Gear Guide1

7/24/2019 Gear Guide1

1/57

Elementary

Information on

Gears

The Role Gears are PlayingGears are some of the most important elements used in machinery. There are few mechanical devices that do

not have the need to transmit power and motion between rotating shafts. Gears not only do this most

satisfactorily, but can do so with uniform motion and reliability. In addition, they span the entire range of

applications from large to small. To summarize:

1.Gears offer positive transmission of power.

2.Gears range in size from small miniature instrument installations, that measure in only several

millimeters in diameter, to huge powerful gears in turbine drives that are several meters in ameter.

3.Gears can provide position transmission with very high angular or linear accuracy, such as used

in servomechanisms and precision instruments.

4.Gears can couple power and motion between shafts whose axes are parallel, intersecting or skew.

5.Gear designs are standardized in accordance with size and shape which provides for

widespread interchangeability.

This introduction is written as an aid for the designer who is a beginner or only superficially knowledgeable about gearing. It

provides fundamental, theoretical and practical information. When you select KHK products for your applications

please utilize it along with KHK3009 catalog.


2/57

Table of Contents

1 ..........................GearTypesandTerminology3

1.1 Type of Gears ...................................3

1.2Symbols and Terminology.........................5

2Gear Trains.........................................................7

2.1Single - Stage Gear Train..............................7

2.2Double - Stage Gear Train............................8

3 ...........................................InvoluteGearing9

3.1Module Sizes and Standards....................9

3.2The Involute Curve....................................10

3.3 Meshing of Involute Gearing.................11

3.4The Generating of a Spur Gear................11

3.5Undercutting.............................................12

3.6Profile Shifting...........................................12

4 ..........CalculationofGearDimensions13

4.1Spur Gears...................................................134.2Internal Gears...........................................18

4.3Helical Gears............................................21

4.4Bevel Gears..............................................28

4.5Screw Gears.............................................34

4.6Cylindrical Worm Gear Pair.......................36


3/57

1Gear Types and Terminology

Elementary Information on Gears

1.1 Type of gears

In accordance with the orientation of axes, there are three

categories of gears:

1. Parallel axes gears

2. Intersecting axes gears

3. Nonparallel and nonintersecting axes gears

Spur and helical gears are the parallel axes gears. Bevel

gears are the intersecting axes gears. Screw or crossed

helical gears and wor gears handle the third categor!.

"a#le 1.1 $ists the gear t!pes per axes orientation.

Table 1.1 Types of gears and their categories

Categories of gears Types of gears Efficiency(%)

Spur gear

Spur rac%

Parallel axes Internal gear&'.( ) &&.*

gears +elical gear

+elical rac%

ou#le helical gear

Intersecting axesStraight #evel gear

Spiral #evel gear &'.( ) &&.(

gears-erol #evel gear

Nonparallel and or gear 3(.( ) &(.(nonintersecting

Screw gear /(.( ) &*.(axes gears

0lso, included in ta#le 1.1 Is the theoretical efficienc! range of

the various gear t!pes. "hese figures do not include #earing and

lu#ricant losses. 0lso, the! assue ideal ounting in regard to

axis orientation and center distance. Inclusion of these realistic

considerations will downgrade the efficienc! nu#ers.

(1) Parallel Axes Gears

(a) Spur Gear

"his is a c!lindrical shaped gear

in which the teeth are parallel to

the axis. It has the largest

applications and, also, it is the

easiest to anufacture.

Fig.1.1

Spur

gear(b) Spur Rack

"his is a linear shaped

gear which can esh

with a spur gear with

an! nu#er of teeth.

"he spur rac% is a

portion of a spur gear

with an infinite radius.

(c) Internal Gear

"his is a c!lindricalshaped gear #ut with the

teeth inside the circular

ring. It can esh with a

spur gear. Internal gears

are often used in

planetar! gear s!stes

and also in gear

couplings.

(d) Helical Gear

"his is a c!lindrical

shaped gear with

helicoid teeth. +elical

gears can #ear ore load

than spur gears, and

wor% ore uietl!. "he!

are widel! used in

industr!. 0 negative is

the axial thrust force the

helix for causes.

(e) Helical Rack

"his is a linear shaped gear

which eshes with a helical

gear. 0gain, it can #e

regarded as a portion of a

helical gear with infinite

radius.


4/57

(f) Double Helical Gear

"his is a gear with #oth lefthand and righthand helical

teeth. "he dou#le helical for #alances the inherent thrust

forces.

Fig.1.2 Spur rack

Fig.1.3 Internal gear

and spur gear

Fig.1.4 Helical gear

Fig.1.5 Helical rack

Fig.1.6 Double helical gear


5/57


(2) Intersecting Axes Gears

(a) Straight Bevel Gear

"his is a gear in which the teeth

have tapered conical eleents

that have the sae direction as

the pitch cone #ase line

generatrix4. "he straight #evel

gear is #oth the siplest to

produce and the ost widel!

applied in the #evel gear fail!.

(b) Spiral Bevel Gear

"his is a #evel gear with a

helical angle of spiral teeth. It is

uch ore coplex to

anufacture, #ut offers a higher

strength and lower noise.

(c) Zerol Bevel Gear

Fig.1.7 Straight bevel gear

Fig.1.8 Spiral bevel gear

(b) Screw Gear

(Crossed Helical Gear)

0 pair of c!lindrical gears used

to drive nonparallel and non

intersecting shafts where the

teeth of one or #oth e#ers ofthe pair are of screw for.

Screw gears are used in the

co#ination of screw gear 5 screw

gear, or screw gear 5 spur gear.

