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M. Mt 1
6th International Scientific and Expert Conference TEAM 2014
Technique, Education, Agriculture & Management
Kecskemt, November 10-11, 2014
THE MICRO-GEOMETRIC MODEL OF THE TOOTHFLANKS OF A CYLINDRICAL
GEAR WITH
ARCHIMEDEAN SPIRAL SHAPED TOOTHLINE M. Mt1,2
1Department of Mechanical Engineering, Faculty of Technical and
Human Sciences Tirgu-Mures, Sapientia University of Cluj-Napoca,
Romania
2Faculty of Agriculture and Engineering, College of Nyregyhza,
Hungary * Corresponding author e-mail: [email protected]
Abstract The present paper discusses a peculiar aspect of the
real meshing phenomenon that appears by cutting of a special type
of cylindrical gear having curved teeth. Here the flank line is
derived from an Archimedean spiral. Using this tooth line-shape the
load capacity of the gear pair will be significantly increased
while the torque transmission is realized contacting a concave and
a convex surface. The phenomenon of the meshing involves a special
generating rack that simultaneously executes a pulsing motion and a
slow tangential feed. The real meshing phenomenon is realized by
the partial surfaces that appear as trails of the cutting edges
during relative motion of the edge reported to the cut gear. The
paper shows the mathematical model and the density, the
displacement and the shape variation of these surfaces during the
cutting cycle.
Keywords: Gear, meshing, generating surface, geometry, curved
tooth
1. Introduction Cylindrical gear transmission is the most
frequently used in the machine industry (about 95%). Despite of
their very wide application range two types were widespread
applied: the right teethed and the helical teethed cylindrical gear
pair. The load capacity and the performance of the transmission can
be set at desired parameters by optimizing the profile correction
and addendum modification values as well as by localizing the
contact patch. However, external gears contacts on convex tooth
flanks. It is proved that load capacity can be improved if the
torque is transmitted between a concave and a convex surface. One
solution is given by the Wildhaber-Novikov gear pair, where a
concave and a convex tooth surface contacts when torque is
transmitted. This type of gear has proved its advantage regarding
the load capacity but presents also disadvantages because its
sensibility to the axis distance variation. However, this type of
gear were widely studied and optimized [1, 2] but its production
costs still remain high.
External cylindrical gear pair with curved teeth denotes a
solution where concave and convex tooth surfaces contact in order
to increase the load capacity, but the manufacturing costs
including the costs of the tooling and the technological system
setting remain at classical gear manufacturing costs level. The
principle of generating is deduced from Oliviers first principle
[3]. The coupling toothflanks are theoretically generated by the
same generating surface. The principle of generating is shown on
figure 1.
Os
ws
w1
ws
Os
Rs
O1
O2
w1
w2
Rs
m x
1m
x2
mq
P1B
C
H
A
a0
L L
L-L
s
Figure 1. The principle of generating
The theoretical generating surfaces are the carrying surfaces of
the cutting edges. Each
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2 THE MICRO-GEOMETRIC MODEL OF THE TOOTHFLANKS OF A CYLINDRICAL
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6th International Scientific and Expert Conference TEAM 2014
Technique, Education, Agriculture & Management
Kecskemt, November 10-11, 2014
cutters profile is identical with the profile of the involute
gear generating rack. The profiles are disposed in radial planes
and guided on an Archimedean spiral. In the lower image of figure 1
the sketch of the milling head is presented. It presents 3 groups
of 5 cutters for each. The cutters are disposed on identical
Archimedean spirals. When the milling head rotates, a moving rack
appears that cuts the teeth of the workpiece whose axis of rotation
passes through O1. Details regarding the mathematics of the surface
meshing process are given in [4, 5]. Focusing on figure 1 it
is to observe that the milling head realizes 30 Z
Archimedean generating surface-systems each one materializing a
tooth of a rack with curved teeth. Each one group of cutters
operate in different tooth spaces realizing the convex (left sided)
and the concave (right sided) toothflank. For one rotation of the
milling head the manufactured
gear rotates with an angle comprising 0Z angular
pitches. This relative motion involves a coupling between a
theoretical rack segment and the cut gear. The superposition of the
tangential feed on the main cutting motion leads to the model of
the pulsing rack. Pulsing motion is a high velocity speed of the
rack till it cuts in the space. When the analyzed space turns again
in the working position the pulsing rack is moved in another
position, finally the whole space will be generated.
