FME_Ch14

Hamrock • Fundamentals of Machine Elements

Chapter 14

Just stare at the machine. There is nothing wrong with that. Just live with it for a while. Watch it the way you watch a line when fishing and before long, as sure as you live, you’ll get a little nibble, a little fact asking in a timid, humble way if you’re interested in it. That’s the way the world keeps on happening. Be interested in it.

Robert Pirsig, Zen and the Art of Motorcycle Maintenance


Spur Gears

Figure 14.1 Spur gear drive. (a) Schematic illustration of meshing spur gears; (b) a collection of spur gears.


Helical Gears

Figure 14.2 Helical gear drive. (a) Schematic illustration of meshing helical gears; (b) a collection of helical gears.


Bevel Gears

Figure 14.3 Bevel gear drive. (a) Schematic illustration of meshing bevel gears; (b) a collection of bevel gears.


(a) (b)

Worm Gears

Figure 14.4 Worm gear drive. (a) Cylindrical teeth; (b) double enveloping; (c) a collection of worm gears.


Tooth profile

Pitch circle

Whole depth, ht

Addendum, a

Dedendum, b

Root (tooth)Fillet

Topland

Pitch point

Pitch circleBase circle

Pressureangle,

Line ofaction

Circular pitch, pc

Base diameter, dbg

Workingdepth, hk

Clearance, cr

Circular tooththickness

Chordal tooththickness

Gear

Pinion

Centerdistance, cd

Pitch

dia

met

er, dg

Pitch (pd )

Root diameter

r bp

r p

ropOutside diameter, d

op

rbg

rg

rog

Figure 14.5 Basic spur gear geometry.

Spur Gear Geometry


Outside circle

Flank

Face

Tooththickness

Pitchcircle

Widthof space

Circular pitch

Bot

tom

land

Top la

ndFace w

idth

Clearance Filletradius Clearance

circle

Dedendumcircle

Addendum

Dedendum

Figure 14.6 Nomenclature of gear teeth.

Gear Teeth


2 3 4 5

6 7 8 9 10 12 14 16

22

1

Figure 14.7 Standard diametral pitches compared with tooth size.

Standard Tooth Size

Diametral pitch,Class pd, in−1

Coarse 1/2, 1, 2, 4, 6, 8, 10Medium coarse 12, 14, 16, 18Fine 20, 24, 32, 48, 64

72, 80, 96, 120, 128Ultrafine 150, 180, 200

Table 14.1 Preferred diametral pitches for four tooth classes


200

100

70

50

30

20

10.0

7.0

5.0

3.0

2.0

1.0

0.7

0.5

Pow

er t

ransm

itte

d, h

p

0 600 1200 1800 2400 3000 3600

Pinion speed, rpm

100

70

50

30

20

10.0

7.0

5.0

3.0

2.0

1.0

0.7

0.5

Pow

er t

ransm

itte

d, k

W

150

60

15

40

6.0

4.0

1.5

Data for all curves:

gr = 4, NP=24

Ka = 1.0

20° full depth teeth

24 pitch; d=1.00 in (m=1; d=25 mm)

12 pitch; d = 2.00 in (m=2; d=50 mm)

6 pitch; d=4.00 in (m=4; d=100 mm)

3 p

itch

; d =

8.00 in (m = 8; d = 200

Figure 14.8 Transmitted power as a function of pinion speed for a number of diametral pitches.

Power vs. Pinion Speed


MetricCoarse pitch Fine pitch module

Parameter Symbol (pd < 20 in.−1) (pd ≥ 20 in.−1) systemAddendum a 1/pd 1/pd 1.00 mDedendum b 1.25/pd 1.200/pd + 0.002 1.25 mClearance c 0.25/pd 0.200/pd + 0.002 0.25 m

Table 14.2 Formulas for addendum, dedendum, and clearance (pressure angle, 20°; full-depth involute).

Gear Geometry Formulas


Base circle

Pitch circle

Pitch circle

Pitch point, pp

Gear

Base circle

Pinion

a

b

φ

φ

rp

rg

rbg

rbp

ωg

ωp

Figure 14.9 Pitch and base circles for pinion and gear as well as line of action and pressure angle.

