7/30/2019 HES3350 Machine Design, Assignment 2: Gear Specification with Consideration of Bending Fatigue and Surface Du
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SWINBURNE UNIVERSITY OF TECHNOLOGY (SARAWAK CAMPUS)
FACULTY OF ENGINEERING AND INDUSTRIAL SCIENCE
HES3350 Machine DesignSemester 1, 2012
Assignment 2: Gear Specification with Consideration of Bending Fatigue and Surface
Durability
By
Group 9
Stephen, P. Y. Bong (4209168)
Kueh Min Hui (4209974)
Mohd Azmudin Bin Isa @ Danial (4209028)
Jimmy Huong How Lee (4209761)
Lecturer: Dr. Soon Kok Heng
Due Date: 4 pm, 27th
April 2012 (Thursday)
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1. OBJECTIVEThe principal objective of this assignment is to select appropriate spur gears for a two-stage
compound reverted gear train from the gear catalog provided. Apart from that, the computations of
the maximum module, the face width, and the material used for the gears with the consideration of
bending fatigue and surface durability are also the purposes of this assignment as well.
2. INTRODUCTIONWith the operating parameters provided such as the amount of power to be delivered, the number of
teeth for Gear 1, 2, 3 and 4, the input speed, the maximum gearbox height, the clearance and wall
thickness, and the pressure angle; we are required to determine the maximum module and select
appropriate spur gears for the two-stage compound reverted gear train from the gear catalog given
based on the prerequisite stated which is the safety factor should not exceed 2.2. In addition, the
determination of the face width and the material used, and the identification of the most critical gear
are incorporated in this report as well. The spur gears are selected from the gear catalog with the
consideration of bending fatigue and surface durability.
Since the assignment by all means, cannot be tantamount to a real life design project as most of the
selection of operating parameters such as the dynamic factor and the surface factor for the
computations of bending stresses and endurance limit respectively are based on experimental data
or charts provided and appropriate assumptions for instance the surfaces in contact (bore and shaft)
are frictionless and thus there has no power losses through the transmission. By neglecting the
power losses due to friction and surface roughness as a result of various manufacturing
methodology and heat treatment processes, we only centralize our design based on the bending
fatigue and surface durability in the selection of spur gears.
3. MAIN CONTENT3.1 Parameters and Schematic Diagram for the Two-stage Compound Reverted Gear
Train in the Gearbox
The schematic diagram of the gearbox is shown in Fig. 1 below:
Fig. 1: Schematic of the gearbox
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The operating parameters given are tabulated in Table 1 below:
Power to be delivered, 9 kWN1 &N3 42 Teeth
N2 &N4 86 Teeth
Input speed, 1 2300 rpm
Maximum Gearbox Height,H 600 mmClearance and Wall Thickness 20 mm
Pressure angle for spur gear, 20Expected life for 40 hr./week, 50 week/year operation 3.25 years
Table 1: Operating parameters
where the Maximum Gearbox Height,His given by the following equation:
ThicknessWallClearance2
22
412 +++++= m
dddH
where m is the module and d1 = mN1, d2 = mN2 and d4 = mN4.
3.2 Calculations and Selection of Maximum Module and Face WidthBased on the operating parameters given, the module of the gear train can be determined by using
the relation of gears diameters and maximum height of gearbox:
ThicknessWallClearance222
412 +++++= m
dddH
SinceN2 =N4, therefore, d2 = d4 and the equation above can be expressed as:
ThicknessWallClearance22
3
2
ThicknessWallClearance222
21
21
2
++++=
+++++=
mdd
H
mdd
dH
Substituting d1 = mN1 and d2 = mN2 into equation above yields:
22
3
2
Thickness)Wall(Clearance
21++
+=NN
Hm
Since the KHK Gear Catalog does not have spur gears with 86 teeth, therefore, spur gears with 85
teeth has been selected in the analysis. Given that N1 = 42 Teeth,N2 = 85 Teeth,H= 600 mm and
Clearance + Wall Thickness = 20 mm, hence the module can be computed as follows:
Toothmm3.8538=
++
=
++
+=
22
Teeth)85(3
2
Teeth42
mm20mm600
22
3
2
)ThicknessWallClearance(
21 NN
Hm
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Based on the computations of module above, the module employed for the two-stage compound
reverted gear train must not exceed m = 3.85388 mm/Tooth. In addition, according to KHK Gear
Catalog, spur gears with module of 2 have the closest number of teeth (N= 85 Teeth) to the number
of teeth for gears 2 and 4 given in this assignment. As a result, steel spur gears with 85 Teeth and
module 2 has been chosen in the analysis of gear tooth strength and surface durability.
