AGMA 2002-B88

ANSI/AGMA 2002--B88(Errata 1995) Reaffirmed December 2006 American National Standard ANSI/AGMA 2002--B88 Tooth Thickness Specification and Measurement Tooth Thickness Specification and Measurement ANWAGMA 2002-B!%? (Revision of AGMA 231.524975) ITables or other self-supporting sectionsmay be quotedor extractedin their entirety. Credit lines should read: Extracted from AGhJA 2002-B88, Tooth ThicknessSpecificationand Measurement, with the permission of the publisher, the American Gear ManufacturersAssociation, 1500King Street,Suite201, Alexandria, Virginia 22314.1 AGMA Standardsare subject to constantimprovement,revision or withdrawal as dictatedby experience. Any person who refers to AGh!T.ATechnicalPublicationsshouldbe surethat the publicationis the latest available from the Association on the subject matter. ABSTRACT This Standard establishesthe proceduresfor determiningtooth thicknessmeasurementsof external and internal cylindrical involute gearing. It includes equations and calculation proceduresfor the commonly used measuring methods. A specific tooth thickness measurementlimit can be establishedfrom the designthicknessor from another tooth thickness measurement.The procedurescan be enteredwith an establisheddesigntooth thickness,or with actual tooth thickness measurements.The effect of tooth geometricquality variationson tooth thicknessmeasurementsis discussed. Backlash information is provided in an appendix. Copyright Ql988 American Gear ManufacturersAssociation 1500 King Street, Suite 201 Alexandria, Viginia, 223 14 First printing. October, 1988 Second printing, with errata,July 1992 Third printing, with errata,June 1995 ISBN 1-55589-523-9 ANWAGMA ii 2002-B88 ToothThicknessSpecificationand Measurement FOREWORD nhis foreword, footnotes,andappendices,if any,are providedfor informationalpurposesonly and should not be conshued as part of ANSIIAGMA 2002-B88, Tooth Thickness Specification and Measurement.1 This Standard presentscalculation proceduresfor determiningtooth thickness measurementsof external and intemal cylindrical involute gearing.It supersedesAGMA 231.52,Znspection - Pin Measurement Tables for Involute Spur Gears. This Staudardhasbeenpreparedto consolidatepreviouslypublishedAGMA tooth thicknessinformation, to add more information on internal and helical gearsand to add detailson more measurementmethods. Previous AGMA publications havepresentedthis information in tabularform, calculatedfor 1 DP and standard tooth proportions, with adjustmentfactorsfor nonstandardconditions.This Standardis arrangedfor direct calculation of the desired results, to eliminate the intermediatecalculationstepsand interpolationpreviously required The study of tooth thicknessandbacklashproblemshasbeena major interestof geartechniciansthroughout the history of the industry. In the last fifty years, many clarifications and contributionshave beenmade by men such as Buckingham, Candee,Leming, Vogel,and Wildhaber. Their work is consolidatedhere,without further attribution. and the work of more recentcontributorsis addedwhereit improvesthe presentation. The appendicesprovide further information on reasonableallowancesfor backlashandtooth thicknessdeviation. sample calculations, and information on four uncommonmethodsof measurementspecified on some gear drawings. The treatmentof the effects of tooth profile, pitch, lead, andmnout deviationson tooth thiclmessmeasurementis new in this Standard. The information on backlashcontrol is new in an AGMA Standard.It is basedon AGMA Paperp239.14,Assured Backlash Control - The ABC System.[l] The first draft of this revision was madein February1984. This version wasapprovedby theAGMA membershipon October9,1988andasan AmericanNational Standardon October 17,1988. Suggestionsfor the improvementof this Standardwill be welcome. They should be sent to the American Gear Manufacturers Association, 1500King Street,Suite 201, Alexandria,Vi@ia, 22314. ERRATA July, 1992 The following editorial corrections have been made to ANSVAGMA 2002-B88, Tooth Thickness Specification and Measurement, (or@ally printed October 1988). Thesechanges,discoveredafter publication, have been madein the second standardprinting, as shownbelow: PAGE ITEM CHANGE Fig 3-l 10 The position of minimum and maximum backlashis shown on the specified circle, also l/2 specifiedtoleranceand l/2 specificationbandslabeledcorrectly. Fig 3-l 26 The angle Wband the assumedform diameter,Do - 4a, indicated correctly. 29 Eq 8.2 The right handbracketshouldbe at the end, with the full equationreading, f3 = arcinv pnd(tl+ t’-$-p +invf, 1 0% 8.2) N1+N2 32 TableA-l The last valuein the table, for 64 inch center,shouldread 0.058. ERRATA June, 1995 (Additional correctionmadein this printing). 29 Eq 8.2 Changedto transverseplane. $3 = arcinv [ Pdt, + t&z N, +Nz 1 + hvQs 0%. 8.2) El] Numbers in bracketsrefer to the bibliography. ANSIIAGMA ... ln 2002-B88 Tooth Thickness Specification and Measurement 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) ANSIIAGMA iv 2002-B88 ............................................ ...............2 8......... Limitations of Chordal Tooth Thickness ......... Scope ..........8 3............... Standard Pitch Diameter ..................................-........... Chordal Tooth Thickness ..... 17 6..................5 3......2 5.......................................................................6 Circular Tooth Thickness .......................................................9 1 1 Symbols and Terminology ........................ ReferenceSurfaces ................................................................ 23 23 25 7..... Span Measurement 7................................5 4..............................................................................................4 4.....................3 8................2 ......... 4.. Selection of Tooth Thickness .......... Measurement Method Effects ....... Advantages of Chordal Tooth Thickness ...... Base Tooth Thickness ..............................................................................................................................7 3.....2 3........ 5... .........................4 3.................... TotalCompositeVariation Specifying Maximum Tooth Thickness ................................................ Application 3...........1 6.............................. 9 ..... Backlash ............1 5... 13 ‘4....... Measurement Methods .... Tooth Thickness Concepts ....... Gear Geometry Calculations ............ 28 28 28 8 29 2002-B88 .......... .. Specifying Minimum Tooth Thickness ....... Calculation of Chordal Tooth Thickness .. Symbols...6 3.............1 3.....................................2 3......................... MountingSurfaces .....................................................................................................................................................................4 6... Calculation of Span Measurement .................Tooth Thickness Specification and Measurement Table of Contents Title Section Page 1................................................2 7.......... 1 2............................3 3.........3 9 11 11 11 11 12 12 13 13 Advantages of Composite Action Test Measurement Limitations of Composite Action Test Measurement Master Gears ................................3 6................... .......2 4.............. Maximum Generated Tooth Thickness ....................................1 7.1 4............4 ANWAGMA 13 14 14 14 14 15 15 5.................... 8........................................... 23 Advantages of Span Measurement ............................. Terminology and Definitions ...3 8..........1 8... 1 2.......3 4....... Limitations of Measurement by Pins .......................................... pin or BallSizes .............5 Advantages of Measurement by Pins . Limitations of Span Measurement ..... Measurement by Pins 6.................. 15 15 15 ............. ..... Definitions ........................... Effective Tooth Thickness Calculation ........................................ Calculation of Measurement by Pins ....... Backlash Calculations .......2 6.......1 2.... 17 17 18 19 21 ....................... Composite Action Test Measurement .................................... Calculation for Composite Action Test Measurement V ..... ......... ... . .... . ..... ....... . .. . _ ... 20 21 Fig Fig Fig Fig Span Measurement of Tooth Thickness .. .. .......... .. . ........ . . ........ .. .. Span Measurement of Helical Gears ... ... ......... .. ... Limits of Span Measurement in Base Tangent Plane ...... Fig 5-l Fig 5-2 Chordal Tooth Thickness Measurement by Means of a Gear Tooth Caliper .. ... ... Best Pm Size (Internal Spur Gear) ..... .......... .. ..... .. Alphabetical Table of Terms and Symbols.. ..... ...... .. ..... . ...... ... .. . . ...... Composite Action Test Measurement of Tooth Thickness .... . ...... .. 19 Fig 6-3 Fig 6-4 Pin Measurement Spur and Helical ........ ..... ..... 15 16 Addendum and Chordal Tooth Thickness Corrections ............ .................. ... ..... . ...................... ................ Limits of Span Measurement for Internal Gear ... ...... ................... 37 Bibliography .. ..... . .... ...... ..... .. ...... . .. ...... .. ......?.... .... 12 Table 6-1 Table 6-1M Standard Pin Diameters for Various Pitches in Inches ............... ... . ......?SIIAGMA vi 5 10 17 2002:B88 ........41 Tables 2 6 Table 2-l Table 2-2 Alphabetical Table of Symbols and Terms...... ..... .... by Terms .. 31 Alternate Methods of Tooth Thickness Measurement ... ....... Best Pm Size. .. .... .... .................. Wdest (External Gears) ........... ........ ........ Standard Pin Diameters in Millimeters ... ....... ... . Fig 6-l Fig 6-2 Tooth Thickness Measurement Over Pins . . ... ............. ............. Table 3-1 Other Gear Variations Included in Tooth Thickness Measurement .. ...... .... . ................. . ....... 22 22 Figures 5 Fig 2-1 Fig 2-2 Backlash ..... .. . ... 24 24 26 28 Schematic of Composite Action Test Measurement ..... . . by Symbols . .....Tooth Thickness Specification and Measurement Table of Contents (cod) Title Section Page Appendices Appendix A Appendix B Backlash and Tooth Thickness Tolerance ... ..... . 28 30 7-l 7-2 7-3 7-4 Fig 8-l Fig 8-2 A. ...... ..................... Fig 3-l Tooth Thickness . . . ... ..... Circular Tooth Thickness ....... . .. for tolerances.4M) CAUTION: The effect of tooth geometry variations on tooth thickness measurements made by different measuring methods may be significant. Definitions. to permit economical manufacture. Table 2-2 and in the text. The user should not assumethat familiar symbols can be used without a careful study of these definitions.01.01. chordal thickness or composite action test. gear tooth strength. Symbols. Symbols and terminology used in this Standard are shown in Table 2-l and Table 2-2. It is important that tooth CAUTION: thickness measurement limits be reasonable for the specified quality class of the gears. such as span over X teeth. Design Manual for Fine Pitch Gears.! sin Jr. and Abbreviations AGMA 600.3 . Standard for Metric Usage 1 2002-B88 . in smaller type. Terminology and Deftitions 2. AGMA 115. Where equations require a different format or constant for use with SI units. The designed tooth thickness is established from engineering considerations. This Standard provides guidance in the selection of reasonable tooth thickness measurement limits. wherever applicable.4) (Eq 4. 