AGMA 2002-B88

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ANSI/AGMA 2002--B88

(Errata 1995

Reaffirmed December 2006

American National Standard

Tooth Thickness Specificationand Measurement

A N S I / A G M A

2 0 0 2 - - B 8 8

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Tooth Thickness Specification and Measurement

ANWAGMA 2002-B %?

(Revision of AGMA 231.524975)

ITables or other self-supporting sectionsmay be quotedor extracted n their entirety.

Credit ines should read:

Extracted from AGhJA 2002-B88, Toot h ThicknessSpecificationand Measurement, with the permission of the

publisher, the American Gear ManufacturersAssociation, 1500King Street,Suite201, Alexandria,Virginia 22314.1

AGMA Standardsare subject to constant mprovement, evision or withdrawal as dictatedby experience. Any

person who refers to AGh T.A echnicalPublicationsshouldbe sure hat he publication s the atestavailable rom the

Association on the subject matter.

ABSTRACT

This Standard establishes he proceduresor determining ooth hicknessmeasurementsf externaland internal

cylindrical involute gearing. It incl udes equationsand calculationprocedures or the commonly used measuring

methods. A specific tooth thicknessmeasurementimit canbe establishedrom the design hicknessor from another

tooth thickness measurement.The procedures anbe enteredwith anestablished esign ooth hickness,or with actual

tooth thickness measurements. he effect of tooth geometricquality variationson tooth thicknessmeasurementss

discussed. Backlash information is provided n an appendix.

Copyright Ql988

American Gear ManufacturersAssociation

1500 King Street, Suite 201

Alexandria, Viginia, 22314

First printing. October, 1988

Secondprinting, with errata,July 1992

Third printing, with errata,June 1995

ISBN 1-55589-523-9

ANWAGMA

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ToothThicknessSpecificationandMeasurement

FOREWORD

nhis foreword, footnotes,andappendices,f any,areprovided or informationalpurposes nly and shouldnot be

conshued as part of ANSIIAGMA 2002-B88, Tooth Thickness Specification and Measurement.1

This Standardpresentscalculation procedures or determining ooth thickness measurements f external and

intemal cylindrical involute gearing. t supersedes GMA 231.52, nspection -

Pin Measurement Tables for Involute

Spur Gears.

This Staudardhasbeenprepared o consolidate reviouslypublishedAGMA tooth thickness nformation, to add

more information on internal and helical gearsand o add detailson more measurementmethods.

Previous AGMA publicationshavepresented his information n tabular orm, calculated or 1 DP and standard

tooth proportions,with adjustment actors or nonstandardonditions.This Standards arrangedor direct calcula tion of

the desired esults, o eliminate he ntermediate alculationsteps nd nterpolationpreviously required

The study of tooth thicknessandbacklashproblemshasbeena major nterestof gea r echnicians hroughout he

history of the industry. In the last fifty years, many clarifications and contributionshave beenmadeby men such as

Buckingham, Candee, eming, Vogel,andWildhaber.

Their work s consolidated ere,without further attribution. and

the work of more recentcontributors s addedwhere t improves he presentation.

The appendices rovide further nformation on reasonable llowances or backlashand ooth thicknessdeviation.

sample calculations, and nformation on four uncommonmethods f measurementpecifiedon somegear drawings.

The treatmentof the effects of tooth profile, pi tch, lead,andmnout deviationson tooth thiclmessmeasurements

new in this Standard.

The information on backlashcontrol s new in an AGMA Standard.

t is based n AGMA Paperp239.14,Assured

Backlash Control - The ABC System.[l]

The first draft of this revision was made n February1984.

This version wasapproved y theAGMA membership n October ,1988andasan AmericanNational Standard n

October 17,1988.

Suggestions or the improvementof this Standardwill be welcome.

They should be sent o the American Gear

Manufacturers Association, 1500King Street,Suite 201, Alexandria,Vi@ia, 22314.

ERRATA

July, 1992

The following editorial corrections have beenmade to ANSVAGMA 2002-B88, Tooth Thickness Specification and

Measurement, (or@ally printed October 1988).

Thesechanges, iscovered fter publication, havebeenmade n the

secondstandardprinting, as shownbelow:

PAGE

ITEM

CHANGE

10

Fig 3-l

The position of minimum andmaximumbacklash s shown on the specified

circle, also /2 specified olerance nd /2 specificationbands abeledcorrectly.

26

Fig 3-l

The angle Wb nd he assumedorm diameter,Do - 4a, indicated correctly.

29 Eq 8.2

The right handbracketshouldbe at the end, with the ull equation eading,

f3

= arcinv

pndtl+ t’- -p

N1+N2

+invf,

1

0% 8.2)

32 TableA-l The last value n the table, for 64 inch center,should ead 0.058.

ERRATA

June, 1995 (Additional correctionmade n this printing).

29 Eq 8.2

Changed o transverse lane.

3

= arcinv

[

Pdt, + t&z

N, +Nz + hvQs

1

0%. 8.2)

El] Numbers in brackets efer to thebibliography.

ANSIIAGMA

. . .

ln

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PERSONNEL of the AGMA Committee for Inspection And Handbook

Chairman: P. M. Dean, Jr. (Honorary Member)

MEASURING METHODS

Chairman:

R. E. Smith (R. E. Smith Company, Inc. - Consultant)

Editor: W. A. Bradley (Consultant)

ACTIVE MEMBERS

L. E. Andrew (Deceased)

M. Bartolomeo (Pratt Whimey Aircraft)

N. Borja (Arrow Gear Company)

L. Flynt (Consultant)

R. Green (Eaton)

E. Hahlbeck (Milwaukee Gear Company)

J. S. Hamilton (Gear Products Division)

R. Kamminga (Eaton)

I. La in (Gear Motions)

E. Lawson (M M Precision)

J. Leming (Deceased)

D. A. McCarroll (Gleason)

D. R. McViuie (Gear Engineers)

E. R. Sewall (Sewall Gear)

L. J. Smith (Invincible Gear)

H. J. Trapp (Klingelnberg)

D. S. Whimey (Retired)

ASSOCIATE MEMBERS

W. Coleman (Deceased)

J. F. Craig (Cummins Engine)

J. Dykhuizen (Fairfield)

D. L. Friedel (Chicago Gear - D. 0. James)

E. Guenter (MA4G)

J. E. Gutzwiller (Honorary Member)

M. M. Hauser (Litton Precision)

G. Henriot (Engrenages et Reducteurs)

A. J. Lemanski (Sikorsky)

R. L. Lesliee(SPECO Division)

C. F. Lichte (General Motors Corporation)

B. C. Newcomb (Chicago Gear - D. 0. James)

B. Nugent (Xtek, Incorporated)

T. Porter (ITW/Spiroid)

V. Z. Rychlinski (Brad-Foote)

D. Senkfor (Precision Gear Company)

W. L. Shoulders (Deceased)

F. A. Siriarmi (Skidmore Gear)

3. R. Smith (Power Tech International, Incorporated)

P. Starr (Falk Corporation)

M. Tanaka (Nippon Gear)

R. F. Wasilewski (Arrow Gear)

R. D. Wilson (Power Tech International, Incorporated)

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Table of Contents

Section

Title

Page

1. Scope

.....................................................................

1

2. Symbols Terminology and Definitions

...........................................

1

2.1

Symbols and Terminology

............................................. 1

2.2

Definitions ..........................................................

1

3. Application

................................................................

9

3.1

Tooth Thickness Concepts

.............................................

9

3.2

Backlash

...........................................................

11

3.3

MountingSurfaces

...................................................

11

3.4

ReferenceSurfaces ..................................................

11

3.5

TotalCompositeVariation

.............................................

11

3.6

Specifying Maximum Tooth Thickness

..................................

12

3.7

Specifying Minimum Tooth Thickness

...................................

12

3.8

Measurement Method Effects

..........................................

13

3.9

Selection of Tooth Thickness ........................................... 13

‘4. Gear Geometry Calculations

...................................................

13

4.1

Circular Tooth Thickness

.............................................

13

4.2

Standard Pitch Diameter

..............................................

14

4.3

Backlash Calculations

................................................

14

4.4

Effective Tooth Thickness Calculation

...................................

14

4.5

Maximum Generated Tooth Thickness

...................................

14

4.6

Base Tooth Thickness

.................................................

15

5. Chordal Tooth Thickness

.....................................................

15

5.1

Advantages of Chordal Tooth Thickness

..................................

15

5.2 Limitations of Chordal Tooth Thickness ................................. 15

5.3

Calculation of Chordal Tooth Thickness

..................................

15

6. Measurement by Pins

.......................................................

17

6.1

Advantages of Measurement by Pins

.....................................

17

6.2

Limitations of Measurement by Pins

.....................................

17

6.3

Measurement Methods

................................................

18

6.4

pin or BallSizes

....................................................

19

6.5

Calculation of Measurement by Pins

.....................................

21

7. Span Measurement

.........................................................

23

7.1

Advantages of Span Measurement

.......................................

23

7.2 Limitations of Span Measurement ...................................... 23

7.3

Calculation of Span Measurement

.......................................

