Design II MDE 221 Mott Ch 8 and Ch 9 Spur Gears

• Chapter 8: Mott p300

• Kinematics of Gears

• Chapter 9: Mott p363

• Spur Gear Design

MACHINE ELEMENTS IN

MECHANICAL DESIGN

Design II - Kinematics of Gears Ch 8 p 300 – 323 (Mott)

- Spur Gear Design Ch 9 p 365 – 414 (Mott)

Spur gears

Spur gears ("straight-cut gears") are the

simplest and most common type of gear

Helical gears

Helical gears offer a refinement over

spur gears. The leading edges of the

teeth are not parallel to the axis of

rotation, but are set at an angle

Rack and pinion

A rack is a toothed bar or rod that can be

thought of as a sector gear with an

infinitely large radius of curvature.

Gears are toothed, cylindrical wheels used

for transmitting motion and power from one

rotating shaft to another.

http://upload.wikimedia.org/wikipedia/commons/1/14/Gears_animation.gif

http://en.wikipedia.org/wiki/Helix

http://upload.wikimedia.org/wikipedia/commons/d/d1/Helical_Gears.jpg

http://upload.wikimedia.org/wikipedia/commons/6/68/Rack_and_pinion_animation.gif



Bevel gears

Bevel gears are essentially conically

shaped,

Crown gear

A crown gear or contrate gear is a

particular form of bevel gear whose teeth

project at right angles to the plane of the

wheel; in their orientation the teeth

resemble the points on a crown.

Worm gear

A worm is a gear that resembles a screw.

http://upload.wikimedia.org/wikipedia/commons/1/16/Bevel_gear.jpg

http://upload.wikimedia.org/wikipedia/commons/3/34/Crown_gear.png

http://en.wikipedia.org/wiki/Screw

http://upload.wikimedia.org/wikipedia/commons/7/75/Worm_Gear_and_Pinion.jpg



4

Objectives – Design 2: Kinematics of Gears Ref: Mott Ch 8 p300 – 323

• Recognize the main features of spur gears,

helical gears, bevel gears and worm/wormgear

sets.

• Understand the important operating

characteristics of SPUR gears.

• Understand the involute-tooth form and discuss

its relationship to the law of gearing.



5

• Understand the basic functions of the American Gear

Manufacturers Association (AGMA) and identify

pertinent standards developed and published by this

organization.

• No official standard for RSA, therefore use this

textbook for subject purposes, but there are ISO

standards that must be used for actual gear design.

• Define velocity ratio as it pertains to two gears

operating together.

• Specify appropriate numbers of teeth for a mating

pair of gears to produce a given velocity ratio.

• Design 3 – Gear Trains, bevel, helical etc p 323 to

357

Objectives (Concluded)



6

Contents

1. Spur Gears

2. Helical Gears – Design 3

3. Bevel Gears – Design 3

4. Worm Gears – Design 3

5. Gear trains – Design 3

6. The AGMA



7

1. Spur gears • Gears that have teeth that are straight

and arranged parallel to the axis of the shaft

that carries the gear.

• Teeth have involute curves to maintain a constant angular velocity

ratio when two working gears mate.

– Angular velocity can be achieved when a line drawn

perpendicular to the surfaces of two rotating bodies at their point

of contact always crosses the center-line between the two

bodies at the same place.

– The above statement is also known as the law of gearing.

• Speed Reduction Ratio: Produce a change in the speed of rotation

of the driven gear (Gear) relative to the driving (Pinion) gear.

teeth of noN and rpmn whereN

N

n

n

P

G

G

P

http://upload.wikimedia.org/wikipedia/commons/1/14/Gears_animation.gif



8

Above: the involute tooth form

Above: diagram illustrating the

law of gearing.







11

where N and D are the number of teeth and the

pitch diameters of the gears.

-In the USA Diametrical Pitch is mainly used and is the number of

teeth per inch of pitch diameter.

-Metric System (S.I.) uses the module: See Table 8-3 p313

dP.m 425

• Spur gear teeth features: Ref: Mott p309

– Circular pitch:

• Diametral Pitch Ref: Mott p310

(in inches): ( to convert to mm x by 25.4)

– Module (in SI units):

– To convert to module (m) Ref: Mott

p312

(Module is in mm)



12

• Addendum: Ref: Mott p312

• The radial distance from the pitch circle to the outside of a

tooth.

– Dedendum: Ref: Mott p312

• The radial distance from the pitch circle to the bottom of the

tooth space.

addendum.dedendum

ulemodaddendum

251

– Backlash:

Gears are made with the tooth spaces

slightly larger than the tooth thickness,

See p315, the difference is call backlash.

