P. A. Simionescu
Department of Mechanical Engineering,
The University of Tulsa,
600 S. College, Tulsa, OK 74104
D. Beale
Department of Mechanical Engineering,
Auburn University,
202 Ross Hall, Auburn, AL 36849
G. V. Dozier
Department of Computer Science,
Auburn University,
109 Dunstan Hall, Auburn, AL 36849
Teeth-Number Synthesis
of a Multispeed Planetary
Transmission Using an Estimation
of Distribution Algorithm
The gear-teeth number synthesis of an automatic planetary transmission used in automo-
biles is formulated as a constrained optimization problem that is solved with the aid of an
Estimation of Distribution Algorithm. The design parameters are the teeth number of
each gear, the number of multiple planets and gear module, while the objective function
is defined as the departure between the imposed and the actual transmission ratios,
constrained by teeth-undercut avoidance, limiting the maximum overall diameter of the
transmission and ensuring proper spacing of multiple planets. For the actual case of a
3+1 speed Ravigneaux planetary transmission, the design space of the problem is ex-
plored using a newly introduced hyperfunction visualization technique, and the effect of
various constraints highlighted. Global optimum results are also presented.
DOI: 10.1115/1.2114867
Introduction
The wide applicability of planetary gears in the aircraft, mari-
time, and mainly automotive industry particularly as automatic
multi-speed transmissions has brought a great deal of attention to
this topic. The literature on the design of planetary automatic
transmissions covers conceptual design 1–9, kinematic analysis
6,8,10–14, and power flow and efficiency analysis 15–18. Less
work has been done however on the design of multi-speed plan-
etary gears for imposed transmission ratios—the available litera-
ture covers mostly fixed axles transmissions 19–23 and the de-
sign of single-ratio planetary units 20,24–26.
Specific to teeth number synthesis of multi-speed planetary
transmissions are the design variables which must be integers
gear-teeth numbers and the number of multiple planets and the
numerous constraints. These constraints reduce significantly the
feasible domain of the design space, making the synthesis prob-
lem quite difficult to solve. The work published on teeth number
synthesis of multi-speed planetary transmissions is, for the most
part, hand-calculation oriented 27–29, or in the case of computer
implemented approaches, only some of the numerous constraints
were actually considered 30–33.
The constraints imposed on multi-speed planetary transmissions
derive from: a the minimum allowed number of teeth each gear
can have so that undercut does not occur, b the maximum al-
lowed diameter of the whole assembly, c the condition of central
gears having coaxial axes, d the requirement of equally spacing
multiple planets, and e the noninterference condition of neigh-
boring gears.
Aspects like tooth geometry optimization 34 or bearing selec-
tion from the condition of volume and cost minimization and of
satisfying a required design life can also be prescribed early in the
design process. However, since these can be decoupled from the
teeth-number selection problem, it is preferable to be solved as a
subsequent multiobjective optimization problem once a satisfac-
tory solution of the original problem becomes available 35.
The Ravigneaux 3+1 Gear Transmission
Figure 1 shows a planetary transmission of the Ravigneaux type
with three forward and one reverse gears used in automobiles
36,37. A kinematic diagram of the transmission is available in
Fig. 2, where the broad planet gear is shown as two compound
gears 2 and 3. Based on the clutch/brake activation required in
each gear Table 1, it can be shown that in the first and reverse
gears, the planet carrier is immobile and the equivalent transmis-
sion is a fixed-axle one with the following transmission ratios
i
1
= N
6
/N
4
1
and
i
R
=- N
2
N
6
/N
1
N
3
2
In the third gear, the planet carrier, sun gears, and ring gear
rotate together as a whole
i
3
=1 3
i.e., a direct drive, which ensures an increased mechanical effi-
ciency of the transmission.
The second gear configuration is the only case when the trans-
mission works as a planetary gear set. Considering the planet
carrier c immobile, three basic transmission ratios can be defined
as follows
i
16
c
=-
N
2
N
6
N
1
N
3
i
46
c
=
N
6
N
4
i
14
c
=-
N
2
N
4
N
1
N
3
4
Through motion inversion, which converts the planetary gear into
a fixed axle transmission, the following additional relations be-
tween the angular velocities of the sun gears 1 and 4, ring gear 6,
and planet carrier c can be written as
i
16
c
=
2
-
c
6
-
c
i
46
c
=
4
-
c
6
-
c
i
14
c
=
1
-
c
4
-
c
5
Eliminating
c
between any two of the above equations and for
4
= 0, the sought-for transmission ratio of the second gear can be
obtained
i
2
=
N
6
N
1
N
3
+ N
2
N
4
N
1
N
3
N
6
- N
4
6
Contributed by Power Transmission and Gearing Committee of ASME for publi-
cation in the JOURNAL OF MECHANICAL DESIGN. Manuscript received: August 24, 2004;
final manuscript received: April 1, 2005. Assoc. Editor: Teik C. Lim.
108 / Vol. 128, JANUARY 2006 Copyright © 2006 by ASME Transactions of the ASME