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Gears 1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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Page 1: gear rack drive.pdf

Gears

1

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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MECE 304 Mechanical Machine Elements-Gears

LECTURE NOTES- MECE 304 Mechanical Machine Elements

Chapter 8- Gears

(Notes from: Chapter 13, Budynas R.G., Nisbett J.K., Shigley’s Mechanical Engineering Design, Mc Graw

Hill, 8th Edition and special notes)Hill, 8th Edition and special notes)

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13-1 Types of gears

1-Spur gears2- Helical gears3-Bevel gears4-Worm gears

MECE 304 Mechanical Machine Elements-Gears

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Spur Gears

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Helical Gears

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Bevel Gears

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Worm Gears

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Fig 13-5 Nomenclature of a spur gear teeth

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13-2 Nomenclature

The pitch circle is the theoretical circle upon which all calculations are based; its diameter is the pitch diameter

The circular pitch p is the distance from a point on one tooth to a corresponding point on an adjacent tooth.The module m is the ratio of the pitch diameter to the number of teeth. It is the index of tooth size in SI. The diametral pitch P is the ratio of the number of teeth to the pitch diameter. P is used with U.S. units.

MECE 304 Mechanical Machine Elements-Gears

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P is used with U.S. units.Addendum is the radial distance between top land and pitch circleDedendum is the radial distance between pitch circle and bottom land.

P=N/d 13-1

m=d/N 13-2

p=π d/N=π m 13-3

pP=π 13-4

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where P=diametral pitchN=number of teethd= pitch diameter, inm=module, mmd=pitch diameter, mmp=circular pitch

The relation between P and m are as follows

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The relation between P and m are as follows

Diametral pitch=25.4 / Module

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Metric Gears Geometrical Relations (without profile modification)(yellow color are controlling parameters of the gears)

MECE 304 Mechanical Machine Elements-Gears

Parameter Description Formula Type or

unit

Ø Pressure angle — degree

m Module — millimeter

Z Number of teeth — integer

p

Pitch of the teeth,on a straight

generative rack. π * m millimeter

11

S

Circular tooth thickness, measured

on the pitch circle. p / 2 millimeter

ha

Addendum = height of a tooth

above the pitch circle. ha=m millimeter

hf

Dedendum = depth of a tooth

below the pitch circle. (For

m>1.25) hf = m * 1.25 millimeter

r Radius of the pitch circle. m * Z / 2 millimeter

ra Radius of the outer circle. r + ha millimeter

rf Radius of the root circle. r - hf millimeter

rb Radius of the base circle. r * cos( Ø ) millimeter

rc Radius of the root concave corner. 0.38 * m millimeter

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13-3 Congugate action

When the tooth profiles are designed so as to produce constant angular velocity ratio during meshing these are said to have congugate action

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Fig 13-6 When contacting surfaces of cam A and follower B are involute profiles, congugate action produces constant angular velocity ratio

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13-4 Involute properties

MECE 304 Mechanical Machine Elements-Gears

Base circle:The circle on which the involute is generated

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Fig 13-7 (a) generation of an involute, (b) involute action

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13-5 Fundamentals

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Fig 13-8 Construction of a involute curve

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Fig 13-9 Circles of a gear layout

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ω1/ ω2=r2/r1 13-5

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Fig 13-9 Relation of base circle to pitch radius by pressure angle:rb=r cos Ø

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Fig 13-12 Tooth action

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Fig 13-13 Rack and pinion

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pb=pc cos Ø 13-7

where pb is base pitch and pc is circular pitch

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Fig 13-14 Internal gear and pinion

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Addendum circle of internal gear lies inside the pitch circle.

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Compound gears: Compound gears are gears attached to each other and rotating around the same axis.

MECE 304 Mechanical Machine Elements-Gears

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Backlash: Is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth measured on the pitch circles

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13-6 Contact ratio

AP+PB=qt (arc of action)

Contact ratio=mc=qt/p (gears to be designed with mc>1.2) 13-8mc indicates the number of gears in contact or

mc=Lab/p*cosØ

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Fig 13-15 Definition of contact ratio

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13-7 Interference

The contact of tooth profiles that are not congugate are called interference

**In gear interference, the involute tip of the driven gear tends to dig out the non-involute flank of the driver.

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driver.**Interference can be eliminated by using more teeth on the gears or byusing a larger pressure angle. The demand for smaller pinions with fewer teeth favors the use of a 25°pressure angle.**As a rule of thumb, use at least 14 teeth in a pinion.

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13-8 Forming of Gear Teeth

Machining: Milling, Shaping, HobbingFinishing: Shaving, Burnishing, Grinding, Lapping

13-9 Straigth Bevel GearsTransmits motion betweeen intersecting shafts

MECE 304 Mechanical Machine Elements-Gears

24Fig 13-20 Terminology of bevel gears

tan γ =NP/NG

tan Г =NG/NP

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13-10 Parallel Helical Gears

Transmit motion between parallel shafts. Helix angle is the same for the 2 gear set but hands are different (rigth hand and left hand helix)

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Fig 13-21 An involute helicoid

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Parallel Helical Gears

MECE 304 Mechanical Machine Elements-Gears

pn=ptcosψ

px=pt/tanc

Pn=Pt/cosψ

Cosψ=tanØn /tanØt

where

26Fig 13-22 Nomenclature of helical gears

where

px=axial pitch

pt= transverse circular pitch

pn=normal circular pitch

Pn=normal diametral pitch

Pt=transversal diametral pitch

ψ= helix angle

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13-12 tooth systemsIs a standard that specifies the relationship involving addendum, dedendum, working depth, tooth thickness and pressure angle

