Agma 925-a03 - Effect Of Lubrication On Gear Surface Distress

AGMA 925- A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925- A03

Effect of Lubrication on Gear Surface Distress

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

Effect of Lubrication on Gear Surface Distress American AGMA 925--A03 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 quoted or extracted. Credit lines should read: Extracted from AGMA 925--A03, Effect of Lubrication on Gear Surface Distress, with the permission of the publisher, the American Gear Manufacturers Association, 500 Montgomery Street, Suite 350, Alexandria, Virginia 22314.] Approved March 13, 2003

ABSTRACT AGMA 925--A03 is an enhancement of annex A of ANSI/AGMA 2101--C95. Various methods of gear surface distress are included, such as scuffing and wear, and in addition, micro and macropitting. Lubricant viscometric information has been added, as has Dudley’s regimes of lubrication theory. A flow chart is included in annex A, Gaussian theory in annex B, a summary of lubricant test rigs in annex C, and an example calculation in annex D. Published by

American Gear Manufacturers Association 500 Montgomery Street, Suite 350, Alexandria, Virginia 22314 Copyright  2003 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--815--7

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Contents Page

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv 1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Symbols and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 Gear information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 Lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Scuffing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7 Surface fatigue (micro-- and macropitting) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8 Wear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Annexes A B C D

Flow chart for evaluating scuffing risk and oil film thickness . . . . . . . . . . . . . . Normal or Gaussian probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test rig gear data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 39 41 43

Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Distances along the line of action for external gears . . . . . . . . . . . . . . . . . . . . . . 6 Transverse relative radius of curvature for external gears . . . . . . . . . . . . . . . . . 7 Load sharing factor -- unmodified profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Load sharing factor -- pinion driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Load sharing factor -- gear driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Load sharing factor -- smooth meshing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Dynamic viscosity versus temperature for mineral oils . . . . . . . . . . . . . . . . . . . 13 Dynamic viscosity versus temperature for PAO--based synthetic non--VI--improved oils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Dynamic viscosity versus temperature for PAG--based synthetic oils . . . . . . 15 Dynamic viscosity versus temperature for MIL Spec. oils . . . . . . . . . . . . . . . . 16 Pressure--viscosity coefficient versus dynamic viscosity . . . . . . . . . . . . . . . . . 16 Example of thermal network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Contact temperature along the line of action . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Plot of regimes of lubrication versus stress cycle factor . . . . . . . . . . . . . . . . . . 25 Probability of wear related distress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Tables 1 2 3 4 5 6 7

Symbols and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Data for determining viscosity and pressure--viscosity coefficient . . . . . . . . . 12 Mean scuffing temperatures for oils and steels typical of the aerospace industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Welding factors, XW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Scuffing risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Stress cycle factor equations for regimes I, II and III . . . . . . . . . . . . . . . . . . . . 25 Calculation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

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 925--A03, Effect of Lubrication on Gear Surface Distress.] The purpose of this information sheet is to provide the user with information pertinent to the lubrication of industrial metal gears for power transmission applications. It is intended that this document serve as a general guideline and source of information about conventional lubricants, their properties, and their general tribological behavior in gear contacts. This information sheet was developed to supplement ANSI/AGMA Standards 2101--C95 and 2001--C95. It has been introduced as an aid to the gear manufacturing and user community. Accumulation of feedback data will serve to enhance future developments and improved methods to evaluate lubricant related wear risks. It was clear from the work initiated on the revision of AGMA Standards 2001--C95 and 2101--C95 (metric version) that supporting information regarding lubricant properties and general tribological knowledge of contacting surfaces would aid in the understanding of these standards. The information would also provide the user with more tools to help make a more informed decision about the performance of a geared system. This information sheet provides sufficient information about the key lubricant parameters to enable the user to generate reasonable estimates about scuffing and wear based on the collective knowledge of theory available for these modes at this time. In 1937 Harmon Blok published his theory about the relationship between contact temperature and scuffing. This went largely unnoticed in the U.S. until the early 1950’s when Bruce Kelley showed that Blok’s method and theories correlated well with experimental data he had generated on scuffing of gear teeth. The Blok flash temperature theory began to receive serious consideration as a predictor of scuffing in gears. The methodology and theories continued to evolve through the 1950’s with notable contributions from Dudley, Kelley and Benedict in the areas of application rating factors, surface roughness effects and coefficient of friction. The 1960’s saw the evolution of gear calculations and understanding continue with computer analysis and factors addressing load sharing and tip relief issues. The AGMA Aerospace Committee began using all the available information to produce high quality products and help meet its long--term goal of manned space flight. R. Errichello introduced the SCORING+ computer program in 1985, which included all of the advancements made by Blok, Kelley, Dudley and the Aerospace Committee to that time. It became the basis for annex A of ANSI/AGMA 2101--C95 and 2001--C95 which helped predict the risk of scuffing and wear. In the 1990s, this annex formed the basis for AGMA’s contribution to ISO 13989--1. Just as many others took the original Blok theories and expanded them, the Tribology Subcommittee of the Helical Gear Rating Committee has attempted to expand the original annex A of ANSI/AGMA 2001--C95 and 2101--C95. Specifically, the subcommittee targeted the effect lubrication may have on gear surface distress. As discussions evolved, it became clear that this should be a stand alone document which will hopefully serve many other gear types. This should be considered a work in progress as more is learned about the theories and understanding of the various parameters and how they affect the life of the gear. Some of these principles are also mentioned in ISO/TR 13989--1. AGMA 925--A03 was was approved by the AGMA Technical Division Executive Committee on March 13, 2003. 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. iv