25

8. Composite Action Test Measurement

...........................................

28

8.1

Advantages of Composite Action Test Measurement

.......................

28

8.2

Limitations of Composite Action Test Measurement

...........

;

...........

28

8.3

Master Gears ...............................................

..-.....2 8

8.4

Calculation for Composite Action Test Measurement

.......................

29

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Page

Table of Contents cod)

Section

Title

Appendices

Appendix A

Backlash and Tooth Thickness Tolerance

. . . . . . . . . _ . . . . . . . . . . . . . . . . . 31

Appendix B

Alternate Methods of Tooth Thickness Measurement . . . . . . . . . . . . . . . . . . . 37

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...41

Tables

Table 2-l

Table 2-2

Table 3-1

Table 6-1

Table 6-1M

Figures

Fig 2-1

Fig 2-2

Fig 3-l

Fig 5-l

Fig 5-2

Fig 6-l

Fig 6-2

Fig 6-3

Fig 6-4

Fig 7-l

Fig 7-2

Fig 7-3

Fig 7-4

Fig 8-l

Fig 8-2

Alphabetical Table of Symbols and Terms, by Symbols

................

2

Alphabetical Table of Terms and Symbols, by Terms

...................

6

Other Gear Variations Included in Tooth Thickness Measurement

.........

12

Standard Pin Diameters for Various Pitches in Inches

...................

22

Standard Pin Diameters in Millimeters

...............................

22

Backlash

.......................................................

5


..........................................

5

Tooth Thickness

.................................................

10

Chordal Tooth Thickness Measurement by Means of a Gear Tooth Caliper . 15

Addendum and Chordal Tooth Thickness Corrections

..................

16

Tooth Thickness Measurement Over Pins

............................

17

Best Pm Size,

Wdest External Gears) .............................. 19

Pin Measurement Spur and Helical

..................................

20

Best Pm Size Internal Spur Gear)

.................................

21

Span Measurement of Tooth Thickness

..............................

24

Span Measurement of Helical Gears

.................................

24

Limits of Span Measurement in Base Tangent Plane

....................

26

Limits of Span Measurement for Internal Gear

........................

28

Schematic of Composite Action Test Measurement

.....................

28

Composite Action Test Measurement of Tooth Thickness

................

30

A.?.?SIIAGMA

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1. Scope

This Standard establishes the calculation pro-

cedures for determining tooth thickness measure-

ments of external and internal cylindrical involute

gearing.

The information is intended for use by the

gear specifier or manufacturer in establishing val-

ues for tooth thickness measurement limits.

CAUTION:

It is important that tooth

thickness measurement limits be reasonable

for the specified quality class of the gears,

to permit economical manufacture. This

Standard provides guidance in the selection

of reasonable tooth thickness measurement

limits.

The designed tooth thickness is established

from engineering considerations. It is determined

by gear geometry, gear tooth strength, and back-

lash. The methods for establishing designed tooth

thickness for a given application are beyond the

scope of this Standard.

This Standard assumes the designed tooth

thickness is known in cases where the values for

various measuring techniques are to be estab-

lished.

It includes equarions and procedures for the

following measuring methods:

(1) Chordal

(2) Fins (wires, rolls and balls)

(3) SPari

(4) Composite Action Test

This Standard also establishes methods of de-

termining tooth thickness of a gear based upon

measurement limits by means of pins, span,

chordal thickness or composite action test. These

methods are often used to convert a tooth thick-

ness specified by one method, such as over pins to

another more convenient method, such as span

over X teeth.

CAUTION: The effect of tooth geometry

variations on tooth thickness measurements

made by different measuring methods may

be significant. This must be considered if

close control of backlash is required.

When this is necessary the tooth thickness

should be measured by the method speci-

fied on the drawing. Refer to 3.8 for addi-

tional discussion of the problem.

Examples included are for coarse pitch gears.

The same mathematical principles apply to gear

teeth of ah sizes. For information on fine pitch

gears, see AGMA 370.01, Design Manual for

Fine Pitch Gears.

This Standard does not contain tolerances on

tooth thickness.

See AGMA 2000-A@, Gear

Classification and Inspection Handbook - Toier-

antes and Measuring Methods for Unassembled

Spur and Helical Genrs {Including Metric Equivn-

Zents), for tolerances.

AGMA 115.01, Reference Information - Ba-

sic Gear Geometry is a source for the derivations

and detailed explanations of the geomeuical rela-

tionships used here.

AGMA 112.05, Gear Nomenclature (Geome-

try) Terms, Definitions, Symbols and Abbrevia-

tions is a source of definitions of common gear

terms as used in this Standard.

2. Symbols, Terminology and Deftitions

2.1 Symbols and Terminology. Symbols and

terminology used in this Standard are shown in

Table 2-l and Table 2-2.

NOTE: The symbols, terminology, and

definitions used in this Standard may differ

from other AGMA standards. The user

should not assume hat familiar symbols can

be used without a careful study of these

definitions.

SI (Metric) units of measure are shown in pa-

rentheses in Table 2-1, Table 2-2 and in the text.

Where equations require a different format or

constant for use with SI units, a second expression

is shown after the first, indented, in smaller type,

and with “M” included in the equation number.

Example:

7T

Px =

p, sin Jr,

@q 4.4)

(Eq 4.4M)

2.2 Definitions. The terms used, wherever

applicable, conform to the following standards:

ANSI Y10.3 - 1968, Letter Symbols for

Quantities Used in Mechanics of Solids

AGMA 112.05, Gear Nomenclature, Terms,

Definitions, Symbols, and Abbreviations

AGMA 600.01, Standard for Metric Usage

ANSUAGMA

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Symbol

Table 2-l


Terms

units

Where

First Used

a

OC

A ac

B

Bmin

Bf

Bt

C

3X

=&in

D

D’

l

2

D.

4

D

omax

Ds

DW

D2w

F

L

’ best

=XGiX

=min

M

Mm

mn

N

Nl

N2

n

P

nd

PI

PX

Pb

Addendum

Chordal Addendum

Correction to Chordal Addendum

Backlash (Transverse Operating)

Minimum Transverse Backlash

Normal Backlash (feeler gage)

Circular Transverse Backlash

Tightest Center Distance

Maximum Center Dice

Minimum Center Distance

Specified Diameter

Operating Pitch Diameter

Base Circle

Base Circle Diameter of Test Gear

Base Circle Diameter of Master Gear

Tip Diameter of Internal Gear

Outside Diameter

Maximum Outside Diameter

Standard (Generating) Pitch Diameter

Contact Diameter for Best Pin Size

Diameter over/between Two F’ins

Facewidth

Lead

Best Length of Base Tangent

Maxim= Length of Base Tangent

Minimum Length of Base Tangent

Span Measurement

Span Measurement, Modified for Tooth Variation

Nod Module

Number of Teeth in Gear

Number of Teeth in Test Gear

Number of Teeth in Master Gear

Number of Teeth in Pinion

Normal Diametral pitch

Operating Transverse kular Pitch

Axial Pitch

Transverse Base Pitch

in (-4

Eq 5.1

in 64

Eq 5.7

in (->

5.3.1

in (->

4.3

in (->

3.1.4

in c->

4.3

in (=a

Eq 4.7

in b@

3.1.4

in (mm)

Eq 8.3

in h-4

Eq 8.5

in (->

4.1

in (I=4

3.7

in (mm>

3.7

in (-)

8.4.1

in (mm)

8.4.1

in (mm)

6.4

in (=4

6.4

in (=a

5.3

in (=4

Eq 4.6

in (-1

Eq 6.1

in (->

6.5

in (mm)

7.3

in (=>

4.1

in (mm>

Eq 7.7

in (=)

Eq 7.3

in (mm)

Eq 7.1

in (mm>

Eq 7.10

in (->

Eq 7.13

mm

Eq 3.6M

--

3.1.4

--

8.4.1

--

8.4.1

--

3.1.4

in-’

4.1

in (mm)

Eq 3.2

in (=I

4.1

in hd

6.5.1

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Table 2-1 (cant)


Symbol

Terms

‘Linits

Where

First Used

%

%lax

Rm

RlW

RTmax

RTmin

s

‘best

LX

%Grl

sW

t

tm

t

tGmax

‘Pmax

GIlin

h

‘nb

tnR Lc

fR

tt

t ts

tW

fT

vapk

%?