2. The geometric model. Theoretical enveloping of the
toothflanks can be computed using a generalization of the classical
models and methods [1, 6]. In the reality, the meshed toothflank is
the union of the surfaces described by the cutting edges during the
relative motion. In the practice of gear generating there exist
realistic studies that emphasize the real aspect of the cut surface
[7]. However a mathematical model that predicts the possible
arrangement of the surface patches is necessary. Here the shape of
the surface is determined by the position of the cutting edge in
the group and by the position of the milling head meaning the axis
distance between this and the axis of the machined gear. The
simplified model of the relative position uses only the specific
coordinate-frames attached to the elements of the technological
gear (Figure 2). The following frames are used:
The 000 YYX fixed frame;
The sss zyx milling head attached frame;
The 111 zyx machined gears frame.
The reference position of the frames in indicated
with superior zero indices: 1;,000 sizyx iii . The
iizx planes are included in the median section of the cut gear
whose width is denoted with B . P is
the pole of the virtual gear pair consisting of the theoretical
rack and the machined gear.
mx
Aw
O1rd
x1(0)
y1(0)
z1(0)
P
Oxs(0)
ys(0)
zs(0)
X0
Y0
Z0
Rs
D
Figure 2. The used coordinate frames
One group of cutter contains a number of 5sz
cutters. The central one is considered the referential cutter.
Its symmetry point (middlepoint of the toothwidth segment on the
racks pitch line)
is positioned at a distance sR from the milling
heads axis. It is denoted through index 0I . As a logical
consequence the set of the indices is
2;1;0;1;2 I . The first insert with index value 2I has the
shortest radius, and the last the
longest. The pitch of the Archimedean spiral is due
to the gearing dependences 0Z times the standard
rack pitch. The parameter of the spiral line becomes
2/2/ 00 mZmZpsp (1)
Considering that 0/2 Zs is the theoretical
central angle occupied by the cutter group and the
axis sx is set on the reference cutter, the minimal
and the maximal radius become:
22
maxmin
mRpR ssp
ss
(2)
The angular pitch between two consequent cutters of the group
is:
1N
s (3)
Using the index value the angular distance of an arbitrary
cutter to the first cutter of the group is:
II D 2 (4)
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6th International Scientific and Expert Conference TEAM 2014
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Kecskemt, November 10-11, 2014
Using (2) and (4) the reference radius of all cutters can be
written as
Is
I mZR D2
00 (5)
The analysis of the micro-geometry requires a precise relative
positioning of the generating surfaces swept by the edges.
Generating surfaces are considered only within the limit planes of
the cut gear. Starting position of the frames is always considered
that position of the cut gear and the milling head where the
reference point A of the first cutter (index -2) reaches the gear
limit plane of
equation 2/BYs as shown in figure 3. This is
valid for any distance of the milling head's axis that indicates
the position of the rack during the cutting process (figure 1).
O
xs
ys X0
Y0
A
j k
B
B/2
B/2
A
j b
A1
m/2
Os
j b
Figure 3. The angular distance and the
coverage angle of an arbitrary edge
Coverage angle is the value of the angular motion of the milling
head till the analyzed cutters edge passes through the width of the
machined gear.
For example, first edges coverage angle is bj2 .
Reference point A of the arbitrary edge reaches the limit plane
in A1. The coverage angle value is
bj2 . The beginning of the arbitrary edge swept
surface happens after a rotation of the milling head
of value where
0minmin 2arcsin
2arcsin
mZ
BB
(6)
While the milling head rotates it also execute the tangential
feed. Here an analogy with the gear shaping technology is needed.
It can be accepted that the length of the milling heads shift while
the gear executes one complete rotation is equal to the circular
feed s. The feed ratio will here be defined as the length of the
shift that corresponds to a rotation of 1 radian:
1
0*
2 z
Zss
(7)
The gearing ratio is computed considering a perfect rolling
between the centroids of the rack and the gear. Equalizing the
tangential velocities it can be written that
d
sp
s
ssspsdtr
spisprv
*
11
*1
w
wwww (8)
The limit positions of the milling head during the cutting of a
tooth space is approximated with the model of rack-involute gear
pair as shown in figure 4.
