ch3 worm gear design-1
TRANSCRIPT
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3.1 Properties, Classification of Worm Gear Drive
3.2 Worm Gear Nomenclature and Geometrical Features
3.3 Loading and Stress in Worm Gear Drive
3.4 Design of Worm and Wormgear3.5 Structure of Worm Gear drive
Chapter 3 Worm Gear Drive Design
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90
Examples of Worm Gear Drive
Worm gear drive can be used for the power drive under skew axes.
Commonly, worm is active; gear is passive. For speed reduction.
Worm
Wormgear
http://localhost/var/www/apps/conversion/tmp/scratch_1/n137.flc -
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Introduction of Worm/WormGear Drive
Forming of worm
If a helical gear with very few teeth(z1=1) and a great helical angle1,
teeth will form several cycles of helical spline.
The gear shaft 1: Worm. The mating gear 2: Wormgear.
Worm2
2
Wormgear
1
1
Left
helicalworm
Right
helical
worm
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Improvements:Cutting tool is manufactured as the shape of worm. Using generating
cutting method, we can get the wormgear with line contact.
Point contact
Advantages:
A great speed ratio, compact structure, smooth movement, low noise,Self-lockingIndexing gear mechanismi=1000, commonly i=8~80.
Disadvantages:
Low efficiency, teeth of wormgear made from Bronze, high cost.
Line contact
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Worm
TypeHourglass
worm gear drive
Cylinder
worm gear drive
Cylind
er worm
Spiroid
worm gear drive
Hourglass worm
Spiroid worm
3.1 Classification of Worm Gear Drive
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Worm Hourglass worm
Cylinder worm
3.1 Classification of Worm Gear Drive
Spiroid worm
Involute worm
Straight sided normal worm
Archimedes worm
Archimedes worm (ZA)
Archimedes' spiral
2 Single processing
Turning, milling,
tooth shaping,grinding (modified wheel)
Difficult to have hard surface and high
precision, commonly used for small load
transmission.
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3.1 Classification of Worm Gear Drive
Involute worm (ZI)
Worm Hourglass worm
Cylinder worm
Spiroid worm
Involute worm
Straight sided normal worm
Archimedes worm
Involute spline Base circle
Hobbing,
grinding (flat surface wheel)
Easy to have hard surface and high precision, commonly used for high speed
and heavy load transmission.
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dx
prolate involute
Straight sided normal
worm (ZN)
3.1 Classification of Worm Gear Drive
Worm Straight sided normal wormHourglass worm
Cylinder worm
Spiroid worm
Archimedes worm
Involute worm
Worm type will be decided by the manufacturing method, load, rotationalspeed and cost.
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1. Geometrical parameters
3.2 Worm Gear Nomenclature and Geometrical Features
Standard center distance
Radial clearance
Helical of wormgear
Lead angle of worm
Dia. of dedendum circle
Dia. of addendum circle
Dedendum
Addendum
Dia. of pitch circle
WormgearWorm
FormulaSymbolName
d
ah
fh
ad
fd
ca
mqd 1 2 2d mz
mha
mhf 2.1mqda )2(1 mZda )2( 22
mqdf )4.2(1 mZdf )4.2( 22
1arctanz
q
mc 2.0
1 2 20.5( ) 0.5 ( )a d d m q z
Table 3-1 Geometrical parameters of worm/wormgear
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Midplane
2. Engagement conditions
Definition of Midplane:
A plane containing the axis of worm and perpendicular to the axis of wormgear.
In the midplane: mt2=ma1=m , t2=a1= Standard value.
In midplane, worm/wormgear seems gear-track engagement.Engagement conditions:
2
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ZA worm gear a=20 in the axial direction
(1) mis standard value
First series 1, 1.25, 1.6, 2, 2.5 , 3.15, 4, 5, 6.3, 8, 10, 12.5,
16, 20, 25, 31.5, 40
Second series 1.5, 3, 3.5, 4.5, 5.5, 6, 7, 12, 14
Table 3-2 Recommendation of module m
3. Module mand pressure angle
Pressure angle ZN worm gear n=20 in the normal direction
ZI worm gear n=20
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To reduce the number of cutting tools, we define d1 as the standard value.
