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AGMA 917- B97

(Revision of AGMA 370.01 (1973))

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 917- B97

Design Manual for Parallel Shaft Fine- Pitch Gearing

AGMA INFORMATION SHEET (This Information Sheet is NOT an AGMA Standard)

Design Manual for Parallel Shaft Fine- Pitch Gearing American AGMA 917--B97 Gear [Revision of AGMA 370.01 1973] Manufacturers CAUTION NOTICE: AGMA technical publications are subject to constant improvement, Association

revision, or withdrawal as dictated by experience. Any person who refers to any AGMA Technical Publication should be sure that the publication is the latest available from the Association on the subject matter. [Tables or other self--supporting sections may be quoted or extracted. Credit lines should read: Extracted from AGMA 917--B97, Design Manual for Parallel Shaft Fine--Pitch Gearing, with the permission of the publisher, the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, Virginia 22314.]

Approved September 25, 1997

ABSTRACT The rewritten Design Manual for Fine--Pitch Parallel Shaft Gearing is a cookbook style manual on how to design fine--pitch spur and helical gears. All work has been done with an eye towards computerization of the equations and the graphs. In addition, the manual contains such specialized subjects as inspection, lubrication, gear load calculation methods, materials, including a wide variety of plastics. Published by

American Gear Manufacturers Association 1500 King Street, Suite 201, Alexandria, Virginia 22314 Copyright  1997 by American Gear Manufacturers Association Reprinted June 1999 All rights reserved. No part of this publication may be reproduced in any form, in an electronic retrieval system or otherwise, without prior written permission of the publisher.

Printed in the United States of America ISBN: 1--55589--694--4

ii

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 917--B97

Contents Page

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1

Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3

Definitions and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

4

Theory of involute gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

5

Application considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6

Design synthesis and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7

Design for control of backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8

Gear drawings and specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

9

Gear tooth tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

10

Materials and heat treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

11

Manufacturing methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

12

Inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

13

Lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

14

Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

15

Load rating and testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Tables 1

Symbols, terms and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2

Profile shift coefficients for 20° profile angle spur gears . . . . . . . . . . . . . . . . . 43

3

Maximum outside diameter for minimum topland of 0.275/Pnd . . . . . . . . . . . . 44

Figures 1

Basic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2

Principal reference planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3

Planes at a pitch point on a helical tooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4

Power transmission by two pulleys and a crossed belt . . . . . . . . . . . . . . . . . . . 8

5

Point on belt generates involute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6

Involute nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

7

Involute action and speed ratio independent of center distance . . . . . . . . . . . . 9

8

A series of small involute cams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

9

Time = T1: Second pair of teeth just starting engagement . . . . . . . . . . . . . . . 10

10

Time = T2: First pair of teeth just leaving engagement . . . . . . . . . . . . . . . . . . 10

11

Time = T3: One pair of teeth in contact at pitch point . . . . . . . . . . . . . . . . . . . 11

12

Two involute curves showing differences in lengths of corresponding arcs . 12

13

Involute polar angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

14

Helix angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

15

Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

16

Tooth pitches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

17

Principal pitches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

18

Base pitch relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 iii

AGMA 917--B97

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Figures (continued) 19

Gear design flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

20

Preliminary design flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

21

Torque split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

22

Speed decreasing gear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

23

Gear train with idler gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

24

External spur gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

25

Internal spur gear and external spur pinion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

26

External helical gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

27

Spur pinion and face gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

28

Bevel gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

29

Crossed helical gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

30

Worm and wormgear set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

31

Spur rack and pinion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

32

Contact ratio vs. center distance deviation for 20 degree profile angle gears 28

33

Gear tooth as a simple beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

34

Tooth load acting at inscribed parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

35

Shaft alignment deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

36

Standard pitch circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

37

Transverse pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

38

Line of action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

39

Line of contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

40

Transverse backlash is arc PR -- arc PQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

41

Undercut teeth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

42

Effect of profile shift (addendum modification) . . . . . . . . . . . . . . . . . . . . . . . . . 40

43

Line of action for external gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

44

Line of action for internal gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

45

Showing angle at which load bears on tooth . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

46

Adjustable center distance gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

47

Spring loaded center distance gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

48

Adjustable split gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

49

Spring loaded split gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

50

Composite gearing with elastic element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

51

Tapered gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

52

Spring preloaded gearing (for limited rotation) . . . . . . . . . . . . . . . . . . . . . . . . . 57

53

Dual path spring loaded gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

54

Contra--rotating input gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

55

Graphical representations of typical gear errors . . . . . . . . . . . . . . . . . . . . . . . . 63

56

Profile of 10 tooth, 20 DP, 20° PA gear tooth with undercut . . . . . . . . . . . . . . 74

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 iv

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 917--B97

Foreword [The foreword, footnotes, and annexes, if any, in this document are provided for informational purposes only and are not to be construed as a part of AGMA Information Sheet 917--B97, Design Manual for Parallel Shaft Fine--Pitch Gearing.] Although there is a great deal of information about parallel shaft fine--pitch gearing in the literature, it is widely scattered and a considerable number of areas are not well covered. As a result, this manual has been compiled to provide a central source of the best information available on the design, manufacture and inspection of fine--pitch gearing. This manual is a revision of 370.01, Design Manual for Fine Pitch Gearing, 1973. Additions have been made to the design section to broaden the concepts of gear theory and the gear design process. Omitted from this Manual are wormgears, bevel gearing and face gearing which appeared in the original design manual. This information is available in other AGMA Standards. An important feature of this manual is the bibliography to which the user is referred for additional data in each area. Suggestions for improvement of this standard will be welcome. They should be sent to the American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, Virginia 22314.

v

AGMA 917--B97

AMERICAN GEAR MANUFACTURERS ASSOCIATION

PERSONNEL of the AGMA Fine Pitch Gearing Committee Chairman: D. E. Bailey . . . . . . . . . . . . . . Rochester Gear, Inc. Editor: D. Castor . . . . . . . . . . . . . . . . . . . Eastman Kodak Company

ACTIVE MEMBERS M.K. Anwar . . . . . . . . . . . . . . . . . . . . . . . . P. Dean . . . . . . . . . . . . . . . . . . . . . . . . . . . F. R. Estabrook . . . . . . . . . . . . . . . . . . . . . I. Laskin . . . . . . . . . . . . . . . . . . . . . . . . . . . D.A. McCarroll . . . . . . . . . . . . . . . . . . . . . K. Price . . . . . . . . . . . . . . . . . . . . . . . . . . . D.Seger . . . . . . . . . . . . . . . . . . . . . . . . . . . L.J. Smith . . . . . . . . . . . . . . . . . . . . . . . . . R.E. Smith . . . . . . . . . . . . . . . . . . . . . . . . . S. Sundaresan . . . . . . . . . . . . . . . . . . . . . M. Weiby . . . . . . . . . . . . . . . . . . . . . . . . . .

Allied Devices Corporation Retired Retired Consultant The Gleason Works Eastman Kodak Company Perry Technology Corporation Invincible Gear Company R.E. Smith & Company, Inc. Eastman Kodak Company Bison Gear & Engineering Co.

ASSOCIATE MEMBERS A.F.H. Basstein . . . . . . . . . . . . . . . . . . . . . D. Gimpert . . . . . . . . . . . . . . . . . . . . . . . . . K. Gitchel . . . . . . . . . . . . . . . . . . . . . . . . . . G.P. Lamb . . . . . . . . . . . . . . . . . . . . . . . . . J.R. Mihelick . . . . . . . . . . . . . . . . . . . . . . . G.E. Olson . . . . . . . . . . . . . . . . . . . . . . . . . C. Sanderson . . . . . . . . . . . . . . . . . . . . . . D.H. Senkfor . . . . . . . . . . . . . . . . . . . . . . . M. Shebelski . . . . . . . . . . . . . . . . . . . . . . . A. Sijtstra . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Tseytlin . . . . . . . . . . . . . . . . . . . . . . . . . F.C. Uherek . . . . . . . . . . . . . . . . . . . . . . . . A. Ulrich . . . . . . . . . . . . . . . . . . . . . . . . . . . F.M. Young . . . . . . . . . . . . . . . . . . . . . . . .

vi

Crown Gear b.v. Koepfer America Ltd. Universal Tech. Systems, Inc. (Deceased) Lamb & Lamb Rockwell Automation/Dodge Olson Engineering Services Koepfer America Ltd. Precision Gear Company Boeing Precision Gear, Inc. Crown Gear b.v. Esterline Federal Products Corp. Flender Corporation UFE, Inc. Forest City Gear Company

AMERICAN GEAR MANUFACTURERS ASSOCIATION

American Gear Manufacturers Association --

Design Manual for Parallel Shaft Fine--Pitch Gearing

AGMA 917--B97

1.2 Design information Information of the following subjects is supplied as required by the design procedure: -- Analysis of tooth proportions and meshing conditions; --

Inspection;

--

Gear tooth tolerances;

--

Gear blank design.

1.3 Additional design related material

1 Scope This manual provides guidance for the design of fine--pitch gearing of the following types: --

Diametral pitch from 20 to 120;

--

Spur and helical (parallel axis);

--

External, internal and rack forms.

The guidance consists of the following: -- Description of a design procedure in a series of steps; -- Design information -- data values, equations and recommended practices; --

Additional design related material.

The English system of units is used in this manual. 1.1 Design procedure The description of the design procedure covers the following: --

Establishing proportions of the gears;

--

Selecting detailed gear data;

--

Confirming suitability of the tentative design;

--

Controlling backlash;

-- Meeting contact ratio and other gear mesh requirements; --

Specifying gear dimensions and tolerances.

The following background and supplementary information is also supplied: --

Manufacturing methods;

--

Gear material and heat treatment;

--

Lubrication;

--

Bearings.

1.4 Annexes Annex A is a bibliography. 1.5 Limitations The information in this manual is meant to serve only as a guide to the designer of fine--pitch gears. It is not intended that it be the procedure which must be followed in the design of such gears, nor is it implied that using the procedures and data will necessarily result in gears that will meet the requirements in every application. It remains the responsibility of the individual designer to properly evaluate the conditions in the particular application and to make use of prior experience or proper testing to confirm the suitability of the design. 1.6 Tooth form (spur and helical gearing, internal and external) The tooth form of the spur and helical gearing considered in this manual is involute. Unless specifically noted, all external spur and helical designs resulting from the procedures discussed in this manual will be conjugate with standard basic racks. See ANSI/AGMA 1003--G93.

1

AGMA 917--B97

AMERICAN GEAR MANUFACTURERS ASSOCIATION

2 References The following documents contain provisions which, through reference in this text, constitute provisions of the manual. At the time of publication, the editions were valid. All publications are subject to revision, and the users of this manual are encouraged to investigate the possibility of applying the most recent editions of the publications listed:

ANSI/ASME Y14.6--1978 (R1993), Screw Thread Representation, Engineering Drawing and Related Documentation Practice (reaffirmation and redesignation of ANSI/ASME Y14.6--1978).

3 Definitions and symbols 3.1 Definitions

AGMA 203.03, Fine--Pitch On--Center Face Gears for 20--Degree Involute Spur Pinions.

The terms used, wherever applicable, conform to ANSI/AGMA 1012--F90.

AGMA 900--F96, Style Manual for the Preparation of Standards, Information Sheets and Editorial Manuals.

3.2 Symbols and terms

AGMA 906--A94, Gear Tooth Surface Texture with Functional Considerations. AGMA 908--B89, Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth. AGMA 910--C90, Formats for Fine--Pitch Gear Specification Data. ANSI/AGMA 110.04, Nomenclature of Gear Tooth Failure Modes. ANSI/AGMA 1003--G93, Tooth Proportions for Fine--Pitch Spur and Helical Gearing. ANSI/AGMA 1012--F90, Gear Nomenclature, Definitions of Terms with Symbols. ANSI/AGMA 2000--A88, Gear Classification and Inspection Handbook. ANSI/AGMA 2001--C95, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth. ANSI/AGMA 2002--B88, Tooth Thickness Specification and Measurement. ANSI/AGMA 2004--B89, Gear Materials and Heat Treatment Manual. ANSI/AGMA 2005--B88, Design Manual for Bevel Gears. ANSI/AGMA 6034--B92, Practice for Enclosed Cylindrical Wormgear Speed Reducers and Gearmotors. ANSI/AGMA Lubrication.

9005--D94,

Industrial

ANSI/ASME B46.1--1985, Surface Texture.

2

Gear

The symbols and terms used throughout this manual are in basic agreement with the symbols and terms given in AGMA Information Sheet 900--F96, Style Manual for the Preparation of Standards and ANSI/AGMA Standard 1012--F90, Gear Nomenclature, Definitions of Terms with Symbols. In all cases, the first time that each symbol is introduced, it is defined and discussed in detail. NOTE: The symbols and definitions used in this standard may differ from other AGMA standards. The user should not assume that familiar symbols can be used without a careful study of their definitions.

Throughout this manual, the term pinion refers to the member of the meshing pair with the smaller number of teeth without regard to which member is driving. The term gear, as part of a meshing pair, refers to the member with the larger number of teeth. In order to avoid confusion and to achieve consistency, any symbol that is applicable to a specific member, pinion or gear, is given the subscript P for pinion and subscript G for gear. The subscript n is used to distinguish the normal plane from the transverse plane. The subscript t is used to distinguish the transverse plane in those situations when confusion with the normal plane might occur. Any symbol not specifically designated by a subscript n is assumed to be in the transverse plane. The subscripts p, i, b and o represent the terms at the operating diameter, inside diameter, base diameter and outside diameter respectively. The symbols and terms, along with the clause numbers where they are first discussed, are listed in alphabetical order by symbol in table 1.

