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DOI 10.1007/s00170-003-1741-8
ORIGINAL ARTICLE
Int J Adv Manuf Technol (2004) 24: 789–793
Feng Xianying · Wang Aiqun · Linda Lee
Study on the design principle of the LogiX gear tooth profile
and the selection of its inherent basic parameters
Received: 2 January 2003 / Accepted: 3 March 2003 / Published online: 3 November 2004
? Springer-Verlag London Limited 2004
Abstract The development of scientific technology and productivity
has called for increasingly higher requirements of gear
transmission performance. The key factor influencing dynamic
gear performance is the form of the meshed gear tooth profile. To
improve a gear’s transmission performance, a new type of gear
called the LogiX gear was developed in the early 1990s. However,
for this special kind of gear there remain many unknown
theoretical and practical problems to be solved. In this paper, the
design principle of this new type of gear is further studied and
the mathematical module of its tooth profile deduced. The influence
on the form of this type of tooth profile and its mesh
performance by its inherent basic parameters is discussed, and
reasonable selections for LogiX gear parameters are provided.
Thus the theoretical system information about the LogiX gear are
developed and enriched. This study impacts most significantly
the improvement of load capacity, miniaturisation and durability
of modern kinetic transmission products.
Keywords Basic parameter · Design principle · LogiX gear · Minute involute · Tooth profile
1 Introduction
In order to improve gear transmission performance and satisfy
some special requirements, a new type of gear [1] was put forward;
it was named “LogiX” in order to improve some demerits
of W-N (Wildhaver-Novikov) and involute gears.
Besides having the advantages of both kinds of gears mentioned
above, the new type of gear has some other excellent
F. Xianying (_) · W. Aiqun
School of Mechanical Engineering,
Shandong University,
P.R. China
E-mail: FXYing@sdu.edu.cn
Tel.: +86-531-8395852(0)
L. Lee
School of Mechanical & Manufacturing Engineering,
Singapore Polytechnic,
Singapore
characteristics. On this new tooth profile, the continuous concave/
convex contact is carried out from its dedendum to its addendum,
where the engagements with a relative curvature of zero
are assured at many points. Here, this kind of point is called the
null-point (N-P). The presence of many N-Ps during the mesh
process of LogiX gears can result in a smaller sliding coefficient,
and the mesh transmission performance becomes almost
rolling friction accordingly. Thus this new type of gear has many
advantages such as higher contact intensity, longer life and a
larger transmission-ratio power transfer than the standard involute
gear. Experimental results showed that, given a certain
number of N-Ps between two meshed LogiX gears, the contact
fatigue strength is 3 times and the bend fatigue strength 2.5 times
larger than those of the standard involute gear. Moreover, the
minimum tooth number can also be decreased to 3, much smaller
than that of the standard involute gear.
The LogiX gear, regarded as a new type of gear, still presents
some unsolved problems. The development of computer numerical
controlling (CNC) technology must also be taken into consideration
new high-efficiency methods to cut this new type of
gear. Therefore, further study of this new type of gear most
significantly impacts the acceleration of its broad and practical
application. This paper has the potential to usher in a new era in
the history of gear mesh theory and application.
2 Design principle of LogiX tooth profile
According to gear mesh and manufacturing theories, in order to
simplify problem analysis, generally a gear’s basic rack is begun
with some studies [2]. So here let us discuss the basic rack of
the LogiX gear first. Figure 1 shows the design principle of divided
involute curves of the LogiX rack. In Fig. 1, P.L represents
a pitch line of the LogiX rack. One point O1 is selected to form
the angle ?n0O1N1 =α0, P.L ? O1N1. The points of intersection
by two radials O1n0 and O1N1 and the pitch line P.L are N1
and n0. Let O1n0 = G1, extend O1n0 to O_
1 , and make two tangent
basic circles whose centres are O1, O_1 and radii are equal
to G1.. The point of intersection between circle O1 and pitch line
790
Fig. 1. Design principle of LogiX rack tooth profile
P.L is n0. The point of intersection between circle O2 and pitch
line P.L is n1. Make the common tangent g1s1 of basic circle O1
and O_1, then generate two minute involute curves m0s1 and s1m1
whose basic circle centres are O1 and O_1. The radii of curvature
at points m0 and m1 on the tooth profile should be: ρm0 = m0n0,
ρm1 = m1n1, and the centres are met on the pitch line.
Multiple different minute involutes consisting of a LogiX
profile should be arranged for a proper sequence. The pressure
angle of the next minute involute curve m1m2 should have an
increment comparable to its last segment m0m1. The centres of
curvature at extreme points m1, m2, etc. should be on the pitch
line, and the radius of the basic circle is a function of pressure [1]
– it varies from G1 to G2. The condition for joining front and rear
curves is that the radius of curvature at point m1 must be equal
to the radius of curvature just after point m1, and the radius of
curvature at point m2 must be equal to the radius of curvature
just after point m2. Figure 2 shows the connection and process of
generating minute involute curves. According to the above discussion,
the whole tooth profile can be formed.