Worm and wormgear have the same helical direction.
if90
12
1+190Wormgear: righthand helical
Worm: righthand helical1+2
We define the cylindrical surface as the pitch cylinder where s=e.
d1
e s
d2
4. Dia. of worm pitch cylinder, d1
t
t
2
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Table 3-3 Dia. of worm pitch cylinder, d1 and module m mm
Module,
m(mm)
Dia. Pitch
circle
d1/mm
Diameter
coefficient
q
Number of
threads of worm,
z1
Dia. dedendum
circle
df/ mm
m2d1/ mm3
1 18 18.000 1 15.6 18
1.2520 16.000 1 17 31.25
22.4 17.920 1 19.4 35
1.620 12.500 1, 2, 4 16.16 51.2
28 17.500 1 24.16 71.68
2
(18) 9.000 1, 2, 4 13.2 72
22.4 11.200 1, 2, 4, 6 17.6 89.6
(28) 14.000 1, 2, 4 23.2 112
35.5 17.750 1 30.7 142
2.5
(22.4) 8.960 1, 2, 4 16.4 140
28 11.2 1, 2, 4, 6 22 175
(35.5) 14.200 1, 2, 4 29.5 221.9
45 18.000 1 39 281
The ratio of , worm diameter coefficient.q=d1/mUsually q=818
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Module,
m(mm)
Dia. Pitch
circle
d1/mm
Diameter
coefficient
q
Number of
threads of
worm,
z1
Dia.
dedendum
circle
df/ mm
m2d1/ mm3
3.15
(28) 8.889 1, 2, 4 20.4 277.8
35.5 11.270 1, 2, 4, 6 27.9 352.2
(45) 14.286 1, 2, 4 37.4 446.5
56 17.778 1 48.4 556
4
(31.5) 7.875 1, 2, 4 21.9 50440 10.000 1, 2, 4, 6 30.4 640
(50) 12.500 1, 2, 4 40.4 800
71 17.750 1 61.4 1 136
5
(40) 8.000 1, 2, 4 28 1 000
50 10.000 1, 2, 4, 6 38 1 250
(63) 12.600 1, 2, 4 51 1 575
90 18.000 1 78 2 250
6.3
(50) 7.936 1, 2, 4 34.9 1 985
63 10.000 1, 2, 4, 6 47.9 2 500
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Module,
m(mm)
Dia. Pitch
circle
d1/mm
Diameter
coefficient
q
Number of
threads of
worm,
z1
Dia.
dedendum
circle
df/ mm
m2d1/ mm3
6.3(80) 12.698 1, 2, 4 64.8 3 175
112 17.778 1 96.9 4 445
8
(63) 7.875 1, 2, 4 43.8 4 032
80 10.000 1, 2, 4, 6 60.8 5 376
(100) 12.500 1, 2, 4 80.8 6 400140 17.500 1 120.8 8 960
10
(71) 7.100 1, 2, 4 47 7 100
90 9.000 1, 2, 4, 6 66 9 000
(112) 11.200 1, 2, 4 88 11 200
160 16.000 1 136 16 000
12.5
(90) 7.200 1, 2, 4 60 14 062
112 8.960 1, 2, 4 82 17 500
(140) 11.200 1, 2, 4 110 21 875
200 16.000 1 170 31 250
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Module,
m(mm)
Dia. Pitch
circle
d1/mm
Diameter
coefficient
q
No. of threads
of worm,
z1
Dia. dedendum
circle
df/ mm
m2d1/ mm3
16
(112) 7.000 1, 2, 4 73.6 28 672
140 8.750 1, 2, 4 101.6 35 840
(180) 11.250 1, 2, 4 141.6 46 080
250 15.625 1 211.6 64 000
20
(140) 7.000 1, 2, 4 92 56 000
160 8.000 1, 2, 4 112 64 000
(224) 11.200 1, 2, 4 176 89 600
315 15.750 1 267 126 000
25
(180) 7.200 1, 2, 4 120 112 500
200 8.000 1, 2, 4 140 125 000(280) 11.200 1, 2, 4 220 175 000
400 16.000 1 340 250 000
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5. Thread number of worm z1, Teeth number of wormgearz2
z1: Number of helical splines
Worm rotates one revolution(as the track translates z1 teeth), and the
wormgear will rotates z1 teeth.