1. AGMA 112. and backlash.1968. terminology. Px = p. NOTE: The symbols.Toierantes and Measuring Methods for Unassembled Spur and Helical Genrs {Including Metric EquivnZents). The terms used. It includes equarions and procedures for the following measuring methods: (1) (2) (3) (4) Chordal Fins (wires. span.05. 2.Basic Gear Geometry is a source for the derivations and detailed explanations of the geomeuical relationships used here.05.2 Definitions. Definitions. SI (Metric) units of measure are shown in parentheses in Table 2-1. The same mathematical principles apply to gear teeth of ah sizes. This Standard does not contain tolerances on tooth thickness. When this is necessary the tooth thickness should be measured by the method specified on the drawing. conform to the following standards: ANSI Y10. These methods are often used to convert a tooth thickness specified by one method. This must be considered if close control of backlash is required. such as over pins to another more convenient method.01. See AGMA 2000-A@. Gear Nomenclature. rolls and balls) SPari Composite Action Test This Standard also establishes methods of determining tooth thickness of a gear based upon measurement limits by means of pins. The information is intended for use by the gear specifier or manufacturer in establishing values for tooth thickness measurement limits. ANSUAGMA 7T 2. @q 4. Example: This Standard assumes the designed tooth thickness is known in cases where the values for various measuring techniques are to be established. For information on fine pitch gears. Gear Classification and Inspection Handbook . a second expression is shown after the first. Terms. and with “M” included in the equation number. The methods for establishing designed tooth thickness for a given application are beyond the scope of this Standard. see AGMA 370. Symbols and Abbreviations is a source of definitions of common gear terms as used in this Standard. and definitions used in this Standard may differ from other AGMA standards. Gear Nomenclature (Geometry) Terms.8 for additional discussion of the problem. It is determined by gear geometry.1 Symbols and Terminology.Tooth Thickness Specification and Measurement Examples included are for coarse pitch gears. Reference Information . Refer to 3. Letter Symbols for Quantities Used in Mechanics of Solids AGMA 112. indented. Symbols. Scope This Standard establishes the calculation procedures for determining tooth thickness measurements of external and internal cylindrical involute gearing. 3 Eq 8.7 Eq 7.3 Eq 4.4.1 Eq 3.1 Eq 7.1 6.1.1 4.1 Eq 5.1.1 6.7 3.4 6.3 Eq 4.1 Eq 7.1.5 7.4 4.5.1.4 4.3 4.7 3.6 Eq 6. 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 OC A ac B Bmin Bf Bt C 4 Domax Ds DW D2w F L ’ best =XGiX =min M Mm mn N Nl N2 n Pnd PI PX Pb ANSIIAGMA 2 in (-4 in 64 in (-> in (-> in (-> in c-> in (=a in b@ in (mm) in h-4 in (-> in (I=4 in (mm> in (-) in (mm) in (mm) in (=4 in (=a in (=4 in (-1 in (-> in (mm) in (=> in (mm> in (=) in (mm) in (mm> in (-> mm ----in-’ in (mm) in (=I in hd Where First Used Eq 5.4 Eq 8.2 4.4.7 5.13 Eq 3.10 Eq 7.3 3.4.6M 3.1 2002-B88 .7 8.3 Eq 7.3.4 5.Tooth Thickness Specification and Measurement Table 2-l Alphabetical Table of Symbols and Terms.1 8.1 6.1 8.4 8. by Symbols Symbol a units %3X =&in D Terms 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 D’ Operating Pitch Diameter % %l %2 D.1 3.5 4.4. 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.1 3. 3 3.4.7 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 Eq Eq Eq Eq Eq Eq Eq 7.5 4. Sector of k Pitches Total Composite Variation Radial Runout of Reference Diameter Radial Runout Tolerance 3 ‘Linits Where First Used in (-> in @N in o=> in (-> in (-) in (-9 ----in @d in (-) in (mm) in (mm) in (mm) in (=I in (-> in (mm> Eq 6. 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.1 Eq 4.3 6.1 3.1 3.4.12 in b-d in (-> Eq 5.8 Eq 7.1 8.1 Eq 7.11 Eq 8.7 5.6 7.Tooth Thickness Specification and Measurement Table 2-1 (cant) Alphabetical Table of Symbols and Terms.2 Eq 6.4. Circular Maximum Transverse Generated Tooth Thickness Transverse Tooth Thickness at Best Pin Contact Diameter Transverse Tooth Thickness Tolerance Accumulated Pitch Variation.3 5.T ANSI/AGMA Terms 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 Circular Tooth Thickness 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.1 7.5 2002-B88 .7 7.3 8.4 Eq 7.3 3.3 5.1 Eq 7.4.1 Eq 4.8 Eq 6.4.1 8.5.1 3.6 5.3 4.11 8.10 3. by Symbols Symbol % %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.2 8.4 2.4 Eq 8. The normal chordal tooth thickness is the length of the chord subtending a tooth thickness arc in the normal plane.4 3. The circular tooth thickness is the length of arc between two sides of a gear tooth. Backlash.5 Eq 6. or on the line of action. A circle defined by the number of teeth and a specified module or circular pitch. Backlash. Tooth ‘Thickness.1 Eq 8. Bd.3 4. Standard Pitch Circle. B. The user should be familiar with these definitions and symbols before applying this information.1 Eq 3. under static conditions.6 Eq 6.5 6.5 5.5. or axial planes. backlash may be determined variously in the transverse. values for backlash refer to transverse operating backlash.3. 2002-B88 . 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).5. (Reference AGMA 112. Such measurements should be converted to corresponding values in the transverse plane at the operating pitch circle for general comparisons. Normal. on a specified diameter.3 Eq 4.2 6.20 6. As actually indicated by measuring devices. and either in the direction of the pitch circles. Unless otherwise defined it is taken as the transverse circular tooth thickness (see Fig 2-2). If not otherwise identified.--------a-- Where First Used Eq 6. normal.1 Eq 3. by Symbols Symbol VrW W Wbest wC.1 Eq 6.1 with the greatest allowable functional tooth thickness is in mesh with the pinion tooth having its greatest allowable functional tooth thickness.05) Tooth Thickness.7 5.V’best 4 4’ % %n 4s 4W (92 43 9 Jtb *R J’S 7 Correction to Pin Measurement for Runout Pin Diameter in the Calculation Best Pin Size Best Pin Size-Transverse Plane Transverse Pressure Angle Transverse Operating Pressure Angle Normal Profile Angle of the Equivalent Standard Rack Cutter Transverse Pressure Angle at Measuring Diameter Transverse Generating Pressure Angle Transverse Pressure Angle at Best Pin Diameter Pressure Angle at Center of Pin Operating Transverse Pressure Angle in Tight Mesh Helix Angle at a Specified Diameter Base Helix &rgle Helix Angle at Measuring Radius R Helix Angle at Standard Pitch Diameter Normal Angular Thickness The following demons are specifically used in this Standard. Tooth Thickness is the thickness of a gear tooth at a specified diameter.Tooth Thickness Specification and Measurement Table 2-l (cant) Alphabetical Table of Symbols and Terms.7 7. Circular. Tooth Thickness. Chordal.1 Eq 4. Minimum. at the tightest allowable center distance. Minimum backlash is the minimum transverse backlash at the operating pitch circle allowable when the gear tooth ANWAGMA units Terms in in in in --- ~~> (-> (-> (-) . Tooth Thickness Specification and Measurement OPERATING PITCH CIRCLES Fig 2-l Backlash 7 Fii 2-2 ANWAGMA Circular Tooth ‘Ikickness 5 2002-B88 . 7 4. Outside Diameter over/between Two Pins Bf B* B -:.1 Eq 8.4. -1 3 Db2 %l ‘best =maX =min C c”.1 Eq 7.3 Eq 4.Tooth Thickness Specification and Measurement Table 2-2 Alphabetical Table of Terms and Symbols.1.4.6M 3. for Best pin Size Diameter.4 7.3 Eq 4.1 5.7 5.4 2002-B88 .3 5. Tip of Internal Gear Di F Face width Helix Angle at Measuring Radius & Helix Angle at Specified Diameter Helix Angle at Standard pitch Diameter Helix Angle.8 Eq 7.1 6.1 3.1 8. Contact. Maximum Center Distance.4. Transverse.1.4 8. Maximum Length of Base Tangent.1 Eq 7. Best Number of Teeth to be Spanned. Minimum Center Distance. MaxWum Outside Diameter. Best Length of Base Tangent. Minimum Base Circle Base Circle Diameter of Master Gear Base Circle Diameter of Test Gear Base Tangent. Transverse. 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.1. Minimum Length of Center Distance. Normal (feeler gage) Backlash.1.7 8.5 3. Tightest Chordal Addendum Chordal Addendum Correction Factor Diameter.4.3 Eq 8.4 Eq 5.3 6.1 Eq 6.4 3.1 4.5 Eq 3.4 8.3 Eq 7.3.1 7.7 3.4 6.1 Eq 3.” c aC Aac DW DOlllXX DO D2w D Diameter. Specified Diameter.1 Eq 4.3 4. Maximum ANSIIAGMA mn N N2 n N1 S ‘best LX 6 Units 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 Where First Used Eq 5.7 Eq 7.5 4. by Terms Svmbol Terms a B Addendum Backlash (Transverse Operating) Backlash. Circular Backlash. 5.3.7 ‘b2 ‘bm in (mm) in (-> 5.4 Eq 8. of Test Gear 7 Ds P n.10 Eq 7.1 6.7 Eq 4.1 Eq 8. zx %W sW M Mm R7n RTma% RTmin 7 degrees 2002-B88 ( . Transverse Gperating Pressure Angle. Minimum (work gear) Transverse Tooth Thickness. Maximum (work gear) Test Radius. Transverse Base. Best .d P %pk $2 +3 + @W &I 9’ 4s % Units -in (-1 in (=> in (mm> in (mm> in (mm> in (mm) in (mm> in (mm) in-1(=> in in (mm) in (=I --------in @d in (mm) in (=> in (mm> in (mm> in (mm) in (=4 in (=) in (mm) in bd Where First Used Eq 7.4.1 Eq 3.5.Tooth Thickness Specification and Measurement Table 2-2 (cant) Alphabetical Table of Terns and Symbols.20 Eq 6.1 7. Modified for Tooth Variation Test Radius.1 8. Modified for Runout and pitch Variation Tooth Thickness. Transverse at Best Fin Diameter Pressure Angle. Minimum Pin Diameter in the Calculation pin Measurement Correction for Runout Pin Size. Transverse.6 5.5.2 6. Transverse at Measuring Diameter Pressure Angle.3 Eq 3. Accumulated.4 4. Transverse at Best Pin Contact Diameter Span Measurement Span Measurement. Normal. of Master Gear Tooth Thickness. Best Pin Size.Transverse Plane Pitch.2 7. Base.4.5 3. Transverse Base. of the Equivalent Standard Rack Cutter Radial Runout of Reference Diameter Radial Runout Tolerance Radius.2 7.3 6. Angular Tooth Thickness.1 v.3 6. Transverse Base. Master Gear Test Radius.3 Eq 6. by Terms Symbol Terms %i.5 4.4 Eq 7.11 Eq 6. Transverse Pitch Diameter. Gperaung in Tight Mesh Pressure Angle.1 Eq 6. Maximum Measuring Radius over/to One Pin Space Width. Transverse Generating Profile Angle.1 3.1 Eq 7.1 Eq 6. Operating W b w wbest wLest px PN pb D’ Pitch Diameter. NormaL Diametral Pitch. Transverse Pressure Angle.1 Eq 8.13 8. Standard (Generating) Pitch. Base.6 Eq 6.5 5. Operating Transverse Circular Pitch Variation.n Number of Teeth to be Spanned. Sector of k Bitches Pressure Angle at Center of pin Pressure Angle. Normal Pitch.12 ‘bl in bw 8.6 4.5. A-da1 Pitch.3 6.4. The total change in center distance whiIe the gear being tested is rotated one complete revolution during double flank composite action test. Transverse.1 Eq 3. established by the mounting (See 3. I$4 . Design tooth thickness is the thickness estabbshed from engineering consideration of strength. Tolerance Thickness. a span or tooth caliper measurement. of Gear Thickness.7 Tooth Thickness. Effective. . Transverse Maximum. Transverse.1 Eq 5. Tooth Thickness. Tooth Thickness. Base. Total accumulated pitch variation is equal to the algebraic difference between the maximum and minimum values obtained from the summation of successivevalues of pitch variation. Tooth Thickness. The effective tooth thickness is the apparent circuIar thickness at the operating pitch diameter with a mate. Tooth Thickness. Specified Minimum Thickness.8 3. mounting and backlash upon the theoretical tooth thickness. at Operating Pitch Diameter of Pinion Tooth Thickness.10.3). Circular Thickness.1 3. deflection. Transverse.1. Variation. t. t T+ The permissible amount of tooth thickness variation. at & Thickness.3 Eq 5.5 Eq 3.7 4. t t. The tooth thickness as determined by meshing with a specified gear on a caliirated composite action test fixture.1 8. Tooth Thickness. 