Vr

t;T

Normal Base Pitch

Maximum Measuring Radius

Master Gear Test Radius

Radius over/to One Pin

Maximum Test Radius (work gear)

Minimum Test Radius (work gear)

Number of Teeth to be Spanned

Best Number of Teeth to be Spanned

Maximum Number of Teeth to be Spanned

Minimum Number of Teeth to be Spanned

Transverse Space Width at Best Pin Contact Diameter


Transverse Tooth Thickness of the Test Gear at c

Transverse Tooth Thickness of the Master Gear at +c

Transverse Base Tooth Thickness

Transverse Base Tooth Thickness of Test Gear

Transverse Base Tooth Thickness of Master Gear

Transverse Base Tooth Thickness, Modified for Runout

and Pitch Variation

Measured Transverse Normal Chordal Tooth Thickness

Maximum Transverse Tooth Thickness at Operating Pitch

Diameter

Maximum Transverse Tooth Thickness of Gear

Maximum Transverse Tooth Thickness of Pinion

Minimum Specified Transverse Tooth Thickness

Normal Tooth Thickness

Normal Base Tooth Thickness

Normal Tooth Thickness at &

Transverse Tooth Thickness at &

Transverse Tooth Thickness, Circular

Maximum Transverse Generated Tooth Thickness

Transverse Tooth Thickness at Best Pin Contact Diameter

Transverse Tooth Thickness Tolerance

Accumulated Pitch Variation, Sector of k Pitches

Total Composite Variation

Radial Runout of Reference Diameter

Radial Runout Tolerance

in (->

in @N

in o=>

in (->

in (-)

in (-9

--

--

--

--

in @d

in (-)

in (mm)

in (mm)

in (mm)

in (=I

in (->

in (mm>

in b-d

in (->

in (=>

in (mm>

in b4

in (mm>

in (-)

in c->

in (->

in bd

in (mm>

in (mm>

in bd

in (=>

in (-0

in (-9

in (=4

7.3

5.3

8.4.1

Eq 6.11

Eq 8.4

Eq 8.6

7.1

Eq 7.8

Eq 7.4

Eq 7.2

Eq 6.4

2.2

8.4.1

8.4.1

Eq 4.11

8.4.1

8.4.1

Eq 7.12

Eq 5.10

3.7

Eq 3.1

Eq 3.1

Eq 3.3

Eq 4.1

Eq 7.6

Eq 5.3

Eq 5.5

4.1

Eq 4.8

Eq 6.3

3.7

7.3

3.7

5.3

6.5.5

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Table 2-l (cant)


Symbol

V

rW

W

Wbest

wC;V’best

4

4’

n

4s

4W

(92

43

9

Jtb

*R

J’S

7

Where

Terms

units First Used

Correction to Pin Measurement for Runout

in ~~>

Eq 6.20

Pin Diameter in the Calculation in (-> 6.5.1

Best Pin Size

in (->

Eq 6.6

Best Pin Size-Transverse Plane

in (-)

Eq 6.5

Transverse Pressure Angle

--

6.1

Transverse Operating Pressure Angle

--

Eq 3.4

Normal Profile Angle of the Equivalent Standard Rack Cutter

- -

3.7

Transverse Pressure Angle at Measuring Diameter

--

7.3

Transverse Generating Pressure Angle

--

4.5

Transverse Pressure Angle at Best Pin Diameter

--

Eq 6.2

Pressure Angle at Center of Pin

--

6.5.1

Operating Transverse Pressure Angle in Tight Mesh -- Eq 8.1

Helix Angle at a Specified Diameter

--

Eq 4.1

Base Helix &rgle

--

Eq 3.7

Helix Angle at Measuring Radius R

--

5.3

Helix Angle at Standard Pitch Diameter

a-

Eq 4.5

Normal Angular Thickness

--

5.3.1

The following demons are specifically used

in this Standard. The user should be familiar with

these definitions and symbols before applying this

information.

with the greatest allowable functional tooth thick-

ness is in mesh with the pinion tooth having its

greatest allowable functional tooth thickness, at

the tightest allowable center distance, under static

conditions.

Backlash, B. Backlash is the amount by

which the width of a tooth space exceeds the

thickness of the engaging tooth on the operating

pitch circles (see Fig 2-l). As actually indicated

by measuring devices, backlash may be deter-

mined variously in the transverse, normal, or axial

planes, and either in the direction of the pitch cir-

cles, or on the line of action. Such measurements

should be converted to corresponding values in

the transverse plane at the operating pitch circle

for genera l comparisons. If not otherwise identi-

fied, values for backlash refer to transverse oper-

ating backlash.

Standard Pitch Circle. A circle defined by

the number of teeth and a specified module or

circular pitch. (Reference AGMA 112.05)

Tooth Thickness. Tooth Thickness is the

thickness of a gear tooth at a specified diameter.

Unless otherwise defined it is taken as the trans-

verse circular tooth thickness (see Fig 2-2).

Tooth Thickness, Chordal, Normal. The

normal chordal tooth thickness is the length of the

chord subtending a tooth thickness arc in the nor-

mal plane.

Backlash, Minimum, Bd. Minimum back-

Tooth ‘Thickness, Circular. The circular

lash is the

minimum transverse backlash at the op-

tooth thickness is the length of arc between two

erating pitch circle allowable when the gear tooth

sides of a gear tooth, on a specified diameter .

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OPERATING PITCH CIRCLES

Fig 2 l Backlash

7

ANWAGMA

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Circular Tooth ‘Ikickness

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Table 2-2

Alphabetical Table of Terms and Symbols, by Terms

Terms

Svmbol

Units

Where

First Used

Addendum

Backlash (Transverse Operating)

Backlash, Normal (feeler gage)

Backlash, Transverse, Circular

Backlash, Transverse, Minimum

a

B

Bf

B*

B

:,

Base

Base

Base

Base

Base

Base

Circle

Circle Diameter of Master Gear

Circle Diameter of Test Gear

Tangent, Best Length of

Tangent, Maximum Length of

Tangent, Minimum Length of

Center Distance, Maximum

Center Distance, Minimum

Center Distance, Tightest

Chordal Addendum

Chordal Addendum Correction Factor

Diameter, Contact, for Best pin Size

Diameter, MaxWum Outside

Diameter, Outside

Diameter over/between Two Pins

Diameter, Specified

Diameter, Tip of Internal Gear

Face width

-1

3

Db2

l

‘best

=maX

=min

C

c”.”

c

aC

Aac

DW

D

OlllXX

DO

D2w

D

Di

F

Helix Angle at Measuring Radius &

Helix Angle at Specified Diameter

Helix Angle at Standard pitch Diameter

Helix Angle, Base

Lead

Normal ModuIe

Number of Teeth in Gear

Number of Teeth in Master Gear

Number of Teeth in Pinion

Number of Teeth in Test Gear

Number of Teeth to be Spanned

Number of Teeth to be Spanned, Best

Number of Teeth to be Spanned, Maximum

mn

N

N2

n

N1

S

‘best

LX

in (=>

in bd

in (mm)

in (-)

b (=)

in (-1

irl(-l

in (-1

in (-1

in ho

in (mm)

in (-1

in (-)

in (mm)

in (-1

in (-1

in C-1

in (-1

in (-1

in C-1

in (-1

in (-1

bl C-1

--

we

--

--

in (-1

IJllIl

--

--

a-

--

--

--

em

Eq 5.1

4.3

4.3

Eq 4.7

3.1.4

3.7

8.4.1 -

8.4.1

Eq 7.7

Eq 7.3

Eq 7.1

Eq 8.3

Eq 8.5

3.1.4

Eq 5.7

5.3.1

Eq 6.1

5.3

6.4

6.5

4.1

6.4

7.3

5.3

Eq 4.1

Eq 4.5

Eq 3.7

4.1

Eq 3.6M

3.1.4

8.4.1

3.1.4

8.4.1

7.1

Eq 7.8

Eq 7.4

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Table 2-2 (cant)

Alphabetical Table of Terns and Symbols, by Terms

Terms

Symbol

Units

Where

First Used

Number of Teeth to be Spanned, Minimum

Pin Diameter in the Calculation

pin Measurement Correction for Runout

Pin Size, Best

Pin Size, Best - Transverse Plane

Pitch, A-da1

Pitch, Base, Normal

Pitch, Base, Transverse

Pitch Diameter, Operating

Pitch Diameter, Standard (Generating)

Pitch, NormaL Diametral

Pitch, Operating Transverse Circular

Pitch Variation, Accumula ted, Sector of k Bitches

Pressure Angle at Center of pin

Pressure Angle, Transverse, Gperaung in Tight Mesh

Pressure Angle, Transverse

Pressure Angle, Transverse at Best Fin Diameter

Pressure Angle, Transverse at Measuring Diameter

Pressure Angle, Transverse Gperating

Pressure Angle, Transverse Generating

Profile Angle, Normal, of the Equivalent Standard Rack

Cutter

Radial Runout of Reference Diameter

Radial Runout Tolerance

Radius, Maximum Measuring

Radius over/to One Pin

Space Width, Transverse at Best Pin Contact Diameter

Span Measurement

Span Measurement, Modified for Tooth Variation

Test Radius, Master Gear

Test Radius, Maximum (work gear)

Test Radius, Minimum (work gear)

Transverse Tooth Thickness, Angular

Tooth Thickness, Transverse Base, of Master Gear

Tooth Thickness, Transverse Base, Modified for Runout

and pitch Variation

Tooth Thickness, Transverse Base, of Test Gear

%i.n

W

b w

wbest

wLest

px

PN

pb

D’

Ds

P

n,d

P

%pk

2

+3

+

@W

&I

9’