Fig. 2. Connection of minute involute curves
3 Mathematicmodule of LogiX tooth profile
3.1 Mathematic module of the basic LogiX rack
According to the above-mentioned design principle, the curvature
centre of every finely divided profile curve should be located
at the rack pitch line, and the value of the relative curvature at
every point connecting different minute involute curves should
be zero. The design of the tooth profile is symmetrical with respect
to the pitch line, and the addendum is convex while the
dedendum is concave. Thus for the whole LogiX tooth profile, it
can be dealt with by dividing it into four parts, as shown in Fig. 3.
Set up the coordinates as shown in Fig. 4, where the origin of
the coordinates O coincides with the point of intersection m0 between
rack pitch line P.L and the initial divided minute involute
curve.
According to the coordinates set up in Fig. 4, the formation
of initial minute involute curve m0m1 is shown in Fig. 5.
Fig. 3. LogiX rack tooth profile
Fig. 4. Set-up of coordinates
Fig. 5. Formation process of initial minute involute curve m0m1
791
Here: n0n_0 ? O1O_1 , n1n_1 ? O1O_1 , n1n1 ?n0n_0, and the parameters
α0, δ, G1 and ρm0 are given as initial conditions. The
curvature radius of the involute curve at point s1 is ρs1 = G1δ, or
ρs1 = ρm1+G1δ1. Thus the curvature radius and pressure angle
of the minute involute curve at point m1 are as follows:
ρm1 = ρs1?G1δ1 = G1(δ?δ1) (1)
α1 = α0+δ+δ1 . (2)
According to the geometrical relationship, we can deduce:
tg(α0+δ) =
2G1?G1 cos δ?G1 cos δ1
G1 sin δ?G1 sin δ1
=
2?(cos δ+cos δ1)
sin δ?sin δ1
. (3)
Based on Eqs. 1, 2 and 3 and the forming process of the LogiX
rack profile, the curvature radius formula of an arbitrary point on
the profile is deduced: ρmi =ρmi?1+Gi(δ?δi ). When i =k and
ρm0 = 0?, it is expressed as follows:
ρmk = G1(δ?δ1)+G2(δ?δ2)+· · ·+Gk(δ?δk)
=
k
_i=1
Gi(δ?δi) . (4)
Similarly, the pressure angle on an arbitrary k point of the tooth
profile can be deduced as follows:
αk = α0+(δ+δ1)+(δ+δ2)+· · · (δ+δk)
= α0+
k
_i=1
(δ+δi) = α0+kδ+
k
_i=1
δi . (5)
By ni?1ni = Gi(sin δ?sin δi)/ cos(αi?1 +δ), Eq. 5 can be
obtained:
n0nk =
k
_i=1
ni?1ni =
k
_i=1
Gi(sin δ?sin δi )
cos(αi?1 +δ)
. (6)
Thus the mathematical model of the No. 2 portion for the LogiX
rack profile is as follows:
_x1 = n0nk ?ρmk cos αk
y1 = ρmk sin αk
(No. 2) . (7)
Similarly, the mathematical models of the other three segments
can also be obtained as follows:
_x1 =?(n0nk ?ρmk cos αk)
y1 =?ρmk sin αk
(No.1) (8)
_x1 = s?(n0nk ?ρmk cos αk)
y1 = ρmk sin αk
(No.3) (9)
_x1 = s+n0nk ?ρmk cos αk
y1 =?ρmk sin αk
(No.4) . (10)
Fig. 6. Mesh coordinates
of LogiX gear and its basic
rack
3.2 Mathematical module of the LogiX gear
The coordinates O1X1Y1, O2X2Y2 and PXY are set up as shown
in Fig. 6 to express the mesh relationship between the LogiX
rack and the LogiX gear. Here, O1X1Y1 is fixed on the rack, and
O1 is the point of intersection between the rack tooth profile and
its pitch line. O2X2Y2 is fixed on the meshed gear, and O2 is the
gear’s centre. PXY is an absolute coordinate, and P is the point
of intersection of the rack’s pitch line and the gear’s pitch circle.
In accordance with gear meshing theories [3], if the above
model of the LogiX rack tooth profile is changed from coordinate
O1X1Y1 to OXY, and then again to O2X2Y2, a new type of gear
profile model can be deduced as follows:
_x2 =?ρmk cos αk cos ?2 ?(ρmk sin αk ?r2) sin ?2
y2 =?ρmk cos αk sin ?2 +(ρmk sin αk ?r2) cos ?2 .
(11)
Here the positive direction of ?2 is clockwise, and only the model
of the LogiX gear tooth profile in the first quadrant of the coordinates
is given.