Screw gears assure sooth, uiet

operation. +owever, the! are not

suita#le for transission of high

horsepower. Fig.1.11 Screw gear

(4) Other Special Gears

(a) Face Gear

"his is a pseudo#evel gear that

is liited to &(6 intersecting

axes. "he face gear is a

circular disc with a ring of

teeth cut in its side face7 hence

the nae face gear

-erol #evel gear is a special

case of spiral #evel gear. It is a

spiral #evel with a spiral angle

of 8ero. It has the characteristicsof #oth the straight and spiral

#evel gears. "he forces acting

upon the tooth are the sae as

for a straight #evel gear.

(3)Nonparallell andNonintersectingAxes Gears

(a) Worm Gear Pair

or gear pair is the nae

for a eshed wor and

wor wheel.

"he outstanding feature is that

it offers a ver! large gear ratio

in a single esh. It also

provides uiet and sooth

action. +owever, transission

efficienc! is ver! poor.

Fig.1.9 Zerol bevel gear

Fig.1.10 Worm gear pair

(b)Enveloping Worm

Gear Pair

"his wor gear pair uses a

special wor shape in that it

partiall! envelops the wor

wheel as viewed in the direction

of the wor wheel axis. Its #ig

advantage over the standard

wor is uch higher load

capacit!. +owever, the wor

wheel is ver! coplicated to

design and produce.

(c) Hypoid Gear

"his is a deviation fro a

#evel gear that originated as a

special developent for the

autoo#ile industr!. "his

peritted the drive to the rear

axle to #e nonintersecting, and

thus allowed the auto #od! to

#e lowered. It loo%s ver!

uch li%e the spiral #evel

gear. +owever, it is

coplicated to design and is

the ost difficult to produce

on a #evel gear generator.

Fig.1.12 Face gear

Fig.1.13 Enveloping worm gear pair


6/57

Fig.1.14 Hypoid gear


7/57

1.2 Symbols and Terminology

"a#le 1.2 through 1.9 indicate the s!#ols and the terinolog!

used in this catalog. IS B (121:1&&&and IS B(1(2:1&&&cancel

and replace forer IS B(121 s!#ols4 and IS B(1(2

voca#ular!4 respectivel!. "his revision has #een ade to

confor to International Standard 6rgani8ation IS64 Standard.

Table 1.2 Linear dimensions and circular dimensions

Terms Symbols

;entre distance a


8/57


Table 1.5 Others

Terms Symbols

"angential force Ft0xial force Fx


9/57

2Gear Trains

"he o#Eective of gears is to provide a desired otion, either

rotation or linear. "his is accoplished through either a

siple gear pair or a ore involved and coplex s!ste of

several gear eshes. 0lso, related to this is the desired

speed, direction of rotation and the shaft arrangeent.

2.1 Single-Stage Gear Train

0 eshed gear is the #asic for of a singlestage gear train.

It consists of z1 andz2nu#ers of teeth on the driver and

driven gears, and their respective rotations, n1F n2.


"he transission ratio is then:

"ransission ratio G

z2

G

n1

2.14z1 n2

@ear trains can #e classified into three t!pes:

"ransission ratio H 1 , increasing : n1 H n2

"ransission ratio G 1 , eual speeds: n1 G n2

"ransission ratio 1 , reducing : n1 n2

=igure 2.1 illustrates four #asic fors. =or the ver! coon

cases of spur and #evel gear eshes, =igures 2.104 and B4,

the direction of rotation of driver and driven gears are

reversed. In the case of an internal gear esh, =igure 2.1;4,

#oth gears have the sae direction of rotation. In the case of

a wor esh, =igure 2.14, the rotation direction of z2 is

deterined #! its helix hand.

Gear 2 Gear 1 Gear 2 Gear 1

z2,n24 z1,n14 z2,n24 z1,n14

(A) A Pair of spur gears (B) Bevel gears

Gear 2 Gear 1 Right-hand worm gear Left-hand worm gear

z2,n24 z1,n14 z1,n14 z1,n14

(C) Spur gear and internal gearRight-hand worm wheel Left-hand worm wheel

z2 ,n24 z2,n24

(D) Worm gear pair


10/57

Fig. 2.1 Single-stage gear trains


11/57


In addition to these four #asic fors, the co#ination of a rac%

and pinion can #e considered a specific t!pe. "he displaceent

of a rac%, l, for rotation of the ating pinion is:

l Gz1

Jpm 2.2439(

where: pmis the reference pitch

z1is the nu#er of teeth of the pinion.

2.2 Double-Stage Gear Train0 dou#lestage gear train uses two singlestages in a series.

=igure 2.2 represents the #asic for of an external gear

dou#lestage gear train.

$et the first gear in the first stage #e the driver. "hen the

transission ratio of the dou#lestage gear train is:

"ransission


12/57

3Involute Gearing

"he invoute profile is the one ost coonl! used toda!

for geartooth fors that are used to transit power. "he

#eaut! of involute gearing is its ease of anufacture and its

sooth eshing despite the isalignent of center distancein soe degree.

3.1 Module Sizes and Standards?odule mrepresents the si8e of involute gear tooth. "he unit of

odule is . ?odule is converted to pitchp, #! the

factor .

p Gm 3.14

"a#le 3.1 is extracted fro IS B 1/(11&/3which defines

the tooth profile and diensions of involute gears. It

divides the standard odule into three series. =igure 3.1

shows the coparative si8e of various rac% teeth.