4s

%

--

in (-1

in (=>

in (mm>

in (mm>

in (mm>

in (mm)

in (mm>

in (mm)

in (=>

-1

in

in (mm)

in (=I

--

--

--

--

--

--

--

--

v,

zx

%W

sW

M

Mm

R7n

RTma%

RTmin

7

‘b2

‘bm

in @d

in (mm)

in (=>

in (mm>

in (mm>

in (mm)

in (=4

in (=)

in (mm)

in bd

degrees

in (mm)

in (->

‘bl

in bw

Eq 7.2

6.5.1

Eq 6.20

Eq 6.6

Eq 6.5

4.1

7.3

6.5.1

3.7

Eq 4.6

4.1

Eq 3.2

7.3

6.5.1

Eq 8.1

6.1

Eq 6.2

7.3

Eq 3.4

4.5

3.7

5.3

6.5.5

5.3

Eq 6.11

Eq 6.4

Eq 7.10

Eq 7.13

8.4.1

Eq 8.4

Eq 8.6

5.3.1

8.4.1

Eq 7.12

8.4.1

7

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Table 2-2 (cant)

Alphabetical Table of Terms and Smbols, by Terms

Terms

Svmbol

hits

Where

First Used

Tooth Thickness, Base, Transverse

tb

in (->

Eq 4.11

Tooth Thickness, Circular

t

in (mm)

2.2

Tooth Thickness, Normal

h

w=>

Eq 4.1

Tooth Thickness, NormaI, at &

‘nR

in (m-d

Eq 5.3

Tooth Thickness, Normal Base

*nb

in (->

Eq 7.6

Tooth Thickness, Normal Chordal Measured

?rn

in (-1

Eq 5.10,

Tooth Thickness, Transverse, of the Test Gear at +c

5

in (mm>

8.4.1

Tooth Thickness, Transverse, of the Master Gear at +c t 2

in (mm>

8.4.1

Tooth Thickness, Transverse, Specified Minimum

&in

in (=a

Eq 3.3

Tooth Thickness, Transverse, Tolerance

tT

in (mm>

3.7

Tooth Thickness, Transverse

tt

h-l (mm)

4.1

Tooth Thickness, Transverse, at Best Pin Contact Diameter t w in bJJ@

Eq 6.3

Tooth Thickness, Transverse, at &

tR

in (mm>

Eq 5.5

Tooth Thickness, Transverse Maximum, of Gear

fGItU

in (-)

Eq 3.1

Tooth Thickness, Transverse Maximum Generated

r

ts

in (mm)

Eq 4.8

Tooth Thickness, Transverse Maximum, at Operating

&lax

in (-)

3.7

Pitch Diameter

Tooth Thickness,

Transve

se M&um, of Pinion

hWC

in bw

Eq 3.1

Total Composite Variation

&4

in (-)

3.7

Tooth Thickness, Design. Design tooth

thickness is the thickness estabbshed from engi-

neering consideration of strength, deflection,

mounting and backlash upon the theoretical tooth

thickness.

Tooth Thickness, Effective. The effective

tooth thickness is the apparent circuIar thickness

at the operating pitch diameter with a mate, estab-

lished by the mounting (See 3.1.3).

Tooth Thickness, Functional. The tooth

thickness as determined by meshing with a speci-

fied gear on a caliirated composite action test fix-

ture.

Tooth Thickness, Measured. The measured

tooth thickness is the actual value of circular tooth

thickness caIcuIated from a specific measurement

over pins, a span or tooth caliper measurement.

Tooth Thickness, Normal, t, . The circular

tooth thickness in a normal plane.

Tooth Thickness, Tolerance, t T+ The per-

missible amount of tooth thickness variation.

Tooth Thickness, Transverse, t t. The circu-

lar tooth thickness in a transverse plane.

Tooth Thickness, Variation. The variation

from a specified value of normal circular tooth

thickness.

Total Accumulated Pitch Variation, I .

Total accumulated pitch variation is equal to the

algebraic difference between the maximum and

minimum values obtained from the summation of

successive alues of pitch variation, VP , and is the

same as total index variation.

Total Composite Variation, I 4 . The total

change in center distance whiIe the gear being

tested is rotated one complete revolution during

double flank composite action test.

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Tooth Thicfcness Specification and Measurement

3. Application

3.1 Tooth Thickness Concepts. Various concepts

dealing with tooth thickness are discussed within

this Standard.

(1) Design Tooth Thickness

(2) Measured Tooth Thickness

(3) Effective Tooth Thickness

3.1.1 Design Tooth Thickness.

Design

tooth thickness is usually established from engi-

neering considerations of gear geometry, gear

tooth strength, mounting and consideration of

backlash. The methods for establishing design

tooth thickness for given applications are beyond

the scope of this Standard.

This Standard assumes he design tooth thick-

ness is known and the values for various measur-

ing techniques are to be established.

3.1.2 Measured Tooth Thickness. The

measured tooth thickness is used to evaluate the

size of an entire tooth or all of the teeth on a

given gear. It can be based on a few measure-

ments between two points or two very short con-

tact lines. The nature and the location of these

contacts is determined by the type of measure-

ment (pins, span, or tooth caliper). It is custom-

ary to assume that the entire gear is characterized

by the measured data from as few as one or two

measurements.

Depending upon the method of measurement,

variations in tooth a lignment, profile, and pitch

will affect the measured values to varying degrees.

The effects of these variations on the measured

values may either be additive or may cancel one

another, depending on the magnitude of the vari-

ation where&the measurements are made.

There is no way to separate these vat&ions

from the measurement for tooth thickhess. If a

given gear is measured by each of these methods

somewha t different results will be observed.

These differences are due

to

the different tooth

variations that enter each measurement. The dif-

ferences are usually ignored, but, when results are

critical, or backlash is closely controlled, it is nec-

essary to specify the measurement method to be

used.

3.1.2.1 Functional Tooth Thickness.

The functional tooth th ickness is that family of

ANWAGMA

9

tooth thickness values obtained on a composite

action test (double flank) by means of a calibrated

master gear. It is a measurement which encom-

passes he e ffects of element variations in profile,

pitch, tooth alignment, etc., (similar to the con-

cept of maximum material condition). Section 8

explains this measurement method.

3.1.3 Effective Tooth Thickness.

In most

designs it is desirab le to establish the maximum

effective thickness equal to the maximum design

thickness. That is the basis of this Standard.

The effective tooth thickness of a gear will be

different than the measured tooth thickness by an

amount equal to all the combined effects of the

tooth element variation, and mounting, similar to

functional tooth thickness.

It is the final envelope

condition which encompasses all the effects which

must be considered to determine the maximum

material condition (see Fig 3-l). As in the case of

measured tooth thickness, the effects of the tooth

element variations may be additive or may cancel

each other at various angular positions within a

given mesh. It is not possible to segregate the indi-

vidual tooth element variations from the effective

tooth thickness.

3.1.4 Maximum Tooth Thickness, tplax.

The maximum tooth thickness of a gear, meas-

ured on the transverse plane is the thickness it

would have if meshed at the tightest center dis-

tance and

minimum backlash with a perfect,

maximum tooth thickness, mating gear.

The maximum effective tooth thickness is the

thickness of the thickest tooth, with reference to

the mounting surfaces, at the operating pitch di-

ameter with its mating gear. In this Standard,

maximum tooth thickness and maximum effective

tooth thickness are taken as numericall y identical.

The selection of tooth thickness is up to the

designer, but, the following relationship must be

satisfied:

tGmax= p’ - Be- tP max

0 3-1)

where

‘Gmax =

Bmin =

maximum transverse tooth

thickness of gear, at operating

pitch radius, jn (mm)

minimum backlash, transverse,

in (->

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ToothThicknessSpecificationand Measurement

MAxiMUM EFFECTIVE

-IooTHTHIcKNEss

SPECIFIED MINIMUM

TOOTH THICKNESS, ‘&

l/Z SPECIFIEDTOIERANCE

l/2 SPECLFICATION

BAN&m

BLWQC$~ $$;~

* THIS FIGURE IS DRAWN XI’ THE POSlTtON OF TIGHTEST CENTER DISTANCE;

lFCENTERDISTANCEISINCREASEDBACI%UHWILLINCI=ASE.

Fii 3-l Tooth Thickness Tkansverse Plane

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tPmax

= maximum transverse tooth

thickness of pinion, at operating

pitch radius, in (mm)

= transverse circular pitch at

operating (tightest) center

distance, in (mm)

pl= 2n &-

( >

(Eq 3.2)

where

jv =

n

=

c =

number of teeth in the gear

number of teeth in the pinion

tightest center distance, in (mm)

(minimum center distance for

external gears or the maximum

center distance for internal gears)

For gears of standard proportions, operating

at standard center distance, it is common to re-

duce the tooth thickness of each member by one

half the backlash allowance, but it is not a re-

quirement. As long as Eq 3.1 is satisfied, the set

will have the specified minimum backlash.

For gea r sets with nonstandard proportions,

or operating at nonstandard center distances, the

designer has a wide range of choices for t-.

The usual approach is to select a center distance,

then to vary the addendum (tip) diameters of the

gears until an approximate balance of strength rat-

ing is achieved. An attempt is made to keep the

cutting depths of both members equal, assuming

that they are to be cut with the same tool. The

design is then rechecked for tip land width, con-

tact ratio, limit diameter *, root clearance, and

rating before it is finalized.

