§ 10 1 Applications and Types of Gear Mechanisms § 10 2 Fundamentals of Engagement of Tooth Profiles § 10 3 The Involute and Its Properties § 10 4 Terminology.

§10 － 1 Applications and Types of Gear Mechanisms

§10 － 2 Fundamentals of Engagement of Tooth Profiles

§10 － 3 The Involute and Its Properties

§10 － 4 Terminology and Definition of Gears

Chapter 10 Gear MechanismsChapter 10 Gear Mechanisms

§10 － 5 Gearing of Involute Spur Gears

§10 － 6 Introduction to Corrected Gear

§10 － 7 Helical Gears for Parallel Shafts

§10 － 8 Worm Gearing

§10 － 9 Bevel Gears

1) Gear mechanisms are widely used in all kinds of machines

to transmit motion and power between rotating shafts.

2) Circular gears have constant transmission ratio whereas, for

non-circular gears, the ratio varies as the gears rotate.

3) Depending upon the relative shafts positions, circular gear

mechanisms can be divided in to planar gear mechanisms and

spatial gear mechanisms.

4) In this chapter, only circular gears are considered.

一、 Introduction

二、 Types of Gear Mechanisms

§§1010 －－ 1 1 Applications and Types of Gear Mechanisms

parallel shafts

spur gear

helical gear

double-helical gear

interse-cting shafts

bevel gear

helical bevel gear

spiral bevel gear

crossedshafts

Spiral

helical gear

worm and worm wheel

Conjugate Profiles—— Meshing profiles of teeth that can yield a desired transmission ratio are termed conjugate profiles. (i12=ω1/ω2)

一、 Fundamental Law of Gearing

ω2

ω1

O2

O1

2

1

C1

C2

P

n

n

Kvk1

vk2

vk1k2

n

The driving pinion rotates clockwise with angular velocity

ω1 while the driven gear rotates counterclockwise with angular

velocity ω2 . The common normal n-n intersects the center line

O1O2 at point P. the point P is the instant center of velocity of the

gears .

i12 ＝ ω1/ω2 ＝ O2 P /O1P

v12 ＝ O1P ω1= O2 P ω2

The transmission ratio:

fundamental law of gearing:

The transmission ratio of two meshing gears is

inversely proportional to the ratio of two line segments cut

from the center line by the common normal of the tooth

profiles through the contact point.

§§1010 －－ 2 2 Fundamentals of Engagement of Tooth Profiles

pitch circle

ω2

ω1

r1

r2

O2

O1

2

1

C1

C2

P

n

n

K

point P ——the pitch point.

As the center distance O1O2 is constant, the position of the po

int P must be fixed if a constant transmission ratio i12 is required.

This implies that, wherever the teeth

contact, the common normal n-n of the too

th profiles through the contact

point must intersect the center

line at a fixed point P, if a

constant transmission ratio i12 is required.

pitch circle——The loci of P on the motion

planes of both gears are called the circles.

二、 Conjugate Profiles

Meshing profiles of teeth that can yield a desired transmission

ratio are termed conjugate profiles. For circular gears, the conjugate

profiles are those that provide the desired constant transmission rati

o. Generally speaking, for any specific tooth profile, we can find its c

onjugate profile. Theoretically. there is an infinity of pairs of conjug

ate profiles to produce any specific transmission ratio. Nevertheless,

only a few curves have been used as conjugate profiles in practice. A

mong them, involutes are used most widely since gears using involut

es as teeth profiles, or involute gears as they are called, can be manuf

actured and assembled easily.

θk

B

K一、 Generation of Involute

The involute——is the curve generated b

y any point on a string which is unwrappe

d from a fixed cylinder.

t

tGenerating

line

Base circle

O

Ark

rb

二、 Properties of the Involute

2 ） The normal of an involute at any point is tangent to its base circle.3 ） The tangent point B of the generating line with the base circle is the curvatur

e center of the involute at the point K. The length of the segment BK is the radius

of curvature of the involute at the point K.

1 ） AB = BK;

4 ） The shape of an involute depends only on the radius of its base circle.

§§1010 －－ 3 3 The Involute and Its Properties

A1

B1

O1A2

k

K

O2

B2k

O3

B3

∞

∞

5 ） No involute exists inside its base circle.

