AGMA 929-A06 Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius

AGMA 929- A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 929- A06

Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius

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

Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius American AGMA 929--A06 Gear Manufacturers CAUTION NOTICE: AGMA technical publications are subject to constant improvement, revision or withdrawal as dictated by experience. Any person who refers to any AGMA Association 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 referenced. Citations should read: See AGMA 929--A06, Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius, published by the American Gear Manufacturers Association, 500 Montgomery Street, Suite 350, Alexandria, Virginia 22314, http://www.agma.org.] Approved August 22, 2006

ABSTRACT This information sheet supplements ANSI/AGMA 2005--D03 with calculations for bevel gear top land and guidance for selection of cutter edge radius for determination of tooth geometry. It integrates various publications with modifications to include face hobbing. It adds top land calculations for non--generated manufacturing methods. It is intended to provide assistance in completing the calculations requiring determination of top lands and cutter edge radii for gear capacity in accordance with ANSI/AGMA 2003--B97. Published by

American Gear Manufacturers Association 500 Montgomery Street, Suite 350, Alexandria, Virginia 22314 Copyright © 2006 by American Gear Manufacturers Association 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--873--4

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AGMA 929--A06

Contents Page

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Symbols, terminology and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv 1 1 5 8

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Annexes A B C D

Additional equations from ANSI/AGMA 2005--D03 . . . . . . . . . . . . . . . . . . . . . . Stock allowance and standard cutter specifications . . . . . . . . . . . . . . . . . . . . . Spiral bevel example problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypoid example problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18 23 24 31

Tables 1 2 3 4 5

Symbols and terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symbols and terms from ANSI/AGMA 2005--D03, table 9 . . . . . . . . . . . . . . . . . Gear rotation factor, kE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suggested defaults for input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 6 7 7 7

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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 929--A06, Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius.] The Bevel Gearing Committee recognized the need for additional equations to aid in the design of bevel gears. The equations for geometry factors found in the annex of ANSI/AGMA 2003--B97 require detailed information on the proposed cutting tool before a proper calculation can be performed. In addition, the minimum top land thickness is required to aid in determining the maximum case depth allowed on carburized bevel gears. The equations required for these values were not published in AGMA documentation, but could be found, for some cases, in the publications listed in the bibliography of this information sheet. AGMA 929--A06 expands on those equations to include gears manufactured with the face hobbing cutting method. In the case of non--generated gears, the equations in this document may yield different values for pinion top land thicknesses and gear tooth depth at the toe and heel than obtained on some well known commercial software. The pinion top land thickness is reduced by curvature added to the pinion, a natural consequence of the non--generated gear member having no profile curvature on the teeth. For the gear member, the non--generating process cuts a rootline tangent to the gear root cone, a rootline which does not wrap around the root cone as in the generated case. This leaves the toe and heel ends of the tooth slots shallow compared to the generated gear case, and the gear tooth space at the ends of the teeth narrower. The non--generated gear is the imaginary generating gear for the pinion. So the pinion teeth, which fit in the non--generated gear tooth slots, are thinner at the ends than their generated gear counterparts. The cutter edge radii calculated in this document are based on the geometrical conditions present and include a manufacturing gauging flat. Individual blade manufacturers have standard blade edge radii and manufacturing tolerances for their products which should be considered when sourcing non--standard radii. It is recommended to work closely with the blade supplier to ensure design specifications and sourced product specifications are consistent. The first draft of AGMA 929--A06 was made in February, 1999. It was approved by the AGMA Technical Division Executive Committee in August, 2006. Suggestions for improvement of this document will be welcome. They should be sent to the American Gear Manufacturers Association, 500 Montgomery Street, Suite 350, Alexandria, Virginia 22314.

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AGMA 929--A06

PERSONNEL of the AGMA Bevel Gear Committee Chairman: Robert F. Wasilewski . . . . . . . . . . . . . . . . . . . . . . Arrow Gear Company Vice Chairman: George Lian . . . . . . . . . . . . . . . . . . . . . . . . . Amarillo Gear Company

ACTIVE MEMBERS T. Guertin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Kolonko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T.J. Krenzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P.A. McNamara . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Miller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Tsung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

© AGMA 2006 ---- All rights reserved

Liebherr Gear Technology Company Rexnord Geared Products Gleason Corporation Caterpillar, Inc. Dana Spicer Off Highway Products Dana Corporation

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American Gear Manufacturers Association --

Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius

AGMA 929--A06

non--generated manufacturing methods, to achieve compatibility between publications. It is intended to provide assistance in completing the calculations requiring determination of top lands and cutter edge radii in ANSI/AGMA 2003--B97, Rating the Pitting Resistance and Bending Strength of Generated Straight Bevel, Zerol Bevel and Spiral Bevel Gear Teeth. Annexes are provided for additional related information and calculation examples.

2 Symbols, terminology and definitions 2.1 Symbols and terminology

1 Scope This information sheet provides a set of equations for the calculation of bevel gear top land and guidance on cutter edge radius. It integrates the equations in ANSI/AGMA 2005--D03, Design Manual for Bevel Gears, and Gleason publication SD3124B, Formulas for Cutter Specifications and Tooth Thickness Measurements for Spiral Bevel and Hypoid Gears, with modifications to include face hobbing, and additions for the top land calculations for

The equations in this information sheet are written in terms generally used for hypoids. See table 1. For other gears, the nomenclature from ANSI/AGMA 2005--D03, table 9 are used (see table 3). NOTE: Some of the symbols and terminology contained in this document may differ from those used in other documents and AGMA standards. Users of this standard should assure themselves that they are using the symbols, terminology and definitions in the manner indicated herein.

Table 1 -- Symbols and terms Symbol AiG, AiP AmG, AmP AoG, AoP AxG aG, aP aiG, aiP a′iG, a′iP aoG, aoP B bG, bP biG, biP boG, boP bxG, bxP b′oG c

Term Inner cone distance, gear or pinion Mean cone distance, gear or pinion Outer cone distance, gear or pinion Cone distance for involute lengthwise curvature point where normal circular pitch and slot width is a maximum Mean addendum, gear or pinion Inner addendum, gear or pinion Adjusted inner addendum, gear or pinion Outer addendum, gear or pinion Outer normal backlash allowance Mean dedendum, gear or pinion Inner dedendum, gear or pinion Outer dedendum, gear or pinion Dedendum at cone distance AxG, gear or pinion Theoretical outer gear dedendum Clearance

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Units inch inch inch inch inch inch inch inch inch inch inch inch inch inch inch

Where first used Eq 7, Eq 1 Eq 9, Eq 1 Eq 7, Eq 3 Eq 31 Eq 50, Eq 50 Eq 78, Eq 77 Eq 82, Eq 81 Eq 163, Eq 162 Eq 6 Eq 19, Eq 22 Eq 25, Eq 24 Eq 20, Eq 22 Eq 44, Eq 45 Eq 19 Eq 77 (continued)

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Table 1 (continued) Symbol D, d FG, FP FiP FoP FxG kE N, n pm pn Q q qi qo qx RbNG1, RbNP2 RbNG2, RbNP1 RibVG, RibVP RibXG, RibXP RiG, RiP RioG, RioP RNG, RNP RoNG, RoNP R′bNG1, R′bNP2 R′bNG2, R′bNP1 R′iG, R′iP R′NG, R′NP R′oNG, R′oNP R′′bNG1, R′′bNP2 R′′bNG2, R′′bNP1 R′′NG, R′′NP R′′oNG, R′′oNP rc rTG, rTP r1G, r1P r1VG, r1VP r1XG, r1XP r2G, r2P r2RG, r2RP

Term

Units

Pitch diameter, gear or pinion Face width of gear or pinion Hypoid pinion face width from calculation point to inside Hypoid pinion face width from calculation point to outside Distance from mean cone to cone distance at involute curvature Gear rotation factor Number of teeth, gear or pinion Mean circular pitch Mean normal circular pitch Intermediate factor Generating angle at mean Generating angle at inside Generating angle at outside Generating angle at involute curvature Mean normal base radius, convex, gear or pinion Mean normal base radius, concave, gear or pinion Inner base radius -- concave, gear or pinion Inner base radius -- convex, gear or pinion Original inner pitch radius, gear or pinion Inner outside radius, gear or pinion Mean normal pitch radius, gear or pinion Mean normal outside radius, gear or pinion Outer normal base radius, convex, gear or pinion

inch inch inch inch inch

Where first used Eq 140, Eq 139 Eq 25, Eq 2 Eq 1 Eq 3 Eq 38

-- --- -inch inch inch degrees degrees degrees degrees inch inch inch inch inch inch inch inch inch

Eq 52 Eq 79, Eq 79 Eq 27 Eq 27 Eq 35 Eq 9 Eq 13 Eq 11 Eq 39 Eq 143, Eq 142 Eq 144, Eq 141 Eq 90, Eq 87 Eq 89, Eq 88 Eq 76, Eq 75 Eq 86, Eq 85 Eq 140, Eq 139 Eq 146, Eq 145 Eq 160, Eq 159

Outer normal base radius, concave, gear or pinion

inch

Eq 161, Eq 158

New inner pitch radius, gear or pinion Outer normal pitch radius, gear or pinion Outer normal outside radius, gear or pinion Inner normal base radius, convex, gear or pinion

inch inch inch inch

Eq 84, Eq 83 Eq 157, Eq 156 Eq 163, Eq 162 Eq 179, Eq 178

Inner normal base radius, concave, gear or pinion

inch

Eq 180, Eq 177

Inner mean normal pitch radius, gear or pinion Inner normal outside radius, gear or pinion Cutter radius Maximum blade edge radius, gear or pinion Maximum blade edge radius for no running interference, gear or pinion Maximum blade edge radius for no running interference concave, gear or pinion Maximum blade edge radius for no running interference convex, gear or pinion Maximum blade edge radius that can be manufactured, gear or pinion Roughing cutter edge radius, gear or pinion

inch inch inch inch inch

Eq 176, Eq 175 Eq 182, Eq 181 Eq 9 Eq 74, Eq 73 Eq 74, Eq 73

inch

Eq 106, Eq 103

inch

Eq 105, Eq 104

inch

Eq 74, Eq 73

inch

Eq 113, Eq 114 (continued)

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AGMA 929--A06

Table 1 (continued) Symbol r3G, r3P r3VG, r3VP r3XG, r3XP r′2RG, r′2RP r′2VG, r′2VP r′2XG, r′2XP r′′2VG, r′′2VP r′′2XG, r′′2XP SAG, SAP S1 Tmn, tmn Tn tiNG, tiNP tLiNG, tLiNP tLNG, tLNP tLoNG, tLoNP tmP toNG, toNP W WBG, WBP WBRG, WBRP W′e WiG, WiP WLG, WLP WMG, WMP WmG, WmP WoG, WoP WRG, WRP WxG, WxP xiVG, xiVP xiXG, xiXP y Γ, γ ΓR ΔaG, ΔaP