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

PERSONNEL of the AGMA Helical Rating Committee and Tribology SubCommittee Chairman: D. McCarthy . . . . . . . . . . . . . . . . . . . . . . . . . Dorris Company Vice Chairman: M. Antosiewicz . . . . . . . . . . . . . . . . . . The Falk Corporation SubCommittee Chairman: H. Hagan . . . . . . . . . . . . . . The Cincinnati Gear Company

COMMITTEE ACTIVE MEMBERS K.E. Acheson . . . J.B. Amendola . . T.A. Beveridge . . M.J. Broglie . . . . . A.B. Cardis . . . . . M.F. Dalton . . . . . G.A. DeLange . . . D.W. Dudley . . . . R.L. Errichello . . . D.R. Gonnella . . . M.R. Hoeprich . . O.A. LaBath . . . .

The Gear Works--Seattle, Inc. MAAG Gear AG Caterpillar, Inc. Dudley Technical Group, Inc. Exxon Mobil Research General Electric Company Prager, Incorporated Consultant GEARTECH Equilon Lubricants The Timken Company The Cincinnati Gear Co.

G. Lian . . . . . . . . . J.V. Lisiecki . . . . . L. Lloyd . . . . . . . . J.J. Luz . . . . . . . . D.R. McVittie . . . . A.G. Milburn . . . . G.W. Nagorny . . . M.W. Neesley . . . B. O’Connor . . . . W.P. Pizzichil . . . D.F. Smith . . . . . . K. Taliaferro . . . .

Amarillo Gear Company The Falk Corporation Lufkin Industries, Inc. General Electric Company Gear Engineers, Inc. Milburn Engineering, Inc. Nagorny & Associates Philadelphia Gear Corp. The Lubrizol Corporation Philadelphia Gear Corp. Solar Turbines, Inc. Rockwell Automation/Dodge

I. Laskin . . . . . . . . J. Maddock . . . . . J. Escanaverino . G.P. Mowers . . . . R.A. Nay . . . . . . . M. Octrue . . . . . . T. Okamoto . . . . . J.R. Partridge . . . J.A. Pennell . . . . . A.E. Phillips . . . . . J.W. Polder . . . . . E. Sandberg . . . . C.D. Schultz . . . . E.S. Scott . . . . . . A. Seireg . . . . . . . Y. Sharma . . . . . . B.W. Shirley . . . . L.J. Smith . . . . . . L. Spiers . . . . . . . A.A. Swiglo . . . . . J.W. Tellman . . . . F.A. Thoma . . . . . D. Townsend . . . . L. Tzioumis . . . . . F.C. Uherek . . . . . A. Von Graefe . . . C.C. Wang . . . . . B. Ward . . . . . . . . R.F. Wasilewski .

