Spur Gear 2103320 Des Mach Elem Mech. Eng. Department Chulalongkorn University
Spur Gear
2103320 Des Mach Elem Mech. Eng. Department
Chulalongkorn University
Introduction
Gear • Transmit power, rotation
• Change torque, rotational speed
• Change direction of rotation
Friction Gear
Less slip
Small gear “Pinion”
(usually driving)
Large gear “Gear”
+ +
Slip
Gear tooth nomenclature (1)
+ +
Pitch circle
Pitch cylinder
Pitch surface
+
Center distance
Same
transmission
The pitch circle is a theoretical circle
upon which all calculations are usually
based; its diameter is the pitch diameter.
Depend on the types of gears
Center distance
+ +
Gear tooth nomenclature (2)
• Circular pitch (CP) = π D/z • Diametral pitch (DP) = z/D (inch.) • Module (m) = D/z (mm)
D = pitch dia. z = number of teeth
• m = 25.4/DP • CP×DP = p
Module and tooth size
Module sizes shown are converted inch sizes
โมดูลเลือก โมดูลเลือก
อันดับสอง อันดับสอง
0.1 0.15 1.5 1.75
0.2 0.25 2 2.25
0.3 0.35 2.5 2.75
0.4 0.45 3 3.5
0.5 0.55 4 4.5
0.6 0.7 5 5.5
0.8 0.75 6 7
1 0.9 8 9
1.25 10 11
โมดูลนิยม โมดูลนิยม
Standard module Pressure angle Diametral Pitch
For inch Size gears
Module
For metric Size gears 14.5° 20°
1st choice 1st choice 2nd choice 2nd choice
Gear tooth nomenclature (3)
Pressure angle (deg.)
14.5 (FD) 20 (FD) 20 (Stub)
Addendum a m m 0.8m
Dedendum b 1.157m 1.25m m
FD: Full depth
Stub: Tooth is shorter than FD
Tooth height a+b
Working depth a1+a2
Top diameter (Addendum dia.)
Do = D+2a
Root diameter (Dedendum dia.)
Dr = D-2b
Clearance b-a
Involute Curve
Gear profile is designed to be involute curve. With this design the velocity ratio of a
gear pair is constant all the time.
Base cylinder is the circle used to construct an involute curve (not a dedendum circle
and cannot be located with the naked eye
Base cylinder
Involute curve
Construction of
an involute curve Involute curve
Pressure angle (1)
φ = Pressure angle (14.5°, 20°, 25°)
φcospb rr =
• Line of action (pressure line)
line that tangent with the base
circles and pass the pitch point
• During meshing, transmitted
force acts along the line of action
φ
Line of action (Pressure line) is perpendicular to
Contact surface at contact point
Line tangent with the
pitch circle
φ
φ
Pitch circle
Pressure angle (2)
φ
Line of action (Pressure line) is perpendicular to
Contact surface at contact point
Driving
Driven
Base circle
Line tangent with the
pitch circle
φ
φ
Pitch circle
Velocity ratio
ω1(rad/s) n1(rpm)
D1 z1
ω2(rad/s) n2(rpm)
D2 z2
1
2
1
2
2
1
2
1
zz
DD
nnm ====
ωω
ω
+
+
Gear ratio & Center distance (1)
Example Center distance 240 mm
module 3 mm
Velocity ratio 5:1
Pressure angle 20°
1
2
1
2
2
1
2
1
zz
DD
nnm ====
ωω
ω
1
2
1
2
15
zz
DDm ===ω
Gear dia : Pinion dia
5:1
Divide the center distance into 5+1 = 6 parts
Each part = 240/6 = 40 mm
R=200
R=40
240
40
Rpinion = 40, Dia. = 80
z = D/m = 80/3 = 26.6 No. of pinion teeth choose 27 teeth
No. of gear teeth zgear = 5 × zpinion = 5 × 27 = 135
Dia. Pitch D = m×z 81 405
Center distance = (Dpinion+Dgear)/2 243
Dia. Addendum = D+2a = D+2m 87 411
Dia. Base = D×cos(PA) = D×cos(20°) 76.115 380.58
Addendum = m 3
Dedendum = 1.25m 3.75
Tooth depth = a+b 6.75
Tooth thickness = πD/2z = πm/2 4.712
pinion gear
(240)
Gear ratio & Center distance (2)
z = D/m = 80/3 = 26.6 choose 26 teeth
Dia. Pitch D = m×z 78 402
Center distance = (Dpinion+Dgear)/2 240
Dia. Addendum = D+2a = D+2m 84 408
Dia. Base = D×cos(PA) = D×cos(20°) 73.30 377.76
Addendum = m 3
Dedendum = 1.25m 3.75
Tooth depth = a+b 6.75
Tooth thickness = πD/2z = πm/2 4.712
pinion gear
134 Velocity ratio = 134/26 = 5.15
Gear ratio & Center distance (3)