Table 3.1 Standard values of module unit: mm

Series 1 Series 2 Series 3 Series 1 Series 2 Series 3

(.1(.1*

3.*3./*

K(.2 (.2* K.*

(.3*

(.3* *.*

(.K (.K*

9

9.*/

(.* (.** '

(.9&

(.9* 1((./ 11(./* 12

(.'1K

(.& 19

11'

2(1.2*

221.* 1./* 2*

22'

2.2* 32

2.* 392./* K(

3K*

3.2* *(

NOTE:"he preferred choices are in the series order #eginning with 1.

iaetral Pitch P(.P.4, the unit to denote the si8e of the

geartooth, is used in the DS0, the D, etc. "he

transforation fro iaetral PitchP.P.4 to odule mis

accoplished #! the following euation:

m G 2*.K 5P


?1

?1.*

?2

?2.*

?3

?K

?*

?9

?1(

Fig.3.1 Comparative size of various rack teeth


13/57


p

p

52 p#

hh

ahf

d#

d?odule m


14/57

Fig.3.3 The involute curve

10


15/57

3.3 Meshing of Involute Gear

=igure 3.K shows a pair of standard gears eshing together."he contact point of the two involutes, as =igure 3.K shows,slides along the coon tangent of the two #ase circles asrotation occurs. "he coon tangent is called the line ofcontact, or line of action.

0 pair of gears can onl! esh correctl! if the pitches andthe pressure angle are the sae. "hat the pressure angles

ust #e identical #ecoes o#vious fro the following

euation for #ase pitch:

p#Gmcos 3./4

"hus, if the pressure angles are different, the #ase pitches

cannot #e identical.

"he contact length a# shown =igure 3.K is descri#ed as

A$ength of path of contact.

"1

"2


"he contact ratio can #e expressed #! the following euation:

"ransverse ;ontact ratio G$ength of path of contact

abBase

pitchp#

3.'4

It is good practice to aintain a transverse contact ratio of

1.2 or greater.Dnder no circustaces should the ratio drop #elow 1.1.

?odule m and the pressure angleare the %e! ites in the

eshing ofgears.

3.4 The Generating of a Spur Gear

Involute gears can #e readil! generated #! rac% t!pe cutters.

"he ho# is in effect a rac% cutter. @ear generation is also

accoplished with gear t!pe cutters using a shaper or planer

achine.

=igure 3.* illustrates how an involute gear tooth profile is

generated. It shows how the pitch line of a rac% cutter

rolling on a pitch circle generates a spur gear.@ear shapers with pinion cutters can also #e used to

generate involute gears. @ear shapers can not onl! generate

external gears #ut also generate internal gears.

d1d#1 a length of pass of Contact

"1

"2

b

d#2

d2

"1

"2

Fig.3.4 The meshing of involute gear

Rack form tool

2 #

sin

d 2

d d#

"

Fig. 3.5 The generating of a standard spur gear

G 2(L,zG 1(,xG ( 4


16/57

11


17/57


3.5 Undercutting

Dndercutting is the phenoenon that soe of tooth

dedendu is cut #! the edge of a generating tool. In case

gears with sall nu#er of teeth is generated as is seen in

=igure 3.*, undercut occurs when the cutting is ade deeper

than interfering point I. "he condition for no undercutting in

a standard spur gear is given #! the expression:

mmz

sin 2 3.&42

and the iniu nu#er of teeth is:

zG

2 3.1(4sin 2

=or pressure angle 2( degrees, the iniu nu#er of

teeth free of undercutting is 1/. +owever, the gears with 19teeth or under can #e usa#le if its strength or contact ratio

pose an! ill effect.

3.6 Profile Shifting

0s =igure 3.* shows, a gear with 2( degrees of pressure

angle and 1( teeth will have a huge undercut volue. "o

prevent undercut, a positive correction ust #e introduced.

0 positive correction, as in =igure 3.9, can prevent undercut.

Dndercutting will get worse if a negative correction is

applied. See =igure 3./. "he extra feed of gear cutter xm4 in

=igures 3.9 and 3./ is the aount of shift or correction. 0ndxis the profile shift coefficient.

Rack form tool

xm

xm

sin2

d 2

db

d

"

"he condition to prevent undercut in a spur gear is:

m xmzm

sin

23.1142

"he nu#er of teeth without undercut will #e:

z $

21x4

3.124sin 2

"he profile shift coefficient without undercut is:

x $ 1

z

sin 2 3.1342

Profile shift is not erel! used to prevent undercut. It can #e

used to adEust center distance #etween two gears.

If a positive correction is applied, such as to prevent undercutin a pinion, the tooth tip is sharpened.

"a#le 3.2 presents the calculation of top land thic%ness ;rest

width 4.

Table 3.2 The calculations of top land thickness ( Crest width )

No. Item Symbol Formula Example

"ip pressure d#m G 2G 2(LzG 19

1a cos

1 x G M (.3d G 32

angle da dbG 3(.(/(19daG 3/.2aG 39.(9919L

"ip tooth 2xtan M M inv inva4 inv aG (.(&''3*2 thic%ness half 2z z

anglea

radian4inv G(.(1K&(K

aG 1.*&'1*L

(.(2/'&3 radian43 ;rest width sa a.da saG 1.(3/92

Rack form tool

a

%a

a

34

34db

d"

%4

Fig.3.6 Generating of positive shifted spur gear Fig.3.7 The generating of negative shifted spur gear Fig. 3.8 Top land thickness

G 2(L,zG 1(,xG M(.* 4 G 2(L,zG 1(,xG (.* 4 ( Crest width )

12


18/57


4Calculation of Gear Dimensions

"he following should #e ta%en into consideration in due

order at the earl! stage of designing:

To calculate the required strength to deterine the specifications, the aterials to #e used, and the degree of accurac!.To calculate the dimensions in order to provide the necessar! data for the gear shaping.

To calclate the tooth thickness in order to provide the necessar! data for cutting and grinding.To calculate the necessary amount of backlash

to provide the necessar! inforation useful for selecting the proper shaftsTo calculate the forces to be acting on the gearand #earings.