三、 Equation of the Involute

t

t

B

K

O

Ark

θk

rb

αk

rb

αk

vk

cosαk= rb/rk

tanαk= BK/rb=AB/rb

=rb(θk+αk) /rb

=θk+αk

rk= rb/cosαk

θk= invαk= tanαk-αk

The common normal N1N2 to the meshi

ng involute profiles through their contact poi

nt K must be the common tangent to their bas

e circles. The position of this common tangent

remains unchanged as both gears rotate, as d

oes the common normal to the involute profile

s. This results in a fixed pitch point P. Theref

ore, according to the fundamental law of gear

ing mentioned, the transmission ratio will re

main constant.

1. The transmission ratio will remain constant.

三、 Gearing of Involute Profiles

P

O2

O1

ω2

ω1

rb2

N2

N1

K/ C1C2

K

i12=ω1/ω2= O2P/ O1P = constant

2. The direction and magnitude of the reaction force does not change

The reaction force is exerted along the line of action

if there is no friction. As the position of the line of act

ion stays unchanged during motion for an involute ge

ar pair, the direction and magnitude of the reaction f

orce does not change.

N1N2—— trajectory of contact （ line of action ）α’ ——pressure angle

P

O2

O1

ω2

ω1

rb2

N2

N1

K/ C1C2

Kα’

3. the separability of the center distance in involute gearing

△ O1N1P∽△O2N2P

As shown in Fig., the transmission ratio:

i12=ω1/ω2= O2P/ O1P = rb2 /rb1A change in centre distance does not therefore affect the constant transmissio

n ratio of an involute gear pair. This property is called the separability of the

center distance in involute gearing .

rr b

O

pn

Tooth depth ： h= ha+hf

ha

hfh

p

r a

se si

ei

pb

r f

pi一、 Terminology and Definition

Addendum circle ： da 、raDedendum circle ： df 、 r

fTooth thickness ： si

Spacewidth ： ei

Circular pitch ： pi= si + ei

Reference circle: Between the addendum circle and the dedendum circle, there is an important circle which is called the reference circle. Parameters on the reference circle are standardized and denoted without subscripts, such as d, s, e and p.

Addendum ：haDedendum ：hf

Normal pitch ： pn = p

b

Base circle ： db 、 rb

§§1010 －－ 4 4 Terminology and Definition of Gears

二、 Basic ParametersNumber of teeth ： z

Module ： m The module m of a gear is introduced on the reference circle as a basic parameter, which is defined as:

m=p/π (as πd = zp ， then d = zp /) d=mz

0.35 0.7 0.9 1.75 2.25 2.75 (3.25) 3.5 (3.75)

Second 4.5 5.5 (6.5) 7 9 (11) 14 18 22

Series 28 (30) 36 45

Modules of involute cylindrical gears （ GB1357 － 87 ）

0.1 0.12 0.15 0.2 0.25 0.5 0.4 0.5 0.6 0.8

First Series 1 1.25 1.5 2 2.5 3 4 5 6 8

10 12 16 20 25 32 40 50

Sizes of the teeth and gear are proportional to the module m.

m=4

Z=16

m=2

Z=16

m=1

Z=16

Pressure angle ： α

The pressure angle α is taken as a basic parameter

to determine the base circle. The pressure angle α is

also standardized. It is most commonly 20°. Coefficient of addendum: ha

* ， be standardized: ha* ＝

1Coefficient of bottom clearance : c* ， be standardized:c* ＝0.25 z 、 m 、 α、 ha* 、 c* are the fundamental parameters whic

h determine the size and shape of a standard involute gear.三、 Parameters of Gear

Standard gear ： 1 ） m ,α, ha

* , c* are standardized2 ） e = s

4 ） da ＝ d + 2ha ＝（ z ＋ 2 ha* ） m

5 ） df ＝ d － 2hf ＝（ z － 2 ha* － 2c* ） m

6 ） s ＝ e ＝ p / 2 =m / 2

8 ） hf=(ha* +c*)m

7 ） ha=ha*m

3 ） d = mz

三、 Parameters of Gear

B

１ . The Rack

Characteristics1) The involute tooth profile become

s a straight line too and the pressure

angle remains the same at all points

on the tooth profile.

2) The pitch remains unchanged on the refer ence line, tip line or any other line, i. e. pi= p =πm

e sp

pb

A rack can be regarded as a special form of gear with an in

finite number of teeth and its center at infinity. The radii of all ci

rcles be come infinite and all circles become straight lines, such as

the reference line, tip line and root line.

