T-116
13.3.1 Relationship Among the Gears in a Planetary
Gear System
In order to determine the relationship among the
numbers of teeth of the sun gear A (za), the planet gears
B (zb) and the internal gear C (zc) and the number of planet
gears (N) in the system, the parameters must satisfy the
following three conditions:
Condition No. 1:
zc = za + 2 zb
(13-5)
This is the condition necessary for the center distances
of the gears to match. Since the equation is true only for
the standard gear system, it is possible to vary the numbers
of teeth by using profile shifted gear designs.
To use profile shifted gears, it is necessary to match
the center distance between the sun A and planet B gears,
ax1, and the center distance between the planet B and
internal C gears, ax2.
ax1 = ax2
(13-6)
(za + zc)
Condition No. 2:
–––––––– = integer
(13-7)
N
This is the condition necessary for placing planet gears
evenly spaced around the sun gear. If an uneven placement
of planet gears is desired, then Equation (13-8) must be
satisfied.
(za + zc) q
–––––––––– = integer
(13-8)
180
where:
q = half the angle between adjacent planet gears
180
Condition No. 3: zb + 2 < (za + zb)sin(–––––) (13-9)
N
Satisfying this condition insures that adjacent planet
gears can operate without interfering with each other. This
is the condition that must be met for standard gear design
with equal placement of planet gears. For other conditions,
the system must satisfy the relationship:
dab < 2 ax sinq
(13-10)
where:
dab = outside diameter of the planet gears
ax = center distance between the sun and planet gears
Fig. 13-5(a)
Condition No. 1
of Planetary
Gear System
Fig. 13-5(b)
Condition No. 2
of Planetary
Gear System
Fig. 13-5(c)
Condition No. 3
of Planetary
Gear System
B
A
B
zb m
zam
zb m
zc m
C
B
A
B
C
q
B
A
B
C
q
dab
ax