4 Effect on the performance of the LogiX gear by its
inherent parameters and their reasonable selection
Besides the basic parameters of the standard involute rack, the
LogiX tooth profile has inherent basic parameters such as initial
pressure angle α0, relative pressure angle δ, initial basic circle
radius G0, etc. The selection of these parameters has a great influence
on the form of the LogiX tooth profile, and the form
directly influences gear transmission performance. Thus the reasonable
selection of these basic parameters is very important.
4.1 Influence and selection of initial pressure angle α0
Considering the higher transmission efficiency in practical design,
the initial pressure angle α0 should be selected as 0?. But
the final calculation result showed that the LogiX gear tooth profile
cut by the rack tool whose initial pressure angle was equal
to zero would be overcut on the pitch circle generally. Thus the
initial pressure angle α0 cannot be zero. Comparing the relative
double circle-arc gear [3], we can also deduce that the smaller
792
the initial pressure angle α0, the larger the gear number for producing
the overcut. Thus the initial pressure angle α0 should
not only not be zero, but should not be too small, either. From
Eqs. 3, 4 and 5, the influence of α0 on the LogiX tooth profile
can be directly described by Fig. 7. Obviously, increasing the initial
pressure angle will cause the curvature of the LogiX rack
tooth profile to become larger. If the rack selects a larger module
and too small an initial pressure angle α0, its addendum will
become too narrow or even overcut. Thus the LogiX tooth profile
that selects a larger module should select a smaller α0, and
the profile that selects a smaller module should select a larger
α0. Generally, by practical calculation experience, the selected
α0 should be located within a range of 2? ～ 12?, and the larger
the LogiX gear module, the smaller should be its initial pressure
angle α0.
4.2 Influence and selection of initial basic circle radius G0
According to the empirical formula Gi = G0{1?sin(0.6αi )} [1],
there are two parameters affecting the basic circle radius Gi of
the LogiX gear at different positions of tooth profile: one is the
G0 and the other is the initial pressure angle αi . Figure 8 shows
the influence of G0 on the LogiX tooth profile when certain
values of parameter α0 and δ are selected. Obviously, as G0 increases,
the curvature of the new type of gear tooth profile will
become smaller and smaller. Conversely, it will become increasingly
larger as G0 decreases. Thus the new type of rack with
a large module parameter should select a large G0 value, and
one having a small module parameter should select a small G0
value.
4.3 Influence and selection of relative pressure angle δ
Figure 9 shows the variable of the tooth profile affected by the δ
parameter. According to the forming process of the LogiX tooth
Fig. 7. Influence of α0 on
LogiX tooth profile
Fig. 8. Influence of G0 on
LogiX tooth profile
Fig. 9. Influence of δ on
LogiX tooth profile
profile, the smaller the selected parameter δ, the larger the number
of N-Ps meshing on the tooth profile of two LogiX gears.
From Sect. 2.1 the formula describing the relative pressure angle
δk of an arbitrary N-P mk can be deduced as follows:
sin(αk?1+δ)
cos(αk?1 +δ) =
2?(cos δ+cos δk)
sin δ?sin δk
. (12)
By Eqs. 5 and 12, the larger the δ parameter being selected, the
larger will be the δk parameter, and at certain selected values
of the initial pressure angle and maximum pressure angle, the
lower will be the number of N-Ps. By contrast, the smaller
the δ parameter, the larger the number of N-Ps. While δ is
0.0006?, the number of zero points can exceed 46,000. In this
case, selecting a gear module of m = 100, the length of the
micro-involute curve between two adjoining N-Ps will be only
a few microns. That is to say, during the whole meshing process
of the LogiX gear transmission, the sliding and rolling
motions happen alternately and last only a few micro-seconds
from one motion to another between two meshed gear tooth profiles.
The greater the number of N-Ps, the longer the relative
rolling time between two LogiX gears and the shorter the relative
sliding time between two LogiX gears. Thus abrasion of the
gear decreases and its loading capability and life span are improved.
But, considering the restriction of memory capability,
interpolation speed, angular resolution, etc. for the CNCmachine
tool used while cutting this type of gear, the relative pressure
angle selected should not be very small. δ _ 0.0006? is generally
satisfactory.
Table 1. Parameter values selected for LogiX rack at different modules
m(mm) α0 δ G0(mm)
1 10? 0.05? 6000
2 8.0? 0.05? 9500
4 6.0? 0.05? 10000
5 5.0? 0.05? 11000
6 4.0? 0.05? 12000
8 3.2? 0.05? 12024
10 2.8? 0.05? 14000
12 2.6? 0.05? 16500
15 2.5? 0.05? 20024
18 2.4? 0.05? 30036
20 2.4? 0.05? 35000
22 2.3? 0.05? 38000
793
4.4 Reasonable selection example
Based on the above analytical rules for LogiX gear inherent parameter
selection, a reasonable calculation and selection results
for the initial pressure angle and basic circle radius while selecting
different modules at the relative pressure angle δ = 0.05? are
listed in Table 1 for reference. In fact, the practical selections
should be reasonably modified by the concrete cutting conditions
and the special purpose requirement.