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 917--B97

Table 1 -- Symbols, terms and units Symbol ah B Bb Bc BhT Bn BnhT C Cn Cs cG cmin cP DbG DbP DG DiG DoG DoP DP DrG DrP dG dP EG EP eaG eaP F J Kv La Lr mF mG mp N NG NP Pd Pnd pbn pb

Terms Addendum of the cutter Transverse backlash (backlash in the transverse plane) Backlash along the line of action Transverse backlash due to change in center distance Minimum backlash for spur gears Normal backlash (backlash in the normal plane) Minimum backlash for helical gears Operating center distance Nominal center distance Standard center distance Clearance at the root of the gear Minimum clearance at the root Clearance at the root of the pinion Base diameter of the gear Base diameter of the pinion Standard pitch diameter of the gear Inside diameter of the gear Outside diameter of the gear Outside diameter of the pinion Standard pitch diameter of the pinion Root diameter of the gear Root diameter of the pinion Operating pitch diameter of the gear Operating pitch diameter of the pinion Young’s modulus of the gear Young’s modulus of the pinion Roll angle at the start of active profile for the gear Roll angle at the start of active profile for the pinion Face width Geometry factor Dynamic factor Length of approach Length of recess Face contact ratio Gear ratio Transverse contact ratio Number of teeth Number of teeth on the gear Number of teeth on the pinion Transverse diametral pitch Normal diametral pitch Normal base pitch Transverse base pitch

Units inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches inches psi psi degrees degrees inches

inches inches

1/inch 1/inch 1/inch 1/inch

Reference 6.8.1 6.3.9.1 6.3.9.3 6.3.9.1 7.2.1 6.3.9.2 7.2.1 6.3.1 6.5.1 6.2.8 6.8.2 6.8.2 6.8.2 6.2.4 6.2.4 6.2.2 6.5.2 6.5.2 6.5.2 6.2.2 6.8.1 6.8.1 6.3.5 6.3.5 6.9.6 6.9.6 6.7.3 6.7.3 6.9.5 6.9.5 6.9.5 6.7.4 6.7.4 6.7.7 5.3.2 6.7.6.1 5.5.1 5.5.1 5.5.1 6.2.1 4.3 4.2.7 6.2.6 (continued)

3

AGMA 917--B97

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Table 1 (continued) Symbol pt Q RaG RaP RbG RbP RG RiG RoG RoP RP RrG RrP T tnG tnP tinG titG tonG tonP totG totP tpnG tpnP tptG tptP ttG ttP Vcq Wa Wn Wr Wt X+min X - max XG

4

Terms Transverse circular pitch AGMA quality number (5 to 15) Radius at the start of active profile of the gear Radius at the start of active profile of the pinion Base radius of the gear Base radius of the pinion Standard pitch radius of the gear Inside radius of the gear Outside radius of the gear Outside radius of the pinion Standard pitch radius of the pinion Root radius of the gear Root radius of the pinion Torque Normal circular tooth thickness at the standard pitch diameter of the gear Normal circular tooth thickness at the standard pitch diameter of the pinion Normal circular tooth thickness at the inside diameter of the gear Transverse circular tooth thickness at the inside diameter of the gear Normal circular tooth thickness at the outside diameter of the gear Normal circular tooth thickness at the outside diameter of the pinion Transverse circular tooth thickness at the outside diameter of the gear Transverse circular tooth thickness at the outside diameter of the pinion Normal circular tooth thickness at the operating pitch diameter of the gear Normal circular tooth thickness at the operating pitch diameter of the pinion Transverse circular tooth thickness at the operating pitch diameter of the gear Transverse circular tooth thickness at the operating pitch diameter of the pinion Transverse circular tooth thickness at the standard pitch diameter of the gear Transverse circular tooth thickness at the standard pitch diameter of the pinion Total composite variation Axial load (thrust load) Normal load Radial load Tangential load Minimum required amount of profile shift coefficient Maximum allowable amount of negative profile shift coefficient Profile shift coefficient of the gear

Units 1/inch inches inches inches inches inches inches inches inches inches inches inches lb--inch inches

Reference 6.2.5 9.3 6.7.1 6.7.1 6.2.4 6.2.4 6.2.2 6.5.2 6.5.2 6.5.2 6.2.2 6.8.1 6.8.1 6.9.2 6.6.2

inches

6.6.2

inches inches inches inches inches

6.6.5 6.6.5 6.6.5 6.6.5 6.6.5

inches

6.6.5

inches

6.6.4

inches

6.6.4

inches

6.6.4

inches

6.6.4

inches

6.6.3

inches

6.6.3

inches lbs lbs lbs lbs

6.6.6 6.9.3 6.9.1 6.9.4 6.9.2 6.4.4.3 6.4.4.4 6.4.4 (continued)

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AGMA 917--B97

Table 1 (concluded) Symbol XP θ ∆C ∆snG ∆snP ∆t νG νP ÔinG ÔitG Ôn ÔonG ÔonP ÔotG ÔotP Ôpn Ôpt Ôt π σb σc ψ ψb ψiG ψoG ψoP ψp HCR LCR SAP TCT TIF

Terms Profile shift coefficient of the pinion Involute polar angle Change in center distance Tooth thinning coefficient of the gear Tooth thinning coefficient of the pinion Change in tooth thickness due to total composite variation Poisson’s ratio of the gear Poisson’s ratio of the pinion Normal pressure angle at the inside diameter of the gear Transverse pressure angle at the inside diameter of the gear Normal standard pressure angle Normal pressure angle at the outside diameter of the gear Normal pressure angle at the outside diameter of the pinion Transverse pressure angle at the outside diameter of the gear Transverse pressure angle at the outside diameter of the pinion Normal operating pressure angle Transverse operating pressure angle Transverse standard pressure angle Constant, value = 3.1415927 Bending stress Contact stress Helix angle Base helix angle Helix angle at the inside diameter of the gear Helix angle at the outside diameter of the gear Helix angle at the outside diameter of the pinion Operating helix angle High contact ratio Low contact ratio Start of active profile (limit diameter) Total composite tolerance True involute form (form diameter)

4 Theory of involute gearing

Clause 4 contains discussions of what an involute is and why it is the geometric shape of choice for gear teeth. Simply put, involute gears transmit uniform rotary motion from one shaft to another shaft. Gear tooth flanks which transmit uniform rotary motion are said to be conjugate. Clause 4 also attempts to bring together the geometric aspects of the involute gear with the more common mathematical expres-

Units radians inches

inches

degrees degrees degrees degrees degrees degrees degrees degrees degrees degrees psi psi degrees degrees degrees degrees degrees degrees

Reference 6.4.4 4.1.11 6.3.9.1 6.6.1 6.6.1 6.6.6 6.9.6 6.9.6 6.5.5 6.5.5 6.2.3.2 6.5.5 6.5.5 6.5.5 6.5.5 6.3.8 6.3.6 6.2.3.3 6.9.5 6.9.6 4.1.9.2 6.5.4 6.5.3 6.5.3 6.5.3 6.3.7 6.7.6 6.7.6 6.7.1 8.4.5.1 6.4.3

sions that are used to specify a gear as a machine element. Clause 4 presents basic background information which will lay the foundation for the reader to proceed with the gear design. Other clauses of this design manual discuss the selection of gear parameters such as type of gearing to be used, number of teeth, diametral pitch and gear ratio. These parameters are the foundation blocks of a gear design. Clause 6 explains how to complete the design, optimize and fully specify the gear system.

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-- Center distance, operating. In parallel axis gearing, the distance between the gear axes in the plane of rotation. See figure 1.

4.1 Basic geometric considerations The following will clarify the concepts needed to understand the many useful features and characteristics of involute gearing.

-- Gear ratio. The ratio of the rotational speeds of the input and output shafts. See 5.3.2.

-- Axis. The axis, perpendicular to the plane of rotation, and lying in the axial plane, passing through the geometric center of the gear. See figure 1.

-- Pitch point. A point on the line of centers which divides the center distance in the same proportion as the gear ratio. See 6.3.3. -- Pitch line. A line lying in a plane of rotation passing through the pitch point and perpendicular to the line of centers.

-- Plane of rotation. Any plane perpendicular to a gear axis. Also known as a “transverse plane”. See figure 1.

-- Pitch plane. For a pair of gears, a plane perpendicular to the axial plane and tangent to the pitch surfaces. A pitch plane in an individual gear may be any plane tangent to its pitch surface. The pitch plane of a rack or a crown gear is the pitch surface. See figure 2.

-- Gear center. The point of intersection of the gear axis and the plane of rotation. See figure 1. -- Line of centers. A line that connects the centers of two meshing gears in the plane of rotation.

Pitch circle (operating) Pinion axis Pinion center Center distance, C

Pitch point Gear axis Line of centers Gear center

Pitch circle (operating)

Axial plane 90°

Plane of rotation (transverse plane) Figure 1 -- Basic geometry

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-- Axial plane. In a single gear, an axial plane may be any plane containing the axis and a given point. See figure 2. Pitch plane

Transverse plane

Pitch cylinder Axial plane

Figure 2 -- Principal reference planes -- Transverse plane. A plane perpendicular to the axial plane and to the pitch plane. In gears with parallel axes, the transverse plane and the plane of rotation coincide. See figure 3. Normal plane

Transverse planes

Pitch plane

Line normal to tooth surface in normal plane

Pitch point

Figure 3 -- Planes at a pitch point on a helical tooth -- Normal plane. A normal plane is, in general, normal to a tooth surface at a pitch point and perpendicular to the pitch plane. See figure 3. 4.1.1 Involute geartooth interaction If two circles are drawn, one about each of the gear centers and each with a radius from its center to the

AGMA 917--B97

pitch point, both will be tangent to each other and to the pitch line. These circles rolling on each other without slipping will transmit uniform rotary motion from one shaft to another. In order to eliminate slipping of two pitch circles, teeth may be placed on them. The purpose of this clause is to acquaint the reader with the basic fundamentals of geartooth interaction. The concepts presented will demonstrate that when one involute curve drives a mating involute curve, the two curves touch at a point of contact. Moreover, all contact takes place along a straight line called the “line of action”. It can also be shown that the force vector between the teeth is in the direction of the line of action. The cylindrical nature of spur and helical gears has fostered the notion that gears transmit force in a manner similar to friction disks, the teeth being present only to prevent slippage. This view of gear tooth interaction has led to confusion for novice gear designers. Actually gear teeth interact as two cam surfaces, one driving the other. The interaction of the cams is accurately modeled as a pair of pulleys connected by a crossed belt. 4.1.2 Crossed belt theory As shown in figure 4, power and motion can be transmitted from one shaft to another using two pulleys and a crossed belt. (The belt is crossed to give the same direction of rotation as meshing gears in the same application.) A point, P, on the belt generates a curve in space as the pulleys rotate and the belt moves. The pulley is shown in four selected positions, 20 degrees apart. The curve generated by the point on the belt is an involute. The involute has several unique properties that have made it the curve of choice for gear teeth. 4.1.3 Involute cams Figure 5 shows the resulting system if a point on the same belt is used to generate an involute starting from the surface of the other pulley. The first useful property of involute curves is that they transmit uniform rotary motion. That is, if the input shaft rotates with uniform rotary motion, its involute curve driving the involute of the output shaft, the output shaft will also turn with uniform rotary motion. The relative speeds of the two shafts will be determined by the ratio of the diameters of the pulleys.

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Pulley

C2

Belt pulled at this end

Pulley

C2

Involute

Belt pulled at this end

P

P

Involute B1

B1 C1

Rb Base circle disc

Time = T1 Pulley

C1

Rb Base circle disc

Time = T2 Pulley Belt pulled at this end Involute Involute P

C2

P

C2

Belt pulled at this end

B1 C1

Rb

B1 C1

Rb

Base circle disc Time = T3

Base circle disc Time = T4

Figure 4 -- Power transmission by two pulleys and a crossed belt 4.1.4 Involute nomenclature Pulley 1 (Pinion)

Base circle disc 1

Rb1

Involute 1 (Pinion)

P

Involute 2 (Gear) Line of action (belt)

Rb2

Base circle disc 2 Pulley 2 (Gear)

Figure 5 -- Point on belt generates involute

8

Figure 6 shows the system being discussed after the line of centers, the operating pitch circles, and the pitch line have been constructed. The point at which the line of centers intersects the line of action is the pitch point. The circles from which the involutes emanate (represented by the pulleys) are the base circles. The shape of an involute curve is dependent upon the diameter of its base circle. The operating pitch circles have been drawn centered on the pulley centers, passing through the pitch point. Thus, they are tangent to one another at the pitch point. The ratio of the diameters of the operating pitch circles is the same as the ratio of the base circle diameters. The pitch line has been constructed perpendicular to the line of centers, passing through the pitch point. The angle between the pitch line and the line of action is called the operating pressure angle. Operating pressure angle, pressure angle and profile angle will be discussed in more detail later in this information sheet. See 6.2.3.

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Pulley 1 (Pinion)

Base circle 1

Operating pitch circle, dP

Operating pressure angle, φpt

Pitch line

AGMA 917--B97

R b2 d = G R b1 dP

Rb1

Pitch point

Involute 1 (Pinion)

P

Line of action (belt)

Involute 2 (Gear) Line of centers

Rb2 Operating pitch circle, dG Pulley 2 (Gear)

Base circle 2 Figure 6 -- Involute nomenclature

Involute action is not dependent upon center distance. Figure 7 shows the effect of changing center distance. The base circles and involutes are the same in both examples. When the center distance is changed, the angle that the line of action (the belt) makes with the line of centers will change

accordingly. Changing the center distance of the system changes the diameters of the operating pitch circles; however, it does not alter their ratio. Uniform rotary motion is still transmitted and the speed ratio remains the ratio of the base circles.

Pinion

Pinion dP1

RbP

RbP

Involute (Pinion)

Ôpt1 C1

dP2

P Involute (Gear)

Line of Action (Belt) RbG

Gear

Involute (Pinion)

Ôpt2 C2

dG1 R bG d d = G1 = G2 RbP d P1 dP2

P Involute (Gear)

Line of Action (Belt) RbG

dG2

Gear

Figure 7 -- Involute action and speed ratio independent of center distance

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4.1.5 Involute gears One large, single involute cam driving another single involute cam has functional limitations, such as being limited to less than one revolution. Figure 8 demonstrates how multiple, smaller involute cams, spaced evenly around the circumference of the pulleys, may be used to transmit motion in a continuous manner as the driver is rotated. These multi--cammed machine elements are usually referred to by a more common name ... involute gears.

leaving contact. Figure 11 shows the gears rotated to a position where only one pair is in contact. Contact ratio is discussed in 6.7.

