Agma Information Sheet

AGMA 908---B89 (Revision of AGMA 226.01)

April 1989 (Reaffirmed August 1999)

AMERICAN GEAR MANUFACTURERS ASSOCIATION Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth

AGMA INFORMATION SHEET (This Information Sheet is not an AGMA Standard) INFORMATION SHEET Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth AGMA 908---B89 (Revision of AGMA 226.01 1984)

[Tables or other self---supporting sections may be quoted or extracted in their entirety. Credit line should read: Extracted from AGMA Standard 908---B89, INFORMATION SHEET, Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth, with the permission of the publisher, American Gear Manufacturers Association, 1500 King Street, Suite 201, Alexandria, Virginia 22314.] AGMA standards are subject to constant improvement, revision or withdrawal as dictated by experience. Any person who refers to any AGMA Technical Publication should determine that it is the latest information available from the Association on the subject. Suggestions for the 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.

ABSTRACT:

This Information Sheet gives the equations for calculating the pitting resistance geometry factor, I,for external and internal spur and helical gears, and the bending strength geometry factor, J, for external spur and helical gears that are generated by rack---type tools (hobs, rack cutters or generating grinding wheels) or pinion---type tools (shaper cutters). The Information Sheet also includes charts which provide geometry factors, I and J, for a range of typical gear sets and tooth forms.

American Gear Manufacturers Assocation 1500 King Street, Suite 201 Alexandria, Virginia 22314

April, 1989

ISBN: 1---55589---525---5

AGMA ii 908---B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

FOREWORD

[The foreword, footnotes, and appendices are provided for informational purposes only and should not be construed as part of American Gear Manufacturers Association Information Sheet 908---B89, Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth.] This Information Sheet, AGMA 908---B89, was prepared to assist designers making preliminary design studies, and to present data that might prove useful for those designers without access to computer programs. The tables for geometry factors contained in this Information Sheet do not cover all tooth forms, pressure angles, and pinion and gear modifications, and are not applicable to all gear designs. However, information is also contained for determining geometry factors for other conditions and applications. It is hoped that sufficient geometry factor data is included to be of help to the majority of gear designers. Geometry factors for strength were first published in Information Sheet AGMA 225.01, March, 1959, Strength of Spur, Helical, Herringbone and Bevel Gear Teeth. Additional geometry factors were later published in Standards AGMA 220.02, AGMA 221.02, AGMA 222.02, and AGMA 223.01. AGMA Technical Paper 229.07, October, 1963, Spur and Helical Gear Geometry Factors, contained many geometry factors not previously published. Due to the number of requests for this paper, it was decided to publish the data in the form of an Information Sheet which became AGMA 226.01, Geometry Factors for Determining the Strength of Spur, Helical, Herringbone and Bevel Gear Teeth. AGMA 218.01, AGMA Standard for Rating the Pitting Resistance and Bending Strength of Spur and Helical Involute Gear Teeth, was published with the methods for determining the geometry factors. When AGMA 218.01 was revised as ANSI/AGMA 2001---B88, the calculation procedures for Geometry Factors, I and J,were transferred to this revision of the Geometry Factor Information Sheet. The values of I and J factors obtained using the methods of this Information sheet are the same as those of AGMA 218.01. The calculation procedure for I was simplified, but the end result is mathematically identical. Also, the calculation of J was modified to include shaper cutters and an equation was added for the addendum modification coefficient, x,previously undefined and all too often misunderstood. Appendices have been added to document the historical derivation of both I and J. Because an analytical method for calculating the Bending Strength Geometry Factor, J, is now available, the layout procedure for establishing J has been eliminated from this document. All references to geometry factors for bevel gears have been removed. This information is now available in AGMA 2003---A86, Rating the Pitting Resistance and Bending Strength of Generated Straight Bevel, ZEROL Bevel and Spiral Bevel Gear Teeth. The first draft of this Information Sheet, AGMA908---B89, was presented to the Gear Rating Committee in August, 1987. It was approved by the AGMA Gear Rating Committee on February 24, 1989, after several revisions. It was approved for publication by the AGMA Technical Division Executive Committee on April 21,1989.

AGMA iii 908---B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

PERSONNEL of the AGMA Committee for Gear Rating

Chairman: J. Bentley (Peerless---Winsmith) Vice Chairman: O. LaBath (Cincinnati Gear )

ACTIVE MEMBERS

M. Antosiewicz (Falk) J. C. Leming (Arrow Gear) (Deceased) J. D. Black (General Motors/AGT) L. Lloyd (Lufkin Industries) E. J. Bodensieck (Bodensieck Engineering) J. Maddock (Consultant) W. A . B r a d l e y ( AG M A ) D. McCarthy (Dorris) R. Calvert (Morgan Construction) D. R. McVittie (Gear Works --- Seattle) A. S. Cohen (Engranes y Maquinaria) M. W. Neesley (Westech) J. DeMarais (Bison Gear) J. A. Nelson (General Electric) R. Donoho (Clark Equipment) W. P. Pizzichil (Philadelphia Gear) R. J. Drago (Boeing) D. W. Dudley (Honorary Member) J. W. Polder (Maag/NNI Netherlands) R. Errichello (Academic Member) E. E. Shipley (Mechanical Technology) H. Hagan (Nuttall Gear) W. L. Shoulders (Reliance Electric) (Deceased) N. Hulse (General Electric) F. A. Thoma (Honorary Member) H. Johnson (Browning Co.) C. C. Wang (Consultant) R. D. Kemp (Kymmene---Stromberg Santasalo) R. Wasilewski (Arrow Gear)

ASSOCIATE MEMBERS

J. Amendola (MAAG/Artec) D. L. Mariet (Falk) K. Beckman (Lufkin) T. J. Maluri (Gleason) E. R. Braun (Eaton) B. W. McCoy (Marathon Le Tourneau) D. L. Borden (Consultant) D. Moser (Nuttall Gear) A. Brusse (Hamilton) B. L. Mumford (Alten Foundry) G. Buziuk (Brad---Foote) W. Q. Nageli (MAAG) J. Cianci (General Electric) B. C. Newcomb (Chicago Gear --- D. O. James) D. M. Connor (Cummins Engine) G. E. Olson (Cleveland Gear) J. T. Cook (Dresser) J. R. Partridge (Lufkin Industries) E. Danowski (Sumitomo Heavy Industries) A. E. Phillips (Emerson Electric/Browning) R. DiRusso (Kaman) B. D. Pyeatt (Amarillo Gear) A. B. Dodd (NAVSEA System Command) T. Riley (NWL Control System) L. L. Haas (SPECO Division) G. R. Schwartz (Dresser) F. M. Hager (Cummins Engine) A. Seireg (Academic Member) A. C. Hayes (DACA) E. R. Sewall (Sewall Gear) W. H. Heller (Peerless---Winsmith) L. J. Smith (Invincible Gear) G. Henriot (Engrenages et Reducteurs) M. Tanaka (Nippon Gear) R. W. Hercus (F. W. Hercus) H. J. Trapp (Klingelnberg) M. Hirt (Renk) T. Urabe (Tsubakimoto Chain) W. H. Jogwick (Union Carbide) D. A. Wagner (General Motors/AGT) T. Kameyama (Seiki---Kogyosho) R. E. Weider (Clark Equipment) D. L. King (Terrell Gear) L. E. Wilcox (Gleason) P. Losekamp (Xtek) H. Winter (Academic Member) K. Mariager (F. L. Smidth) J. Worek (IMO Delaval)

