CH3 Worm Gear Design-1

7/26/2019 CH3 Worm Gear Design-1

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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


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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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)


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)

Documents

CH3 Worm Gear Design-1