8. Transverse Thickness. VP . Tolerance.6 Eq 5. Transverse Maximum. Tooth Thickness. Transverse. The measured tooth thickness is the actual value of circular tooth thickness caIcuIated from a specific measurement over pins.7 in bw in (-) Eq 3. Design. I$ .4. Functional. Normal Base Thickness. Tooth Thickness.11 2. The circular tooth thickness in a transverse plane. Total Composite Variation tb t h ‘nR *nb ?rn 5 t2 &in tT tt tw tR fGItU r ts &lax hWC &4 hits Where First Used in (-> in (mm) w=> in (m-d in (-> in (-1 in (mm> in (mm> in (=a in (mm> h-l (mm) in bJJ@ in (mm> in (-) in (mm) in (-) Eq 4. Measured. Total Composite Variation. Transverse Maximum Generated Thickness.3 3. NormaI. at Best Pin Contact Diameter Thickness. Total Accumulated Pitch Variation. Transverse. AIISIIAGMA 8 .3 Eq 7. at & Thickness.Tooth Thickness Specification and Measurement Table 2-2 (cant) Alphabetical Table of Terms and Smbols. of the Master Gear at +c Thickness. of the Test Gear at +c Thickness. Transverse. The circular tooth thickness in a normal plane.1 Eq 6. Transverse Thickness. The variation from a specified value of normal circular tooth thickness. Transverse. Transverse M&um. Normal.1 Eq 4.4. Normal Chordal Measured Thickness. by Terms Svmbol Terms Tooth Tooth Tooth Tooth Tooth Tooth Tooth Tooth Tooth Tooth Tooth Tooth Tooth Tooth Tooth Tooth Thickness.2 Eq 4. Normal Thickness. and is the same as total index variation. Design 3. 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). span. but.2 Measured Tooth Thickness. at the operating pitch diameter with its mating gear. in (-> 9 2002-B88 . etc. or tooth caliper). That is the basis of this Standard. depending on the magnitude of the variation where&the measurements are made. If a given gear is measured by each of these methods somewhat different results will be observed. mounting and consideration of backlash. The measured tooth thickness is used to evaluate the size of an entire tooth or all of the teeth on a given gear.1 Design Tooth Thickness. (similar to the concept of maximum material condition). similar to functional tooth thickness. variations in tooth alignment. There is no way to separate these vat&ions from the measurement for tooth thickhess. Section 8 explains this measurement method.1 Functional Tooth Thickness. 3. The methods for establishing design tooth thickness for given applications are beyond the scope of this Standard. when results are critical. The maximum effective tooth thickness is the thickness of the thickest tooth. jn (mm) 3. profile.Tooth Thicfcness Specification and Measurement tooth thickness values obtained on a composite action test (double flank) by means of a calibrated master gear. 3. Various concepts dealing with tooth thickness are discussed within this Standard. tplax. with reference to the mounting surfaces. or backlash is closely controlled. tooth thickness is usually established from engineering considerations of gear geometry. measured on the transverse plane is the thickness it would have if meshed at the tightest center distance and minimum backlash with a perfect. but.1. tooth alignment. Application 3. These differences are due to the different tooth variations that enter each measurement. transverse. it is necessary to specify the measurement method to be used. pitch. the effects of the tooth element variations may be additive or may cancel each other at various angular positions within a given mesh. The functional tooth thickness is that family of ANWAGMA p’ . It can be based on a few measurements between two points or two very short contact lines. The maximum tooth thickness of a gear.Be- Bmin = minimum backlash. In this Standard.1. at operating pitch radius. mating gear.1. maximum tooth thickness and maximum effective tooth thickness are taken as numerically identical. 3. As in the case of measured tooth thickness.4 Maximum Tooth Thickness. and pitch will affect the measured values to varying degrees. and mounting.3 Effective Tooth Thickness. The effects of these variations on the measured values may either be additive or may cancel one another. maximum tooth thickness. It is a measurement which encompassesthe effects of element variations in profile. It is customary to assume that the entire gear is characterized by the measured data from as few as one or two measurements.1 Tooth Thickness Concepts.1.. The nature and the location of these contacts is determined by the type of measurement (pins. designs it is desirable to establish the maximum effective thickness equal to the maximum design thickness.2. the following relationship must be satisfied: tGmax= tP max 0% 3-1) where ‘Gmax = maximum transverse tooth thickness of gear. (1) Design Tooth Thickness (2) Measured Tooth Thickness (3) Effective Tooth Thickness In most 3. 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.1. This Standard assumesthe design tooth thickness is known and the values for various measuring techniques are to be established. gear tooth strength. Depending upon the method of measurement. It is not possible to segregate the individual tooth element variations from the effective tooth thickness. The selection of tooth thickness is up to the designer. The differences are usually ignored. ToothThicknessSpecification and Measurement MAxiMUM EFFECTIVE -IooTHTHIcKNEss SPECIFIED MINIMUM TOOTH THICKNESS. Fii 3-l Tooth Thickness Tkansverse Plane ANSIIAGMA 10 2002-B88 . lFCENTERDISTANCEISINCREASEDBACI%UHWILLINCI=ASE.~ * THIS FIGURE IS DRAWN XI’THE POSlTtON OF TIGHTEST CENTER DISTANCE. ‘& l/Z SPECIFIEDTOIERANCE BAN&m l/2 SPECLFICATION BLWQC$~!$$. For gears of standard proportions. Gear Nomenciuture Definitions of Terms with Symbols. but it is not a requirement. V . 3. root clearance. in a variable center distance fixture. contact ratio. operating at standard center distance. See AGMA 112. 3. An individual gear does not have backlash. reference diameter. Total composite variation. The usual approach is to select a center distance. which form the basis for the calculation or measurement under discussion. It has only a tooth thickness. at the tightest allowable center distance. If the mounting surfaces are finished after the teeth are cut or inspected.2 Backlash. Appendix A covers this subject in more detail. and reference ax& are used to denote surfaces. = number of teeth in the gear = number of teeth in the pinion = tightest center distance.05. An attempt is made to keep the cutting depths of both members equal.1 is satisfied. tooth thickness measurement. in (mm) pl= 2n &( where jv n c > (Eq 3. and minimum tooth thickness specifications. and rating before it is finalized. The expressions reference surface. because they are used as the reference surfaces (tooling points) for all backlash and effective tooth thickness measurements. in (mm) (minimum center distance for external gears or the maximum center distance for internal gears) 3. assuming that they are to be cut with the same tool.Tooth Thickness Specification and Measurement tPmax lash. The amount of backlash which is appropriate for different sizes and classes of gears is discussed in Appendix A.4 Reference Surfaces. misalignments. = maximum transverse tooth thickness of pinion. These surfaces must be specified. then to vary the addendum (tip) diameters of the gears until an approximate balance of strength rating is achieved. the designer has a wide range of choices for t-. actual or hypothetical. it is common to reduce the tooth thickness of each member by one half the backlash allowance. Bmin. the set will have the specified minimum backlash. reference plane. The design is then rechecked for tip land width. bearing runouts. 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 finished gear in the gear assembly.3 Mounting Surfaces. As long as Eq 3.5 Total Composite Variation. an auxihary pair of reference surfaces (trueing registers or proof surfaces) should be specified for tooth inspection. It is not a factor limiting maximum tooth thickness. or operating at nonstandard center distances. This Standard uses the term minimum backkzsh in a carefully defined way. For example. Backlash of a meshed set of gears is governed by the center distance at which the gears are operated and the tooth thickness of each of the gears. This is sometimes referred to as the start or end of contact. and any unknowns. at operating pitch radius.2) The tightest center distance is the minimum center distance for external gears or the maximum center distance for internal gears. but it has an important effect on operating backlash. This is the traditional backlash allowance provided by the designer to allow for deflections.2 uses specified diameter in this way. Minimum back[ *] Limit diameter is the diameter on a gear at which the line of action intersects the maximum addendum circle of the mating gear. under static conditions. limit diameter *. temperature effects. For gear sets with nonstandard proportions. in (mm) = transverse circular pitch at operating (tightest) center distance. 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 thickness. 3. AGMA 2000-A88 provides tables of values for Vcq for gears of various sizes and quality numbers. is the variation in center distance when a tl$ gear is rolled in tight mesh for a complete revolution with an appropriate master gear. chordal measurement uses the outside diameter as a reference surface. The definition of circular tooth thickness in 2. ANSIIAGMA 11 2002-B88 . The manufacturing variation allowance should be a function of the manufacturing method and the actual gear quality (total composite variation and tooth thickness tolerante) . in (mm) (taken from AGMA 2000-A88. in (mm) fT In each of the following sections.6 Specifying Maximum Tooth Thickness. This Standard uses this method. Table 3-1 shows the influence of tooth geometry variations on each measurement method.Tooth Thickness Specification and Measurement each measurement method. particularly in coarse pitch gearing with large backlash allowance. the effect of profile deviation is minimized. Because it is very difficult to measure tooth thickness directly. = tooth thickness tolerance. less an allowance for manufacturing variation. in (mm) Table 3-l Other Gear Variations Included in Tooth Thickness Measurement Method of Measurement Chordal Thickness over-two-pins over-one-pm Variation Concentricity in reference to Gutside Gut-ofDiameter Ys2: Roundness X X X X Span Test Radius with master gear X X Profile pitch Tooth Alignment X’ X’ X’ X’ X X X X X W 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. f Helical gears only. the magnitude of the variations is taken as the maximum for that gear and quality class per AGMA 2000-A88 and the direction is taken so that maximum effective tooth thickness is not exceeded. and applied to the transverse plane) V cq = total composite variation v = transverse operating pressure angle d Tooth thickness is calculated in the transverse plane and specified as the appropriate value for ml D’ 3.4) = operating pitch diameter.2 yLqtanQ: ‘~=t--tT Where variations in tooth geometry or reference surface geometry influence the tooth tbickness measurement. the specified tooth thickness measurement should be adjusted for each specific method of measurement.7 Specifying Minimum Tooth Thickness. in (mm) t max = maximum transverse tooth thickness at operating pitch diameter. and the indirect methods include different effects of tooth variations. but the effects are important in fine pitch gears and where backlash is tightly controlled.3) where tnlin = minimum specified transverse tooth thickness. ml 3. the maximum 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. . usually in the normal plane. The specified minimum tooth thickness is equal to maximum tooth thickness. 