Term Maximum blade edge radius to avoid mutilation, gear or pinion Maximum blade edge radius to avoid mutilation concave, gear or pinion Maximum blade edge radius to avoid mutilation convex, gear or pinion Maximum roughing blade edge radius which can be manufactured, gear or pinion Maximum finishing blade edge radius concave, defined by maximum roughing blade edge radius and minimum stock allowance, gear or pinion Maximum finishing blade edge radius convex, defined by maximum roughing blade edge radius and minimum stock allowance, gear or pinion Maximum finishing blade edge radius concave, gear or pinion Maximum finishing blade edge radius convex, gear or pinion Stock allowance, gear or pinion Crown gear to cutter center distance Mean normal circular thickness at pitch line, gear or pinion Gear mean normal circular thickness without backlash Inner normal circular thickness at pitch line, gear or pinion Inner normal top land, gear or pinion Mean normal top land, gear or pinion Outer normal top land, gear or pinion Mean pinion transverse circular thickness Outer normal circular thickness at pitch line, gear or pinion Finishing point width gear (Unitool and single sided) Finishing blade point, gear or pinion Roughing blade point, gear or pinion Effective gear point width Inner slot width, gear or pinion Minimum slot width, gear or pinion Maximum slot width, gear or pinion Mean slot width, gear or pinion Outer slot width, gear or pinion Roughing point width, gear or pinion Slot width at AxG (maximum gear or pinion slot) Limit tooth height for interference concave, gear or pinion Limit tooth height for interference convex, gear or pinion Amount of fillet mutilation permitted Pitch angle, gear or pinion Gear root angle Change in inner addendum, gear or pinion

Units inch

Where first used Eq 74, Eq 73

inch

Eq 136, Eq 133

inch

Eq 135, Eq 134

inch

Eq 111, Eq 112

inch

Eq 118, Eq 115

inch

Eq 117, Eq 116

inch

Eq 122, Eq 119

inch

Eq 121, Eq 120

inch inch inch

Eq 65, Eq 66 Eq 34 Eq 47, Eq 48

inch inch inch inch inch inch inch

Eq 47 Eq 189, Eq 188 Eq 192, Eq 190 Eq 154, Eq 152 Eq 173, Eq 171 Eq 49 Eq 170, Eq 169

inch inch inch inch inch inch inch inch inch inch inch inch inch inch degrees degrees inch

Table 2 Eq 69, Eq 70 Eq 67, Eq 68 Eq 52 Eq 59, Eq 60 Eq 61, Eq 62 Eq 63, Eq 64 Eq 50, Eq 51 Eq 55, Eq 56 Eq 65, Eq 66 Eq 57, Eq 58 Eq 102, Eq 97 Eq 101, Eq 99 Eq 129 Eq 76, Eq 75 Eq 17 Eq 80, Eq 79 (continued)

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Table 1 (continued) Symbol ΔbiG ΔboG ΔbxG ΔcG, ΔcP Δf Δqi Δqo Δqx ΔW′G, ΔW′P Δφ2 δG, δP ηi ηo ηx η1 ρiVG, ρiVP ρiXG, ρiXP ξ Σb Σbi Σbo Σbx Σφ φBXG φiV φiX φ1 φ2 φ1TG, φ2TP φ2TG, φ1TP φ′1TG, φ′2TP φ′2TG, φ′1TP φ′′1TG, φ′′2TP φ′′2TG, φ′′1TP ψG, ψP ψiG, ψiP ψoG, ψoP ψxG ΩVG, ΩVP ΩXG, ΩXP

4

Term

Units

Depth reduction on non--generated gear at inside Depth reduction on non--generated gear at outside Depth reduction on non--generated gear at involute curvature Change in clearance, gear or pinion Width of blade flat Increment in generating angle at inside Increment in generating angle at outside Increment in generating angle at involute curvature Difference between minimum slot width and blade point, gear or pinion Normal tilt of finishing cutter non--generated Dedendum angle of gear or pinion Generating angle at inside Generating angle at outside Generating angle at involute curvature, for face hobbing Second auxiliary angle Inner profile radius of curvature concave, gear or pinion Inner profile radius of curvature convex, gear or pinion Angle between gear root plane and plane in which taper is specified Sum of pinion and gear mean dedendums Sum of pinion and gear inner dedendums Sum of pinion and gear outer dedendums Sum of pinion and gear dedendums at cone distance AxG Included pressure angle Inside blade angle gear cutter (non--generating) Inner pinion pressure angle -- concave Inner pinion pressure angle -- convex Normal pressure angle, concave, pinion Normal pressure angle, convex, pinion Pressure angle at tip of tooth, convex gear or pinion Pressure angle at tip of tooth, concave gear or pinion Pressure angle at tip of tooth, outer convex gear or pinion Pressure angle at tip of tooth, outer concave gear or pinion Pressure angle at tip of tooth, inner convex gear or pinion Pressure angle at tip of tooth, inner concave gear or pinion Mean spiral angle, gear or pinion Inner spiral angle, gear or pinion Outer spiral angle, gear or pinion Spiral angle at cone distance AxG Intermediate blade mutilation value concave, gear or pinion Intermediate blade mutilation value convex, gear or pinion

inch inch inch

Where first used Eq 18 Eq 17 Eq 42

inch inch degrees degrees degrees inch

Eq 7, Eq 6 Eq 111 Eq 16 Eq 15 Eq 41 Eq 128, Eq 127

degrees degrees degrees degrees degrees degrees inch inch degrees

Eq 8 Eq 19, Eq 22 Eq 14 Eq 12 Eq 34 Eq 10 Eq 96, Eq 93 Eq 95, Eq 94 Eq 53

inch inch inch inch degrees degrees degrees degrees degrees degrees degrees degrees degrees degrees

Eq 28 Eq 30 Eq 29 Eq 46 Eq 5 Eq 8 Eq 91 Eq 92 Eq 5 Eq 5 Eq 150, Eq 149 Eq 151, Eq 148 Eq 167, Eq 166 Eq 168, Eq 165

degrees degrees

Eq 186, Eq 185 Eq 187, Eq 184

degrees degrees degrees degrees inch2

Eq 9, Eq 139 Eq 7, Eq 6 Eq 7, Eq 6 Eq 32 Eq 132, Eq 129

inch2

Eq 131, Eq 130

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2.2 Definitions

2.2.2 Bevel gear cutting method terminology

This clause provides supplemental definitions for bevel gear cutter and cutting method terminologies referred to in this information sheet. The list of definitions given here is not intended to be all inclusive. For more detailed information regarding a particular cutter design or cutting method, the user is encouraged to consult the cutter manufacturer’s data or machine tool manufacturer.

completing. A machining process where the tooth space is completed in one machining setup. Unlike single side, single setting, or fixed setting methods where the tooth space roughing and finishing are carried out in separate machining setups.

2.2.1 Bevel gear cutter terminology

Formate. A term describing non--generated gear members whose teeth have surfaces of revolution, and straight profiles in their normal sections. Pinions generated to run with such gears are called Formate pinions.

alternate blade. Any multiple blade face cutter having successive blades that cut on opposite sides of a tooth space. blade flat. A flat land on the end of blades required for blade manufacturing control. blade point. The length across the end surface of a blade or tool (measured along a radius of a circular cutter), bounded by the extension of the cutting edge and non--cutting edge of the blade. blade, inside. A blade of a circular face cutter with a cutting edge that produces the convex side of a tooth surface. blade, outside. A blade of a circular face cutter with a cutting edge that produces the concave tooth surface. coast side. The side of a tooth flank that is in contact with the opposite flank of the mate when the gear set is driven in the reverse direction. drive side. The side of a tooth flank that is in contact with the opposite flank of the mate when the gear set is driven in the forward direction. effective gear point width. Effective point width is one half of the difference of the outside minus the inside point diameters of a cutter. The effective point width is not equal to the slot width produced on the part when indexing motion occurs between drive and coast side generation. For spread blade cutters, the effective point width equals the actual cutter point width. When indexing accounts for part of the slot width, the effective point width can be negative. fillet mutilation. Fouling of tooth fillet by the non--cutting side of the blade tip. This is caused by a tool edge radius or a blade point which is too large. point width. One half of the difference between the inside and outside point diameters of an alternate blade cutter.

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cutter tilt. A change in the relative position between workpiece and cutter, measured as an angle in the normal, or axial, or both planes of the workpiece.

non--generated method. A gear cutting process where the tooth space is machined without generating motion (see Formate). The tooth surfaces are formed by the sweep of a straight--sided cutting tool and thus have straight profiles in any normal section. single setting. A finishing method which is a variation of the spread blade method. It is used on wide face width gears to avoid having two cutter blades cutting in the same slot at the same time. single side. A cutting method which uses an alternate blade cutter to separately cut the profiles on each side of a tooth space, with independent machine settings. spread blade. A cutting method which uses a circular face cutter with alternate inside and outside blades to cut both sides of a tooth space at the same time. Unitool. A method for producing pairs of spiral bevel, zerol bevel or hypoid gears using the same single face mill cutter for both members. The cutters used with this method are designated as Unitool cutters. Versacut. A process which requires very few cutters to accommodate a wide variety of spiral bevel gears. The cutters used with this method are designated as Versacut cutters.

3 Input data The equations in this information sheet are intended to be as general as possible. Many of the calculations require knowledge of the specific cutting method used in manufacture. Supplemental

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AGMA 929--A06

definitions to be used with the terms described in ANSI/AGMA 2005--D03 for proper input data selection based on the manufacturing method used can be found in 2.2. The tables in this clause describe the data necessary to complete the calculations of this information sheet. In most cases the values can be obtained from the calculations detailed in ANSI/AGMA 2005--D03. Many of those calculations are duplicated in annex A. Additional tables are provided here to supplement those calculations. 3.1 Input variables Table 2 contains all the input variables necessary for the calculations used in this information sheet.

Table 2 -- Input variables Symbol AiG AmG AmP AoG aG aP aoG aoP B bG bP c D d FG FP N n pm rc SAG SAP Tn W y Symbol

6

Term Inner gear cone distance Gear mean cone distance Pinion mean cone distance Gear outer cone distance Gear mean addendum Pinion mean addendum Outer gear addendum Outer pinion addendum Outer normal backlash allowance Gear mean dedendum Pinion mean dedendum Clearance Gear pitch diameter Pinion pitch diameter Gear face width Pinion face width Gear number of teeth Pinion number of teeth Mean circular pitch Cutter radius Stock allowance, gear Stock allowance, pinion Gear mean normal circular thickness without backlash Gear finishing point width (Unitool and single setting) Amount of fillet mutilation permitted Term

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Gear pitch angle Gear root angle Pinion pitch angle Width of blade flat Dedendum angle of gear Dedendum angle of pinion Inside blade angle, gear cutter (non--generating) Concave pinion normal pressure φ1 angle Convex pinion normal pressure φ2 angle (always negative) Gear mean spiral angle ψG Pinion mean spiral angle ψP Inner gear spiral angle ψiG Inner pinion spiral angle ψiP Outer gear spiral angle ψoG Outer pinion spiral angle ψoP Additional input variables for face hobbing Q Intermediate factor S1 Crown gear to cutter center distance Generating angle at inside ηi Generating angle at outside ηo Second auxiliary angle η1 Additional input variables for hypoids FiP Hypoid pinion face width from calculation point to inside FoP Hypoid pinion face width from calculation point to outside Γ ΓR γ Δf δG δP φBXG

3.2 Variable substitutions The calculations in this information sheet are written in terms of hypoid gears. Many of the variables for non--hypoid bevels are subscripted the same for pinion and gear. Table 3 provides a means for proper substitution of these variables into the calculations for hypoids. 3.3 Gear rotation factor Table 4 provides for the proper selection of the input variable kE based on the cutting method to be used in manufacture. 3.4 Additional input data Table 5 contains some suggested default values for variables not described in ANSI/AGMA 2005--D03. Other values may be used if experience dictates.

