Planetary gear trains
Gear train - tasks
Gear train
motor/engine
driven link (links) from which required forces and motions are obtained
in
M
in
out
M
out
If efficiency = 1
out out
in
in
M
M =
Gear box – example 1
Gear box – example 2
h a gear pair
helical h 0
a 0
worm h 0
a = /2
bevel (conical) h = 0
a 0 h 0 cylindrical a = 0
gear type a h
Gear train types
helical h 0
a 0
worm h 0
a = /2
bevel (conical) h = 0
a 0
cylindrical h 0
a = 0
gear type a h
Planetary gear train definition
Most of simple and compound gear trains have the restriction that their gear shafts may rotate in bearings fixed to the frame.
If one or more shafts rotate around another
shaft a gear train is called a planetary (or
epicyclic) gear train
Planetary gear nomenclature
A simple planetary gear
Planetary gear box of the power split device
Simple planetary gear train (obtained from unmovable axes train)
carrier
FRAME → CARRIER
1 → FRAME
FRAME → CARRIER 2 → FRAME
Simple planetary gear train (obtained from unmovable axes train)
Properties of planetary gear train
Interesting trajectories of planet gear points
Gears and other parts must be manufactured in very high accuracy → COSTS !!!
Large velocity ratio (for compact gear train)
A few motors can drive one machine Ability to transfer large forces (and power)
One motor can drive few links (car differentials)
One planet gear Ability to transfer large forces (power)
Planet gear 3 Planet gear 1
Planet gear 2
3 gear pairs take part in force transfer Ability to transfer large forces (power)
Mass 87 kg Mass 1400 kg
Gears with unmovable gear axes
Planet gear trains
420x320 610x520 850x510 1150x600
The same power and ratio !
Compare
One motor can drive few links (two wheels)
engine
po-line.sam
Planetary mechanism – trajectory (1)
po-stop.sam
Planetary mechanism – trajectory (2)
Planetary mechanism – trajectory (3)
po-ham.sam
Examples of trajectories
Examples of trajectories
Velocity ratio External gear
2 2
1 1
v v
R R
=
=
( ) 1
1 2 1
2 2
1
= = −
z z R
R
z
2 R = m
Velocity ratio Internal gear
2 2
1 1
v v
R R
=
=
( ) 1
1 2 1
2 2
1
= = +
z z R
R
Analytical method
Idea of analytical method
1 3 2
J
1 J
3
J
1 2
Gear train seen from carrier
Revolutions in frame (gear 3)
Revolutions seen from carrier J
gear 1 n1 n1J = n1 - nJ
gear 2 n2 n2J = n2 - nJ
gear 3 n3 = 0 n3J = n3 - nJ
Carrier J nJ 0 30
min rev s
1
=
n
3
J
1
2
( )
iJ s
J u
sJ
uJ
= f z
−
= −
( ) 1
1 3
1
= −
−
−
z z
J
J
0
3
=
( ) 1
1 3 3
1
= −
−
−
z z
J
J
( ) ( )
+
−
=
=
•
=
−
= −
1 1
2 3 1
2
3 2 2
1 3
1 3
1
z z z
z
J J J
J J
J J
J
+
= 1
1 3
1
z
z
J
50
; 99
; 51
; 101 :
numbers tooth
Assume z1 = z2 = z3 = z4 =
5049 1 51
99
50 1 101
3
= −
−
=
J
( ) ( ) 1 1
3 4 2
1 1
3
= + +
−
−
z z z
z
J
J
„seen” from the carrier J:
1
= 0
Since:
2 3
4 1
3
1
z z
z z
J
−
=
Then:
1 3
2 4
J
?
3
=
J
Graphical method (Velocity analysis)
1 2
J
A B
2
J
M
1 2
J
A B
2
J
vB=ABJ M
1 2
J
A B
2
J
vB=ABJ M
S21
1 2
J
A B
2
J
vB=ABJ M
S21
=2 vB R2 2 2
=ABRJ
1 2
J
A B
2
J
vB=ABJ M
S21
2
=S M vM 21
=2 vB R2 2 2
=ABRJ
2
2
R
J
AB
=
2
1
R
R AB = +
( )
2 2 1
2
R
R
J
R +
=
2
2 1
2
2 1
2 1 2
1
mz
mz
J
mz
+
=
( )
2 2 1
2
z
z
J
z +
=
1 2
J
A B
2
J
M
1
C D
1 1
2
J
A B
2
J
vB=ABJ M
1
1 C=R v C
D
1
vD=vC
1 1
2
J
A B
2
J
vB=ABJ M
S20
1
1 C=R v C
D
1
vD=vC
1 1
2
J
A B
2
J
vB=ABJ M
S20
2
=S M vM 20
1
1 C=R v C
D
1
vD=vC
1
v
B.
1
J→
D C
1
v v
.
2 → =
frame) (0
.
3 S
20−
Two driving gears (gear 1 and carrier)
Planetary gear train – graphical method
1 2
J
A B
2
C D
1
3
1 2
J
A B
2
J
C D
1
3
1 2
J
A B
2
J vB=ABJ
C D
1
3
1 2
J
A B
2
J vB=ABJ
C D
1
2
3
S23
1 2
J
A B
2
J vB=ABJ
C D
1
2
3
S23
2 D=2R
v 2
1 2
J
A B
2
J vB=ABJ
1
C D
1
2
3
S23
2 D=2R
v 2
v =vC D
J B = AB v
2
1 R
R AB = +
2
2 R
vB
= J
R R
R
2 2 1 2
= +
2
2R2
D =
v vD = 2(R1 + R2)J
D
C v
v = vC = 2(R1 +R2)J
1
1 R
vC
=
1 2 1 1
) (
2
R R R J
= +
R J
R
+
=
1 3
1 1
J B =(R1+R2) v