5 Conclusions
The following conclusions were made based on the findings presented
in this paper.
1. Two-dimensional meshing transmission models of LogiX
gears were deduced by further analysis of its forming
principle.
2. The influence on the LogiX gear tooth profile and its performance
by the gear’s own basic parameters such as initial
pressure angle, initial basic circle radius and relative pressure
angle was discussed and their reasonable selection was given.
3. The theoretical system of the LogiX gear was developed and
the mathematical basis for generating the LogiX tooth profile
by modern CNC technology was established. The characteristics
of the LogiX gear, which are different from those of the
ordinary standard involute gear, can have broad application
and most significantly impact the improvement of carrying
capacity, miniaturisation and longevity of kinetic transmission
products.
References
1. Komori T, Arga Y, Nagata S (1990) A new gear profile having
zero relative curvature at many contact points. Trans ASME 112(3):
430–436
2. Xutang W (1982) Gear meshing theory. Machinery Industry Press,
Beijing
3. Jiahui S (1994) Circle-arc gears. Machinery Industry Press, Beijing
6 Nomenclature
α0 initial pressure angle
αi pressure angle at contact point mi
δ parameter of pressure angle
ρs1 radius of curvature of gear tooth profile at contact point s1
ρmi radius of curvature of gear tooth profile at contact point mi
ρm1 radius of curvature of gear tooth profile at contact point
m1
G0 initial radius of basic circle in tooth profile
Gi radius of basic circle of mi point in gear tooth profile
?2 rotation angle of LogiX gear meshing with basic LogiX
rack
r2 radius of basic circle of LogiX gear meshing with basic
LogiX rack
m model of gear
z gear tooth number
s gear tooth thickness at pitch circle; here, i is an optional
number
DOI 10.1007/s00170-003-1741-8
ORIGINAL ARTICLE
Int J Adv Manuf Technol (2004) 24: 789–793
Feng Xianying · Wang Aiqun · Linda Lee
Study on the design principle of the LogiX gear tooth profile
and the selection of its inherent basic parameters
Received: 2 January 2003 / Accepted: 3 March 2003 / Published online: 3 November 2004
? Springer-Verlag London Limited 2004
Abstract The development of scientific technology and productivity
has called for increasingly higher requirements of gear
transmission performance. The key factor influencing dynamic
gear performance is the form of the meshed gear tooth profile. To
improve a gear’s transmission performance, a new type of gear
called the LogiX gear was developed in the early 1990s. However,
for this special kind of gear there remain many unknown
theoretical and practical problems to be solved. In this paper, the
design principle of this new type of gear is further studied and
the mathematical module of its tooth profile deduced. The influence
on the form of this type of tooth profile and its mesh
performance by its inherent basic parameters is discussed, and
reasonable selections for LogiX gear parameters are provided.
Thus the theoretical system information about the LogiX gear are
developed and enriched. This study impacts most significantly
the improvement of load capacity, miniaturisation and durability
of modern kinetic transmission products.
Keywords Basic parameter · Design principle · LogiX gear · Minute involute · Tooth profile
1 Introduction
In order to improve gear transmission performance and satisfy
some special requirements, a new type of gear [1] was put forward;
it was named “LogiX” in order to improve some demerits
of W-N (Wildhaver-Novikov) and involute gears.
Besides having the advantages of both kinds of gears mentioned
above, the new type of gear has some other excellent
F. Xianying (_) · W. Aiqun
School of Mechanical Engineering,
Shandong University,
P.R. China
E-mail: FXYing@sdu.edu.cn
Tel.: +86-531-8395852(0)
L. Lee
School of Mechanical & Manufacturing Engineering,
Singapore Polytechnic,
Singapore
characteristics. On this new tooth profile, the continuous concave/
convex contact is carried out from its dedendum to its addendum,
where the engagements with a relative curvature of zero
are assured at many points. Here, this kind of point is called the
null-point (N-P). The presence of many N-Ps during the mesh
process of LogiX gears can result in a smaller sliding coefficient,
and the mesh transmission performance becomes almost
rolling friction accordingly. Thus this new type of gear has many
advantages such as higher contact intensity, longer life and a
larger transmission-ratio power transfer than the standard involute
gear. Experimental results showed that, given a certain
number of N-Ps between two meshed LogiX gears, the contact
fatigue strength is 3 times and the bend fatigue strength 2.5 times
larger than those of the standard involute gear. Moreover, the
minimum tooth number can also be decreased to 3, much smaller
than that of the standard involute gear.
The LogiX gear, rega