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Table 3 -- Symbols and terms from ANSI/AGMA 2005--D03, table 9

Symbol AmG AmP AoG AoP FG FP φ1 φ2 ψG ψoG ψP

Equivalent symbol from ANSI/AGMA 2005--D03 Am Am Ao Ao F F φ φ ψ ψo ψ

Term Gear mean cone distance Pinion mean cone distance Gear outer cone distance Pinion outer cone distance Gear face width Pinion face width Pressure angle -- drive side Pressure angle -- coast side, set φ2 to --φ1 Gear mean spiral angle Outer gear spiral angle Pinion mean spiral angle

Table 4 -- Gear rotation factor, kE Generating method Spread blade Gleason and Klingelnberg Oerlikon not using a roughing blade Oerlikon using a roughing blade Straight bevels and planing generator

Face hobbing

Face milling X

kE 0.0 1.0 1.0 1.3

X X X

2 W mG pn

Unitool and single setting

X

2 W mG − W  pn

Versacut and standard single side

X

2W mG + b Gtan φ 1 − tan φ 2 pn

Table 5 -- Suggested defaults for input data

φBXG

Term Inside blade angle of gear cutter (non--generated gear only)

SAP

Stock allowance, pinion

inch

SAG

Stock allowance, gear

inch

y Δf

Amount of fillet mutilation permitted Width of blade flat Cutter radius (straight bevels) Cutter radius (all other bevels)

inch inch

Symbol

rc

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Units degrees

inch

Suggested default Average pressure angle For completing, use 0.000 For rough and finish, use actual or see annex B For completing, use 0.000 For rough and finish, use actual or see annex B 0.002 0.008 10000 See ANSI/AGMA 2005--D03

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4 Calculations

Face hobbing qo = ηo Generating angle at inside

4.1 General geometry

Face milling

Inner pinion cone distance Hypoids

q i = arctan (1)

A iP = A mP − F iP For all other bevels

(2)

A iP = A mP − 0.5 F P Outer pinion cone distance



r c cos ψ iG A iG − r c sin ψ iG



(13)

Face hobbing qi = ηi

(14)

Increment in generating angle Outside

Hypoids

(15)

Δq o = q − q o

A oP = A mP + F oP

(3)

Inside Δq i = q i − q

For all other bevels A oP = A mP + 0.5 F P

(4)

Included pressure angle Σφ = φ 1 − φ 2

(5)

Change in clearance

Outside

B A iP cos ψ iP 1 Δc P = A oP cos ψ oP sin φ 1 − sin φ 2

(6)

B A iG cos ψ iG 1 Δc G = A oG cos ψ oG sin φ 1 − sin φ 2

2

tan Δφ 2

2 r c cos 2 ψ G

(7)

For non--generated gears normal tilt of finishing cutter (8)

Δb iG =

A mG − A iG

2

tan Δφ 2

2 r c cos2 ψ G

+

A iG sin 2 Δq i 2 tan Γ R (18)

Theoretical outer gear dedendum b′ oG = b G + A oG − A mG tan δ G

(19)

Outer gear dedendum b oG = b′ oG − Δb oG

Face milling



(9)

b oG = b′ oG

(21)

Outer pinion dedendum

Face hobbing q = η1 + ψG

(10)

Generating angle at outside

Hypoid b oP = b P + F oP tan δ P

(22)

All other bevels

Face milling r c cos ψ oG A oG − r c sin ψ oG

(20)

Generated gear

r c cos ψ G q = arctan A mG − r c sin ψ G



A oG sin 2 Δq o 2 tan Γ R

Non--generated gear

Generating angle at mean



+

Inside

If ΔcP is greater than 0.75 c, or if ΔcG is greater than 0.75 c, reduce B or increase the clearance c.

Δφ 2 = φ BXG − φ 1

Δb oG =

AoG − AmG

(17)

Gear

q o = arctan

(16)

Generated gears have slightly deeper teeth at the toe and heel ends than non--generated gears caused by the relative rolling action between cutter and work piece. For non--generated gear depth reduction:

Pinion

8

(12)



b oP = b P + 0.5 F P tan δ P (11)

(23)

Inner pinion dedendum b iP = b oP − F P tan δ P

(24)

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Inner gear dedendum

For the third and subsequent trials iterate.

Non--generated gear b iG = b′ oG − F G tan δ G − Δb iG

(25)

Generated gear b iG = b oG − F G tan δ G

(26)

Mean normal circular pitch p n = p m cos ψ G

(27)

Sum of pinion and gear outer dedendums (29)

Sum of pinion and gear inner dedendums Σb i = b iP + b iG

Cone distance for involute lengthwise curvature point where normal circular pitch and slot width is a maximum for face milling (31)

Spiral angle at cone distance AxG (face milling)

 

rc ψ xG = arcsin A xG

(32)

For face hobbing iterate for cone distance and spiral angle at involute lengthwise curvature point, AxG,where normal circular pitch is a maximum. Initial cone distance at involute lengthwise curvature point (33)

A xG = A mG

Generating angle at involute curvature for face hobbing η x = arccos

A 2 + S 21 − r 2c xG

2 A xG S 1

(34)

Spiral angle at cone distance AxG for face hobbing ψ xG = arctan

A xG − Q cos η x Q sin η x

(35)

Change AxG, until

 ηx − ψ

xG

 ≤ 0.0001

(36)

For the second trial make A xG = A xG + 0.0001 inch

© AGMA 2006 ---- All rights reserved

F xG = A xG − A mG

(38)

Face milling



r c cos ψ xG A xG − r c sin ψ xG



(39)

Face hobbing qx = ηx

(40)

Increment in generating angle at involute curvature (41)

Δq x = q − q x (30)

A xG = A 2mG − 2 A mG r c sin ψ G + 2 r 2c

Distance from mean cone to cone distance at involute curvature

q x = arctan (28)

Σb o = b oP + b oG

If AiG < AxG < AoG, calculate FxG, qx, Δqx, ΔbxG, bxP and Σbx; otherwise, set these items to zero.

Generating angle at involute curvature

Sum of pinion and gear mean dedendums Σb = b P + b G

AGMA 929--A06

Depth reduction on gear at involute curvature Non--generated gear Δb xG =

F 2 tan Δφ 2 xG

2 r c cos 2 ψ G

A xG sin 2 Δq x 2 tan Γ R

(42)

Generated gear Δb xG = 0

(43)

Gear dedendum at cone distance AxG (44)

b xG = b G + F xG tan δ G − Δb xG Pinion dedendum at cone distance AxG

(45)

b xP = b P + F xG tan δ P

Sum of pinion and gear dedendums at cone distance AxG (46)

Σb x = b xP + b xG 4.2 Slot width specifications

calculations

and

blade

4.2.1 Slot width calculations, mean The following calculations for the mean normal circular thickness for the gear and pinion are modifications to the calculations in table 9 of ANSI/AGMA 2005--D03. Backlash has been added and the backlash is divided evenly between the two members. If the mean normal circular tooth thicknesses are otherwise available, bypass equations 47 and 48. Mean normal gear circular thickness T mn = T n −

(37)

+

A mG A oG

B cos ψ G

Σφ2  cos ψ

2 cos

(47) oG

9

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Outer pinion slot width

Mean normal pinion circular thickness t mn = p n − T mn −

A mG A oG

B cos ψ G

Σφ2  cos ψ

cos

A cos ψ oG W oP = oG p − Σb otan φ 1 − tan φ 2 A mG cos ψ G n oG

(48)

Mean pinion transverse circular thickness t mP =

t mn cos ψ G

(49)

− W oG +

(56) Σφ 2 If AiG < AxG < AoG, calculate WxG and WxP, otherwise, set these items to zero cos

Mean gear slot width

Slot width at AxG, gear (maximum gear slot width)

W mG = p n − T mn − Σb tan φ 1 − tan φ 2 2 Σφ (50) − a P − a G tan 2

W xG = W′ e 1 −

 

Mean pinion slot width W mP = p n − Σb tan φ 1 − tan φ 2 +

A mG cos ψ G A oG cos ψ oG

B − W mG Σφ cos 2

 



kE pn 2 4.2.2 Slot width calculation, outer W′ e = W mG −

+ A mG − A xG

×

(52)

× tan φ 1 − tan φ 2 tan ξ

 

Σφ cos 2

A xG cos ψ xG A oG cos ψ oG

(57)

Slot width at AxG, pinion (maximum pinion slot width)

(53)

ξ = δG All other cutting methods. ξ=0

(54)

Outer gear slot width

A cos ψ xG W xP = xG p − Σb xtan φ 1 − tan φ 2 A mG cos ψ G n A cos ψ xG − W xG + xG A oG cos ψ oG





A iG cos ψ iG A + iG A mG cos ψ G A mG

× t mP cos ψ iG −

A iG cos ψ iG b A mG cos ψ G G

× tan φ 1 − tan φ 2 +

B Σφ cos 2

 

×

+ A mG − A oGtan φ 1 − tan φ 2 tan ξ (55)

(58)

Inner gear slot width W iG = W′ e 1 −

A cos ψ oG × t mP cos ψ oG − oG A mG cos ψ G

B Σφ cos 2

4.2.3 Slot width calculation, inner



A cos ψ oG A 1 − oG + oG A mG cos ψ G A mG

× b Gtan φ 1 − tan φ 2 +

B

(51)

Single side and Versacut methods



A xG cos ψ xG A mG cos ψ G

× b G tan φ 1 − tan φ 2 +

Angle between gear root plane and plane in which taper is specified:

W oG = W′ e



A xG cos ψ xG A + xG A mG cos ψ G A mG

× t mP cos ψ xG −

Effective gear point width

10

B

A iG cos ψ iG A oG cos ψ oG

B + A mG − A iG Σφ cos 2

 

× tan φ 1 − tan φ 2 tan ξ

(59)

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Pinion

Inner pinion slot width W iP =

A iG cos ψ iG p − Σb i tan φ 1 − tan φ 2 A mG cos ψ G n

− W iG +

A iG cos ψ iG A oG cos ψ oG

AGMA 929--A06

 

(60)

4.2.4 Slot widths, minimum and maximum

(70)

Gear W BG = W LG − S AG

(71)

Pinion

Minimum gear slot width W LG = minimum of W oG, W mG, or W iG

W LP + 0.001 2

Maximum for rough and finish methods

B

Σφ cos 2

W BP =

W BP = W LP − S AP (61)

(72)

4.2.6 Standard blade point

Minimum pinion slot width W LP = minimum of W oP, W mP, or W iP

(62)

Maximum gear slot width W MG = maximum of W oG, W xG, W iG, or W mG (63) Maximum pinion slot width W MP = maximum of W oP, W xP, W iP, or W mP (64) 4.2.5 Point width and blade point calculations Roughing cutter calculations for processes with separate rough and finish operations and cutters: Roughing point width Gear W RG = W LG − S AG

(65)

Pinion W RP = W LP − S AP

(66)

Minimum roughing blade point Gear W BRG =

W RG + 0.001 2

(67)

W RP + 0.001 2

(68)

Bevel gear blades are either sharpened on the rake face or on the blade profile. For face sharpened blades, the blade point and edge radius are formed by the blade manufacturer. For each distinct blade point, a separate cutter blade would be required. A common practice is to consolidate similar sized blade points into standard ones. A table of standard blade points is included in annex B for reference. When standard blade point is used, choose one that satisfies the blade point equations given in 4.2.5. Note that there may be more than one standard blade point that would satisfy the blade point equations. For profile sharpened blades, the required blade point and edge radii are formed at sharpening. Standardization of blade point is generally not an issue. The choice of blade point may affect maximum edge radius that can be manufactured, see 4.3.2, and maximum edge radius for avoiding mutilation, see 4.3.3. In general, a larger blade point allows a larger edge radius to be manufactured. However, the larger blade point may reduce the maximum edge radius for avoiding mutilation. It might be possible to select a blade point, standard or otherwise, in an attempt to balance the maximum edge radius that can be manufactured and the maximum edge radius for avoiding mutilation.