3. AISUAGMA 12 2002-B88 . These differences are often ignored. 3. = where tt 3. A tight tooth thickness tolerance should not be used unless absolutely necessary. cos Jrb = 1 4. 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.7) m.05. in (mm) cos ‘PC Db = (Eq 3.t. “) A method to calculate maximum backlash from tolerances for center distance. 3.l.Tooth Thickness Specification and Measurement d=2C ml and measurement method is not critical and the most convenient method can be used. allowing a larger range of tooth thickness tolerance or operating backlash will not affect the performance or load capacity of gears. tooth thickness. increases as the quality level decreases and as the size increases. maximum backlash. In those cases where maximum backlash must be closely controlled.8 Measurement Method Effects. tt . cos & (Eq 3. maxim= backlash does not affect the function or smoothness of transmission motion. increases with size and lower quality. Values are tabulated in AGMA 2000-A88.9 Selection of Tooth Thickness. tightest center distance base circle diameter. and measurement methods must be carefully specified. degrees ‘b ‘b = arc sin ‘i$= a-f @q 3. the selection of tooth thickness [*] t. Usually. Gear Geometry Calculations The tooth thickness tolerance (allowance for tool wear or adjustment of the cutting machine) is a function of pitch and quality number. tn= tt = helix angle at the specified diameter 112. and total composite variation is included in Appendix A. At any specified diameter at or above the base circle: t. and tooth thickness tolerance. since it has a strong influence on manufacturing cost.6) cos Qb Db = N m. (Eq 3. B a. 3-5) In many applications. or in the plane normal to the helix angle at the reference circle (normal plane). and effective tooth thickness variation is not the main consideration in the selection of gear quality. a careful study of these factors must be made and the gear quality class. see 9.. since total composite variation. Jr For complete discussion. 4. Tooth thickness calculations are usually made in the transverse plane and tooth thickness measurements are made in the normal plane. It may be necessary to speci@ a higher quality class to hold maximum backlash within the desired limits. &.1) = tooth thickness in transverse plane = tooth thickness in normal plane For spur gears. The magnitude and direction of adjustments for variations in tooth or reference surface geometry are taken so that tooth thickness is decreased by inaccuracies inherent in each measurement method. t =..1 Circular Tooth Thickness. and may allow more economical manufacturing. Circular tooth thickness may be specified in the plane of rotation. degrees [ *] = base helix angle. Bmax. For any given value of minimum backlash. (transverse plane).01 of AGMA ANSIIAGMA tt cos Jr 0% 4.7M) For spur gears.6M) cos$ where Pnd = normal diametral pitch mn = normal module +c = normal profile angle of the equivalent standard rack cutter. Under these conditions. center distance tolerance. 13 2002-B88 . ANSI/AGMA 14 2002-B88 .the eHedivetooththickmsisalsothemaximnmmatedial cwditiontoorhthiclrnessatthegewratingdiameters = helixangleatslimdardpiti diameter % =wb action.Backlashisspecified in the transverse plane. = arctan X where D = Px = axial pitch [*I. with Pndandpx give= Fur a detailedtext on gv seeAGMA115.4 is usedas the basicdimensioaIfthegezus~atstandardcenterdistance. [***I The diswssion of shortpitch cnltersis beyondthe scopeof this Standard.inanassembledgearsetistheckamnceorplaybetweentheteethof themeshinggears.in (mm) 4.is: Forspnrgears. Asmndardpitchdiametts = ter(generatingpitChdiameter).inv $s] (3%4. I@~~ Sheet-Basic Gear Geum~.B.Itistheamountbywhichthewidthof ~v#‘+~v #$I (Eq4.10) [*I This calculationis basedon standardgearhobbingpractke.4w ?& = .D.is0w~nlatedaC- cordingtothestand2u. 0% 4. in (mm) the speci&ddiameter.05 for morei&rmation.Bf.InthisStandard. [**I See 8.cos~=l Forexuxnalgears: .16 of AGMA ll2. Backlashmaybemeasnredintheuansvtxsep~e.4) where lu. t.1.7) Db coq) ameterdeteminedin 3.dpitchofthegear~gtooL[**] N ‘nd =% ~2 ‘d Ds[(+) + hv #‘. #s = ec Forintemalgears: tts= DsK+9- 43 BaWashcalrnlations Ba&iash. The memnmgeneratedtooththickn~(tooththicknessat thesmndasdpitchdiameter).. where Bt = cilcdartransversebac~iIl(mm) biad&shmeaslnednormaltotootil Bf = snrface (feelergage)in the planeof Zm sin Bf Bt = = nansvedsediametipitch = = tmfimttmaansversegewratingtooth thickn~in (mm) = tramesegen~gpressareangle arc*(-) sin d$ 0% 4. in (mm) P*=$- ml431 = lead of gearor machineguide L cow When the helix angleat the standardpitch diameterisgivexx Px = P& 1z sin 9s -IL % - (Eq 4.Tooth ThicknessSpeciii&on and Measurement q thetoothspaceexceedsthe tooththickns of theengaging tooth on the opesathgpitch circles (seeFig 2-l).4 EfRctiveToothThicknesCal~n..asmeasmedbyafeelergage..9) ws *b Fmspmgears.# 42 StandardPitchDhune~.~the mnmal planealongthe operatingpitch cylinderor normal to the toothsurihcein the planeof action.8) where mi4.6) Its % % pd Themaxi- (maximnmgetkemedtooth thidmss).[***l % = aTcsk(y) Ds= m4. = arCti( > PXtld (445) 45 Maximum (&mated Tooth Thickness.01(Rl988). is: For external gears: or (Eq 4.12) 5.6 Base Tooth Thickness.2) (Eq 5.2M) = addendum.3 Calculation of Chordal Tooth Thickness. This puts the point of measurement at approximately half the working depth. rather than on the flats. u =0. The maximum expected reference radius. they should be used. it may be assumed to be equal to the allowable runout of the teeth. in (mm) 2002-B88 . = transverse tooth thickness at base circle. even if the gear has a nonstandard nominal outside diameter.8 The theoretical addendum.Tooth Thickness Specification and Measurement 5. in (mm) For internal gears: or (Eq 4.1 Advantages of Chordal Tooth Thickness. The vernier gear tooth caliper. The base tooth thickness. Its portability and its simplicity are its principal advantages. If the actual oufside diameter and the runout of the outside diameter at the point of measurement are known. For coarse pitches and small numbers of teeth. is affected by variations in the outside diameter. used in subsequent calculations. The tooth caliper requires an experienced operator.8 mn where a The tooth thickness caliper cannot be used for internal gears. equal to half the maximum outside diameter plus half the allowable xunout. to minimize the effect of profile deviation. The addendum bar setting is usually based on a standard addendum. Chordal Tooth Thickness 5. If the nmout of the outside diameter is not known. because the anvils make contact with the tooth flank on their comers. Instruments are not available for very large or very small teeth. is the basis for calculation.11) tb = q) [[$$)+hvd] where t. Fig 5-1.2 Limitations of Chordal Tooth Thickness. the addendum must be corrected and the chordal tbkkness must be calculated (see Fig u=1 for full depth teeth pnd Q=m 5-2). o. is a portused to measure the able hand held Went thickness of external gear teeth. for stub teeth pnd Q = 0.001 in (25 w). The instrument is hard to read consistently with a deviation less than 0. ANSIIAGMA n 15 @q 5-l) (Eq 5lM) 0% 5. taper and runout of the blank since the outside diameter is used as a reference surface for the caliper. 4. Fig 5-l Chordal Tooth Thickness Measurement by Means of a Gear Tooth Caliper 5. (may be assumed to be equal to allowable runout of the gear teeth per AGh4A 2000-A88). the chordal thickness measurement. u c. it is significant for coarse pitches and low numbers of teeth. 53..7) COS 6) 5.4) For spur gears. cos(2Da)] 0% 5-5) &= + (cos2 4fR> 2 Rmax > (Eq 5n 16 = measured normal chordal tooth thickness. the AhSIlAGh4A .2 Chordal Correction. Although this difference is frequently ignored. instead of the assumed value.. in (mm) Prnaz +vr)-a R-= where Domax VT (Eq 5.9) 5. in (mm) t m= 2R- (COSJT R ) sin where If the maximum runout of the outside diameter to the mounting diameter is specified. =C ac m Corrections specified value of runout should be used. total indicator movement. a NORMAL PLANE Fig 5-2 Addendum and Chordal m ‘nR = tR cos *R *R = maximum normal tooth thickness. outside diameter size. Since the tooth caliper measures on a straight chordal line. is slightly less than the distance along the arc of the reference circle.3) where *nR Tooth Thickness =a+Aac = or (+)-R- 0% 5. in (=) = helix angle at measuring radius.Tooth Thickness Specification and Measurement CHORDAL ADDENDUM ADDENDUM. a correction.1 Addendum Correction. [k -~v(~c nxz3ximut-nmeasuring radius. cos eR = 1 where 7 = normal angular thickness fR = transverse tooth thickness at R. To calculate the chordal addendum. must be made for the height of the chord spanned by the tooth caliper. and gear tooth runout must be carefully controlled when tooth thickness is controlled by tooth caliper measureI ment.3. t. 5. outside diameter runout. in (mm) 2002-B88 .. radians 5-8) Since they have such a large influence on gear tooth thickness measurement. (Es 54 = maximum outside diameter = radial runout of reference diameter. in (mm> 7 -= 2 tR=2R. Aczc . 3. makes it easy to detect small changes in tooth thickness. Internal helicals cannot be measured with pins and are usually measured with balls. The measurement over rolls should Measurements over one pin or ball. 6.2 Limitations of Measurement by Pins. the tooth will appear to be thicker than it is. Both are difficult setups.3 Specifying Chordal Tooth Thickness If the outside diameter of the Measurement.0001.. if their outside diameters are less than the maximum. or rack teeth. Measurements are not affected by outside diameter deviation or by runout of the outside diameter. 