AGMA iv 908---B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

Table of Contents Section Title Page 1. Scope 1.1 Pitting Resistance Geometry Factor, I ...... 1 1.2 Bending Strength Geometry Factor, J ...... 1 1.3 Tables ...... 1 1.4 Exceptions...... 1 1.5 Bending Stress in Internal Gears...... 1

2. Definitions and Symbols 2.1 Definitions...... 2 2.2 Symbols ...... 2

3. Basic Gear Geometry 3.1 Contact Ratios...... 6 3.2 Minimum Length of the Lines of Contact...... 6 3.3 Load Sharing Ratio, m N ...... 6 3.4 Operating Helix Angle, ψr ...... 6 3.5 Operating Normal Pressure Angle, Ônr ...... 6

4. Pitting Resistance Geometry Factor, I 4.1 Pitting Resistance Geometry Factor Calculation...... 7 4.2 Operating Pitch Diameter of Pinion, d ...... 7 4.3 Radii of Curvature of Profiles at Stress Calculation Point...... 7 4.4 Helical Overlap Factor, Cψ ...... 7

5. Bending Strength Geometry Factor, J 5.1 Virtual Spur Gear...... 8 5.2 Pressure Angle at the Load Application Point...... 8 5.3 Generating Rack Shift Coefficient...... 9 5.4 LoadAngleandLoadRadius...... 9 5.5 Tool Geometry...... 10 5.6 Generating Pressure Angle...... 12 5.7 Algorithm for Determining the Critical Point...... 13 5.8 Iteration Convergence...... 14 5.9 Radius of Curvature of Root Fillet...... 15 5.10 Helical Factor, Ch ...... 15 5.11 Stress Correction Factor, Kf ...... 16 5.12 Helix Angle Factor, Kψ ...... 16 5.13 Tooth Form Factor, Y ...... 16

6. Determining Addendum Modification Coefficients 6.1 Generating Rack Shift Coefficients...... 16 6.2 Sum of the Addendum Modification Coefficients for Zero Backlash...... 16 6.3 Tooth Thinning for Backlash...... 17 6.4 Addendum Modification Coefficients...... 17

AGMA v 908---B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

Table of Contents (cont) Section Title Page 7. Geometry Factor Tables 7.1 Using the Tables...... 17 7.2 Whole Depth...... 18 7.3 Outside Diameter...... 18 7.4 Type of Gearing...... 18 7.5 Center Distance...... 18 7.6 Tooth Thickness Backlash Allowance...... 18 7.7 Undercutting...... 19 7.8 Top Land...... 19 7.9 Cutter Geometry...... 19 7.10 Axial Contact Ratio...... 19

Bibliography ...... 53

Appendices Appendix A Original Derivation of AGMA Geometry Factor for Pitting Resistance, I ...... 55 Appendix B ANSI/AGMA 2001---B88 Pitting Resistance Formula Derivation...... 61 Appendix C Explanation of the AGMA Gear Tooth Strength Rating Derivation For External Gears...... 65 Appendix D Selection of Shaper Cutter Geometry...... 69 Appendix E Derivation of Helical Overlap Factor, Cψ ...... 71 Appendix F High Transverse Contact Ratio Gears...... 73

Ta ble s Table 2---1 Symbols Used in Equations...... 2 Table 5---1 Limiting Variation in Action for Steel Spur Gears for Load Sharing...... 8 Ta ble s I and J FACTORS...... 20

Figures Fig3---1 TransversePlaneViewofTheLineofAction...... 5

Fig5---1 LoadAngleandLoadRadius...... 9 Fig 5---2 Pressure Angle Where Tooth Comes to Point...... 10 Fig 5---3 Shaper Cutter with Protuberance...... 11 Fig 5---4 Involute Drawn Through Point “S”...... 12 Fig 5---5 Pressure Angle Where Cutter Tooth Comes to a Point...... 12 Fig 5---6 Angle to Center, S, of Tool Tip Radius (Effective Cutter)...... 12 Fig 5---7 Critical Point of Maximum Bending Stress...... 13 Fig5---8 ShaperCutterGeneration...... 13 Fig 5---9 Iteration Function...... 14 Fig 5---10 Oblique Contact Line...... 15 F i g 5 --- 1 1 H e l i c a l Fa c t o r , Ch ...... 15

Fig 7---1 Undercutting Criteria...... 19

AGMA vi 908---B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

1. Scope thickness due to protuberance below the active profile is handled correctly by this method. The procedures in this Information Sheet de- (7) The root profiles are stepped or irregular. scribe the methods for determining Geometry The J factor calculation uses the stress correction Factors for Pitting Resistance, I, and Bending factors developed by Dolan and Broghamer[l]. Strength, J. These values are then used in con- These factors may not be valid for root forms junction with the rating procedures described in which are not smooth curves. For root profiles AGMA 2001-B88, Fundamental Rating Factors which are stepped or irregular, other stress correc- and Calculation Methods for Involute Spur and tion factors may be more appropriate. Helical Gear Teeth, for evaluating various spur (8) Where root fillets of the gear teeth are and helical gear designsproduced using a generat- produced by a process other than generating. ing process. (9) The helix angle at the standard (refer- 1.1 Pitting Resistance Geometry Factor, I. A ence) diameter* is greater than 50 degrees. mathematical procedure is described to determine the Geometry Factor, I, for internal and external In addition to these exceptions, the following gear sets of spur, conventional helical and low ax- conditions are assumed: ial contact ratio, LACR, helical designs. (a) The friction effect on the direction of force is neglected. 1.2 Bending Strength Geometry Factor, J. A (b) The fillet radius is assumed smooth (it is mathematical procedure is described to determine actually a series of scallops). the Geometry Factor, J, for external gear sets of spur, conventional helical and low axial contact 1.5 Bending Stress in Internal Gears. The ratio, LACR, helical design. The procedure is Lewis method [2] is an accepted method for cal- valid for generated root fillets, which are pro- culating the bending stress in external gears, but duced by both rack and pinion type tools. there has been much research [33 which shows that Lewis’ method is not appropriate for internal 1.3 Tables. Several tables of precalculated Ge- gears. The Lewis method models the gear tooth ometry Factors, I and J, are provided for various as a cantilever beam and is most accurate when combinations of gearsets and tooth forms. applied to slender beams (external gear teeth with 1.4 Exceptions. The formulas of this Informa- low pressure angles), and inaccurate for short, tion Sheet are not valid when any of the following stubby beams (internal gear teeth which are wide conditions exist: at their base). Most industrial internal gears have thin rims, where if bending failure occurs, the fa- (1) Spur gears with transverse contact ratio tigue crack runs radially through the rim rather less than one, mp c 1.0. than across the root of the tooth. Because of their (2) Spur or helical gears with transverse con- thin rims, internal gears have ring-bending tact ratio equal to or greater than two, mp 2 2.0. stresses which influence both the magnitude and Additional information on high transverse contact the location of the maximum bending stress. Since ratio gears is provided in Appendix F. the boundary conditions strongly influence the (3) Interference exists between the tips of ring-bending stresses, the method by which the teeth and root fillets. internal gear is constrained must be considered. (4) The teeth are pointed. Also, the time history of the bending stress at a (5) Backlash is zero. particular point on the internal gear is important (6) Undercut exists in an area above the theo- because the stresses alternate from tension to retical start of active profile. The effect of this un- compression. Because the bending stressesin in- dercut is to move the highest point of single tooth ternal gears are influenced by so many variables, contact, negating the assumption of this calcula- no simplified model for calculating the bending tion method. However, the reduction in tooth root stress in internal gears can be offered at this time. [ ] Numbers in brackets refer to the bibliography. * Refer to AGMA 112.05 for further discussion of standard (reference) diameters.