Pitch and Base Circles


A4

A3

A2

A1C1

C2

B1

B2

B3

B4

C4

C3

A0

Base circle

0

Involute

Figure 14.10 Construction of the involute curve.

Involute Curve


Construction of the Involute Curve

1. Divide the base circle into a number of equal distances, thus constructing A0, A1, A2,...

2. Beginning at A1, construct the straight line A1B1, perpendicular with 0A1, and likewise beginning at A2 and A3.

3. Along A1B1, lay off the distance A1A0, thus establishing C1. Along A2B2, lay off twice A1A0, thus establishing C2, etc.

4. Establish the involute curve by using points A0, C1, C2, C3,... Gears made from the involute curve have at least one pair of teeth in contact with each other.


Arc of approach qa Arc of recess qr

PBA

aOutside circle

Pitch circle Outside circle

Motion

b

Lab

Line of action

Figure 14.11 Illustration of parameters important in defining contact.

Contact Parameters


Figure 14.12 Details of line of action, showing angles of approach and recess for both pinion and gear.

Line of Action

Lab =√r2op− r2bp+

√r2og− r2bg− cd sin!

Cr =1

pc cos!

[√r2op− r2bp+

√r2og− r2bg

]− cd tan!

pc

Length of line of action:

Contact ratio:


Base circle

Base circle

Pitch circle

Pitch circle

0

Backlash

Pressure line

Backlash

φ

Figure 14.13 Illustration of backlash in gears.

Backlash

Diametralpitch Center distance, cd, in.pd, in.−1 2 4 8 16 3218 0.005 0.006 — — —12 0.006 0.007 0.009 — —8 0.007 0.008 0.010 0.014 —5 — 0.010 0.012 0.016 —3 — 0.014 0.016 0.020 0.0282 — — 0.021 0.025 0.0331.25 — — — 0.034 0.042

Table 14.3 Recommended minimum backlash for coarse-pitched gears.


Gear 1

(N1 teeth)

Gear 2

(N2 teeth)

(+) (–)1

2

r1

r2

1

2

Gear 1

(N1)

Gear 2

(N2)

r1

r2

Figure 14.14 Externally meshing gears.

Meshing Gears

Figure 14.15 Internally meshing gears.


N2

N1

N1

N2 N5

N6

N7 N8N3 N4

Figure 14.16 Simple gear train.

Gear Trains

Figure 14.17 Compound gear train.


Only pitchcircles of gears shown

Shaft 1

Input

Shaft 2

Shaft 3

Shaft 4

Output

A

B

NA = 20

NB = 70

NC = 18

ND = 22

NE = 54

A

B

C

D

E

C

D

E

Example 14.7

Figure 14.18 Gear train used in Example 14.7.


RingR R

P

P

S

P

SA

P

Planet

Arm

Sun

(a) (b)

Planetary Gear Trains

Figure 14.19 Illustration of planetary gear train. (a) With three planets; (b) with one planet (for analysis only).

!ring−!arm

!sun−!arm

=−NsunNring

!planet−!arm

!sun−!arm

=− Nsun

Nplanet

Nring = Nsun+2Nplanet

Zp =!L−!A

!F−!A

Important planet gear equations:


Rel

ativ

e co

st

10

100

1

0.5

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

DIN quality number

(4)(6) (5)(7)891011121315(17)

14(16)

AGMA quality index

Special methods

Production grinding

Shaving

Gear shaper-hobbing

0.0

0005 i

n.

0.0

0010

0.0

005

0.0

010

0.0

10

0.0

15

Gear Quality

Figure 14.20 Gear cost as a function of gear quality. The numbers along the vertical lines indicate tolerances.

Application Quality index, Qv

Cement mixer drum driver 3-5Cement kiln 5-6Steel mill drives 5-6Corn pickers 5-7Punch press 5-7Mining conveyor 5-7Clothes washing machine 8-10Printing press 9-11Automotive transmission 10-11Marine propulsion drive 10-12Aircraft engine drive 10-13Gyroscope 12-14

Pitch velocity Quality index, Qv

ft/min m/s0-800 0-4 6-8800-2000 4-10 8-102000-4000 10-20 10-12> 4000 > 20 12-14

Table 14.4 Quality index Qv for various applications.