αα

h ah

f

四、 The Rack and Internal Gears

p b

N

α

s e

hha

hf

pB

1) The teeth are distributed on the internal surface of a hollow

cylinder. The tooth of an internal gear takes the shape of the

tooth space of the corresponding external gear, while the

tooth space of an internal gear

takes the shape of tooth of the

corresponding external gear.2) df > d > da

da ＝ d - 2ha

df ＝ d + 2hf

2. Internal Gears

3) To ensure that the profile of the to

oth on the top is an involute curve, da

>db .

Characteristics:

O

r f

r

r ar b

一、 Proper Meshing Conditions for Involute Gears

rb2

r2

O2

ω2

rb1

r1

rb1

r1

O1

ω1

p b2

p b1

rb2

r2

O2

ω2

pb1> pb2

pb1= pb2

P

N1

N2

B2B1

O1

ω1

p b1

p b2

§§1010 －－ 5 5 Gearing of Involute Spur Gears

To maintain the proper meshing of two pairs of profiles at the

same time, the normal distances of the teeth on both gears must

be the same. pb1= pb2

m1cosα1=m2cosα2

m1 = m2 = m

α1=α2 =α

The proper meshing condition for involute gear

s: the modules and pressure angles of two meshing

gears should be the same.

rb2

r2

O2

ω2

rb1

r1

O1

ω1

p b2

p b1

P

N1

N2

B2B1

To obtain zero backlash of a gear pair:

r 2

O2

r 1

O1

ω1

ω2

P

N1

N2

rb1

r a1

r f2

a

Standard mounting

Zero backlash

C=C*m

C

r f1

二、 Center Distance and Working Pressure Angle of a Gear Pair

1. There are two requirements in designing a gear pair.

1) The backlash should be zero to prevent shock between the gears.

s’1= e’

2 s’2= e’

1

2) The bottom clearance should take the standard value

c=c*m2. Standard(reference) center distance

working center distance a’=r’1+ r’2

reference center distance a = r1+ r2

If two gears are mounted with the reference center distance, then ：

' '1 1 1 1

' '2 2 2 2

/ 2

/ 2

s e s e m

s e s e m

'

' '1 2 0s e

O2

rb2

ω2

r a2

O1

ω1

rb1

r a1r 1

r 2

P

N1

N2

a

α’

f1 2

* * * * =( )

a

a a

c h h

h c m h m c m

3. Center distance a and working pressure angle α’

1) Standard mounting(a’ = a) The reference circles coincide with their pitch circles. r’

1=r1 r’2=r2 α’=α c=c*m

2)Nonstandard mounting(a’ >a) The reference circles do not coincide with their pitch circles.r’

1> r1 r’2> r2 α’>α c’>c*m

r 1’ =

r 1

α’ =α

r 2’ =

r 2

rb2

O2

ω2

O1

ω1

rb1

a’

α’

P

N1

N2

r 2

r’ 2

r 1

r’ 1

α’ >α

α’>α

r’ 2 >

r 2

r 1’ >

r 1

rb1 ＋ rb2 = (r1’+r2

’)cosα’ rb1 ＋ rb2 = （ r1 ＋ r2 ） cosα a’cosα’= a cosα

N1

r a1

N2

α’=α

v2 2

O1

1r f1

ω1

r 1

P

Meshing of a rack and pinion

1 ） Standard mounting

2 ） Nonstandard mounting

The pitch line of the rack coincides

with its reference line :

r1’ = r1 ， α’ ＝ α

The pitch line of the rack does not coincides with its reference line :

r1’ = r1 ， α’ ＝ α

As mentioned above, α’ ＝ α, and r ' ＝ r are charac

teristics of rack and pinion gearing and differ from those

of two spur gears.

三、 Mating Process of a Pair of Gears and Continuous Transmission Condition

N1

O1

r b1

P

r b2

ω2

ω1

O2

r a2

N2

r a1

B2

B1

B1 ——meshing ends at point B1

B2 ——meshing begins at point B2

B1B2 ——the actual line of action

N1N2 ——the theoretical line of action

N1 、 N 2 ——meshing limit points

1. Mating Process of a Pair of Gears

p b

B 1B 2

2. Continuous Transmission Condition In order to get a continuous motion transmission, the second pair of teeth must have meshed before the first pair moves out of contact.