Consultant The Gear Works -- Seattle, Inc. ISPJAE Consultant UTC Pratt & Whitney Aircraft CETIM Nippon Gear Company, Ltd. Lufkin Industries, Inc. Univ. of Newcastle--Upon--Tyne Rockwell Automation/Dodge Delft University of Technology Det Nordske Veritas Pittsburgh Gear Company The Alliance Machine Company University of Wisconsin Philadelphia Gear Corporation Emerson Power Transmission Invincible Gear Company Emerson Power Trans. Corp. IIT Research Institute/INFAC Dodge F.A. Thoma, Inc. NASA/Lewis Research Center Dodge Flender Corporation MAAG Gear AG 3E Software & Eng. Consulting Recovery Systems, LLC Arrow Gear Company

COMMITTEE ASSOCIATE MEMBERS M. Bartolomeo . . A.C. Becker . . . . E. Berndt . . . . . . . E.J. Bodensieck . D.L. Borden . . . . M.R. Chaplin . . . . R.J. Ciszak . . . . . A.S. Cohen . . . . . S. Copeland . . . . R.L. Cragg . . . . . T.J. Dansdill . . . . F. Eberle . . . . . . . L. Faure . . . . . . . . C. Gay . . . . . . . . . J. Gimper . . . . . . T.C. Glasener . . . G. Gonzalez Rey M.A. Hartman . . . J.M. Hawkins . . . G. Henriot . . . . . . G. Hinton . . . . . . . M. Hirt . . . . . . . . . R.W. Holzman . . R.S. Hyde . . . . . . V. Ivers . . . . . . . . A. Jackson . . . . . H.R. Johnson . . . J.G. Kish . . . . . . . R.H. Klundt . . . . . J.S. Korossy . . . .

New Venture Gear, Inc. Nuttall Gear LLC Besco Bodensieck Engineering Co. D.L. Borden, Inc. Contour Hardening, Inc. Euclid--Hitachi Heavy Equip. Inc. Engranes y Maquinaria Arco SA Gear Products, Inc. Consultant General Electric Company Rockwell Automation/Dodge C.M.D. Charles E. Gay & Company, Ltd. Danieli United, Inc. Xtek, Incorporated ISPJAE ITW Rolls--Royce Corporation Consultant Xtek, Incorporated Renk AG Milwaukee Gear Company, Inc. The Timken Company Xtek, Incorporated Exxon Mobil The Horsburgh & Scott Co. Sikorsky Aircraft Division The Timken Company The Horsburgh & Scott Co.

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AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

SUBCOMMITTEE ACTIVE MEMBERS K.E. Acheson . . . J.B. Amendola . . T.A. Beveridge . . M.J. Broglie . . . . . A.B. Cardis . . . . . R.L. Errichello . . . D.R. Gonnella . . . M.R. Hoeprich . .

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The Gear Works -- Seattle, Inc. MAAG Gear AG Caterpillar, Inc. Dudley Technical Group, Inc. Exxon Mobil Research GEARTECH Equilon Lubricants The Timken Company

G. Lian . . . . . . . . . D. McCarthy . . . . D.R. McVittie . . . . A.G. Milburn . . . . G.W. Nagorny . . . B. O’Connor . . . . D.F. Smith . . . . . . K. Taliaferro . . . .

Amarillo Gear Company Dorris Company Gear Engineers, Inc. Milburn Engineering, Inc. Nagorny & Associates The Lubrizol Corporation Solar Turbines, Inc. Rockwell Automation/Dodge

AMERICAN GEAR MANUFACTURERS ASSOCIATION

American Gear Manufacturers Association --

Effect of Lubrication on Gear Surface Distress

1 Scope This information sheet is designed to provide currently available tribological information pertaining to oil lubrication of industrial gears for power transmission applications. It is intended to serve as a general guideline and source of information about gear oils, their properties, and their general tribological behavior in gear contacts. Manufacturers and end--users are encouraged, however, to work with their lubricant suppliers to address specific concerns or special issues that may not be covered here (such as greases). The equations provided herein allow the user to calculate specific oil film thickness and instantaneous contact (flash) temperature for gears in service. These two parameters are considered critical in defining areas of operation that may lead to unwanted surface distress. Surface distress may be scuffing (adhesive wear), fatigue (micropitting and macropitting), or excessive abrasive wear (scoring). Each of these forms of surface distress may be influenced by the lubricant; the calculations are offered to help assess the potential risk involved with a given lubricant choice. Flow charts are included as aids to using the equations. This information sheet is a supplement to ANSI/ AGMA 2101--C95 and ANSI/AGMA 2001--C95. It has been introduced as an aid to the gear manufacturing and user community. Accumulation of feedback data will serve to enhance future developments and improved methods to evaluate lubricant related surface distress.