O1
x1(0)
A B
mx
P
P1
C
DT
E
rb
rdra
Pitch line
Rolling line
z1(0)
x1
z1
x1'
z1'
Beginning of meshing End of
meshing
Figure 4. The limit positions of the meshing
Using figure 4 the 11j and 12j values can be
easily computed. The complete length of the feed shift is given
by
1211 jj D dr (9)
Finally the geometrical dependence of the moving frames must be
determined. Lets consider the frames shown in Figure 5.
O1
x1(0)
zs
x*s
Os
z1(0)
X0
z1'
Start of rolling
Z0
O
X0
Y0
O
y*s
O*s
xs
x1'
Rsrd j 11
O1
x1'y1
z1'
Reference position
0z*s
x*sy*s
A
j b
Start of first cut
x*1z*1
z*1 x*1
B/2
Figure 5. The relative positions of the frames when first cutter
starts.
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4 THE MICRO-GEOMETRIC MODEL OF THE TOOTHFLANKS OF A CYLINDRICAL
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6th International Scientific and Expert Conference TEAM 2014
Technique, Education, Agriculture & Management
Kecskemt, November 10-11, 2014
Comparing the frames shown in figure 2 with those shown in
figure 5 it can be concluded that the mills frame must be rotated
counterclockwise by an
angle bss j 2/ while point A of the first cutter
reaches the gears limit plane. Due to the rolling it
must be also shifted with the distance ss *
0 .
Frame of cut gear is also rotated ccw. by
ssi j 111 . If another cutter is considered than
the milling head must be rotated clockwise with a supplementary
setting angle computed from (4) and (6):
0minmin 2arcsin
2arcsin
mZ
BB
i
isi
DD
(10)
Supplementary rotation is needed to correct the
gears position too, reaching sisi i 11 . These
are the start angular values when the cutters work at the
beginning of the rolling segment. After n revolutions of the cut
gear the mills axis is shifted
right with a distance *2 sns . In order to avoid
the senseless complication of the model lets
consider s a continuous variable. For this
position of the mills axis the cut gear must be
rotated clockwise with ds r/ . Based on the
statements above the angular positions of the frames for an
arbitrary cutter and an arbitrary rolling position can be computed
as follows:
dssiis
sibss
rixx
xx
/,
2/,
11110
11
0
j
j
j
(11)
For an arbitrary position of the cutting edge during the
sweeping of the generating surface,
considering principal parameter the rotation sj of
the milling head the expressions (11) become the general
form
ssdssiis
ssibsss
irixx
xx
jj
jj
j 1111110
11
0
/,
2/,
(12)
The transformation matrix between the mills and the cut gears
frame is:
1000
sincossinsincossin
00cossin
cossinsincoscoscos
010111
010111
1zx
xx
ss
ss
ss
ss
ss
s
M
(13)
where constants of the fourth column are computed as
follows:
11
01
110
1
*0110
cossin
sincos
jj
ARz
ARx
srx
s
s
ssds
(14)
3. The equations of the cutting edges The equations of the
cutting edges relative to the frame of the milling head are
computed using figure 6. The equations are written first in the
auxiliary frame 222 zyx , followed by a rotation. The
concave and the convex sided edges can be
A10
A
A1,-2
Os
Os
ys
x2
z2
Kv
Kx
xs
x2
y2
A
spiral
Figure 6. The cutting edges in the frame of the milling
head.
written using a single equation system.
0
0
uuz
pRuauy
pRuaux
T sps
spss
KxKv
2
2, sintg;
costg;
: a
a
(15)
The concave flanks parametric equations will be
obtained for 1 , as appropriate the convex flank results for 1
.