O1

N2

N1

K

O2

ω2

ω1

B1

B2

The condition of continuous motion transmission is ： B1B2≥pb

Contact ratio: = B1B2/pb

Theoretically, if ＝ 1, a pair of gears ca

n transmit continuously. Considering the man

ufacture tolerance, the contact ratio should

be larger than 1. Actually, the contact ratio sho

uld be equal to or larger than the permissible c

ontact ratio[]. []

N1

N2

O1

rb1

rb2

O2

P

Equations of Contact Ratio

r a1

B1

αa2

αa1

'

'

B2

ra2

εα ＝ [z1(tanαa1-tanα’)　　 +z2(tanαa2-tanα’)]/2π

εα ＝ B1B2/pb ＝ (PB1+P B2) /πmcosα PB1 ＝ B1 N1-PN1 ＝ rb1tanαa1-

rb1tanα’

＝ z1mcosα(tanαa1-tanα’ )/2PB2 ＝ z2mcosα(tanαa2-tanα’ )/2

The value of the contact ratio indicates the average number of tooth pairs in contact during a cycle to share the load. The higher the contact ratio, the greater the average number of tooth pairs to share the load and the higher the capacity of the gear set to transmit the power.

=1.46

1.46 pb

B1 B2

Two pairs Two pairsOne pair

0.46 pb0.54 pb0.46 pb

pb

pb

CD

2 ） The curvature radius of the tooth profile and the tooth thickness of the pini

on on the dedendum circle are less than those of the gear. The strength of the pi

nion is much lower than that of the gear, and contact time of the pinion is more t

han that of the gear.

Standard gears enjoy interchangeability and are widely use

d in many kinds of machines. However, they also have some disad

vantages.1 ） It is not fit that a’≠a. When a’<a , the pair of gears can not be installed at

all. When a’>a, the backlash will increase and the contact ratio will decrease.

3 ） When z< zmin ， undercutting will occur.

Basecircle

Referencecircle

Cutter interference——In a generating process, it i

s sometimes found that the top of the cutter enters the

profile of the gear and some part of the involute profil

e near the root portion is removed.

一、 Standard gears have some disadvantages

§§1010 －－ 6 6 Introduction to Corrected Gear

To improve the performance of gears, addendum modification is employed.

二、 Manufacturing Methods of Involute Profiles

1. Cutting of Tooth Profiles

pinion-shaped shaper cutter rack-shaped shaper cutter

The cutting motion is the reciprocation of the cutter while the feed is the movement of the cutter toward the blank. The blank should retreat a little as the cutter goes back to prevent scraping on the finished flank by the cutter.

Gear hobbing

c*m

Reference line

rb’

N1’

P

α

rbrr a

N1

O1

O1’

O1’’

Reference circle

Gear blank

h a* m

B1

B2

v

N1’’

rb’’

Involute

2. Cutting a Standard Gear with Standard Rack-shaped Cutter

e = s = p / 2 ha= ha

* m; hf =(ha*+ c*)m;

The reference line of the c

utter should be tangent to

the reference circle of the gea

r

1 ） The addendum line of the cutter does not exceed the limit

point N1’’ of the line of action, cutter interference will not

occur.

2 ） Cutter interference will occur if the addendum line of the cutter passes the limit point N1

’’ of the line of action.

3. Minimum Teeth Number of Standard Gear Without Undercutting

To prevent cutter interference, the point B2 should not pass p

oint Nl , ： PN1≥PB2

PN1=rsinα=mzsinα/2 PB2=ha*m/sinα=mzsinα/2

*

min 2

2

sin

hz

4. Methods to Avoid Undercutting

1 ） Decrease the coefficient of addendum depth ha*

ha* zmin

ha* the transmission characteristics will be influenced and

the cutter will not be standard.

There are several methods to avoid undercutting ：

The cutter will be standard.

The method commonly used to eliminate u

ndercutting is to cut the gears with profile-sh

ifted, i.e., with unequal addendum and deden

dum teeth.

3 ） Corrected gear

Therefore, parameters m , , ha* , c* , of th

e corrected gear remain the same as those of

standard gears, but s≠e ， the gear is called

corrected gear (profile-shifted gear).