Pinion W BRP =

Finishing blade point Minimum for completing manufacturing methods Gear W BG =

W LG + 0.001 2

© AGMA 2006 ---- All rights reserved

(69)

Cutters for the Unitool method use standard blades exclusively. For each available cutter diameter, the corresponding cutter blades have standard blade point and edge radius. When a cutter diameter is chosen, the blade point and tool edge radii are also defined. Consult the manufacturer for available cutter information.

11

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4.3 Cutter edge radius calculation

Inner pinion outside radius

The cutter edge radius for a bevel gear should not exceed the radius that would cause any of the following conditions:

R ioP = R iP + a iP Inner gear outside radius R ioG = R iG + a iG

-- unable to manufacture radius on blade, see 4.3.2;

Concave

-- fillet mutilation, see 4.3.3.

Convex

Maximum pinion blade edge radius

R ibXP = R iP cos φ 2 (73)

(74)

4.3.1 Cutter edge radius for no running interference Virtual gear calculations

R iP =

cos2 ψ iP

(75)

R iG =

cos 2 ψ iG

(76)

Pinion inner addendum a iP = b iG − c

(78)

Change in inner pinion addendum n cos Γ Δa P = Δc G n cos Γ + N cos γ N cos γ n cos Γ + N cos γ

(80)

(81) (82)

New inner pinion pitch radius R′ iP = R iP − Δa P

12

  R ibVP R′ iP

(91)

 

(92)

φ iV = arccos

φ iX = arccos

R ibXP R′ iP

Inner pinion profile radius of curvature

à iVP = R′ iP sin φ iV −

R2ioG − R2ibXG + R′iG sin φiV (93)

Convex

R2ioG − R2ibVG + R′iG sin φiX (94)

Convex à iXG = R′ iG sin φ iV −

R2ioP − R2ibVP + R′iP sin φiV (95)

Concave à iVG = R′ iG sin φ iX −

R2ioP − R2ibXP + R′iP sin φiX (96)

Limit tooth height for interference, pinion, concave (83)

New inner gear pitch radius R′ iG = R iG − Δa G

Inner pinion pressure angle

Inner gear profile radius of curvature

Adjusted inner gear addendum a′ iG = a iG + Δa G

(90)

R ibVG = R iG cos φ 2

(79)

Adjusted inner pinion addendum a′ iP = a iP + Δa P

Concave

à iXP = R′ iP sin φ iX −

Change in inner gear addendum Δa G = Δc P

(89)

R ibXG = R iG cos φ 1

Concave (77)

Gear inner addendum a iG = b iP − c

Inner gear base radius

Convex

Original inner gear pitch radius A iG tan Γ

(88)

Concave

Original inner pinion pitch radius A iP tan γ

(87)

R ibVP = R iP cos φ 1

Convex

Maximum gear blade edge radius r TG = minimum of r 1G, r 2G, or r 3G

(86)

Inner pinion base radius

-- running interference, see 4.3.1;

r TP = minimum of r 1P, r 2P, or r 3P

(85)

Generated gear x iVP = R ibVP cos φ 1 + Ã iVP sin φ 1 − R iP + b iP

(84)

(97)

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Non--generated gear x iVP = c − Δc P

(98)

Limit tooth height for interference, pinion, convex Generated gear x iXP = R ibXP cos φ 2 − Ã iXP sin φ 2 − R iP + b iP (99) Non--generated gear x iXP = c − Δc P

(100)

Limit tooth height for interference, gear x iXG = R ibXG cos φ 1 + Ã iXG sin φ 1 − R iG + b iG (101) Concave

Maximum blade edge radius for no running interference on pinion tooth Concave x iVP 1 − sin φ 1

(103)

x iXP 1 + sin φ 2

(104)

Convex

Maximum blade edge radius for no running interference on gear tooth Convex x iXG r 1XG = 1 − sin φ 1 x iVG 1 + sin φ 2

(105)

(106)

Maximum blade edge radius for no running interference on pinion tooth if, r1VP < r1XP r 1P = r 1VP

(107) (108)

Maximum blade edge radius for no running interference on gear tooth if, r1VG < r1XG r 1G = r 1VG

© AGMA 2006 ---- All rights reserved

4.3.2.1 Roughing Maximum gear roughing blade edge radius which can be manufactured W BRG − Δf sec φ 1 − tan φ 1

r′ 2RG =

(111)

r′ 2RP =

W BRP − Δf sec φ 1 − tan φ 1

(112)

r 2RG = maximum of r′ 2RG or 0.010 inches (113) Pinion roughing cutter edge radius r 2RP = maximum of r′ 2RP or 0.010 inches (114) 4.3.2.2 Finishing The following four r′ finishing cutter calculations are each defined by the roughing blade edge radius and stock allowance for cutting processes that use separate rough and finish operations and cutters: Maximum pinion finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance

r′ 2VP =

S AP

2 sec φ 1 − tan φ 1

+ r 2RP

(115)

+ r 2RP

(116)

Convex r′ 2XP =

S AP

2 sec φ 2 + tan φ 2

Maximum gear finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance Convex

otherwise, r 1P = r 1XP

4.3.2 Cutter edge radius which can be manufactured

Concave

Concave r 1VG =

(110)

Gear roughing cutter edge radius

x iVG = R ibVG cos φ 2 − Ã iVG sin φ 2 − R iG + b iG (102)

r 1XP =

otherwise, r 1G = r 1XG

Maximum pinion roughing blade edge radius which can be manufactured

Convex

r 1VP =

AGMA 929--A06

(109)

r′ 2XG =

S AG

2 sec φ 1 − tan φ 1

+ r 2RG

(117)

+ r 2RG

(118)

Concave r′ 2VG =

S AG

2 sec φ 2 + tan φ 2

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

The following four r′′ calculations are as defined by the finishing cutter:

If any of the following Ω values are negative, reduce WBP or WBG accordingly:

The values of r′′ are never negative; if necessary reduce Δf:

Pinion intermediate blade mutilation value

Maximum pinion finishing blade edge radius Concave W BP − Δf r″ 2VP = sec φ 1 − tan φ 1

(119)

W BP − Δf sec φ 2 + tan φ 2

(120)

Convex Ω XG = y 2 + 2yΔW′ G sec φ 1 − tan φ 1 (131) Concave

Convex

Ω VG = y 2 + 2 yΔW′ Gsec φ 2 + tan φ 2 (132)

W BG − Δf sec φ 1 − tan φ 1

(121)

W BG − Δf sec φ 2 + tan φ 2

(122)

Concave r″ 2VG =

Ω XP = y 2 + 2y ΔW′ P sec φ 2 + tan φ 2 (130) Gear intermediate blade mutilation value

Maximum gear finishing blade edge radius

r″ 2XG =

Ω VP = y 2 + 2y ΔW′ P sec φ 1 − tan φ 1 (129) Convex

Convex r″ 2XP =

Concave

Maximum pinion blade edge radius to avoid mutilation Concave

Maximum pinion blade edge radius which can be manufactured

r 3VP =

Rough and finish

Convex

r 2P = minimum of r′ 2VP, r′ 2XP, r′′ 2VP, or r′′ XP (123)

r 3XP =

y + ΔW′ Psec φ 2 + tan φ 2 + Ω XP

sec φ 2 + tan φ 2

(124)

Maximum gear blade edge radius which can be manufactured Rough and finish

sec φ 1 − tan φ 1

y + ΔW′ Gsec φ 2 + tan φ 2 + Ω VG

sec φ 2 + tan φ 2

(125)

Concave

(126)

r 3VG =

Completing

Difference between pinion minimum slot width and finishing blade point ΔW′ P = W LP − W BP

(127)

Difference between gear minimum slot width and finishing blade point ΔW′ G = W LG − W BG

14

(128)

(134)

Convex r 3XG =

4.3.3 Cutter edge radius to avoid mutilation

2

Maximum gear blade edge radius to avoid mutilation

r 2G = minimum of r′ 2VG, r′ 2XG, r′′ 2VG, or r′′ XG

r 2G = minimum of r′′ 2VG, or r′′ 2XG

(133)

y + ΔW′ Psec φ 1 − tan φ 1 + Ω VP

Completing r 2P = minimum of r′′ 2VP, or r′′ 2XP

2

2

(135)

y + ΔW′ Gsec φ 1 − tan φ 1 + Ω XG

sec φ 1 − tan φ 1

2

(136)

Therefore, maximum pinion blade edge radius to avoid mutilation r 3P = minimum of r 3VP, or r 3XP

(137)

Therefore, maximum gear blade edge radius to avoid mutilation r 3G = minimum of r 3VG, or r 3XG

(138)

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 929--A06

4.4 Top land formulas

Concave

4.4.1 Mean top lands

φ 2TG = arccos



Mean normal pinion pitch radius A mP 0.5 d R NP = 2 cos γ cos ψ P A oP A mG 0.5 D 2 cos Γ cos ψ G A oG

(151)

Mean normal pinion top land (139)

Generated t LNP =

Mean normal gear pitch radius R NG =



R bNG2 R oNG

(140)

Rt

mn NP

+ invφ 1 − invφ 1TP + inv− φ 2



− invφ 2TP R oNP

(152)

NOTE: “inv” is the mathematical function involute. It is the tangent of the angle minus the angle in radians.

Mean normal pinion base radius

Non--generated

Concave R bNP1 = R NP cos φ 1

(141)

Convex (142)

R bNP2 = R NP cos φ 2 Mean normal gear base radius

Rt

mn NP

+ invφ 1 − invφ 1TP + inv− φ 2



− invφ 2TP R oNP − − tan φ 2  −

Convex (143)

R bNG1 = R NG cos φ 1

t LNP =

RT

mn NG

T

mn

+ a P tan φ 1

+ invφ 1 − invφ 1TG





(153)

t LNG = T mn − a Gtan φ 1 − tan φ 2

(154)

+ inv −φ 2 − invφ 2TG R oNG Mean normal gear top land

Concave (144)

R bNG2 = R NG cos φ 2 Mean normal pinion outside radius R oNP = R NP + a P

(145)

Mean normal gear outside radius

Non--generated Generated t LNG =

Generated R oNG = R NG + a G

(146)



T mn + invφ 1 − invφ 1TG + inv−φ 2 R NG

− inv φ 2TG

R

oNG

(155)

4.4.2 Outer top lands

Non--generated R oNG = R NG − a P

(147)

Pressure angle at tip of pinion tooth (mean)

R′ NP =

Concave





R φ 1TP = arccos bNP1 R oNP

Outer normal pinion pitch radius 0.5 d cos γ cos 2 ψ oP

(156)

Outer normal gear pitch radius (148)

R′ NG =

0.5 D cos Γ cos2 ψ oG

(157)

Outer normal pinion base radius

Convex

RR  bNP2

φ 2TP = arccos

oNP

Concave (149)

R′ bNP2 = R′ NP cos φ 2

Convex





R bNG1 R oNG

© AGMA 2006 ---- All rights reserved

(158)

Convex

Pressure angle at tip of gear tooth (mean)

φ 1TG = arccos

R′ bNP1 = R′ NP cos φ 1

(159)

Outer normal gear base radius (150)

Convex R′ bNG1 = R′ NG cos φ 1

(160)

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Non--generated

Concave R′ bNG2 = R′ NG cos φ 2

(161) t LoNP =

Outer normal pinion outside radius R′ oNP = R′ NP + a oP

(162)

