gears

1. All figures taken from Design of Machinery, 3rd ed. Robert Norton 2003
MENG 372
Chapter 9
Gears

2. Rolling Cylinders
• Gear analysis is based on rolling cylinders
• External gears rotate in opposite directions
• Internal gears rotate in same direction

3. Gear Types
• Internal and external gears
• Two gears together are called a gearset

4. Fundamental Law of Gearing
• The angular velocity ratio between 2 meshing gears
remains constant throughout the mesh
• Angular velocity ratio (mV)
• Torque ratio (mT) is mechanical advantage (mA)
in
out
in
out
out
in
T
out
in
out
in
in
out
V
d
d
r
r
ω
ω
m
d
d
r
r
ω
ω
m










v ωr

in in out out
ω r ω r

Input
Output

5. Involute Tooth Shape
• Shape of the gear tooth
is the involute curve.
• Shape you get by
unwrapping a string
from around a circle
• Allows the fundamental
law of gearing to be
followed even if center
distance is not
maintained

6. Meshing Action

7. Contact Geometry
• Pressure angle (f): angle between force and motion

8. Fundamental Law of Gearing
• The common normal of the tooth profiles, at all
contact points within the mesh, must always pass
through a fixed point on the line of centers, called
the pitch point

9. Change in Center Distance
• With the involute tooth form, the fundamental law
of gearing is followed, even if the center distance
changes
• Pressure angle
increases

10. Backlash
• Backlash – the clearance between mating teeth
measured at the pitch circle
• Whenever torque changes sign, teeth will move
from one side of contact to another
• Can cause an error in position
• Backlash increases with increase in center
distance
• Can have anti-backlash gears (two gears, back
to back)

11. Gear Tooth Nomenclature
• Circular Pitch, pc=pd/N
• Diametral Pitch (in 1/inch), pd=N/d=p/pc
• Module (in mm), m=d/N

12. Interference and Undercutting
• Interference – If there are too few pinion teeth, then
the gear cannot turn
• Undercutting – part of the pinion tooth is removed in
the manufacturing process
For no
undercutting
f
(deg)
Min #
teeth
14.5 32
20 18
25 12

13. Gear Types
• Spur Gears
• Helical Gears (open or crossed)
• Herringbone Gears
• Worm Gears
• Rack and Pinion
• Bevel Gears

14. Spur Gears
• Straight teeth
• Noisy since all of the
tooth contacts at one
time
• Low Cost
• High efficiency (98-
99%)

15. Helical Gears
• Slanted teeth to smooth contact
• Axis can be parallel or crossed
• Has a thrust force
• Efficiency of 96-98% for
parallel and 50-90% for crossed

16. Crossed Helical Gears

17. Herringbone Gears
• Eliminate the thrust force
• 95% efficient
• Very expensive

18. Rack and Pinion
• Generates linear motion
• Teeth are straight (one way to cut a involute form)

19. • Worm gear has one or two teeth
• High gear ratio
• Impossible to back drive
• 40-85%
efficient
Worm Gears

20. Bevel Gears
• Based on rolling cones
• Need to share a common
tip

21. Other Gear Types
• Noncircular gears – give a
different velocity ratio at
different angles
• Synchronous belts and
sprockets – like pulleys
(98% efficient)

22. Simple Gear Trains
• Maximum gear ratio of 1:10 based on
size constraints
• Gear ratios cancel each other out
• Useful for changing direction
• Could change direction with belt
in
in
out
ω
N
N
ω
N
N
N
N
N
N
N
N
ω
6
2
6
5
5
4
4
3
3
2



23. Compound Gear Trains
• More than 1 gear on a shaft
• Allows for larger
gear train ratios
2 4
3 5
out in
N N
ω ω
N N
  
 
   
  

24. Compound Train Design
in
ω
out
ω
2
3 4
5
2 4
3 5
in out
N N
ω ω
N N
  

  
  
If N2=N4 and N3=N5
2
2
3
in out
N
ω ω
N
 

 
 
2
3
2
in
out
ω N
ω N
 
  
 
Reduction ratio
2 stages
Will be used to determine the no. of
stages given a reduction ratio