4. The equations of the swept surfaces Using the elements of the
model presented above, the surfaces swept by the edges can
considered as the simple infinity of edges in the frame of the
cut gear determined by the values of parameter sj
. The equations are written using matrix (13) and equations (15)
within the matrix transformation
ss rMr 11 (16)
5. Computer simulation results The simulation was realized in
MathCad 15 environment, for the following initial values:
module 5m mm;
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M. Mt 5
6th International Scientific and Expert Conference TEAM 2014
Technique, Education, Agriculture & Management
Kecskemt, November 10-11, 2014
rack profile angle 200a ;
teeth number 411 z ;
profile correction 15.0x ;
gear width 40B mm;
spiral reference radius 80sR mm;
number of cutter groups 30 Z ;
number of cutters in the group 5sz ;
circular feed 3.0s mm/rev.workpiece. The length of the rolling
segment is 40.702 mm on which a 9 segment equidistant division was
considered. This marks 10 equidistant points including the ends of
the rolling length. Each point present a possible position of the
milling heads axis when the generating process begins with the
first edge. The program computes the coordinates depending on the
cutters index value and the milling head position. Two types of
representations were done. The first type shows the succession of
the surfaces generated successively by each cutter of the group.
The second representation tries to answer the successive positions
of the surfaces swept by the same cutter, during the linear motion
of the milling head due to the tangential feed. First type of
representation is shown in figures 7-12. Figures 7, 9 and 11
contain the surfaces swept by the edges from the concave side,
while figures 8, 10 and 11 those that appear on the convex
side.
x1
y1
z1
kv1
kv2kv3
kv4
kv5
Figure 7. The swept surfaces on the concave side at the
beginning of the rolling distance.
On any of the figures 5..1, iikv denotes the
concave surface swept by the edge of the cutter i ,
where 1i marks the first cutter of the group. In
analogy 5..1, iikx is the notation for the convex
side.
x1
y1
z1
kx1
kx2
kx3
kx4
kx5
Figure 8. The swept surfaces on the convex side at the beginning
of the rolling distance.
x1
y1
z1
kv1
kv2
kv3
kv4
kv5
Figure 9. The swept surfaces on the concave side at the middle
of the rolling distance.
x1y1
z1
kx1
kx2kx3 kx4
kx5
Figure 10. The swept surfaces of the convex side at the middle
of the rolling distance
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6 THE MICRO-GEOMETRIC MODEL OF THE TOOTHFLANKS OF A CYLINDRICAL
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6th International Scientific and Expert Conference TEAM 2014
Technique, Education, Agriculture & Management
Kecskemt, November 10-11, 2014
x1
y1
z1
kv1
kv2
kv3
kv4
kv5
Figure 11. The swept surfaces of the concave side at the end of
the rolling distance
x1y1
z1
kx1
kx2
kx3
kx4
kx5
Figure 12. The swept surfaces of the convex side at the end of
the rolling distance
Analyzing figures 7-12 it is to conclude that the considered
surfaces present a very various arrangement depending on the
position of the milling heads axis during the rolling. Near the
ends of the rolling segment the surfaces are distanced, but in the
middle zone of the rolling segment the distance between them
becomes significantly small. Generally it can be admitted that
surfaces swept by two consequent edges doesnt intersect. It is not
possible to write an approximant for the meshing surface using the
intersection curves between the consequent swept surfaces. In some
situations it exist the intersection between two consequent
surfaces but this cannot be considered
x1
y1
z1
Kv(-2;0)
Kv(-2;3)
Kv(-2;6)
Kv(-2;9)
Kv(-2;12)
Kv(-2;16)
Kv(-2;18)
Figure 13. The successive surfaces swept by the concave edge of
the first cutter of the
group for different rolling positions
x1
y1
z1
Kx(-2;0)
Kx(-2;3)
Kx(-2;6)
Kx(-2;9)
Kx(-2;12)
Kx(-2;16)
Kx(-2;18)
Figure 14. The successive surfaces swept by the convex edge of
the first cutter of the group
for different rolling positions
as a certainty. A similar arrangement of the surfaces can be
observed when analyzing the second representation. The step was set
on 3 mm, corresponding to 10 complete rotations of the workpiece.
The results of the computing are presented in figures 13-18. The
annotations for the surfaces are similar to the precedent, but they
are completed with a parenthesis, where first number signify the
index of the cutter that was used in the formulas above, and the
second the distance covered by the milling head due to the
tangential
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M. Mt 7
6th International Scientific and Expert Conference TEAM 2014
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Kecskemt, November 10-11, 2014
x1
y1
z1
Kv(0;3)
Kv(0;6)
Kv(0;9)
Kv(0;12)
Kv(0;16)
Kv(0;18)Kv(0;0)
Figure 15. The successive surfaces swept by the concave edge of
the central cutter of the
group for different rolling positions
x1
y1
z1
Kx(0;0)
Kx(0;3)
Kx(0;6)
Kx(0;9)
Kx(0;12)
Kx(0;16)
Kx(0;18)
Figure 16. The successive surfaces swept by the convex edge of
the central cutter of the
group for different rolling positions
feed. For example, 12,2kv signify the surface swept by the
concave (here: the right sided) edge
of the first cutter in the group ( 2I ).
