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(1) Features of helical gear
Contact line leans at an angle with respect to the axis of gear (Left/Right)
There are more teeth engaged simultaneously, a greater contact ratio
Smooth transmission, low noise, high load capacity
Normal module mn1=mn2
Normal pressure angle n1=n2
Helical angle 1=
2
(2) Engagement conditions
1. Introduction
2.7.3 Design of helical gear
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Directions of force component
Ft, Fr
Axial force Fa
Identical to the spur gear
For driving gear, Fa1can be decided by Right/left hand rule.
Decided by Direction of helical lineand rotational direction.
Left hand for left hand helical, Right hand for right hand helical.
If bending direction of fingers matches with the rotational direction,
tip of thumb points to the direction of Fa1.
Ft1
Fr1Fa1
Driving
gear
For driven gear, direction of Fa2is opposite to that of Fa1.
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Relationship between force components
n1 n2F F r1 r 2F F a1 a 2F F t1 t 2F F
Example
n1
n2
1
2
Top view
Right hand
helical
Left hand
helical
Fa2Fa1
Ft1
Ft2
Fr2
Fr1
n2
n1
Side view
Fa1
Fa2 Ft1Ft2
Fr2
Fr1
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Principles:
3. Contact strength of helical gear
Module = Normal module of helical gear mn
Pressure angle = Normal pressure angle of helical gear n
Number of teeth = Virtual number of teeth of helical gear zv= z /cos3
Normal force = Normal force of helical gear Fn
2
1
1cos
ddv
3
11
cos
TTv
cos
bbv u
z
z
z
zu
v
v
v
1
2
1
2
(1) Considering the strength of an equivalent virtual spur gear
(2) Contact line has an angle with respect to axis, which is advantageous to
reduce contact stress. So we introduce a coefficient of helical angle Z.
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For spur gear
For helical gear
1H E H 2
1
2 ( 1)KT uZ Z Z
bd u
1H E H 2
1
2 ( 1)KT uZ Z Z Zbd u
cosZ Coefficient of
contact ratio
Under same condition, H_Helical < H_Spur
So, Helical gear has a greater load capacity than spur gear.
Coefficient of local
area at pitch point
(Fig. 2-18)
Coefficient of helical angle
0.75 ~ 0.88Z
1
1
cos
nm z
d
b
The greater number of teeth, the less value of Z.
tt
b
HZ
tancos
cos22
Coefficient of elasticity
(Table 2-15)
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1H E H HP1 HP22
1
2 1min[ , ] (MPa)
KT uZ Z Z Z
bd u
For checking contact strength
H E 2 13
1
HP d
2 1( ) (mm)Z Z Z Z KT udu
HP1 HP2min[ , ] d 1b d
For designing diameter of pitch circle
See Table 2-14
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For design of geometries
H E 2 13
1
HP
Z Z Z Z 2 1( ) (mm)
d
KT ud
u
Recalculating d1, d2, accurate to the three decimal places
First computing d1 , then specifying other geometries:
Number of pinion teeth z1(soft surface, closed gear drive20~40hard
surface, closed gear drive or open gear drive1725)
Initial helical angle , most commonly in range of 10~ 15
Computing mn= d1cos/z1Rounded up to a basic value and mn1.5 forpower driving
Computing center distance a=(d1+ d2)/2=mn(z1+ z2) / (2cos)Rounded up to
an integer
Recalculating= cos1 [mn(z
1+ z
2) / 2a ]Accurate to second
add
2
21
cos
nzm
dAttention: Satisfying
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For checking the bending strength:
1F1 Fa1 Sa1 FP1
1 n
2(MPa)
KT
Y Y Y Y bd m
2
1 Fa Sa3n 2
d 1 FP
2 cos (mm)
KT Y Y Y Ym
z
1F2 Fa2 Sa2 FP2
1 n
2(MPa)
KTY Y Y Y
bd m
The greater value
between pinion and gear
For designing the normal module:
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2.7.4 Design of bevel gear
Gear drive with vertically intersecting axes
Teeth on a conical shaped surface
Uneven distribution of load along the tooth width
b
We only investigate bevel gear of straight teeth,
and axis angle is 90
.