Often, z1=1 2 4 6
z2= i z1 To avoid under cutting
z226
Commonly z280
z2 Length of wormStiffness and precisionStructure dimension
Table 3-4 Recommendation of z1and z2
Speed ratio i 5 7~15 14~30 29~82
z1 6 4 2 1
z2 29~31 29~61 29~61 29~82
z2: Teeth number of wormgear
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6. Lead angle of worm
Expanding the pitch cylinder, we have
= z1 pa1/d1 =mz1/d1tan1=l/d1 =z1/q
d1
1
1
d1
l
pa1
1
1, lead angle of worm
1, helical angle of worm
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, one of the factors affecting efficiency of worm gear drive.For power drive, a bigger value of is needed.
That means a bigger value of z1, or a smaller dia. of pitch circle of worm.
When >30, the efficiency will not increase obviously.
When >45, the efficiency will go down slowly.
For worm gear set with self-locking requirements,
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7. Speed ratio iand teeth number ratio u
Speed ratio:
If a bigger iis needed, we can have z1=1, but low efficiency.For applications of power drive, we have z1=2, or 4.
z1
z2= ---
n2
n1i = --- = u ----teeth number ratio
8. Standard center distance of worm gear drive
a =(d1+d2 )/2= m(q+ z2) /2
40 50 63 80 100 125 160 (180) 200
(225) 250 (280) 315 (335) 400 (450) 500
Recommendation of center distance
9 M difi ti f d i
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9. Modification of worm gear drive
Purposes:
(1) To adjust center distance
(2) To adjust speed ratio
(3) To increase load capacity
(4) To avoid under cutting
Forming method:
Like the modification of gear drive, modification of worm gear drive is
achieved by changing the radial position of cutting tool with respect to
the wormgear blank.
Af ter modif ication:
For worm: As the worm can be regarded as the hobbing tool, thedimensions of worm will not change, but the pitch circle will change
a little.
For wormgear: Addendum circle, dedendum circle and tooth
thickness will change, but the pitch circle will not change.
m m
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e. Modification, x2>0,
z2'z2, a'=a
Fig. 3-6 Modifications of worm gear drive
O2
P
d'1=d
1
a
O2
a'
P
d1
d1
x2m
x2m
x2m
x2m
d1
d1
P
a'
O2
Center line of worm
x2m
d1 d
1
d1
d1
x2m
O2 O2
a
P PCenter line of worm
b. Modification, x2
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Functions of modification:
Standard worm gear drive
a=(d1+d2)/2=m(q+z2)/2
(1) Modification to adjust center distance
a'=a+x2m=m(q+z2+2x2)/2
x2=a'/m-(q+z2)/2Coefficient of modification -0.5
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10. Rotational speed of wormgear
1
2
2v
(1) Left hand for left hand helical, Right hand for right hand helical.
(2) Bending direction of fingers matches with the rotational direction.(3) Tip of thumb points to the oppositedirection of v2.
1
2
2v
11 C l l ti f t i
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Example:
Worm: module m=4mm,
number of threads z1=2,
Dia. of pitch circle d1=40mm;
Worm gear: teeth number z2=39;
Try to find diameter coefficient q, lead angle and center distance a.
11. Calculations of geometries
Solution:
(1) by q=d1/m ,q=40/4=10
(2) by tan=z1/q, tan=2/10=0.2, =11.3099(1118'36'')
(3) Center distance a=0.5m(q+ z2)=0.54(10+39)=98mm
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Discussions:
(1) If non-standard center distance is allowable, we have a=98mm.