No. of pinion teeth
No. of gear teeth
Gear train
Pin Pout
Increase gear ratio Change direction of rotation
No. of teeth of gear G2 dose not affect
to the ratio
z1 z2
z3
z5
z4
z6
ωi
ωo
mω = ωi/ωo
mω = (z2/z1) (z4/z3) (z6/z5)
Planetary gear train
Epicyclic gear train, Planetary gear train
• Sun gear
• Planet carrier
• Ring gear
Each part can be set to be input,
output or fixed
Various gear ratio
can be obtained
All parts rotate around the center point
Sun gear
Planet carrier
Planetary gear train
Epicyclic gear train, Planetary gear train
Zs = 60
Zp = 20
ZR = 100
Fixed member
Input
(rpm)
Planet (rpm)
Output (rpm)
Ratio (ωi/ωo)
Ring fix Carrier
9 36
Sun
24 0.375
Sun fix Carrier
9 36
Ring 14.4
0.625
Carrier fix
sun 9
27 Ring 5.4
1.667
Planetary gear train
http://www.diseno-art.com/encyclopedia/terms/automatic_transmission.html
Gear manufacture
Methods of machining
Form milling : used mostly for large gears. A
milling cutter that has the shape of the tooth
space is used.
Shaping : frequently 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.
Casting : used most often to make blanks for
gears which will have cut teeth. Possible to
use to make toothed gears with little or no
machining
Milling cutter Hobbing
Shaping
VDO
Gear quality (1)
Quality in gearing is the precision of the individual gear teeth and the precision with
which two gears rotate in relation to one another.
• Runout
• Tooth-to-tooth spacing
• Profile
Schematic diagram of a typical
gear rolling fixture Chart of gear-tooth errors of a typical gear
when run with a specific gear in a rolling
fixture.
Gear quality (2)
• Gear quality is specified by AGMA as quality numbers.
• Quality numbers range from 5 to 15 with increasing precision (AGMA)
• Besides AGMA standard, there are also other standards (JIS, DIN, ISO-similar to DIN)
Selected values for total composite tolerance Table of Gear Precision Grade
Gear quality (3)
Recommended AGMA quality numbers
Pitch line speed:
Velocity at the pitch circle v
pitch circle
Materials
Steel:
• medium-carbon steel is usually used
• Due to low surface endurance capacity, heat treatment (Flame hardening, Induction
hardening, Carburizing, Nitriding) is required
• Surface finishing (grinding) is probably done after heat treatment to obtain high
precision gear
Cast iron:
Cheap, ASTM grade 20, 30, 40, 50, 60 are normally used
surface fatigue strength is higher than bending fatigue strength
Quieter than steel due to the damping property of cast iron
Ductile or nodular cast-iron is probably used to increase strength of gear
Gear force analysis (1)
Pinion
Gear
Base circle Pitch circle
Pressure line
φ
Pinion
Pressure line
φ
F
Fb
Fr
Gear
Pressure line φ
F
Fb
Fr
FBD
Neglect sliding friction in FBD
Gear force analysis (2)
φcosFFb =
φsinFFr =
rFrFT bb ⋅=⋅=
Pinion
Pressure line
φ
F
Fb
Fr
Gear
Pressure line φ
F
Fb
Fr
FBD
( ) ( ) ωωω ⋅⋅=⋅⋅== rFrFTP bb
From FBD
Transmitted torque can be calculated from
Transmitted power can be calculated from
rb = base radius
r = pitch radius
Gear tooth strength
Photoelasic pattern of stresses in a spur gear tooth
Max. stress
position
Contact point Tooth root
Failure Pitting
Crack
Design
principle
Contact stress
Hertzian stress
Bending stress
Lewis equation
Gear tooth bending stress (1)
Fb
L
t
b
Lewis equation
Assumptions
1. Gear tooth is considered as a cantilever beam
2. All transmitted force acts at the tooth tip of a tooth
pair
3. Fr is neglected
4. Sliding friction is neglected
23
6)12/()2/(
btLF
bttLF
IMc bb ===σ
LbtFb 6
2σ=
Gear tooth bending stress (2)
For constant bending stress
22 const.)(6
ttFbL
b
=
=
σ
The relation between L and t is parabola.