2 ） Increase the pressure angle of cutter

rb This procedure will reduce the active length and t

he contact ratio will reduce too, which will also lead to rougher, no

isier gear operation and the cutter will not be standard.

zmin

α

N1

α

O1

P Q

ha*

m

xm

xminm

xm

5. Corrected gear

Modification distance （ xm ）—— In c

utting the corrected gear, the rackshape

d cutter is located a distance xm from th

e position used for cutting the standard

gear.

x ——modification coefficient

α

N1

α

O1

P Q

h a* m

xm

xminm

xm

Positive modification( x>0) ——The cutter is placed further away

from the position for cutting a standard gear.

positive modification gear

Negative modification( x<0) ——The cutter is placed towards the

axis of the blank. negative modification gear

三、 Geometric Dimensions of Corrected Gears

1. Geometric dimensions are identical with that of the standard gear

d = mz

db = mzcos

p = m

2. Geometric dimensions are not identical with that of the standard gear

K J

I

xm

xm

Pitch line of cutter

α

B2

Reference line of cutter

KI J

πm/2

Reference circle

PN1

O1

α

rb

1 ） Tooth thickness and spacewidth

( )a2 2

2 2 t nKJm

s x m

( )a2 2

2 2 t nKJm

e x m

Base circle

2 ） Addendum and dedendum* * * *

f ( )a axmh h m c m h c mx * *

a ( )a axmh h m h mx *

a ( )ar r h x m Positive modification gear x>0

Reference circle

Standard gear x ＝ 0 Negative modification gear

x<0

四、 Gearing of a Corrected Gear Pair1. Proper meshing conditions and condition of continuous transmission

Proper meshing conditions ： m1= m2 α1=α2

Condition of continuous transmission ： []

2. Centers distance of a pair corrected gear

1) Gearing equation without backlash

To keep zero backlash for a corrected gear pair, the following relations should hold, as in the case of standard gears, i.e., sl'= e2' , s2'= el'

, therefore,

p' ＝ s'1+ e'

1 ＝ s'2+ e'

2 ＝ s'1+ s'

2

' 1 2

1 2

2 tan ( )

( )2

x xinv inv

z z

' 1 2

1 2

2 tan ( )

( )2

x xinv inv

z z

(x1+x2) ’ The two pitch circles will not overlay on the two reference circles

acos＝ acos a’ a

2 ） Shifting coefficient of centers distance y

Difference of the centers distance a’ with standard centers distance a ： ym = a’- a

1 2'

( ) cos1

2 cos

z zy

Analysis

y——Shifting coefficient of centers distance

3 ） Shifting coefficient of addendum depth y

Clearance be

standard ：

With no backlash ：

If two gears mating with no backlash and remaining standard clearance, therefore

Problem ： (x1+x2) > y if x1+ x2≠0 a' > a''

yxxy )( 21

ymzzm

ymaa )(2 21

a'=a'' y=x1+x2

mxxzzm

mxxrr

mxchmcmxhrrhrchrrcra

aa

fafa

)()(2

)(

)()(

21212121

2***

1*

21

221121

Solution ： No backlash can be assured, the depth of addendum

circle is decreased.

myxhymxmmhhAddendum aaa )( ** ：

3. Types of Corrected Gear Pairs

（ 1 ） Standard transmission （ x1+ x2 ＝ 0 ， and x1 ＝x2 ＝ 0 ）

Types of corrected gear pairs can be divided into three types

by the sum of the shifting coefficients（ x1+ x2） .

z1 > zmin , z2 > zmin

As x1+x2 = 0 and the above three equations

a’ = a , ’= ， y = 0 ， y = 0

The pinion should be positive corrected gear( x1 >0) ； the gear

should be negative corrected gear （ x2<0 ） .

Two gears should not be undercutting ： z1 + z2 ≥ 2zmin

（ 2 ） Zero transmission (height shifting gears transmission ） x1+ x2 ＝ 0 ， and x1 ＝ -x2≠0

Since gears are positive corrected gear, the strengths of two gears increase. But the contact ratio decreases since the working pressure angles decrease.

（ 3 ） Angle shifting gear transmission ( x1+x2≠0 )

1 ） Positive transmission （ x1+x2 > 0 ） As x1+x2 > 0 and the above three equations

a’ > a , ’ > ， y > 0 ， y > 0 As x1+x2 > 0 z1+z2 < 2 zmin

2 ） Negative transmission （ x1+x2 < 0 ）

As x1+x2 < 0 and the above three equations

a’ < a , ’ < ， y < 0 ， y > 0

AS x1+x2 < 0 ， therefore z1+z2 > 2 zmin

This transmission is contrary to positive transmission. Since gears are negative corrected the strengths of the two gears decrease.But the contact ratio increases since the working pressure angle decrease.

