speed and revolving speed unchanged, replace radial speed
vg with
A ball-end 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 r1 = 2 mm, the radius of groove arc CD of r2
= 4 mm, the length of the cutting edge strip of l = 1 mm, the clearance
angles of cutting strip of αe = 6° and αE
= 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, z1∈[-6,-1.042] mm, z1∈[0,11.860]
mm, the distance between the origins O and O1 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 ball-end 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 Sg of grinding wheel |
| Fig. 10: | Te actual produced ball-end 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 ball-end taper milling cutter cutting edge curve. Figure 9 shows the feeding amount Sg of grinding wheel. Figure 10 shows the actual produced ball-end taper milling cutter.
CONCLUSIONS
This study presents a modeling method that can be used to design and manufacture a ball-end taper milling cutter by using a simple two-axis NC machine. From the related models of design and NC machining and the simulated results, it is known that the general models for the ball-end taper cutter are presented in this study. The validity of machining this kind of cutter by two-axis NC machining is verified by example. The surface shape around the interfacial cross-sectional 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 ball-end 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 O1 |
| b | = | Spiral parameter |
| E | = | Coefficients of first fundamental form (= ru2) |
| F | = | Coefficients of first fundamental form (= ru•rφ) |
| f(u) | = | Radius function |
| G | = | Coefficients of first fundamental form (= rφ2) |
| 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 |
| r1 | = | Equation of the sphere |
| r1 | = | Radius of arc BC |
| r2 | = | Equation of the cone |
| r2 | = | Radius of arc CD |
| re | = | Sectional profile of grinding wheel |
| rj | = | Contact curve between the profile of the grinding wheel and the spiral groove |
| r* | = | Equation of the helical groove |
| Sg | = | Radial displacement of grinding wheel |
| T | = | Pitch of spiral |
| vg | = | Radial feeding speed of grinding wheel |
| = | Actual modified radial feeding speed of grinding wheel | |
| vz | = | Axial feeding speed of grinding wheel |
| z1, z2 | = | Parametric variable of z coordinate |
| α | = | Angle of taper |
| αe | = | Clearance angle of cutting strip |
| αE | = | Angle of straight line EF to the X-axis |
| φ | = | Helical angle |
| φ1 | = | Angular parameter of circle |
| φ2 | = | Angular parameter of circle |
| θ | = | Angular parameter of revolving surface |
| = | Angular parameter of revolving surface | |
| θ0 | = | Initial value of parameter φ |
| θ1 | = | Angular parameter of the sphere |
| θ2 | = | Angular parameter of the cone |
| γ | = | Rrake angle of groove |
| λ | = | Actual modified radial feeding speed parameter of grinding wheel |
| λ1 | = | Length parameter |
| λ2 | = | Length parameter |
| λ3 | = | Length parameter |
| μ1 | = | Length parameter |
| μ2 | = | Length parameter |
| μ3 | = | Length parameter |
| ψ | = | Angular parameter |
| σ | = | Coordinate system attached to the pinion cutter |
| σ1 | = | Coordinate system attached to the grinding wheel |
| σ´ | = | Temporary coordinate system; σ rotates 45° around z-axis |
| ω | = | Revolving angular velocity of the cutter |