Tooth tip area has more material than parabola,
hence the failure will occur at the tooth root (BED)
From triangle relation
LbtFb 6
2σ=
Lt
tx 2/2/
=x
tL4
2
=
Hence bypppxbFb σσ =
=
32
p : circular pitch
y : Lewis form factor
Y : Lewis form factor
P : Diametral pitch
m : module
PbYFb
σ=
bYmFb σ=
bYPFb=σ
bYmFb=σ
No Teeth Load Near Tip of Teeth Load at Near Middle of Teeth
14 1/2 deg 20 deg FD 20 deg Stub 25 deg 14 1/2 deg 20 deg FD Y y Y y Y y Y y Y y Y y
10 0,176 0,056 0,201 0,064 0,261 0,083 0,238 0,076 11 0,192 0,061 0,226 0,072 0,289 0,092 0,259 0,082 12 0,21 0,067 0,245 0,078 0,311 0,099 0,277 0,088 0,355 0,113 0,415 0,132 13 0,223 0,071 0,264 0,084 0,324 0,103 0,293 0,093 0,377 0,12 0,443 0,141 14 0,236 0,075 0,276 0,088 0,339 0,108 0,307 0,098 0,399 0,127 0,468 0,149 15 0,245 0,078 0,289 0,092 0,349 0,111 0,32 0,102 0,415 0,132 0,49 0,156 16 0,255 0,081 0,295 0,094 0,36 0,115 0,332 0,106 0,43 0,137 0,503 0,16 17 0,264 0,084 0,302 0,096 0,368 0,117 0,342 0,109 0,446 0,142 0,512 0,163 18 0,27 0,086 0,308 0,098 0,377 0,12 0,352 0,112 0,459 0,146 0,522 0,166 19 0,277 0,088 0,314 0,1 0,386 0,123 0,361 0,115 0,471 0,15 0,534 0,17 20 0,283 0,09 0,32 0,102 0,393 0,125 0,369 0,117 0,481 0,153 0,544 0,173 21 0,289 0,092 0,326 0,104 0,399 0,127 0,377 0,12 0,49 0,156 0,553 0,176 22 0,292 0,093 0,33 0,105 0,404 0,129 0,384 0,122 0,496 0,158 0,559 0,178 23 0,296 0,094 0,333 0,106 0,408 0,13 0,390 0,124 0,502 0,16 0,565 0,18 24 0,302 0,096 0,337 0,107 0,411 0,131 0,396 0,126 0,509 0,162 0,572 0,182 25 0,305 0,097 0,34 0,108 0,416 0,132 0,402 0,128 0,515 0,164 0,58 0,185 26 0,308 0,098 0,344 0,109 0,421 0,134 0,407 0,13 0,522 0,166 0,584 0,186 27 0,311 0,099 0,348 0,111 0,426 0,136 0,412 0,131 0,528 0,168 0,588 0,187 28 0,314 0,1 0,352 0,112 0,43 0,137 0,417 0,133 0,534 0,17 0,592 0,188 29 0,316 0,101 0,355 0,113 0,434 0,138 0,421 0,134 0,537 0,171 0,599 0,191 30 0,318 0,101 0,358 0,114 0,437 0,139 0,425 0,135 0,54 0,172 0,606 0,193 31 0,32 0,101 0,361 0,115 0,44 0,14 0,429 0,137 0,554 0,176 0,611 0,194 32 0,322 0,101 0,364 0,116 0,443 0,141 0,433 0,138 0,547 0,174 0,617 0,196 33 0,324 0,103 0,367 0,117 0,445 0,142 0,436 0,139 0,55 0,175 0,623 0,198 34 0,326 0,104 0,371 0,118 0,447 