Time = T1

Point of contact (T1=T)

Base pitch

Pitch point

Involute 2 cam surfaces

Line of centers

Point of contact (T1=T)

Line of action

Base circle

Base circle

Base circle Direction of force vector

Base circle

Involute 1 cam surfaces

Figure 9 -- Time = T1: Second pair of teeth just starting engagement

Figure 8 -- A series of small involute cams Not every pulley can be used as the basis for a gear. Certain criteria must be met. The circumference of the base circle must be divisible into an integer number of segments, so that an integer number of teeth can be constructed around it. With very few exceptions, gears contain integer numbers of teeth in order to rotate continuously. This division of the circumference defines another useful gearing concept. The length of each equal segment of the circumference of the base circle is called the base pitch (see 4.2.6). Physically, the base pitch is the distance between successive teeth around the gear and along the line of action. Two involute gears will mesh together if they possess equal base pitches.

Time = T2

Point of contact (T2 = T) Base pitch Line of centers Pitch point Point of cont (T2 = T) Base circle

4.1.6 Contact ratio Figure 9 shows that the base pitch is the distance along the line of action between successive involute flanks. In figure 9, one pair of teeth is in contact and the next pair is just beginning contact. Figure 10 shows the gears rotated until the initial pair is just

10

Base c Direction of force vector

Figure 10 -- Time = T2: First pair of teeth just leaving engagement

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theoretically vary as subsequent tooth pairs roll through mesh. However, the loading on a given tooth will vary depending on whether the load is carried by a single pair or by multiple pairs. Later discussion will show that the actual forces on the flanks vary because of frictional effects due to the rolling and sliding nature of involute action.

Time = T3

Point of contact (T2 = T)

AGMA 917--B97

Base pitch Line of centers

Pitch point Base circle Base circle Direction of force vector

Figure 11 -- Time = T3: One pair of teeth in contact at pitch point Contact ratio can be thought of as the average number of teeth in contact during one mesh cycle. It is very important to have a contact ratio greater than one, ensuring that a driving pair comes into mesh before the previous pair leave mesh. If contact ratio is less than one, the action between the gears is not occurring on involute flanks. Contact ratio less than one can result in non--uniform rotary motion being transmitted, noise and high stresses in the gears and other components of the system. Contact ratio is a key parameter in most gear designs and will be discussed in depth in subsequent clauses (see 6.7.5). 4.1.7 Properties of involute gear action Figures 9, 10 and 11 show an involute gear pair in mesh. The three positions shown demonstrate how the point of contact between the flanks of the gear teeth follows the straight line path of the line of action. It can be shown that the direction of the force vector between the two flanks is also in the direction of the line of action. This force is equal to and in the same direction as the force that will be transmitted in the crossed belt that comprised the first model of the system. Note the transmitted torque in the system does not

4.1.8 Rolling and sliding of involute gears Figure 12 shows two involute curves divided into corresponding lengths of arc which will pass through mesh at the same time. Notice that the length of the segment along an involute is short near the base circle and gets progressively longer farther out the involute. As two external involute teeth enter mesh, contact is near the root of the tooth on the driver and near the tip on the driven. The driver involute section is much shorter than the driven, yet both pass through mesh in the same amount of time. There is relative sliding between the tooth flanks. As the pair approach the pitch point, (in arc 12) the lengths of their respective involute arcs approach equality. Sliding is much less and at the pitch point pure rolling exists. Near the end of a mesh cycle, the situation is the reverse of the beginning, short segments on the driven and long on the driver. The result, however, is the same ---- significant relative sliding. Skilled gear designers attempt to minimize the sliding in gear meshes in order to increase efficiency and prolong the life of the gears. 4.1.9 Base pitch, diametral pitch and standard pitch diameter In general, the spacing of the driving surfaces of the teeth on a gear is called the “pitch” (see 4.2). This spacing may be measured on any circle, but one selected must be specified. Prior to the introduction of involute gearing, cog wheels, cycloidal and other tooth forms were used for clocks and machines. The nomenclature already in use for these early forms of gearing, terms such as pitch diameter and circular pitch, remained after involute gearing became more popular. Over time, the mathematics of the base pitch system of gear specification was almost completely replaced by the mathematically simpler “diametral pitch” system that is used today in the United States. The nomenclature of the diametral pitch system contains many of the same terms as the preinvolute systems, a fact that can

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provide confusion for novice gear designers. The remainder of this design manual uses the diametral pitch system to design gears. The mathematical link between the two methods (base pitch and diametral pitch) that describes the involute gear is

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the following: Diametral Pitch =

π cosprofile angle Base Pitch

Position of minimum sliding is the one shown here where arc 12 equals arc 12′. For rotation in either direction, sliding action will increase until the contact point reaches the base circle. Figure 12 -- Two involute curves showing differences in lengths of corresponding arcs

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It is clear that both a diametral pitch and a profile angle (see 4.1.9.1) must be specified in order to determine a unique base pitch. The prior discussion of involute theory has been based upon two dimensional geometries. The above expression is true for geometry in the transverse plane (see figures 1 and 2). A spur gear is a two dimensional profile projected into the third dimension just as a circle is projected to create a cylinder. A helical gear (see 4.1.9.2) is a two dimensional geometry that is rotated about its center as it is projected, forming a gear in the shape of a spiral. With the concept of helical gears comes the concept of a normal plane, a plane perpendicular to the flanks of the involute teeth (see figure 3), that is not the same plane as the transverse plane. In spur gearing, the normal plane and the transverse plane are the same. Normal diametral pitch is related to transverse diametral pitch by the expression presented in 6.2.1. In order to control the cost of gear cutting tooling, the industry standardized on the list of fine--pitch normal diametral pitches that appears in 4.3 as well as a very few standard normal profile angles. Normal diametral pitch hobs can be used to cut both spur and helical gears simply and inexpensively. It is important to understand that the design process that follows in this manual involves selecting a normal diametral pitch and normal profile angle, usually based upon performance criteria. Usually the specifications that appear on gear drawings contain only parameters in the normal plane because they relate to the cutter used to make the gear. However, the true gear action occurs in the transverse plane and therefore accurate analysis of a gear design requires analysis of the motion in the transverse plane. A thorough analysis clearly contains the elements of the base pitch system that is the root of involute gearing as described in the beginning of this manual. Dividing the number of teeth in the full circumference of a gear by its transverse diametral pitch results in a quantity called the standard pitch diameter (see 6.2.2). The standard pitch circle intersects the involute at the point where the normal pressure angle is equal to the normal profile angle of the cutter used to generate the tooth (see 4.1.9.1). The tooth thickness (see 6.6) is generally specified at the standard pitch diameter. The following clauses of this design manual will explain the

AGMA 917--B97

diametral pitch system and demonstrate its use. The user should always remember, despite numerous references to circles and diameters, that involute gears operate along a line of action, not around a circle! 4.1.9.1 Pressure angle, operating pressure angle and profile angle Many terms in common usage possess multiple meanings. Pressure angle is one of those terms. Figure 13 shows one usage of pressure angle, as a variable angle related to an arbitrary point on an involute curve. The pressure angle in this case is different for each point on an involute curve. This meaning is useful only when describing the geometric aspects of a single involute gear tooth. Point

P

Involute

r

Pressure angle

A Polar angle

θ Ô B rb O

Figure 13 -- Involute polar angle Figure 6 illustrates that the term operating pressure angle, Ôpt , is the angle between the line of action and perpendicular to the line connecting the centers of two meshed gears. Figure 7 demonstrates that the operating pressure angle changes when the center distance changes for the same pair of base circles. As was previously noted, a pressure angle must be specified in conjunction with a diametral pitch in order to uniquely specify the base pitch of an involute gear system. Confusion sometimes accompanies this seemingly contradictory concept of a specified constant pressure angle that changes with center distance.

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The AGMA has attempted to reduce the confusion by introducing the term profile angle to gearing nomenclature. In general, the profile angle is the angle between a line tangent to an involute gear tooth at the point of intersection between the involute and the standard pitch diameter and a radial line emanating from the center of the gear. This profile angle can be found elsewhere in gear geometry. Referring again to figure 13, the profile angle has the same magnitude as the pressure angle if the point on the involute is the intersection of the standard pitch diameter and a radial line from the center of the base circle. The profile angle at the standard pitch diameter has the same magnitude as the flank angle of a rack type cutter used to make a gear by the generating process. See Colbourne [1] for a description of gear generating. As was noted previously, a gear generating hob specification must include both a diametral pitch and a pressure angle. This usage of pressure angle is where employing the term profile angle is encouraged. Profile angle should be used to describe individual gears, gears not in mesh. Pressure angle should be used to describe the angle of the line of action when gears are in mesh. The pressure angle is equal to the profile angle when gears are operating in mesh on their standard center distance. If they are mounted at a center distance different from standard, then their operating pressure angle is numerically different from the profile angle to which they were manufactured. 4.1.9.2 Helix angle and helical gears A helical gear has the appearance of a spur gear that has been twisted about its axis. In actual manufacture, helical gears can be cut on hobbing machines just as are spur gears. The same normal diametral pitch hobs are used. The hobbing machine is set up with a specific feed rate in relation to the index ratio of the work spindle in order to cut the desired spiral (see Townsend [5] for more information on the hobbing process). The helix angle, ψ, of a helical gear is the angle between the axis of the gear and the element of the flank that intersects the standard pitch cylinder, see figure 14. Helical gears are normally specified by proportions measured in the plane normal to the gear tooth

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profile. This is primarily because normal profile dimensions correspond directly to the hob used to manufacture the gear. However, during the design of the gear, much attention is paid to the proportions of the gear measured in the transverse plane. The transverse plane is the plane perpendicular to the axis of the gear.

Helix Tooth

Helix angle Axis

Figure 14 -- Helix angle Clause 6 presents the equations necessary to calculate gear parameters for both spur and helical gears. It is clear from the equations presented that a spur gear is mathematically a helical gear with zero helix angle. 4.1.10 The involute and the rack A driving involute can also drive a rack. Note that as the diameter of a base circle grows, the curvature of the involute described by it increases in radius. A rack may be considered a sector of a gear of infinite radius. Thus the rack teeth of the involute system of gearing have straight line profile elements, (straight--sided teeth). This has great advantages in the manufacture of gears in that many of the types of tools used to cut and finish gear teeth can employ straight line elements. Some hobs, rack cutters and grinding wheels are examples. Such straight--line elements can be more accurately formed and inspected than can complex curves. A single hob can be used to cut gears of any number of teeth, all of which will mesh properly with each other. 4.1.11 The involute function Figure 13 shows the geometric features of the involute curve. Previous portions of clause 4 described the involute as the path of a point on a belt as it is transported around a pulley system. The involute is also the curve produced, as a trace on a fixed background, by a point (knot) in a taut string unwound from a circle. The circle is the base circle

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of the involute, just as was the pulley in the previous visualization. The equation of the involute function is: θ = inv Ô = tan Ô -- Ô where Ô

...(1)

is pressure angle at a given radius measured in radians.

4.2 Pitch Pitch is the distance between a point on one tooth and the corresponding point on an adjacent tooth. It is a dimension measured along a curve in the transverse, normal or axial directions. The use of the single word “pitch” without qualification may be confusing, and for this reason specific designations are preferred, e.g., transverse circular pitch, normal base pitch, axial pitch. See figure 15.

AGMA 917--B97

4.2.4 Normal circular pitch Normal circular pitch is the circular pitch in the normal plane and also the length of arc along the normal pitch helix between helical teeth. See figure 16. 4.2.5 Axial pitch Axial pitch is the linear pitch in an axial plane and in a pitch surface. In helical gears, axial pitch has the same value at all diameters. The term axial pitch is preferred to the term linear pitch.

Transverse circular pitch Axial pitch

Pitch

Axis Normal circular pitch

Circular pitch Figure 16 -- Tooth pitches 4.2.6 Base pitch Figure 15 -- Pitch 4.2.1 Circular pitch Circular pitch is the arc distance along the pitch circle between corresponding profiles of adjacent teeth. See figure 15.

Base pitch in an involute gear is the pitch on the base circle or along the line of action. Corresponding sides of involute gear teeth are parallel curves and the base pitch is the constant and fundamental distance between them along a common normal in a plane of rotation, (transverse plane). See figures 17 and 18. Base pitch

4.2.2 Transverse circular pitch Transverse circular pitch is the circular pitch in the transverse plane. See figure 16.

Circular pitch

4.2.3 Transverse operating circular pitch Transverse operating circular pitch is the arc distance along the operating pitch circle between corresponding profiles of adjacent teeth. This parameter is used to calculate backlash because backlash is defined on the operating pitch diameters.

Base pitch Base circle Base tangent Figure 17 -- Principal pitches

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AGMA 917--B97

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4.2.7 Normal base pitch Normal base pitch in an involute helical gear is the base pitch in the normal plane. It is the normal distance between parallel helical involute surfaces on the plane of action in the normal plane, and it is the length of arc on the normal base helix. It is a constant distance in any helical involute gear. See figure 18. Helical rack

Base pitch

and racks, have a basic tooth form which, when cutting a gear on a generating machine, produces involute gear teeth. Although a hob, for example, can generate gears having any desired number of teeth, it can only produce a single normal diametral pitch and normal profile angle. Since several tools may be needed to produce a job lot of gears, it is generally desirable to select a diametral pitch for which most gear shops are likely to have tooling. This may avoid the need to purchase special tooling. Thus the following have become recommended normal diametral pitches, Pnd . 20, 24, 32, 40, 48, 64, 72, 80, 96, 120 4.4 Center distance

Normal base pitch Axial base pitch Figure 18 -- Base pitch relationships 4.2.8 Axial base pitch Axial base pitch is the base pitch of helical involute tooth surfaces in an axial plane. See figure 18. 4.3 Diametral pitch Diametral pitch is not a pitch in the same sense as the preceding pitches. It represents the size of the tooth. The larger the numeric value of the diametral pitch, the smaller the size of the gear tooth. Diametral pitch is related to circular pitch by the following: π Diametral pitch = Circular Pitch It is customary to discuss the size of a given gear in terms of its diametral pitch rather than its circular pitch. Many fine--pitch gears are produced by means of generating tooling. Even gears produced by molding, casting or stamping are intended to have teeth which are the same as if they were generated. Gear generating tools, such as hobs, shaper cutters

16

The center distance, the distance between two gear axes, is a basic dimension for meshing a pair of gears. It is controlled by the frame (casing or gear box) and the bearings that support the gears. Two values are of particular interest: the minimum operating center distance which can occur when the sum of all tolerances accumulates to the minimum functional value, and the maximum operating center distance which occurs when all tolerances act in the opposite sense. Items that comprise the center distance tolerance in designs using rolling element bearings include such factors as: -- The maximum and minimum bearing center distances; -- The clearances between the outer races and the bores; --

Outer race concentricity;

--

Bearing internal clearance;

--

Eccentricity of the inner race;

-- Concentricity between race journals and the gear bores; --

Clearance between inner races and shafts.