According to KHK Gear Catalog, Steel Spur Gears with 42 and 85 Teeth with module of 2 haveface widths of 20 mm (See KHK Gear Catalog in Appendix):
mm20Toothmm20
Teeth42146)pp.Catalog,Gear(KHK
31=
=
== b
m
NN
mm20Toothmm20
Teeth85156)pp.Catalog,Gear(KHK
42=
=
== b
m
NN
Therefore, b1 = b2 = b3 = b4 = 20 mm.
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3.3 Analysis of Gear Tooth Strength3.3.1 Bending Fatigue Analysis by Using Steel Spur Gear with Module 2For m = 2 mm/Tooth,
( )( )( )( )
( )( )
( )( ) mm170Teeth58Toothmm2
mm84Teeth42Toothmm2
mm170Teeth58Toothmm2mm84Teeth42Toothmm2
44
33
22
11
===
===
===
===
mNd
mNd
mNdmNd
The gear ratio for a pair of spur gears (pinion and gear) is given by:
p
g
g
p
d
d=
where
g = Angular velocity of gearp = Angular velocity of piniondg = Diameter of gear
dp = Diameter of pinion
By using the relation of gear ratio above, the angular velocity of gear 2 can be obtained:
rpm1136.47=
=
== mm170
mm48rpm2300
2
112
1
2
2
1
d
d
d
d
Since gear 2 and gear 3 are located on the same shaft, therefore,
rpm1136.47== 23
The speed ratio for double-reduction gear train is given by:
rpm561.55=
=
==
8585
4242rpm2300
42
3114
31
42
4
1
NN
NN
NN
NN
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Stage 1 (Gear 1 & Gear 2)
Pitch Line Velocity,60000
dV=
sm10.12=
===
===1Stage
222
11
1
sm12.0160000
rpm)7mm)(1136.4170(
60000:2Gear
sm12.0160000
rpm)mm)(230084(
60000:1GearV
dV
d
V
Tangential Forces, Ft= /V
N889.33=
===
===
1Stage,
1Stage2
1Stage1
N33.889sm10.12W9000:2Gear
N33.889sm10.12W9000:1Gear
t
t
tF
VWF
VWF
Gear Tooth Bending Stress, movt KKK
bJm
F=
From Figure 14-6 (Budynas, R. G. & Nisbett, J. K., pp. 753), the Geometry Factor for spur
gears of 42 Teeth and 85 Teeth areJ1, 30.2725 andJ2,40.295.
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minft1992.13===m.30430
ft1
min1
s60
s
m12.10sm12.101StageV
Velocity (Dynamic) Factor,78
78 VKv
+= (Highest precision, shaved and ground)
1.2539=+
=+
==78
13.199278
78
78 1StageVKK vv
From TABLE 15.1 (Juvinall, R. C. & Marshek, K. M., pp. 615), the Overload Correction
Factor, Ko = 1 (Uniform source of machinery and power)
From TABLE 15.2 (Juvinall, R. C. & Marshek, K. M., pp. 616), the Mounting Factor,
Km = 1.3 (Accurate mounting, small bearing clearance, minimum deflection)
MPa132.99=== )3.1)(1)(2539.1(Tooth)mm)(2mm)(0.272520(
N33.889
11
1Stage,
1 mov
tKKK
mJb
F
MPa122.85=== )3.1)(1)(2539.1(Tooth)mm(2mm)(0.295)20(
N33.889
22
1Stage,
2 mov
tKKK
mJb
F
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Stage 2 (Gear 3 & Gear 4)
Pitch Line Velocity,60000
dV=
sm4.8895=
===
===2Stage
444
33
3
sm9985.460000
rpm)mm)(561.55170(
60000:4Gear
sm9985.460000
rpm)7mm)(1136.484(
60000:3GearV
dV
d
V
Tangential Forces, Ft= /V
N1800.54=
===
===
2Stage,
2Stage4
2Stage3
N54.1800sm4.9985W9000:4Gear
N54.1800sm4.9985W9000:3Gear
t
t
tF
VWF
VWF
Gear Tooth Bending Stress, movt KKK
bJm
F=
From Figure 14-6 (Budynas, R. G. & Nisbett, J. K., pp. 753), the Geometry Factor for spur
gears of 42 Teeth and 85 Teeth areJ1, 30.2725 andJ2,40.295.