3.2 Backlash. The amount of backlash which is

appropriate for different sizes and classes of gears

is discussed in Appendix A.

An individual gear does not have backlash. It

has only a tooth thickness. Backlash of a meshed

set of gears is governed by the center distance at

which the gears are operated and the tooth thick-

ness of each of the gears.

This Standard uses the term minimum back-

kzsh in a carefully defined way. Minimum back-

lash, Bmin, is the minimum transverse backlash

allowable on the operating pitch circle when the

gear tooth with the greatest allowable effective

tooth thickness is in mesh with the pinion tooth

having its greatest allowable effective tooth thick-

ness, at the tightest allowable center distance, un-

der static conditions. This is the traditional back-

lash allowance provided by the designer to a llow

for deflections, misalignments, bearing runouts,

temperature effects, and any unknowns.

The tightest center distance is the minimum

center distance for external gears or the max imum

center distance for internal gears.

3.3 Mounting Surfaces. Mounting surfaces are

the surfaces (usually two) which determine the

axis of rotation and axial location (usually a plane

perpendicular to the axis of rotation) of the fin-

ished gear in the gear assembly. These surfaces

must be specified, because they are used as the

reference surfaces (tooling points) for all backlash

and effective tooth thickness measurements. If the

mounting surfaces are finished after the teeth are

cut or inspected, an auxihary pair of reference

surfaces (trueing registers or proof surfaces)

should be specified for tooth inspection.

3.4 Reference Surfaces. The expressions refer-

ence surface, reference diameter, reference

plane, and reference ax& are used to denote sur-

faces, actual or hypothetical, which form the basis

for the calculation or measurement under discus-

sion. For example, chordal measurement uses the

outside diameter as a reference surface. The defi-

nition of circular tooth thickness in 2.2 uses speci-

fied diameter in this way.

3.5 Total Composite Variation. Total compos-

ite variation, V

, is the variation in center dis-

tance when a tl gear is rolled in tight mesh for a

complete revolution with an appropriate master

gear, in a variable center distance fixture. It is not

a factor limiting

maximum tooth thickness, but it

has an important effect on operating backlash,

tooth thickness measurement, and minimum tooth

thickness specifications. Appendix A covers this

subject in more detail. AGMA 2000-A88 pro-

vides tables of values for Vcq for gears of various

sizes and quality numbers.

[ *] Limit diameter is the diameter on a gear at which the line of action intersects the maximum

addendum circle of the mating gear. This is sometimes referred to as the start or end of contact.

See AGMA 112.05, Gear Nomenciuture Definitions of Terms with Symbols.

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3.6 Specifying Maximum Tooth Thickness.

Because it is very difficult to measure tooth thick-

ness directly, and the indirect methods include

different effects of tooth variations, the specified

tooth thickness measurement should be adjusted

for each specific method of measurement. These

differences are often ignored, particularly in

coarse pitch gearing with large backlash allow-

ance, but the effects are important in fine pitch

gears and wher e backlash is tightly controlled. Ta-

ble 3-1 shows the influence of tooth geometry

variations on each measurement method.

Where variations in tooth geometry or refer-

ence surface geometry influence the tooth tbick-

ness measurement, the magnitude of the vari-

ations is taken as the

maximum for that gear and

quality class per AGMA 2000-A88 and the direc-

tion is taken so that

maximum effective tooth

thickness is not exceeded.

In each of the following sections, the maxi-

mum tooth thickness in the transverse plane at the

minimum operating pitch diameter maximum for

internal) will be used as a basis for calculating the

specified

maximum value for each measurement

method.

Tooth thickness is calculated in the transverse

plane and specified as the appropriate value for

each measurement method, usually in the normal

plane.

3.7 Specifying Minimum Tooth Thickness.

The specified

minimum tooth thickness is equal to

maximum tooth thickness, less an allowance for

manufacturing variation. The manufacturing vari-

ation allowance should be a function of the manu-

facturing method and the actual gear quality total

composite variation and tooth thickness toler-

ante) .

‘~=t--tT

- 2 yLqtanQ:

ml 3.3)

where

tnlin

= minimum specified transverse tooth

thickness, in mm)

t

max

= maximum transverse tooth thickness

at operating pitch diameter, in mm)

fT

= tooth thickness tolerance, in mm)

taken from AGMA 2000-A88, and

applied to the transverse plane)

V

cq

= total composite variation

v

= transverse operating pressure angle

d

ml 3.4)

D’ = operating pitch diameter, in mm)

Table 3-l

Other Gear Variations Included in Tooth Thickness Measurement

Variation

Method of

Measurement

Concentricity in reference to

Tooth

Ys2:

Gutside

Gut-of-

Profile pitch Alignmen t

Diameter

Roundness

Chordal Thickness

X

X’

over-two-pins X X’ X

over-one-pm

X X

X’

X

Span

X’ X

W

Test Radius with master gear X

X

X

X

X

* If pin size or number of teeth spanned is selected to locate the measurement at one half the working

depth from the addendum tip) circle, the effect of profile deviation is minimized.

This Standard uses this

method.

f Helical gears only.

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d=2C

ml 3-5)

tightest center distance

base circle diameter, in (mm)

Db =

cos ‘PC

cos Qb

(Eq 3.6)

Db =

N m, cos &

cos

(Eq 3.6M)

where

P

nd

= normal diametral pitch

mn

= normal module

+c

= normal profile angle of the equivalent

standard rack cutter, degrees [ *]

‘b

= base helix angle, degrees

‘b

= arc sin

@q 3.7)

‘i = a-f m;l., “)

(Eq 3.7M)

For spur gears, cos Jrb = 1

The tooth thickness tolerance (allowance for

tool wear or adjustment of the cutting machine) is

a function of pitch and quality number. Values

are tabulated in AGMA 2000-A88.

3.8 Measurement Method Effects. The effect

of measurement method on the specified value of

minimum tooth thickness is discussed in Sections

5 through 8 as it applies to each measurement

method. The magnitude and direction of adjust-

ments for variations in tooth or reference surface

geometry are taken so that tooth thickness is de-

creased by inaccuracies inherent in each measure-

ment method.

3.9 Selection of Tooth Thickness. Usually,

maxim= backlash does not affect the function or

smoothness of

transmission motion, and effective

tooth thickness variation is not the main consid-

eration in the selection of gear quality. Under

these conditions, the selection of tooth thickness

and measurement method is not critical and the

most convenient method can be used.

In many applications, allowing a larger range

of tooth thickness tolerance or operating backlash

will not affect the performance or load capacity of

gears, and may allow more economical manufac-

turing. A

tight

tooth thickness tolerance should

not be used unless absolutely necessary, since it

has a strong influence on manufacturing cost.

For any given value of minimum backlash,

B

a,

and tooth thickness tolerance, t =, maxi-

mum backlash, Bmax,

increases as the quality

level decreases and as the size increases, since to-

tal composi te variation, &, increases with size

and lower quality.

In those cases where

maximum backlash must

be closely controlled, a careful study of these fac-

tors must be made and the gear quality class, cen-

ter distance tolerance, and measurement methods

must be carefully specified. It may be necessary

to speci@ a higher quality class to hold maximum

backlash within the desired limits.

A method to calculate maximum backlash

from tolerances for center distance, tooth thick-

ness, and total composite variation is included in

Appendix A.

4. Gear Geometry Calculations

4.1 Circular Tooth Thickness. Circular tooth

thickness may be specified in the plane of rota-

tion, (transverse plane), tt , or in the plane normal

to the helix angle at the reference circle (normal

plane),t,.

Tooth thickness calculations are usually made

in the transverse plane and tooth thickness meas-

urements are made in the normal plane.

At any specified diameter at or above the base

circle:

t,=

tt cos

where

tt =

t, =

Jr =

Jr

0% 4.1)

tooth thickness in transverse plane

tooth thickness in normal plane

For spur gears, tn= tt

helix angle at the specified diameter

[*] For complete discussion, see 9.01 of AGMA 112.05.

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Tooth Thickness peciii&on andMeasurement

q = arctan

X

where

D =

Px =

P*=$-

L =

the speci&ddiameter, in (mm)

axial pitch [*I. in (mm)

ml431

lead of gearor machine uide

When he helix angleat the standard itch

diameterisgivexx

Px =

1z

P& sin 9s

Zm

-IL

% -

sin ?&

where

% = helixangleatslimdardpiti

diameter

lu, =

arCti(

PXtld

>

%

= aTcsk(y)

(Eq 4.4)

0% 4.4w

(445)

Forspnrgears,cos~=l

42

StandardPitchDhune~. Asmndardpitchdiame-

ter(generatingpitChdiameter),D,.is0w~nlatedaC-

cordingtothestand2u.dpitchofthegear~gtooL[**]

Ds= N ~2

‘nd =% ‘d

mi4.6)

pd

= nansvedsediametipitch

43 BaWashcalrnlations Ba&iash,B,inanassem-

bledgearsetistheckamnceorplaybetweentheteethof

themeshinggears.Itistheamountbywhichthewidthof

the oothspace xceedshe ooth hickns of theengag-

ing tooth on the opesathg itch circles (seeFig 2-l).