A common error committed by the novice designer is to underestimate the buildup of tolerances and the effect the resulting change in center distance has on the operation of gearing. This effect can be significant in fine--pitch gearing. For example, a total center distance tolerance of 0.010” may be trivial in a system using 20 diametral pitch gearing and completely unacceptable in a system using 64 diametral pitch gears. Careful analysis and control of center distance variation is imperative in fine-pitch gear design.

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An outstanding advantage of the involute as a profile shape is its ability to transmit uniform motion independently of center distance (see 4.1.1.). It is the only curve suitable for gearing that has this characteristic. This gives the designer great freedom in creating a spur or helical gear design. Center distance is generally categorized as standard, enlarged or reduced. Enlarged and reduced center distance designs are achieved by making design adjustments to tooth proportions. See clause 6 for a detailed discussion of center distance.

AGMA 917--B97

the designer to a set of gears that meet the load requirements (bending strength and surface strength) and the contact ratio requirements of the system. In the detailed design stage, the tooth geometry is determined in order to optimize the gear system for its application. The specific tooth modifications will vary depending upon the type of gear system being designed. At this level of the design, it is helpful to consult an experienced gear designer and/or the gear design references listed at the end of this information sheet. 5.2 Limitations

5 Application considerations 5.1 Principal gear functions Gears are used to transmit power and/or motion from one shaft to another. If their principal function is to transmit power, they are called power gears. If their principal function is to transmit smooth motion, they are called smooth motion gears. If their principal function requires minimal backlash, they are called zero backlash gears. Since there are usually major differences in the basic requirements of each of these systems, the design emphasis will usually be different for each system. It is, however, quite possible that each of these systems may have some design requirements that are common to each other (example: must be quiet and have adequate load capacity). A gear design flow chart is shown in figure 19. This flowchart should help guide the designer through the various steps of the design process. It is not intended to be a detailed road map for every gear application, but rather a general overview of some of the important things to consider when designing gear systems. As shown in the flowchart, at the first step of the design process the functional requirements of the system should be determined. This will establish a “perspective” as one proceeds through the rest of the steps. Many design decisions will have to be made as one proceeds through the different phases of design. Realizing the functional requirements of the system early will help in making those decisions. After determining the type of gear system that is required, one proceeds down through the column on the flowchart. The steps up to and including the preliminary design analysis help guide

The information in this design manual is meant to serve only as a guide to the designer of fine--pitch gear systems. It is not implied that using the procedures will necessarily result in gears that will meet the requirements in every application. It remains the responsibility of the individual designer to properly evaluate the conditions in the particular application and to make use of prior experience and/or proper testing to confirm the suitability of the design. 5.3 Design parameters Gear design is a decidedly iterative process of estimation, followed by calculation of gear parameters, followed by analysis to assure no engineering rules have been violated. The initial estimate is usually difficult for the novice gear designer. Flow charts (figures 19 and 20) attempt to provide insight into the complexity of how to choose diametral pitch, number of teeth, and number of stages in the gear system. The designer must determine the external boundary conditions before the type of gears (spur, helical, worm etc.) can be determined. This clause lists some of the external boundary conditions that should be considered for gear applications. It is the responsibility of the designer to consider any other boundary conditions that may be relevant to the particular application. 5.3.1 Loads To determine the loading conditions on the gear system, consideration should be given to the following: -- The power rating of the prime mover, its overload potential, and the uniformity of its output torque. The overload potential is especially important if there is a possibility of a lockup in the system which could cause the motor to stall.

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-- The output load requirement, including the following: -- Normal output load; -- Peak loads and their duration;

--

Frequency of severe loading or stalling;

-- Inertia loads arising from acceleration or deceleration of the complete system.

Determine functional requirements • • • • •

Design parameters

Loads Space constraints Life requirements Speed ratio Operating environment

Preliminary estimate of gears

• • • • • •

Bending stress Contact stress Pitting resistance Fatigue strength Lubrication

Number of teeth Diametral pitch Face width Helix angle Material/heat treat Quality class

Is analysis OK

False

True Gear Assembly

• Mounting/alignment • Housing strength/deflection • Environment

Power gear systems

Smooth motion gear systems

Position control gear systems

Detailed design optimize considering:

Detailed design optimize considering:

Detailed design optimize considering:

• • • • •

Minimum volume Fatigue strength Pitting resistance Noise Cost

Gear specifications • Data block • Blank design

• • • • • • •

Transmission error Contact ratio Tooth modification Minimum volume Fatigue strength Noise Cost

Gear specifications • Data block • Blank design

Figure 19 -- Gear design flowchart

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• • • • • • • •

Total angular backlash Contact ratio Transmission error Tooth modification Minimum volume Fatigue strength Noise Cost

Gear specifications • Data block • Blank design

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Given:

• Speed requirements • Center distance • Assume 20 deg. profile angle at this stage • Torque requirement

AGMA 917--B97

Determine preliminary estimate of:

Determine gear train ratio from speed requirements

• • • • •

Add another stage to the gear train if possible

yes

Is stage ratio > 5

no Calculate operating pitch diameters from center distance & gear ratio

Number of teeth in pinion Number of teeth in gear Diametral pitch Helix angle Face width

Refer to clause 6.3.5

Calculate tangential force from torque and operating pitch diameter Spur gears: assume geometry factor = 0.4 Helical gears: assume geometry factor = 0.45

Refer to clause 6.9.5

Preliminary estimate of material based on: Duty cycle Cost considerations Lubrication requirements Noise considerations Determine failure criteria and the appropriate mechanical properties based on application

Refer to clause 5.3

If space constraints permit F = 0.5 DP can be used for a starting point Spur gear: Helix angle = 0.0 Helical gear: Initial choice for helix angle = 15 -- 20 deg.

Pd min.

(Coarsest)

Pd max.

Determine limits for diametral pitch Pd

Calculate Pd (coarsest) from operating pitch diameter such that pinion gear has min. of 9 teeth

See clause 6.9.5

(Finest)

Calculate Pd (max., finest) using Lewis Form approximation for bending stress

Is Pd max < Pd min

Change material and/or heat treatment

False

True

Pd min

20 24

Pd max

32

40

48

64

72

Coarsest Advantages

80

120

96

Finest

Lowest bending stress Fewest teeth in pinion & gear Min. change in contact ratio as center distance deviates from nominal

Disadvantages

Lowest contact ratio at nominal center distance Addendum modifications should be done to pinions with fewer than 24 teeth to eliminate undercutting

Lowest tooth mesh frequency

Advantages

Highest contact ratio at nominal center distance

Disadvantages

Highest bending stress Max. change in contact ratio as center distance deviates from normal

Highest tooth mesh frequency

Select a diametral pitch between the maximum and mimimum limits based on the specific application and the general guidelines previously mentioned Calculate NP and NG from operating pitch diameter and diametral pitch refer to clause 6

Figure 20 -- Preliminary design flowchart

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-- The additional loads induced by couplings or other interconnections between the prime mover, gear train and driven load; -- Some gear systems have multiple outputs driven by a common input. In these cases, the torque required at each output is determined first. Torque requirements at each stage are then calculated being careful to add together the torques at the gear interfaces where the torque path splits. In this way, the torque is determined for each gear in the path back to the prime mover. An example of a torque split system can be seen in figure 21. From this analysis, a basic design load for the gear system as well as suitable overload factors can be determined. Factors to account for desired life and other safety factors will also result from this analysis. The actual loads may vary from the predicted loads due to inefficiencies of the system. It is recommended that torque measurements be made to verify actual loads before the design is finalized. Clause 15 provides more information on recommended testing procedures.

5.3.2 Gear train ratio The gear train ratio is the ratio of the angular speed of the first (driving) shaft to the last (driven) shaft. A gear train may consist of several pairs of meshing gears. Each meshing gear pair consists of a driver and a driven gear. The ratio of the speed of the driver to the driven gear, of any pair, is called the stage ratio. Since there must be an integer number of teeth on each gear, the achievable stage ratios are limited to ratios of integer numbers. The gear train ratio is equal to the product of the individual stage ratios. The required gear ratio will often influence the selection of the type of gearing, since each type of gearing has its own stage ratio limitations.

Torque required output 1

n Gear Ratio : (m Gt ) = n 1 L where: n1 is rotation speed of first shaft; nL is rotation speed of last shaft. mGt = (mGs1) (mGs2) (mGs3) (mGs4)...(mGs(L-- 1) )

-- When designing parallel axis spur or helical gear systems, a stage ratio limitation of 5:1 should be considered for the following reasons: --

To minimize the specific sliding ratios;

--

To increase the mesh efficiency;

-- To achieve more balanced strength between pinion and gear; --

Torque required output 2

Motor pinion

Figure 21 -- Torque split

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To minimize the size of the gear.

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The 5:1 stage ratio limitation is a general “rule of thumb”, not an exact design criterion. Addendum modifications (profile shift modifications) can be made to the pinion to improve its strength and reduce its specific sliding ratio. If the gear designer/engineer requires stage ratios greater than 5:1, more detailed information can be found in gear texts such as “On the Geometry of External Spur Gears” by T.W. Khiralla [2]. An experienced gear engineer should be consulted if the required stage ratio is significantly higher than 5:1.

AGMA 917--B97

Idler gears, which are both driver and driven gears, do not affect the overall gear train ratio. They do, however, change the direction of rotation. See figure 23 for an illustration of a gear system with an idler gear. 5.3.3 Space constraints Space constraints can also limit the type of gear that can be used. Some design parameters that can be considered as space constraints are listed below. The examples listed are not meant to be limiting or all inclusive.

-- A speed decreasing gear train should consist of only speed decreasing gear stages when possible. (The driven gear has more teeth than the driver.) An example can be seen in figure 22. -- A speed increasing gear train should consist of only speed increasing gear stages when possible. (The driven gear has less teeth than the driver.) Care must be taken when designing speed increasing drives for maximum recess action. The driven gear should have a sufficient number of teeth so that it can be modified (reduced) without causing excessive undercutting. Reference [2] contains more information about recess action gearing. NOTE: It is sometimes necessary to mix speed decreasing and speed increasing gear stages to achieve an exact ratio. However, this will usually result in larger gears and/or more gear stages.

60T

20T

--

The dimensions of the design volume;

-- Existing center distance specifications (for example: several gear systems that can be assembled into a common gear box to provide different gear ratios); -- Making use of the same gear in more than one location in the gear system; -- Reverted gear trains, where the output shaft and input shaft are co--linear; -- The shaft angle orientation. The possibilities are parallel, nonparallel and intersecting, and nonparallel and nonintersecting. Each of these conditions will determine the type of gearing that can be used. The type of gearing that is appropriate for each of these conditions is discussed in 5.4.

60T

20T

Driver n1, rpm ccw

Stage Ratio cw m Gs nn1 = N 2 N1 2 where: n1 is speed of driving shaft, rpm; n2 is speed of driven shaft, rpm; N1 is number of teeth in driving gear of stage; N2 is number of teeth in driven gear of stage.

Driven n2, rpm ccw

Figure 22 -- Speed decreasing gear system

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60T 40T 20T

Driver n1 rpm ccw Idler Gear

Gear Train Ratio = 40 × 60 = 60 = 3 20 40 20

Driven nL rpm ccw

Figure 23 -- Gear train with idler gear 5.3.4 Operating environment The operating environment may affect the performance of the gear system. It is extremely important that the gear system be designed to operate in the environment in which it will be used. Some types of gears have advantages over others in certain operating environments. Some important considerations of the operating environment are: -- Enclosed gear system. The gears are mounted in an enclosed gearbox. They are usually protected from external contamination and can often be oil lubricated and cooled. This is an ideal situation for gears to operate; -- Open gear system. The gears are mounted on shafts in the open. They often experience outside contamination such as: --

Moisture;

--

Corrosive environment;

--

Abrasive wear debris;

--

Dirt, dust, etc.

