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All figures taken from Design of Machinery, 3rd ed. Robert Norton 2003
MENG 372Chapter 9
Gears
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)
in
out
in
out
out
inT
out
in
out
in
in
outV
d
d
r
r
ω
ωm
d
d
r
r
ω
ωm
v ωr
in in out outω r ω r
Input
Output
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 (): 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=d/N• Diametral Pitch (in 1/inch), pd=N/d=/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
(deg)
Min # teeth
14.5 32
20 18
25 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 (98-99%)
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 gear has one or two teeth
• High gear ratio
• Impossible to back drive
• 40-85%
efficient
Worm Gears
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
• 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
inout
ωN
N
ωN
N
N
N
N
N
N
Nω
6
2
6
5
5
4
4
3
3
2
Compound Gear Trains
• More than 1 gear on a shaft• Allows for larger
gear train ratios
2 4
3 5out in
N Nω ω
N N
Compound Train Designinω
outω
2
3 4
5
2 4
3 5in out
N Nω ω
N N
If N2=N4 and N3=N5
2
2
3in out
Nω 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
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
4164.13180
5.6461803
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
179.678914
793
33
2
180 5.646N
N
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
18014
84
14
702
We could have used:180=2x90=2x2x45=2x2x5x9=4x5x9or 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
Commercial three stage reverted compound train
5
4
3
2
N
N
N
N
ω
ω
in
out
3 5
2 4
18N 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
6N
N
5
4
3N
N
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 3
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
Axis of symmetry
Sun (external)
Ring (internal)bearing
teeth
Compound Epicycloidal Gear Train
• Which picture is this?
Tabular Method For Velocity Analysis
• Basic equation: gear=arm+gear/arm
• Gear ratios apply to the relative angular velocitiesGear# gear= arm gear/arm Gear
ratio
Example
Given:Sun gear N2=40 teethPlanet gear N3=20 teethRing gear N4=80 teetharm=200 rpm clockwisesun=100 rpm clockwise
Required:Ring gear velocity ring
Gear# gear= arm+ gear/arm
2
3
4
N2=40, N3=20, N4=80arm= -200 rpm (clockwise)sun= -100 rpm (clockwise)
Tabular Method For Velocity Analysis
Sign convention:Clockwise is negative (-)Anti-clockwise is positive(+)
40
20
20
80
Gearratio
-200
-200
-200
-100 100
-200- 400
-50-250
4= - 250 rpm
Tabular Method For Velocity Analysis
• N2=40, N3=20, N4=30, N5=90
• arm=-100, sun=200
Gear# gear= arm gear/arm Gear ratio
Gear# gear= arm+ gear/arm Gear ratio
#2 200 -100 300
-4020#3 -100 -600
1#4 -100 -6003090#5 -300 -100 -200
Equation Method For Velocity Analysis
• N2=40, N3=20, N4=30, N5=90
• arm=-100rpm, sun=200
gearsdriven ofproduct
gearsdriver ofproduct
armin
armout
ω
ω
30010018
12300
(20)(90)
(-40)(30)
100200
100
out
outω