AGMA 925--A03

It was clear from the work on the revision of standard ANSI/AGMA 2001--C95 (ANSI/AGMA 2101--C95, metric version) that supporting information regarding lubricant properties and general tribological understanding of contacting surfaces would aid in understanding of the standard and provide the user with more tools to make an informed decision about the performance of a geared system. One of the key parameters is the estimated film thickness. This is not a trivial calculation, but one that has significant impact on overall performance of the gear pair. It is considered in performance issues such as scuffing, wear, and surface fatigue. This information sheet provides sufficient information about key lubricant parameters to enable the user to generate reasonable estimates about surface distress based on the collective knowledge available. Blok [1] published his contact temperature equation in 1937. It went relatively unnoticed in the U.S. until Kelley [2] showed that Blok’s method gave good correlation with Kelley’s experimental data. Blok’s equation requires an accurate coefficient of friction. Kelley found it necessary to couple the coefficient of friction to surface roughness of the gear teeth. Kelley recognized the importance of load sharing by multiple pairs of teeth and gear tooth tip relief, but he did not offer equations to account for those variables. Dudley [3] modified Kelley’s equation by adding derating factors for application, misalignment and dynamics. He emphasized the need for research on effects of tip relief, and recommended applying Blok’s method to helical gears. In 1958, Kelley [4] changed his surface roughness term slightly. Benedict and Kelley [5] published their equation for variable coefficient of friction derived from disc tests. The AGMA Aerospace Committee began investigating scuffing in 1960, and Lemanski [6] published results of a computer analysis that contains data for 90 spur and helical gearsets, and formed the terms for AGMA 217.01 [7], which was published in 1965. It used Dudley’s modified Blok/Kelley equation and included factors accounting for load sharing and tip relief.

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AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

The SCORING+ computer program [8] was released in 1985. It incorporated all advancements made by Blok, Kelley, Dudley and AGMA 217.01. In addition, it added several improvements including: -- Helical gears were analyzed by resolving the load in the normal plane and distributing the normal load over the minimum length of the contact lines. The semi--width of the Hertzian contact band was calculated based on the normal relative radius of curvature; -- Derating factors for application, misalignment and dynamics were explicit input data; -- Options for coefficient of friction were part of input data, including a constant 0.06 (as prescribed by Kelley and AGMA 217.01), a constant under user control, and a variable coefficient based on the Benedict and Kelley equation. SCORING+ and AGMA 217.01 both use the same value for the thermal contact coefficient of BM = 16.5 N/[mm⋅s0.5⋅K], and they calculate the same contact temperature for spur gears if all derating factors are set to unity. Annex A of ANSI/AGMA 2101--C95 and ANSI/ AGMA 2001--C95 was based on SCORING+ and included methods for predicting risk of scuffing based on contact temperature and risk of wear based on specific film thickness. This information sheet expands the information in annex A of ANSI/AGMA 2101--C95 and ANSI/AGMA 2001--C95 to include many aspects of gear tribology.

2 References The following standards contain provisions which are referenced in the text of this information sheet. At the time of publication, the editions indicated were valid. All standards are subject to revision, and parties to agreements based on this document are encouraged to investigate the possibility of applying the most recent editions of the standards indicated. ANSI/AGMA 2001--C95, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth ANSI/AGMA 2101--C95, Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth (Metric Edition) ANSI/AGMA 1010--E95, Appearance of Gear Teeth -- Terminology of Wear and Failure ISO 10825:1995, Gears -- Wear and Damage to Gear Teeth -- Terminology

3 Symbols and units The symbols used in this document are shown in table 1. NOTE: The symbols and definitions used in this document may differ from other AGMA standards.