Table 13-1 Standard and Commonly Used Tooth Systems for Spur Gears

MECE 304 Mechanical Machine Elements-Gears

Tooth System Pressure Angle Ø deg. Addendum a Dedendum b

Full depth 20 1/Pd or m 1.25/Pd or 1.25m 1.35/Pd or 1.35m

22 1/P or m 1.25/P or 1.25m

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221/2 1/Pd or m 1.25/Pd or 1.25m 1.35/Pd or 1.35m

25 1/Pd or m 1.25/Pd or 1.25m 1.35/Pd or 1.35m

Stub 20 0.8/Pd or 0.8m 1/Pd or m

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13-12 tooth systems

Table 13-2 Tooth sizes in general uses

MECE 304 Mechanical Machine Elements-Gears

Diametral Pitch

Coarse 2, 21/2, 21/2, 3, 4, 6, 8, 10, 12, 16 Fine 20, 24, 32, 40, 48, 64, 80, 96, 120, 150, 200

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Fine 20, 24, 32, 40, 48, 64, 80, 96, 120, 150, 200

Modules

Preferred 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40, 50 Next Choice 1.125, 1.375, 1.75, 2.25, 2.75, 3.5, 4.5, 5.5, 7, 9, 11, 14, 18, 22, 28, 36, 45

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13-12 tooth systemsTable 13-4 Standard Tooth Proportions for Straigth Helical Gears

MECE 304 Mechanical Machine Elements-Gears

Quantity Formula Quantity Formula Addendum (a)

[ ]n

n

mP

00.1

External gears:

Dedendum (b) [ ]n

n

mP

25.125.1

Standard center dis.

2

dD +

Pinion pitch diameter

ψψ coscos

npp mN

P

N

Gear outside diameter

D+2a

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ψψ coscosnP

diameter

Gear pitch diameter

ψψ coscos

nG

n

G mN

P

N

Pinion outside diameter

d+2a

Normal arc tooth

thickness

−−

22

n

n

n

n

Bm

B

π

Gear root diameter Pinion root diameter Internal gears:

D-2b d-2b

Pinion base diameter d cosØt Center distance

2

dD −

Gear base diameter D cosØt Inside diameter D-2a Base helix angle tan-1[tanψ cosØt Center distance D+2b

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13-13 Gear trains

If a pinion 2 is driving a gear 3 thann3=(N2/N3)n2=(d2/d3)n2 13-29

where n=rpmN=number of teethd=pitch diameter

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Fig 13-27 A gear train

.

numberstoothdrivenofproduct

numberstoothdrivingofproducte

....

....= 13-30

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nL=e*nF 13-31nL=speed of last gear, nF=speed of first gear

Notes:**Gear ratio between two gears should not exceed 6 (may be up to 10 in certain cases).

This should also be extended to train values of gear trains such that:

e ≤ 35 for a 3-gear system

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e ≤ 35 for a 3-gear systeme ≤ 150 for a 4-gear system

** Gear ratio should not be an integer number. Otherwise the same set of teeth will come in contact repeatedly, thus generating an unwanted wear

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Planetary Gear trains: Gear trains in which some of gear axis rotate about axis of the other gears

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Fig 13-30 A planetary gear train

Fig 13-31 A gear train on the arm of a planetary gear train

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Planetary Gear trains:

From figure 13-31n23=n2-n3 and n53=n5-n3

If we divide the 2 equations side by side

MECE 304 Mechanical Machine Elements-Gears

3553

nn

nn

n

n

−=

33

3223 nnn −

3

3

nn

nne

F

L

−= 13-32

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13-14 Force analysis-Spur Gears

Power (H) transmitted is :H=T*ω 13-33

To convert to customary units:

V=π d*n 13-34(V=π d*n/12)

Where V= pitch line velocity mm/sec (ft/min)d=gear diameter mm (inch)

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d=gear diameter mm (inch)n=gear speed rev per sec (rpm)

Wt=(60000H)/π d*n 13-36(Wt=33000*H/V)

where Wt=transmitted load kN (lbf)H=power kw (hp)d=gear diameter, mm n=speed rpm(V=pitch line velocity, ft/min)

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Force analysis-Spur Gears

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Fig 13-32 Spur gears free body diagrams

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13-15 Force analysis Straigth Bevel gears

Wt=T/rav 13-37where T is the torque, rav is the pitch radius at the midpoint of tooth for the gear under consideration

Wr=Wt*tanØ *cosγWa=Wt*tanØ *sinγ 13-38

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Fig 13-35 Bevel gear tooth forces (see also Fig 13-20)

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13-16 Force analysis Helical Gearing

MECE 304 Mechanical Machine Elements-Gears

Wr = W sinØn

Wt = W cosØn cosΨ

Wa = W cosØn sinΨ

where W= total force

W =radial component

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Fig 13-37 Tooth forces acting on a rigth hand helical gear

Wr=radial component

Wt=tangential component (transmitted load)

Wa = axial component (thrust load)

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Notes

**Helical gears subject the shaft bearings both radial and axial loads

(see following figures for the direction of axial loads)

**The initial contact of helical gear teeth is a point which extends into a line as the teeth come into more engagement. It is this gradual

engagement of teeth and the smooth transfer of load from one tooth to another which give helical gears the ability to transmit heavy loads at high speeds.

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high speeds.

**When the thrust load is objectionable, it may be desirable to use double helical gears (herringbone)

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Direction of Axial Force in Helical Gear Systems

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