Form Cutting

Figure 14.21 Form cutting of teeth. (a) A form cutter. Notice that the tooth profile is defined by the cutter profile. (b) Schematic illustration of the form cutting process. (c) Form cutting of teeth on a bevel gear.

Gear

blank

Form

cutter

(a) (b) (c)


Pinion-Shaped Cutter

Figure 14.22 Production of gear teeth with a pinion-shaped cutter. (a) Schematic illustration of the process; (b) photograph of the process with gear and cutter motions indicated.


Gear Hobbing

Figure 14.23 Production of gears through the hobbing process. (a) A hob, along with a schematic illustration of the process; (b) production of a worm gear through hobbing.

Gearblank

Gearblank

Hob

Top view

(a)

Hob

Helical gear

Hob rotation

(b)

(b)


10150120 200 250 300 350 400 450

20

100

150

200

250

300

350

400

30

40

50

60

All

ow

able

ben

din

g s

tres

s n

um

ber

, S

b,

MP

a

ksi

Brinell hardness, HB

Grade 1

Through-hardenedThrough-h

ardened

Grade 2

Nitrided

Nitrided

Material Grade Allowable bending stress number

Through-hardened steels 12

Nitriding through- hardened steels

12

MPa

0.703 HB + 1130.533 HB + 88.3

0.0823 HB + 12.150.1086 HB + 15.89

ksi

0.0773 HB + 12.80.102 HB + 16.4

0.568 HB + 83.80.749 HB + 110

Allowable Bending Stress

Figure 14.24 Effect of Brinell hardness on allowable bending stress number for steel gears. (a) Through-hardened steels. Note that the Brinell hardness refers to the case hardness for these gears.


Material designation Grade Typical Allowable bending stress, σall,b Allowable contact stress, σall,b

Hardnessa lb/in.2 MPa lb/in.2 MPaSteel

Through-hardened 1 — See Fig. 14.24a See Fig. 14.252 — See Fig. 14.24a See Fig. 14.25

Carburized & hardened 1 55-64 HRC 55,000 380 180,000 12402 58-64 HRC 65,000b 450b 225,000 15503 58-64 HRC 75,000 515 275,000 1895

Nitrided and through- 1 83.5 HR15N See Fig. 14.24b 150,000 1035hardened 2 — See Fig. 14.24b 163,000 1125

Nitralloy 135M and 1 87.5 HR15N See Fig. 14.24b 170,000 1170Nitralloy N, nitrided 2 87.5 HR15N See Fig. 14.24b 183,000 1260

2.5% Chrome, nitrided 1 87.5 HR15N See Fig. 14.24b 155,000 10702 87.5 HR15N See Fig. 14.24b 172,000 11853 87.5 HR15N See Fig. 14.24b 189,000 1305

Cast IronASTM A48 gray cast Class 20 — 5000 34.5 50,000-60,000 345-415

iron, as-cast Class 30 174 HB 8500 59 65,000-75,000 450-520Class 40 201 HB 13,000 90 75,000-85,000 520-585

ASTM A536 ductile 60-40-18 140 HB 22,000-33,000 150-230 77,000-92,000 530-635(nodular) iron 80-55-06 179 HB 22,000-33,000 150-230 77,000-92,000 530-635

100-70-03 229 HB 27,000-40,000 185-275 92,000-112,000 635-770120-90-02 269 HB 31,000-44,000 215-305 103,000-126,000 710-870

BronzeSut > 40, 000 psi 5700 39.5 30,000 205

(Sut > 275GPa)Sut > 90, 000 psi 23,600 165 65,000 450

(Sut > 620GPa)

Allowable Bending and Contact Stress

Table 14.5 Allowable bending and contact stresses for selected gear materials.


Material Grade Allowable bending stress number

MPa ksi

Niralloy 12

0.594 HB + 87.760.784 HB + 114.81

0.0862 HB + 12.730.1138 HB + 16.65

2.5% Chrome 123

0.7255 HB + 63.890.7255 HB + 153.630.7255 HB + 201.91

0.1052 HB + 9.280.1052 HB + 22.280.1052 HB + 29.28Grade 1 - Nitralloy

250 300 350275 325

100

200

300

400

500

All

ow

able

ben

din

g s

tres

s n

um

ber

, S

b,

MP

a


Grade 3 - 2.5% Chrome


Grade 2 - Nitralloy


70

20

30

40

50

60

ksi

Allowable Bending Stress

Figure 14.24 Effect of Brinell hardness on allowable bending stress number for steel gears. (b) Flame or induction-hardened nitriding steels. Note that the Brinell hardness refers to the case hardness for these gears.