1. Features of bevel gear drive
Significant vibration and noise, only for low speed transmission, v5m/s
Tooth profile: Straight teeth, helical teeth, curved teeth
Geometrical factors on large end are standard
For convenience of computing, we assume
Normal force Fn, a concentration force on the pitch cone at the midface of teeth.
Strength of bevel gearStrength of virtual spur gear at the midface of teeth.
0.5b
Fn
Large end
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2. Geometries of straight bevel gear and its virtual spur gear
1) Geometries of straight bevel gear
Module at the larger end m is standard value,Dia. of pitch circle
2211 ; mzdmzd
Ratio of teeth number uz2 / z1d2 / d1
Cone angle of pitch circle
uu 21 tan;/1tan
Cone distance2
1 15.0 udR Coefficient of
tooth widthRb/R
Dia. of pitch circle at the midface of teeth
1Rm1 )5.01( dd
Module at the midface of teeth mm )5.01( Rm
dm1
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Ft2=Ft1 Fr2=Fa1 Fa2=Fr1
n2
Fr2
Fa2
Fr1Ft1
Fa1
n1
Ft2
Example problem
1
2
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Force checking
For design
1H H E HP2 3
R R 1
4
0.85 (1 0.5 )
KTZ Z
d u
2H E 13
1 2
HP R R
4( )
0.85 (1 0.5 )
Z Z KTd
u
4. Contact fatigue strength of bevel gear
Considering the equivalent virtual spur gear at the midface of teeth.
Ignoring the influence of contact ratio.
Effective face width
beH0.85b
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Discussions
ZE, ZH, HP, identical to the process of spur gear
Often u 5, limiting the diameter of bigger bevel gear, easier to manufacture
After specifying d1 , other geometries needs to be decided
Initialz
1
Computing m=d1/z1,
rounded up to standard value
Calculating d1= mz1, z2, d2, u , et al2
1 15.0 udR
Dont round up R !
Often b1
b2, easier to assemble and coincide the vertex
mm)5.01(85.0
43
2
RR
1
2
HP
HE1
u
KTZZd
For design
Dia. of pitch circle at the large
end of pinion
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Spur gear:
5. Bending fatigue strength of bevel gear
1
1F Fa Sa3 2
d
2KTY Y Y
m z
1F Fa Sa FP2 3 2 2
1
4
(1 0.5 ) 1R R
KTY Y
m z u
1 Fa Sa3
2 2 2FP1
4
(1 0.5 ) 1R R
KT Y Ym
z u
For checking
For design Virtual number of teeth v /cosz z
Bigger value
In the like manner, considering the bending fatigue strength of virtual gear.
Ignoring the influence of contact ratio, using the geometries of virtual gear,depending on
A d bl d i d i b l t i l t
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Example:
A double gear reducer is driven by an electrical motor.
Inputting power P17kW.
Inputting rotational speed n11000r/min.
Speed ratio of the first stage i4.6
Unidirectional, slight impact.
Life expectancy 10 years, 8h per day.
Try to design the first gear drive.
1. Requirements
Medium speed and medium load.
2. Design plan
Plan A: Helical gear
Either soft surface or hard surface is OK.
Using helical gear for smooth movement.
Plan B: Helical gear
45 steel, soft surface
45 steel, hard surface
First gear drive
Second gear drive
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3. Design process
1
Material and heat treatment
45 45Gear
45 45Pinion
H.T. of pinion H & T
Norm.