6. from the mounting diameter. To avoid rejecting gears which are at the minimum tooth thickness.Pins and balls form point contacts on helical gears. This is a common method of tooth thickness inspection. Measurements are affected by deviations in pitch and profile. External helical gears with odd numbers of teeth should be measured with balls or with three pins between parallel planes. Balls must be held exactly in the plane of rotation.Pins on spur gears form line contacts .Balls on spur gears form point contacts . whiie the Jpecial calibrated rack-tooth wedges may not be at hand. gear is under the maxim= size. a pair of calibrated wedges. ANSIIAGMA 17 2002-B88 . To avoid accepting gears which are thicker than tmax. Therefore. Pins or balls afford a method of measuring tooth thickness of gears of any diameter within the capacity of available micrometers (see Fig 6-l). because of the limited contact. The following should be noted: . make a much more reliable measurement for tooth ihickness than do rolls. the fact that the measurement over a pin is a function of t/tan +. show the effects of runout in the gear teeth and approximate the measurement of functional tooth thickness. Measurement by Pins 6. it is also necessaryto calculate the minimum chordal tooth thickness from the maximum outside diameter. by Earle Buckingham [2] Measurements over rolls on helical gears are very diff?cuIt to make with any great degree of accuracy unless definite precautions are taken. If tight control of tooth thickness is required. The following is quoted from Analytical Mechanics of Gears. deflection. The amplifying effect of pin or ball measurement. the maximum chordal tooth thickness must be calculated from the maximum outside diameter. the size and concentricity of the outside diameter must also be tightly controlled.Tooth Thickness Specification and Measurement 5.1 Advantages of Measurement by Pins.e. the variation of the measurement among several operators may exceed 0. i. In many cases. However rolLs are often available when needed. Micrometers are often used to measure the [23 Xumbers in brackets refer to the bibliography. This procedure wiIl allow some thin gears to be accepted. can cause variation in readings and will vary with gaging pressure. Fig 6-l Tooth Thickness Measurement overPbls dimension over pins.001. a difficult task. Even though the micrometers may be graduated to 0. but here the two balls must be definitely aligned in respect to the face of the gear blank. wire or ball”. “Pin” is used in this text for ‘pin. is probably closer to the functional tooth thickness. the measurement is made across the high points of two properly sired pins placed in diametrically opposed tooth spaces. the measurement technique is similar to that on spur gears. OdS. there are measurement setups using pins or balls for which there are cor[ *] To approximate the maximum functional tooth thickness. One method uses three pins instead of two. and the largest used. Ball-point micrometers may be used. For all types of spur and helical gears. and the two bail points may be held against this same surface plate. or diametrically opposite to each other. Large micrometers are required for large gears. as previously stated. which is best measured by double flank testing. when combined with the radius of the reference surface corresponds to the calculated radius over one pin. When properly positioned for measurement.Tooth Thickness Specification and Measurement responding geometrically exact calculation meth- be made between parallel flat surfaces and not with a micrometer alone. The other method uses a single pin and is suitable for any external helical gear. The third pin is placed alongside one of the others so that the pair will be diametrically opposite the single pin. For external spur gears with even numbers of teeth. With odd numbers of teeth.?.3 Measurement Methods. and any measured values so obtained are useless for any purpose. For example. A.?SI/AGMA 18 2002-B88 . the gear blank may be laid flat on a surface plate. When the rolls are held in position on the gear by two parallels. but is limited to gears whose face widths are greater than their axial pitches. 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. 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. If an attempt is made to measure odd numbers of teeth over the rolls directly with a micrometer. is positioned to contact the single pin at a location halfway along the axial pitch distance. when odd numbers of teeth are involved. Where balls are used. repeated measurements must be made. Although the two pins are not parallel. The measurement over the single pin is made relative to the gear center line or relative to the bore or other concentric cylindrical reference surface on the gear. 6. For helical gears with odd numbers of teeth. the tooth spaces used are those nearest to diamenically opposed. Multiple readings taken around the gear should be averaged to find the mean. Measurement over pins can also be performed on medium and small external helical gears. the two rolls will be on opposite sides of the gear. In the case of spur gears with odd numbers of teeth. whatever the face width and whether the number of teeth is odd or even. Measurements made over two pins or balls do not show functional tooth thickness. all three pins will be in line contact with their respective parallel anvil surfaces. one or both rolls will be tipped away from the correct plane of measurement. whether the number of teeth in the gear is odd or even. The mean value should be used in comparison of readings. there are two techniques with geometrically exact calculation methods. The wide anvil is positioned to span the axial pitch distance between the two adjacent pins and the second anvil. it is possible to position the anvils on a conventional micrometer so as to measure at diametrically opposite points. When the gear has an even number of teeth. This method also requires the use of a micrometer or other measuring instrument with an anti of size greater than the axial pitch. The measurement is twice the calculated radius over one pm. the calcuiation 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. representing either. The calculations are made using a disc of infinitesimal thickness. The maximum reading. which need not be as wide. This measurement. 3 The arc space width at tbis diameter is: SW= To calculate the best pin size for external gears. 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. The arc tooth thickness at this diameter is: tw= Dw[i$)-Ww] It would extend above the outside diameter of an external gear or below the inside diameter of an internal gear. in (mm) +W = pressure angle at best pin diameter DW tW = transverse tooth thickness at best pin diameter. ( > ‘rrDW 7 -tw 0% 6.est= best pin sire . In the case of helical gears with odd numbers of teeth. where Do = contact diameter for best pin sire.4) The best disc diameter is: (Eq 6.4 Pin or Ball Sizes. in (mm) Fig 6-2 Best pin Size.Tooth Thickness Specification and Measurement The best disc diameter would contact the tooth profile at DW All two pin measurements can be made with the pins replaced by balls of the same size. 0%6. This minimizes the effect of profile deviation. in (mm) SECTION IN TRANSVERSE PLANE Wt. ANSIIAGMA For internal gears (see Fig 6-4): DW=Di+ 19 2a 0% 6-7) 2002-B88 . The ideal (best) sire pin or ball would contact the tooth profiles at half their working depth. w’ best (External Gears) This calculation is based on a disc of infinitesimal thickness in the transverse plane (see Fig 6-3). in (=> = outside diameter.5) This value must be converted to the normal plane by rotation in the plane tangent to the base circle through the center of the disc (see Fig 6-3). For external gears: DW= Do-2a (Eq 6-1) The pressure angle at this diameter is: 6. and would not touch the root of the tooth space. in (-> wbee best pin sire. in (mm) sW = transverse space width at best ‘pin diameter.transverse plane. This measurement is the same as that for spur gears with odd numbers of teeth. it is also possible to measure over two balls. see Fig 6-2. <? _..” . .<.\...<.Tooth Thickness Specification and Measurement EXTERNAL INTERNAL I .: TOOTH CENTERLINE q W qJ cow* :..g ..’.c ..\G‘ )’ &f’ 8’ I .<:+‘.a k LJ I TRANSVERSE PLANE Fii 6-3 Pin Measurement Spur and Helical ANSIIAGMA 20 2002-B88 .#’ ..$j I t ’ . it may not be available.1 Radius Over One Pin.6. Bin diameters have been standardized so that sets of pins for common pitches can be used. plus or minus the pm diameter. to the root radius. ANWAGMA 21 2002-B88 . and that it will prouude past the tip circle of the gear. in (mm) Conversely %=2-[A&] c 1 % = Db in~+~+~ -- (Eq 6. however.5 Calculation for Measurement by Pins. W+Jem must be converted to a ball diameter.5. Dimensions over or between pins can be calculated for any pm diameter.5. Further information on standard pin sizes can be found in manufacturer’s catalogs. This is done after the specified radius measurement over pins is calculated per Eq. [33 6. The pin size specified should be the next largest selected from Table 6-1.Tooth Thickness Specification and Measurement space.[($)+ww] where pb = transverse base pitch 0% 6. For esernal gears: fa_ mv+2= Db + W Db cos lip P Fig 6-4 Best Pin Size (Internal spur Gear) [‘I (Eq 6-N Db where Di = tip diameter of internal gear tw=D. 6. Figure 6-3 shows the general case for external and internal gears. in (mm) = pin diameter. or from a table of pins known to be available to the gear manufacturer.8) R 1w =A 2coE42 +tv 2 (Eq 6. P(best9 using Eq 6.11) where 0% 6-9) $2 RIW W For internal helical gears.12) W cos Jr b (Eq 6. and the pin measurement radius. This is the theoretical best pin size. by comparing the pm measurement to the tip radius. Table 6-l Iisrs some commonly used pm sizes. The size selected should be checked to be sure that it will not bottom in the root of the tooth [ *] = pressure angle at center of pm = radius over or to one pin.13) The inverse function is usually found by iteration. The final pin diameter selected is labeled W in the calCUMOIIS. 6. 576 1.280 0.25 3.768 0.864 0.210 0.320 0.25 4.240 9 10 11 12 14 0.5 8 9 10 10.5 6 6.216 0.280 0.640 0.25 2.160 0.192 0.75 3 3.152 0.560 1.1920 0.140 0.75 4 4.420 0.432 0.12343 0.108 0.240 0.960 0.5 3.18666 0.120 0.680 1.840 0.168 0.144 0.288 0.728 1.13714 16 18 0.17454 0.480 0.672 0.120 0.384 0.5 7 7.25 Abstracted ANSIIAGMA 5.336 0.09333 0.6912 0.10667 1 1 l/2 2 2 l/2 3 Table 6-1M Standard Pin Diameters in Millimeters 2 2.096 0.105 0.5 11 12 14 15 16 18 20 22 25 28 30 35 40 45 50 from DIN 3977[4] 22 2002-B88 .728 w= Pnd Pitch For Long-Addendum PilliOnS w= - 1.21333 0.27428 0.Tooth Thickness Specification and Measurement Table 6-l Standard Pin Diameters for Various Pitches in Inches Diametral Pnd For Standard Internal Gears 1.920 1.5 2.120 0.1728 0.15709 0.680 w= Pnd For Standard External Gears 1.24686 0.920 Pnd 1.5 5 5.3456 0.16273 0. the effects of runout are included and no correction is necessary. The effect of allowable pitch deviation is much smaller than allowable nmout. This method is particularly useful for large gears.rT 2 3 W Db RIW =~COS+ 2 -2 (Eq 6. Span Measurement where 49 = = correction to pin measurement for runout. If the pins make contact near the mid-point of the active profile.5. from AGIvLA 2000-A88.16.5. For internal spur gears the measurement is made between teeth. S. For external gears with odd numbers of teeth: D2w (Eq 6. No unusual skill is required to make the measurement. Measurements are not affected by outside diameter deviations or by runout of the outside diameter (see Figs 7-l and 7-2). and the distance measured is (S+l) normal base pitches minus one normal base tooth thickness. D2w = dimension over or between two pins.Tooth Thickness Specification and Measurement ignored. in (mm) 7. so it can be ANSJIAGMA = allowable runout of gear teeth. except with very low numbers of teeth and other unusual cases.5. For external gears.17) 7. Span measurement cannot be applied when a combination of high he& angle and narrow face width For internal gears with odd numbers of teeth: A‘L cos Tr .2 Radius To One Pin.1). If D2 w is known from measurements: arc cos (Eq 6.5.5 Correction for Tooth Deviations. over a number of teeth.Ob cos +2 cos = + w ( 2N > 0% 6. M.3 Dimension Over Two Pins.20) 23 2002-B88 .4 for further information on helical gears.4 Dimension Between Two Pins.w ( 2N > D2w = cos + 2 0% 6-W 6. 