AGMA 1 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

2. Definitions and Symbols resistance and bending strength formulas are shown in Table 2-l. 2.1 Definitions. The terms used, wherever ap- NOTE: The symbols, definitions and termi- plicable, conform to the following standards: nology used in this Standard may differ ANSI Y10.3-1968, Letter Symbols for Quan- from other AGMA standards. The user tities Used in Mechanics of Solids should not assumethat familiar symbols can AGMA 112.05 Gear Nomenclature (Geome- be used without a careful study of these try) Terms, Definitions, Symbols, and Abbrevia- definitions. tions Units of measure are not shown in Table 2-l AGMA 600.01 Metric Usage because the equations are in terms of unity nor- 2.2 Symbols. The symbols used in the pitting mal module or unity normal diametral pitch. Table 2-l Symbols Used in Equations Svmbols Terms Where First Used Bn normal operating circular backlash Eq 6.7 c standard center distance Eq 6.4 c42, --- 4 distances along line of action (Fig 3-l) Eq 3.11-3.16 c nly cn4*cn6 distances along line of action of virtual spur gear Eq 5.15-5.17 ‘h helical factor Eq 5.69 c, operating center distance Eq 3.7 helical overlap factor Eq 5 4.1 D al3 Da2 addendum diameter, pinion and gear Eq 7.1-7.2 d pinion operating pitch diameter Eq 4.1 F effective face width Eq 3.20 H parameter for stress correction factor Eq 5.72 ha0 nominal tool addendum Eq 5.36 hF height of Lewis parabola Eq 5.62 I pitting resistance geometry factor Eq 4.1 J bending strength geometry factor Eq 5.1 Jl adjusted geometry factor Eq 7.6 JS geometry factor from table Eq 7.6 Kf stress correction factor Eq 5.1 helix angle factor Eq 5.77 ? parameter for stress correction factor Eq 5.72 L min minimum length of contact lines Eq 3.21 M parameter for stress correction factor Eq 5.72 “F axial contact ratio Eq 3.20 “G gear ratio Eq 3.1 “N load sharing ratio Eq 3.24 mn normal module Eq 7.9M transverse contact ratio Eq mP 3.18 n virtual tooth number Eq 5.2

nO virtual tooth number of tool Eq 5.29 n1 pinion tooth number Eq 3.1 n2 gear tooth number Eq 3.1

AGMA 2 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

Table 2-l (cant) Symbols Used in Equations Symbols Terms Where First Used

na fractional part of mF Eq 3.22 nC tool tooth number Eq 5.29 nr fractional part of mp Eq 3.22 Pnd normal diametral pitch Eq 7.5 pb transverse base pitch Eq 3.8 PN normal base pitch Eq 3.9 px axial pitch Eq 3.19 R1’ R2 standard pitch radii, pinion and gear Eq 3.2-3.3 Rbl, Rb2 base radii, pinion and gear Eq 3.5-3.6 Rbc base radius of tool Eq 5.34 Rc standard pitch radius of tool Eq 5.33 %21 mean radius of pinion Eq 4.3 R01, Ro2 addendum radii, pinion and gear, internal and external Eq 3.12, 3.15 Roc outside radius of tool Eq 5.36 rn> ‘722 reference pitch radii of virtual spur gear Eq 5.3, 5.12 r;; generating pitch radius of virtual spur gear Eq 5.51 ‘no reference pitch radius of virtual tool Eq 5.30 GO generating pitch radius of virtual tool Eq 5.52 GO radius to center “S” of tool tip radius Eq 5.39 ‘na , ‘na2 virtual outside radii Eq 5.5, 5.14 ‘nb ’ ‘nb2 virtual base radii Eq 5.4, 5.13 ‘izbo virtual base radii of tool Eq 5.31 ‘nL virtual load radius Eq 5.28 SF tooth thickness at critical section Eq 5.72 ‘n reference normal circular tooth thickness Eq 5.20 ‘n 19‘ n2 reference normal circular tooth thickness, pinion and gear Eq 6.1-6.2 Sna tooth thickness at outside diameter Eq 7.9 ‘no reference normal circular tooth thickness of tool Eq 5.35 Sns standard tooth thickness, thinned for backlash Eq 7.6 uS stock allowance per side of tooth Eq 5.37 x addendum modification coefficient at zero backlash Eq 5.19 xl’ x2 addendum modification coefficient, pinion and gear Eq 6.5 generating rack shift coefficient xs Eq 5.19 xO addendum modification coefficient of tool Eq 5.35 xgl, xg2 generating rack shift coefficient, pinion and gear Eq 6.1-6.2 Y tooth form factor Eq 5.1 Y iteration function Eq 5.63 Y’ derivative of iteration function Eq 5.64 z active length of line of action Eq 3.17

AGMA 3 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

Table 2-1 (cant) Symbols Used in Equations

Symbols Terms Where First Used

%z angle of surface, normal Eq 5.53 %l iteration angle Eq 5.65 b2 angle between tangent to fillet and tooth center line Eq 5.59 As, amount gear tooth is thinned for backlash Eq 5.19 60 amount of protuberance, tool Eq 5.38 Sao amount of effective protuberance, tool Eq 5.38 % ordinate of gear fillet curve Fig 5-8 ‘7nF ordinate of critical point “F” Eq 5.61 en angular displacement of gear Eq 5.57 8no angular displacement of tool Eq 5.56 KFKS distance from pitch point to points “F” and “S” Eq 5.54, 5.55 xns angle to center “S” of tool tip radius Eq 5.47 kno auxiliary angle locating point “S” Eq 5.53 En abscissa of gear fillet curve Fig 5-8 E nF abscissa of critical point “F” Eq 5.60 Ply p2 radii of curvature of profiles at point Eq 4.1 of contact stress calculation pmls pm2 radii of curvature of profile at mean radius Eq 4.8 pao tool tip radius Eq 5.39 G radius of curvature of fillet curve Eq 5.66 PF minimum radius of curvature of fillet curve Eq 5.68 4 standard transverse pressure angle Eq 3.4 +n standard normal pressure angle Eq 3.4 4); generating pressure angle Eq 5.48 +nL load angle Eq 5.22 pressure angle at radius where gear tooth is pointed cpnp Eq 5.22 pressure angle at radius where tool tooth is pointed 9w Eq 5.43 4 nr operating normal pressure angle Eq 3.28 cpns pressure angle at point “S” on tool Eq 5.40 4nW pressure angle at load application point Eq 5.10 cbr operating transverse pressure angle Eq 3.7 * standard helix angle Eq 3.2 base helix angle Eq 3.10 4 f operating helix angle Eq 3.27 0 angle of inclination of helical contact line Eq 5.70