0,142 0,44 0,14 0,553 0,176 0,628 0,2 35 0,327 0,104 0,373 0,119 0,449 0,143 0,443 0,141 0,556 0,177 0,633 0,201 36 0,329 0,105 0,377 0,12 0,451 0,144 0,446 0,142 0,559 0,178 0,639 0,203 37 0,33 0,105 0,38 0,121 0,454 0,145 0,449 0,143 0,563 0,179 0,645 0,205 38 0,333 0,106 0,384 0,122 0,455 0,145 0,452 0,144 0,565 0,18 0,65 0,207 39 0,335 0,107 0,386 0,123 0,457 0,145 0,454 0,145 0,568 0,181 0,655 0,208 40 0,336 0,107 0,389 0,124 0,459 0,146 0,457 0,145 0,57 0,181 0,659 0,21 43 0,339 0,108 0,397 0,126 0,467 0,149 0,464 0,148 0,574 0,183 0,668 0,213 45 0,34 0,108 0,399 0,127 0,468 0,149 0,468 0,149 0,579 0,184 0,678 0,216 50 0,346 0,11 0,408 0,13 0,474 0,151 0,477 0,152 0,588 0,187 0,694 0,221 55 0,352 0,112 0,415 0,132 0,48 0,153 0,484 0,154 0,596 0,19 0,704 0,224 60 0,355 0,113 0,421 0,134 0,484 0,154 0,491 0,156 0,603 0,192 0,713 0,227 65 0,358 0,114 0,425 0,135 0,488 0,155 0,496 0,158 0,607 0,193 0,721 0,23 70 0,36 0,115 0,429 0,137 0,493 0,157 0,501 0,159 0,61 0,194 0,728 0,232 75 0,361 0,115 0,433 0,138 0,496 0,158 0,506 0,161 0,613 0,195 0,735 0,234 80 0,363 0,116 0,436 0,139 0,499 0,159 0,509 0,162 0,615 0,196 0,739 0,235 90 0,366 0,117 0,442 0,141 0,503 0,16 0,516 0,164 0,619 0,197 0,747 0,238 100 0,368 0,117 0,446 0,142 0,506 0,161 0,521 0,166 0,622 0,198 0,755 0,24 150 0,375 0,119 0,458 0,146 0,518 0,165 0,537 0,171 0,635 0,202 0,778 0,248 200 0,378 0,12 0,463 0,147 0,524 0,167 0,545 0,173 0,64 0,204 0,787 0,251 300 0,38 0,122 0,471 0,15 0,534 0,17 0,554 0,176 0,65 0,207 0,801 0,255
Rack 0,39 0,124 0,484 0,154 0,55 0,175 0,566 0,18 0,66 0,21 0,823 0,262
Lewis Form Factor
Y, y
Load near tip of teeth
• Total transmitted force
acts at the tooth tip
(actually at this point
double teeth meshing
occurs, hence the force
acting on a tooth pair is
reduced)
• Safer design
Load near middle of teeth
• Total transmitted force
acts at the position near
the middle of teeth (single
tooth meshing occurs)
• Close to the actual
condition
Gear tooth bending stress (3)
Lewis form factor, Y or y 1. Y and y increase when no. of teeth is increased
2. More Y or y, gear tooth can withstand more Fb
3. Pinion has lower no. of gear teeth than gear. Pinion can
withstand low load than gear.