6.16) = . ‘b .18) 6. in (=> Equation 6. For internal gears with even numbers of teeth see Eq. along a line tangent to the base cylinder. If the measurements are made with two pins. For internal gears: inv+ ‘b Db 2--- N 2 If pin measurements are made as a radius to one pin from the mounting diameter. the effects of nmout should be calculated and D2 w adjusted accordingly. deviation effects are minimized. the distance measured is the sum of (S-l) normal base pitches. The measurement can sometimes be made without stopping the gear cutting machine. plus the normal thickness of one tooth at the base cylinder. 6.1 Advantages of Span Measurement. See 6.17 applies to helical gears with odd numbers of teeth when measured over balls. in (mm) The direction of the correction reduces the allowable tooth thickness (see 3. For external gears with even numbers of teeth: = 2R1W D2w VrT 0% 6.‘b = (Eq 6.2 Limitations of Span Measurement.14) The amount of correction is: V Vrw = . because it does not require auxiliary balls or pins and a smaller micrometer or caliper can be used. This method utilizes a vernier caliper or a disc micrometer to measure the distance.15) where vrW 6. which is similar to measuring a diameter. 7. / .+ ’ .$”.... . The effects of profile deviation are greatly reduced if the measurement is made at half the working height of the teeth. Readings are influenced by deviations in base pitch.$ \ \$ ..~.G. \ Fig 7-l .<. ../I.*p <~f>$. and lead.. This can be overcome to some extent by making measurements on the machine when gears are stacked in cutting.\.Tooth Thickness Specification and Measurement prevent the caliper from spanning a sufficient number of teeth./ v \ \\ / .P.L i . . or by using a disc micrometer.. . accumulated pitch over S teeth.. ‘-.: .r\.S .$Pr \“!. tooth profile./ \ Span Measurement of Tooth Thickness RMAL PLANE BASE TANGENT PLANE BASE TANGENT BASE CIRCLE xx- Fig 7-2 Span Measurement of Helical bears ANWAGMA 24 2002-B88 . the face width is too narrow for span measurement at this helix angle.Tooth Thickness Specification and Measurement or. i L kin =integer portion of p%p S-= F L-= Lmax= #D/ =integer %X3X portion of rTjFtnb maximum number of teeth to be spanned = face width. If smax is not greater than or equal to Smin.). Span measurement cannot be used for internal helical gears or for pitches which are too fine for the anvils of the measuring instrument to enter the tooth space. so it does not measure functional tooth thickness.I ‘nb = $ cos $. of the ends ‘of the teeth. (Eq7.-bl sin%-b It does not have the amplifying effect of pin measurement.25/P& (0. rounding up will place the caliper contact above half the working height. can be a range. in (mm) maximum length of base tangent PN s = normal base pitch. + 1 I plane. integer portion of Readings are erroneous if attempted on a portion of the tooth flank which has been modified from true involute form. The best span measurement.4a)‘- n [~-(~. and rounding down will place the caliper contact below half the working height.3 Cakulation of Span Measurement.7) to the nearest integer. S is also limited by the limit diameter of the gear.3) ?)2-D. 7. -t b 25 2002-B88 . in (mm) = number of teeth to be spanned %b = nornial base tooth thickness. bounded by the outside diameter and the face width of the gear (see Fig 7-3). When rotding Shea (Eq. Lmin = d (D. The range of S is limited by the size of the plane which is tangent to the base cylinder. For gears with exrra tip relief or extra fillet clearance rounding may be favored one way or the other.25 mn>. if helical. in (mm) 0% 7. Db2 77 [ 7.125 m. [+k-p~] sin I& Span measurement does not show the effect of runout.2) S&Z _ ‘nb whichever is least- The following calculation also limits the contact between the measuring instrument and the gear so that no contact occurs within 0.4) ANSI/AGMA minimum number of teeth to be spanned minimum length of base tangent 0% 7-l) + (Eq 7. of the outside diameter or 0. in (-1 + .3M) 1] (Eq 7.n bd 2 =-= JiygZ pN where Se= 03 7.125/P& (0. i. is made where the base tangent plane intersects the teeth at approximately half their working height. S. Xumber of teeth to be spanned for eternal gears. which must be greater than or equal to 2. . Sbest .5M) plane. l>pb 1 -- 4pnd - 1 gp.Tooth Thichess S@ficaion and Measurement *A-& 1-c ) 3 3 2 (s.d f BASETANGENTPLAIQ ’ 1 D-0 4Pnd TRmwERSEPLANE Fig 7-3 Limits of Span Measurement in Base Tangent Plane ANSUAGMA 26 2002-B88 . <=S.10) Tbelimitdiameterisapproximateby where Le= = spanmeasurement.05179) L. S-= (Eq 7. in (mm) nansversebasetooththickn~ modifiedforrunoutaudpitch deviatio&in (mm) [email protected] (mm) uausvmepressmeangleat measuringdiameter >(Eq-17. Numberofteethtobespmedforintemalspur gears(seeFig 7-4). + t&b~b 0% 7.11) ThisvalueofMistheoreticaLItshouldbewrreued fortbeeffectoftoothgeometrydeviaiionsaudrunoutby decrea&gthebasetoothtJ&Amess.= M =[(Sbest- l)Pb+ fpwb (Di +4aj2- (Eq 7..12. best length of basetangent bestmlmberofteethtobespduned 6S.8) Lw= arc cm [ Do:zn Lh +t& pb ‘bm = %pk = Mm = #Jrn = >@-1l7W1 * +‘& -1 pb = RadialRunout Toleranceof gear teeth. VrT ]approximatelY 0% 7.17) (Di +2a)2- integerportionof (Eq 7.19) 1 l$Srnin$Sbp&S- 11 0% 721) (Eq7J2) calcnlatespm M =(Sbest+1 )pb-tb 0% 723) Toccmectthisthecmxkalspanfortooth g~metrydeviations..Tooth ThicknessSpecifk&on and Measurement +I$ Lbest= -2a)‘- For externalgears Db” em= = roimdedvalueof %est [~&-Q em = arccosI: DiF2a sr& The effectsof leadandprofile deviationshaveheen ignoredsincetheyareusuallymuchsmallerthauruuout andpitch. see AppendixEof AGMA 2000-A88. tbm mustbe calculated per Eq 7. in (mm) spanmeasure~modi6edfor tooth deviations.fiom AGMAZOOCU88.y&k cos #rn Di+4a dn L-z 0% 7.Thevalueofallowabletolerancesfor eachgear size and quality numberis obtainedfrom AGMA 2000-A88.12) -h$.in (mm) M x Ob Fb=N = cosqb *bm = fb . Mm = <&t +I& .14) Forintemalgeats +J 0% 7.tbm ‘@I7241 [*I valnesof kppkare not givenin AGUA 200&A88.sectorof k pitches[*J. ANSIIAGMA 27 24K%B88 .13) & ct = roundedvalueof %est K where (Eq 7.18) Db” + fh pb hill= integerportionof K tan #m . andare a designersoption (seeAppendixA).v..15) where sm= ] approximately ml 7.16) Dl (Eq 7. . Carefuldesigntouseexistingtoolingcansavesomeof this expense. since testing machinescapable of more than twenty inch centerdistanceare rarely available. Appendix A and AGMA 2NN438 for a detaileddescriptcm. withoutinterference. Fii 8-l Schematicof CompositeAction Test Measurement ANSIIAGMA 28 2002-B88 . Wherethesizeoftheworkpermitsandthetooling canbejnstii5e43acompositeactiontes& testmdiusmezsuremen&isthebestmethodto inspecttooththiclmess.In specialcircmnstancestestingcanbeaccomplishedinplace on the cuttingmache. SpeciaIattemionmustbepaidtomountingsmfaces. Refer to AGIHA 2000for detaikd calibrationins&uctions. Thepropordonsofthemastergearmustbechecked forpropermtxhingwiththeworkgeartobesurethat co~~placeneartothetipand~einvolnteform diametm. Small lot producersencounter@@cant tooling cosrsinnsingthetestradiusmeth~sincespecial [email protected] Adwmlages of Come Action Test MeasnreiiUlCti0WIltOOththiClUl~. toassure&atthetestperformedisreprexntativeofthe gearasitwillbehstalled. 83 MasterGears Mastergeatssuitableforchecking mostspurgearsateavailableinsizesandtoothproportions standardized by their manuktmers (seeAGMA 2OO&A88).This is much f&sterthanmaEngmnhiplemeasurem entswithtbeorher methods.See Fig S-1. Compositeaction test measurementinspea every tooth of the wo& gear in one opedation. CompositeAction Test Measurement 8.2 Limitatious of CompositeAction Test Measurement. ThismethodislimitedtomediumandsmaUer gears. 8. Specialmachinesoratlachmentsarerequiredforintemal gears. Testmacbksmustbecarefnlly&bratf&particulady for fine pitch and high quality gears.Tooth ThicknessSpecificationand Measurement TRANSVERSEPLANE Fig 74 Limits of Span Measurementfor Internal Gear 8. me& Thismedxximeasures sinceit iuchxiesthe effectsof all tooth deviations.Thetooththicknessofrhesemastergearsis madeequalorclosetoonehalfofthecircnkpitchatthe standardpitchdiam~. 8. from which the operating transverse pressure angle can be found.4) R T max = C max − R m where Rm = master gear test radius.Tooth Thickness Specification and Measurement Master gears are usually marked with a test radius which is the radius at which they would mesh with a standard mating gear having a tooth thickness at DS of (π DS/2N). φ3  φ 3 = arc inv Special master gears are often required for spur gears with nonstandard proportions. ANSI/AGMA 29 2002--B88 . in (mm) The maximum test radius. The tolerance band for composite action test or test center distance must allow the full deviation of the total composite tolerance plus the tooth thickness tolerance.1 and Cmin. D b1 N 1 + N 2 N1 2 cos φ 3 (Eq. This requires a very accurate master gear.3) where Cmax = maximum center distance. in (mm) N1 = number of teeth in the test gear N2 = number of teeth in the master gear. in (mm) (Eq. the sum of their tooth thicknesses on their operating pitch circles is equal to the circular pitch on that circle. 8. Accuracy requirements for master gears are included in AGMA 2000--A88.2) Pd C max = If two gears are in tight mesh. The “trace for maximum gear” represents a gear which has a tooth at the maximum effective thickness.4. (Eq.8. with the fundamental tooth thickness equations. The calculation method assumes that the errors in the master gear are too small to affect the test results. RT max. Master gears must be made very accurately since any deviation in the master gear is added. in (mm) tT [*] 2 tan φ 3 (Eq. Both components vary with the test gear size and quality. If greater accuracy is required.8. φ 3 = arc inv = transverse operating pressure angle in tight mesh = maximum transverse base tooth thickness of test gear. yield a series of simultaneous equations. in (mm) 8. to the deviations in the work gear.1.6) ________________ [*] The use of φ3 for the minimum pressure angle is an approximation.2 Minimum Test Radius.4. in (mm) t2 = transverse tooth thickness of the master gear at φs . b2 b1 b b2 tb2 =transverse standard diametral pitch t1 = maximum transverse tooth thickness of the test gear at φs .1) where tb1 (Eq. 8. These relationships.4 Calculation for Composite Action Test Measurement. in the test results.8.8. Figure 8--2 illustrates a typical composite action test chart. the operating pitch diameters of the two gears must be in proportion to the numbers of teeth. iterating for a final value.1 Maximum Test Radius.8. if precision gears are to be measured. The maximum test radius is based on the maximum effective tooth thickness as defined in 3. b1  All measurements are in the transverse plane 8. The following method applies to external gears. AGMA 2000--A88 provides appropriate values. recalculate. is: (Eq. Also. in (mm) Cmin = minimum center distance R T min = C min − R m Db2 = base circle diameter of master gear. using Eq.5) where Db1 = base circle diameter of test gear. tmax.4. in (mm) C min = C max − V cq − = transverse base tooth thickness of master gear. t D+ +t D− p  P d t 1 + t 2 − π + inv φ s N1 + N2 where Helical gears usually require special master gears. .:.