Table 1 -- Symbols and units Symbol

Description

A aw B BM BM1, BM2 b bH

Dimensionless constant Operating center distance Dimensionless constant Thermal contact coefficient Thermal contact coefficient (pinion, gear) Face width Semi--width of Hertzian contact band

CA ... CF CR

Distances along line of action Surface roughness constant

c cM1, cM2 Di d

Parameter for calculating ηo Specific heat per unit mass (pinion, gear) Internal gear inside diameter Parameter for calculating ηo

 i

avgx

2

Units -- -mm -- -N/[mm s0.5K] N/[mm s0.5K] mm mm mm -- --- -J/[kg K] mm -- --

Where first used Eq 61 Eq 4 Eq 61 6.2.3 Eq 84 Eq 23 Eq 57 4.1.2 Eq 85 Eq 69 Eq 89, 90 4.1.2 Eq 69 (continued)

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Table 1 (continued) Symbol

Description

E1, E2 Er Ft (Ft)nom Fwn G g Hc

Modulus of elasticity (pinion, gear) Reduced modulus of elasticity Actual tangential load Nominal tangential load Normal operating load Materials parameter Parameter for calculating ηo Dimensionless central film thickness

h hc

Thickness of element measured perpendicular to flow Central film thickness

 i

 i

hmin K KD Km Ko Kv k ksump Lx Lmin mn n1 N na nr P P(x) p pbn pbt px Q Q(x) R avgx

Ra1x, Ra2x Rqx Rqx avg Rq1x, Rq2x r1, r2 ra1, ra2 rb1, rb2 rw1 Sf s

Minimum film thickness Flash temperature constant Combined derating factor Load distribution factor Overload factor Dynamic factor Parameter for calculating α Parameter for calculating θM Filter cutoff of wavelength x Minimum contact length Normal module Pinion speed Number of load cycles Fractional (non--integer) part of εβ Fractional (non--integer) part of εα Transmitted power Probability of survival Pressure Normal base pitch Transverse base pitch Axial pitch Tail area of the normal probability function Probability of failure Average of the average values of pinion and gear roughness Average surface roughness (pinion, gear) at Lx Root mean square roughness at Lx Arithmetic average of Rq1x and Rq2x at Lx Root mean square roughness at Lx (pinion, gear) Standard pitch radius (pinion, gear) Outside radius (pinion, gear) Base radius (pinion, gear) Operating pitch radius of pinion Contact time Parameter for calculating α

Units N/mm2 N/mm2 N N N -- --- --- --

Where first used Eq 58 Eq 57 Eq 42 Eq 40 Eq 43 Eq 65 Eq 69 Eq 65

m mm

Eq 59 Eq 75

mm -- --- --- --- --- --- --- -mm mm mm rpm cycles -- --- -kW -- -N/mm2 mm mm mm -- --- -mm

Eq 102 Eq 84 Eq 41 Eq 41 Eq 41 Eq 41 Eq 74 Eq 91 Eq 77 Eq 25 Eq 2 Eq 33 Fig 14 Eq 25 Eq 25 Eq 40 8.2.2 Eq 64 Eq 10 Eq 9 Eq 11 Eq B.2 8.2.2 Eq 87

Eq 78 Eq 79 Eq 99 Eq 99 Eq 2, 3 Eq 19, 16 Eq 6, 7 Eq 4 ms (sec ¢10--3) Eq 97 -- -Eq 74 mm mm mm mm mm mm mm mm

(continued)

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AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Table 1 (continued) Symbol T U(i) u v ve

i

Description Absolute temperature Speed parameter Gear ratio (always ≥ 1.0) Velocity Entraining velocity

v r1 , v r2 Rolling (tangential) velocity (pinion, gear)  i  i vs Sliding velocity i

Units K -- --- -m/s m/s

Where first used Eq 61 Eq 65 Eq 1 Eq 59 Eq 39

m/s

Eq 36, 37

m/s

Eq 38

m/s -- -N/mm -- --- --

Eq 35 Eq 65 Eq 44 Eq 96 4.3

mm -- --- --- --- -mm2/N degrees degrees degrees degrees degrees degrees degrees radians radians -- --- -mPa⋅s mPa⋅s mPa⋅s mPa⋅s mPa⋅s mPa⋅s °C

Eq 21 7.5 Eq B.3 Eq 1 Eq 1 Eq 64 Eq 5 Eq 5 Eq 14 Eq 8 Eq 2 Eq 12 Eq 13 Eq 29 Eq 28 Eq 22 Eq 23 Eq 59 Eq 64 Eq 64 Eq 67 Eq 70 Eq 71 Eq 92

vt W(i) wn XW XΓ

Operating pitch line velocity Load parameter Normal unit load Welding factor Load sharing factor

Z ZN ZQ z1 z2 α αn αt αwn αwt β βb βw ξ(i) ξA ... ξE εα εβ η ηatm ηP ηM η1, η2 η40, η100 θB