It is important to understand the type of contamination that can be present so that the proper lubrication can be specified. Lubrication is often a necessity, but occasionally, in open gear system applications, external oils or grease can actually increase the wear rate by trapping contamination or wear debris in the gear mesh. It is often necessary to increase the operating

22

backlash to allow space for the debris. It is recommended that gear systems operate in an enclosure if the possibility of outside contamination exists; -- Operating temperature of the gears. Overlooking or underestimating the effect of temperature on a gear system can have disastrous consequences. Aircraft gearing must often function from --50°F to +300°F. In general, it is good practice to consider the possibility of lubrication and cooling for any gear system. Elevated temperatures can contribute to premature failure, especially when the gears are made from plastics. The mechanical properties of plastics (tensile strength, flexural modulus, fatigue strength, etc.), vary with temperature and in some plastics are significantly lower at elevated temperatures. When plastic gears are operated at elevated temperatures, the designer must account for the effects the elevated temperatures have on the mechanical properties. In many applications plastic gears are run without external lubrication. The higher coefficient of friction of unlubricated plastic gear mesh combinations, combined with high speed operation can generate enough heat to significantly increase the operating mesh temperature. Additionally, thermal expansion coefficients of plastics (K¢10--5), are typically an order of magnitude higher than of steels (K¢10--6). Therefore, adequate backlash must

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be provided to compensate for the increased tooth thickness at elevated temperature. Some applications require special tooth compensation when manufactured at ambient temperature to ensure the tooth form will be correct at the elevated operating temperature. The gear designer is advised to fully research the mechanical properties of any plastic considered for a gear design, particularly with respect to the plastics performance at elevated or reduced temperature. 5.3.5 Life requirements The life requirements of the gear system will determine the allowable bending stress (fatigue), and surface stress (pitting) of each of the gears in the system. Some factors that can affect the allowable stresses are: -- Number of load cycles. This will be used to determine the bending fatigue and surface fatigue endurance limits for the material being used. Note that non--ferrous materials, such as plastics, bronze, etc., do not have true endurance limits. The designer must know the allowable stress for the number of cycles required. Note that the number of load cycles is not always the same as the number of revolutions; an idler gear is an example; -- Reliability of the gear system. The higher the required reliability of the system, the lower the allowable stresses. Note that the reliability of the system includes the reliability of the gear train, which is the product of the reliabilities of the successive gear stages. The reliabilities of the stages need not be equal, and in general are not equal; e.g., the reliability of a very low speed stage could be higher than that of a very high speed stage; -- One--way vs. reversing drive. The endurance limit for a reversing drive is lower than for a one--way drive because the root of the tooth experiences complete stress reversals. Complete stress reversals also occur in idler gears, even when they are driven in one direction, because opposite tooth flanks transmit the load. The designer can obtain more information about the effect of complete stress reversals on fatigue life by consulting various machine design texts;

AGMA 917--B97

-- Operating temperature. The operating temperature can affect the life of a gear system by reducing the tensile strength and fatigue strength of the material. The relationship between temperature and mechanical properties need to be known for the specific material being used. This information should be obtained from the material supplier when possible. Testing should be done prior to finalizing the design; -- Miscellaneous effects. Many other factors can affect the life of the gear system. For example, the type and accuracy of the mounting could affect the load distribution across the gear face, which may increase the bending stress. The designer must consider any other factors which may adversely affect the life of the gear system. 5.3.6 Lubrication requirements The method of lubrication depends on the environment in which the gears will operate. Some designs provide an oil--tight enclosure with suitable shaft seals. This is generally a very desirable way to provide lubrication for gears. In other designs the surrounding equipment may be absolutely intolerant of external lubrication in any form. Gears in this type of design must be designed to operate in a non--lubricated condition. The lubrication requirements of the gear system need to be understood before the type of gear and gear material can be determined. There are three general categories of gear lubricants: liquids (oils), semisolids (grease), and solids (graphite, molybdenum disulphide, polytetrafluoroethylene (PTFE)). Oil and grease are applied externally; however, some solid lubricants such as PTFE can be molded into a plastic gear at the time of manufacture. Oil lubrication of gears has the following advantages over grease: -- Oil has a greater range of operating speeds and temperatures; -- Oil is more effective in conducting heat from the gear teeth to the housing; -- Oil will saturate all areas of contact and is more effective in removing wear debris; --

Easier to fill and drain a reservoir using oil.

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Grease lubrication of gears has the following advantages over oil:

Pinion

-- Under heavy loads and slow speeds, grease can provide a lubricating film whereas under the same conditions oil cannot; -- Grease has much slower spreading and evaporation rates than oil; -- Grease is easier to seal from leakage in enclosed gear drives.

Gear

Some features of solid lubricants that make them suitable for special applications are: -- Solids can be used in open gearing where oil or grease would collect contaminates;

Rack

-- Solids resist being squeezed out of gear teeth under high pressure; --

Figure 24 -- External spur gears

Solids do not evaporate;

-- Solids can be used in applications where oil or grease contamination can not be tolerated. Refer to clause 13 of this Design Manual and ANSI/AGMA 9005--D94 for more information on lubrication of gear systems. 5.4 Types of gearing After the external boundary conditions are known, the selection of the type of gearing can be made. Each type of gearing has unique features that make it more suitable for some applications and less suitable for others. The following paragraphs give a summary of each and list some advantages and disadvantages. Several references, such as “Analytical Mechanics of Gears”, by Earle Buckingham [3], contain more detailed information about the various types of gearing. Brief descriptions are given for each type of gearing in this clause, but only spur and helical gear systems are covered in this Design Manual. 5.4.1 Gearing on parallel axes 5.4.1.1 Spur gears A spur gear is a gear cut from a cylindrical blank, with teeth parallel to the gear axis. Spur gears can be either external, with the teeth projecting away from the center, or internal, with the teeth projecting toward the center. A set of external spur gears can be seen in figure 24. An internal spur gear with an external spur pinion can be seen in figure 25.

24

Figure 25 -- Internal spur gear and external spur pinion

Advantages of spur gears: --

Transmit no axial (thrust) force;

--

Relatively low manufacturing cost;

--

Relatively low assembly costs.

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Disadvantages of spur gears: -- Lower load carrying capacity than helical gears of same size; -- Relatively low contact ratio (usually less than 2.0); -- Larger change in mesh stiffness during a complete mesh cycle; --

Often noisier than helical gears.

5.4.1.2 Helical gears A helical gear is a gear cut from a cylindrical blank, with teeth that are on helices about the axis of the gear. The twist direction of the helix is designated as right hand if it twists clockwise when viewed along its axis. The twist direction is designated as left hand if it twists counter--clockwise when viewed along its axis. External helical gear pairs are of opposite hands, while external -- internal helical gear pairs are of the same hand. Figure 26 shows a set of external helical gears.

AGMA 917--B97

Disadvantages of helical gears: --

Transmit axial (thrust) forces;

--

Somewhat higher manufacturing cost.

5.4.2 Gearing on nonparallel shafts 5.4.2.1 Face gearing Face gearing consists of a spur or helical pinion and a disk--like gear. The axes are usually at right angles and may be intersecting or nonintersecting. The face gear resembles a rack wrapped into a circle. Figure 27 shows a spur pinion and face gear with right angle intersecting axes. There are some practical limitations to the allowable stage ratios when using face gears. Generally ratios less than 1.5:1 should not be attempted with face gears. This type of gearing is not included in this Design Manual. For further information, see AGMA 203.03, Fine--Pitch On--Center Face Gears for 20--Degree Involute Spur Pinions. Offset

Pinion on center

Pinion off center

Figure 27 -- Spur pinion and face gear Figure 26 -- External helical gears Advantages of helical gears: -- Higher load carrying capacity than spur gears of same size; --

Higher contact ratio can be obtained;

-- Less change in mesh stiffness during a complete mesh cycle; -- Generally quieter than spur gears if the active face width is greater than one axial pitch.

5.4.2.2 Bevel gearing Bevel gearing is conical in form. Bevel gears can carry higher loads than crossed helical gears and face gears of the same size. The center distance and shaft alignments in two directions must be very accurate to ensure proper load distribution across the tooth. The following tooth element shapes can be used: -- Straight tooth elements which if extended, would pass through the intersection point of the axes (see figure 28a);

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-- ZEROLR curved tooth elements which follow the same general direction as straight teeth (see figure 28b). -- Spiral tooth elements, which are curved and oblique (see figure 28c).

These types of gearing are not included in this Design Manual. For further information, see ANSI/AGMA 2005--B88, Design Manual for Bevel Gears. 5.4.3 Gearing on nonparallel, nonintersecting shafts 5.4.3.1 Crossed--axis helical gearing (formerly called spiral gearing)

Straight bevel gears

Skew bevel gears

(a) -- Straight tooth bevel elements

Spiral bevel gears

Crossed helical gears are helical gears mounted on axes which are skewed. The teeth may be of the same or opposite hands, and the helix angles can be equal or unequal. The use of crossed helical gears should be limited to low torque applications because the teeth make point contact with each other. It is recommended that an experienced gear designer be consulted when designing gear systems using crossed helical gears. Crossed--axes gearing is not included in this Design Manual. Figure 29 shows a set of crossed helical gears.

ZEROL bevel gears

(b) -- Curved tooth bevel elements Figure 29 -- Crossed helical gears Left--hand

5.4.3.2 Worm gearing

Right--hand

Spiral bevel gears

Spiral bevel pinions

Left--hand

Right--hand

(c) -- Spiral tooth bevel elements Figure 28 -- Bevel gearnig

26

This type of gearing uses a worm to drive a wormgear. The worm is cylindrical in form with the teeth shaped like screw threads. The worm can have one or more teeth (also called starts). The designer should be aware that the torque ratio is not equal to the reduction ratio in worm gearing, as it is in parallel axis spur and helical gear systems. This is because the pitch diameters are not in the same ratio as the number of teeth. The efficiencies of worm drives are dependent upon the coefficient of friction between the worm and wormgear as well as the lead angle of the worm (helix angle of wormgear for 90 degree shaft angles). This type of gearing is

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not covered by this Design Manual. Information on fine--pitch wormgearing may be found in ANSI/ AGMA 6034--B92. Figure 30 shows a worm and wormgear set. Cylindrical worm

Enveloping wormgear

Cylindrical (non--enveloping) wormgear

Figure 30 -- Worm and wormgear set 5.4.3.3 Rotary to linear motion gearing This type of gearing is made up of a rack and pinion and is used to convert rotary motion into linear motion, or vice--versa. When two gears mesh, the smaller of the two is referred to as the pinion, and the larger is usually referred to as the gear. Any gear meshed with a rack is considered smaller than the rack, since the rack is part of a gear with an infinite number of teeth (it has an infinite pitch diameter). Therefore, it is common to speak of a rack and pinion. A spur rack, which has its teeth at right angles to the direction of its motion, is used with a spur pinion (see figure 31). A helical rack, which has its teeth oblique to the direction of its motion, is used with a helical pinion.

Figure 31-- Spur rack and pinion

AGMA 917--B97

5.5 Preliminary estimate of gears After the external boundary conditions are known and the type of gearing has been determined, a preliminary estimate of the size of the gears can be made. The amount of power or torque that a gear can transmit is a function of the size of the gears, their type and the materials from which they are made. The size of each gear is defined by its diametral pitch, number of teeth and face width. The preliminary design is only the first step of the total design process. A design should not be considered complete until it has been analyzed and tested. In some design problems, a preliminary design is needed in order to establish some size requirements for the complete equipment. Such a preliminary design may not have the benefit of adequate knowledge of loads or other constraints imposed by the final design of the equipment. In these cases, the preliminary design must be reviewed and revised in light of new information. 5.5.1 Number of teeth The number of teeth selected for each member (pinion and gear) will have a considerable effect on the performance of the gears in service. The following should be considered when selecting the numbers of teeth: -- Integers. The number of teeth in each gear must be an integer. This restricts the actual ratio that a given pair of gears may have; -- Undercutting. Gears with less than 24 teeth (for 20 degree profile angles) usually require addendum modifications (profile shift) to avoid undercut or undesirable contact close to the base circle. Undercutting reduces the tooth’s strength and often reduces the contact ratio. See clause 6 for additional information on calculating contact ratio for gears with undercut. Contact close to the base circle increases the ratio of sliding to rolling which reduces the efficiency of the mesh; -- Hunting tooth action. In some gear applications, it may be desirable to provide hunting tooth action; i.e., using gears whose tooth numbers have a greatest common factor of 1. This technique minimizes the number of times a specific pair of teeth mesh with each other. This is done to reduce repeatability of individual meshes and promote uniform wear.

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lowers the highest point of single tooth contact so that when a single tooth is carrying the load it occurs lower on the tooth flank. This reduces the bending moment on the tooth which reduces the deflection.

5.5.2 Diametral pitch There are several factors which need to be considered when selecting the diametral pitch for gear systems. Some of these factors are listed below: NOTE: These comparisons are valid for gears of a constant base diameter (i.e., if the diametral pitch is doubled, the number of teeth is doubled so that the base circle diameter remains the same).

--

Load carrying capacity (beam strength).

It is important to remember that a gear tooth acts as a cantilever beam. It has a decidedly non-uniform cross section and its load point varies, but when loaded it deflects like a cantilever beam; -- The bending stress in the root area of the tooth is approximately proportional to diametral pitch (e.g., if the diametral pitch is doubled, the bending stress is doubled). The reason it is not exactly proportional to diametral pitch is that a change in contact ratio also changes the radius of the highest point of single tooth contact; -- Even though the mesh stiffness equation shows that the mesh stiffness is independent of diametral pitch, the mesh stiffness actually increases slightly as the diametral pitch is increased. This occurs because the contact ratio is slightly higher for the finer pitch gears. The higher contact ratio

--

Contact ratio. -- The contact ratio increases as the diametral pitch increases (at the nominal center distance); -- The tolerance on the center distance, to maintain an acceptable contact ratio, must decrease as the diametral pitch increases to maintain the same contact ratio. Figure 32 also shows contact ratio as a function of center distance deviation for various diametral pitches.

-- To fit the design envelope (pitch diameters of gears). After the number of teeth have been chosen to give the required ratio, the diametral pitch can be determined by knowing the allowable size of the pitch diameters. The designer must ensure that the gears have adequate load carrying capacity when determining diametral pitch in this way. The gear industry has standardized on several diametral pitches. Although it is sometimes necessary to design gears with nonstandard diametral pitches, it is usually recommended that standard diametral pitches be used when possible because tooling and inspection gears are readily available.

2.0

Transverse contact ratio

1:1 Ratio 5:1 Ratio

120 DP

80 DP

40 DP

20 DP

1.0 Center distance deviation NOTE: The numbers on the chart were generated for a specific center distance.

Figure 32 -- Contact ratio vs. center distance deviation for 20 degree profile angle gears

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

The standard fine--pitch normal diametral pitches are:

AGMA 917--B97

contact ratio be at least 2.0 to achieve good helical overlap.

5.5.3 Face width

-- The transverse contact ratio is defined the same as for spur gears, using transverse dimensions;

Face width is the length of the teeth in the axial plane. The following factors should be considered when determining the face width of gears:

-- The face contact ratio is a measurement of the helical overlap and is equal to the face width divided by the axial pitch.

20, 24, 32, 40, 48, 64, 72, 80, 96, 120

NOTE: The following comparisons are made assuming there is uniform load distribution across the face width.