Backlashmaybemeasnredintheuansvtxsep~e,~the

mnmalplanealong he operating itch cylinderor nor-

mal to the toothsurihcen theplaneof action,asmeas-

medbyafeelergage,InthisStandard,Backlashisspeci-

fied n the transverselane.

Bt =

Bf

= .Bf,

cow =wb Db coq)

m4.7)

where

Bt

= cilcdartransversebac~iIl(mm)

Bf

= biad&shmeaslnednormaltotootil

snrface feelergage) n the planeof

action, n (mm)

4.4 EfRctiveToothThicknesCal~n. Themaxi-

ameterdetemined n 3.1.4 s usedas the basicdimen-

sioaIfthegezus~atstandardcenterdistance,the

eHedivetooththickmsisalsothemaximnmmatedial

cwditiontoorhthiclrnessatthegewratingdiameters

(maximnm etkemedooth thidmss).[***l

45

Maximum (&mated Tooth Thickness. The

memnmgeneratedtooththickn~(tooththicknessat

thesmndasdpitchdiameter),

is:

Forexuxnalgears:

.

tts = Ds[(+) + hv ‘- inv $s]

(3%4.8)

where

Its

= tmfimttmaansversegewratingtooth

thickn~in (mm)

%

= tramesegen~gpressareangle

%

sin d$

= arc*(-)

ws *b

Fmspmgears, s = ec

Forintemalgears:

0% 4.9)

tts= Ds +9-

~v ‘+~v $I (Eq4.10)

[*I

This calculation s based n standard earhobbing ractke, with Pndandpx give=

Fur a

detailed ext on gv seeAGMA 15.01 Rl988), I@~~ Sheet-Basic

Gear Geum~.

[**I

See 8.16

of AGMA ll2.05 for more &rmation.

[***I

The diswssion of shortpitch cnlters s beyond he scope f this Standard.

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4.6 Base Tooth Thickness. The base tooth

thickness, used in subsequent calculations, is:

For external gears:

or (Eq 4.11)

tb = q) [[ )+hvd]

where

t,

= transverse tooth thickness at base

circle, in (mm)

For internal gears:

or (Eq 4.12)

5. Chordal Tooth Thickness

5.1 Advantages of Chordal Tooth Thickness.

The vernier gear tooth caliper, Fig 5-1, is a port-

able hand held

Went used to measure the

thickness of external gear teeth. Its portability

and its simplicity are its principal advantages.

5.2 Limitations of Chordal Tooth Thickness.

The tooth caliper requires an experienced opera-

tor, because the anvils make contact with the

tooth flank on their comers, rather than on the

flats.

The instrument is hard to read consistently

with a deviation less than 0.001 in (25 w). In-

struments are not available for very large or very

small teeth.

For coarse pitches and small numbers of

teeth, the addendum must be corrected and the

chordal tbkkness must be calculated (see Fig

5-2).

The theoretical addendum, o, is affected by

variations in the outside diameter, taper and

runout of the blank since the outside diameter is

used as a reference surface for the caliper.

The tooth thickness caliper cannot be used

for internal gears.


5.3 Calculation of Chordal Tooth Thickness.

The addendum bar setting is usually based on a

standard addendum, even if the gear has a non-

standard nominal outside diameter. This puts the

point of measurement at approximately half the

working depth, to

minimize the effect of profile

deviation.

Fig 5-l Chordal Tooth Thickness

Measurement by Means of a Gear

Tooth Caliper

The maximum expected reference radius,

equal to half the maximum outside diameter plus

half the allowable xunout, is the basis for calcula-

tion.

If the actual oufside diameter and the

runout of the outside diameter at the point of

measurement are known, they should be used. If

the nmout of the outside diameter is not known, it

may be assumed to be equal to the allowable

runout of the teeth.

u = 1 for full depth teeth

pnd

@q 5-l)

Q=m

n (Eq 5lM)

u =0.8 for stub teeth

pnd

0% 5.2)

Q = 0.8 mn

(Eq 5.2M)

where

a = addendum, in (mm)

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ADDENDUM. a

CHORDAL

ADDENDUM

NORMAL PLANE

Fig 5-2 Addendum and Chordal Tooth

Thickness Corrections

‘nR =

tR cos *R

m 5.3)

where

*nR

= maximum normal tooth thickness,

in (=)

*R

= helix angle at measuring radius,

For spur gears, cos eR = 1

m 5.4)

fR

= transverse tooth th ickness at R,,

in (mm>

tR=2R,, [k -~v(~c cos(2Da) ]

0 5-5)

&=

nxz3ximut-n easuring radius, in (mm)

R-=

where

D

omax

VT

Prnazr)-a

Es 54

= maximum outside diameter

= radial runout of reference diameter,

total indicator movement, (may be

assumed to be equal to allowable

runout of the gea r teeth per

AGh4A 2000-A88), in (mm)

If the maximum runout of the outside diame-

ter to the mounting diameter is specified, the

specified value of runout should be used, instead

of the assumed value.

53.1 Addendum Correction. To calculate

the chordal addendum, u c, a correction, Aczc ,

must be made for the height of the chord spanned

by the tooth caliper.

=C

=a+Aac or

0 5.7)

ac = (+)-R- COS 6)

(Eq 5-8)

where

7

= normal angular thickness

7

-=

+ (cos2fR

2

2 Rmax

>

, radians (Eq 5.9)

Since they have such a large influence on gear

tooth thickness measurement, outside diameter

size, outside diameter runout, and gear tooth

runout must be carefully controlled when tooth

thickness is controlled by tooth caliper measure-

ment.

I

5.3.2 Chordal Correction. Since the tooth

caliper measures on a straight chordal line, the

chorda l thickness measurement, t,, is slightly

less than the distance along the arc of the refer-

ence circle. Although this difference is frequently

ignored, it is significant for coa rse pitches and low

numbers of teeth.

t

m= 2R-

(COSJT

R

) sin

where

5n

= measured normal chordal tooth

thickness, in (mm)

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5.3.3 Specifying Chordal Tooth Thickness

Measurement.

If the outside diameter of the

gear is under the

maxim= size, the tooth will ap-

pear to be thicker than it is. To avoid accepting

gears which are thicker than tmax, the maximum

chordal tooth thickness must be calculated from

the maximum outside diameter. To avoid reject-

ing gears which are at the min imum tooth thick-

ness, it is also necessary o calculate the minimum

chordal tooth thickness from the maximum out-

side diameter. This procedure wiIl allow some

thin gears to be accepted, if their outside diame-

ters are less than the

maximum. If tight control of

tooth thickness is required, the size and concen-

tricity of the outside diameter must also be tightly

controlled.

6. Measurement by Pins

6.1 Advantages of Measurement by Pins. Pins

or balls afford a method of measuring tooth thick-

ness of gears of any diameter within the capacity

of available micrometers see Fig 6-l). Measure-

Fig 6-l Tooth Thickness Measurement

ments are not affected by outside diameter devia-

overPbls

tion or by runout of the outside diameter.

dimension over pins. Even though the microme-

Measurements over one pin or ball, from the ters may be graduated to 0.0001, the variation of

mounting diameter, show the effects of runout in the measurement among several operators may

the gear teeth and approximate the measurement

exceed 0.001.

of functional tooth thickness.

Balls must be held exactly in the plane of ro-

The amplifying effect of pin or ball measure-

tation; a difficult task.

ment; i.e., the fact that the measurement over a

Internal helicals cannot be measured with pins

pin is a function of t/tan +, makes it easy to detect

and are usually measured with balls.

small changes n tooth thickness.

External helical gears with odd numbers of

This is a common method of tooth thickness

teeth should be measured with balls or with three

inspection.

pins between parallel planes. Both are difficult

6.2 Limitations of Measurement by Pins. Meas-

setups.

urements are affected by deviations in pitch and

The following is quoted from Analytical Me-

profile.

chanics of Gears, by Earle Buckingham [2]

The following should be noted:

Measurements over rolls on helical gears

- Pins on spur gears form line contacts

are very diff?cuIt to make with any great de-

- Balls on spur gears form point contacts

gree of accuracy unless definite precautions

- Pins and balls form point contacts on helical

are taken. In many cases, a pair of cali-

gears.

brated wedges, or rack teeth, make a much

more reliable measurement for tooth ihick-

Therefore, deflection, because of the limited

ness than do rolls. However rolLs are often

contact, can cause variation in readings and will

available when needed, whiie the Jpecial

vary with gaging pressure.

calibrated rack-tooth wedgesmay not be at

Micrometers are often used to measure the

hand. The measurement over rolls should

[23 Xumbers in brackets refer to the bibliography.


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be made between parallel flat surfaces and

not with a micrometer alone. When the

rolls are held in position on the gear by two

parallels, the two rolls will be on opposite

sides of the gear, or diametrically opposite

to each other, whether the number of teeth

in the gear is odd or even.