75150 200 250 300 350 400 450

100

600

700

800

900

1000

1100

1200

1300

125

150

175A

llow

able

conta

ct

stre

ss n

um

ber,

Sc, M

Pa

ksi


Grade 1

Grade 2

1400

2.22 HB +200 (MPa)

0.322 HB + 29.1 (ksi)Sc = {

Grade 2:

2.41 HB +237 (MPa)

0.349 HB + 34.3 (ksi)Sc = {

Grade 1:

Allowable Contact Stress

Figure 14.25 Effect of Brinell hardness on allowable contact stress number for two grades of through-hardened steel.


Number of load cycles, N

102 103 104 105 106 107 108 109 1010

Str

ess

cycl

e fa

ctor,

Yn

2.0

1.0

0.9

0.8

0.7

0.6

0.5

3.0

4.0400 HB: Y

N = 9.4518 N-0.148

Case Carb.: YN = 6.1514N

-0.1192

250 HB: YN = 4.9404 N-0.1045

Nitrided: YN = 3.517 N-0.0817

160 HB: YN = 2.3194 N-0.0538

YN = 1.3558 N-0.0178

YN = 1.6831 N-0.0323

Stress Cycle Factor

Figure 14.26 Stress cycle factor. (a) Bending stress cycle factor YN.


Str

ess

cycl

e fa

ctor,

Zn

2.0

1.1

1.0

0.9

0.8

0.7

0.6

0.5

Number of load cycles, N

102 103 104 105 106 107 108 109 1010

Nitrided

Zn = 1.249 N-0.0138

Zn = 2.466 N-0.056

Zn = 1.4488 N-0.023

1.5

Stress Cycle Factor

Figure 14.26 Stress cycle factor. (a) pitting resistance cycle factor ZN.


Reliability Factor

Table 14.6 Reliability factor, KR.

Probability of Reliability factora,survival, percent KR

50 0.70b

90 0.85b

99 1.0099.9 1.2599.99 1.50

a Based on surface pitting. If tooth breakageis considered a greater hazard, a largervalue may be required.

b At this value plastic flow may occur ratherthan pitting.


Har

dnes

s ra

tio f

acto

r,C

H

1.16

1.14

1.12

1.10

1.08

1.06

1.04

1.02

1.00180 200 250 300 350 400

Brinell hardness of gear, HB

Ra,p = 0.4 µm

(16 µin.)

Ra,p = 0.8 µm

(32 µin.)Rap = 1.6 µm (64 µin.)

For Rap > 1.6,

use CH = 1.0

Hardness Ratio Factor

Figure 14.27 Hardness ratio factor CH for surface hardened pinions and through-hardened gears.


Loads on Gear Tooth

Figure 14.24 Loads acting on an individual gear tooth.

Pitch circle

φWr

Wt

W


rf

t

x

(a) (b)

Wt

Wt

Wr

W

a

φ

bw

t

l

l

Loads and Dimensions of Gear Tooth

Figure 14.29 Loads and length dimensions used in determining tooth bending stress. (a) Tooth; (b) cantilevered beam.


Bending and Contact Stress Equations

Lewis Equation

AGMA Bending Stress Equation

!t =Wtpd

bwY

!t =WtpdKaKsKmKvKiKb

bwYj

Hertz Stress

AGMA Contact Stress Equation

pH = E ′(W ′

2!

)1/2

!c = pH(KaKsKmKv)1/2


Lewis Form Factor

Table 14.7 Lewis form factor for various numbers of teeth (pressure angle, 20°; full-depth involute).

