# All figures taken from Design of Machinery, 3rd ed. Robert Norton 2003

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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

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 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 (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 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 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 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

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

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

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.

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

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

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

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
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

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

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

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