4. Calculation is done at pinion bY
PFb=σ
bYmFb=σ
The effect of Fr 1. Compressive stress at B side increase, but tensile stress at
D side decrease.
2. Most material can withstand compressive force more than
tensile force. Fatigue will occur at tensile side.
3. The presence of Fr will increase strength of gear tooth,
since it help to reduce the stress in tensile side
AGMA Stress Equation (bending)
American Gear Manufacturers Association (AGMA) proposed the method to design gear
based on the Lewis equation (bending stress consideration)
bYmFb=σ
Lewis Equation AGMA Equation (bending)
vBmsOJ
tt KKKKK
mFYWs =
Y JYYJ : Geometry factor is Y that include the effect of root fillet stress-concentration factor
KO : Overload factor
Ks : Size factor
Km : Load-distribution factor
KB : Rim thickness factor
Kv : Dynamic factor
K >= 1.0 (Always)
σ
bFts
tWb F
Geometry factor (YJ)
Bending strength geometry factor YJ for full depth teeth spur gears
vBmsOJ
tt KKKKK
mFYWs =
20° pressure angle 25° pressure angle
Overload factor (KO)
Suggested overload factors, KO
Driven Machine
Power source Uniform Light shock
Moderate
shock
Heavy
shock
Uniform 1.00 1.25 1.50 1.75
Light shock 1.20 1.40 1.75 2.25
Moderate shock 1.30 1.70 2.00 2.75
Power sources Uniform: Electric motor, Constant-
speed gas turbine
Light shock: Water turbine, variable-
speed drive
Moderate shock: Multicylinder engine
Driven machine Uniform: Continuous-duty generator
Light shock: Fans and low speed centrifugal pump, variable-duty
generators, uniform loaded conveyors
Moderate shock: High-speed centrifugal pumps, reciprocating pumps
and compressors, heavy duty conveyors, machine tool drives
Heavy shock: Rock crushers, punch press drivers
vBmsOJ
tt KKKKK
mFYWs =
Size factor (Ks)
Suggested size factors, Ks
Diametral pitch, Pd Metric module, m Size factor, Ks
>=5 <=5 1.00
4 6 1.05
3 8 1.15
2 12 1.25
1.25 20 1.40
vBmsOJ
tt KKKKK
mFYWs =
Load dist. factor (Km)
Used to reflect nonuniform distribution of load across the line of contact.
If load is uniformly distributed Km = 1.0
Causes of nonuniform distribution
1. Gear tooth error
2. Misalignment, Eccentricity
3. Deformation of gear, shaft, bearing,
housing
4. Clearance between shaft, gear, bearing,
housing
5. Deformation from the temperature
6. Gear tooth modification (crowning, end
relief)
The method to reduce Km (min = 1.0)
1. Use high quality gear (high quality
number)
2. Narrow face widths
3. Locate gear at the center between two
bearings
4. Short shaft spans between bearing
5. Large shaft diameters (high stiffness)
6. Rigid stiff housings
7. High precision, small clearance on all
drive components
vBmsOJ
tt KKKKK
mFYWs =
Load dist. factor (Km)
Load distribution factor (Km) can be
calculated by
( )emapmpfmcm CCCCCK ++= 0.1
Cmc = 1 for uncrowned teeth
Cmc = 0.8 for crowned teeth
Cmc = lead correction factor
Cpm = pinion proportion modifier
Cpm = 1 S1/S < 0.175
Cpm = 1.1 S1/S ≥ 0.175
Cpf = pinion proportion factor
025.010
−=d
FCpf
Fd
FCpf 0125.00375.010
+−=
in 1 ≤F
in 17 1 ≤< F
d, dp = pinion diameter
vBmsOJ
tt KKKKK
mFYWs =
Load dist. factor (Km)
Load distribution factor (Km) can be
calculated by
( )emapmpfmcm CCCCCK ++= 0.1
Cma = mesh alignment factor
Ce = mesh alignment correction factor
Ce = 0.8 for gearing adjusted at assembly, compatibility is improved by lapping
Ce = 1 for all other conditions
2CFBFACma ++=
Condition A B C
Open gearing 0.247 0.0167 -0.765(10-4)
Commercial, enclosed units 0.127 0.0158 -0.093(10-4)
Precision, enclosed units 0.0675 0.0128 -0.926(10-4)
Extraprecision enclosed gear
units
0.00360 0.0102 -0.822(10-4)
* Face width F in Inches
vBmsOJ
tt KKKKK
mFYWs =
Rim thickness factor (KB) vBmsOJ
tt KKKKK
mFYWs =
Used when the rim thickness is not sufficient to provide full support for the tooth root.