-- Load carrying capacity. The bending stress is inversely proportional to face width. Therefore, when everything else is held constant, as the face width is doubled the bending stress is halved. Note that when calculating load carrying capacity the active face width should be used. The active face width is the actual face overlap between the pinion and gear. Axial misalignment of the gears and rounding or chamfering of the tooth may reduce the active face width to a value less than the minimum face width of the pair. Usually the face width of one of the gears is made wider to allow for axial misalignment and still maintain full face contact. The increase in face width is usually made to the pinion because it requires less additional material. For additional information on the load rating of gear teeth, refer to ANSI/AGMA 2001--C95, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth; -- Face width--to--diameter ratio. To achieve more uniform tooth contact along the face, the ratio of face width to diameter should usually be held to below 2.0. If the alignment of the pinion and gear teeth cannot be closely controlled, the ratio should be held well below this limiting value. The alignment is influenced by gear accuracy, housing accuracy, shaft deflections, gear blank rigidity and the accuracy and rigidity of the gear mounting features; --

Contact ratio. -- Spur Gears: Face width does not affect the contact ratio of spur gears; -- Helical Gears: The total contact ratio is the sum of the transverse contact ratio and the face contact ratio. If face contact ratio is less than 1, no advantage is gained from helical action. It is suggested that the face

The helical overlap is proportional to the face width for a given diametral pitch and helix angle (axial pitch). In some applications, it is advantageous to design face width and helix angle combinations which result in integer face contact ratios. The resulting contact lines are of constant length, yielding more uniform mesh stiffness. Uniform mesh stiffness may reduce vibration which would result in smoother operation of the gears. 5.5.4 Helix angle The following factors should be considered before choosing a helix angle for a gear system: -- The higher the helix angle the higher the axial component (thrust) that is transmitted; -- The maximum strength of a helical gear occurs between a helix angle of 10 to 20 degrees. See AGMA 908--B89; -- The higher the helix angle, the higher the face contact ratio, for a given face width; -- The higher the helix angle, the larger the pitch diameter of the gear, for a given number of teeth and normal diametral pitch; -- Index gearing for machine tools. Consult gear manufacturer for manufacturability; -- Availability of cutter and helix guide for gear shaping. 5.5.5 Profile angle The fine--pitch gear industry has standardized on 20 degree profile angles for most applications. The use of 14.5 degree profile angle gears has declined in the past several years due in large part to the increased strength and reduced sliding characteristic of 20 degree profile angle gears. The contact ratio is higher for 14.5 degree profile angle gears than for 20 degree profile angle gears of the same diametral pitch and numbers of teeth. In some applications this increase in contact ratio

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could allow enough additional center distance deviation to allow for assembly tolerances. 5.5.6 Material and heat treatment In order to make a preliminary estimate of the size required for a set of gearing, it is necessary to establish the general type of material to be used, such as steel, aluminum, bronze or plastics. In power gearing, where strength and pitting resistance is of major importance, steel is often selected because it can be heat treated. In position control gearing, wear resistance is of prime importance. The material combinations are chosen to minimize tooth wear. In some applications there is an added constraint of operation without lubrication. In these applications the selection of the materials is very critical to the reliability of the system. Depending on the required loads, plastics may be a better choice than steel when the gears must operate without external lubricants. For power gear systems, or any gear system that is transmitting high loads, the heat treatment is vital to the performance of the system. The following four general ranges of tooth hardness may be considered when making a preliminary design: --

low through hardened (150 -- 210 BHN);

--

medium through hardened (210 -- 420 BHN);

--

surface hardened (50 -- 62 Rc);

-- through hardened to 40--62 Rc when employing alloy or tool steels, martensitic stainless, etc. NOTE: For very fine--pitch gears, care should be taken so that the case depth from the heat treatment does not go through the entire tooth. See also clause 10.

5.5.7 Manufacturing method There are many processes available to manufacture gears. There are many factors which collectively determine the manufacturing process that should be used. Some of these factors are: -- Process must be capable of meeting design quality parameters. -- Process must make parts for acceptable cost. -- Process must be capable of meeting the schedule demands of the part. It is important to consider the process to be used early in the design process because each process

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has unique design requirements. Refer to clause 11 for more information about the various manufacturing processes. It is important for the gear designer to understand that correct gear blank design is essential to gear quality. A precision gear blank is required to make a precision gear. See clause 8 for a detailed discussion of gear blanks. 5.6 Engineering analysis of preliminary design Once the preliminary design has been determined, it is necessary to analyze the design to ensure that it will meet the load capacity and contact ratio requirements. Refer to clause 6 for the equations for calculating bending stress, contact stress and contact ratio. If the analysis of the preliminary design reveals that it would not be adequate for the application, the designer should revise the original design. The cycle of design followed by analysis should be repeated until the preliminary analysis verifies that the design is adequate. This clause contains a brief synopsis of some of the types of engineering analysis that should be done at the preliminary design stage. The primary objective of the preliminary design analysis is to ensure that the gears have adequate strength for the application. More detailed analysis will be done later during the detailed design stage (refer to clause 6). At the detailed design stage the specific tooth geometry and meshing parameters will be determined to optimize the design for its particular application. The reader is also referred to the publications in annex A for additional information. 5.6.1 Bending stress Bending stress can be determined using a variety of techniques. Each method has some advantages and disadvantages. -- Lewis form factor. This method treats a gear tooth as a stubby cantilever beam. Figure 33 maps the geometry of the gear tooth to that of a cantilever beam. This method usually predicts conservative (higher) than actual bending stress for unmodified gears. This method evaluates a normalized measure called the Lewis form equation based on the dimensions of the beam. Machine design text books give a detailed explanation of this method. This method does not account for load sharing and evaluates stresses at the root by applying the load at the tip of the gear tooth.

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AGMA 917--B97

W W

F

l t rf

a x

l

(b)

(a)

t

Figure 33 -- Gear tooth as a simple beam -- AGMA geometry factor method. This method is also based on construction. However, the load is applied at the highest point of single tooth contact for spur gears and inscribes a parabola inside the gear tooth. Figure 34 illustrates the construction. The point where the parabola is tangent to the tooth profile is called the critical point and is where the maximum stress occurs. This method evaluates a normalized measure called the geometry factor which accounts for the stress concentration at the root of the gear tooth and the load sharing between the mating gears. For helical gears, the load is applied at the tip of the gear tooth. Clause 6 uses this technique to evaluate bending stresses at the root of the gear tooth. C L Gear tooth Vertex

Load

Inscribed parabola Critical point

Figure 34 -- Tooth load acting at inscribed parabola -- Finite element model. This method uses the finite element method to model the actual

gear tooth geometry. This method will provide more accurate results than the Lewis form factor, especially for modified gear tooth geometry. The effects of thin rim gears can also be analyzed using this method. The tooth model is usually loaded at the highest point of single tooth contact (HPSTC). -- Boundary element model. This method, like the finite element method, will provide more accurate results than the Lewis form factor, especially for modified gear tooth geometry. 5.6.2 Contact stress Contact stress occurs when two bodies come in contact under a force. The involutes of external gear pairs can be approximated as a cylinder on a cylinder. The involutes of internal--external gear pairs can be approximated as a cylinder inside a cylinder. The Hertzian (contact) stress can be calculated at any point of contact by knowing the radius of curvature of each involute at the point of contact. The two most common places to calculate contact stresses are: -- Pitch diameter. At the pitch diameter the contact stress is less than the maximum, but it is the point where pitting usually occurs. Pitting is a surface fatigue failure due to repetitions of high contact stress. Pitting occurs near the pitch diameter because the relative sliding between the pinion and gear changes direction as the contact passes through the pitch point. This change in

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sliding creates frictional subsurface shear stresses which can eventually remove material and form the surface cavities known as pitting; -- Lowest point of single tooth contact. This is the point where the contact stress is at its maximum. It occurs here because the radii of curvature of the involutes are at their most opposite extremes, when the tooth is under its maximum load (when there is only one pair of teeth carrying the load). For external gear pairs, the contact on the driver is near its root, (relatively small radius of curvature) and the contact on the driven gear is near its tip (relatively large radius of curvature). If the contact stresses exceed the surface endurance strength of the material, surface failure will result. 5.6.3 Contact ratio Contact ratio can be visualized as the average number of tooth pairs in contact during the mesh cycle. For example, if the contact ratio for a gear mesh is 1.75, then two tooth pairs will be in contact 75% of the time and only one tooth pair will be in contact 25% of the time. In general, gear meshes with more tooth pairs in contact exhibit smoother operation. For most applications, it is recommended that the contact ratio be at least 1.4. Contact ratio is a function of center distance. Contact ratio decreases as the center distance increases from the nominal center distance. It is important to determine the allowable deviation from the nominal center distance that will maintain an acceptable contact ratio for the application. Finer diametral pitch gears have a smaller allowable center distance deviation. It is important to consider the expected center distance tolerance stack up, of the entire system, early in the design process. The designer should then design the system so the contact ratio is acceptable throughout the complete center distance tolerance zone. 5.6.4 Fatigue strength Many machine design texts, such as “Mechanical Engineering Design” by J.E. Shigley [6], contain information on how to determine fatigue strengths, or endurance limits, for ferrous materials. It is important to realize that the analytical approaches do not yield absolutely accurate results. The results should only be used as a guide, as something that

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indicates what is important and what is not important, in designing to avoid fatigue failure. Many materials, such as plastics, aluminum and bronze do not have true endurance limits so they cannot be designed to have infinite life. Fatigue curve information must be obtained from the material supplier and verified by testing in order to design the gears to meet life requirements. It is extremely important to confirm the design by conducting a testing program on the materials that will be used. Heat treatment of ferrous materials will increase fatigue strength. Refer to clause 15 for more information on load rating and testing procedures. 5.6.5 Surface durability Surface durability, also known as pitting resistance, is the capacity to resist the kind of failure which results from repeated surface or subsurface stresses. See ANSI/AGMA 2001--C95 for more information on surface durability ratings for spur and helical gears. These rating methods assume the design provides adequate lubrication. Inadequate lubrication can lead to other modes of surface failure (wear), which are not covered by these rating methods. The load rating procedure in ANSI/AGMA 2001--C95 is not suitable for every fine--pitch application. The rating procedure is based primarily on experience with coarse pitch gears. As with fatigue strength, available data on material properties are limited to the more traditional gear materials. Heat treatment of ferrous materials will increase surface durability. Material property information should be obtained from the material supplier and testing should be done to confirm the design. 5.6.6 Gear system assembly Before the detailed design can be performed, the designer must consider the method of mounting to be used. Proper installation of the gear system is essential for achieving good performance. Some items of consideration should be: -- Mounting of gears on the shafts. There are several methods used to mount gears to shafts. Many designs offer various degrees of precision, cost, reliability and ease of assembly. The designs can be classified into two main types: removable fastenings and permanent fastenings.

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Removable fastenings pinning clamping set screws

key way spline taper and screw

Permanent fastenings: press fit shrink fit molded assembly cementing compounds

staking pegging riveting spinning

For more information on mounting gears refer to “Precision Gearing, Theory and Practice” by Michalec [7]; -- Shaft alignment. A deviation in the alignment of shafts is composed of two components: in--plane deviation and out--of--plane deviation. The in--plane deviation is measured in the common plane of axes and out--of--plane deviation is measured in the plane (skew plane) perpendicular to the common plane of axes. Figure 35 shows the in--plane and out--of--plane deviations for two shafts A and B;

Shaft A

Shaft B

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-- Gear--box housing. The housing must be designed to remain sufficiently rigid during operating conditions. It is possible to encounter a resonant condition, where the natural frequency of the complete system assembly coincides with an operating frequency. It is beyond the scope of this Information Sheet to provide detailed information on vibration control and analysis. It is suggested that the designer consult someone experienced in this field or a text on the subject. 5.7 Application considerations The preliminary design step helped to determine the estimated values for parameters such as number of teeth, diametral pitch, pressure angle and helix angle. The detailed design step, that will be discussed in clause 6, will optimize the specific tooth geometry in order to meet the functional requirements. As mentioned previously, the three main types of gear systems are power gears, smooth motion gears and zero backlash gears. This clause will discuss some of the specific requirements of each of these systems. It is important to keep in mind that many applications have requirements that overlap into more than one of these classifications. 5.7.1 Design emphasis for power gear systems

Plane of axes In--plane deviation

Shaft B

Shaft A

Out--of--plane deviation Plane of axes Figure 35 -- Shaft alignment deviations -- Couplings. The coupling must have some degree of flexibility to accommodate the types of misalignment mentioned previously. It must be able to transmit torque, yet limit the forces on machine components such as shafts and bearings that result from misalignment. However, it is important to understand the effects the coupling has on smooth motion when misalignment is present;

As the name suggests, the primary function of power gear systems is to transmit power (speed and torque). This does not mean that the gears transmit only heavy loads, but that greater importance is given to transmitting power than to transmitting uniform motion. The design process for power gears emphasizes adequate size, suitable materials, appropriate heat treat procedures and proper lubrication to maximize gear life. Designers usually evaluate the resistance of the gears to pitting and bending fatigue. It is important to note that the designers must also recognize life--cycle costs, noise and other related parameters during the design process. 5.7.2 Design emphasis for smooth motion gears In addition to the requirements of power gear systems, smooth motion gears are designed to have low transmission error. Transmission error is defined as the deviation of the position of the driven gear, for a given angular position of the driving gear, from the position that the driven gear would occupy if the gears were geometrically perfect and infinitely

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stiff. Transmission error is usually measured as a linear error along the line of action.

-- Runout in the gear blank mounting arbor or tool arbor or both;

Generally, transmission error in gear meshes has the following two components:

--

Variations in the hob;

--

Tooth--spacing error in the shaper cutter;

-- Once per revolution of the gear(or pinion) variation;

-- Position errors in the gear generator’s indexing gear train;

--

-- Effect of approximating the involute profile with generated straight cuts;

Once per tooth variation.