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minft983.96===m.30430
ft1
min1
s60
s
m9985.4sm9985.42StageV
Velocity (Dynamic) Factor,78
78 VKv
+= (Highest precision, shaved and ground)
1.1844=+
=+
==78
96.98378
78
78 1StageVKK vv
From TABLE 15.1 (Juvinall, R. C. & Marshek, K. M., pp. 615), the Overload Correction
Factor, Ko = 1 (Uniform source of machinery and power)
From TABLE 15.2 (Juvinall, R. C. & Marshek, K. M., pp. 616), the Mounting Factor,
Km = 1.3 (Accurate mounting, small bearing clearance, minimum deflection)
MPa269.19=== )3.1)(1)(2539.1(Tooth)mm)(2mm)(0.272520(
N54.1800
33
2Stage,
3 mov
tKKK
mJb
F
MPa248.73=== )3.1)(1)(2539.1(Tooth)mm(2mm)(0.295)20(
N54.1800
44
2Stage,
4 mov
tKKK
mJb
F
By comparing the four bending stresses above, the maximum bending stress is max = 3 = 269.19
MPa.
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The gear in the gear train which is subjected to the highest tooth bending stresses is the one which
has the largest possibility to fail in advance. Based on the stresses computed above, max = 269.19MPa and therefore the critical gear is Gear 3.
Refined Analysis of Gear-Tooth Bending Strength Endurance Limit
msKKKCCCSS trSGLnn =
whereun SS 5.0=
Since the Steel Spur Gears are made of S45C Steel with a maximum Brineel Hardness of 194 HB,
therefore, the tensile strength, Su, can be determined by using the relation:
Su = Kh(HB) = (3.45)(194) MPa = 669.3 MPa (Kh = 3.45 in SI Units)
( ) MPa65.334MPa3.6695.05.0 === un SS
Load Factor, CL = 1 (Bending loads)
)5(1
in.Teeth7.12
mm25.4
in.1mm170
Teeth58
in.Teeth7.12
mm25.4in.1mm84
Teeth42
in.Teeth7.12
mm25.4
in.1mm170
Teeth58
in.Teeth7.12
mm25.4
in.1mm84
Teeth42
4
44
3
33
2
22
1
11
>=
=
==
=
==
=
==
=
==
PC
D
NP
D
NP
D
NP
D
NP
G
From TABLE 15.3 (Juvinall, R. C. & Marshek, K. M., pp. 615), the Reliability Correction
Factor, Kr= 1 (50% Reliability)
Temperature Factor, Kt= 1 (Assumed the temperature lower than 160 F)
Mean Stress Factor, Kms = 1 (Gear 3 is idler)
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From FIGURE 8.13 (Juvinall, R. C. & Marshek, K. M., pp. 301), the Surface Factor,
CS= 0.76 (Machined or cold-drawn)
Therefore, the endurance limit is computed as follows:
MPa254.334=
=
=
1)76)(1)(1)(MPa)(1)(0.(334.65
msKKKCCCSS trSGLnn
Safety Factor, 0.9448===MPa19.269
MPa334.254SF
max
nS
Since the safety factor obtained above is less than 2.2 in which the pre-requisites is not
satisfied. Therefore, Steel Spur Gear with module of 2 cannot be employed in the double-reduction gear train. As a result, Steel Spur Gear with module of 3 has been selected for the
application.
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3.3.2 Bending Fatigue Analysis by Using Steel Spur Gear with Module 3 (N2 =N4 = 90)For m = 3 mm/Tooth,
( )( )
( )( )
( )( )
( )( ) mm270Teeth09Toothmm3
mm126Teeth42Toothmm3
mm270Teeth09Toothmm3
mm126Teeth42Toothmm3
44
33
22
11
===
===
===
===
mNd
mNd
mNd
mNd
The gear ratio for a pair of spur gears (pinion and gear) is given by:
p
g
g
p
d
d=
where
g = Angular velocity of gearp = Angular velocity of piniondg = Diameter of gear
dp = Diameter of pinion
By using the relation of gear ratio above, the angular velocity of gear 2 can be obtained:
rpm1073.33=
=
==
mm270
mm261rpm2300
2
112
1
2
2
1
d
d
d
d
Since gear 2 and gear 3 are located on the same shaft, therefore,
rpm1073.33== 23
The speed ratio for double-reduction gear train is given by:
rpm500.89=
=
==
0909
4242rpm2300
42
3114
31
42
4
1
NN
NN
NN
NN
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Stage 1 (Gear 1 & Gear 2)
Pitch Line Velocity,60000
dV=
sm15.17=
===
===1Stage
222
11
1
sm17.1560000
rpm)3mm)(1073.3270(
60000:2Gear
sm17.1560000
rpm)mm)(2300126(
60000:1GearV
dV
d
V
Tangential Forces, Ft= /V
N593.28=
===
===
1Stage,
1Stage2
1Stage1
N28.593sm15.17W9000:2Gear
N28.593sm15.17W9000:1Geart
t
tF
VWF
VWF
Gear Tooth Bending Stress, movt KKK
bJm
F=
From Figure 14-6 (Budynas, R. G. & Nisbett, J. K., pp. 753), the Geometry Factor for spur
gears of 42 Teeth and 85 Teeth areJ1, 30.2725 andJ2,40.2975.