Bb m
K 242.2ln6.1=
1=bK mB ≥ 1.2
mB < 1.2
t
RB h
tm =
Dynamic factor (Kv) vBmsOJ
tt KKKKK
mFYWs =
To compensate the effect of vibration, dynamic unbalance that will increase load that
the gear tooth must withstand especially at high velocity.
B
tv A
vCAK
+=
)0.1(5650 BA −+=667.0)12(25.0 vQB −=
m/sin for 200
ft/minin for 1
t
t
vC
vC
=
=
Qv = Gear quality number
vt = Pitch line velocity
Maximum recommended pitch line velocity:
2
2
max,)]3([
CQAv v
t−+
=
Selection of material (bending stress)
vBmsOJ
tt KKKKK
mFYWs =
R
Natat KSF
Yss⋅
=′<
AGMA Equation (bending)
(Load that the gear must withstand)
Adjusted Allowable Bending
Stress Numbers
(Depend on the material property)
sat : Allowable bending stress
YN : Bending strength stress cycle number
KR : Reliability factor
SF : factor of safety (design decision)
KR : Reliability factor
Reliability KR
0.90, one failure in 10 0.85
0.99, one failure in 100 1.00
0.999, one failure in 1000 1.25
0.9999, one failure in 10000 1.50
SF : factor of safety (design decision)
• To compensate any uncertainty in analysis,
material property, or error in manufacturing
• Most of factors are included in AGMA
equation, hence the recommended SF is
around 1.00-1.50
Allowable bending stress, Sat
Allowable bending stress number for
through-hardened steels
Allowable bending stress number for
nitriding steel gears
Bending strength stress cycle number, Yn
LnqN 60= N : expected number of cycles of loading
L : design life in hours
n : rotational speed of the gear (rpm)
q : number of load applications per revolution
Bending strength stress cycle number, Yn
Application Design life (h)
Domestic appliances 1,000-2,000
Aircraft engines 1,000-4,000
Automotive 1,500-5,000
Agricultural equipment 3,000-6,000
Elevators, industrial fan, multipurpose gearing 8,000-15,000
Motors, industrial blowers, general industrial machines 20,000-30,000
Pumps and compressors 40,000-60,000
Critical equipment in continuous 24-h operation 100,000-200,000
Gear tooth contact stress (1)
r1
r2
Driving Driven
r1
r2
W
W
+
+
+
+
r1
r2
W
W
F
• The contact of gear teeth can be modeled as the contact of two cylinders
• r is radius of involute curve at the contact point (not the radius of pitch cylinder)
• Radius r is changed along the meshing position
• W is force in the direction of pressure line and
• Stress at contact point can be calculated by Hertzian stress equation
Base circle
rWrWT tb ⋅=⋅= tWW =φcos
Gear tooth contact stress (1)
+
+
rp
rG
W
W
F
2/1
2221
21
21
])1()1([)/1/1(
−+−
+=
EEFrrW
ννπσ
Hertzian stress can be calculated from
( ) 2sin2 φGdr =
( ) 2sin1 φPdr =
φcostWW =
Define Cp (Elastic coefficient) 2/1
2221
21 ])1()1([
1
−+−
=EE
Cp ννπ
2/111
sincos2
+=
GP
tp ddF
WCφφ
σ
2/11
sincos2
+=
G
G
P
tp m
mFd
WCφφ
σ
2/1
=
IFdWC
P
tpσ
12sincos
+=
G
G
mmI φφ
PGPGG ddZZm ==
Define I (Geometry factor)
Basic equation related closely
with AGMA for contact stress
AGMA Stress Equation (contact)
2/1
=
IFdWC
P
tpσ
Hertzian stress AGMA Equation (Contact)
2/1
= vmsO
P
tpc KKKK
IFdWCs
KO : Overload factor
Ks : Size factor
Km : Load-distribution factor
Kv : Dynamic factor
Can be found as same as
the bending stress cal. 12
sincos+
=G
G
mmI φφ
2/1
2221
21 ])1()1([
1
−+−
=EE
Cp ννπ
โดย
dP : Pitch diameter (pinion)