With odd num-

bers of teeth, one roll may make contact

near one edge of the gear while the other

roll makes contact near the opposite edge

of the face width.

If an attempt is made to

measure odd numbers of teeth over the rolls

directly with a micrometer, one or both

rolls will be tipped away from the correct

plane of measurement, and any measured

values so obtained are useless or any pur-

pose.

Ball-point micrometers may be used, but

here the two balls must be definitely aligned

in respect to the face of the gea r blank.

For example, the gear blank may be laid

flat on a surface plate, and the two bail

points may be held against this same sur-

face plate. Where balls are used, when odd

numbers of teeth are involved, the calcuia-

tion of the actual chordal measurement

must include the offset condition or position

in exactly the same way as the calculations

are made for spur gears with odd numbers

of teeth.

Large micrometers are required for large

gears.

Measurements made over two pins or balls do

not show functional tooth thickness.

Multiple readings taken around the gear

should be averaged to find the mean. The mean

value should be used in comparison of readings.

The maximum reading, as previously stated, is

probably closer to the functional tooth thickness,

which is best measured by double flank testing.

6.3 Measurement Methods. It is important to

use a measurement over pins [‘I setup for which

there is a suitable calculation method relating the

measurement to the tooth thickness.

For all types

of spur and helical gears, there are measurement

setups using pins or balls for which there are cor-

responding geometrically exact calculation meth-

OdS.

For external spur gears with even numbers of

teeth, the measurement is made across the high

points of two properly sired pins placed in diamet-

rically opposed tooth spaces.

In the case of spur

gears with odd numbers of teeth, the tooth spaces

used are those nearest to diamenically opposed.

Measurement over pins can also be per-

formed on medium and small external helical

gears. When the gear has an even number of

teeth, the measurement technique is similar to

that on spur gears. Although the two pins are not

parallel, it is possible to position the anvils on a

conventional micrometer so as to measure at dia-

metrically opposite points.

For helical gears with odd numbers of teeth,

there are two techniques with geometrically exact

calculation methods. One method uses three pins

instead o f two, but is limited to gears whose face

widths are greater than their axial pitches. This

method also requires the use of a micrometer or

other measuring

instrument with an anti of size

greater than the axial pitch. The third p in is

placed alongside one of the others so that the pair

will be diametrically opposite the single pin. The

wide anvil is positioned to span the axial pitch dis-

tance between the two adjacent pins and the sec-

ond anvil, which need not be as wide, is posi-

tioned to contact the single pin at a location half-

way along the axial pitch distance. When properly

positioned for measurement, all three pins will be

in line contact with their respective parallel anvil

surfaces. The measurement is twice the calcu-

lated radius over one pm.

The other method uses a single pin and is suit-

able for any external helical gear, whatever the

face width and whether the number of teeth is

odd or even. The measurement over the single

pin is made relative to the gear center line or rela-

tive to the bore or other concentric cylindrical ref-

erence surface on the gear. This measurement,

when combined with the radius of the reference

surface corresponds to the calculated radius over

one pin.

To approximate the maximum functional

tooth thickness, repeated measurements must be

made, and the largest used.

[ ] “Pin” is used in this text for ‘pin, wire or ball”. The calculations are made using a disc of

infinitesimal thickness, representing either.

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All two pin measurements can be made with

the pins replaced by balls of the same size.

In the

case of helical gears with odd numbers of teeth, it

is also possible to measure over two balls. This

measurement is the same as that for spur gears

with odd numbers of teeth. All measurements

over balls have the special requirement that the

two balls be located with their centers in a single

plane perpendicular to the axis of the gear.

6.4 Pin or Ball Sizes. The ideal (best) sire pin

or ball would contact the tooth profiles at half

their working depth. This minimizes the effect of

profile deviation.

The best disc diameter would contact the

tooth profile at DW

For external gears:

DW=

Do-2a

(Eq 6-1)

The pressure angle at this diameter is:

The arc tooth thickness at this diameter is:

It would extend above the outside diameter of

tw= Dw[i )-Ww]

0% .3

an external gear or below the inside diameter of

an internal gear, and would not touch the root of

The arc space width at tbis diameter is:

the tooth space.

‘rrDW

To calculate the best pin size for external

SW= 7 -tw

0% 6.4)

gears, see Fig 6-2.

( >

The best disc diameter is:

SECTION IN TRANSVERSE PLANE

Fig 6-2 Best pin Size, w’ best

(External Gears)

This calculation is based on a disc of infini-

For internal gears (see Fig 6-4):

tesimal thickness in the transverse plane (see Fig

6-3).

DW=Di+ 2a

0% 6-7)


(Eq 6.5)

This value must be converted to the normal

plane by rotation in the plane tangent to the base

circle through the center of the disc (see Fig 6-3).

where

DW

= contact diameter for best pin sire,

in (=>

Do

= outside diameter, in (mm)

+W

= pressure angle at best pin diameter

tW

= transverse tooth thickness at best

pin diameter, in (mm)

sW

= transverse space width at best ‘pin

diameter, in (mm)

Wt;est=

best pin sire -

transverse plane,

in (->

wbee best pin sire, in (mm)

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INTERNAL

TOOTH

CENTERLINE

q

W

t

J cow*

:.g

.$j

I

’ .a

LJ

EXTERNAL

I I

.,” . .

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Fig 6-4 Best Pin Size

(Internal

spur Gear)

where

Di

= tip diameter of internal gear

tw=D,[( )+ww]

0% 6.8)

0% 6-9)

For internal helical gears, W+Jemmust be con-

verted to a ball diameter, P(best9 using Eq 6.6.

This is the theoretical best pin size; however , it

may not be available.

The pin size specified should be the next larg-

est selected from Table 6-1, or from a table of

pins known to be available to the gear manufac-

turer. The size selected should be checked to be

sure that it will not bottom in the root of the tooth


space, and that it will prouude past the tip circle

of the gear. This is done after the specified radius

measurement over pins is calculated per Eq. 6.5,

by comparing the pm measurement to the tip ra-

dius, and the pin measurement radius, plus or mi-

nus the pm diameter, to the root radius.

Dimensions over or between pins can be cal-

culated for any pm diameter. Bin diameters have

been standardized so that sets of pins for common

pitches can be used. Table 6-l Iisrs some com-

monly used pm sizes. Further information on

standard pin sizes can be found in manufacturer’s

catalogs. [33

6.5 Calculation for Measurement by Pins. The

final pin diameter selected is labeled W in the cal-

CUMOIIS.

6.5.1 Radius Over One Pin. Figure 6-3

shows the general case for external and internal

gears.

For esernal gears:

fa_

mv+2= Db +

W P

Db cos lip

Db

[‘I (Eq 6-N

where

pb

= transverse base pitch

R

=A +tv

1w 2coE42 2

(Eq 6.11)

where

2

= pressure angle at center of pm

RIW

= radius over or to one pin, in (mm)

W

= pin diameter, in (mm)

Conversely

% 2-[A&]

(Eq 6.12)

%

= Db in~+~+~ --

c 1

W

cos Jr

(Eq 6.13)

b

[ *] The inverse function is usually found by iteration.

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Table 6 l

Standard Pin Diameters for Various Pitches in Inches

Diametral Pitch

P

nd

For Standard

For Standard

For Long Addendum

External Gears

Internal Gears

PilliOnS

1.728

1.680

1.920

w=

w=

w=

P

nd

Pnd Pnd

1

1.728

1.680

1.920

1 l/2

1.152

1.120

1.280

2

0.864

0.840

0.960

2 l/2

0.6912

0.672

0.768

3

0.576

0.560

0.640

9

0.1920

0.18666

0.21333

10

0.1728

0.168

0.192

11

0.15709

0.16273

0.17454

12

0.144

0.140

0.160

14

0.12343 0.120

0.13714

16

0.108

0.105

0.120

18

0.096 0.09333

0.10667

0.432

0.420 0.480

0.3456

0.336

0.384

0.288

0.280

0.320

0.24686

0.240

0.27428

0.216 0.210 0.240

Table 6 1M

Standard Pin Diameters in Millimeters

2 5.5

2.25

6

2.5

6.5

2.75

7

3

7.5

3.25

8

3.5

9

3.75 10

4 10.5

4.25 11

4.5

12

5

14

5.25

15

Abstracted from DIN 3977[4]

16

18

20

22

25

28

30

35

40

45

50

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6.5.2 Radius To One Pin.

For internal gears:

inv+ 2---

‘b

2 N Db

‘b - ‘b -

=

(Eq 6.14)

3

Db

W

RIW =~COS+

2

-2

(Eq 6.15)

6.5.3 Dimension Over Two Pins.

For external gears with even numbers of

teeth:

D2w = 2R1W

0 6.16)

For external gears with odd numbers of teeth:

D2w =

Ob

- cos = + w

cos +2

( >

N

0 6.17)

where

D2w =

dimension over or between two pins,

in (=>

Equation 6.17 applies to helical gears with

odd numbers of teeth when measured over balls.

See 6.4 for further information on helical gears.

If D2 w is known from measurements:

49 =

arc cos

(Eq 6.18)

6.5.4 Dimension Between Two Pins.

For internal gears with even numbers of teeth

see Eq. 6.16.