The once per revolution variation is due to accumulated pitch variation. The once per tooth variations are due to variations in profile, pitch, tooth thickness and tooth alignment. The above two components affect the positional accuracy of the drive train. In addition, the once per tooth component is related to the noise and vibration characteristics of the gear train. The main sources of transmission error can be classified into the two following categories: -- Variations mounting;

during

manufacturing

and

-- Deflections of gear teeth, shafts, bearings and housing. Parameters such as transverse contact ratio and face contact ratio influence load sharing between the gear teeth and thus affect the deflections of the gear teeth. Deflections of shafts, bearings and housings can cause an uneven load distribution across the face of the gear tooth. Calculating transmission error based on the above gear parameters is beyond the scope of this design manual. However, this clause will discuss those variations that contribute to transmission error. Total transmission error for a gear train can be calculated by assuming that the total transmission error is the sum of the transmission error of each mesh. However, such a technique yields very high transmission error values. Multi--mesh gear trains should be analyzed using statistical techniques to determine a more realistic value. The sources of transmission error are discussed in 5.7.2.1 and 5.7.2.2, and techniques for minimizing transmission error are briefly discussed in 5.7.2.3. 5.7.2.1 Manufacturing variation Some of the possible sources of these errors are: -- Eccentric mounting of the gear blank or the generating tool or both;

34

--

Vibration and chatter of the machine tool;

-- Deflection due to the work mass and cutting forces; --

Deformations of the gear blank;

--

Non--homogeneous gear blank material;

--

Differential temperature effects;

--

Slippage of the blank on the arbor;

-- Errors in the dressing of the wheel profile for profile ground gears. 5.7.2.2 Assembly variation Shaft misalignment causes uneven load distribution and higher tooth fillet stresses in gears. Uneven load distribution results in larger transmission error. Misalignment between shafts can cause the shaft couplings to transmit non--uniform motion. Another contributor of transmission error comes from mounting runout that causes the gear true center to be displaced from the center of rotation. Some of the sources of runout are: --

Clearance between gear bore and shaft;

--

Runout of the shaft;

-- Eccentricity of the rotating race in the ball bearing. 5.7.2.3 Minimizing transmission error Minimizing the once per revolution component of transmission error is best achieved by minimizing accumulated pitch error. Minimizing the once per tooth component is achieved by controlling tooth form. Gear meshes with larger contact ratios exhibit lower tooth to tooth transmission error because they have more tooth pairs sharing the load. Hence it is recommended that gear designers maximize contact ratio when designing gears for smooth motion. Tooth modifications (tip relief and root relief) and tooth lead modifications (crowning) are techniques that have been applied successfully under certain conditions to minimize transmission error. When designed for a specific load condition, these are

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suitable for only that load condition. It is recommended that an experienced gear designer be consulted when designing gears with profile and lead modifications. 5.7.3 Zero backlash gears In addition to the requirements of power gear systems, zero backlash gear meshes are designed to have low backlash. Backlash is necessary to prevent tight mesh and interference due to manufacturing and assembly variations in a gear. In some applications, backlash needs to be controlled to achieve accurate angular positioning of machine components. Many techniques have been developed to control backlash in a gear mesh. Clause 7 of this Design Manual deals with those techniques for controlling backlash in a gear mesh.

6 Design synthesis and analysis 6.1 Introduction Clause 5 introduced the gear designer to basic considerations for the early conceptualization of a gear design. The selection of type of gearing and preliminary values for number of teeth, diametral pitch, helix angle and number of stages may be very simple or exceedingly complex. The function of the design and the size of the design space influence the options available to the designer. Clause 5 presented general guidelines and the most basic procedures but could not offer a complete cookbook approach because the number of options available at the preliminary stage is far too large. Therefore, it is assumed that first estimates of number of teeth, diametral pitch, pressure angle, helix angle and number of stages will be made using clause 5 as well as other sound mechanical engineering practices. Clause 6 presents procedures by which the designer can determine, refine and analyze the details that comprise a complete gear design. This optimization of the design may result in only minute modifications to the original concept or it may identify the need for major redesign. In either case, the proper function and life of the gear system depends upon the optimization being done with knowledge and thoroughness.

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The information presented in clause 6 is rigorous mathematically and may be somewhat intimidating to the novice gear designer. It may be beneficial to remember that an involute gear tooth is a combination of four geometric sections: top land, flanks, trochoid or root fillet and root circle. Within certain limits, these sections are independent of one another and each can be varied without necessarily affecting the others. For example, within certain limits, the tip diameter can be made larger or smaller without changing the tooth thickness. Standards, such as ANSI/AGMA 1003--G93, Tooth Proportions for Fine--Pitch Spur and Helical Gearing, define relationships between the tooth sections based upon the concept of a basic rack. However, without violating the precepts of the standard, the designer has freedom to optimize the design by specifying gear parameters. The specifics of tip diameter, tooth thickness and root diameter are what gear optimization is all about. Clause 6 presents the procedure for accomplishing that optimization. The procedure is presented in a logical order for making the calculations, i.e., the inputs needed for a given calculation have been calculated previously. Most gear practitioners use a form of spreadsheet to perform these calculations, and the sequence of clause 6 will make constructing a spreadsheet a relatively simple undertaking. 6.2 Standard gear parameters 6.2.1 Transverse diametral pitch (Pd ) This is a measure of the size of a gear tooth in the plane of rotation. It is the number of gear teeth in one inch of standard pitch diameter. The larger this value, the smaller is the size of the gear tooth. Transverse diametral pitch is given by: Pd = Pnd cos ψ

...(2)

6.2.2 Standard pitch diameter for pinion and gear (DP, DG ) This diameter is equal to the ratio of the number of teeth to the transverse diametral pitch. The circle with this diameter is called the standard pitch circle. Figure 36 illustrates the standard pitch circle. DP =

NP N = P P nd cos ψ Pd

NG N DG = = G P nd cos ψ Pd

...(3)

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6.2.4 Base diameter (DbP, DbG ) Base diameter is the diameter of the base circle associated with the involute profile (figure 36). It is equal to the standard pitch circle diameter multiplied by the cosine of the transverse pressure angle at the standard pitch diameter.

Standard pitch radius Base radius Outside radius

Root radius

DbP = DP cos Ôt DbG = DG cos Ôt ...(5) A discussion on the base diameter and involutometry can be found in clause 4. 6.2.5 Transverse circular pitch (pt ) This is the circular arc distance measured along the standard pitch circle between two corresponding points of adjacent teeth. It is equal to the circumference of the standard pitch circle divided by number of teeth. It is also equal to the distance between corresponding points of two adjacent teeth in the cutter at its pitch line. pt = π ...(6) Pd

Figure 36 -- Standard pitch circle 6.2.3 Profile angle and pressure angle The term profile angle is used in conjunction with the cutter used to manufacture the gear by a generating process. The term pressure angle is used in conjunction with the operating considerations of the meshing gears. See 4.1.9.1.

6.2.6 Transverse base pitch (pb )

6.2.3.1 Profile angle

pb = pt cos Ôt ...(7) Figure 37 illustrates the transverse standard circular pitch and transverse base pitch.

The profile angle of a rack--type cutter (hob) is the angle between a line perpendicular to a pitch line of the rack and the profile of a rack tooth. For a pinion--type cutter (shaper cutter), at the point of intersection of the pitch circle and the tooth profile, the angle between a line normal to the pitch circle and a line tangent to a tooth profile is defined as the profile angle. 6.2.3.2 Normal standard pressure angle (Ôn ) Normal standard pressure angle is the profile angle of the cutter used to manufacture the gear by the generation process. 6.2.3.3 Transverse standard pressure angle (Ôt ) Transverse standard pressure angle is the angle between a line in the transverse plane tangent to the involute at the standard pitch diameter and a line passing through the center of the gear and the tangency point. The transverse standard pressure angle is given by:



tan Ô n Ô t = tan−1 cos ψ

36



...(4)

This is the circular arc distance measured along the base circle diameter between two corresponding points of the adjacent teeth. It is equal to the base circle circumference divided by the number of teeth.

Transverse circular pitch

Transverse base pitch B

A

D

C Base radius Standard pitch radius

Figure 37 -- Transverse pitch 6.2.7 Normal base pitch (pbn ) Normal base pitch in an involute helical gear is the base pitch in the normal plane. In order for two helical gears to mesh without interference, the two gears must have the same normal base pitch. Normal base pitch is the normal distance between parallel helical involute surfaces on the plane of

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action in the normal plane or in the length of the arc on the normal base helix. pbn = pt cos ψ cos Ôn 6.2.8 Standard center distance (Cs )

...(8)

DP + DG 2 6.3 Gear mesh related parameters

at the pitch point. At all other contact points along the line of action, both sliding and rolling occur. In figure 38, point P is the pitch point. 6.3.4 Line of contact

Standard center distance is the sum of the standard pitch circle radii of the pinion and the gear and is given by: Cs =

AGMA 917--B97

...(9)

Line of contact is the line or curve along which the two mating tooth surfaces are tangent to each other at any instant of time. For spur gears, the line of contact is parallel to the axis of rotation whereas in helical gears, the line of contact is inclined to the axis of rotation. Figure 39 shows the line of contact.

6.3.1 Operating center distance (C) Operating center distance (C) is the distance between the axis of the pinion and the axis of the gear. The operating center distance is controlled by the frame (casing or gear box) and the bearings that support the gears. Clause 5 presented a brief discussion on the geometric parameters that influence center distance tolerance. One of the main advantages of the involute profile is its ability to transmit uniform rotary motion independently of the changes in center distance.

Tangent plane Helical line of contact

6.3.2 Line of action The line tangent to base circles of the pinion and the gear in the transverse plane is called the line of action or the base tangent line. It is the locus of points in the transverse plane where the pinion tooth makes contact with the gear tooth. Figure 38 illustrates the line of action.

Base circle (pinion)

Base circle (gear)

Operating pressure angle

P Line of centers

Line of action

Driver

Driven

Figure 38 -- Line of action 6.3.3 Pitch point Pitch point is a function of the actual operating center distance and not a function of the standard center distance. It is the point where the line joining the center of the pinion to the center of the gear intersects the line of action. Pure rolling occurs only

Spur line of contact Figure 39 -- Line of contact 6.3.5 Operating pitch diameter (dP, dG ) The diameter of the circle in the transverse plane passing through the pitch point with its center at the center of the gear or pinion is called the operating pitch diameter. dP =

2N PC NP + NG

(external gears) ...(10a)

2N GC dG = NP + NG 2N PC NG − NP 2N GC dG = NG − NP dP =

(internal gears) ...(10b)

The operating pitch radii of the pinion (rP ) and of the gear (rG ) are equal to one half of the corresponding operating pitch diameter. The operating pitch diameter is used to calculate the tangential load on the gear. For a given mesh, the operating pitch diameters of the pinion and the gear are tangent to each other. Note that the standard pitch diameters

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of the pinion and the gear are not necessarily tangent to each other and they are tangent to each other only when they are equal to the corresponding operating pitch diameters. 6.3.6 Operating transverse pressure angle (Ôpt ) The angle between the line of action and the line joining the center of the pinion and the center of the gear is called the operating pressure angle. This is equal to the arc cosine of the ratio between the base diameter and the operating pitch diameter. Figure 38 shows the operating transverse pressure angle. φ pt=cos –1

Dd  (external and internal gears) bP P

....(11)

6.3.7 Operating helix angle (ψp ) The operating helix angle is used to calculate the component of the transmitted force in the axial direction. The operating helix angle is the helix angle at the operating pitch radius and is given by: ψ p = tan−1

d Dtan ψ P

(external and internal gears)

P

...(12)

6.3.8 Operating normal pressure angle (Ôpn ) This is the operating pressure angle in the normal plane and is given by: Ôpn = tan - 1 (tan Ôpt cos ψp ) (external and internal gears)

...(13)

The operating normal pressure angle describes the direction of the force transmitted between the contacting gear tooth surfaces. In a given design, an increase in the operating pressure angle leads to an increase in the separating force.

Backlash can be observed by holding one of the mating gears fixed and moving the other gear from a contact on one side of a tooth to a contact on the other side. One can measure backlash in a gear mesh in the following three ways: --

Transverse backlash;

--

Normal backlash;

--

Backlash along the line of action.

Even though the values for the backlash measured in the above three ways are different, they are all related to each other mathematically. 6.3.9.1 Transverse backlash Transverse backlash (B) is defined as the difference between the transverse circular tooth space of the gear measured at its operating pitch circle and the transverse circular tooth thickness of the pinion measured at its operating pitch circle. In figure 40, arc PQ is the transverse circular tooth thickness of the pinion at the operating pitch circle and arc PR is the transverse circular tooth space of the gear. The difference between the arcs PR and PQ is the transverse backlash for the given mesh. Backlash is usually achieved by decreasing the circular tooth thickness of the two mating gears. During machining, the cutter is fed deeper into the gear blank to decrease gear circular tooth thickness. Transverse backlash for both external and internal gears is given by the following equation: B=

πd πd P − t ptP − t ptG = G − t ptP − t ptG NP NG ...(14)

Pinion

Operating pitch circle (pinion)

6.3.9 Backlash Backlash is a property of the gear mesh and not of an individual gear. It is the clearance between the meshing teeth. In most applications, it is not detrimental to have backlash in a gear mesh. On the other hand, backlash is necessary to accommodate manufacturing variations in gears and the relative change in size of gears and their casings due to the thermal and other environmental effects. In gears used in position control systems, backlash may be detrimental. For controlling backlash in such systems, refer to clause 7 of this manual.

38

Q R Operating pitch circle (gear)

P

Gear

Figure 40 -- Transverse backlash is arc PR -- arc PQ

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Equations for evaluating the tooth thickness at the operating pitch circle of the pinion (tptP ) and the gear (tptG ) are given in 6.6.4.

profile. Its shape and size depend upon the manufacturing method employed.

The amount of backlash should meet the requirements of the application. This amount should enable gears to run freely when the mesh is operating under worst case conditions (at the shortest possible operating center distance, worst condition of humidity, temperature and geometrical tooth variations).

In gears that are manufactured by a generation process, the fillet curve is called a trochoid. During the generation process, the tip corner of the cutter tooth generates the trochoid and the flank of the tooth generates the involute profile. The trochoid is always tangent to the root circle and in gear teeth without undercut, tangent to the involute.