To consider what kind of lubrication is necessary and appropriate.

a

"he explanation is given, hereafter, as to ites necessar! for

the design of gears. "he calculation of the dientions coes

first. "he dientions are to #e calculated in accordance with thed2 da2fundaental specifications of each t!pe of gears. "he processes d#2

of turning etc. are to #e carried out on the #asis of that data.d

1 d#1 df2

"1 "2

4.1 Spur Gears d

f1

(1) Standard Spur Gear=igure K.1 shows the eshing of standard spur gears. "he

eshing of standard spur gears eans reference circles of

two gears contact and roll with each other. "he calculation

forulas are in "a#le K.1.

da1

Fig.4.1 The meshing of standard spur gears G 2(L,z1G 12,z2G 2K,x1Gx2G ( 4

Table 4.1 The calculation of standard spur gears

No. Item Symbol FormulaExample

Pinion Gear

1 ?odule m 3

2


19/57

13


20/57


0ll calculated values in "a#le K.1 are #ased upon given

odule m4 and nu#er of teeth z 1 and z2 4. If instead

odule m4, center distance a4 and speed ratio i4 are given,

then the nu#er of teeth ,z1andz2, would #e calculated with

the forulas as shown in "a#le K.2.

Table 4.2 The calculation of number of teeth


1 ?odule m 3

2 ;enter distance a *K.(((

3 "ransission ratio i (.'

4 Su of No. of teeth z1Mz22a

39m

5 Nu#er of teeth zz1Mz2 i z1Mz24

19 2(i M1 i M1

Note that the nu#er of teeth pro#a#l! will not #e integer

values #! calculation with the forulas in "a#le K.2. In

that case, it will #e necessar! to resort to profile shifting or

to eplo! helical gears to o#tain as near transission

ratio as possi#le.

14


21/57


(2) Profile Shifted Spur Gear

=igure K.2 shows the eshing of a pair of profile shifted

gears. "he %e! ites in profile shifted gears are the

operating wor%ing4 pitch diaeters d'4 and the wor%ingoperating4 pressure angle '4.

"hese values are o#taina#le fro the odified center

distance and the following forulas:

d'1 G 2az1

z1 Mz2

d'2 G 2az2

K.14z1 Mz2

d#2d#1 M 1

' Gcos

2a

a

d'2 da2

d'1d2

d#2

d1 d#1df2

"1 ' "2

df1 '

In the eshing of profile shifted gears, it is the operating pitch

circle that are in contact and roll on each other that portra!s gear

action. Fig. 4.2 The meshing of profile shifted gears

"a#le K.3 presents the calculation where the profile shiht G 2(L,z1G 12,z2G 2K,x1G M (.9,x2G M (.39 4

coefficient has #een set atx1andx2at the #eginning. "his

calculation is #ased on the idea that the aount of the tip and

root clearance should #e (.2* m.

Table 4.3 The calculation of profile shifted spur gear (1)


Pinion (1) Gear (2)

1?odule m 3

2


22/57

15


23/57


"a#le K.K is the inverse forula of ites fro K to ' of

"a#le K.3.

Table 4.4 The calculation of profile shifted spur gear (2)


1 ;enter distance a *9.K&&&

2 ;enter distance y a z1Mz2 (.'333odification coefficient m 2

cos1

cos

3 or%ing pressure angle ' 2yM 1

29.(''9z1

Mz2

Su of profile shift

4 coefficient x1Mx2z1Mz24 inv ' inv 4

(.&9((2 tan

5 Profile shift coefficient x (.9((( (.39((

"here are several theories concerning how to distri#ute the su

of profile shift coefficient x1Mx24 into pinion x14 and gear x24separatel!. BSS British4 and IN @eran4 standards are the

ost often used. In the exaple a#ove, the 12 tooth pinion was

given sufficient correction to prevent undercut, and the residual

profile shift was given to the ating gear.

16


24/57

(3) Rack and Spur Gear"a#le K.* presents the ethod for calculating the esh of a

rac% and spur gear.

=igure K.314 shows the the eshing of standard gear and a

rac%. In this eshing, the reference sircle of the gear

touches the pitch lin of the rac%.

=igure K.324 shows a profile shifted spur gear, with positive


correctionxm, eshed with a rac%. "he spur gear has a larger

pitch radius than standard, #! the aount xm. 0lso, the pitch

line of the rac% has shifted outward #! the aountxm.

"a#le K.* presents the calculation of a eshed profile

shifted spur gear and rac%. If the profile shift coefficientx1is

(, then it is the case of a standard gear eshed with the rac%.

Table 4.5 The calculation of dimensions of a profile shifted spur gear and a rack


Spur gear Rack

1 ?odule m 3

2


25/57

G 2(L,z1G 12,x1G ( 4

Fig.4.3(2) The meshing of profile shifted spur gear and rack G 2(L,z1G 12,x1G M (.9 4

17


26/57


4.2Internal Gears

(1)Internal Gear Calculations

=igure K.K presents the esh of an internal gear and external

'gear. 6f vital iportance is the wor%ing pitch diaeters d'4 andwor%ing pressure angle '4. "he! can #e derived fro centerdistance a'4 and >uations K.34. "1

'a

z1d'1G 2a z2z1 "2

z2 K.34d'2G 2a z2z1

1 d#2d#1' G cos

2a

d#2

da2

d2

df 2

"a#le K.9 shows the calculation steps. It will #ecoe a

standard gear calculation ifx1Gx2G (.Fig.4.4 The meshing of internal gear and external gear

Table 4.6 The calculation of a profile shifted internal gear and externl gear (1)

G 2(L,z1G 19,z2G 2K,x1Gx2G M (.* 4


External gear Internal gear

1 ?odule m 3

2


27/57


If the center distance a4 is given, x1 and x2 would #e

o#tained fro the inverse calculation fro ite K to ite '

of "a#le K.9. "hese inverse forulas are in "a#le K./.