(2) If non-standard center distance is not allowable, we have to
adjust center distance to a standard value a'=100mm. Modification
to adjust center distance:
(3) If we want to adjust the speed ratio 20, z2'=40. Modification to
adjust speed ratio:
Then, x2=a'/m-(q+z2)/2=100/4-(10+39)/2=0.5.
Then we have x2=(z2-z2')/2=0.5.
12 Pre ision rade of orm dri e
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Precision
grade
Applications Manufacturing
process
Roughness
Ra/m
Allowable
sliding speed
Grade 6
Indexing
mechanisms for
machine tool
Grinding and
polishing;
Carburizing
Worm: 0.4
Gear: 0.4
>10m/s
Grade 7Conveyor with
medium precision
Grinding and
polishing;
Carburizing
Worm: 0.4-0.8
Gear: 0.4-0.8
10m/s
Grade 8Low line speed, not
important situation
Turning or
worm gear
milling
Worm: 0.8-1.6
Gear: 1.6
5m/s
Grade 9Low speed or hand
appliance
Turning or
worm gear
milling
Worm: 1.6-3.2
Gear: 3.2
2m/s
12. Precision grade of worm drive
Table 3-6 Precision grade of worm drive
13 Efficiency of worm drive
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13. Efficiency of worm drive
Power loss of worm drive
Friction loss
Oil resistance lossBearing friction loss
Efficiency of closed gear drive = 123
1Efficiency in mesh, decided by precision grade;
2Churning loss;
3Bearing efficiency.
Mostsignificant
factor
For speed reduction:
1tan
tan e
e-- Equivalent angle of friction, which relates to the materials of
worm and wormgear, precision grade and relative sliding speed.
We can findein Table 3-7.
For speed increase:
1
tan
tan
e
(1) Equivalent angle of friction
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3.00 0.028 136 0.035 2 0.045 235
4.00 0.024 122 0.031 147 0.04 2175.00 0.022 116 0.029 140 0.035 2
8.00 0.018 102 0.026 129 0.03 143
10.0 0.016 055 0.024 122
15.0 0.014 048 0.020 109
24.0 0.013 045
Table 3-7 Equivalent angle of friction
Wormgaer material Tin bronze Tin-free bronze Grey cast iron
HB of worm(Steel) HRC 45 other HRC 45 HRC 45 other
Sliding speed fv e fv e fv e fv e fv e
0.01 0.11 6 17 0.12 651 0.18 1012 0.18 1012 0.19 1045
0.10 0.08 4 34 0.09 543 0.13 724 0.14 758 0.16 905
0.50 0.055 309 0.065 343 0.09 509 0.09 509 0.1 543
1.00 0.045 235 0.055 309 0.07 4 0.07 4 0.09 509
2.00 0.035 2 0.045 235 0.055 309 0.07 4
vs m/s
(1) Equivalent angle of frictione V
arctan f
(2) Relative sliding speed
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(2) Relative sliding speed
vs=v1/cos v1Linear speed of worm pitch circle;
vsRelative sliding speed, in Fig. 3-8.
Fig. 3-8 Estimation of sliding speed
14 Self locking of worm drive
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14. Self-locking of worm drive
Self-locking:
For speed reduction applications, worm can drive the wormgear, but, if
torque is applied to the wormgear shaft, the worm does not turn.
Self-locking condition:
lead angle equivalent angle of frictione
Efficiency of worm drive with self-locking
0.5
Table 3-8 Estimation of worm drive efficiency,
Threads of
worm z1
1
(self-locking)
1 2 4 6
0.4 0.65-0.75 0.75-0.82 0.82-0.85 0.85-0.95
3 3 Loading and Stress in Worm Gear Drive
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1. Force analysis of worm drive
Ft2Fr2
Fa2
Ft1
Fr1
Fa1
2
Normal force Fncan be decomposed into 3 components
Tangential force
Ft
Axial forceFa
Radial forceFr
Between worm and wormgear, we have,Ft1= -Fa2
Fr1= -Fr2
Fa1= -Ft2
=2000T1/ d1
=2000T2/ d2
= Ft2tan
T1, T2are the torques acting on worm and wormgear, and
T2= T1i
1
Ft1, opposite to the linear speed
Fa1, Right/Left Hand Rule
Fr1
, to the center
d1, d2Pitch diameters of worm and wormgear;
--Pressure angle on pitch diameter of worm.