~:.::.Tooth Thickness Specification and Measurement INCREASING u C” .: ~:~lJNCREASING ” f n ONE EVOLUTION iMORK GEAR Fig 8-2 Composite Action Test Measurement of Tooth Thickness ANSIlAGMA 30 2002-B88 .::~~~:i::::j:i.:.:. videfolz Anindividualgeardoesnothave~ithas (1) defkctionsof housings. in an assembledgearsetis A4.andaminimnm attheminimumcen~dallowing~thedesired requimmentcalculatedJudgmentand experiencxate minimum ba&bsh.with staudardtip diameters.sincethemant&ict5rerusuallymaksa reductionin tooththicknm to allow for backlash.MMmumbacklash.MaXimUmTOOthThi&lIeSs.B. nation of lubricantand swelling of nonmetaUicgear TheamoImtofbacJda&reqireddependsonthe size of the gears. suchas allowancefor comamimostadveneopemtiugcondition. describedinDlN3967 El. ANSVAGMA 31 2002-B88 . etc.This is the traditional butitiscalc&edandspe&iedintheuansvetseplane ‘ %ackhh allowance”providedby the designerto proorindleplaneofaction. A3. [l] Numbersin bracketsrefer to the bibliography.Tooth ThicknessSpeciCcationand Measurement Appendix A Backlash and Tooth Thickness Tolerance Cl] m Appenaa is not apart of ANSXJAGMA2OEI-B88.Backkhinameshisgovemed (2) misalignmentsof gear axes dne to housing bythecenterdismnceatwhichthegearsareopemtedand deviatiousandbearhgcthe effectivetooththicknessof eachof the gears.Bmin.MinimnmBacklash.ifthe thicknessofagearisdetenk&inzcordancewithEq fbors listed aboveare conuolled.center temperature.] For gearsintendedto operateat standardcenter distance.ThisAppendixprovidesarationalmeth~ to selectgeartooth thicknesstolerancesand miuimnm backk&Italsoprovidesamethodtocalcnlate maximnmexpectedbac~inagearmesh. mollllting and the The Europan treatmentof back&h allowanceis application. Backlashmay be measmedinthenorrnalplanearalongthelineofaction.mder staticconditions. when it can be (7) centrifugalgrowth of rofating elements measm&toassuresnflicientkklashunderloadatthe (8) otherbctors.the theoreticalor basictooththicknessiscustomarilyequaltoonehalfthe circdar pitch on the standardpitch circle.but is includedfor informationputposesonly. Tooth thicknessvariationsrednce requidtoassesstheminimum~reqoiremem.MaXimlImkWth Thevalueofmhlimmnbacklashcanbesmall.tooth thickna tolerances. effedivetooththichessatthetigh~allowablecenta d&tame. AZ.is thecl~cebetweentheteethofthemeshinggears.load.asifthegearwereinmeshwithaperktmatinggear e&uue&byanalyzingthetolerances.Backlash. sincetheworstcasetoledzmcesarenotlikelytocoincide. Unless 0therwisespecifieQtheactnalmaximumtooththickness onanunassembledgearwillnsuzdlybelessthanthe Worekalvahz. Backlash.shaftsand bearings onlyatooththickness. (3) skewof gear axesdne to housingdeviations Somebackkshshouldbepresentinallgearmeshes.1. Adequatebackkh should be distauce and mate&ddiffexnce) present dming static conditions.ToothThicknessSpec@cationand Measurement. themaximumtooththichmsfiromthemaximmnvahte.Itis the minimum barkhsh allowablewhen the gear tooth theamountbywhichthewidthofthetoothspace withthegrrastaUowableeffectivetooththickne~isin exceedsthetooththicWssoftheengagingtoothonthe meshwith a matingtooth havingits greaEstallowable operatingpitch circles(SeeFig A-l). their quality.using minimum bacldash. andbearingclearances (4) mountingerrorssuchas shaft eccentricity Itisrequkedtoassmzthatthenortdrivingsidesofthe (5) bealiIlgrlm0~ teethdonotmakeco~ Backkhinagivenmesh (6) tempemtmeefkcts(afunctionoftemperatme variesdmingoperationasaresultofchangesinspe& differences betweenhousingand gear elements.Each factor can be 3. andincreaseALPurpose. suggestedvaluesfor minimllm back&h me inclnded.center distancetoleranceand gear tooth quality tolerances. ooo5c + =4ld 3 .007 -- 0.62 0.016 0.5 Tooth ThicJmessTolerance. NOTE: C must be au absokte value.82 - 1.hamesvariation asafnnuionoftotalcomposite tolmce for the apprqxbqnalitymxm~fiom tlWdX’ 0.cosuch things as balaubg bendiug suen~slidingvelocity. 035 0.in pd 18 12 8 5 3 2 1 l/4 A5.06+ omo5c &AU + o-03 m s O$AlW mn 50 100 200 400 800 1.008 0.0024+0.028 0.014 0. .021 - ANSUAGMA - 16 0. = 0.025 32 0.l5).c.C.020 0.014 0. Table A-l Minimum Backlash.14 0.wi& typical fz4xllm~ ulumbmag tolelau~ fixhousings.andriteminimnmbacklashBBmin.andcuaingdepths.c.18 --- 0.033 64 - fe iscaMatedbys&mctingatooththidmess tolerance seleued hm the table 63 in AGMA 2OW488 (Eq5XL5.010 0.40 .034 0.14 or5.007 0. A. The sum of fGand fpis control& m the center 2 4 8 0.16 0.41 050 0. Mmunum Bac&lash.0420. for Coarse Pitch Gean (miNimetervaWs) MinimumcenterDisrance.009 0.operaring atpitchlinespeedslessthau3OOOfpm(15m/s).l Basic Tolerance tIIalcnlation. shaftsandbwings.016 0.058 32 2002-B88 .Tooth ThicknessSpecScationand Measmement LINE OF ACTION Fig A-l Feeler Gage Backlash Measurement (Normal Plane) Table A-l shows conswative valuesof minimum ba&lashfixf~usgeaninfermushousings.00 1.andanallowaacefor tootht.600 15 2 3 5 8 12 18 0. Thevalues usedinthenmnexMexampksarechosenforsome arbilmyobje&e.006 0. Ba for Coaxse Pitch Gears (inch values) distance.80 0.012 0.17 020 026 035 - 022 025 031 0. Bd.40 052. Bmiu=0. Table A-l(M) .13 0.1.005 0.mm ThevaluesfoandinTableA-lmaybecak&ted from Eq Al. Itdinimumcen~Distaece..70 - 0.010 0. rmax is calculated in accordaue with Eq 3.The valnesfortooththi&essarenormaRychosenbythe designer.006 0. 397 D’ 5. Table A-4 Preliminary Calculation for AGMA Quality Q12-C Data Item Qn Table A-2 Typical Input Data (Dimensions in inches) Data Item n.1 AGMA 2000 AGMA 2000 Eq 3.391 Db 5.6 Eq 3.01529 Q9-B 6. Maximum Backlash. The maximum backtooth thicklash in a gear set is the sum of Bat ness tolerance.498 *b a% 27” 26 87 28” 59 30 2 8 70 71 Gear -. 0.360 26 0.7 Eq 3.153 -. ANWAGMA 33 2002-B88 .003 0. * Note: For the purpose of the mathematical example the values have been calculated to 16 significant figures and the results rounded to five decimal places.001 6 AGMA 2000 Vcq 0.030 % px Qn F tPII.003 tmin 0. are meshed at the loosest allowable center distance. The theoretical maximum backlash occurs when two perfect gears. but the practice should be discouraged.001 9 AGMA 2000 &in 0.828 6 22.3) The allowance for total composite variation is sometimes ignored. selected from AGMA 2000-A88. the data might be modified as shown in Table A-4 [ ‘1.801 19.5 Eq 3.1 Eq 3.773 -.- Notes 38 13 29 2 3 77 Ql2-C 0.161 0.2 Eq A.Tooth Thickness Specification and Measurement tm =tmax-tT .-.WC 19.005 0.162 44 Eq 3. The loosest center distance is the maximum for external gears or the minimum for internals. is given in Table A-3 [ ‘3.-. 0% 3. The values which would be used for other inspection methods are calculated in Table A-5 [ ‘I. since it may lead to interference in the assembled unit.187 PI 0.3 Eq 7.0.1 Eq 3.3600 Gin %X Data Item Gear 197 Gear Pinion l$ 3. The minimum specified tooth thickness should also be reduced.165 59 tT 0.3 A52 Specifications For Tooth Thickness Measurement.31. made to the minimum tooth thickness specification. An example calculation sequence of fmin and tmax for an AGMA Quality Q9-B gearset. so that the tooth thickness tolerance.538 Bmin 0. and the effects of gear tooth geometry variation.363 0 0.010 Designer’s option Bmin *IMX 0. Table A-3 Cakulations for Quality Q9-B Pinion Notes The values of tmax and fmin calculated above are the values which would be measured by a composite action test.001 4 0. The maximum tooth thickness specified for any inspection method should be reduced to be sure that the effects of runout and other tooth cutting variations on the inspection results do not increase the maximum effective tooth thickness.017 ‘Gmax --rT 0.4 Eq 3. manufactured to AGMA Quality Ql2-C.806 Do 2% Prelimimry AGMA 9. Normal practice would be to round to four decimal places. the effect of center distance variation. as listed in Table A-2.272 33.353 pb 0.11 A6.003 Vcq 0. is available for economical gear manufacture.N Pinion 34 Pnd Both 6 20 3. and is not used up by the other tolerances implied by the quality class. if the parts are near the maximum allowable variation.2 vcqtan $ If this gearset were used in a high speed drive.001 6 0. 5 5 7 6 2.360 0 0.8 Eq 5.8 Eq 7.258 0.288 16.773 0.884 78 6.002 23 25 24 12.047 0.4 Eq 6.905 80 33.352 09 3346 47 6. and v < vrT a@ 2002-B88 .933 0.002 00 6.397 26 0.1 AGMA 2000-A88 Eq 5.001 35 33.702 83 6.16 6.099 12.692 95 0.11 6.238 0.11 Eq 4.943 93 -- Table 6-1 Eq Eq Eq Eq -- 6.175 0.704 18 -- 0.10 Eq 7.3 Eq 5.001 7 Values of V AiSSI/AGMA 64 88 9 7 a# are not in AGMA 2000-A88. w correction for nmout corrected Dzwmax corrected D2Wmjn 67 7 68 96 66 rad 72 69 66 37 19 67 0 33 95 42 rad 92 15 91 49 56 Eq 5.5 6.258 0.255 0.885 0.244 0.756 0.898 2.242 0.262 0.892 0. ‘Rznax *R fnRmax OC htmax rRIlliIl t mmin Pin Measurement: ’ w RlWmax RIWmin D2Wmax (even) D2Wmax (odd) D2Wmin (theoretical) V.184 0.167 0.10 Section 6.004 Eq 7.244 0.004 16.6 Eq 5.165 0.11 6.Tooth Thickness Specification and Measurement Table A-5 Example Calculations for Tooth Thickness Measurements (Dimensions in inches) Data Item Preliminary D’ LX tmin Qn Db Pinion Gear Notes Data from Tables A-2 and A-3 5.438 65 33.14 AGMA 2000-A88 r*1 by.17 6.953 43 16.353 70 Q9-B 5.20 Span Measurement: Gin 22 MmaX Mmin COS+m VTT V a+ [‘] 0.444 49 *bmax *bin 13 29 77 38 32 35 Eq 4.793 0.272 0.004 0.11 Chordal Measurement: a ST R.5. %pk = 5 (1 + -$- ).092 0.886 78 0.828 87 0.153 Q9-B 31.002 3..166 0.10 Eq 5.4 and 0.691 60 33.161 0.786 0.4 Eq 5.5 Eq 5.5 Eq 5.5.10 Eq 7.2 Eq 7. approximate 34 19 32 3 0 0 Eq 7.903 80 33.251 0.384 3.166 0.248 0. 890 00 ‘brn max tbm min Notes Gear Pinion 0.742 82 Eq 8.441 57 0.024 01 16.054 81 18.093 12.783 10 Eq 8.13 7. tani Example: Using the values for the example gearsets of Tables A-2 to A-4. individual elements of tooth and center distance variation. (Eq. since the parts are of opposite hands. the maximum acceptable value must be carefully chosen to allow reasonable manufacturing tolerances for each part in the assembly.047 48 18. is used as an acceptance criterion.019 2 in.5 RTmin 3.895 76 M mmin 2.12 7. for the Ql2-C set %MX This example emphasizes the importance of quality number. a careful study of each element of maximum backlash must be made and a quality class selected which will limit tooth variations as necessary.752 30 Eq 8.809 83 0. particularly a unit with multiple stages. 