Backlash is also a function of center distance. The amount of transverse backlash (Bc ) due to a change in center distance (∆C) is approximated by the following equation:

6.4.1.2 Formed fillet

Bc = 2 ∆C tan Ôpt (external and internal gears) ...(15) When two gears mesh at their low points of total composite variation, the change in center distance (∆C) is equal to half the sum of the total composite variation of the gear and the pinion. For gears with significant total composite variation, the increase in the amount of backlash due to this change in center distance is quite significant. The total transverse backlash is equal to the sum (B + Bc ). 6.3.9.2 Normal backlash (Bn ) Normal backlash is the difference between the normal circular tooth space of the gear measured at its operating pitch circle and the normal circular tooth thickness of the pinion measured at its operating pitch circle. The relation between the normal backlash and the transverse backlash is given by: Bn = B cos ψp (external and internal gears) ...(16) 6.3.9.3 Backlash along line of action (Bb ) Backlash along the line of action is the component of the transverse backlash measured along the line of action and is given by: Bb = B cos Ôpt (external and internal gears) ...(17) 6.4 Tooth fillet based geometric parameters

6.4.1.1 Generated fillet

In some gear manufacturing processes, such as form milling, form grinding or injection molding, the fillet shape is transferred directly from the tool. It is often most convenient to make the fillet portion of the tool in the shape of a circular arc. This arc is generally made tangent to the root circle and the involute profile. The radius of the arc is usually selected to provide the maximum tooth bending strength without the risk of interference with the tip of the mating tooth. In gears that must have undercut profiles to provide sufficient clearance, it may not be possible to approximate the undercut profile with a single arc. In such cases, a series of arcs can be used, but only after checking for possible interference. 6.4.2 Undercutting A gear tooth is said to be an undercut tooth when any part of the trochoid lies inside a line drawn tangent to the involute profile at the point of intersection of the involute and the trochoid. In figure 41, the gear tooth profile shown is that of an undercut tooth as the trochoid lies inside a line drawn tangent to the involute at True Involute Form (TIF). One needs to be concerned about undercut when the pinion or the gear has a small number of teeth. In most applications, some amount of undercut can be tolerated. In some cases, designers intentionally choose gear cutters that produce an undercut gear tooth to provide relief in the root of the gear tooth for the grinding or the shaving process. Severe undercutting can result in considerable weakening of the tooth and loss of tooth action.

6.4.1 Tooth fillet

6.4.3 Radius at true involute form (RtifP, RtifG )

Tooth fillet refers to the portion of the gear tooth profile that connects the root circle to the involute

TIF represents the point of intersection of the fillet and the involute. Radius at TIF is the radial distance

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from the center of the gear to the point of intersection of the fillet and involute. Figure 41 illustrates the radius at TIF. Non--conjugate action and interference is seen when tooth contact occurs below TIF. Khiralla [2] presents a method to evaluate the radius at TIF for gears manufactured by hobbing and shaping methods. Tangent to involute at point G

G

Radius at TIF Figure 41 -- Undercut teeth 6.4.4 Profile shift (addendum modification) Figure 42 illustrates another useful characteristic of involute gear teeth. It is possible, using a cutter of given diametral pitch and profile angle, to cut the same number of teeth into gear blanks of various outside diameters. The resulting gears, despite the fact that they appear to be of different sizes, all have the same base pitch and therefore will transmit uniform rotary motion if run together, providing contact ratios and clearances are adequate. This blank diameter modification is called profile shift or addendum modification. 6.4.4.1 Definitions The numerical value of profile shift is equal to the amount by which the nominal center distance of a

X = 0.5 X = 0.0

10 tooth 20¥ PA

gear pair must be changed in order to accommodate the modified gear. The profile shift coefficient is the product of the profile shift and the normal diametral pitch. Unmodified (or standard) gears have a nominal center distance that is equal to one--half the sum of their standard pitch diameters. Standard gears are said to have zero profile shift. Gears with positive profile shift (called enlarged gears) result when the generating pitch line (hob tool) or pitch diameter (shaper cutter) is held at a larger radius than standard when machining the gear. Gears with negative profile shift (called reduced gears) result when this generating radius is less than standard. Figure 42 shows the change in the gear tooth form due to a positive profile shift coefficient of 0.5, resulting in enlarged teeth, for two profile angles on a 10--tooth pinion. Note that the enlarged tooth form shows little or no undercut, which is one of the primary justifications for profile shift in gear designs. The addendum of a gear is the radial distance from the standard pitch diameter to the outside diameter. Since the outside radius of the gear is usually increased by the same amount as the profile shift, the addendum increases for an enlarged gear. For a time, it was common to refer to enlarged gears and reduced gears as long and short addendum gears respectively. Hence, the term addendum modification is used to describe this process of enlargement and reduction. However, because it is not necessary to make the outside diameter any specific dimension, the amount of change in the outside diameter may or may not correspond exactly to the amount of the profile shift.

X = 0.5 X = 0.0

10 tooth 14.5¥ PA

Figure 42 -- Effect of profile shift (addendum modification)

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The cutter may be fed in or held out with respect to the gear blank in order to cut less or greater tooth thickness. Some gear experts have proposed that an additional term called rack shift be introduced to define the amount of cutter feed in or hold out associated with profile shift coupled with tooth thickness change for backlash alteration. Since this proposal was not adopted at the time of this writing, this Design Manual will use the most rigorous and simple definition for profile shift. Modification of tooth thickness is considered a separate topic (see 6.6). 6.4.4.2 Reasons for using profile shift Avoiding undercut. Pinions with small number of teeth are usually enlarged to avoid undercut. For 20 degree profile angle gears, gears with fewer than 24 teeth should be designed with positive profile shift. However, note that there is a maximum limit to which the gear tooth can be enlarged before the top land thickness (tooth thickness of the gear at its outside diameter) becomes zero. The top land thickness should be at least 0.275/Pnd . Balancing bending strength. When the shape of the trochoid of the pinion is considerably different from that of the gear, the pinion tooth may be weaker than the gear tooth. In using this to balance the bending strength of the two mating members, positive profile shift is applied to the pinion and negative profile shift to the gear. This makes the pinion tooth stronger and gear tooth weaker resulting in balanced bending strength. In some cases, designers balance bending fatigue strength instead of the bending strength because the pinion tooth experiences more load cycles than the gear tooth. Balanced specific sliding. Specific sliding is the ratio of gear tooth sliding velocity to its rolling velocity. The amount of specific sliding influences the amount of wear on the tooth surface of the pinion and the gear. The extreme specific sliding at each end of the path of contact should be balanced in order to minimize wear of the gear teeth. In speed reducing external gears, balanced specific sliding is achieved by enlarging the pinion and mating it with either a reduced or standard gear. In speed increasing external gear meshes, balancing the specific sliding by enlarging the pinion usually results in a lower percentage of recess action (see 6.7.4). It is necessary to compromise between

AGMA 917--B97

balanced sliding and recess action in speed increasing gear meshes. Khiralla [2] presents a detailed discussion on the specific sliding calculations for external gears. 6.4.4.3 Minimum positive profile shift The minimum amount of positive profile shift is based upon the following two criteria: -- When engaged with any mating external gear or rack, the lowest contact point on the pinion profile should be above any undercutting of the involute by the generating action of a standard tool; -- When engaged with any standard mating external rack, the lowest contact point on the pinion profile should be located above an initial portion of the involute. The initial portion is avoided because it is often difficult to manufacture accurately. The initial portion corresponds to the first five degrees of roll. Based upon the above two criteria, the minimum value for a positive profile shift coefficient (enlargement) is given by: X +min =



MAX 0.0, 1.05 −



N sin Ô tsin Ô t − cos Ô t tan 5° 2 cos ψ

...(18)

6.4.4.4 Maximum negative profile shift The teeth of a gear containing a large number of teeth, mating with a pinion with profile--shifted enlarged teeth, can be reduced to increase the percentage of recess action in the mesh. The amount of negative profile shift should be chosen carefully and mesh contact below 5 degrees of roll should be avoided. The maximum value for a negative profile shift coefficient (tooth reduction) is given by: X – max =



MIN 0.0, 1.05 −



N sin Ô t(sin Ô t − cos Ô t tan 5°) 2 cos ψ ...(19)

Table 2 is the result of charting equations 18 and 19 as a function of number of teeth for 20 degree profile angle gears. It is clear that when the gear has 24 or more teeth, the lowest point of contact with a standard rack is above 5 degrees of roll with no tooth enlargement required. In fact, in a gear with a

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large number of teeth, a significantly high value of negative profile shift coefficient (tooth reduction) may be used without contact occurring below 5 degrees of roll. However, use of the maximum allowable reduction may produce a tooth that has unacceptably low bending strength. Therefore, it is recommended that less than 50% of the negative profile shift allowed in table 2 be used for most designs. When reducing gears with large numbers of teeth, engineers normally choose to reduce the gear by the same amount the pinion was enlarged. If more than 50% of the reduction in table 2 is thought to be appropriate, the design must be carefully analyzed for bending strength, specific sliding ratio and contact ratio. The amount of negative profile shift should never exceed that indicated in table 2. In all cases, tool data and clearances with mate must be evaluated. Remember that it is not necessary to reduce gears with large numbers of teeth. In fact, it is detrimental to reduce gears in speed increasing drives as it increases the percentage of approach action (see 6.7.4). 6.5 Additional gear parameters 6.5.1 Nominal center distance In applications where it is necessary to design for a fixed value of standard center distance, if one enlarges the pinion by applying a positive profile shift, then one has to reduce the gear by applying an equal but negative profile shift. Such designs are called Long and Short Addendum Designs (for enlarged pinions and reduced gears). In applications where there is no need to maintain standard center distances, one can enlarge the pinion without reducing the gear. Such gear designs are called designs with spread centers (enlarged center distance system). The nominal center distance for a gear mesh is given by: Cn =

NP + NG X + X G (external gear + P 2P nd cos ψ P nd meshes) ...(20a)

Cn =

NG − NP X − X P (internal gear + G 2P nd cos ψ P nd meshes) ...(20b)

Note that for external gears when XP = - XG , the nominal center distance equals standard center

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distance. The actual operating center distance should be greater than or equal to the nominal center distance. Operating the gears at center distances smaller than the nominal value could lead to a tight mesh condition and interference. 6.5.2 Outside and inside diameter The outside diameter on the external gears and the inside diameter on the internal gears are normally related to the dedendum of the cutter when generation cutters of topping variety (one that cuts the outside and the inside diameter) are used. However, when such tooling is not used, the outside and inside diameter can be made without any relation to the dedendum of the cutter. The standard outside diameter and standard inside diameter are given by: D oP = DP +

2X P + 2 P nd P nd

D oG = DG +

2X G + 2 (for external gears) P nd P nd ...(21b)

D iG = DG +

2X G – 2 (for internal gears) P nd P nd ...(21c)

...(21a)

For spur pinions with small numbers of teeth (9 to 12), one would have to design gears with an outside diameter smaller than the value given by the above equations because the opposite involute profiles of a gear tooth intersect inside of, or too close to, this value (low top land thickness). In such cases, the outside diameter must be reduced to a size such that there is sufficient top land. The top land thickness of the gear should be at least 0.275/Pnd . Table 3 gives the maximum outside diameter for pinions with fewer than 15 teeth for 20 degree profile angle gears. For 20 degree profile angle spur gears, table 3 shows that maximum outside diameter for pinions with fewer than 13 teeth is lower than the standard value. 20 degree profile angle spur gears with 13 teeth or more can be designed with standard value for the outside diameter. The standard value for the addendum coefficient of the cutting tools is 1.2 + (0.002)(Pnd ). In order to maintain sufficient clearance at the root of the gear tooth, designers are not encouraged to choose outside diameters greater than the standard value given by equations (21 a, b and c).

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AGMA 917--B97

Table 2 -- Profile shift coefficients for 20° profile angle spur gears1) Number of teeth 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Profile shift coefficient2) Minimum Maximum enlargement reduction 0.6501 0.0000 0.6057 0.0000 0.5613 0.0000 0.5168 0.0000 0.4724 0.0000 0.4280 0.0000 0.3836 0.0000 0.3391 0.0000 0.2947 0.0000 0.2503 0.0000 0.2058 0.0000 0.1614 0.0000 0.1170 0.0000 0.0725 0.0000 0.0281 0.0000 0.0000 --0.0163 0.0000 --0.0607 0.0000 --0.1052 0.0000 --0.1496 0.0000 --0.1940 0.0000 --0.2385 0.0000 --0.2829 0.0000 --0.3273 0.0000 --0.3718 0.0000 --0.4162 0.0000 --0.4606 0.0000 --0.5050 0.0000 --0.5495 0.0000 --0.5939 0.0000 --0.6383 0.0000 --0.6828 0.0000 --0.7272 0.0000 --0.7716 0.0000 --0.8160 0.0000 --0.8605 0.0000 --0.9049 0.0000 --0.9493 0.0000 --0.9938 0.0000 --1.0382 0.0000 --1.0826

Number of teeth 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

Profile shift coefficient Minimum Maximum enlargement reduction 0.0000 --1.1271 0.0000 --1.1715 0.0000 --1.2159 0.0000 --1.2603 0.0000 --1.3048 0.0000 --1.3492 0.0000 --1.3936 0.0000 --1.4381 0.0000 --1.4825 0.0000 --1.5269 0.0000 --1.5714 0.0000 --1.6158 0.0000 --1.6602 0.0000 --1.7046 0.0000 --1.7491 0.0000 --1.7935 0.0000 --1.8379 0.0000 --1.8824 0.0000 --1.9268 0.0000 --1.9712 0.0000 --2.0157 0.0000 --2.0601 0.0000 --2.1045 0.0000 --2.1489 0.0000 --2.1934 0.0000 --2.2378 0.0000 --2.2822 0.0000 --2.3267 0.0000 --2.3711 0.0000 --2.4155 0.0000 --2.4599 0.0000 --2.5044

NOTES: 1) This table gives the minimum required positive profile shift coefficient (enlargement) and the maximum allowable reduction for 20 degree profile angle gears. Equations 18 and 19 may be used to generate similar tables for gears having profile angles other than 20 degrees. A thorough understanding of 6.4.4 is recommended before using this table. 2) Profile shift coefficients are dimensionless. Divide values by normal diametral pitch to obtain profile shift in inch units.

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Table 3 -- Maximum outside diameter for minimum topland of 0.275/Pnd Number of teeth 9 10 11 12 13 14 15

Pressure angle = 20 degrees Profile shift coefficient Max OD1) 0.6501 12.0144 0.6057 13.0256 0.5613 14.0304 0.5168 15.0296 0.4724 15.9448 0.4280 16.8560 0.3836 17.7672

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