SUBSCRIPTS 0 tool n normal or virtual spur gear 1 pinion r operating or running 2 gear - absence of a subscript indicates transverse

AGMA 4 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

3. Basic Gear Geometry -1 (Rb;; Rbl) $r= cos 0% 3.7) The following equations apply to spur and where heIica1 gears where spur gearing is a particular = operating center distance case with zero helix angle. Where double signs cr are used (e.g., ? ), the upper sign applies to exter- Transverse base pitch, pb nal gears and the lower sign applies to internal 2qRbl gears. The equations are derived in terms of unity pb= nl @q 3.8) normal module (mn = 1 .O) or unity normal diametral pitch and are valid for any consistent set Norma1 base pitch, pN of units. All angles are given in terms of radians, pN = 7T cos +n 0% 3.9) unless otherwise specified. Base helix angle, qb The following variables must be made dimensionless by dividing with the normal module, mn, qb= cos-’( PN> or multiplying with the normal diametral pitch, ‘b Pnd, (See AGMA 112.05 for definitions of mn Figure 3-l is a view of the line of action in the or Pnd ). The variables to be adjusted are C, , transverse plane. The lengths, C 1 through C 6, F, R. lf R. 2, Rot , R,, ha0 , 8,) pa,, and As,. are derived from Fig 3-l. See 1.4 item (6), refer- encing exceptions regarding gear tooth undercut. Gear ratio, mG t n2 “G = n1 where n2 = gear tooth number = pinion tooth number n1 Standard (reference) pitch radius, RI

(Eq 3.2) where Jr = standard helix angle Standard (reference) pitch radius, R2

R2 = Rl mG 0% 3.3) Standard transverse pressure angle, +

+ =tan-‘(2 ) 0% 3.4) where +n = standard normal pressure angle* Pinion base radius, Rbl Rbl = Rl cos C$I (Eq 3.5) Gear base radius, Rb2

Rb2 = Rbl “G 03 3.6) Fig 3-l Transverse Plane View of The Operating transverse pressure angle, 4, Line of Action

* For a complete discussion, see 9.01 of AGMA 112.05 Gear Nomenclature (‘Geometry) Terms, Definitions, Symbols, and Abbreviations

AGMA 908-B89 Geometry Factors for Determinirg the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

Sixth distance along line of action, C6 where F = effective face width at m, = 1.0 (Eq 3.11) ‘6 = Cr sin f#r For spur gears, mF = 0.0 First distance along line of action, Cl 3.2 Minimum Length of the Lines of Contact. Cl = 2 [Cs- (Ro2’- Rb;ps] (Eq 3.12) For spur gears with mp < 2.0 the minimum length of contact lines, Lmir, is: where R 02 = addendum radius of gear, for Lmin =F (Eq 3.21) internal or external gears For helical gears, two cases must be considered: Third distance along line of action, C3 Case I, for na 5 1 - 72, r mp F-nun, px c3= “6 (Eq 3.13) L = (Eq 3.22) *G + i min cos $, Fourth distance along line of action, C4 Case II, for n, > 1 - n, c4 = cl+ Pb (Eq 3.14) mp F-U - na>U - n,> px L Fifth distance along line of action, C5 min = (Eq 3.23) 2 0.5 cg = (Ro\-Q) (Eq 3.15) where nr = fractional part of mp where = fractional part of mF = addendum radius of pinion na R 01 Example: Second distance along line of action, C2 for a contact ratio, mp, of 1.4, then n, = 0.4

‘2 = ‘5 -Pb (Eq 3.16) 3.3 Load Sharing Ratio, mN Active length of line of contact, 2 z= cg -cl (Eq 3.17) For helical gears: F Distance C2 locates the lowest point of single (Eq 3.24) *N= L tooth contact (LPSTC) and distance C, locates min the highest point of single tooth contact (HPSTC) , For spur gears with mp< 2.0, (Eq 3.21) gives where C, , are values for mn = 1.0. R. 1 and Ro2 Le= F, therefore: 3.1 Contact Ratios. mN= 1.0 (Eq 3.25) Transverse contact ratio, mp For LACR helicals, (mF < 1.0)) load sharing is z (Eq 3.18) accomodated by CJ~, therefore: “P = pb mN= 1.0 (Eq 3.26) Axial pitch, px 3.4 Operating Helix Angle, eT LL (Eq 3.19) Px = sin Jr (Eq 3.27) Axial contact ratio, mF 3.5 Operating Normal Pressure Angle, +nT F (Eq 3.20) *F= px +nr = sin1 (cos Jtb sin $ > (Eq 3.28)

AGMA 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

4. Pitting Resistance Geometry Factor, I R02 = addendum radius, gear, internal or external The pitting resistance geometry factor, I, is a Radius of curvature of the pinion profile at dimensionless number. It takes into account the the point of contact stress calculation, p, effects of: = (R;l- R;l)0.5 ;Eq 4.4) (1) radii of curvature Pl (2) load sharing where (3) normal component of the transmitted load Rbl = base radius, pinion 4.1 Pitting Resistance Geometry Factor Radius of curvature of the gear profile at the Calculation. point of contact stress calculation, p2

P2 = CgT P1 (Eq 4.5) 0% 4-l) where ‘6 = sixth distance along line of action (see Eq 3.11) where 4.3.2 Spur and Low Axial Contact Ratio +r = operating transverse pressure angle Helical Gears. For spurs and LACR helicals = helical overlap factor (See 4.4 and % Appendix E) (mF < 1.0) the radii of curvature are calculated at the LPSTC d = pinion operating pitch diameter = c2 (Eq 4.6) = load sharing ratio Pl mN where = radius of curvature of pinion profile p1 c2 = second distance along line of action at point of contact stress calculation (see Eq 3.16) = radius of curvature of gear profile at p2 = C6TP1 0% 4.7) point of contact stress calculation p2 4.4 Helical Overlap Factor, C+* 4.2 Operating Pitch Diameter of Pinion, d. 2 G For LACR helicaI gears (mF 5 1.0) d = cm 4.2) 0.5 mG+ 1 pm1pm2 z 0% 4.8) where ‘1 ‘2 pN )I “G = gear ratio where Z = active length of line of action 4.3 Radii of Curvature of Profiles at Stress = normal base pitch Calculation Point PN 4.3.1 Conventional Helical Gears. For Radius of curvature of the pinion profile at conventional helical gears (mF > 1.0) the radii of the mean radius of the pinion, pm1 curvature are calculated at the mean radius or middle of the working profile of the pinion where: Pml’

