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