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8/10/2019 CH3 Worm Gear Design-2
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3.4 Design of Worm and Wormgear
Pitting, Scuffing and Wear first occur on the surface of wormgear.
Contact strength of tooth surface and bending strength of tooth.(1) Contact strength of tooth surface
For design:
2
2
1 2
HP 2
15000m d KT
z
For Checking: 2H HP2
1 2
9400E A V
TZ K K K
d d
T2-- Torque acting on wormgear, N.m;
K-- Coefficient of load, commonly in range of 1.0-1.4.
If well-distributed load, v23m/s and higher precise thanGrade 7, a smaller value. Otherwise, a bigger value.
HP -- Allowable contact stress of wormgear,
decided by material of wormgear.
1. Strength of wormgear
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For tin-free bronze, brass and cast iron, HP in Table 3-11;
For tin bronze, HP='HPZvsZN
'HPAllowable contact stress if load cycle number NL=107,
in Table 3-10.ZvsCoefficient of sliding speed, Fig. 3-10.
ZNCoefficient of contact fatigue life, Fig. 3-11.
Fig. 3-10 Coefficient of sliding speed
Injection
Churning
Zvs
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Fig. 3-11 Coefficient of contact/bending fatigue life
For cast iron
For bronze and brass
ZNo
rYN
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Table 3-11 Allowable contact stress of tin-free bronze, HP
Material of
wormgearM. of worm
Sliding speed vs, m/s
0.25 0.5 1 2 3 4 6 8
Al-
bronze
ZCuAl10Fe3
Steel,
Hardening - 250 230 210 180 160 120 90
ZCuAl10Fe3
Mn2
Steel,
Hardening- 250 230 210 180 160 120 90
BrassZCuZn38Mn2
Pb2
Steel,
Hardening-- 215 200 180 150 135 95 75
Grey
cast
iron
HT150,
HT200
Steel,
H.T.160 130 115 90 -- -- -- --
HT250Steel,
H.T./Hardening140 110 90 70 -- -- -- --
ZECoefficient of Elasticity, Table 3-12.
Material of
worm
Material of wormgear
Cast
tin bronze
Cast
Al bronze
Cast
Al brass
Grey cast
iron
Nodular cast
iron
Steel 155 156 157 162 182
Table 3-12 Coefficient of Elasticity
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Table 3-13 Overload Coefficient
Power sourceDriven machine
Uniform Moderate shock Heavy shock
Electrical motor,
Steam turbine0.8-1.25 0.9-1.5 1.0-1.75
Multi cylinder
engine0.9-1.5 1.0-1.75 1.25-2.0
Single cylinderengine
1.0-1.75 1.25-2.0 1.5-2.25
KAOverload coefficient, Table 3-13.
KVDynamic load coefficient
If v2
3m/s, KV
=1.0-1.1;
If v2>3m/s, KV=1.1-1.2.
KLoad distribution coefficient
If uniform load, K=1.0;
If variable load, K=1.1-1.3.
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(2) Bending strength of wormgear tooth
For design: 21 2 FS
FP 2
600m d KT Y
z
For Checking: 2 A V F FS FP
1 2
666T K K K Y Y
d d m
FPAllowable bending stress of wormgear, N/mm2, which
relates to the material of worm gear tooth, satisfying,FP= 'FPYN
'FP Allowable bending stress if number of load cycles
NL=106, Table 3-10.
YNCoefficient of bending fatigue life, Fig. 3-11.
YFSCombined coefficient of tooth profile, YFS=YFaFSa,
in Table 2-20 and Table 2-21,
Equivalent teeth number ze2=z2/cos3.
YCoefficient of lead angle, Y=1- /120.
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2. Stiffness of worm
Surface hardness of worm is high, but worm shaft will deform at
the engagement point under the radial force and tangential force.
A great deflection will cause unbalanced load or interference,and damage the engagement of worm and wormgear.
For Checking:
2 2
1 1 3
1 P48
t rF Fy L y
EI
Ft1Tangential force on worm;Fr1Radial force on worm;
EModulus of elasticity of worm material, if a steel worm,
E=2.07
105N/mm2;
IMoment of inertia at the critical section, mm4,
I=df14/64, where, df1dia. of dedendum circle;
L Span between supporting points, decided by structure design,
initial value L=0.9d2, d2-dia. of pitch circle of wormgear;
ypAllowable deflection, in the range of yp=(0.0010.0025)d1,
d1diameter of pitch circle of worm;
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3. Thermal equilibrium of worm drive
For the self-locking worm drive, the efficiency is as low as 50%.
Under closed and continuous driving conditions, there will be
overheatto deteriorate the lubrication.
Finally, it will cause failure.
In worm drive, the power of heat loss is
1 1sP P P1- Inputting power;
- Total efficiency.
Radiating power of in unit time is
1 2cP kA t t
k- Thermal conductivity,
14-17.5 W/(m2
.C) under good ventilation,8.7-10.5 W/(m2.C) under poor ventilation;
ACooling area, estimating by A=0.33(a/100)1.73, ais the
center distance between worm and worm gear, mm.
t1
Allowable temperature of lubrication, commonly 95;
t2Environment temperature, commonly 20.
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11 2
195
Pt t
kA
Worm drive must under the allowable temperature, so
If the thermal equilibrium can not be satisfied, Some measures
need to be adopted to increase heat dissipation .
Cooling waterVentilator Circulating lubrication
Fan
Oil Lubricating oil Oil pump
Cooler
Filter
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Example ProblemTry to design the worm drive for hoister.
