all figures taken from design of machinery, 3 rd ed. robert norton 2003 meng 372 chapter 9 gears
TRANSCRIPT
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All figures taken from Design of Machinery, 3rd ed. Robert Norton 2003
MENG 372Chapter 9
Gears
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Rolling Cylinders• Gear analysis is based on rolling cylinders
• External gears rotate in opposite directions
• Internal gears rotate in same direction
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Gear Types
• Internal and external gears
• Two gears together are called a gearset
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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
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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
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Meshing Action
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Contact Geometry• Pressure angle (): angle between force and motion
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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
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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
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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)
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Gear Tooth Nomenclature• Circular Pitch, pc=d/N• Diametral Pitch (in 1/inch), pd=N/d=/pc• Module (in mm), m=d/N
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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
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Gear Types
• Spur Gears
• Helical Gears (open or crossed)
• Herringbone Gears
• Worm Gears
• Rack and Pinion
• Bevel Gears
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Spur Gears
• Straight teeth
• Noisy since all of the tooth contacts at one time
• Low Cost
• High efficiency (98-99%)
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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
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Crossed Helical Gears
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Herringbone Gears
• Eliminate the thrust force
• 95% efficient
• Very expensive
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Rack and Pinion
• Generates linear motion
• Teeth are straight (one way to cut a involute form)
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• Worm gear has one or two teeth
• High gear ratio
• Impossible to back drive
• 40-85%
efficient
Worm Gears
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Bevel Gears
• Based on rolling cones• Need to share a common
tip
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Other Gear Types
• Noncircular gears – give a different velocity ratio at different angles
• Synchronous belts and sprockets – like pulleys (98% efficient)
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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
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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
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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
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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
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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.
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Manual Transmission
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Manual Synchromesh Transmission
Based on reverted compound gears
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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
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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
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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
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Planetary Gearset with Fixed Ring
Planetary Gearset with Fixed Arm
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Planetary Gearset with Ring Gear Output
• Two inputs (sun and arm) and one output (ring) all on concentric shafts
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Different Epicyclic Configurations
Gear plots are about axis of rotation/symmetry
Axis of symmetry
Sun (external)
Ring (internal)bearing
teeth
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Compound Epicycloidal Gear Train
• Which picture is this?
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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
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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
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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
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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
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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