Table 4.7 The calculation of profile shifted internal gear and external gear (2)


1 ;enter distance a 13.19'3

2;enter distance

ya

z2z1

(.3'&K3odification coefficient m 2

cos

cos12y

31.(&3/L3 or%ing pressure angle 'z2z1 M 1

ifference of profile shift z2 4 x2x1

z14 inv ' inv 4

(.*coefficient 2tan

5 Profile shift coefficient x ( (.*

Pinion cutters are often used in cutting internal gears and

external gears. "he actual value of tooth depth and root

diaeter, after cutting, will #e slightl! different fro the

calculation. "hat is #ecause the cutter has a profile shift

coefficient. In order to get a correct tooth profile, the profile

shift coefficient of cutter should #e ta%en into consideration.

(2) Interference In Internal Gears

"hree different t!pes of interference can occur with internal

gears:

a4 Involute Interference,

#4 "rochoid Interference, and

c4"riing Interference.

a4Involute Interference"his occurs #etween the dedendu of the external gear and the

addendu of the internal gear. It is prevalent when the nu#er

of teeth of the external gear is sall. Involute interference can

#e avoided #! the conditions cited #elow:

z11

tana2K.K4

z2 tan'

where a2 is the pressure angle at a tip of the internal geartooth.

1d#2a2 G cos K.*4

da2

' : wor%ing pressure angle

'G cos 1 z2z14 mcos K.942a

>uiation K.*4 is true onl! if the tip diaeter of the internal

gear is #igger than the #ase circle:

da2 d#2 K./4

=or a standard internal gear, where G 2(L, >uation K./4is valid onl! if the nu#er of teeth isz2 3K.

(b) Trochoid Interference

"his refers to an interference occurring at the addendu of

the external gear and the dedendu of the internal gear

during recess tooth action. It tends to happen when the

difference #etween the nu#ers of teeth of the two gears is

sall. >uation K.'4 presents the condition for avoiding

trochoidal interference.

!1z1

M inv' inva2 !2 K.'4z2

+ere

1 ra22

ra12

a2

a1inv'!1 G cos M inv2ara1K.&42 2 2

!2 G cos1 a Mra2 ra12ara2

wherea1 is the pressure angle of the spur gear tooth tip:

1 d#1

G cos K.1(4a1 da1d#2

G cos1

a2da2

In the eshing of an external gear and a standard internal

gear G 2(L, trochoid interference is avoided if thedifference of thenu#er of teeth,z1z2, is larger than &.


28/57

19


29/57


(c) Trimming Interference

"his occurs in the radial direction in that it prevents pulling the

gears apart. "hus, the esh ust #e asse#led #! sliding the

gears together with an axial otion. It tends to happen when the

nu#ers of teeth of the two gears are ver! close. >uation

K.114 indicates how to prevent this t!pe of interference.

z2

!1 M inva1inv' !2 M inva2inv' 4z1 K.114

!1 G sin1 1cosa1 5 cosa242

" 1z15z24 2K.124

cosa2 5 cosa1421!2 G sin 1 2

1" z25z14

"his t!pe of interference can occur in the process of cutting

an internal gear with a pinion cutter. Should that happen,there is danger of #rea%ing the tooling.

"a#le K.'14 shows the liit for the pinion cutter to prevent

triing interference when cutting a standard internal gear,

with pressure angle#

G 2(L, and no profile shift, i.e.,x( G (.(

Table 4.8(1)The limit to prevent an internal gear

x(Gx2G(From trimming interference #( G2(L

z( 1* 19 1/ 1' 1& 2( 21 22 2K 2* 2/z2 3K 3K 3* 39 3/ 3' 3& K( K2 K3 K*

z( 2' 3( 31 32 33 3K 3* 3' K( K2

z2 K9 K' K& *( *1 *2 *3 *9 *' 9(

z( KK K' *( *9 9( 9K 99 '( &9 1((z2 92 99 9' /K /' '2 'K &' 11K 11'

"here will #e an involute interference #etween the internal

gear and the pinion cutter if the nu#er of teeth of the

pinion cutter ranges fro 1* to 22 z(G 1* to 224.

"a#le K.'24 shows the liit for a profile shifted pinion

cutter to prevent triing interference while cutting a

standard internal gear. "he correction x(4 is the agnitude

of shift which was assued to #e:x(G (.((/*z(M (.(*.

Table 4.8(2) The limit to prevent an internal gear

G 2(L ,x2G(from trimming interference #(z( 1* 19 1/ 1' 1& 2( 21 22 2K 2* 2/

x( (.192* (.1/ (.1//* (.1'* (.1&2* (.2 (.2(/* (.21* (.23 (.23/* (.2*2*z2 39 3' 3& K( K1 K2 K3 K* K/ K' *(

z( 2' 3( 31 32 33 3K 3* 3' K( K2

x( (.29 (.2/* (.2'2* (.2& (.2&/* (.3(* (.312* (.33* (.3* (.39*z2 *2 *K ** *9 *' *& 9( 9K 99 9'

z( KK K' *( *9 9( 9K 99 '( &9 1((

x( (.3' (.K1 (.K2* (.K/ (.* (.*3 (.*K* (.9* (.// (.'z2 /1 /9 /' '9 &( &* &' 11* 139 1K1

"here will #e an involute interference #etween the internal

gear and the pinion cutter if the nu#er of teeth of the

pinion cutter ranges fro 1* to 1& z(G 1* to 1&4.