Active length of line of action Stress cycle factor Normal probability density function Number teeth in pinion Number teeth in gear (positive) Pressure--viscosity coefficient Normal generating pressure angle Transverse generating pressure angle Normal operating pressure angle Transverse operating pressure angle Helix angle Base helix angle Operating helix angle Pinion roll angle at point i along the line of action Pinion roll angle at points A ... E Transverse contact ratio Axial contact ratio Dynamic viscosity Viscosity at atmospheric pressure Viscosity at pressure P Dynamic viscosity at gear tooth temperature θM Dynamic viscosity at temperature θ1, θ2 Dynamic viscosity at 40°C, 100°C Contact temperature

θB max θ fl

Maximum contact temperature Flash temperature

°C °C

Eq 93 Eq 84

θfl max θfl max, test θM θM, test

Maximum flash temperature Maximum flash temperature of test gears Tooth temperature Tooth temperature of test gears

°C °C °C °C

Eq 91 Eq 96 Eq 69 Eq 96

(i)

 i

 i

(continued) 4

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Table 1 (concluded) Symbol θoil θS θS

Oil inlet or sump temperature Mean scuffing temperature Method of calculating scuffing temperature, θS

°C °C -- --

Where first used Eq 91 Eq 94 Annex A

θ1, θ2 λmin λ 2b

Temperature at which η1, η2 was measured Specific film thickness Specific film thickness at point i with a filter cutoff wavelength of 2bH

°C -- --- --

Eq 70 Eq 104 Eq 76

λM1, λM2 λW&H my mm

Heat conductivity (pinion, gear) Wellauer and Holloway specific film thickness Mean value of random variable y Mean coefficient of friction

N/[s K] -- --- --- --

Eq 89, 90 Eq 102 6.5.5 Eq 84

mmet mm const mλ min ν ν1, ν2 ν40, ν100 ρ ρM1, ρM2 ρ1 , ρ2

Method for approximating mean coefficient of friction Mean coefficient of friction, constant Mean minimum specific film thickness Kinematic viscosity Poisson’s ratio (pinion, gear) Kinematic viscosity at 40°C, 100°C Density Density (pinion, gear) Transverse radius of curvature (pinion, gear)

-- --- -mm mm2/s -- -mm2/s kg/m3 kg/m3 mm

Annex A Eq 85 Eq 109 Eq 60 Eq 58 Eq 62 Eq 60 Eq 89, 90 4.1.5

ρn

Normal relative radius of curvature

mm

Eq 32

Transverse relative radius of curvature

mm

Eq 31

Composite surface roughness for filter cutoff wavelength, Lx Standard deviation of the minimum specific film thickness Composite surface roughness adjusted for a cutoff wavelength equal to the Hertzian contact width Shear stress Angular velocity (pinion, gear)

mm mm mm

Eq 77 Eq 109 Eq 76

met

H  i

 i

i

ρr

i

i

 i

σx σλ min σ 2b

H  i

τ ω1, ω2

Description

4 Gear information

N/mm2 rad/s

Eq 59 Eq 33, 34

Standard pitch radii z1 mn 2 cos β r2 = r 1 u

r1 =

4.1 Gear geometry This clause gives equations for gear geometry used to determine flash temperature and elastohydrodynamic (EHL) film thickness. The following equations apply to both spur and helical gears; spur gearing is a particular case with zero helix angle. Where double signs are used (e.g., ¦), the upper sign applies to external gears and the lower sign to internal gears. 4.1.1 Basic gear geometry

(2) (3)

Operating pitch radius of pinion r w1 =

aw u1

(4)

Transverse generating pressure angle α t = arctan

tancosαβ  n

(5)

Base radii

Gear ratio z u = z2 1

Units

(1)

r b1 = r 1 cos α t r b2 = r b1 u

(6) (7)

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Transverse operating pressure angle

 

r α wt = arccos r b1 w1

(8)

Transverse base pitch p bt =

2 π r b1 z1

αwt

(9)

ra2

rb2

Normal base pitch

aw

p bn = π m n cos α n

HPSTC

(10)

Z

Axial pitch px =

π mn sin β

(11)

Base helix angle

 

p β b = arccos pbn bt

A CA

(12)