Outer normal gear outside radius Generated R′ oNG = R′ NG + a oG R′ oNG = R′ NG − a oP

R′ bNP1 R′ oNP

(165)

t

−



  oNG + a oP tan φ 1 − tan φ 2

t oNG

R′ NG

+ invφ 1 − invφ′ 1TG + inv−φ 2

 R′  oNG

(172)

Generated t LoNG =

Convex φ′ 2TP = arccos

R′ bNP2 R′ oNP

(166)

Pressure angle at tip of gear tooth (outer)

R′ φ′ 1TG = arccos bNG1 R′ oNG

(168)

(173)

(174)

(175)

Inner mean normal gear pitch radius A iG 0.5 D cos Γ cos 2 ψ iG A oG

(176)

Inner normal pinion base radius Concave (177)

Convex (170)

R″ bNP2 = R″ NP cos φ 2

(178)

Inner normal gear base radius

Outer normal pinion top land

Convex R″ bNG1 = R″ NG cos φ 1

Generated

R′ NP

A iP 0.5 d 2 cos γ cos ψ iP A oP

R″ bNP1 = R″ NP cos φ 1

t oNG = b oP tan φ 1 − tan φ 2 + W oP

t oNP

R″ NP =

R″ NG =

Outer normal gear circular thickness at pitch line

 

oNG

Inner mean normal pinion pitch radius

(169)

B − Σφ cos 2

 R′

− invφ′ 2TG

4.4.3 Inner top lands

t oNP = b oG tan φ 1 − tan φ 2 + W oG

 

+ invφ 1 − invφ′ 1TG + inv−φ 2

(167)

Outer normal pinion circular thickness at pitch line

B Σφ cos 2

t oNG

R′ NG

t LoNG = t oNG − a oGtan φ 1 − tan φ 2

Concave R′ φ′ 2TG = arccos bNG2 R′ oNG



Non--generated

Convex

(179)

Concave + invφ 1 − invφ′ 1TP + inv−φ 2

− invφ′ 2TP

16

−

oNP

Outer normal gear top land

φ′ 1TP = arccos



 R′

− invφ′ 2TG

Concave

t LoNP =

+ invφ 1 − invφ′ 1TP + inv−φ 2

(164)

Pressure angle at tip of pinion tooth (outer)

−

t oNP

R′ NP

− invφ′ 2TP

(163)

Non--generated



 R′

R″ bNG2 = R″ NG cos φ 2

(180)

Inner normal pinion outside radius oNP

(171)

R″ oNP = R″ NP + a iP

(181)

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Inner normal gear thickness at pitch line

Inner normal gear outside radius Generated

t iNG = b iPtan φ 1 − tan φ 2 + W iP − (182)

R″ oNG = R″ NG + a iG Non--generated

(183)

R″ oNG = R″ NG − a iP Pressure angle at tip of pinion tooth (inner)



cos ψ iG cos ψ oG



B Σφ cos 2

 

(189)

Inner normal pinion top land

t LiNP =

R″ bNP1 R″ oNP

φ″ 1TP = arccos

×

A iG A oG

Generated

Concave (184)



t iNP + invφ 1 − invφ″ 1TP + inv−φ 2 R″ NP

− invφ″ 2TP

 R″

oNP

(190)

Non--generated

Convex





R″ bNP2 R″ oNP

φ″ 2TP = arccos

(185)

t LiNP =

Convex

−



φ″ 1TG = arccos



t iNP + invφ 1 − invφ″ 1TP + inv−φ 2 R″ NP

− invφ″ 2TP

Pressure angle at tip of gear tooth (inner)



R″ bNG1 R″ oNG

(186)

−

t



 R″

oNP

  iNG + a iP tan φ 1 − tan φ 2

t iNG

R″ NG

+ invφ 1 − invφ″ 1TG + inv−φ 2

 R″ 

− invφ″ 2TG

Concave



φ″ 2TG = arccos



R″ bNG2 R″ oNG

(187)

Inner normal pinion thickness at pitch line t iNP = b iGtan φ 1 − tan φ 2 + W iG − cos ψ iP × cos ψ oP

AGMA 929--A06

B Σφ cos 2

 

© AGMA 2006 ---- All rights reserved

A iP A oP (188)

oNG

(191)

Inner normal gear top land Generated t LiNG =



t iNG

R″ NG

+ invφ 1 − invφ″ 1TG + inv−φ 2

− invφ″ 2TG

 R″

oNG

(192)

Non--generated t LiNG = t iNG − a iG tan φ 1 − tan φ 2

(193)

17

AGMA 929--A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Annex A (informative) Additional equations from ANSI/AGMA 2005--D03 [This annex is provided for informational purposes only and should not be construed as a part of AGMA 929--A06,

Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius.]

A.1 Purpose This annex is to provide the user additional equations that are in or derived from those in ANSI/AGMA

2005--D03. The additional symbols used in the equations are defined in table A.1. Also see table 1.

Table A.1 -- Symbols Symbol E NS Nc RoG Z γi γoo γR ΔΣ εi εi’ εo εo’ η1 ζi ζo

Term Hypoid offset Number of blade groups Number of crown gear teeth Outside gear pitch radius Gear pitch apex beyond crossing point Pinion inside pitch angle Outer pinion pitch angle Pinion root angle Shaft angle departure from 90° Pinion offset angle in axial plane at inside Pinion offset angle in pitch plane at inner end Pinion offset angle in face plane Pinion offset angle in pitch plane at outer end Second auxiliary angle Intermediate angle Intermediate angle

Units in (mm) -- --- -in (mm) in (mm) deg (rad) deg (rad) deg (rad) deg (rad) deg (rad) deg (rad) deg (rad) deg (rad) deg (rad) deg (rad) deg (rad)

inner gear spiral angle (face hobbing)

A.2 Common equations inner gear cone distance A iG = A mG − 0.5 F G

(A.1)

inner gear spiral angle (face milling)



ψ iG = arcsin

2A mG r c sin ψ G − A 2 2A iG r c

mG

+ A2

iG



(A.2) intermediate variable (face hobbing) Q=

S1 N 1+ S Nc

(A.3)



18

A 2 + S 21 − r 2c iG 2A iG S 1



ψ iG = arctan





A iG − Q cos η i Q sin η i

(A.5)

generating angle at outside (face hobbing)



η o = arccos

A 2 + S 21 − r 2c oG

2 A oG S 1



(A.6)

outer gear spiral angle (face hobbing)

generating angle at inside (face hobbing) η i = arccos

First used Eq. A.12 Eq. A.3 Eq. A.3 Eq. A.17 Eq. A.12 Eq. A.14 Eq. A.20 Eq. A.23 Eq. A.12 Eq. A.13 Eq. A.15 Eq. A.19 Eq. A.21 Eq. A.8 Eq. A.12 Eq. A.18

ψ oG = arctan



A oG − Q cos η o Q sin η o



(A.7)

second auxiliary angle (face hobbing) (A.4)

cos η 1 =

A mG cos ψ G N c + N S S1 Nc

(A.8)

© AGMA 2006 ---- All rights reserved

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 929--A06

inner pinion spiral angle

A.3 Spiral bevels inner pinion spiral angle equals inner gear spiral angle ψ iP = ψ iG

(A.9)

outer pinion spiral angle equals outer gear spiral angle ψ oP = ψ oG

(A.10)

A.4 Hypoids inside gear pitch radius (A.11)

R iG = A iG sin Γ



ζ i = arctan E tan ΔΣ cos Γ A iG − Z cos Γ

(A.12)

(A.17)

R oG = A oG sin Γ intermediate angle





ζ o = arctan E tan ΔΣ cos Γ A oG − Z cos Γ

E cos ζ i sin Γ ε i + ζ i = arcsin A iG − Z cos Γ



pinion outer offset angle in axial plane





E cos ζ o sin Γ A oG − Z cos Γ

outer pinion pitch angle

(A.13)

pinion offset angle in pitch plane at outer end



sin ε ε′ o = arcsin cos γ o oo



γ i = arcsinsin ΔΣ sin Γ + cos ΔΣ cos Γ cos ε i (A.14)

outer pinion spiral angle

pinion offset angle in pitch plane at inner end

ψ oP = ψ oG + ε′ o

 

© AGMA 2006 ---- All rights reserved

(A.19)

(A.20)

pinion inside pitch angle

sin ε ε′ i = arcsin cos γi i

(A.18)

γ oo = arcsinsin ΔΣ sin Γ + cos ΔΣ cos Γ cos ε o

pinion inner offset angle in axial plane



outside gear pitch radius

ε o + ζ o = arcsin

intermediate angle



(A.16)

ψ iP = ψ iG + ε′ i

(A.15)

(A.21)

(A.22)

pinion dedendum angle δP = γ − γR

(A.23)

19

AGMA 929--A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Annex B (informative) Stock allowance and standard cutter specifications [This annex is provided for informational purposes only and should not be construed as a part of AGMA 929--A06,

Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius.]

B.1 Purpose This annex provides information on typical stock allowance and standard cutter specifications. B.2 Stock allowance For rough and finish cutting methods, if the information on the stock allowance is not available, the following default stock allowance can be used for calculating point width and blade point (see 4.2.5). Table B.1 -- Default stock allowances (inch)

Consult the manufacturer for accurate cutter information.

Table B.2 -- Standard blade specifications, face milled (inch) Point width range

Blade point

0.015 -- 0.015

0.010

0.020 -- 0.020

0.012

0.025 -- 0.025

0.015

Pinion

Gear

0.030 -- 0.035

0.020

Pd < 3.0

0.025

0.030

0.040 -- 0.045

0.025

3.0 ≤ Pd < 7.0

0.025

0.020

0.050 -- 0.055

0.030

7.0 ≤ Pd < 10.0

0.010

0.010

0.060 -- 0.070

0.040

10.0 ≤ Pd

0.000

0.000

0.080 -- 0.090

0.050

0.100 -- 0.120

0.065

0.130 -- 0.130

0.080

0.160 -- 0.160

0.100

0.170 -- 0.170

0.110

0.190 -- 0.190

0.125

0.210 -- 0.210

0.150

Diametral pitch

B.3 Standard blade point table The following table of standard blade points is used for some of the face milling cutting methods discussed in AGMA 929--A06. The table should not be construed to be all inclusive.

20

© AGMA 2006 ---- All rights reserved

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 929--A06

Annex C (informative) Spiral bevel example problem [This annex is provided for informational purposes only and should not be construed as a part of AGMA 929--A06,

Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius.]