Interference

Involute interference

!1

!2

Interference

Trochoid interference

!1

!2

Interference

Fig.4.5 Involute interference and trochoid interference Fig.4.6 Trimming interference


30/57

20


31/57

4.3 Helical Gears

0 helical gear such as shown in =igure K./ is a c!lindrical gear in

which the teeth flan% are helicoid. "he helix angle in reference

c!linder is $, and the displaceent of one rotation is thelead,pz.

"he tooth profile of a helical gear is an involute curve fro

an axial view, or in the plane perpendicular to the axis. "he

helical gear has two %inds of tooth profiles T one is #ased on

a noral s!ste, the other is #ased on an transverse s!ste.

Pitch easured perpendicular to teeth is called noral pitch,pn.p

0nd ndivided #! is then a noral odule, mn.

mn Gpn

K.134%


ho# if odule mn and pressure angle #n are constant, no

atter what the value of helix angle $.

It is not that siple in the transverse s!ste. "he gear ho#

design ust #e altered in accordance with the changing of

helix angle $, even when the odule mtand the pressure

angle #tare the sae.6#viousl!, the anufacturing of helical gears is easier with

the noral s!ste than with the transverse s!ste in the

plane perpendicular to the axis.

In eshing helical gears, the! ust have the sae helix

angle #ut with opposite hands.

"he tooth profile of a helical gear with applied noral odule,

mn, and noral pressure angle#n#elongs to a noral s!ste.

In the axial view, the pitch on the reference is called the

transverse pitch, pt. 0nd pt divided #! is the transverseodule, mt.

mt Gpt

%K.1K4

"hese transverse odule mtand transverse pressure angle#t

are

the #asic configuration of transverse s!ste helical gear.

In the noral s!ste, helical gears can #e cut #! the sae gear

px

dreferencecircle

dReferencediameter

%

Lengthof

t

p

p

n

&Helix angle

p8G %d 5 tan&

Lead

Fig.4.7 Fundamental relationship of a helical gear (Right-hand)


32/57

21


33/57


(1) Normal System Helical GearIn the noral s!ste, the calculation of a profile shifted

helical gear, the wor%ing pitch diaeter d' and transverse

wor%ing pressure angle 'tis done per >uations K.1*4. "hatis #ecause eshing of the helical gears in the transverse

plane is Eust li%e spur gears and the calculation is siilar.

d'1 G 2az1

z1 Mz2

z2d'2 G 2az1 Mz2 K.1*4

d#1Md#2 1

G cos't 2a

"a#le K.& shows the calculation of profile shifted helical

gears in the noral s!ste. If noral profile shift

coefficientsxn1,xn2are 8ero, the! #ecoe standard gears.

Table 4.9 The calculation of a profile shifted helical gear in the normal system (1)


Pinion Gear

1 Noral odule mn 3

2 Noral pressure angle n 2(L3


34/57


If center distance, a, is given, the noral profile shift

coefficientsxn1andxn2can #e calculated fro "a#le K.1(. "hese

are theinverse euations fro ites K to 1( of "a#le K.&.

Table 4.10 The calculations of a profile shifted helical gear in the normal system (2)No. Item Symbol Formula Example

1 ;enter distance a 12*

2

;enter distance

y

a

z1Mz2

(.(&/KK/odification coefficient mn 2cos&

"ransverse wor%ing cost 1 &

3 pressure angle 't cos 2y cos M 1 23.1129Lz1

Mz2

Su of profile shiftz1

4 coefficient xn1Mxn2Mz24 inv 't inv t4

(.(&'(&2tan n

5 Noral profile shift coefficient xn (.(&'(& (

"he transforation fro a noral s!ste to a transverse

s!ste is accoplished #! the following euations:

xtGxncos&

mtG

mn

cos &1 tann

G tantcos&

K.194

23


35/57


(2) Transverse System Helical Gear

"a#le K.11 shows the calculation of profile shifted helical gears

in a transverse s!ste. "he! #ecoe standard ifxt1Gxt2G (.

Table 4.11 The calculation of a profile shifted helical gear in the transverse system (1)


Pinion Gear

1 "ransverse odule mt 3

2 "ransverse pressure angle t 2(L3


36/57


(3) Sunderland Double Helical Gear

0 representative application of transverse s!ste is a dou#le

helical gear, or herring#one gear, ade with the Sunderland

achine.

"he transverse pressure angle, t, and helix angle, &, arespecified as 2(L and 22.*L, respectivel!.

"he onl! differences fro the transverse s!ste euations of

"a#le K.11 are those for addendu and tooth depth.

"a#leK.13 presents euations for a Sunderland gear.

Table 4.13 The calculation of a double helical gear of SUNDERLAND tooth profile


Pinion Gear

1 "ransverse odule mt 3

2 "ransverse pressure angle t 2(L3


37/57


(4) Helical Rack

Uiewed in the transverse plane, the eshing of a helical rac% and

gear is the sae as a spur gear and rac%. "a#le K.1K presents the

calculation exaples for a ated helical rac% with noral

odule and noral pressure angle. Siilaril!, "a#le K.1*

presents exaples for a helical rac% in the transverse s!ste

i.e., perpendicular to gear axis4.

Table 4.14 The calculation of a helical rack in the normal system


Gear Rack

1 Noral odule mn 2.*

2 Noral pressure angle n 2(L

3


38/57

Table 4.15 The calculation of a helical rack in the transverse system

No. Item Symbol Formula

1 "ransverse odule mt

2 "ransverse pressure angle t

3


39/57


4.4 Bevel GearsBevel gears, whose pitch surfaces are cones, are used to

drive intersecting axes. Bevel gears are classified according

to their t!pe of the tooth fors into Straight Bevel @ear,

Spiral Bevel @ear, -erol Bevel @ear, S%ew Bevel @ear etc.