Case hardening
H.T. of gear
Table 2-11, middle valueHB of pinion 230HBS 50HRC
190HBS 50HRCHB of gear
1170Contactstress
1limH 580Fig. 2-242limH 550 1170
Case hardening
HB of pinion > HB of gear
Table 2-11
2Allowable contact stress and bending stress
Fig. 2-26
Bending
stress
1limF 220 340Fig. 2-30
Fig. 2-282limF 210 340
Design contents Design principles Plan A Plan B
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load cycle 1N )830010(100016060 11 tanN 14.4108
2N iNN /12
3.13108300 day per year
Coefficient of
contact life
1NZ
Fig. 2-271
2NZ 1
Coefficient of
bending life
1NY
Fig. 2-321
1NY 1
Design content Design principles Plan A Plan B
Min. safety
coefficient
minHS Table 2-14 1
Table 2-14minFS 1.4
Allowable
contact stress
1HP minH1N1limH1HP / SZ 580 1170
2HPminH2N2limH2HP / SZ 550 1170
Allowable
bending stress
1FP minF1NST1limF1FP / SYY 314 486
2FP minF1NST1limF1FP / SYY 300 486
2STY
Pitting not
allowable
Line 1
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3Specifying initial parameters
12Initial helical angle
Precision grade Table 2-1 Grade 8
Coe. of face width d Table 2-14 0.9 0.5
No. of teeth 1z Closed, soft z120~40
Closed, hard z1
17~25 26
Modification
vA KKKKK 1.898 2.234
1.25 1.25Where
K
K
Assuming the linear
speed v9m/s
2z
21z2z1iz14.6 119 97
Load coefficient
0 0
AK
Table 2-7, slight impact
vK Fig. 2-6, Kv1.02~1.2 1.1 1.1
Table 2-9, soft1~1.2hard1.1~1.35 1.15 1.25
Design content Design principles Plan A Plan B
K Table 2-8 K1.2~1.4 1.2 1.3
In the range of 10~15
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4Design and calculation
Principles
Coe. of elasticityEZ
2.45
0.8Coe. of contact ratio
HZCoe. of local area
189.8
Z
Table 2-15
ZZ
0.75~0.880.989
HP2
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Normal module nm
71.986mm
222.36mmaCent distance
162350N.mm
11o33'21' '
26
12cos986.71cos
1
1n
z
dm
Accurate helical
angle
Torque 1T1000
17109550109550
3
1
13
1
n
PT
Dia. of Pitch circle
Design content Design principles Plan A
2.71mm
Rounded up to 3
12cos2
)11926(3
cos2
)( 21n
zzma
Rounded up to
222
2222
)11926(3cos
2
)(cos 121n1
a
zzm
3
2
16.4
16.4
9.0
162350898.12
550
989.08.045.28.189
d
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Face width
v
b272mm
Checking bending
strength
79.61mm
1vz 27.65
60000100061.79
100060
11
ndv
Virtual teeth
number
Accurate dia. of
pitch circles1d "'zm 213311cos/263cos/
o
1n
4.17m/sOK!
3v cos
zz
364.39mm2d "'213311cos/1193 o
Satisfyingd1d2/2a
65.7161.799.01d2 db
b178mmmm)10~5(21 bb
Line speed
2vz 126.54
1FaY 2.6Coe. of tooth
profile 2FaY 2.2
Design content Design principles Plan A
Fig. 2-20
FPSaFa
1n
1
F
2YYYY
dbm
KT
Rounded upb
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Coe. of contact ratio
Y 0.89
Checking bending
stress
1.61
92.0~85.0Y
Coe. of stress
modification
Sa1Y
1.82Sa2Y
85.0~65.0Y
Coe. of helical angle
The endclosed, soft face
89.075.061.16.2
61.79372
162350898.12
Design contents Design principles Plan A
Fig. 2-21
Y 0.75
Bigger, bigger Y
F1Bending
strength OK!MPa100 MPa3141HP
61.16.2
82.12.2100
Sa1Fa1
Sa2Fa21F
YY
YY
F2Bending
strength OK!