Lewis LewisNumber form Number formof teeth factor of teeth factor

10 0.176 34 0.32511 0.192 36 0.32912 0.210 38 0.33213 0.223 40 0.33614 0.236 45 0.34015 0.245 50 0.34616 0.256 55 0.35217 0.264 60 0.35518 0.270 65 0.35819 0.277 70 0.36020 0.283 75 0.36122 0.292 80 0.36324 0.302 90 0.36626 0.308 100 0.36828 0.314 150 0.37530 0.318 200 0.37832 0.322 300 0.382


Spur Gear Geometry Factors

01512 20 25 30 40 60 80 275

.10

.20

.30

.40

.50G

eom

etry

fac

tor,

Yj

Number of teeth, N

Number of

teeth in

mating gear.

Load considered

applied at

highest point

of single-tooth

contact.

10001708550352517

Load applied at

tip of tooth

∞125

Figure 14.30 Spur gear geometry factors for pressure angle of 20° and full-depth involute profile.


Application and Size Factors

Table 14.8 Application factor as function of driving power source and driven machine.

Driven MachinesPower source Uniform Light shock Moderate shock heavy shock

Application factor, Ka

Uniform 1.00 1.25 1.50 1.75Light shock 1.20 1.40 1.75 2.25Moderate shock 1.30 1.70 2.00 2.75

Diametral pitch, pd, Module, m,in.−1 mm Size factor, Ks

≥ 5 ≤ 5 1.004 6 1.053 8 1.153 12 1.25

1.25 20 1.40

Table 14.9 Size factor as a function of diametral pitch or module.


Load Distribution Factor

where

Cmc ={1.0 for uncrowned teeth0.8 for crowned teeth

Km = 1.0+Cmc(CpfCpm+CmaCe)

Ce =

0.80 when gearing is adjusted at assembly0.80 when compatability between gear teeth is improved by lapping1.0 for all other conditions


0.60

0.50

0.40

0.30

0.20

0.10

0.00

b w/d p

= 2.00

1.50

1.00

0.50

For bw/dp < 0.5 use

curve for bw/dp = 0.5

Pin

ion p

roport

ion f

acto

r, C

pf

0.70

0 200 400 600 800 1000

Face width, bw (mm)

400 10 20 30

Face width, bw (in.)

Pinion Proportion Factor

Figure 14.31 Pinion proportion factor Cpf.

Cpf =

bw

10dp−0.025 bw ≤ 25 mm

bw

10dp−0.0375+0.000492bw 25 mm< bw ≤ 432 mm

bw

10dp−0.1109+0.000815bw− (3.53×10−7)b2w432 mm< bw ≤ 1020 mm


S

S/2S1

Pinion Proportion Modifier

Figure 14.32 Evaluation of S and S1.

Cpm ={1.0,(S1/S) < 0.1751.1,(S1/S)≥ 0.175


0 200 400 600 800 1000

Face width, bw (mm)

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.10

0.00

Mes

h a

lig

nm

ent

fact

or,

Cma

Face width, bw (in)

403020100

Open gea

ring

Commercial enclosed gear u

nits

Precision enclosed gear units

Extra-precision enclosed gear units

Cma = A + Bbw + Cbw

2

If bw is in inches:

Open gearing

Commercial enclosed gears

Precision enclosed gears

Extraprecision enclosed gears

0.247

0.127

0.0675

0.000380

0.0167

0.0158

0.0128

0.0102

-0.765 10-4

-1.093 10-4

-0.926 10-4

-0.822 10-4

Condition A B C

Open gearing

Commercial enclosed gears

Precision enclosed gears

Extraprecision enclosed gears

0.247

0.127

0.0675

0.000360

6.57 10-4

6.22 10-4

5.04 10-4

4.02 10-4

-1.186 10-7

-1.69 10-7

-1.44 10-7

-1.27 10-7

Condition A B C

If bw is in mm:

Mesh Alignment Factor

Figure 14.33 Mesh alignment factor.


Dynam

ic f

acto

r, K

v

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

Pitch line velocity, m/s

0 10 20 30 40 50

Qv = 5 Qv = 6

Qv = 7

Qv = 8

Qv = 9

Qv = 10

Qv = 11

"Very accurate" gearing

0

Pitch line velocity, ft/min

750050002500

Dynamic Factor

Figure 14.34 Dynamic factor as a function of pitch-line velocity and transmission accuracy level number.

FME_Ch14

Documents

allowable

brinell hardness

brinell hardness

form cutting

spur gears

case hardness

base circle

form cutter