"he eshing of #evel gears eans pitch cone of two gears

contact and roll with each other.$etz1andz2#e pinion and gear tooth nu#ers7 shaft angle

+7 and reference cone angles ,1and ,27 then:

tan,1 Gsin +

z2

M cos +z1 K.2(4sin +

tan,2 G z1

M cos +z2

@enerall!, shaft angle +G &(L is ost used. 6ther angles

=igure K.'4 are soeties used. "hen, it is called #evelgear in nonright angle driveO. "he &(L case is called #evel

gear in right angle driveO.

m 1z

hen +G &(L, >uation K.2(4 #ecoes:

z1,1

1

G tan z2

z2, 1

z12 G tan

?iter gears are #evel gears with +G &(L andz1transission ratioz25z1G 1.

K.214

Gz2. "heir

,1

,2

+

z2m

Fig. 4.8 The reference cone angle of bevel gear

R

b

dv2

2 R

,2

,1

Rv1

d1

Fig. 4.9 The meshing of bevel gears

=igure K.& depicts the eshing of #evel gears.

"he eshing ust #e considered in pairs. It is #ecause the

reference cone angles ,1and ,2are restricted #! the gear ratio

z15z 2. In the facial view, which is noral to the contact line

ofpitch cones, the eshing of #evel gears appears to #e

siilar to the eshing of spur gears.

28


40/57

(1) Gleason Straight Bevel Gears0 straight #evel gear is a siple for of #evel gear having

straight teeth which, if extended inward, would coe together

at the intersection of the shaft axes. Straight #evel gears can #e

grouped into the @leason t!pe and the standard t!pe.

In this section, we discuss the @leason straight #evel gear.

"he @leason ;opan! defined the tooth profile as: toothdepth hG 2.1'' m7 tip and root clearance cG (.1''m7 and

wor%ing depth h'G 2.(((m.

"he characteristics are:

Design specified profile shifted gears:In the @leason s!ste, the pinion is positive shifted

and the gear is negative shifted. "he reason is to

distri#ute the proper strength #etween the two gears.

?iter gears, thus, do not need an! shifted tooth profile.

The tip and root clearance is designed to be parallel:"he face cone of the #lan% is turnd parallel to the rootcone of the ate in order to eliinate possi#le fillet

interference at the sall ends of the teeth.

"a#le K.19 shows the iniu nu#er of teeth to prevent

undercut in the @leason s!ste at the shaft angle +G &(L.

Table 4.16 The minimum numbers of teeth to prevent undercut


Rb

i a

d d d

,&(L

,a

) ha

)#h

hf

!! af

,, ,af

Fig. 4.10 Dimentions and angles of bevel gears

Pressure angle Combination of number of teethz15z2

(14.5L) 2&52& and higher 2'52& and higher 2/531 and higher 2953* and higher 2*5K( and higher 2K5*/ and higher

20L 19519 and higher 1*51/ and higher 1K52( and higher 1353( and higher

(25L) 13513 and higher

"a#le K.1/ presents euations for designing straight #evel

gears in the @leason s!ste. "he eanings of the

diensions and angles are shown in =igure K.1( a#ove. 0ll

the euations in "a#le K.1/ can also #e applied to #evel

gears with an! shaft angle.

"he straight #evel gear with crowning in the @leason s!ste

is called a ;oniflex gear. It is anufactured #! a special

@leason ;oniflexO achine. It can successfull! eliinate

poor tooth contact due to iproper ounting and asse#l!.


41/57

29


42/57


Tale 4.17 The calcultions of straight bevel gears of the gleason system


Pinion (1) Gear (2)

1 Shaft angle + &(L2 ?odule m 3

3


43/57


(2) Standard Straight Bevel Gears

0 #evel gear with no profile shifted tooth is a standard straight

#evel gear. "he applica#le euations are in "a#le K.1'.

Table 4.18 Calculation of a standard straight bevel gears


Pinion (1) Gear (2)

1 Shaft angle + &(L2 ?odule m 3

3


44/57


(3) Gleason Spiral Bevel Gears

0 spiral #evel gear is one with a spiral tooth flan% as in =igure

K.12. "he spiral is generall! consistent with the curve of a cutter

with the diaeter dc. "he spiral angle $is the angle #etween a

generatrix eleent of the pitch cone and the tooth flan%. "he spiral

angle Eust at the tooth flan% center is called ean spiral angle $.

In practice, spiral angle eans ean spiral angle.

0ll euations in "a#le K.21 are dedicated for the

anufacturing ethod of Spread Blade or of Single Side

fro @leason. If a gear is not cut per the @leason s!ste,

the euations will #e different fro these.

"he tooth profile of a @leason spiral #evel gear shown here

has the tooth depth hG 1.'''m7 tip and root clearance cG

(.1''m7 and wor%ing depth h' G 1./((m. "hese @leason

spiral #evel gears #elong to a stu# gear s!ste. "his is

applica#le to gears with odules m 2.1.

"a#le K.1& shows the iniu nu#er of teeth to avoid

undercut in the @leason s!ste with shaft angle +G &(L andpressure angle #nG 2(L.

dc

&

R

b b52

b52

,

Rv

Fig.4.12 spiral bevel gear (Left-hand)

Table 4.19 The minimum numbers of teeth to prevent undercut &G 3*L

Pressure angle Combination of numbers of teethz15z220L 1/51/ and higher 1951' and higher1*51& and higher 1K52( and higher 13522 and higher 12529 and higher

If the nu#er of teeth is less than 12, "a#le K.2( is used to

deterine the gear si8es.