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minft2991.13===m.30430
ft1
min1
s60
s
m17.15sm17.151StageV
Velocity (Dynamic) Factor,78
78 VKv
+= (Highest precision, shaved and ground)
1.3043=+
=+
==78
13.199278
78
78 1StageVKK vv
From TABLE 15.1 (Juvinall, R. C. & Marshek, K. M., pp. 615), the Overload Correction
Factor, Ko = 1 (Uniform source of machinery and power)
From TABLE 15.2 (Juvinall, R. C. & Marshek, K. M., pp. 616), the Mounting Factor,
Km = 1.3 (Accurate mounting, small bearing clearance, minimum deflection)
MPa41.02=== )3.1)(1)(3043.1(Tooth)mm)(3mm)(0.272530(
N93.285
11
1Stage,
1 mov
tKKK
mJb
F
MPa37.57=== )3.1)(1)(3043.1(Tooth)mm)(3mm)(0.297530(
N93.285
22
1Stage,
2 mov
tKKK
mJb
F
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Stage 2 (Gear 3 & Gear 4)
Pitch Line Velocity,60000
dV=
sm7.079=
===
===2Stage
444
33
3
sm079.760000
rpm)mm)(500.89270(
60000:4Gear
sm079.760000
rpm)3mm)(1073.3126(
60000:3GearV
dV
d
V
Tangential Forces, Ft= /V
N1271.37=
===
===
2Stage,
2Stage4
2Stage3
N37.1271sm7.079W9000:4Gear
N37.1271sm7.079W9000:3Geart
t
tF
VWF
VWF
Gear Tooth Bending Stress, movt KKK
bJm
F=
From Figure 14-6 (Budynas, R. G. & Nisbett, J. K., pp. 753), the Geometry Factor for spur
gears of 42 Teeth and 85 Teeth areJ1, 30.2725 andJ2,40.295.
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minft23.23===m.30430
ft1
min1
s60
s
m079.7sm079.72StageV
Velocity (Dynamic) Factor,78
78 VKv
+= (Highest precision, shaved and ground)
1.0304=+
=+
==78
23.2378
78
78 2StageVKK vv
From TABLE 15.1 (Juvinall, R. C. & Marshek, K. M., pp. 615), the Overload Correction
Factor, Ko = 1 (Uniform source of machinery and power)
From TABLE 15.2 (Juvinall, R. C. & Marshek, K. M., pp. 616), the Mounting Factor,
Km = 1.3 (Accurate mounting, small bearing clearance, minimum deflection)
MPa69.44=== )3.1)(1)(0304.1(Tooth)mm)(3mm)(0.272530(
N37.1271
33
2Stage,
3 mov
tKKK
mJb
F
MPa63.61=== )3.1)(1)(0304.1(Tooth)mm(3mm)(0.295)30(
N34.1271
44
2Stage,
4 mov
tKKK
mJb
F
By comparing the four bending stresses above, the maximum bending stress is max = 3 = 69.44
MPa.
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The gear in the gear train which is subjected to the highest tooth bending stresses is the one which
has the largest possibility to fail in advance. Based on the stresses computed above, max = 269.19MPa and therefore the critical gear is Gear 3.