For internal gears with odd numbers of teeth:

A‘L cos

D2w = cos

Tr - w

+

( >

N

2

0 6-W

6.5.5 Correction for Tooth Deviations. If

the pins make contact near the mid-point of the

active profile, deviation effects are minimized.

The effect of allowable pitch deviation is

much smaller than allowable nmout, so it can be

ignored, except with very low numbers of teeth

and other unusual cases.

If pin measurements are made as a radius to

one pin from the mounting diameter, the effects

of runout are included and no correction is neces-

sary. If the measurements are made with two pins,

the effects of nmout should be calculated and

D2 w adjusted accordingly.

The amount of correction is:

V

V

rT

rw = -

2

(Eq 6.20)

where

v

rW

= correction to pin measurement for

runout, in (mm)

VrT

= allowable runout of gear teeth, from

AGIvLA 2000-A88, in (mm)

The direction of the correction reduces the

allowable tooth thickness (see 3.1).

7. Span Measurement

7.1 Advantages of Span Measurement. This

method utilizes a vernier caliper or a disc mi-

crometer to measure the distance, M, over a num-

ber of teeth, S, along a line tangent to the base

cylinder.

For external gears, the distance measured is

the sum of (S-l) normal base pitches, plus the

normal thickness of one tooth at the base cylin-

der.

For internal spur gears the measurement is

made between teeth, and the distance measured is

(S+l) normal base pitches minus one normal base

tooth thickness.

Measurements are not affected by outside di-

ameter deviations or by runout of the outside di-

ameter (see Figs 7-l and 7-2).

This method is particularly useful for large

gears, because it does not require auxiliary balls or

pins and a smaller micrometer or caliper can be

used. The measurement can sometimes be made

without stopping the gear cutting machine.

No unusual skill is required to make the meas-

urement, which is similar to measuring a diameter.

7.2 Limitations of Span Measurement. Span

measurement cannot be applied when a combina-

tion of high he& angle and narrow face width

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prevent the caliper from spanning a sufficient

Readings are influenced by deviations in base

number of teeth. This can be overcome to some

pitch, accumulated pitch over S teeth, tooth pro-

extent by making measurements on the machine

file, and lead. The effects of profile deviation are

when gears are stacked in cutting, or by using a

greatly reduced if the measurement is made at

disc micrometer.

half the working height of the teeth.

r

/

‘-.$P \“’

.+ ..;*- /I,,,

$,S~ ,

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Readings are erroneous if attempted on a por-

tion of the tooth flank which has been modified

from true involute form.

Span measurement does not show the effect

of runout, so it does not measure functional tooth

thickness.

Span measurement cannot be used for inter-

nal helical gears or for pitches which are too fine

for the anvils of the measuring instrument to enter

the tooth space.

It does not have the amplifying effect o f pin

measurement.

7.3 Cakulation of Span Measurement. Xum-

ber of teeth to be spanned for eternal gears, S,

can be a range.

The range of S is limited by the size of the

plane which is tangent to the base cylinder,

bounded by the outside diameter and the face

width of the gear (see Fig 7-3). S is also limited

by the limit diameter of the gear.

The following calculation also limits the con-

tact between the measuring instrument and the

gear so that no contact occurs within 0.125/P&

(0.125 m,), of the outside diameter or 0.25/P&

(0.25 mn>,

of the ends ‘of the teeth.

Lmin = d (D, - 4a)‘- Db2

0% 7-l)

kin

=integer portion of

p%p + ;I

03 7.2)

S&Z 2

=-= JiygZ

0% 7.3)

Lmax= D/ ?)2-D; (Eq7.3M)

%X3X

integer portion

of rTjFtnb + 1]

(Eq 7.4)

or, if helical, integer portion of

i

[+k-p~] -t

sin I&

n

b

77

L

[

[~-(~;-bl _ nb

sin%-b

+ 1 (Eq 7.5M)

pN

I

whichever is least-

where

Se=

S-=

F =

L-=

PN =

s =

%b =

minimum number of teeth to be

spanned

maximum number of teeth to be

spanned

face width, in (mm)

maximum length of base tangent

plane, i.n bd

minimum length of base tangent

plane, in (mm)

normal base pitch, in (mm)

number of teeth to be spanned

nornial base tooth thickness,

in (-1

‘nb =

$ cos $,

If smax

is not greater than or equal to Smin,

which must be greater than or equal to 2, the face

width is too narrow for span measurement at this

helix angle.

The best span measurement, Sbest , is made

where the base tangent plane intersects the teeth

at approximately half their working height. When

rotding Shea

(Eq. 7.7) to the nea rest integer

rounding up will place the caliper contact abov

half the working height, and rounding down wil

place the caliper contact below half the working

height. For gears with exrra tip relief or extra fille

clearance rounding may be favored one way o

the other.

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Tooth Thichess S@ficaion and Measurement

3

*

3

1-c-

(s- l>pb

1

--

4pnd

-

’ 1

D--

0 4Pnd

TRmwERSEPLANE

1

f

gp,d

BASETANGENTPLAIQ

Fig 7-3 Limits of Span Measurement in Base Tangent Plane

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Tooth ThicknessSpecifk&on andMeasurement

Lbest

+I$ -2a)‘-

Db”

%est

roimdedvalueof

[~&-Q +J

where

Lw=

sm=

0% 7.8)

best engthof base angent

bestmlmberofteethtobespduned

sr& 6S,, 1

1

(Eq 7.19)

hill

integerportionof

K

Lh +t&

pb

> 1

1

@l7W

%est

rounded alueof

K

* + &

pb

1 1

1

0% 721)

l$Srnin$Sbp&S-

(Eq7J2)

calcnlatespm

M

=(Sbest+1

)pb-tb

0% 723)

Toccmectthisthecmxkalspanfortooth

g~metrydeviations, tbm mustbe calculated

per Eq 7.12.

Mm

=

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Tooth Thickness pecificationndMeasurement

TRANSVERSEPLANE

Fig 74 Limits of Span Measurement or Internal Gear

8. CompositeAction Test Measurement

8.l Adwmlagesof Come Action Test Measnre-

me Thismedxximeasures

iUlCti0WIltOOththiClUl~,

since t iuchxies he effectsof all tooth deviations.

See

Fig S-1, AppendixA and AGMA 2NN438 for a de-

taileddescriptcm.

Wherethesizeoftheworkpermitsandthetooling

canbejnstii5e43acompositeactiontesestmdiusmezs-

uremen is hebestmethod o inspecttooththiclmess.

Composite ction estmeasurementnspea every

tooth of the wo gear n one opedation.

This s much

f sterthanmaEngmnhiplemeasurem

ntswithtbeorher

methods.

Fii 8-l Schematic f Composite ction

Test Measurement

8.2 Limitatious of CompositeAction Test Measure-

ment. ThismethodislimitedtomediumandsmaUer

gears,since testingmachinescapable of more than

twenty nch center istance re arely available. n spe-

cialcircmnstancestestingcanbeaccomplishedinplace

on the cuttingmache.

Small ot producers ncounter@@cant tooling

cosrsinnsingthetestradiusmeth~sincespecial

mountingkturesandmastergearsareoftenrec@ed.

Carefuldesigntouseexistingtoolingcansavesomeof

this expense.

SpeciaIattemionmustbepaidtomountingsmfaces,

toassure atthetestperformedisreprexntativeofthe

gearasitwillbehstalled.

Specialmachinesoratlachmentsarerequiredforin-

temalgears.

Testmacbksmustbecarefnlly bratf particu-

lady for fine pitch and high quality gears.

Refer to

AGIHA 2000 or detaikd calibration ns uctions.

83 MasterGears Mastergeatssuitableforchecking

mostspurgearsateavailableinsizesandtoothpropor-

tions standardizedy their manuktmers (seeAGMA

2OO A88).Thetooththicknessofrhesemastergearsis

madeequalorclosetoonehalfofthecircnkpitchatthe

standardpitchdiam~.

Thepropordonsofthemastergearmustbechecked

forpropermtxhingwiththeworkgeartobesurethat

co~~placeneartothetipand~einvolnteform

diametm, without nterference.

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2002--B88 ANSI/AGMA 29

Master gears are usually marked with a test radius

which is the radius at which they would mesh with a stan-

dard mating gear having a tooth thickness at DS of

(π DS /2 N ).

Special master gears are often required for spur

gears with nonstandard proportions.

Helical gears usually require special master gears.

Mastergears must bemade very accurately sinceany

deviationin themaster gear is added,in thetest results, to

the deviations in the work gear. Accuracy requirements

for master gears are included in AGMA 2000--A88.

8.4 Calculation for Composite Action Test Measure-

ment. The following method applies to external gears.

8.4.1 Maximum Test Radius. The maximum test

radiusis based on themaximum effective tooth thickness

as defined in 3.1.4. The calculation method assumes thatthe errorsin the master gearare too small toaffect the test

results.This requiresa veryaccuratemaster gear, if preci-

sion gears are to be measured.

If two gears are in tight mesh, the

AGMA 2002-B88

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