Symbol

Description

Equation

Type of gears Face hobbing or face milling Generated or non--generated Completing or rough and finish, pinion Completing or rough and finish, gear

Variables Spiral Face milled Generated Completing Completing

Units

2.69971 3.19971 3.19971 3.69971 0.06259 0.19043 0.08122 0.24734 0.00500 0.22206 0.09422 0.03163 6.96429 2.50000 1.00000 1.00000 39.00000 14.00000 0.48518 table 5 table 5 table 5 0.14817 table 5 table 5 70.25316 63.75967 table 5 19.74684 6.49349 2.13424 20.00000 --20.00000 35.00000 35.00000 33.94561

inch inch inch inch inch inch inch inch inch inch inch inch inch inch inch inch -- --- -inch inch inch inch inch inch inch degrees degrees inch degrees degrees degrees degrees degrees degrees degrees degrees

Table 2 AiG AmG AmP AoG aG aP aoG aoP B bG bP c D d FG FP N n pm rc SAP SAG Tn W y Γ ΓR Δf γ δG δP φ1 φ2 ψG ψP ψiG

Inner gear cone distance Gear mean cone distance Pinion mean cone distance Gear outer cone distance Gear mean addendum Pinion mean addendum Outer gear addendum Outer pinion addendum Outer normal backlash allowance Gear mean dedendum Pinion mean dedendum Clearance Gear pitch diameter Pinion pitch diameter Gear face width Pinion face width Gear number of teeth Pinion number of teeth Mean circular pitch Cutter radius Pinion stock allowance Gear stock allowance Gear mean normal zero backlash circular thickness at pitch line Gear finishing point width Amount of fillet mutilation permitted Gear pitch angle Gear root angle Width of blade flat Pinion pitch angle Dedendum angle of gear Dedendum angle of pinion Concave pinion normal pressure angle Convex pinion normal pressure angle Gear mean spiral angle Pinion mean spiral angle Inner gear spiral angle

© AGMA 2006 ---- All rights reserved

21

AGMA 929--A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description Equation Inner pinion spiral angle ψiP Outer gear spiral angle ψoG Outer pinion spiral angle ψoP Additional input variables for face hobbing (Table 2) Q Intermediate factor S1 Crown gear to cutter center dist Second auxiliary angle η1 Generating angle at outside ηo Generating angle at inside ηi Additional input for hypoids FoP Hypoid pinion face width from calculation point to outside FiP Hypoid pinion face width from calculation point to inside Table 4 Method Cutting method kE Gear rotation factor Table 5 Inside blade angle gear cutter (non--generated) φBXG SAP Pinion stock allowance SAG Gear stock allowance y Amount of fillet mutilation permitted Width of blade flat Δf rc Cutter radius 4 Calculations 4.1 General geometry AiP Inner pinion cone distance (eq 1 or 2) AoP Outer pinion cone distance (eq 3 or 4) Included pressure angle (eq 5) Σφ Change in pinion clearance (eq 6) ΔcP Change in gear clearance (eq 7) ΔcG Normal tilt of finishing cutter (non--generated) (eq 8) Δφ2 q Generating angle at mean (eq 9 or 10) qo Generating angle at outside (eq 11 or 12) qi Generating angle at inside (eq 13 or 14) Increment in generating angle at outside (eq 15) Δqo Increment in generating angle at inside (eq 16) Δqi Depth reduction on non--generated gear at (eq 17) ΔboG outside Depth reduction on non--generated gear at (eq 18) ΔbiG inside b′oG Theoretical outer gear dedendum (eq 19) boG Outer gear dedendum (eq 20 or 21) boP Outer pinion dedendum (eq 22 or 23) biP Inner pinion dedendum (eq 24) biG Inner gear dedendum (eq 25 or 26) pn Mean normal circular pitch (eq 27) Sum of pinion and gear mean dedendums (eq 28) Σb Sum of pinion and gear outer dedendums (eq 29) Σbo Sum of pinion and gear inner dedendums (eq 30) Σbi

22

Variables 33.94561 36.84576 36.84576

Units degrees degrees degrees

Not applicable Not applicable Not applicable Not applicable Not applicable

inch degrees degrees degrees

Not applicable Not applicable

degrees degrees

Spread blade 0.00000 Not applicable 0.00000 0.00000 0.00200 0.00800 4.50000

degrees inch inch inch inch inch

2.69971 3.69971 40.00000 0.00553 0.00553 Not applicable 80.47339 74.46243 87.13405 6.01095 6.66066 Not applicable

inch inch degrees inch inch degrees degrees degrees degrees degrees degrees inch

Not applicable

inch

0.27897 0.27897 0.11285 0.07559 0.16515 0.39744 0.31628 0.39182 0.24074

inch inch inch inch inch inch inch inch inch

© AGMA 2006 ---- All rights reserved

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description AxG Cone distance for involute lengthwise curvature point where normal circular pitch and slot width is a maximum Spiral angle at cone distance AxG ψxG Generating angle at involute curvature ηx FxG Distance from mean cone to cone distance at involute curvature qx Generating angle at involute curvature Increment in generating angle at involute Δqx curvature Depth reduction on non--generated gear at ΔbxG involute curvature bxG Gear dedendum at cone distance AxG bxP Pinion dedendum at cone distance AxG Sum of pinion and gear dedendums at cone Σbx distance AxG 4.2 Slot width calculations and blade specifications 4.2.1 Slot width calculations, mean Tmn Mean normal gear circular thickness tmn Mean normal pinion circular thickness tmP Mean pinion transverse circular thickness WmG Mean gear slot width WmP Mean pinion slot width W′e Effective gear point width 4.2.2 Slot width calculation, outer Angle between gear root plane and plane in ζ which taper is specified WoG Outer gear slot width WoP Outer pinion slot width WxG Slot width at AxG, gear (maximum gear slot width) WxP Slot width at AxG, pinion (maximum pinion slot width) 4.2.3 Sloth width calculation, inner WiG Inner gear slot width WiP Inner pinion slot width 4.2.4 Sloth widths, minimum and maximum WLG Minimum gear slot width WLP Minimum pinion slot width WMG Maximum gear slot width WMP Maximum pinion slot width 4.2.5 Point width and blade point calculations WRG Gear roughing point width WRP Pinion roughing point width WBRG Minimum gear roughing blade point WBRP Minimum pinion roughing blade point WBG Gear finishing blade point WBP Pinion finishing blade point

© AGMA 2006 ---- All rights reserved

AGMA 929--A06

Equation (eq 31)

Variables 5.84984

Units inch

(eq 32 or 35) (eq 34) (eq 38)

50.28674 Not applicable 0.00000

degrees degrees inch

(eq 39 or 40) (eq 41)

0.00000 0.00000

degrees degrees

(eq 42 or 43)

0.00000

inch

(eq 44) (eq 45) (eq 46)

0.00000 0.00000 0.00000

inch inch inch

(eq 47) (eq 48) (eq 49) (eq 50) (eq 51) (eq 52)

0.14581 0.24691 0.30142 0.08997 0.08194 0.08997

inch inch inch inch inch inch

(eq 53 or 54)

0.00000

degrees

(eq 55) (eq 56) (eq 57)

0.08997 0.07906 0.00000

inch inch inch

(eq 58)

0.00000

inch

(eq 59) (eq 60)

0.08997 0.07840

inch inch

(eq 61) (eq 62) (eq 63) (eq 64)

0.08997 0.07840 0.08997 0.08194

inch inch inch inch

(eq 65) (eq 66) (eq 67) (eq 68) (eq 69 or 71) (eq 70 or 72)

0.08997 0.07840 0.04599 0.04020 0.04599 0.04020

inch inch inch inch inch inch

23

AGMA 929--A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description 4.3 Cutter edge radius calculations rTP Maximum pinion blade edge radius rTG Maximum gear blade edge radius 4.3.1 Cutter edge radius for no running interference RiP Original inner pinion pitch radius RiG Original inner gear pitch radius aiP Pinion inner addendum aiG Gear inner addendum Change in inner pinion addendum ΔaP Change in inner gear addendum ΔaG a′iP Adjusted inner pinion addendum a′iG Adjusted inner gear addendum R′iP New inner pinion pitch radius R′iG New inner gear pitch radius RioP Inner pinion outside radius RioG Inner gear outside radius RibVP Inner pinion base radius, concave RibXP Inner pinion base radius, convex RibXG Inner gear base radius, convex RibVG Inner gear base radius, concave Inner pinion pressure angle, concave φiV Inner pinion pressure angle, convex φiX ρiVP Inner pinion profile radius of curvature, concave ρiXP Inner pinion profile radius of curvature, convex ρiXG Inner gear profile radius of curvature, convex ρiVG Inner gear profile radius of curvature, concave xiVP Limit tooth height for interference pinion, concave xiXP Limit tooth height for interference pinion, convex xiXG Limit tooth height for interference gear, convex xiVG Limit tooth height for interference gear, concave r1VP Maximum blade edge radius for no running interference pinion, concave r1XP Maximum blade edge radius for no running interference pinion, convex r1XG Maximum blade edge radius for no running interference gear, convex r1VG Maximum blade edge radius for no running interference gear, concave r1P Maximum blade edge radius for no running interference on pinion r1G Maximum blade edge radius for no running interference on gear

24

Equation

Variables

Units

(eq 73) (eq 74)

0.04063 0.05425

inch inch

(eq 75) (eq 76) (eq 77) (eq 78) (eq 79) (eq 80) (eq 81) (eq 82) (eq 83) (eq 84) (eq 85) (eq 86) (eq 87) (eq 88) (eq 89) (eq 90) (eq 91) (eq 92) (eq 93) (eq 94) (eq 95) (eq 96) (eq 97 or 98)

1.40824 10.92822 0.13352 0.04396 0.00063 0.00490 0.13415 0.04885 1.40761 10.92333 1.54176 10.97218 1.32331 1.32331 10.26917 10.26917 19.92929 19.92929 0.33882 0.33882 3.41201 3.41201 0.02674

inch inch inch inch inch inch inch inch inch inch inch inch inch inch inch inch degrees degrees inch inch inch inch inch

(eq 99 or 100) (eq 101) (eq 102) (eq 103)

0.02674 0.05377 0.05377 0.04063

inch inch inch inch

(eq 104)

0.04063

inch

(eq 105)

0.08171

inch

(eq 106)

0.08171

inch

(eq 107 or 108)

0.04063

inch

(eq 109 or 110)

0.08171

inch

© AGMA 2006 ---- All rights reserved

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description 4.3.2 Cutter edge radius which can be manufactured 4.3.2.1 Roughing r′2RG Maximum gear roughing blade edge radius which can be manufactured r′2RP Maximum pinion roughing blade edge radius which can be manufactured r2RG Gear roughing cutter edge radius r2RP Pinion roughing cutter edge radius 4.3.2.2 Finishing r′2VP Maximum pinion finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance, concave

AGMA 929--A06

Equation

Variables

Units

(eq 111)

0.05425

inch

(eq 112)

0.04599

inch

(eq 113) (eq 114)

0.05425 0.04599

inch inch

(eq 115)

0.04599

inch

r′2XP

Maximum pinion finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance, convex

(eq 116)

0.04599

inch

r′2XG

Maximum gear finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance, convex

(eq 117)

0.05425

inch

r′2VG

Maximum gear finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance, concave

(eq 118)

0.05425

inch

Maximum pinion finish blade edge radius concave r′′2XP Maximum pinion finish blade edge radius convex r′′2XG Maximum gear finish blade edge radius, convex r′′2VG Maximum gear finish blade edge radius, concave r2P Maximum pinion blade edge radius that can be manufactured r2G Maximum gear blade edge radius that can be manufactured 4.3.3 Cutter edge radius to avoid mutilation Difference between pinion minimum slot width ΔW′P and finishing blade point Difference between gear minimum slot width ΔW′G and finishing blade point Pinion intermediate blade mutilation value, ΩVP concave Pinion intermediate blade mutilation value, ΩXP convex Gear intermediate blade mutilation value, ΩXG convex Gear intermediate blade mutilation value, ΩVG concave r3VP Maximum pinion blade edge radius to avoid mutilation, concave

(eq 119)

0.04599

inch

(eq 120)

0.04599

inch

(eq 121) (eq 122)

0.05425 0.05425

inch inch

(eq 123 or 124)

0.04599

inch

(eq 125 or 126)

0.05425

inch

(eq 127)

0.03820

inch

(eq 128)

0.04399

inch

(eq 129)

0.00011

inch2

(eq 130)

0.00011

inch2

(eq 131)

