fme_ch14
TRANSCRIPT
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
Hamrock • Fundamentals of Machine Elements
Spur Gears
Figure 14.1 Spur gear drive. (a) Schematic illustration of meshing spur gears; (b) a collection of spur gears.
Hamrock • Fundamentals of Machine Elements
Helical Gears
Figure 14.2 Helical gear drive. (a) Schematic illustration of meshing helical gears; (b) a collection of helical gears.
Hamrock • Fundamentals of Machine Elements
Bevel Gears
Figure 14.3 Bevel gear drive. (a) Schematic illustration of meshing bevel gears; (b) a collection of bevel gears.
Hamrock • Fundamentals of Machine Elements
(a) (b)
Worm Gears
Figure 14.4 Worm gear drive. (a) Cylindrical teeth; (b) double enveloping; (c) a collection of worm gears.
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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:
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
N2
N1
N1
N2 N5
N6
N7 N8N3 N4
Figure 14.16 Simple gear train.
Gear Trains
Figure 14.17 Compound gear train.
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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:
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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)
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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)
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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
Brinell hardness, HB
Grade 3 - 2.5% Chrome
Grade 2 - 2.5% Chrome
Grade 2 - Nitralloy
Grade 1 - 2.5% Chrome
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.
Hamrock • Fundamentals of Machine Elements
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
Brinell hardness, HB
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.
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
Loads on Gear Tooth
Figure 14.24 Loads acting on an individual gear tooth.
Pitch circle
φWr
Wt
W
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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
Hamrock • Fundamentals of Machine Elements
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.
Hamrock • Fundamentals of Machine Elements
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.