25. Compound Train Design
• Design train with gear ratio of 180:1
• Two stages have ratio too large
• Three stages has ratio
• At 14 teeth
actual ratio is
• OK for power
transmission;
not for phasing
4164
.
13
180 
5.646
180
3

Pinion Teeth * ratio Gear teeth
12 5.646 67.7546
13 5.646 73.4008
14 5.646 79.0470
15 5.646 84.6932
16 5.646 90.3395
179.6789
14
79
3







3
3
2
180 5.646
N
N
 
 
 
 

26. Compound Train Design: Exact RR
•Factor desired ratio:
180=22x32x5
• Want to keep each ratio
about the same (i.e.
6x6x5)
• 14x6=84
• 14x5=70
• Total ratio
180
14
84
14
70
2













We could have used:
180=2x90=2x2x45=2x2x5x9=4x5x9
or 4.5x6x(20/3) etc.

27. Manual Transmission

28. Manual Synchromesh Transmission
Based on reverted compound gears

29. Reverted Compound
Train
• Input and output shafts
are aligned
• For reverted gear trains:
R2+R3=R4+R5
D2+D3=D4+D5
N2+N3=N4+N5
• Gear ratio is
Commercial three stage
reverted compound train
5
4
3
2
N
N
N
N
ω
ω
in
out


30. 3 5
2 4
18
N N
N N
  

  
  
Design a reverted compound gear train
for a gear ratio of 18:1
18=3x6 N3=6N2, N5=3N4
N2+N3=N4+N5=constant
N2+6N2=N4+3N4=C
7N2=4N4=C
 Take C=28, then N2=4, N4=7
 This is too small for a
gear! Choose C=28x4=112 (say)
• N2=16, N3=96,
• N4=28, N5=84
3
2
6
N
N
 

 
 
5
4
3
N
N
 

 
 

31. Planetary or Epicyclic Gears
• Conventional gearset has one DOF
• If you remove the ground at gear 3, it has two DOF
• It is difficult to access w3

32. Planetary Gearset
with Fixed Ring
Planetary Gearset
with Fixed Arm

33. Planetary Gearset with Ring Gear
Output
• Two inputs (sun and arm) and one output (ring)
all on concentric shafts

34. Different Epicyclic Configurations
Gear plots are about axis of rotation/symmetry
Axis of
symmetry
Sun (external)
Ring (internal)
bearing
teeth

35. Compound Epicycloidal Gear Train
• Which picture is this?

36. Tabular Method For Velocity Analysis
• Basic equation: wgear=warm+wgear/arm
• Gear ratios apply to the relative angular velocities
Gear# wgear= warm wgear/arm Gear
ratio

37. Example
Given:
Sun gear N2=40 teeth
Planet gear N3=20 teeth
Ring gear N4=80 teeth
warm=200 rpm clockwise
wsun=100 rpm clockwise
Required:
Ring gear velocity wring

38. Gear# wgear= warm+ wgear/arm
2
3
4
N2=40, N3=20, N4=80
warm= -200 rpm (clockwise)
wsun= -100 rpm (clockwise)
Tabular Method For Velocity Analysis
Sign convention:
Clockwise is negative (-)
Anti-clockwise is positive(+)
40
20

20
80

Gear
ratio
-200
-200
-200
-100 100
-200
- 400
-50
-250
w4= - 250 rpm

39. Tabular Method For Velocity Analysis
• N2=40, N3=20, N4=30,
N5=90
• warm=-100, wsun=200
Gear# wgear= warm wgear/arm Gear
ratio
Gear# wgear= warm+ wgear/arm Gear
ratio
#2 200 -100 300
-40
20
#3 -100 -600
1
#4 -100 -600
30
90
#5 -300 -100 -200

40. Equation Method For Velocity
Analysis
• N2=40, N3=20, N4=30, N5=90
• warm=-100rpm, wsun=200
gears
driven
of
product
gears
driver
of
product



arm
in
arm
out
ω
ω
w
w
 
 
300
100
18
12
300
(20)(90)
(-40)(30)
100
200
100










out
out
ω
w