0.00013

inch2

(eq 132)

0.00013

inch2

(eq 133)

0.08013

inch

r′′2VP

© AGMA 2006 ---- All rights reserved

25

AGMA 929--A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description r3XP Maximum pinion blade edge radius to avoid mutilation, convex r3XG Maximum gear blade edge radius to avoid mutilation, convex r3VG Maximum gear blade edge radius to avoid mutilation, concave r3P Maximum pinion blade edge radius to avoid mutilation r3G Maximum gear blade edge radius to avoid mutilation 4.4 Top land formulas 4.4.1 Mean top lands RNP Mean normal pinion pitch radius RNG Mean normal gear pitch radius RbNP1 Mean normal pinion base radius, concave RbNP2 Mean normal pinion base radius, convex RbNG1 Mean normal gear base radius, convex RbNG2 Mean normal gear base radius, concave RoNP Mean normal pinion outside radius RoNG Mean normal gear outside radius Pressure angle at tip of pinion tooth, concave φ1TP Pressure angle at tip of pinion tooth, convex φ2TP Pressure angle at tip of gear tooth, concave φ1TG Pressure angle at tip of gear tooth, convex φ2TG tLNP Mean normal pinion top land tLNG Mean normal gear top land 4.4.2 Outer top lands R′NP Outer normal pinion pitch radius R′NG Outer normal gear pitch radius R′bNP1 Outer normal pinion base radius, concave R′bNP2 Outer normal pinion base radius, convex R′bNG1 Outer normal gear base radius, convex R′bNG2 Outer normal gear base radius, concave R′oNP Outer normal pinion outside radius R′oNG Outer normal gear outside radius Pressure angle at tip of pinion tooth outer, φ′1TP concave Pressure angle at tip of pinion tooth outer, φ′2TP convex Pressure angle at tip of gear tooth outer, convex φ′1TG Pressure angle at tip of gear tooth outer, φ′2TG concave toNP Pinion outer normal circular thickness at pitch line toNG Gear outer normal circular thickness at pitch line

26

Equation (eq 134)

Variables 0.08013

Units inch

(eq 135)

0.08990

inch

(eq 136)

0.08990

inch

(eq 137)

0.08013

inch

(eq 138)

0.08990

inch

(eq 139) (eq 140) (eq 141) (eq 142) (eq 143) (eq 144) (eq 145) (eq 146 or 147) (eq 148) (eq 149) (eq 150) (eq 151) (eq 152 or 153) (eq 154 or 155)

1.71177 13.28366 1.60853 1.60853 12.48256 12.48256 1.90220 13.34625 32.26164 32.26164 20.72564 20.72564 0.07175 0.09992

inch inch inch inch inch inch inch inch degrees degrees degrees degrees inch inch

(eq 156) (eq 157) (eq 158) (eq 159) (eq 160) (eq 161) (eq 162) (eq 163 or 164) (eq 165)

2.07384 16.09347 1.94878 1.94878 15.12291 15.12291 2.32118 16.17469 32.90619

inch inch inch inch inch inch inch inch degrees

(eq 166)

32.90619

degrees

(eq 167) (eq 168)

20.77605 20.77605

degrees degrees

(eq 169)

0.28773

inch

(eq 170)

0.15589

inch

© AGMA 2006 ---- All rights reserved

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description tLoNP Outer normal pinion top land tLoNG Outer normal gear top land 4.4.3 Inner top lands R′′NP Inner mean normal pinion pitch radius R′′NG Inner mean normal gear pitch radius R′′bNP1 Inner normal pinion base radius, concave R′′bNP2 Inner normal pinion base radius, convex R′′bNG1 Inner normal gear base radius, convex R′′bNG2 Inner normal gear base radius, concave R′′oNP Inner normal pinion outside radius R′′oNG Inner normal gear outside radius Pressure angle at tip of pinion tooth inner, φ′′1TP concave Pressure angle at tip of pinion tooth inner, φ′′2TP convex Pressure angle at tip of gear tooth inner, convex φ′′1TG Pressure angle at tip of gear tooth inner, φ′′2TG concave tiNP Inner normal pinion thickness at pitch line tiNG Inner normal gear thickness at pitch line tLiNP Inner normal pinion top land tLiNG Inner normal gear top land

© AGMA 2006 ---- All rights reserved

AGMA 929--A06

Equation (eq 171 or 172) (eq 173 or 174)

Variables 0.05345 0.09614

Units inch inch

(eq 175) (eq 176) (eq 177) (eq 178) (eq 179) (eq 180) (eq 181) (eq 182 or 183) (eq 184)

1.40824 10.92822 1.32331 1.32331 10.26916 10.26916 1.54176 10.97217 30.87230

inch inch inch inch inch inch inch inch degrees

(eq 185)

30.87230

degrees

(eq 186) (eq 187)

20.62140 20.62140

degrees degrees

(eq 188) (eq 189) (eq 190 or 191) (eq 192 or 193)

0.20617 0.12940 0.08972 0.09731

inch inch inch inch

27

AGMA 929--A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Annex D (informative) Hypoid example problem [This annex is provided for informational purposes only and should not be construed as a part of AGMA 929--A06,

Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius.]

Symbol

Description

Equation

Type of gears Face hobbing or face milling Generated or non--generated Completing or rough and finish, pinion Completing or rough and finish, gear

Variables Hypoid Face milled Non--generated Completing Completing

Units

4.04868 4.84349 5.06560 5.64868 0.08460 0.27708 0.10327 0.33072 0.00600 0.31520 0.12902 0.04442 10.77070 3.45020 1.60000 1.83628 45.00000 11.00000 0.64475 table 5 table 5 table 5 0.18870 table 5 table 5 72.43638 68.18198 table 5 16.75611 4.25440 1.27592 15.44545 --24.55455 30.02788 47.99638 23.83723

inch inch inch inch inch inch inch inch inch inch inch inch inch inch inch inch -- --- -inch inch inch inch inch inch inch degrees degrees inch degrees degrees degrees degrees degrees degrees degrees degrees

Table 2 AiG AmG AmP AoG aG aP aoG aoP B bG bP c D d FG FP N n pm rc SAP SAG Tn W y Γ ΓR Δf γ δG δP φ1 φ2 ψG ψP ψiG

28

Inner gear cone distance Gear mean cone distance Pinion mean cone distance Gear outer cone distance Gear mean addendum Pinion mean addendum Outer gear addendum Outer pinion addendum Outer normal backlash allowance Gear mean dedendum Pinion mean dedendum Clearance Gear pitch diameter Pinion pitch diameter Gear face width Pinion face width Gear number of teeth Pinion number of teeth Mean circular pitch Cutter radius Pinion stock allowance Gear stock allowance Gear mean normal zero backlash circular thickness at pitch line Gear finishing point width Amount of fillet mutilation permitted Gear pitch angle Gear root angle Width of blade flat Pinion pitch angle Dedendum angle of gear Dedendum angle of pinion Concave pinion normal pressure angle Convex pinion normal pressure angle Gear mean spiral angle Pinion mean spiral angle Inner gear spiral angle

© AGMA 2006 ---- All rights reserved

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description Equation Inner pinion spiral angle ψiP Outer gear spiral angle ψoG Outer pinion spiral angle ψoP Additional input variables for face hobbing (Table 2) Q Intermediate factor S1 Crown gear to cutter center distance Second auxiliary angle η1 Generating angle at outside ηo Generating angle at inside ηi Additional input for hypoids FoP Hypoid pinion face width from calculation point to outside FiP Hypoid pinion face width from calculation point to inside Table 4 Method Cutting method kE Gear rotation factor Table 5 Inside blade angle gear cutter (non--generated) φBXG SAP Pinion stock allowance SAG Gear stock allowance y Amount of fillet mutilation permitted Width of blade flat Δf rc Cutter radius 4 Calculations 4.1 General geometry AiP Inner pinion cone distance (eq 1 or 2) AoP Outer pinion cone distance (eq 3 or 4) Included pressure angle (eq 5) Σφ Change in pinion clearance (eq 6) ΔcP Change in gear clearance (eq 7) ΔcG Normal tilt of finishing cutter (non--generated) (eq 8) Δφ2 q Generating angle at mean (eq 9 or 10) qo Generating angle at outside (eq 11 or 12) qi Generating angle at inside (eq 13 or 14) Increment in generating angle at outside (eq 15) Δqo Increment in generating angle at inside (eq 16) Δqi Depth reduction on non--generated gear at (eq 17) ΔboG outside Depth reduction on non--generated gear at (eq 18) ΔbiG inside b′oG Theoretical outer gear dedendum (eq 19) boG Outer gear dedendum (eq 20 or 21) boP Outer pinion dedendum (eq 22 or 23) biP Inner pinion dedendum (eq 24) biG Inner gear dedendum (eq 25 or 26) pn Mean normal circular pitch (eq 27) Sum of pinion and gear mean dedendums (eq 28) Σb Sum of pinion and gear outer dedendums (eq 29) Σbo Sum of pinion and gear inner dedendums (eq 30) Σbi

© AGMA 2006 ---- All rights reserved

AGMA 929--A06

Variables 44.98259 36.53169 51.88843

Units degrees degrees degrees

Not applicable Not applicable Not applicable Not applicable Not applicable

inch degrees degrees degrees

0.84044 0.82960

degrees degrees

Spread 0.00000 22.00000 0.00000 0.00000 0.00200 0.00800 4.50000

degrees inch inch inch inch inch

4.23600 5.90604 40.00000 0.00723 0.00718 6.55455 56.36854 50.60129 61.55194 5.76725 5.18341 1.28684

inch inch degrees inch inch degrees degrees degrees degrees degrees degrees inch

0.99546

inch

0.37510 0.35264 0.14774 0.10684 0.23870 0.55821 0.44422 0.50038 0.34554

inch inch inch inch inch inch inch inch inch

29

AGMA 929--A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description AxG Cone distance for involute lengthwise curvature point where normal circular pitch and slot width is a maximum Spiral angle at cone distance AxG ψxG Generating angle at involute curvature ηx FxG Distance from mean cone to cone distance at involute curvature qx Generating angle at involute curvature Increment in generating angle at involute Δqx curvature Depth reduction on non--generated gear at ΔbxG involute curvature bxG Gear dedendum at cone distance AxG bxP Pinion dedendum at cone distance AxG Sum of pinion and gear dedendums at cone Σbx distance AxG 4.2 Slot width calculations and blade specifications 4.2.1 Slot width calculations, mean Tmn Mean normal gear circular thickness tmn Mean normal pinion thickness in gear normal plane tmP Mean pinion transverse circular thickness WmG Mean gear slot width WmP Mean pinion slot width W′e Effective gear point width 4.2.2 Slot width calculation, outer Angle between gear root plane and plane in ζ which taper is specified WoG Outer gear slot width WoP Outer pinion slot width WxG Slot width at AxG, gear (maximum gear slot width) WxP Slot width at AxG, pinion (maximum pinion slot width) 4.2.3 Slot width calculation, inner WiG Inner gear slot width WiP Inner pinion slot width 4.2.4 Slot widths, minimum and maximum WLG Minimum gear slot width WLP Minimum pinion slot width WMG Maximum gear slot width WMP Maximum pinion slot width 4.2.5 Point width and blade point calculations WRG Gear roughing point width WRP Pinion roughing point width WBRG Minimum gear roughing blade point WBRP Minimum pinion roughing blade point

30

Equation (eq 31)

Variables 6.49194

Units inch

(eq 32 or 35) (eq 34) (eq 38)

43.88132 Not applicable 0.00000

degrees degrees inch

(eq 39 or 40) (eq 41)