Operating helix angle



EAP

pbt

LPSTC

SAP

pbt

B

C

CB CC



E

ra1 CD

rb1

tan β β w = arctan cos α b wt

D

CE

CF

(13)

Normal operating pressure angle α wn = arcsincos β b sin α wt

(14)

4.1.2 Distances along the line of action Figure 1 is the line of action shown in a transverse plane. Distances Cj are measured from the interference point of the pinion along the line of action. Distance CA locates the pinion start of active profile (SAP) and distance CE locates the pinion end of active profile (EAP). The lowest and highest point of single--tooth--pair contact (LPSTC and HPSTC) are located by distances CB and CD, respectively. Distance CC locates the operating pitch point. CF is the distance between base circles along the line of action. (15)

C F = a w sin α wt



C A = C F − r 2a2 − r 2b2

(16)

4.1.3 Contact ratios Transverse contact ratio ε α = pZ

bt

nr

Axial contact ratio --

for helical gears

ε β = pb

x

na --

for spur gears

ε β = 0.0

(24)

Minimum contact length

CF CC = u1 C D = C A + p bt

L min =

for helical gears, case 1, where 1 − n r  ≥ n a ε αb − n a n r p x cos β b

(25)

for helical gears, case 2, where 1 − n r  < n a

(18)

--

(19)

L min =

C B = C E − p bt

(20)

--

Z = CE − CA

(21)

L min = b

0.5

(23)

is fractional (non--integer) part of εβ.

-(17)

(22)

is fractional (non--integer) part of εα.

D NOTE: For internal gears r a2 = i . 2

C E = r 2a1 − r 2b1

6



0.5

Figure 1 -- Distances along the line of action for external gears

ε α b − 1 − n a1 − n r  p x cos β b

(26)

for spur gears (27)

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Transverse relative radius of curvature

4.1.4 Roll angles

Ã1 Ã2 i i Ãr = Ã  i 2  Ã1

Pinion roll angles corresponding to the five specific points along the line of action shown in figure 1 are given by: Cj ξj = r

(31)  i

Normal relative radius of curvature

(28)

b1

i

Ãr Ãn =

where

(32)

cos β b

i

j = A, B, C, D, E

 i

and à n are the equivalent radii of cylinders i riding on a flat plate that represent the gear pair curvatures in contact along the line of action. Ãr

4.1.5 Profile radii of curvature Transverse radii of curvature Figure 2 shows the transverse radii of curvature, Ã 1

 i

4.2 Gear tooth velocities and loads i

and à 2 , of the gear tooth profiles at a general i

contact point defined by the roll angle, ξ(i), where (i) is any point on the line of action from A to E (see figure 1).

Rotational (angular) velocities πn 1 30 ω1 ω2 = u Operating pitch line velocity ω1 =

ω 1 r w1 1000 Rolling (tangential) velocities vt =

ω1 Ã1 Ã 1 Ã2  i  i Ãr = Ã2  Ã 1 i  i

v r1 =

ω2 Ã2

 i

 i

v r2 =

 i

(35)

(37)

1000

 i

(34)

(36)

1000

 i

(33)

Sliding velocity (absolute value) Ã1

Ã2

 i

r b1



v s = v r1 − v r2 i i i

 i



(38)

Entraining velocity (absolute value) ξ(i)



v e = v r1 + v r2 i  i i

CF



(39)

Nominal tangential load

F t Figure 2 -- Transverse relative radius of curvature for external gears

(29)

where ξ A ≤ ξ (i) ≤ ξE i

P = 1000 v t

(40)

Combined derating factor K D = K o Km Kv

(41)

where

à 1 = r b1ξ i i

Ã2 = CF  Ã1

nom

(30) i

Ko

is overload factor;

Km

is load distribution factor;

Kv

is dynamic factor.

See ANSI/AGMA 2101--C95 for guidance in determining Ko, Km and Kv factors.

7

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Actual tangential load

For modified tooth profiles

F t = F t nom K D

(42)

Normal operating load F wn =

Ft

(43)

cos α wn cos β w

Normal unit load wn =

F wn L min

(44)

4.3 Load sharing factor The load sharing factor accounts for load sharing between succeeding pairs of teeth as influenced by profile modification, and whether the pinion or gear is the driving member. By convention, the load sharing factor is represented by a polygonal function on the line of action with magnitude equal to 1.0 between points B and D (see figure 3). The load sharing factor is strongly influenced by profile modification of the tooth flanks of both gears. On the other hand, profile modifications are chosen such that load sharing follows a desired function. The following equations give the load sharing factor for unmodified tooth profiles, and for three typical cases of profile modifications.

