MENG 372
Chapter 9
Gears
All figures taken from Design of Machinery, 3rd ed. Robert Norton 2003
Rolling Cylinders
• Gear analysis is based on rolling cylinders
• External gears rotate in opposite directions
• Internal gears rotate in same direction
Gear Types
• Internal and external gears
• Two gears together are called a gearset
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
v  ωr
ωinrin  ωout rout
Output
ωout
rin
din
mV 


ωin
rout
d out
ωin
rout
d out
mT 


ωout
rin
din
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
Meshing Action
Contact Geometry
• Pressure angle (f): angle between force and motion
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
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
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)
Gear Tooth Nomenclature
• Circular Pitch, pc=pd/N
• Diametral Pitch (in 1/inch), pd=N/d=p/pc
• Module (in mm), m=d/N
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
20
25
32
18
12
Gear Types
•
•
•
•
•
•
Spur Gears
Helical Gears (open or crossed)
Herringbone Gears
Worm Gears
Rack and Pinion
Bevel Gears
Spur Gears
• Straight teeth
• Noisy since all of the
tooth contacts at one
time
• Low Cost
• High efficiency (9899%)
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
Crossed Helical Gears
Herringbone Gears
• Eliminate the thrust force
• 95% efficient
• Very expensive
Rack and Pinion
• Generates linear motion
• Teeth are straight (one way to cut a involute form)
Worm Gears
•
•
•
•
Worm gear has one or two teeth
High gear ratio
Impossible to back drive
40-85%
efficient
Bevel Gears
• Based on rolling cones
• Need to share a common
tip
Other Gear Types
• Noncircular gears – give a
different velocity ratio at
different angles
• Synchronous belts and
sprockets – like pulleys
(98% efficient)
Simple Gear Trains
N 2 N3 N 4 N5
ωout 
ωin
N3 N 4 N5 N 6
N2

ωin
N6
• 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
Compound Gear Trains
• More than 1 gear on a shaft
• Allows for larger
gear train ratios
ωout
  N 2   N 4 


 ωin
 N3  N5 
Compound Train Design
ωin
2
4
3
5
2 stages
ωout
 N 2  N 4 
ωin 

  ωout
 N3  N5 
If N2=N4 and N3=N5
ωin  N3 


ωout  N 2 
2
 N2 
ωin 
  ωout
 N3 
2
Reduction ratio
Will be used to determine the no. of
stages given a reduction ratio
Compound Train Design
•
•
•
•
Design train with gear ratio of 180:1
Two stages have ratio 180  13.4164too large
 N3  3
3
Three stages has ratio 180  5.646

  180  5.646
 N2 
At 14 teeth
Pinion Teeth
* ratio
Gear teeth
actual ratio is
3
 79 
   179.6789
 14 
• OK for power
transmission;
not for phasing
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
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
2
 70  84 
    180
 14  14 
We could have used:
180=2x90=2x2x45=2x2x5x9=4x5x9
or 4.5x6x(20/3) etc.
Manual Transmission
Manual Synchromesh Transmission
Based on reverted compound gears
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
ωout N 2 N 4

ωin
N3 N5
Commercial three stage
reverted compound train
Design a reverted compound gear train
for a gear ratio of 18:1
 N3  N5 
18=3x6
N3=6N2, N5=3N4


  18
 N 2  N 4 
N2+N3=N4+N5=constant
 N5 
 N3 
N2+6N2=N4+3N4=C

6  N 3
7N2=4N4=C
 4
 N2 
 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
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
Planetary Gearset
with Fixed Ring
Planetary Gearset
with Fixed Arm
Planetary Gearset with Ring Gear
Output
• Two inputs (sun and arm) and one output (ring)
all on concentric shafts
Different Epicyclic Configurations
Gear plots are about axis of rotation/symmetry
bearing
Ring (internal)
teeth
Axis of
symmetry
Sun (external)
Compound Epicycloidal Gear Train
• Which picture is this?
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
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
Tabular Method For Velocity Analysis
Sign convention:
N2=40, N3=20, N4=80
warm= -200 rpm (clockwise) Clockwise is negative (-)
wsun= -100 rpm (clockwise) Anti-clockwise is positive(+)
Gear#
wgear=
warm+
2
-100
-200
100
3
- 400
-200
-200
4
-250
-200
-50
w4= - 250 rpm
wgear/arm Gear
ratio
40

20
20

80
Tabular Method For Velocity Analysis
• N2=40, N3=20, N4=30,
N5=90
• warm=-100, wsun=200
Gear# wgear= wwarm
arm+ wgear/arm Gear
ratio
#2
200
-100
300
-40
20
#3
-100
-600
1
#4
-100
-600
30
90
#5
-300 -100
-200
Equation Method For Velocity
Analysis
ωout  w arm product of driver gears

ωin  w arm product of driven gears
• N2=40, N3=20, N4=30, N5=90
• warm=-100rpm, wsun=200
ωout   100 (-40)(30)

200  100 (20)(90)
 12
w out  300
 100  300
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