– Centre Distance:

222

PGPG mNmNDDC

Ref: Mott p314

Ref: Mott p315

Ref: Mott p316



13

– Pressure angle

• The pressure angle is the angle between the tangent to the pitch

circles and the line drawn normal to the surfaces of the gear

tooth.

cosDDb



14

• Contact ratio – for info. Ref: Mott p317

– Indicates the average number of teeth in contact during the

transmission of power.

– Recommended: 1.2 minimum.

Note: See Example Problem 8-1 Mott p318



15

• Interference between mating spur gear teeth: Ref: Mott p320

– Important that there is NO interference between teeth.

– See table 8-6 for values.

– Undercutting, cutting away of the material at the base (fillet or root) of the tooth

relieve interference, BUT does weaken the tooth design. Ref: Mott p321

– NOTE: Ref: Mott p321

For 20º, full-depth, involute system, using no few than 18 teeth will

ensure that no interference occurs. Therefore, for most Design 2 assumptions use 18 or 19 teeth (18 Preferable)

• Velocity Ratio: The ratio of rotational speed of the input gear to that

of the output gear for a single set of gears.

• Pitch Line Speed: of the gear and pinion are the same.

P

G

G

P

P

G

P

G

G

P

G

P

size

size

speed

speed

D

D

N

N

n

nVR

60

DnVt



16

• Compute the forces exerted on gear teeth as they rotate and transmit power.

• Understand various methods for manufacturing gears and levels of precision and quality to which they can be produced.

• Design 3 - Specify a suitable level of quality for gears according to the use to which they are to be put.

• Design 3 - Describe suitable metallic materials from which to make the gears, in order to provide adequate performance for both strength and pitting resistance.

Objectives – Design 2: Spur Gear Design Ref: Mott p365 – 414 Ch 9



17

Objectives for Ch 9 (Continued)

• Use the standards of the American Gear

Manufacturers Association (AGMA) as the basis

for completing the design of the gears.

• Use appropriate stress analyses to determine

the relationships among the applied forces, the

geometry of the gear teeth, the precision of the

gear teeth and other factors specific to a given

application, in order to make final decisions

about those variables.



18

Objectives for Ch 9 (Concluded)

• Design 3 - Perform the analysis of the tendency for the contact stresses exerted on the surfaces of the teeth to cause pitting of the teeth, in order to determine the adequate hardness of the gear material that will provide an acceptable level of pitting resistance for the reducer.

• Design 3 - Complete the design of the gears, taking into consideration both the stress analysis and the analysis of pitting resistance. The result will be a complete specification of the gear geometry, the material fro the gear and the heat treatment of the material.

Pitting: Small particles removed from the surface of the tooth face because of the high contact stresses.



19

Contents

1. Gear Manufacture

2. Materials Used

3. Force and Stress Analysis

4. Design of Spur Gears



20

1. Gear Manufacture Ref: Mott p370 (INFO)

• Small gears

– frequently made from wrought plate and bar.

– Dimensions are machined to precision.

• Large gears

– Frequently fabricated from components.



21

• Gear teeth: Ref: Mott p371 (INFO)

– Machined by various methods.

– Most popular methods –

• Form milling: mainly used for large gears. A milling cutter that

has the shape of the tooth space is used.

• Shaping: usually used for internal gears. Cutter used

reciprocates on a vertical spindle.

• Hobbing: similar process to milling except that both the

workpiece and the cutter rotate in a coordinated manner.

Milling cutter Shaping for small gears

Hobbing



22

• Gear quality Ref: Mott p373

• Composite variation: allowable amounts of variations of the

actual tooth form from the actual tooth form

– Specified by AGMA as quality numbers.

– Quality numbers range from 5 to 15 with increasing

precision.

Above: Schematic diagram of a

typical gear rolling fixture

Right: Chart of gear-tooth errors

of a typical gear when run with a

specific gear in a rolling fixture.



23

– Table below shows the tolerance of composite variation



24

• Recommended quality numbers by AGMA.

• ISO standards 1328-1-1995 has own quality

numbers. See Mott p 375 for comparison.



25

2. Materials Used – Info: Design 3

• Main consideration when selecting

materials:

1. Producibility of the gear

2. Strength and pitting resistance

3. Weight

4. Appearance

5. Corrosion resistance

6. Noise

7. Cost



26

• Steel gear material:

– Through-hardened steels

– Case-hardened steels

• Flame hardened/ induction hardening

• Carburizing

• Nitriding

• Iron and bronze gear materials

– Cast irons

– Bronze

• Phosphor or tin, manganese bronze, aluminum

bronze, silicon bronze.



