----------------------------------------------------------------------------------..
The European Rack
Shift Coefficient ··x"
for Americans
Don McViittie
Ge.ar Englinee.rs, Inc .• Sea.ltle. WA
Basic Rack
Introduction
and equations in this article
The definition
The ue of dimensionless factors to describe
seems to have a strong
are ba ed on a "basic rack" ill which addendum
appeal 10 gear engineers ..The stress factors I and
and dedendum are mea ured from a reference
J, for instance. are well established in AGMA
line located where the tooth thieknes
literature. The use of the rack shift coefficient
space width on the reference line are equal, The
gear 'loath geometry
and the
"x" to describe nonstandard gear proportions i . basic rack repre ent the tooth form in the nor-
common in Europe, but is not as commonly used
ma] plane of a gear with an infinite number of
in the United States. When it is encountered in
teeth .. The normal module of the basic rack is
the Eu ropean literature or in the operating manu-
equal to thenormal
circular pitch divided by 7t.
als for imported machine tools, it can be a source
The normal diametral pitch of the basic rack is
of confusion to the American engineer.
equal to 7t divided by the normal circular pitch.
The basic rack repre ents the theoretical. gear
What follows is intended to provide a source
"1{"
tooth form, not the form of the cutting 11001. No
European standards and pa-
allowance i. made for backlash, fini hing stock,
for the background
factor as
II
edin
pers. The addendum
and derivations
modification,
of the
rack shift,
or manufacturing
method.
Q
The
or profile shift factor bas several mathematical definitions
documents
in the U.S. Most European
use a specific definition,
a theoretical
"zero backlash"
right me h at the nominal
(See McVittie,
based on
gear pair in
center
distance.
tandard
20
norma] pres ure angle
ba ic rack of ISO S3 is commonly
document
used. This
is va]id for that basic rack and for
any other basic rack which meets the criteria
of Fig. I.
86 FTM 1 for discussion.)
Addendum
Modification
The addendum
Factor
modification
factor
"x"
(Profitverscniebungsfaktor,
"profile shift fac-
tor" in German) represent
the distance, in tight
mesh, from the reference line of the basic rack
to the reference circle of the gear (rack shift or
profile sHift) for normal module
diametral pitch
=
e
1.0 or normal
LO.
Sum of X Faetees
The European practice is to define the sum of
Hg. 1 -
34
GEI\.R
Basic rack definition - normal plane.
TECfffllOI.O(lV·
Xl
+
"'2'
(Ix) for a theoretical gear pair which
operates in tight
mesh (has. no backlash) on the
nominal. operating center distance.
Table of Symbols
The basic equation can be derived from the
basic tooth thickness involute geometry equa-
ISO
tions and the requirement. that the sum. of the
c
transverse
at the operating
d
pitch diameter is equal to the transverse circular
IE
pitch at that diameter ..
hal
tooth thicknesses
Definition
Genter Distance
Gliearan,ee (ha - hfl
Diameter
Tooth Thinning for Backlash
Addendum
Dedendurn
Tip Shortening Factor
Backlash
Module
Circular Pi,tch
Tooth Thickness
Addendum Modification Factor
d
a
hf
k
iny Ctwl. + illY ut
tan ,un
(]) DIN 3992 Eq 9
B
m
m
p
p
t
s
x
tan un
p
cos
(2) DIN 3992 Eq 6
z 1 +z
AGMA
C
a,
a
Pressure Angle
Helix Angle
III
p.
'I'
Table of Subscripts
2
Subscript
(nonel
2
(3) DIN 3992 Eq 5
At Reference
a
Diameter
At Addendum (Tipl Diameter
At Base Cylinder Diameter
b
f
At Root Diameter
n
NormallPllane
o
ToollDimensions
t
w
Transverse Plane
At Working: Diameter
Y
At Any (Undefilledl D'iameter
3992, and lSOrrR 4467 for further information
1
on choice ofx factors) according to the operating
2
Pinion
Gear or iRack
(4)
X Factor for Each Gear
Th.e values of x for gear and pinion are
chosen somewhat arbitrarily (See Maag. DlN
conditions
and gear ratio, so that their total is
equal to LX. The theoretical
addendum
(tip)
diameters and tooth thicknesses of the two gears
in the gear pair are defined by their
X
factors,
tion is to be measured must be specified, since
there is no recognized
convention.
Working Group (WG)2 ofISorrC60
is con-
sidering a draft technical report. DTRI006412,
(5)
containing
tables which recommend
that the
tooth thinning for backlash, called "upper allow1t
- -0 )o'm
"
ance of size", Essn' be a function of the pitch
diameter of each part. The values are measured
(6) ISO DTR 1006412 Eq 6 ..4
normal to the helix angle in the reference cylin-
-
Sn =(·-+2·oxotanu
2··
..
der. The values can be converted as follows:
The actual. addendum. diameters and tooth
The transverse circular allowance, Esst' is:
thicknesses ar,e then adjusted (usually reduced)
(7)
to control backlash and! tip to root clearance,
Backlash
Allowance
A common convention
facturersis
among gear manu-
to reduce the normal tooth thick-
ness of each member
by the same amount,
which may be a value in um or a function of
module, such as ..024· mn. This maintains the
same cutting
maximizes
depth for both members
contact ratio, The direction
mal, transverse,
The normal allowance in the base tangent
reference
Don IMcVitti,e
plane, Ebsn' (normal to the tooth surface) is
~sn
= E SI.· cos ~
• co
ilb
is President of Gear Engineers, tnc., Seattle, WA.
