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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

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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) 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 (f): 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=pd/N Diametral Pitch (in 1/inch), pd=N/d=p/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 f (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 Gears Worm gear has one or two teeth High gear ratio**

Impossible to back drive 40-85% efficient

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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

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**Compound Gear Trains More than 1 gear on a shaft Allows for larger**

gear train ratios

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**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

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

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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 We could have used: 180=2x90=2x2x45=2x2x5x9=4x5x9 or 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

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**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

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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 w3

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

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**Compound Epicycloidal Gear Train**

Which picture is this?

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**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

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**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

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**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

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**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

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**Equation Method For Velocity Analysis**

N2=40, N3=20, N4=30, N5=90 warm=-100rpm, wsun=200

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