Table 4.20 Dimentions for pinions with number of teeth less than 12

Number of teeth in pinion z1 6 7 8 9 10 11

Number of teeth in gear z2 3K and 33 and 32 and 31 and 3( and 2& andhigher higher higher higher higher higher

Working depth h' 1.*(( 1.*9( 1.91( 1.9*( 1.9'( 1.9&*

Tooth depth h 1.999 1./33 1./'' 1.'32 1.'9* 1.''2

Gear addendum ha2 (.21* (.2/( (.32* (.3'( (.K3* (.K&(

Pinion addendum ha1 1.2'* 1.2&( 1.2'* 1.2/( 1.2K* 1.2(*

Tooth thickness of 30 (.&11 (.&*/ (.&/* (.&&/ 1.(23 1.(*3

40 (.'(3 (.'1' (.'3/ (.'9( (.''' (.&K'gear

s2 50 - (./*/ (./// (.'2' (.''K (.&K9

60 - - (./// (.'2' (.''3 (.&K*Normal pressure angle n 2(L

Spiral angle & 3*L) K(LShaft angle + &(LNOTE:0ll values in the ta#le are #ased onmG 1.

32


45/57


"a#le K.21 shows the calculations of spiral #evel gears of the

@leason s!ste

Table 4.21 The calculations of spiral bevel gears of the Gleason system


Pinion Gear

1 Shaft angle &(L

2 ?odule m 3


4 ?ean spiral angle 3*L

5 Nu#er of teeth and spiral hand z 2( $4 K(


46/57

33


47/57


4.5 Screw Gears

Screw gearing includes various t!pes of gears used to drive

nonparallel and nonintersecting shafts where the teeth of one

or #oth e#ers of the pair are of screw for. =igure K.1K

shows the eshing of screw gears.

"wo screw gears can onl! esh together under the conditionsthat noral odules mn14 and mn24 and noral pressure

angles an1, an24 are the sae.

$et a pair of screw gears have the shaft angle and helix

angles !1and!2:

If the! have the sae hands, then:

G1M2

If the! have the opposite hands, then:

G12or+G21

K.224

If the screw gears were profile shifted, the eshing would

#ecoe a little ore coplex. $et '1, '2 represent the

wor%ing pitch c!linder7

Gear 1

(Right-hand) (Left-hand)

1

2 21

Gear 2

(Right-hand)

Fig.4.14 Screw gears of nonparallel and nonintersecting axes

If the! have the sae hands,

then: G'1M'2

If the! have the opposite hands,

then: G'1'2orG'2'1

K.234

"a#le K.22 presents euations for a profile shifted screw gear

pair. hen the noral profile shift coefficients

xn1Gxn2G (, the euations and calculations are the sae as for

standard gears.

34


48/57


Table 4.22 The equations for a screw gear pair on nonparallel and

Nonintersecting axes in the normal system


Pinion Gear

1 Noral odule mn 3


3


49/57

35


50/57


4.6 Cylindrical Worm Gear Pair

;!lindrical wors a! #e considered c!lindrical t!pe gears with

screw threads. @enerall!, the esh has a &(6 shaft angle. "he

nu#er of threads in the wor is euivalent to the nu#er of teeth

in a gear of a screw t!pe gear esh. "hus, a onethread wor is

euivalent to a onetooth gear7 and twothreads euivalent to two

teeth, etc.


51/57



52/57

37


53/57


(2) Normal Module System Worm Gear Pair

"he euations for noral odule s!ste wor gears are

#ased on a noral odule, mn, and noral pressure angle, an

G 2(L. See "a#le K.29.

Table 4.26 The calculations of normal module system worm gear pair


Worm Worm Wheel

1 Noral odule mn 3

2 Noral pressure angle n 2(L4

3 No. of threads, No. of teeth z ou#le


54/57

38


55/57

(b) Recut With Hob Center Position Adjustment.

"he first step is to cut the wor wheel at standard center

distance. "his results in no crowning. "hen the wor wheel

is finished with the sae ho# #! recutting with the ho# axis

shifted parallel to the wor wheel axis #! X =h. "his results

in a crowning effect, shown in =igure K.1'.

(c) Hob Axis Inclining=#From Standard Position.

In standard cutting, the ho# axis is oriented at the proper angle

to the wor wheel axis. 0fter that, the ho# axis is shifted

slightl! left and then right, =#, in a plane parallel to the worwheel axis, to cut a crown effect on the wor wheel tooth.

"his is shown in =igure K.1&. 6nl! ethod a4 is popular.

?ethods #4 and c4 are seldo used.


(d)Use a Worm with a Larger Pressure Angle than the

Worm Wheel.

"his is a ver! coplex ethod, #oth theoreticall! and

practicall!. Dsuall!, the crowning is done to the wor

wheel, #ut in this ethod the odification is on the wor.

"hat is, to change the pressure angle and pitch of the worwithout changing #ase pitch, in accordance with the

relationships shown in >uations K.2*:

pxcosxGpx' cosx' K.2*4

In order to raise the pressure angle fro #efore change, ax,

to after change, ax, it is necessar! to increase the axial pitch,

px, to a new value,px, per >uation K.2*4. "he aount of

crowning is represented as the space #etween the wor and

wor wheel at the eshing point 0 in =igure K.21. "his

aount a! #e approxiated #! the following euation:

h

h

Fig.4.18 Offsetting up or down

$$

0ount of crowning k pxpx' d1 K.294

2px'

where d1 :


56/57

39


57/57


"a#le K.2' shows an exaple of calculating wor crowning.

Table 4.28 The calculation of worm crowning


1 0xial odule mx' NOTE: "hese are the 3

data #efore2 Noral pressure angle n' 2(Lcrowning.3 Nu#er of threads of wor z1 2

4 ' ;? 34 6@;3;6#8 8;A;3

$et anin >uation K.2/4 #e 2(L, then the condition:

Ft1 > ' :;88 $6ABcos2(L sin"T %cos"/ > '

=igure K.22 shows the critical liit of selfloc%ing for lead

Documents

Gear Guide1