3 max = p’ -zGmin.435 73 2. Neither occurrence is likely in practice.309 61 3. Any tooth variations due to manufacmring will decrease the mardmum expected backAXWAGMA = 0.365 69 rad Eq 8.6 NOTE: The example is calculated as if the same master gear were used for both parts.781 12.773 62 Eq 8. If maximum backlash must be controlled. GX cmin The maximum expected backlash is a funcand the statistical distribution of the tion of B. This would not be true in practice. A-2) = minimum transverse tooth thickness of gear = minimum transverse tooth thickness of pinion = maximum center distance = minimum center distance %XDC = 0.Tooth Thickness Specification and Measurement Table A-5 (cod) Example Calculations for Tooth Thickness Measurements (Dimensions in inches) Data Item Span Measurement: (cant) M mmax 0..174 53 rad 4.016 68 16.087 05 08 99 12 Composite Action Test: N2 Eq Eq Eq Eq 7.4 %. if backlash is to be limited.034 4 in.788 0.13 Master 24 TEETH 0.030 80 ‘b2 Db2 *s % Rm 0. Experience and judgment are required to estimate reasonable values. for the Q9-B set When maximum backlash of an assembled unit. The maximum theoretical backlash will also occur when two teeth. made to the minimum effective tooth thickness. 35 2002-B88 . coincide while operating at the loosest center distance.3 RTmax 3.425 76 rad %X 5.12 7.061 70 2.1 +3 0.tpmin+ [ c --C&J2 where rGIitl tPIIYill lash.in 5. Tooth Tlickness Specification and Measurement (This page has been left blank) A!!SIJAGMA 36 2002-B88 . so they require an allowance for eccentricity. and is constructed for a specific pitch and pressure angle. which is an advantage over pins. including standard block dimensions and a calculation method. Fig B-l Female Block 0% B-1) 0% B-2) Fig B-2 37 Male Block 2002-B88 . and are seldom specified on new drawings. made to the exact tooth thickness and normal pressure angle of the standard rack. in = n/2&d *ns = normal generating tooth thickness of the gear. [6] B2. Standard proportions for these blocks are shown in figures Figs B-l and B-2.4 Calculation for Measuring Blocks. Measurements made over blocks are not affected by deviations in blank geometry.1 Advantages of Measuring Blocks. B2. Tooth Comparator. B2. Each set consists of three blocks. These blocks are in effect theoretical rack teeth of standard proportions. RBl=T “+~+(h~~) RB2’ 9 ANWAGMA + g+(ynT) (3 B-3) for odd number of teeth (Eq B-4) OB = RB1 + RB2 where t nt = normal tooth thickness of standard rack at the reference line. in DB . Coordinate Measuring Machine (CMM) and other alternate tooth thickness measurement methods. B2. They are more expensive than pins. and the combination of one male and one female for gears with an odd number of teeth. Measuring blocks cannot be used on internal teeth. Two male blocks are used on gears with an even number of teeth. in RB2 = Radius over female block. Gear measuring blocks can be used on external spur and helical gears.2 Limitations of Measuring Blocks. Measuring Blocks.3 Measuring Block Sets. in RB1 = Radius over male block. but is included for information purposes only. Tooth Thickness Specification and Measurement. B2.] for even number of teeth DB = 2RB1 Bl. two males and one female.Tooth Thickness Specification and Measurement Appendix B Alternate Methods of Tooth Thickness Measurement [This Appendix is not a part of AGMA 2002-B88. in = Measuring Dimension. Purpose. This Appendix provides infoxmation on the Measuring Block. Block measurements are independent of the mounting diameter of the part. The blocks rest firmly on the lines of action of helical gears without rocking. Measurements over blocks have a similar amplifving factor to measurements over pins. 1 The instrument stylus should have asmalltipradius. This method of gear tooth thickness measurement is based on using a CNC Gear Measuring . Spur Gear /‘ STANDARD PITCH DIAMETER BASE DIAMETER RB1 ” I x = fns- tnf 2 tan f& ts = -?zL =\ cos tlrs I Fig B-4 Measuring Block Engagement. except with very low numbers of teeth and other unusual cases. v. the effects of runout are included and no correction is necessary. The effect of allowable pitch deviation is much smaller than allowable runout.tnr 2& i ht = * nd Fig B-3 Measuring Block Engagement. .Instrument equipped with a high resolution rotary table and measuring stylus which can be moved to a known position.5 Correction for Tooth Deviations. CNC Gear Tooth Thickness Measurement. where 3-B If block measurements are made as a radius to one block from the mounting diameter. the effects of runout should be calculated and 0~ adjusted accordingly.zi:i .I‘d.A L BASE DIAMETER ! / i x= I tns. in (mm) (Eq B-5) B3. The amount of correction is: v.. B3.T ANSIIAGMA = allowable runout of gear teeth. .Tooth Thickness Specification and Measurement I I I I I ‘1 -L Pi?. Helical Gear The direction of the correction reduces the allowable tooth thickness (see 3.1.a. 38 2002-B88 ./&.rB y-VrT = correction to block measurement for nmout. If the measurements are made with two blocks. B-2. from AGMA 2000-A88.r-. so it can be ignored.1). in (mm) B3.1 Alternate Methods. 2. The anvils of the comparator are equivalent to the sides of the generat- . separate comparators for ea+-i pressure an#e.2 The instrument tip contact is to be known relative to the center of the rotary table. B4.6 The gear is rotated back until the opposite tooth flank (from step B3. The gear tooth comB5. 81. -ILINE B3.8.4 The rotary position is recorded in radians. B3. 00 . The requirement for an accurate outside diameter to be used as a reference surface. II B3.3 The gear is rotated until the tooth flank contacts the probe and is at radial position on the probe. undersize.2. B3.1 )-tip&meter (Eq B-6) R(% = Circular tooth thickness at measuring Radius R = Angular position in radians 8 T= B4.7 according the following equation: B5. The principle can be understood from Fig B-6. B4. and pitch line of rack is at a distance equal to the pitch radius from the centerline of the gear.2. Gear tooth comparator measurements can be made on large external gears without the use of large micrometers.7 The new rotary position for the opposite flank is recorded in radians.2.2 Limitations of Tooth Comparator. parator compares the thickness of a standard rack with the sample gear tooth.Tooth Thickness Specification and Measurement B3. The gear to be measured is B3. This method is seldom specified for new gear designs. but is seen on many older drawings.2.5 The probe is moved into the next space for the opposite tooth flank measurement and positioned to the measuring radius.2. Tooth Comparator. It is not limited by helix angle or face width.1 Comparator layout of the space from the basic rack for gear to be measured is made to a suitable scale depending on capacity of the instrument and gear pitch.2. taper. Gear tooth comparator measurements are affected by all deviations.3 The projection of the gear tooth must fall within the tolerance lines shown. and oversize in the reference outside diameter of the gear. An optical comparator is best suited for fine pitch gears (see Fig B-S).2 l/2 TOOTH THTCKNESS TOLERANCE General Method of Measurement. into the space adjacent to the tooth to be measured. and precision setting blocks for each pitch an’d pressure angle can outweigh the advantages. A scale of at least 20 to 1 is recommended.2. such as runout. ANSVAGMA 39 2002-B88 .1 mounted concentric to the rotary table.2. Outside diameter can also be checked by tolerance lines in root of rack. Fig B-5 Optical Comparator Measurement B3. the method has the same ease of measurement and amplifying factor as measurement over pins or balls.2. Optical Comparator.2 Position layout and gear to be measured so that the centerline of the rack space and the centerline of the gear tooth are coincident. If the outside diameter is accurately known. B5. B5.2 The probe is moved to the measuring radius.1 Advantages of Tooth Comparator. and difference in angular positions in steps B3.2.3) is contacted with the probe and at the radial position.1. R. B3.4 and B3. TOLERANCE LINES AT OUTSIDE DIAMETER (OPTIONAL) / PROJECTED I/ B3. B4. B3.8 The tooth thickness can now be computed from the measuring radius.3 Comparators. measured from Ds. B5. is v/2Pnd. Tooth thickness is specified as thick (minus reading) or thin (plus reading). outside diameter size and nmout and gear tooth nmout must be carefully controlled. /’ / To correct for actual outside diameter and to account for runout: Ah = A- i Fig B-6 Measurement of Tooth Thickness by Means of a Gear Tooth Comparator 0% B-8) ‘0” SETTING AGAINST STANDARD Do . The anvils of the instrument always make contact with the tooth at the standard pitch diameter. the tooth thickness can be calculated. [7] If the outside diameter is known.+ 2a-Do-)+ 1.4 Calculation. The instrument is then placed on the gear tooth to be measured.POSITION OF STANDARD TOOTH WITH STANDARD Do + = THIN X hr =. because they have such a large influence on gear tooth thickness measurements. The dial indicator is set to zero with the anvils against a standard setting block (having an addendum of l/Pnd and a tooth thickness of “/2Pnd). See Fig B-7. The theoretical indicator deflection. is 11 Pnd and the tooth thickness at D. A- : . directly as read from the instrument indicator. the indicator reads zero when the addendum.[D. When the instrument is set. The anvils contact the tooth flanks such that the centerline of the tooth at the generating pitch circle is the measurement point and the indicator reads the difference between the actual addendum and the standard addendum. + x)] h = hr = STANDARD ADDEND1JM Ah = INDICATOR TRAVEL = THICK I h.Tooth Thickness Specification and Measurement ing rack tooth and have the same profile angle.+2 (h. \ -. is: /’ r( ‘1 BASE DIAMETER 0% B-7) 2bT CD. When tooth thickness is controlled by comparator measurement. Ds.2fnstan Qc Fig B-7 ANSIIAGMA arator Measurement Variations 40 2002-B88 . A. 1979.. References MAAG Gear Book . Vol. 1958. Oct.C. Switzerland-methods of Inspection. Eng. 1958. American Machinist.. Gen. Measuring the Tooth Thickness of Helical Involute Gears. Mass. Tooth Thickness Principles. Werner F. Rochester.lS-16. Works Engineers. Detroit.HilI Book Co.. New York. 259. Book Production by Denham & Co. Copyright 1961.. Watertown. M ich. Dr. Oct. 28-31.. A. Backlash.Semi-Annual Oct. Reprinted. Set Head.14 Oct. Earle. Zahorski. Inc. [7] Farrel Gear Manual for Herringbone. Ott. Bulletin 6 IOA... A G M A Paper 239.Form No. NY . Tooth Thickness Allowances. W . [4] DIN 3977 . Assured Backlash Control .. Analytical Mechanics of Gears. Precision Laboratory.D. Farrel-Birmingham Co.MichaIec. Copyright 1945 by M . Div.Wheel Co. Ltd. PleasantvilIe. Zurich. [2] Buckingham. 59 No. The Van Keuren Company.The ABC System. George.ucnu”r.. Eng. M ichigan Tool Co. NY [3] Van Keuren Precision Measuring Tools ... IL. E. 1979. 1949. Fall Tech Meeting. 18.. pp 361-369. 1923. Helical and Spur Gears. Ihinois Tool Works. Chicago. Involutometry and Trigonometry. Leonard J. A!!SI/AGMA 41 2002-B88 . . USA.T. Detroit.. GULU I. V S M 15535. American Machinist.I ““Ul L IllLNIGDD apw.Measuring Element Diameters for the Radial or Diametral Dimension for Testing Tooth Thickness of Cylindrical Gears.System of Gear Firs.*buacy bI*AcIac Bibliography [l] Smith.07. Wilhaber.Handbook No. [6] IlJinois Gear Measuring Blocks . 26-29.. July 12. A G M A Paper 239.Calculation and Manufacture of Gears and Gear Drives for Designers and M A A G Gear . Ball Measurement of Helical Gears. Printed by Ann Arbor Press. Analysis and Comparison of Total Composite Error and Position Error in Gears. McGraw . Vogel.. Ph. 1939. N-Y. Tolerances. [j] DLN 3967 . 36. .ALD(ANDRIA.PuBusHEDBY ABlERlOAN GEAR WIUFAcWREFtSm~~~~ lSOOlGNGSTREET. 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