Spur gear

Helical gear

Properties: Tooth profiles go into and out of contact al

ong the whole facewidth at the same time ； Sudden loading and sudden unloading on t

eeth ； Vibration and noise are produced.

Properties:The tooth surfaces of two engaging helical gears

contact on a straight line inclined to the axes of the

gears ；The length of the contact line changes gradually from

zero to maximum and then from maximum to zero ； The loading and unloading of the teeth become

gradual and smooth.

§§1010 －－ 7 7 Helical Gears for Parallel Shafts

一、 Basic Parameters of Helical Gears

There are two sets of parameters for a helical gear.One set is on the

transverse plane and the other set on the normal plane.

The parameters on the normal plane are the standard values.

To make use of the formulae for spur gear, the parameter in the equations for

spur gears should be replaced by those on the transverse plane of helical gears.

Therefore, it is necessary to set up relationships between both sets of

parameters.

1. Helix angleβ

righthanded lefthanded

β β

helix angle （ β）—— is the helix angle on the reference cylinder.

（一） Basic Parameters of Helical

2. Normal module mn and transverse module mt

B

β pt β

πd

p n

costn pp

costn mm 3. Normal pressure angle n and transverse pressure angle t

''

'

tan tanba

ca

ab

acnt

''baab cos ' acca

costan an t tn

4. Coefficient of addendum （ h*an 、 h*

at ） and coefficient

of bottom clearance(c*n 、 c*

t)hf=(h*

an+cn*)mn = (h*

at+ct*)mt ha=h*

anmn = h*atmt

cos**naat hh cos**

nt cc

( 二） Sizes of helical gear

Reference diameter ： cos/nt zmzmd

Center distance: cos2/)(2/)( 2121 zzmdda n

Modification coefficient ：

cosnt xx

二、 Gearing of a pair of helical gears

1. Proper Meshing Conditions for Helical Gears

21 nn mm

)gear external21 （

21 nn

）（ gear internal21

or

21 tt mm

21 tt

)gear external21 （

）（ gear internal21

2. Contact Ratio for a Helical Gear Pair

B

B2

B2△L

βb βb

B1

B1

B1

B1

B

B2

B2

L

Spur gear ：

Helical gear ：

The contact ratio of a helical gear pair is much higher than that of a spur gear pair.

btp

L

p

L

b

btbtbt p

ΔL

p

L

p

ΔLL

)]tan(tan)tan(tan[2

1 '22

'11 tattat zz

transverse contact ratio

nntt

t

m

B

p

B

αp

B

sin

cos/

cos/sin

cos

costg

bt

b

bt p

B

p

L tg

is the face contact ratio or overlap ratio.

三、 Virtual Number of Teeth for Helical Gears

Virtual gear——the tooth profile of the spur gear is equivalent to t

hat of a helical gear on the normal plane. The spur gear is called t

he virtual gear of the helical gear. The number zv of teeth of the virt

ual gear is called the virtual number of teeth （ zv ） .

a

r

b

22

2

cos

1)

cos(

r

r

r

b

a

3

22

cos

coscos

2

z

m

zm

m

d

mz

n

t

nnv

The minimum number of teeth of the standard

helical gear without cutter interference ： zmin=zvmincos3β

四、 The main advantages and disadvantages of helical gears

1. Main advantages ：

１） Better meshing properties.

２） A much higher total contact ratio.

３） Being more compact means of mechanical power transmission.

2. Main disadvantages ：The helix angle results in a thrust load in addition to the usual tangential and separating loads. Fa=Ft tg Fa

βFn Ft

β

aF

erringbone gear

β ＝ 8° ～ 20°

一、 Worm Gearing and its Characteristics

Worm gear drives are used to transmit motion and power between non inte

rsecting and non-parallel shafts, usually crossing at a right angle. ＝ 90

1) Smooth silent operation as screw drives.

2) Greater speed reduction in a single

step. This means compact designs.

3) If the lead angle of a worm is less than the friction

angle, the back-driving is self-locking.

4) Lower efficiency due to the greater relative sliding speed . The friction loss ma

y result in overheating and serious wear. There fore, brass is usually used as the

material for the worm wheel to reduce friction and wear.