Refined Analysis of Gear-Tooth Bending Strength Endurance Limit
msKKKCCCSS trSGLnn =
whereun SS 5.0=
Since the Steel Spur Gears are made of S45C Steel with a maximum Brineel Hardness of 194 HB,
therefore, the tensile strength, Su, can be determined by using the relation:
Su = Kh(HB) = (3.45)(228) MPa = 786.6 MPa (Kh = 3.45 in SI Units)
( ) MPa3.393MPa86.675.05.0 === un SS
Load Factor, CL = 1 (Bending loads)
)5(1
in.Teeth4667.8
mm25.4
in.1mm270
Teeth58
in.Teeth4667.8
mm25.4in.1126mm
Teeth42
in.Teeth4667.8
mm25.4
in.1mm270
Teeth09
in.Teeth4667.8
mm25.4
in.1mm126
Teeth42
4
44
3
33
2
22
1
11
>=
=
==
=
==
=
==
=
==
PC
D
NP
D
NP
D
NP
D
NP
G
From TABLE 15.3 (Juvinall, R. C. & Marshek, K. M., pp. 615), the Reliability Correction
Factor, Kr= 1 (50% Reliability)
Temperature Factor, Kt= 1 (Assumed the temperature lower than 160 F)
Mean Stress Factor, Kms = 1 (Gear 3 is idler)
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From FIGURE 8.13 (Juvinall, R. C. & Marshek, K. M., pp. 301), the Surface Factor,
CS= 0.74 (Machined or cold-drawn)
Therefore, the endurance limit is computed as follows:
MPa291.042=
=
=
1)74)(1)(1)(MPa)(1)(0.(393.3
msKKKCCCSS trSGLnn
Safety Factor, 4.19===MPa9.446
MPa042.291SF
max
nS
Since the safety factor is equal 4.19 (SF > 2.2), therefore, the pre-requisite is attained and Steel Spur
Gears with module 3 and face width of 30 mm has been selected from the KHK Gear Catalog forthe double-reduction gear train.
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3.4 Analysis of Surface Durability of Tooth SurfaceThe critical gear (Gear 3) has been selected for the analysis of surface durability as it is subjected to
maximum bending stress.
Surface Fatigue Stress,mov
p
t
p
g
g
p
p
gp
t
HKKK
Ibd
FC
E
v
E
vb
ddF
=
+
+
=22
11cos
sin
2
sin
2
564.0
Gear 3: Ft3 = 1271.37 N, b = 30 mm = 0.03 m, dp = d3 = 126 mm = 0.126,R = dg/dp = 270/126 =
2.14. From previous calculations, Kv = 1.0304, Ko = 1 and Km = 1.3.
From TABLE 15.4b (Juvinall, R. C. & Marshek, K. M., pp. 621), the Surface Factor, the Values of
Elastic Coefficient Cp for Spur Gears is Cp = 191 ( )Pa10191MPa 3= .
Geometry Factor, 1024.0114.2
14.2
2
20cos20sin
12
cossin=
+
=
+=R
RI
Surface Fatigue Stress, MPa123.8===m)(0.1024)m)(0.126(0.03
(1)(1.3)N)(1.0304)37.1271(mov
p
t
HKKK
IbdF
Surface Fatigue Strength, SH= SfeCLiCR
The angular velocity of gear 3, 3 = 1073.33 rpm and given that the expected life for 40 hr/week,and 50 week/year operation is 3.25 years. Therefore, the Surface Fatigue Life (Cycle) is given by:
Cycle102.4hr1
min60years25.3
year
week50
week
hr40
min
rev1073.33LifeFatigueSurface
8=
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From FIGURE 15.27 (Juvinall, R. C. & Marshek, K. M., pp. 624), the Life Factor, CLi = 0.83 for a
surface fatigue life of 4.2(108) Cycle.
From TABLE 15.6 (Juvinall, R. C. & Marshek, K. M., pp. 625), the Reliability Factor, CR = 1.25
for a reliability of 50%.
From TABLE 15.5 (Juvinall, R. C. & Marshek, K. M., pp. 624), the relation for the computation of
the Surface Hardness, Sfe, is Sfe = 2.8(Bhn) 64 MPa.
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Since the maximum Brineel Hardness of the Steel Spur Gear is 228 HB or 228 Bhn, therefore, the
surface hardness can be determined by using the relation above:
Sfe = 2.8(228 Bhn) 64 MPa = 574.4 MPa
Thus, the Surface Fatigue Strength SHcan be computed as follows:
SH= (574.4 MPa)(0.83)(1.25) = 595.94 MPa
Checking of Safety Factor for Surface Durability:
4.8===MPa123.9
MPa94.595SFFactor,Safety
H
HS
Since the Safety Factor SF = 4.8 (> 2.2), therefore, the selection and utilization of Steel Spur Gear
with module of 3 from the KHK Gear Catalog will not results in any operating failure. As a verdict,
based on the analysis of bending fatigue and surface durability of gear tooth strength above and asin previous section, it is ascertain that the Steel Spur Gear with Module 3 from the KHK Gear
Manufacturer can be employed in the compound-reverted gear train. 2 models have been selected
from the KHK Gear Catalog due to the same size of pinions and gears.
Gear 1 & Gear 3 (Pinion):
42TeethofNumber3ModuleGearSpurS45C:Steel
423SS=
Gear 2 & Gear 4 (Gear):
90TeethofNumber3ModuleGearSpurS45C:Steel
903SS=
The material used for the spur gears is Steel S45C with Brineel Hardness of 228 HB or 228 Bhn.
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