3.3 Loading, and Stress in Worm Gear Drive
S ifi ti f f di ti
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Specification of force direction
1
2
p
Ft2 Fa1
Fr1
Fr2
2
2) For worm gear: directions of Ft2,2are same; and Ft2=-Fa1
4)Fr1, Fr2pointing to the center, and Fr1=-Fr2.
3) For worm: directions of Ft1,1are opposite; and Ft1=-Fa2
1) For worm: directions of T1,1are same
T
T
Ft1Ft1
Fa2 Fa2
Front view Side view
2 Failure type of worm drive
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2. Failure type of worm drive
(1)The most common failure of worm drive: scuffing, pitting and wear
A great sliding speed on the wormgear tooth face;
Worms diameter is much smaller than the length, so strengthand stiffness of worm may cause failure.
(2)Design principles of worm drive
* By calculating stress on critical section of worm shaft
* By contact strength of wormgearstooth surface* By bending strength of wormgearstooth
(only z2>80 or negative modification)
To avoid scuffing and pitting:
To avoid weakness of worm strength:
* By checking deflection if a great span between supportsTo avoid weakness of worm stiffness:
* By checking thermal equilibriumTo avoid overheat of worm drive:
(3)Materials of worm drive
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(3)Materials of worm drive
Requirements:
strength, stiffness, abrasive resistance, anti-wear and anti-scuffing.
Worm is long and thin, so carbon steel and alloy steel is most commonly used.
By heat treatment
Hard surface worm
Hardened and tempered
worm
Grinding and polishing
Easy to manufacture, and good for
short impact
Material trademark Heat treatment Hardness Roughness Ra/m
45,40Cr, 42SiMn, 40CrNi,38SiMnMo
Case hardening HRC=45-55 1.6-0.8
15CrMn, 20CrMn, 20Cr,
20CrNi
Carburizing HRC=58-63 1.6-0.8
45 H.T. HB
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wormgear
Tin bronze vs 12m/s, continuous drive
Al bronze vs 10m/s, with hard surface worm
Brass Low sliding speed
Grey iron Very low sliding speed, vs 2m/s
Material of
wormgear
Cast
process
vs,
m/s
B
N.mm-2
'HP
N.mm-2
'FP
N.mm-2
HB of worm
surface
One-
sided
load
Both-
sided
loadHB350 HRC>45
Tin
bronze
ZCuSn10
P1
Sand
mould
12 220 180 200 51 32
Metal
mould
25 310 200 220 70 40
ZCuSn5P
b5Zn5
Sand
mould
10 200 110 125 33 24
Metalmould
12 250 135 150 40 29
Table 3-10 Common materials of wormgear and allowable stress
Table 3-10 continued
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Material of
wormgear
Cast
process
vs,
m/s
BN.mm-2
'HPN.mm-2
'FPN.mm-2
HB of worm
surface
One-
sided
load
Both-
sided
loadHB350 HRC>45
Al
bron
ze
ZCunAl10Fe
3
Sand
mould
10 490 82 64
Metal
mould
540 90 80
ZCuAl10Fe3
Mn2
Sand
mould
10 490 -- --
Metal
mould
540 100 90
Brass ZCuZn38Mn
2Pb2
Sand
mould
10 245 62 56
Metal
mould
345 -- --
Grey
cast
Iron
HT150 2 150 40 25
HT200 2-5 200 48 30HT250 2-5 250 56 35
Homework-10
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Homework-101. Specify the rotational speed direction of worm or worm gear.
2. Decide the force component directions of worm and wormgear at meshing point.
Worm is the driving components.
n11
2
n2
2 1
n1 1
2
n2
Specify the helical direction ofworm and worm gear
(1)(2)
(3)