27

3. Force and Stress Analysis – NB

Design 2

• Forces in gear teeth

Forces on gear teeth.

Wt = force exerted by the pinion teeth on the gear teeth.

Wn = total force transferred from one tooth to the mating tooth.

Wr = vertical component of the total force acting radially on the gear

tooth.



28

- Tangential force

D

PWt

2

t

tv

PW

where P = power provided

ω = angular velocity

vt = pitch line velocity

- Radial force

- Normal force

tantr WW

cos

tn

WW

Dn

PW:Therefore t

60

60

n2



• Stresses Ref: Mott p385

– Lewis equation – to calculate stress at the base of the

involute profile.

– Acts like a cantileaver OVt

OVdt

t K.K.FYm

WK.K.

FY

PW

where Wt = tangential force

Pd = diametral pitch of the tooth where Pd=1/m (module in meters)

F = face width of the tooth

Y = Lewis form factor (dependent on the tooth form, the

pressure angle, the diametral pitch, the number of teeth in the

gear, and the place where Wt acts.

m=Module in meters

Design 2: Lewis including Kv: Velocity Factor p392 with Figure 9-21

Ko: Overload Factor p388/9 with Table 9-5

Nominal Face Width F=12 x m p408

Design Power p409 with Figure 9-27

NOTE: only for Design 2 because Lewis does not take into account stress

concentration that exists in the fillet of the tooth



30

– Bending stress – DESIGN 3 (Info)

vBmsodt

t KKKKKFJ

PWs

where Ko = overload factor for bending strength

Ks = size factor for bending strength

Km = load distribution factor for bending strength

KB = rim thickness factor

Kv = dynamic factor for bending strength

J = geometry factor

Selection of gear material based on bending stress

Necessary to keep bending stress < allowable stress.

Valid only if temperature < 121.1°C, 107 cycles of tooth

loading, reliability of 99% and safety factor of 1.00.



31

4. Design of Spur Gears

• Overall objectives:

– Be compact and small

– Operate smoothly and quietly

– Long life

– Low cost

– Easy to manufacture

– Be compatible with other elements in the

machines.



32

• General guideline:

– Identify input speed of pinion and the desired

output.

– Choose the type of material –Design 3

– Specify the overload factor.

– Specify a trial value for the diametral pitch.

– Specify face width

– Compute/ specify the factors needed to

determine the bending stress and contact

stress.

– Iterate design process to seek for more optimal

designs – Design 3



33

• Example Problem 9-5 Mott p 410

• Example Problem 9-6 Mott p 413

• Exercises p444 no 1,2,3,36,37 and 38



Spur Gears:

Gear Terminology

1-Pitch Circle Diameter (PCD or D)

The diameter of the pitch circle or the diameter of

the discs, if driven by friction alone, would have

the same ratio as the pair of gears

PCD

(D)

2-Module (m): Diameter in mm / no. of teeth = D/t

(t = no. of teeth)

3-Circular Pitch (Pc): Distance from any point on

the one tooth to the corresponding point on the

adjacent tooth, measured on the pitch circle.

CC

C

Pm and m.P

m teeth of no

D but

teeth of no

D.P

http://upload.wikimedia.org/wikipedia/commons/f/f6/Action_line.jpg



4-Addendum: The pitch of the tooth above

(outside) the pitch circle

Addendum = Module

5-Dedendum: The portion of tooth BELOW the

pitch circle.

Dedendum = 1.157 x Module

6-Whole Depth:

(1 x Module) + (1.175 x Module) = 2.157 x Module

7- Clearance = 0.157 x M

8- Working Depth = Whole Depth – Clearance

= 2.157m – 0.157m = 2 x M

9-Pitch Point: Where the gear touch

Pitch point

Line of action



10-Base Circle: Circle from which

the involute curve is drawn.

Involute Curve – Gear tooth profile

11-Line of Action: the line normal

(perpendicular ┴) to a pair of mating

profiles at their point of contact.

12-Pressure angle (θ): The angle

between the line of action and the

common tangent to the base

circles.