He is a past president oj
(8)
AGMA and Chairman oj
the U, S. Technical Advi-
which can aha be expressed as
and
(nor-
(9)
circle, or base tan-
gent plane) I.D which tile tooth thickness reduc-
The resulting transverse circular backlash at
sory Group for Intefila·
tional Gear Standards. He
is a licensed professional
engineer in the Stare oj
Wa,shingIOIi.
JULY/AUGUST
1993
35
the working diameter is at function of allowance,
center distance, and tooth accuracy. (See AGMA
2002 for more information.)
Tip Sbortening for Clearance
When Lx. > 0, the tips of external gear pairs
should be shortened to maintain standard tip to
root clearance.
The reduction :in clearance is
skiving, or grinding, a more detailed study is
required to estimate the fim hed root diameter.
(See Appendix E, Sec. E60f AGMA 2[8.01 for
more informatlon.)
Convention. for Signs
For external gear, the value of x i positive
thickne s is increased arid the
whenthetooth
often ignored for small value. of Ix, but for
value of Ix is positive when the center distance
larger values the addendum
i greater than standard.
by k·
hould be shertened
The same convention can be used for inter-
mo'
nal gears jf the sign of the center distance is
k=
considered
negative. "Long addendum"
inter-
aal gears have a negative x. This convention is
+ Zz [inV a"..,l - inv ~
--.'
2
.
tan ~
I
ZI
( cos '0;
----··----1
co I~ cos <Xwt -
)]
(0) Maag Eq 68
but is not. universal.
common
Intemal
'Gears
The equations in thi article are arranged for
external gears. With a few exceptions, they can
Tip diameters of intern aJ gear pairs should be
be u ed for internal gears if the internal. diam-
checked for clearance and interference with cut-
eters, center distance, and number of teeth are
for signs must
ters and mates by calculation of actual cutting
made negative. The convention
and mating conditions.
be checked carefully. One trap i divi ion by a
Actual Root Diameter
and. Cleaeance
negative value to. calculate
all involute
func-
The European method doesn't. calculate the
tion, which must be positive, h is good pro-
actual root diameter of gear which are thinned.
gramming practice to take the absolute value of
the quotient before calculating the angle from
the iavolute function. I.
for backlash
by feeding the cutler to greater
depth. When the actual root diameters are calculated, the addendum diameter required for
standard clearance can be calculated more accurately from Eq .. 11.
Appendix
~so
::
-
Derivations
Esst • cos ~ • CD
of Equation
9
P
. b
Essn
'
= --cos
P • cos '0;. • cos I~b
The root diameter at maximum tooth thickness can be calculated as follows:
Il
Po
r.t
Pbn
cos ... =····p·- co "'b=':"-
cosa
d(",d.-2.(h-x
+
.• m
ao
1I
Equation 12 is ba ed
Essn
2. tan ~ • cos
011.
.)i02)
the cutter addendum, hao is measured as shown
ill Fig .. 1 for the basic rack, If the gear i to be
finished in a second operation,
"
/
.........
t;a:
<,
I
- lRefer,enCIi Diameter
i
I
Fig. 2 -
36
G.EAR
Addendum modification
TECHNOLOGY
as by shaving.
........
~r...... '" ....
Diameter
"
Pbn
n
=-Pn
Pbl
cos a =I
PI
(9)
cos~
fi
the assumption that.
Pbt
t
References:
American Gear Manufacturers As oclatien AGMA
908. Geometry Factors/or Determining the Pilling Resistance and Bending
Teeth.
SIre/lgth
of Spur and Helical
Gear
DIN 3992. Profilverschiebung bei Stimrddem mil
AUSstnller:;alrnnng.
ISO 53. Cylindrical Gears for General and Heavy
Engineering - Basic Rack.
ISO/DTR 10064/4,. Part 2. Inspection Related to Radial Composite Deviations, Runout, Tooth Thickness, and
Backlash.
]SOfTR 4467. Addendum ModiJjcariol10/
the Teetb
0/ Cylindrical Gears /01' Speed·Reducing and SpeedIncreasing Gear Pairs.
Lorenz. Gear Curling Tools. 1961.
Maag. Maag Gear Book. 1990.
McVittie.Don.
"De cribing Nonstandard Gears-c-An
Alternative to the Rack Shift Coefficient." AGMA 86 fTM
L ALso in Gear Technology, Vol. 5, No. I, January!
February, J 988, p, lOff.
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