MPa3002HPMPa96
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22.44
Design by
bending
strength
Virtual number
of teeth
mmcos2
3
FP
SaFa
d
2
1
2
1
n
YY
z
YYKTm
v1z o33
1v1 12cos/21cos/ zz
103.65v2z o33
2v2 12cos/97cos/ zz
1FaY 2.76Coe. of tooth
form 2FaY 2.22Fig. 2-20
1.55Coe. of stress
modification
Sa1Y
1.79Sa2YFig. 2-21
Design contents Design principles Plan B
Coe. of contact
ratio
Y 0.8992.0~85.0Y
85.0~65.0Y
Coe. of helical
angle
Y 0.8The more teeth,
the less value
Closed, hard
surface
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Comparingbending strength
0088.0
486
55.176.2
FP1
Sa1Fa1
YY
0082.0486
79.122.2
FP2
Sa2Fa2
YY
P
G
Considering
pinion
Normal modulus
nm 3
FP
SaFa
d
2
1
2
1
cos2
YY
z
YYKT
32
o2
0088.05.021
89.08.012cos162350234.22
mm70.2
165.87mmaCenter distance
12cos2
)9721(75.2
cos2
)( 21n
zzma
Rounded to 166
Design contents Design principles Plan B
Rounded up, mn2.75mm
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12o12'6' 'Accurate helical
angle1662
)9721(75.2cos
2
)(cos 121n1
a
zzm
59.08mmAccurate pitch
diameter1d "'zm 61212cos/2175.2cos/
o
1n
272.92mm2d "'61212cos/9775.2 o
Face width
v
b230mm
60000
100008.59
100060
11
ndv
3.09m/s
OK!
54.2908.595.01d2 db
b
1
35mmmm)10~5(21bb
Linear speed
b
Checking
contact stress MPa
12
HP21
1
HEH
u
u
bd
KTZZZZ
Coe. of elasticity EZ
2.45HZ
Coe. of local area
189.8Table 2-15 MPa
Fig. 2-18
Design contents Design principles Plan B
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0.82Coe. of contact ratio
Z
Z
Helical gear Z0.75~0.88
0.989HP2HP11170MPa 1170MPa
u Speed reducer ui 4.6
"'Z 61212coscos
Coe. of helical angle
Allowable contact stressHP
Ratio of teeth
Design contents Design principles Plan B
Checking contact
stress
u
u
bd
KTZZZZ
122
1
1HE
H
6.4
6.5
08.5930
162350234.22989.082.045.28.189
2
MPa1170MPa1089 HP
Conclusion: contact strength is OK!
The end( closed, hard surface)
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Comparison of two design plan
45 45Material
Heat treatment Case
Hard.
P: H&T
G:Norm.
Helical angle
No. of teeth26 21
79.61 59.08Pitch diameter
mm
1d
2d364.39 272.92
3 2.75Modulusmm
Face width
mm
78 35
Center distancemm 222 166
Design contents Plan A Plan B
12o12'6' '11o33'21' '
1z
2z
119 97
nm
1b
2b 72 30
a
Analysis:
Plan A: soft surface, easy
to manufacture, low cost.But structure size larger.
Plan B: hard surface,
complicated tomanufacture, high cost.
But smaller structure size,
light weight, high
strength.
Plan A for common use
Plan B: for great load
and compact size
2 8 Other Considerations of Gear Drive Design
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1. Efficiency of gear drive
2.8 Other Considerations of Gear Drive Design
Power loss of gear drive
Friction loss
Oil resistance loss
Bearing friction loss
Efficiency of closed gear drive = 123
1Efficiency in mesh, decided by precision grade;
2Churning loss;
3Bearing efficiency.
Average efficiency of closed cylinder gear drive, 0.96-0.99;
Average efficiency of closed bevel gear drive, 0.94-0.98;
Most
significant
2 Lubrication of gear drive
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2. Lubrication of gear drive
Teeth in mesh Relative sliding Friction and wearing
Functions of lubrication:
Cooling
Anti-corrosion
Vibration absorbingNoise reduction
Types of
lubrication
12 15m/sv
15m/sv
Churning
Injection
Oil sump
3 Structure design
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By strength design:
Such as center distance,
module, helical angle
To specify pattern and size of spoke and hub
To specify geometrical parameters
By structure design
(decided by dia. of addendum circle)(1) Gear shaft
The distance from dedendum to root of gear e
Cylinder geare
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(2) Solid gear
Dia. of addendum circle da200mm
Forged
gear blank
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(3) Web
Dia. of addendum circle da200~500mm
Forged
gear blank
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4. Spoke
Dia. of addendum circle da500mm
cast iron or cast steel
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Homework-9
Specify the helical direction of Gear 1 and Gear 2. Mark the directions of each force components.
Tips: Both Gear 2 and Gear 3 exert axial forces on Shaft II. We can
ff t th ft if th h li l di ti
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