AGMA 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

5. Bending Strength Geometry Factor, J Virtual base radius, rnb r =r cos Ô (Eq 5.4) The bending strength geometry factor, J,isadi- nb n n mensionless number. It takes into account the ef- Virtual outside radius, rna fects of: r =r +R --- R (1) shape of the tooth na n o 1 1 (Eq 5.5) (2) worst load position For spur gears, the actual geometry is used (3) stress concentration n = n 1 (Eq 5.6) (4) load sharing between oblique lines of r = (Eq 5.7) contact in helical gears n R 1 r = R (Eq 5.8) Both tangential (bending) and radial (compres- nb b1 sive) components of the tooth load are included. This rna = R (Eq 5.9) analysis applies to external gears only. o1 5.2 Pressure Angle at the Load Application Point. The J factor calculation procedure must be re- Spur gears develop the most critical stress when load peated for both the pinion and the gear using the ap- is applied at the highest point of the tooth where a propriate dimensions for each. single pair of teeth is carrying all of the load. Spur Y Cψ gears having variations that prevent two pairs of J = (Eq 5.1) Kf m N teeth from sharing the load may be stressed most heavily when the load is applied at the tip. Table 5---1 where has been used in previous standards to establish the Y = tooth form factor (See 5.13) variation in base pitch between the gear and pinion, = helical overlap factor (See 4.4) Cψ which determines whether or not load sharing exists K = stress correction factor (See 5.11) in steel spur gears. Values in excess of those shown in f Table5---1requiretheuseoftiploading. m = load sharing ratio (See 3.3) N Ta b l e 5 --- 1 LimitingVariationinActionforSteelSpurGears It is recognized that an anomaly exists when cal- for Load Sharing culating the J factor for LACR gears where the value (Variation in Normal Base Pitch) obtained may be greater than a conventional helical Maximum Allowable Variation in gear. For this reason, it is recommended that the J Number inches (mm), When Teeth Share Load factor be calculated for both the LACR condition of Load per Inch of Face (per mm of face) and as a conventional helical gear, using a value for F Pinion Teeth 500 lb 1000 lb 2000 lb 4000 lb 8000 lb which is slightly greater than p .Theresultingcon- (90 N) (175 N) (350 N) (700 N) (1400 N) x servative value should be used unless otherwise justi- 15 0.0004 0.0007 0.0014 0.0024 0.0042 fied. (0.01) (0.02)(0.04) (0.06) (0.11) 5.1 Virtual Spur Gear. The following analysis is 20 0.0003 0.0006 0.0011 0.0020 0.0036 (0.01) (0.02)(0.03) (0.05) (0.09) based on the work of Errichello [4] [5] [6]. Helical gears are considered to be virtual spur 25 0.0002 0.0005 0.0009 0.0017 0.0030 gears with the following virtual geometry: (0.01) (0.01)(0.02) (0.04) (0.08) Virtual tooth number, n For helicalgears and spur gears that are analyzed n1 where the load is applied at the tip of the tooth, the n= (Eq 5.2) cos# ψ pressure angle at load application point, ,isgiv- ÔnW Standard (reference) pitch radius of virtual spur en by: gear, r 0.5 n 2 rna n tan Ô = − 1 (Eq 5.10) r = (Eq 5.3) nW rnb  n 2  

AGMA 8 908---B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

For spur gears, where the highest bending stress at zero backlash occurs when the load is at the highest point of single = amount gear tooth is thinned for ∆ sn tooth contact (HPSTC), the pressure angle is given backlash by: C s +∆ s --- π /2 4 (Eq 5.11) x n n (Eq 5.20) tan Ô = r = nW nb 2 tan Ôn where Equation 5.11 may also be used for LACR heli- s = normal circular tooth thickness C n cal gears, but distance 4 must be based on the virtu- measured on the Standard (reference) al spur gear. The following equations are developed pitch cylinder from analogy with Eqs 3.3, 3.6, 3.11, 3.12, 3.14, 5.5 π s = +2 x tan Ô (Eq 5.21) and 5.11. n 2 g n Standard (reference) pitch radius of virtual spur 5.4 Load Angle and Load Radius. F i g u r e 5 --- 1 d e - gear, r fines the load angle, Ô ,andtheloadradius,r . n2 nL nL rn2 = rn mG (Eq 5.12) The load is shown applied at an arbitrary point “W”, Virtual base radius, such that: rnb2 Ô tan --- i n v Ô r nb2 = rnb mG (Eq 5.13) nL = ÔnW np (Eq 5.22) where Virtual outside radius, r na2 Ô = pressure angle at radius where gear np r = r + R --- R (Eq 5.14) na2 n2 o2 2 tooth is pointed. see Fig 5---2 Sixth distance along line of action, ,ofvirtual Cn6 C spur gear LOAD C =(r +r )tan Ô (Eq 5.15) Ô n6 nb2 nb nr nL First distance along line of action, ,ofvirtual W Cn1 spur gear 0.5 2 2 C = C − r − r (Eq 5.16) inv n1  n6  na2 nb2  Ô np Fourth distance along line of action, ,ofvirtual tan Ô Cn4 nW spur gear C = C +p (Eq 5.17) n4 n1 N ÔnW r nL The pressure angle at load application point, ÔnW Cn4 Ô tan Ô = (Eq 5.18) nL nW rnb 5.3 Generating Rack Shift Coefficient. The generating rack shift coefficient, x , applies to the complete- g r nb ly finished teeth. It includes the rack shift for addendum modification plus the rack shift for thinning the gear teeth to obtain backlash: ∆ sn x g = x --- (Eq 5.19) 2tanÔn where x = addendum modification coefficient Fig5---1LoadAngleandLoadRadius

