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

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

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

23. Compound Gear Trains More than 1 gear on a shaft Allows for larger
gear train ratios

24. Compound Train Design 2 If N2=N4 and N3=N5 3 4 Reduction ratio 5
Will be used to determine the no. of stages given a reduction ratio
2 stages

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
Pinion Teeth
* ratio
Gear teeth 12
5.646 13 14 15 16

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
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

30. Design a reverted compound gear train for a gear ratio of 18:1
18=3x 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

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
bearing
Sun (external)
Ring (internal)
teeth
Axis of symmetry

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. Tabular Method For Velocity Analysis
N2=40, N3=20, N4=80
warm= -200 rpm (clockwise)
wsun= -100 rpm (clockwise)
Sign convention:
Clockwise is negative (-)
Anti-clockwise is positive(+)
Gear#
wgear=
warm+
wgear/arm 2 3 4
Gear
ratio
-100
-200
100
- 400
-200
-250
-50
w4= 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
-600 1 #4 30 90 #5
-300
-200

40. Equation Method For Velocity Analysis
N2=40, N3=20, N4=30, N5=90
warm=-100rpm, wsun=200