It is to conclude that the arrangement of the surfaces depends
on the position of the cutter in the group, and differs from the
concave to the convex side. Analyzing figure 13 it is to
concludeThat surfaces swept by the concave edge of the first cutter
in the group are quasi linear disposed considering any
intersections at the exit side, when edges leave the space of the
cut gear. But the convex edge of the same cutter produces more
distanced surfaces. Comparing figures 15 and 16 a similarity appear
regarding the arrangement, while surfaces are interlocked.
x1
y1
z1
Kv(2;3)
Kv(2;6)
Kv(2;9)
Kv(2;12)
Kv(2;16)
Kv(2;18)
Kv(2;0)
Figure 17. The successive surfaces swept by the concave edge of
the last cutter of the
group for different rolling positions
x1
y1
z1
Kx(2;0)
Kx(2;3)
Kx(2;6)
Kx(2;9)
Kx(2;12)
Kx(2;16)
Kx(2;18)
Figure 18. The successive surfaces swept by the convex edge of
the last cutter of the group
for different rolling positions
Finally, comparing figures 17 and 18 it can be noticed that last
cutters edges produce similar arrangement as shown in figures 13
and 14. In opposition to those, here surfaces swept by the convex
side are closer arranged that at the concave side.
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8 THE MICRO-GEOMETRIC MODEL OF THE TOOTHFLANKS OF A CYLINDRICAL
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6th International Scientific and Expert Conference TEAM 2014
Technique, Education, Agriculture & Management
Kecskemt, November 10-11, 2014
6. Conclusion Computer simulation results demonstrates that the
enveloping surfaces manifold is more complex in comparison with
classical situations (gear shaping with rack, with cutter, gear
hobbing). Due to the two component based kinematic of the cutting
tool (first the pulsing rack simulated by the cutter group and the
second the tangential motion of the milling head) the virtual rack
that realizes finally the meshing changes its dimensions, the form
and the curvature of the generating surface. As a consequence
situations of ante and posterior undercut can appear. For example,
lets consider that a curve segment situated on the holder surfaces
of the cutting edges (this is an Archimedean surface, [4]) fulfills
the law of gearing [1, 3, 6, 7] when the milling head is located in
some place of the rolling line. Due to the changing of the relative
position of the edge swept surfaces during the tangential feed the
meshing segment that was calculated before can be eliminated during
the next cutting process. Or, it can be considered a segment of
curve that fulfills the low of gearing but it was cut down in a
precedent cutting phase. Despite of this inconvenience the
Archimedean spiral curved teeth match the requirements of correct
coupling due to the law of Willis and Oliviers first principle
regarding the common generating surfaces. The correct calculus of
the tooth flank form requires besides the applying of the law of
gearing a numerical checking of the existence of the theoretically
computed points. This can be performed through using the surfaces
swept by edge as limit surfaces of the tooth space. Here a
CAD-program is indispensable.
7. Acknowledgement This research was supported by the European
Union and the State of Hungary, co-financed by the European Social
Fund in the framework of TMOP-4.2.4.A/2-11/1-2012-0001 National
Excellence Program. A kutats a TMOP-4.2.4.A/2-11/1-2012-0001
azonost szm Nemzeti Kivlsg Program Hazai hallgati, illetve kutati
szemlyi tmogatst biztost rendszer kidolgozsa s mkdtetse
konvergencia program cm kiemelt projekt keretben zajlott. A projekt
az Eurpai Uni tmogatsval, az Eurpai Szocilis Alap trsfinanszrozsval
valsul meg.
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[2] S.M. Nacy, M.Q.Abdullah, M.N. Mohammed Generation of Crowned
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[6] I. Dudas, K. Banyai, G.Varga Simulation of meshing of worm
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