AGMA 9 908---B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

equal a large number such as nc = 10 000. When exact cutter dimensions are not known, refer to Appendix D. Helical gears are considered to be generated by a virtual shaper cutter with the following virtual geometry: Virtual tooth number of tool, no

no =- nC (Eq 5.29) co9 Jr where

nC = tool tooth number r n Standard (reference) pitch radius of virtual tool, ‘no

r nO no =- (Eq 5.30) 2 Virtual base radii of tool, rnbo

‘nbo = ‘no cos 4n (Eq 5.31) For spur gears, the actual cutter geometry is \ used no = nc Fig 5-2 Pressure fAngle Where Tooth (Eq 5.32) Comes to Point r no = R, (Eq 5.33) Sn inv+ = ‘“V+n + z, (Eq 5.23) np where but Rc = standard pitch radius of tool (Eq 5.24) irW+n = tan+ n- ‘72 ‘nbo = Rbc (Eq 5.34) 2r, = n (Eq 5.25) where Rbc = base radius of tool il-lVC$ = tan4n- 9, +% (Eq 5.26) np Figure 5-3 shows a shaper cutter with protu- Substituting this value in Eq 5.22 gives: berance, 8,. A tool without protuberance is a particular case for which So = 0. The center of 4JnL = f.=+nw - t-bn + +n- 2 (Eq 5 27j the tip radius, point “S”, is located by radius Equation 5.27 gives the load angle for any r i. and angle X,/2. The nominal tool adden- load position specified by tan +nW. dum is hcLo. The reference addendum related to From Fig 5-1, the virtual load radius, rnL, is: the virtual radius, rno , is ha0 + X0, where X0 is the addendum modification coefficient corre- ‘nb (Eq 5.28) ‘nL = cos OnL sponding to the present sharpening condition of the cutter. The addendum modification coeffi- 5.5 Tool Geometry. The following analysis is cient of the tool, X0, relates the actual normal cir- based on pinion type generating tools commonly cular tooth thickness of the tool, sno, to the referred to as shaper cutters. However, the nominal value of m/2. If sno is known from method applies equally well to rack-type generat- measurements of the tool, x0 may be calculated ing tools by letting the tooth number of the tool from:

AGMA 10 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

-n/2 where x0 = Sno (Eq 5.35) 2 tan 4, Roc = outside radius of tool where where sno , u s, R,,, and Rcare values for S no = reference normal circular tooth mn = 1.0. thickness of tool Finishing hobs usually have sno = ?r/2 in NOTE: x0 is positive when sno > 7~12 (cor- which case x0 = 0. A pre-grind hob which has responding to a new shaper cutter), or teeth thinner than n/2 to provide stock allowance negative when sno < n/2 (corresponding to for grinding usually has a tooth thickness of: a used shaper cutter). Near the mid-life of 7-r (Eq 5.37) the cutter, its tooth thickness equals rr/2 sno = Iii-- 2% and x0 = 0.0. where

uS = stock allowance per side of the gear tooth Since us is removed during grinding, the basic rack corresponding to the finished gear teeth is used for the analysis; i.e., let sno = n/2 and re- duce the amount of protuberance: x0 = 0

6 a0 = 60 - us cos +n (Eq 5.38) -l-r where no 8a0 = amount of effective protuberance, tool = amount of protuberance, tool 60

from Fig 5-3 rsno = rno + ‘a0 + ‘0 - %o (Eq 5.39) where TSno = radius to center “S” of tool tip radius Pa0 = tool tip radius Figure 5-4 shows an involute drawn through TOOL TOOTH 6 point “S”. The pressure angle at point “S” on 6 TOOL SPACE tool, +ns, is: Fig 5-3 Shaper Cutter with Protuberance 9 ns = cos -l(T) (Eq 5.40) The nominal tool addendum is defined as the inv+ns = tan +rs - +ns (Eq 5.41) addendum where the normal circular tooth thickness of the tool equals the nominal value of r/2. The reference circular tooth thickness of the If the outside radius of the tool is known from cutter is: measurements, the nominal tool addendum, ha0 , ‘no = z2 + 2x, tan+, (Eq 5.42) may be calculated from: In Fig 5-5, +nPo is the pressure angle where h -R,-x0 , 00 = Rot (Eq 5.36) the cutter tooth comes to a point. It is given by:

AGMA 11 908-B89 Geometry Factors for Deterrniniqg the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

Sno b-W+ = inv+,t- npo 2rno but: hv+n = tan C& - +n (Eq 5.44)

2rno = no (Eq 5.45) ‘no inv~$~~~= tan+, - +n +- (Eq 5.46) no from Fig 5-6 xns G&J - PC20> - = in@ npo - inv9ns + 2 ‘do (Eq 5.47) where tip radius x ns = angle to center “S” of tool

Fig 5-5 Pressure Angle Where Cutter Tooth Comes to a Point

P a0 t- nbo

Fig 5-4 Involute Drawn Through Point ‘73” 5.6 Generating Pressure Angle. The generating pressure angle, 4; , depends on the virtual center distance between the cutter and the gear which is determined by xg and x0. The generating pressure angle, 4; , is obtained from:

2 (“g’ x0 Itan +n iIlV$;; = invc$n+ Fig 5-6 Angle to Center, S, of Tool Tip n + no (Eq 5.48) Radius (Effective Cutter)

AGMA 12 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

Arc involute may be solved by iteration.[5] From Fig 5-8 the auxiliary angle locating point Let the first trial value for 4; be: “S”, pno, is:

0.33 CQ = (3 inv $; ) (Eq 5.49) ~~~ = cos-I(“” ;;an) - cr, (Eq 5.53) This trial value is successively improved upon using: where invdi + ~$;li- tan+;i $((i+l) = +;;i + (Eq 5.50) Qn = angle of surface normal tan 2 (pii 5.6.1 Generating Pitch Radii. The generating pitch radius of virtual spur gear, ri, is:

(Eq 5.51)

The generating pitch radius of virtual tool, Al;, is:

(Eq 5.52)

r 5.7 Algorithm for Determining the Critical I no Point. Figure 5-7 shows the critical point of maximum bending stress located at the intersec- tion of the Lewis parabola and the gear tooth fillet. To locate this point, we consider the relative motion between the shaper cutter and the generated gear tooth. Figure 5-8 shows the shaper cutter generating an arbitrary point “F” on the gear tooth fillet. From the law of conjugate gear tooth action, point “F” lies on a line which extends from the generating pitch point “P” through the center of the tool tip radius, point “S”. The fillet coordinates are best expressed by selecting the angle on as the independent parameter. Then for an= IT/~, generating starts at the lowest point on the fillet and proceeds up the fillet as on is dimin- ished corresponding to clockwise rotation of the tool and counter-clockwise rotation of the gear.

GEAR ‘TOOTH VERTEX CENTERLINE

Fig 5-7 Critical Point of Maximum Bending Stress Fig 5-8 Shaper Cutter Generation