F : Face width
Elastic coefficient, Cp
2/1
= vmsO
P
tpc KKKK
IFdWCs
2/1
2221
21 ])1()1([
1
−+−
=EE
Cp ννπ1 - Pinion
2- Gear
Subscript:
Geometry factor, I 2/1
= vmsO
P
tpc KKKK
IFdWCs
12sincos
+=
G
G
mmI φφ PGPGG ddZZm ==
Selection of material (Contact stress)
R
HNacac KSF
CZss⋅
=′<
AGMA Equation (contact)
(Load that the gear must withstand)
Adjusted Allowable Contact
Stress Numbers
(Depend on the material property)
sac : Allowable contact stress
ZN : Pitting resistance stress cycle number factor
CH : Hardness ratio factor
KR : Reliability factor
SF : factor of safety (design decision)
KR : Reliability factor
Reliability KR
0.90, one failure in 10 0.85
0.99, one failure in 100 1.00
0.999, one failure in 1000 1.25
0.9999, one failure in 10000 1.50
SF : factor of safety (design decision)
2/1
= vmsO
P
tpc KKKK
IFdWCs
As same as
bending case
• To compensate any uncertainty in analysis,
material property, or error in manufacturing
• Most of factors are included in AGMA
equation, hence the recommended SF is
around 1.00-1.50
Allowable contact stress, Sac
Repeatedly Applied Contact
Strength Sc at 107 Cycles and
0.99 Reliability for Steel Gears
ANSI/AGMA 2001-D04.
Contact-fatigue strength Sc at 107 cycles
and 0.99 reliability for through-hardened
steel gears.
ANSI/AGMA 2001-D04 and 2101-D04.
Pitting resistance stress cycle number factor, ZN
LnqN 60= N : expected number of cycles of loading
L : design life in hours
n : rotational speed of the gear (rpm)
q : number of load applications per revolution
As same as bending case
Hardness ratio factor, CH
• Pinion is smaller and rotate faster
than gear
• If the surface of pinion is harder than
gear, the capacity of pitting
resistance is increase
• HBP : Brinell hardness of pinion
• HBG : Brinell hardness of gear
• CH is used for gear only (does not
use for pinion calculation)
• If the hardness of pinion and gear
are equal, CH = 1
Design Guidelines
1. Gear ratio should be less than 1:6 (Too much gear ratio will bring about interference
problem and large gear will cause the weight and size problem).
2. If gear ratio more than 1:6 is required, multi-stages gear reduction should be used.
3. Recommended face width is around 8m < F < 16m (commonly use 12m).
4. Very large face width will bring about the alignment and load distribution problem.
5. At the same center distance, the gear pair with small teeth + more no. of teeth is
quieter than the gear pair with large teeth + less no. of teeth.
6. Small no. of teeth make the compact gear pair, but the interference must be checked.
Design Guidelines
Data for all curves:
mG = 4, NP = 24
KO = 1.0, Class 1 service
20° full-depth teeth
Example
A gear pair is to be designed to transmit 15 kW of power to a large meat grinder in a
commercial meat processing plant. The pinion is attached to the shaft of an electric motor
rotating at 575 rpm. The gear must operate at 270-280 rpm. The gear unit will be enclosed
and of commercial quality. Commercially hobbed (quality number 5), 20°, full depth, involute gears are to be used in the metric module system. The maximum center distance is to be 200
mm. Specify the design of the gears. [Ex.9-6 Machine Elements in Mechanical Design. Robert L. Mott]