0.00000 0.00000

degrees degrees

(eq 42 or 43)

0.00000

inch

(eq 44) (eq 45) (eq 46)

0.00000 0.00000 0.00000

inch inch inch

(eq 47) (eq 48)

0.18575 0.36656

inch inch

(eq 49) (eq 50) (eq 51) (eq 52)

0.42339 0.13956 0.09886 0.13956

inch inch inch inch

(eq 53 or 54)

0.00000

degrees

(eq 55) (eq 56) (eq 57)

0.14151 0.10221 0.00000

inch inch inch

(eq 58)

0.00000

inch

(eq 59) (eq 60)

0.14115 0.10369

inch inch

(eq 61) (eq 62) (eq 63) (eq 64)

0.13956 0.09886 0.14151 0.10369

inch inch inch inch

(eq 65) (eq 66) (eq 67) (eq 68)

0.13956 0.09886 0.07078 0.05043

inch inch inch inch

© AGMA 2006 ---- All rights reserved

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description WBG Gear finishing blade point WBP Pinion finishing blade point 4.3 Cutter edge radius calculations rTP Maximum pinion blade edge radius rTG Maximum gear blade edge radius 4.3.1 Cutter edge radius for no running interference RiP Original inner pinion pitch radius RiG Original inner gear pitch radius aiP Pinion inner addendum aiG Gear inner addendum Change in inner pinion addendum ΔaP Change in inner gear addendum ΔaG a′iP Adjusted inner pinion addendum a′iG Adjusted inner gear addendum R′iP New inner pinion pitch radius R′iG New inner gear pitch radius RioP Inner pinion outside radius RioG Inner gear outside radius RibVP Inner pinion base radius, concave RibXP Inner pinion base radius, convex RibXG Inner gear base radius, convex RibVG Inner gear base radius, concave Inner pinion pressure angle, concave φiV Inner pinion pressure angle, convex φiX ρiVP Inner pinion profile radius of curvature, concave ρiXP Inner pinion profile radius of curvature, convex ρiXG Inner gear profile radius of curvature, convex ρiVG Inner gear profile radius of curvature, concave xiVP Limit tooth height for interference pinion, concave xiXP Limit tooth height for interference pinion, convex xiXG Limit tooth height for interference gear, convex xiVG Limit tooth height for interference gear, concave r1VP Maximum blade edge radius for no running interference pinion, concave r1XP Maximum blade edge radius for no running interference pinion, convex r1XG Maximum blade edge radius for no running interference gear, convex r1VG Maximum blade edge radius for no running interference gear, concave r1P Maximum blade edge radius for no running interference on pinion r1G Maximum blade edge radius for no running interference on gear

© AGMA 2006 ---- All rights reserved

AGMA 929--A06

Equation (eq 69 or 71) (eq 70 or 72)

Variables 0.07078 0.05043

Units inch inch

(eq 73) (eq 74)

0.05069 0.08248

inch inch

(eq 75) (eq 76) (eq 77) (eq 78) (eq 79) (eq 80) (eq 81) (eq 82) (eq 83) (eq 84) (eq 85) (eq 86) (eq 87) (eq 88) (eq 89) (eq 90) (eq 91) (eq 92) (eq 93) (eq 94) (eq 95) (eq 96) (eq 97 or 98)

2.54922 15.28824 0.19428 0.06242 0.00051 0.00672 0.19479 0.06914 2.54871 15.28153 2.74350 15.35066 2.45715 2.31868 14.73610 13.90567 15.40361 24.52927 0.43599 0.90042 3.51569 5.93591 0.03719

inch inch inch inch inch inch inch inch inch inch inch inch inch inch inch inch degrees degrees inch inch inch inch inch

(eq 99 or 100) (eq 101) (eq 102) (eq 103)

0.03719 0.09066 0.06530 0.05069

inch inch inch inch

(eq 104)

0.06363

inch

(eq 105)

0.12356

inch

(eq 106)

0.11173

inch

(eq 107 or 108)

0.05069

inch

(eq 109 or 110)

0.11173

inch

31

AGMA 929--A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description 4.3.2 Cutter edge radius which can be manufactured 4.3.2.1 Roughing r′2RG Maximum gear roughing blade edge radius which can be manufactured r′2RP Maximum pinion roughing blade edge radius which can be manufactured r2RG Gear roughing cutter edge radius r2RP Pinion roughing cutter edge radius 4.3.2.2 Finishing r′2VP Maximum pinion finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance, concave

Equation

Variables

Units

(eq 111)

0.08248

inch

(eq 112)

0.05574

inch

(eq 113) (eq 114)

0.08248 0.05574

inch inch

(eq 115)

0.05574

inch

r′2XP

Maximum pinion finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance, convex

(eq 116)

0.05574

inch

r′2XG

Maximum gear finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance, convex

(eq 117)

0.08248

inch

r′2VG

Maximum gear finishing blade edge radius defined by maximum roughing blade radius and minimum stock allowance, concave

(eq 118)

0.08248

inch

Maximum pinion finish blade edge radius, concave r′′2XP Maximum pinion finish blade edge radius, convex r′′2XG Maximum gear finish blade edge radius, convex r′′2VG Maximum gear finish blade edge radius, concave r2P Maximum pinion blade edge radius that can be manufactured r2G Maximum gear blade edge radius that can be manufactured 4.3.3 Cutter edge radius to avoid mutilation Difference between pinion minimum slot width ΔW′P and finishing blade point Difference between gear minimum slot width ΔW′G and finishing blade point Pinion intermediate blade mutilation value, ΩVP concave Pinion intermediate blade mutilation value, ΩXP convex Gear intermediate blade mutilation value, ΩXG convex Gear intermediate blade mutilation value, ΩVG concave r3VP Maximum pinion edge radius to avoid mutilation, concave

(eq 119)

0.05574

inch

(eq 120)

0.06603

inch

(eq 121) (eq 122)

0.08248 0.09770

inch inch

(eq 123 or 124)

0.05574

inch

(eq 125 or 126)

0.08248

inch

(eq 127)

0.04843

inch

(eq 128)

0.06878

inch

(eq 129)

0.00015

inch2

(eq 130)

0.00013

inch2

(eq 131)

0.00021

inch2

(eq 132)

0.00018

inch2

(eq 133)

0.10767

inch

r′′2VP

32

© AGMA 2006 ---- All rights reserved

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description r3XP Maximum pinion edge radius to avoid mutilation, convex r3XG Maximum gear edge radius to avoid mutilation, convex r3VG Maximum gear edge radius to avoid mutilation, concave r3P Maximum pinion blade edge radius to avoid mutilation r3G Maximum gear blade edge radius to avoid mutilation 4.4 Top land formulas 4.4.1 Mean top lands RNP Mean normal pinion pitch radius RNG Mean normal gear pitch radius RbNP1 Mean normal pinion base radius, concave RbNP2 Mean normal pinion base radius, convex RbNG1 Mean normal gear base radius, convex RbNG2 Mean normal gear base radius, concave RoNP Mean normal pinion outside radius RoNG Mean normal gear outside radius Pressure angle at tip of pinion tooth, concave φ1TP Pressure angle at tip of pinion tooth, convex φ2TP Pressure angle at tip of gear tooth, concave φ1TG Pressure angle at tip of gear tooth, convex φ2TG tLNP Mean normal pinion top land tLNG Mean normal gear top land 4.4.2 Outer top lands R′NP Outer normal pinion pitch radius R′NG Outer normal gear pitch radius R′bNP1 Outer normal pinion base radius, concave R′bNP2 Outer normal pinion base radius, convex R′bNG1 Outer normal gear base radius, convex R′bNG2 Outer normal gear base radius, concave R′oNP Outer normal pinion outside radius R′oNG Outer normal gear outside radius Pressure angle at tip of pinion tooth outer, φ′1TP concave Pressure angle at tip of pinion tooth outer, φ′2TP convex Pressure angle at tip of gear tooth outer, convex φ′1TG Pressure angle at tip of gear tooth outer, φ′2TG concave toNP Pinion outer normal circular thickness at pitch line toNG Gear outer normal circular thickness at pitch line

© AGMA 2006 ---- All rights reserved

AGMA 929--A06

Equation (eq 134)

Variables 0.08832

Units inch

(eq 135)

0.14445

inch

(eq 136)

0.11903

inch

(eq 137)

0.08832

inch

(eq 138)

0.11903

inch

(eq 139) (eq 140) (eq 141) (eq 142) (eq 143) (eq 144) (eq 145) (eq 146 or 147) (eq 148) (eq 149) (eq 150) (eq 151) (eq 152 or 153) (eq 154 or 155)

3.45071 20.41456 3.32609 3.13865 19.67727 18.56839 3.72779 20.13748 26.84382 32.65236 12.27271 22.76775 0.10537 0.12372

inch inch inch inch inch inch inch inch degrees degrees degrees degrees inch inch

(eq 156) (eq 157) (eq 158) (eq 159) (eq 160) (eq 161) (eq 162) (eq 163 or 164) (eq 165)

4.72947 27.64037 4.55866 4.30177 26.64212 25.14074 5.06019 27.30965 25.72501

inch inch inch inch inch inch inch inch degrees

(eq 166)

31.77535

degrees

(eq 167) (eq 168)

12.69412 22.98885

degrees degrees

(eq 169)

0.39368

inch

(eq 170)

0.20414

inch

33

AGMA 929--A06

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Symbol Description tLoNP Outer normal pinion top land tLoNG Outer normal gear top land 4.4.3 Inner top lands R′′NP Inner mean normal pinion pitch radius R′′NG Inner mean normal gear pitch radius R′′bNP1 Inner normal pinion base radius, concave R′′bNP2 Inner normal pinion base radius, convex R′′bNG1 Inner normal gear base radius, convex R′′bNG2 Inner normal gear base radius, concave R′′oNP Inner normal pinion outside radius R′′oNG Inner normal gear outside radius Pressure angle at tip of pinion tooth inner, φ′′1TP concave Pressure angle at tip of pinion tooth inner, φ′′2TP convex Pressure angle at tip of gear tooth inner, convex φ′′1TG Pressure angle at tip of gear tooth inner, φ′′2TG concave tiNP Inner normal pinion thickness at pitch line tiNG Inner normal gear thickness at pitch line tLiNP Inner normal pinion top land tLiNG Inner normal gear top land

34

Equation (eq 171 or 172) (eq 173 or 174)

Variables 0.08697 0.12842

Units inch inch

(eq 175) (eq 176) (eq 177) (eq 178) (eq 179) (eq 180) (eq 181) (eq 182 or 183) (eq 184)

2.58275 15.28824 2.48947 2.34918 14.73609 13.90566 2.77703 15.09396 26.30451

inch inch inch inch inch inch inch inch degrees

(eq 185)

32.22794

degrees

(eq 186) (eq 187)

12.50133 22.88704

degrees degrees

(eq 188) (eq 189) (eq 190 or 191) (eq 192 or 193)

0.31092 0.17681 0.13357 0.13105

inch inch inch inch

© AGMA 2006 ---- All rights reserved

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 930--A05

Bibliography

The following documents are either referenced in the text of AGMA 929--A06, Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius, or indicated for additional information.

Gleason Bevel & Hypoid Gear Terminology, Publication No. SD 4053A--1971--C&T, Gleason Works. Formulas for Cutter Specifications and Tooth Thickness Measurements for Spiral Bevel and Hypoid Gears, SD 3124B, Jan 1982, Gleason Works.

© AGMA 2005 ---- All rights reserved

35

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