AGMA 13 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

Distance from pitch point to point “S”, KS , is: hF = height of Lewis parabola Differentiating Eq 5.63 gives: KS = 5; sin an - 5; sin (an + I+, ) 2hF (Eq 5.54) Y‘ = - KF sin pn COS2 Pn The distance from pitch point to point “F”, KF, is: +- nO -1 n KF= KS- P,, (Eq 5.55) 1 2 CnF tan fin - ‘7nF - 3% KF and KS are vectors and may be positive or (3082pn negative. n sin cy, a, - The angular displacement of tool, eno, is: - ‘no tan Can + P-no> 1 x = pno - y + $ ( l~~p~n) - 0 no (Eq 5.56) (Eq 5.64) 0 The gear rotation angle is: where Yl = derivative of iteration function 8, = ? en0 (Eq 5.57) Assuming an initial approximation for CY, = ~14, where it is successively improved upon by using Newton’s Method of iteration. ‘n = angular displacement of gear Y O!nl= CXn-y’ (Eq 5.65) en = : (p,,- +%$) (Eq 5.58) On each iteration, cr, is set equal to on1 and The slope of the line tangent to the fillet at Eq 5.53 through Eq 5.65 are iterated until Iy[ in point “F” is: Eq 5.63 is a negligible tolerance. P, =O! n-8n (Eq 5.59) 5.8 Iteration Convergence. Equation 5.63, ex- pressing the function y = f ( CYn ), is plotted in Fig where 5-9 for a typical case. By selecting Q!n =7r/4 as @n = angle between tangent to fillet and the initial approximation, rapid convergence to tooth center line the proper root is obtained, usually within 3 to 5 The coordinates of critical point “F” are: iterations. This choice for the initial value pre- vents convergence to the incorrect root which ex- The abscissa of critical point “F”, gnF ists closer to on = 0. This incorrect root corre- EnF = r-” sin en + KF cos 8, (Eq 5.60) sponds to the case where the Lewis parabola is inverted, opening upward rather than downward. The ordinate of critical point “F”, qnF 15 l-i&Z = r; cod, + KF sin @, (Eq 5.61) 10 Let Y = f( :WJ hF = ‘nL- qnF (Eq 5.62) 5

For point “F” to be on the Lewis parabola, the following equation must be satisfied: 0 Y = 2hFtan$,-gnF = 0 (Eq 5.63) -5 0 n/4 where % Y = iteration function Fig 5-9 Iteration Function

AGMA 14 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

5.9 Radius of Curvature of Root Fillet. The addendum beyond the loaded portion (see Fig radius of curvature of the fillet curve, Pb, at any 5-10) point defined by CYn is given by:

Ks’ P; = pa0 + N n (Eq 5.66) rn rno sin an r; f rio -KS

KS = rlo - rn: (Eq 5.67)

The J factor uses the minimum radius of cur- NOTE: Full Buttressing Exists When vature which occurs at the point where the fillet F, 2 One Addendum curve is tangent to the root circle, where Q!n = rr/2 Fig 5-10 Oblique Contact Line and pno = 0. For Spur and LACR Helical Gears(m. 5 1.0)) a Subsituting, the minimum radius of curvature of unity value is used, fillet curve, pF, is: (r;lo- rio I2 Ch = 1.0 (Eq 5.69) PF = pa* + I, ,, rn ‘no _ (r” N I, no - ‘nos ) For Conventional Helical Gears, when mF > 1.0 Tn+ ‘no (Eq 5.68) 5.10 Helical Factor, Ch. The heiical factor, (Eq 5.70) is the ratio of the root bending moment pro- ch=lp(l: &;l”.’ ‘h’ duced by tip loading to the root bending moment L -1 produced by the same intensity of loading applied where along the oblique helical contact line. It is based w = angle of inclination of helical on the work of Wellauer and Seireg [7]. contact line in degrees If the worst condition of load occurs where w = tan-’ (tan* sin+,) (Eq 5.71) full buttressing exists, the value of Ch may be in- Equation 5.70 is valid for $r < 50”. creased by 10 percent. Full buttressing exists when the face of the tooth extends at least one ch VaheS can be taken from Fig 5-l 1

2.0

1.8 +n =30°

492 =22O 1.6 +n = 150

Helix Angle, Q Fig 5-11 Helical Factor, ch

AGMA 15 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

The helical factor, Ch, is based on the work 6. Determining Addendum Modification of Wellauer and Seireg [ 71. Coefficients

5.11 Stress Correction Factor, Kf. The stress In order to use Section 5, xl and x2 must be correction factor, Kf , includes the effects of stress determined. If these values are not known, this concentration and load location, based on the Section provides a method for determining them. work of Dolan and Broghamer [ 11. It is given by: 6.1 Generating Rack Shift Coefficients. If the normal circular tooth thicknesses are known, the Kf = u +($“($” (Eq 5.72) generating rack shift coefficients are found from Eqs 6.1 and 6.2. where SF = tooth thickness at critical section

= 2blF (Eq 5.73) s = n2 'i 0% 6.2) $7 = minimum radius of curvature of ‘g2 = ’ (\ 2tan$, > fillet curve = Xg2 hF = height of Lewis parabola =xg H = 0.331 - 0.436+, (Eq 5.74) where generating rack shift coefficient, L = 0.324 - 0.492+, (Eq 5.75) “gl = pinion M = 0.261 + 0.5454, (Eq 5.76) generating rack shift coefficient, xg2 = where gear +n = standard normal pressure angle cxg = sum of generating rack shift coefficients reference normal circular Note: In order to calculate an accurate ‘nl = tooth thickness, pinion (see Eq 5.21) value for Kf, the significant decimal places in Eq 5.74 through 5.76 are neces- ‘n2 = reference normal circular tooth sary. The resulting values of H, L and M thickness, gear (see Eq 5.21) may be rounded to two decimal places. 6.2 Sum of Addendum Modification Coeffi- 5.12 Helix Angle Factor, Q. The helix angle cients for Zero Backlash. Although the amount of tooth thinning applied to each gear may be un- factor, KQ, depends on the type of gear. known, the sum of the addendum modification For Spur and LACR Helical Gears(mF I 1.0)) a coefficients for the gear pair, CX, can be estab- unity value is used, K,J, = 1.0 lished. c (inv +r - inv +) For Conventional Helical Gears, when mF > 1.0 cx = tan C$I (Eq 6.4)

K* = cos qrr COSQ (Eq 5.77) cx = x2 f Xl (Eq 6.5)

5.13 Tooth Form Factor, Y. The tooth form c =R2 2 Rl 0% 6.6) factor, Y, is calculated by: where Kdr X 1 = addendum modification coefficient, Y = (Eq 5.78) pinion

X 2 = addendum modification coefficient, 1 gear

AGMA 16 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

C = standard center distance criteria listed in 7.1 through 7.10 were used in cal- Rl = standard pitch radius, pinion culating the table values. R2= standard pitch radius, gear The following paragraphs and equations use dimensionless numbers to describe the gear ge- 6.3 Tooth Thinning for Backlash. It is usually ometry. To make actual measurements dimensionless, they are converted to ratios by multiply- impossible to determine the ratio A% l’As?l.2 that was used for existing gears. The following ing them by diametral pitch, dividing them by analysis is based on common practice where module, or comparing them to a v (3.1416) circu- As,, = As,,, in which case: lar pitch at the standard pitch diameter. Any consistent system of units can be used for this conver- 2 &tan 4k cx B, = f ( - Zxg) 0% 6.7) sion, see Section 3. C 7.1 Using the Tables. Each table of I and J values was generated for a specific tool form (ba- A.snl= As,, = 0% 6.8) sic rack) defined by whole depth factor, normal pressure (profile) angle and tool edge (tip) radius. where Each tool form was used to generate 66 tables of Bn = normal operating circular backlash values: G = operating center distance For spur gears: As,,= tooth thinning for backlash, pinion Loaded at Tip x1 = x2 = 0 As,,= tooth thinning for backlash, gear x1 = 0.25, x2 = -0.25 Xl = 0.50, xa = -0.50 6.4 Addendum Modification Coefficients. The Loaded at HPSTC addendum modification coefficients, xl and x2, x1 =x2 = 0 can be established from Eq 6.9 and 6.10. Xl = 0.25, x2 = -0.25 Xl = 0.50, x2 = -0.50

