gear guide1
TRANSCRIPT
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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.
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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
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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.
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(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
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Elementary Information on Gears
(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
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Fig.1.14 Hypoid gear
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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
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Elementary Information on Gears
Table 1.5 Others
Terms Symbols
"angential force Ft0xial force Fx
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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.
Elementary Information on Gears
"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
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Fig. 2.1 Single-stage gear trains
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Elementary Information on Gears
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
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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
Elementary Information on Gears
?1
?1.*
?2
?2.*
?3
?K
?*
?9
?1(
Fig.3.1 Comparative size of various rack teeth
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Elementary Information on Gears
p
p
52 p#
hh
ahf
d#
d?odule m
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Fig.3.3 The involute curve
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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
Elementary Information on Gears
"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
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Elementary Information on Gears
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 )
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Elementary Information on Gears
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
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Elementary Information on Gears
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
No. Item Symbol Formula Example
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.
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Elementary Information on Gears
(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)
No. Item Symbol FormulaExample
Pinion (1) Gear (2)
1?odule m 3
2
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Elementary Information on Gears
"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)
No. Item Symbol Formula Example
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.
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(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
Elementary Information on Gears
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
No. Item Symbol FormulaExample
Spur gear Rack
1 ?odule m 3
2
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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
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Elementary Information on Gears
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
No. Item Symbol FormulaExample
External gear Internal gear
1 ?odule m 3
2
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Elementary Information on Gears
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)
No. Item Symbol Formula Example
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 &.
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Elementary Information on Gears
(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
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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%
Elementary Information on Gears
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)
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Elementary Information on Gears
(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)
No. Item Symbol FormulaExample
Pinion Gear
1 Noral odule mn 3
2 Noral pressure angle n 2(L3
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Elementary Information on Gears
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
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Elementary Information on Gears
(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)
No. Item Symbol FormulaExample
Pinion Gear
1 "ransverse odule mt 3
2 "ransverse pressure angle t 2(L3
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Elementary Information on Gears
(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
No. Item Symbol FormulaExample
Pinion Gear
1 "ransverse odule mt 3
2 "ransverse pressure angle t 2(L3
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Elementary Information on Gears
(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
No. Item Symbol FormulaExample
Gear Rack
1 Noral odule mn 2.*
2 Noral pressure angle n 2(L
3
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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
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Elementary Information on Gears
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.
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(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
Elementary Information on Gears
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!.
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Elementary Information on Gears
Tale 4.17 The calcultions of straight bevel gears of the gleason system
No. Item Symbol FormulaExample
Pinion (1) Gear (2)
1 Shaft angle + &(L2 ?odule m 3
3
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Elementary Information on Gears
(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
No. Item Symbol FormulaExample
Pinion (1) Gear (2)
1 Shaft angle + &(L2 ?odule m 3
3
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Elementary Information on Gears
(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.
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Elementary Information on Gears
"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
No. Item Symbol FormulaExample
Pinion Gear
1 Shaft angle &(L
2 ?odule m 3
3 Noral pressure angle n 2(L
4 ?ean spiral angle 3*L
5 Nu#er of teeth and spiral hand z 2( $4 K(
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Elementary Information on Gears
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.
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Elementary Information on Gears
Table 4.22 The equations for a screw gear pair on nonparallel and
Nonintersecting axes in the normal system
No. Item Symbol FormulaExample
Pinion Gear
1 Noral odule mn 3
2 Noral pressure angle n 2(L
3
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Elementary Information on Gears
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.
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Elementary Information on Gears
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Elementary Information on Gears
(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
No. Item Symbol FormulaExample
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
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(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.
Elementary Information on Gears
(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 :
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Elementary Information on Gears
"a#le K.2' shows an exaple of calculating wor crowning.
Table 4.28 The calculation of worm crowning
No. Item Symbol Formula Example
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