§§1010 －－ 8 8 Worm Gearing

Cylindrical worms

Enveloping worms

spiroids

Archimedes worm

Involute helicoid worms

Arc-contact wormsTypes of Worms

二、 Types of Worms

三、 Proper Meshing Conditions for Worm Drives mid-plane ： The transverse plane of a worm wheel passing through the axis of the worm The engagement between a worm and a worm wheel on the mid-plane c

orresponds to that of a rack and pinion

Proper Meshing Conditions ： The modules and pressure angles of the worm and worm wheel on the mid-

plane should be equal to each other.

mmm tx 21

21 tx

21

The directions of both helices should be the same.

)90( 0

四、 Main Parameters and Dimensions for Worm Drives

2. The module

The series of modules for worms is somehow different from those for gears.

3. The profile angle of worm (pressure angle)

Archimedes worm ： 20º

In power transmission ： 25º

In indexing devices ： 5º or 2º

1. The number of teeth

The number of threads on the worm z1 : usually, z1＝ 1 ~ 10 ，

the recommended value of z1:

z1＝ 1、 2、 4、 6。

The number of teeth on the worm gear z2 is determined according

to the speed ratio and the selected value of z1. For power

transmission, z2＝ 29 ~ 70.

4. The lead angleγ1 of the worm

1

1

1

11

11tan

d

mz

d

pz

d

l x

5. reference diameter

The mid-diameter d1 of worm ： the mid-diameter d1 of

the worm is standardized.

The reference diameter d2 of worm wheel ： d2 = mz2

21 rra

6. The center distance a of the worm gear pair

一、 Introduction to Bevel Gears

Bevel gears are used to transmit motion and power between intersecting shafts. The teeth of a bevel gear are distributed on the frustum of a cone. The corresponding cylinders in cylindrical gears become cones, such as the reference cone, addendum cone and dedendum cone. The dimensions of teeth on different transverse planes are different. For convenience, parameters and dimensions at the large end are taken to be standard values.

The shaft angle of a bevel gear pair can be any required value.

In most cases, the two shafts intersect at a right angle.

1. Characteristics of Bevel Gears

§§1010 －－ 9 9 Bevel Gears

2. Types and Applications or Bevel Gears

Bevel Gears

Straight bevel gears ：

Helical bevel gears ：

Spiral bevel gears ：

are most widely used as they are easy

to design and manufacture.

operate smoothly and easy to design .

operate smoothly and have higher load

capacity.

r2

O2

O1

rv1

r1

δ1

P 2

1

=90°δ2

∑

Crown gear

P

P1

二、 Back Cone and Virtual Gear of a Bevel Gear

Crown gear ----d 2 = 90 ， the surface of the reference cone becomes a plane.

Back cone——the cone , the element of which crosses the large end of a bevel gear and is perpendicular to the element of the reference cone.

Virtual gear of the bevel gear ： mv = m ； αv = α ； rv= r

The tooth profile of the virtual gear is almost the same as that of the bevel g

ear at the large end. Virtual number of teeth zv ： The tooth number of the virtual gear

r2

O2

O1

rv1

r1

δ1

P 2

1

=90°δ2

∑

Crown gear

P

P1

Virtual number of teeth zv

2cos2cosv

v

mzmzrr

cos

zzv

The engagement of bevel gears The engagement of spur gears

Proper Meshing Conditions ： m1=m2 , α1=α2

The contact ratio of the bevel gear set. The virtual number of teeth zv should not be less than the minimum number of teeth o

f the virtual gear. zmin=zvmincosδ

三、 Parameters and Dimensions of Bevel Gears

The most dimensions of bevel gears are measured at the large end being standardized.

1. The reference diameter is

11 sin2 Rd 22 sin2 Rd

2. The transmission ratio of a gear pair is

1

2

1

2

1

2

2

112 sin

sin

d

d

z

zi 2112 tgctg i （∑＝ 90° ）

R—Outer cone distance δ—Reference cone angle

δa—Addendum cone angle

b—Face width

da—Addendum diameter

df—dedendum diameter

b

Rd1

δa1

δa2

da2

d2

df 2

δ2

δ1 d1 , d2—Reference diameter

Transmission ratio ： i12 ＝ ω1 / ω2

When ∑ ＝ 90° ，

＝ z2 /z1

＝ r2 / r1

＝ sinδ2 /sinδ1

＝ cotδ1 i12 ＝ tanδ2∑=90°

δ2 +δ1 ＝ 90°

2 hahf

O

θf

1

δ1r1

r2

δ2R