A

R

Rb

70°

Pressure

Angle 20°

90°

90°

20°

20°

Ψ

Ψ

70° P

Pitch Point

B

Common

Tangent to

base

circles

LINE OF

ACTION

=PLL

r rb

Driver

Ø200mm PITCH

Circle

Base Circle

Base Circle

Ø300mm PITCH

Circle

AnglePressure

20 withsPCD' 300 and 200

of gears meshing 2 Consider

:GeometryBasic

o

cosRR

20 cos AP AB

AB

AP 20 cos

radius circle Base AB

radius circle pitch AP

: ABPTriangle Consider

b

Ft

NOTE: PLL is tangent

to the Base Circle

NOTE: Ft

is tangent

to the

PCD



Construction of an involute curve:

e.g. construct an involute curve for gear teeth of 20° pressure angle on a 300mm PCD gear:

PCR = 150mm = R

Base Rad = 150 cos 20 = 140.95 = Rb

1-Divide base circle into segments –

the smaller the angle, the greater the

accuracy – in this example 10°.

2-Draw tangents as shown.

3-Calculate distance from A to B

measured along the base circle:

4-Mark off B1 = 24.6mm

C1= 2(24.6) = 49.2mm

D3 = 3(24.6) = 73.8mm

E4 = 4(24.6) = 98.4mm

6.2495.140180

10bL

mm6.24360

1095.1402

360

10R2 b



Tooth

Thickness

meshing.allow to - themselves teeth the as same

the is teeth the between space the :reason

2t

360

teeth of no.t



Rbase

Pinion – the smaller of the two meshing gears

(small letters)

Gear – the larger of the two meshing gears

(capital letters)

Addendum – the height from the pitch circle

to the tip of the tooth.

Dedendum – the height from the pitch circle to

the root of the tooth

Circular pitch:

the distance from a point on one tooth to the

same point on the next tooth measured along a

pitch circle:

mP

mT

D or mTD but

T

DP

C

C

Standard modules: 1; 1.25; 1.5; 2; 2.5; 3; 4; 5; 6;

8; 10; 12; 16; 20; 25; 32; 40; 50.

Standard proportions for 20° full depth involute

teeth are:

addendum = module

dedendum = 1.157 x module



Strength of Gear Teeth:

Cantilever Method: this method considers the gear tooth to be as a cantilever with 2/3 of the pitch

line load acting on the tip of the tooth.

stress bending

angle pressure

load line pitch PLL

tooth of breadth b

tooth of height h

cos

FtPLL

PLL

Ftcos

radiusFtTorque

CC 0.48Pt and 0.7Ph

:used be can ionsapproximat following the available not are t"" and h"" for values If

ns)calculatio in PLL of place in Ft use references some :(note

bt

4hPLL

bt

6hPLL

3

2

12

bt

)2

t()hPLL

3

2(

I

My

2

2

3AA

Fs = Separating force

Ft = Tangential force



Pitch line load PLL may be found from:

Power = Force x V

= Ft x V

Or Power = PLL x V (depending on which radius you are using)

Where power is in Watts and velocity (V) (Ft Velocity) is:

Rev/Min N

metres in PCD D :Where

60

DNV

Example: Calculate the

load at the tip of a gear

tooth given the following:

Power Transmitted = 10 kW

N = 600 rpm

PCD of gear = 140mm

N 613.081.3

2at tip Load

N56.2419cos

:But

Load) l(Tangentia 64.2273

398.4

10000

/398.460

60010140

60

3

LL

TLL

T

T

T

P

FP

NF

V

PowerF

VFPower

smDN

V

Ft Tangent

to PCR

PCR

Base Circle

Radius

PLL

Tangent

to Base

Circle

N 613.081.3

2at tip Load

Load) Line(Pitch 55.2419

133.4

10000

/133.460

6001056.131

60

56.13120cos140

20cosCircle Base

:elyAlternativ

3

LL

LL

LL

LL

P

NP

V

PowerP

VPPower

smDN

V

mm

PCD



Lewis Formula: y.P.b..kFt CV

factorvelocity k m or T

D pitch Circular P

designed being gear on teeth of number T tooth of widthface b

factor) (tooth factor Lewis y tooth in induced stress Bending

VC

)648.0

(124.0

: teeth5.14

Ty

o

Tooth Factors: (y)

)T

912.0(154.0y

:teeth 20o

Velocity Factors: (kV)

m/s 20 V V6.5

6.5k

m/s 20 - 10 V6

6k

m/s 10 V V3

3k

V

V

V

Pitch Circle Velocity:

Diameter Circle Pitch D

(rev/min) Speed N

:gears both to common is V :NOTE

60

DNV

Note: The Lewis formula must be applied to the weaker of the two meshing gears. The

weaker of the two is the one with the smaller σ x y product and the basis for the design.

Documents

Design II MDE 221 Mott Ch 8 and Ch 9 Spur Gears