As7ll For helical gears: 0% 6.9) 10 degree standard helix angle xl = ‘gl + 2 tan+ n Xl =x2=0

Xl = 0.25, x2 = -0.25 Asrl2 x2 = Xg2 +_ 2tan+ (Eq 6.10) Xl = 0.50, x2 = -0.50 n 15 degree standard helix angle

Xl =x2=0 7. Geometry Factor Tables x1 = 0.25, x2 = -0.25 Xl = 0.50, xa = -0.50 The following tables provide the Geometry 20 degree standard helix angle Factor for Pitting Resistance, I, and the Geometry x1 =x2 = 0 Factor for Bending Strength, J, for a range of Xl = 0.25, x2 = -0.25 typical pairs of gears. The tables were prepared x1 = 0.50, xa = -0.50 by computer, programmed in accordance with Section 5. The values were rounded to two sig- 25 degree standard helix angle nificant figures. The tables cover various combi- Xl =x2=0 nations of helix angle, pressure angle, whole xl = 0.25, x2 = -0.25 depth, tool edge radius, tooth load point and ad- Xl = 0.50, xp = -0.50 dendum modification. The Tables do not cover all 30 degree standard helix angle

tooth forms, pressure angles, and pinion or gear Xl =x2=0

modifications, and are not applicable to all gear Xl = 0.25, x2 = -0.25

designs. In addition to the basic geometry, the Xl = 0.50, xa = -0.50

AGMA 17 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

To obtain values for I and J, enter the table nl+ n2 c = - (Eq 7.3) for the appropriate whole depth factor tool, helix 2cos Jr angle, loading condition and addendum modifica- where tion factor and select values for the numbers of C = standard center distance. pinion and gear teeth. If the exact values for your gearset are not listed, the calculation method of For this center distance the sum of the adden- Section 5 is recommended. Interpolation is not dum modification coefficients is zero (See 5.3 for recommended. definitions).

A “U” in the tables indicates a gear tooth x1+x2 = 0 0% 7.4) combination which should be avoided due to undercutting. where A “T” in the tables indicates a gear tooth = addendum modification coefficient, combination which should be avoided due to x1 pinion pointed pinion teeth. x2 = addendum modification coefficient, 7.2 Whole Depth. Whole depth is expressed in gear the Tables as a “whole depth factor”, the whole depth of a basic rack for 1 normal diametral pitch 7.6 Tooth Thickness Backlash Allowance. or 1 normal module. The actual generated depths The values in the tables are calculated based on a are slightly greater due to tooth thinning for back- backlash allowance. The circular tooth thick- lash. nesses for the pinion and the gear are each thinned by an amount, A sn, shown in Eq 7.5. 7.3 Outside Diameter. The tabulated values are calculated for gears having an outside diameter As,= 0.024 = 0.024 0% 7.5) (addendum diameter) equal to (in terms of mn pnd = 1.0): If the gears being evaluated have different D -CL + 2 (1 + x1) minimum tooth thicknesses, the Bending Strength al = cos Jr 0% 7.1) Geometry Factor, J, can be approximated using Eq 7.6. The Pitting Resistance Geometry Factor, Da2 = A+2 (1 +x2) 03 7.2) cos jr I, is unaffected by variations in tooth thickness. where 2 nl = pinion tooth number Jl snl 0% 7.6) n2 = gear tooth number -T-= (- sns > = standard helix angle, degrees * where Oal = pinion addendum diameter Jl = adjusted geometry factor 42 = gear addendum diameter J, = geometry factor from table

7.4 Type of Gearing. The tables apply to ‘nl = adjusted circular tooth thickness external gears only. An analytical method for Sns = standard tooth thickness, thinned determining the Bending Strength Geometry Fac- per Eq 7.5 tor, J, for internal gears is beyond the scope of this Standard. Example: From the table at the top of page 32 for 20” 7.5 Center Distance. The tables apply to pressure angle spur gears, loaded at the highest gearsets that operate on a standard center dis- point of single tooth contact, the J factor for a 21 tance. This center distance is the tight mesh tooth pinion operating with a 35 tooth gear is center distance for gears not yet thinned for found to be 0.34. The table is based on a circular backlash. See 7.6. tooth thickness of:

AGMA 18 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

Tr - - 0.024 = 3* _ 0.024 = 1.547 7.9 Cutter Geometry. The hob geometry used 2 in the calculation of I and .I is as follows: (from Sections 7 and 7.3) n For a 10 normal diametral pitch gear or pin- C = 10 000 ion, the equivalent circular tooth thickness would S = 1.5708 be: no xO = 0.0 z= 0.155 8O = 0.0 10

If a value for J for a 0.0 10 inch thinner pin- where n = tool tooth number ion, having a circular thickness of C 0.155 - 0.010 = 0.145 inch Sno = reference normal circular tooth is required, the approximate value is: thickness of tool 2 = addendum modification coefficient 0.34 (f+g = 0.30 %O > 0% 7.7) of tool

A 6.5% reduction in tooth thickness reduces J 8O = amount of protuberance by 12%.

7.7 Undercutting. The tables do not include geometry factors when an undercutting condition exists in either of the two gears. This condition . HOB TOOTH can be evaluated using Eq 7.8 and Fig 7-l where ---7- the generating rack shift coefficient, xg, must be Xgmint equal to or greater than the expression in Eq 7.8.

h - paO(l-sin+n)- 4 sina C$ $min = a0 n (Eq 7.8) where ha0 = nominal tool addendum = tool tip radius Pa0 n = pinion or gear tooth number

7.8 Top Land. The tables do not include geometry factors when either the pinion or gear tooth top land is less than the value expressed in Eq 7.9. Fig 7-l Undercutting Criteria 0.3 ‘namin 2 p 0% 7.9) nd 7.10 Axial Contact Ratio. The I and J factors s ? 0.3m, na min (Eq 7-W for helical gears are calculated using an axial con- where tact ratio, mF9 equal to 2.0. When the axial contact ratio is other than 2.0, the resulting values for Sna = tooth thickness at outside diameter, in (mm) I and J may be reduced by as much as 10%.

AGMA 19 908-B89 Geometry Factors for Determining the Pitting Resistance and Bending Strength of Spur and Helical Gear Teeth

I AND J FACTORS FOR:! 14.5 DEG. PRESSURE ANGLE 2.157 WHOLE DEPTH FACTOR 0.0 DEG. HELIX ANGLE 0.024 TOOTH THINNING FOR BACKLASH 0.157 TOOL EDGE RADIUS LOADED AT TIP EQUAL ADDENDUM (x12= x =0)