Given:
(1) Inputting power of worm P1=7.5KW;
(2) Inputting rotational speed n1=1450r/min;(3) Speed ratio i=20;
(4) Load on driven machine is uniform;
(5) Slight shock from power source;
(6) Life expectancy is 5 years, 300 days/year, 8 hours/ day;(7) Worm is arranged below the wormgear;
(8) Small batch production;
Solution
(1) Worm type, precision grade and materialConsidering small inputting power, and medium inputting rotational
speed, we can choose ZA worm, and precision grade Grade 8.
Material of worm: Steel 45, Case hardening, Hardness HRC=45-50.
Material of wormgear: ZCuSn10P1, sand mould casting process.
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(2) Teeth of worm and wormgear
Speed ratio i=20, by Table 3-4, z1=2, z2=iz1=40.
Rotational speed of worm gear n2=n1/i=1450/20=72.5r/min.
(3) Allowable contact stress of worm gearMaterial of worm gear is tin bronze, so HP='HPZvsZN
By Table 3-10, 'HP=200N/mm2.
By Fig. 3-8, the estimated sliding speedv
s=7m/s, churning lubrication.By Fig. 3-10, Coefficient of sliding speed Zvs=0.87.
Wormgear rotates in unidirectional direction, No. of load cycle on
wormgear is
NL=60n2th=60
1
72.5
5
300
8=5.22
107
By Fig. 3-11, Coefficient of life ZN=0.8.
HP='HPZvsZN= 200N/mm2
0.87
0.8=139.2N/mm2
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(4) Designing by contact stress
Load coefficient K=1.1, by Table 3-8, the estimated efficiency of
worm drive =0.8, then the torque on worm gear is
T2= T1i =9550 P1/n1i =9550 7.5/1450 20 0.8=790.3 N.m2
2
1 2
HP 2
15000m d KT
z
By
2
2 3
1
150001.1 790.3 6309.12mm
139.2 40m d
By Table 3-3, m2d1is about 6400 mm3, then we have m=8mm,
d1=100mm, q=12.5.
(5) Calculating Geometrical parameters
Dia. of pitch circle of wormgear d2=mz2=840=320mm.
Lead angle of worm tan =z1/q=2/12.5=0.16, then =9.09.
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Face width of wormgear
2 2 0.5 1 2 8 0.5 12.5 1 mm=66.788mmb m q
We can set b2=68mm.
The center distance between worm and wormgear
a=0.5(d1+d2)=0.5(100+320)mm=210mm
(6) Linear speed ,and efficiency of worm gear
Linear speed of wormgear
v2=d2n2/(601000)=320725/(601000)=1.21m/s
Relative sliding speed
vs=v1/cos =d1n1/(60
1000)
cos 9.09=7.69m/sBy Table 3-7, Equivalent angle of frictione=1 03 =1.05 , we
have
1
tan tan 9.090.894
tan tan 9.09 1.05e
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Efficiency of churning 2=0.96, Efficiency of rolling bearing
3=0.99, we have
= 123=0.894
0.96
0.99=0.85
Approximating to the estimated
value
(7) Checking the contact strength
Torque on the worm gear
T2= T1i =9550 P1/n1i=9550 7.5/1450 20 0.85=839.7N.m
By Table 3-12, coefficient of elasticity ZE=155.
By Table 3-13, overload coefficient KA=1.
As v2=1.21m/s
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666T K K K
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2 A V
F FS
1 2
FP
666
666 790.3 1.05 14.04 0.92 8.03
100 320 8
T K K K Y Y
d d m
Bending strength is OK.
(9) Checking stiffness of worm
Tangential force
6
1
1
1
7.52 9.55 10
2 1450987.93N
100t
TF d
Radial force
3
2
r1 a
2
2 2 839.7 10tan tan 20 1910.2N
320
TF
d
Span between supporting points2
0.9 0.9 320 288L d
Moment of inertia at the critical section
44
6 4f1 100 2.5 8
2.01 10 mm64 64
dI
Allowable deflexion
10.001 0.001 100 0.1mmPy d
W h th d fl ti f
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We have the deflection of worm2 2
1 1 3
1
2 2
3
5 6
48
987.93 1910.2
28848 2.1 10 2.01 10
0.0025mm
t r
P
F Fy L
EI
y
Stiffness of worm is OK.
(10) Calculating the thermal equilibrium
Efficiency of worm drive =0.85, thermal conductivity
k=15W/(m2), temperature of environment t2=20,
Cooling area, estimated by
A=0.33(a/100)1.73=0.33(210/100)1.73=1.191m2
1
1 2
1 7500 1 0.85
20 82.97 9515 1.191
P
t tkA
Thermal equilibrium is OK.
(11) Calculating other geometrical parameters (Omitted)
(12) Designing structure (Omitted)
3 5 S f W G d i
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Worm structure:
Commonly Worm in a form of shaft worm shaft
3.5 Structure of Worm Gear drive
1) Non-tool withdrawal groove, by milling the worm teeth
2) Tool withdrawal groove, by turning the worm teeth
L
df1
L
df1
T. W. G. T. W. G.
W
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ormgear tooth width angle 90~130
Thickness of wormgear c 1.6m+1.5 mm
Width of flange B 0.75da 0.67 da
Dia. of wormgear addendum de2 da2 +2m da2 +1.5m da2 +2m
o. of thread z1 1 2 4
Wormgear structure
Unit wormgear Combined
by interference fit
de2
de2
de2
de2
BB B B
c cc
Combined
by bolt connection
Combined
by casting
Set screw 4~8, =2~3mm
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