(2) Gears

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Chapter 10

Gears

Transmits

Rotary motionTorques

Why use teeth? Why not frictionrollers?

Teeth need to be specially shaped toallow smooth engagement.

Involute curves are ideal.

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Types of Gears:

Spur Gear Rack and Pinion

Internal Gear Helical Gear

Bevel GearHerrin bone Gear

Miter Gear Worm Gear

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Spur Gear Geometry:

Involute toothUnwind a string from a base circleTrack the path of the string end

Pitch Diameter (d)Size of equivalent friction rollers

Pitch circle

Pitch circle

Line of Centers

Base Circle

Involutetooth profile

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Number of Teeth (N)Must be an integer value

Diametral Pitch (Pd)size of a the gear tooth

d

NPd =

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Pressure Angle ()

Standard values: = 140, 200, 250

Mating gears must have same

pressure angle and diamtral pitch.

Pitch Circle

Line of Centers

Pitch Line

Pressure Angle,

Pressure LinePitch Circle

Base Circle

Base Circle

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Dedendum Circle

Addendum C

Dedendum (b)

ddendum (a)

Face Width (F)

Circular Pitch (p)

Tooth Thickness

Pitch Circle

Flank

Face Fillet Radius

Tip Radius

Pitch Diameter (d)

Others Features

Circular Pitch p = d/NBase circle db=d cos These features are standardized for

interchangeability:

Addendum a=1/PdDedendum b=1.25/PdFace Width F=12/Pd

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Example:A 200, full-depth, involute spur gear

with 18 teeth has a diametral pitch of12. Determine:

Outside (addendum) diameter. Root (dedendum) diameter. Standard face width. Base circle diameter. Circular pitch.

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Line of Centers

Pitch Line

Pressure Angle,

Contact Line

Driven Gear

Driver Gear

(pinion)

123 BaseCircle

Base

Circle

Pitch

Circle

Pitch

Circle

Addendum

Circle

AddendumCircle

Center Distance, c

Mating Spur Gears :

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Center Distance (c)

cexternal gears=r1 + r2

cinternal gears=r2 r1

Example:

Two mating external 4-pitch, spurgears have 14 and 42 teeth. Determinethe center distance.

Pitch circle

Pitch circle

Line of Centers

cexternal

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Contact Ratio (mp) Average number of teeth in

contact at any instant.

b

pp

Zm =

Base pitch (pb):

2

2

1

1 coscos

N

d

N

dpb

==

Length of contact path (Z):

sin)cos()(

sin)cos()(

1

2

1

2

11

2

2

2

2

22

rrar

rrarZ

++

+=

Larger values have smootherload transfer

Recommend 1.4 1.5Example:

Two mating external 4-pitch, spurgears have 18 and 42 teeth. Determine

the contact ratio.

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Interference Gears with too few teeth. Top of one gear digs into

base of the other.

Use table 10.4 to check. = 140 = 200 =250

Number ofPinion teeth

Maximumnumber of

gear teeth

Number ofpinion teeth

Maximumnumber of

gear teeth

Number ofpinion teeth

Maximumnumber of

gear teeth

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Gear Kinematics:

Velocity Ratio (VR)Gear Ratio

2

1

=VR

+ same direction- opposite direction

3:1 or three to one means VR = 3

Commonly written :

1

2

1

2

2

1

N

N

d

dVR ===

2

1

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Example:

A 5-in gear, rotating at 1725 rpm, cw,

meshes with a gear having a pitchdiameter of 20 in. Determine the

velocity of the driven gear.

An 8-pitch, 18 tooth gear rotates at1200 rpm, ccw. It meshes with a gear

with a pitch diameter of 6 in.

Determine the velocity of the drivengear.

A 10-pitch, 16 tooth gear mates with a

gear having a pitch diameter of 6 in.

The driven gear rotates at 300 rpm, cw.Determine the velocity of the driver

gear.

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Pitch Line Velocity (vt)

vt= r11 = r22

watch units:

vt= d n/12

d (in)n (rpm)

vt (fpm)

The pitch line speed determines:

lubrication needed. quality required.

(accuracy & surface finish)

21

1

2

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Gear SelectionNeed to decide:

P Pressure AngleP Suitable diametral pitchP Number of teeth on each gear

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Number of Teeth for

Commercially Available, Stock Gears

32 Diametral Pitch12 16 20 28 36 48 64 80 112

14 18 24 32 40 56 72 96 128

24 Diametral Pitch12 18 24 30 42 54 72 96 144

15 21 27 36 48 60 84 120

20 Diametral Pitch12 16 24 35 50 80 100 160

14 18 25 40 60 84 120 180

15 20 30 45 70 90 140 20016 Diametral Pitch

12 16 24 32 48 64 96 16014 18 28 36 56 72 128 192

15 20 30 40 60 80 144

12 Diametral Pitch12 15 20 28 42 60 84 120 168

13 16 21 30 48 66 96 132 192

14 18 24 36 54 72 108 144 216

10 Diametral Pitch

12 16 24 30 45 55 80 120 20014 18 25 35 48 60 90 140

15 20 28 40 50 70 100 160

8 Diametral Pitch12 16 22 32 44 60 80 11214 18 24 36 48 64 88 120

15 20 28 40 56 72 96 128

6 Diametral Pitch12 16 24 33 48 66 96

14 18 27 36 54 72 10815 21 30 42 60 84 120

5 Diametral Pitch12 16 24 30 45 70 110 160

14 18 25 35 50 80 120 180

15 20 28 40 60 100 140

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Example:A set of gears must be selected that

transmit power from a 5 hp motor at1200 rpm to a grinding wheel at 310

(10) rpm. Select a set of appropriategears.

Example:A set of gears must be selected that

transmit power from a 20 hp engine at

2000 rpm to an air compressor at 660( 10) rpm. These gears must fit into a

housing with a center distance of 6.00in. Select a set of appropriate gears.

Example

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An escalator is being designed to

transport 6000 people per hour.

Assume an average weight of 175 lbper person. Determine the required

power to operate the escalator.

15 ft

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Gear Trains:

Several gear pairs are placed in

series.

Why?

Train Value (TV)out

in

TV

=

TV = (VR)1 (VR)2 (VR)3

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ExampleThe gear train shown is used with an

input speed of 1200 rpm, cw. Giventhe following properties:

N1=24, N2= 48 & Pd = 16

N3=24, N4= 48 & Pd=12

N5=15, N6= 35 & Pd = 10

Determine the output velocity and

center distance of the gear train.

in

1

2

3

4

5

6

out

1 2

3 4

5 6

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ExampleThe gear train shown is used with an

input speed of 1200 rpm, cw. Giventhe following properties:

N1=24 N2= 36 & Pd = 12

D3=2.0 in. & Pd=10 N4= 40

N5=16 & Pd=8 D6= 6.0 inDetermine the output velocity and

center distance of the gear train.

in

out

1

2

3

45

6

4

1

2

63

5

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Gear Train Design Select the appropriate ratios for

each gear pair.

Factor the TV into suitableratios.

Example:Design a gear train with a train value

of +400:1. From interference criteria,no gear should have fewer than 15

teeth and due to size restrictions, no

gear should have more than 75 teeth.

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Worm Gears:

Able to obtain high ratios inlimited space.

Shafts are perpendicular andnon-intersecting

Self-locking 40-50% efficient

Worm Gear Kinematics:

worm

gear

N

NVR ==

2

1

2

1

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Example:Determine the output speed of the gear

drive, when the worm is driven at 1200rpm.

in=1200 rpm1

2

34

5

N2= 18

N3=60

N4=15

N5=45out = ?