If adequate tip and root relief is designed for high load capacity, and if the pinion drives the gear (see figure 4): XΓ = 6 7 i





ξ  i − ξ A ξB − ξ A

for ξ A ≤ ξ i < ξ B (48)

X Γ = 1 for ξ B ≤ ξ i ≤ ξ D i XΓ = 1 + 6 7 7 i



ξ E − ξ  i ξ E − ξD



(49)

for ξ D < ξ i ≤ ξ E (50)

1 6 7

1 7 A

B

D

E

Figure 4 -- Load sharing factor -- pinion driving

For unmodified tooth profiles If there is no tip or root relief (see figure 3): XΓ = 1 + 1 3 3 i



ξ i − ξ A ξB − ξA



for ξ A ≤ ξ i < ξ B (45)

X Γ = 1 for ξ B ≤ ξ i ≤ ξ D i

XΓ = 1 + 1 3 3 i



ξ E − ξ  i ξ E − ξD



(46)

for ξ D < ξ i ≤ ξ E (47)

If adequate tip and root relief is designed for high load capacity, and if the pinion is driven by the gear (see figure 5): XΓ = 1 + 6 7 7 i



ξ i − ξ A ξB − ξA



for ξ A ≤ ξ i < ξ B (51)

X Γ = 1 for ξ B ≤ ξ i ≤ ξ D i XΓ = 6 7 i



ξE − ξ i ξE − ξD



(52)

for ξ D < ξ i ≤ ξE (53)

1 1

2 3

6 7

1 3

A

B

D

E

Figure 3 -- Load sharing factor -- unmodified profiles 8

1 7 A

B

D

E

Figure 5 -- Load sharing factor -- gear driving

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For smooth meshing

5 Lubrication

If adequate tip and root relief is designed for smooth meshing (see figure 6):

5.1 Viscometric information

XΓ = i

ξ  i − ξ A ξB − ξA

(54)

for ξ A ≤ ξ i < ξ B

X Γ = 1 for ξ B ≤ ξ i ≤ ξ D i XΓ = i

ξ E − ξ i ξE − ξ D

(55) (56)

for ξ D < ξ i ≤ ξ E

1

A

B

D

Lubricants are commonly referred to by their base type, for example mineral or synthetic, and their viscosity, usually in relation to a defined viscosity grade. Viscosity is one of the basic and very important properties of a lubricant and is used extensively in tribological calculations. Viscosity is a bulk property of a fluid, semi--fluid or semi--solid substance that causes it to resist flow. In addition to the basic composition and structure of the material, viscosity decreases with increasing temperature and increases with increasing pressure. For a liquid under shear, the rate of deformation or shear rate is proportional to the shearing stress. This relationship is Newton’s law, which essentially states that the ratio of the stress to the shear rate is a constant. That constant is viscosity. Dynamic viscosity, η, sometimes referred to as absolute viscosity, is defined by equation 59.

E

Figure 6 -- Load sharing factor -- smooth meshing

η = 10 9 τ dv dh

 

(59)

where: 4.4 Hertzian contact band The semi--width of the rectangular contact band is given by: 0.5

8 XΓi wn Ãni b H =   i  π Er  i

Ãn

i

Er

τ

is shear stress, N/mm2;

v

is velocity, m/s;

h

is thickness of an element measured perpendicular to the flow, m;

dv is dh

known as the rate of shear [s --1] and

sometimes listed as γ.

is load sharing factor (see 4.3); is normal unit load, N/mm (see equation 44);

wn

is dynamic viscosity, mPa•s;

(57)

where XΓ

η

is normal relative radius of curvature, mm (see equation 32); is reduced modulus of elasticity given by:



Er = 2

1 − ν 21 E1

+



1 − ν 22 E2

−1

(58)

where ν1, ν2 is Poisson’s ratio (pinion, gear); E1, E2 is modulus of elasticity, gear).

N/mm2

(pinion,

Lubricants used in industry today, however, have their viscosity measured by capillary viscometers which provide a kinematic viscosity. Kinematic viscosity, ν, is the ratio of dynamic viscosity, η, to the density, ρ, at a specified temperature and pressure (see equation 60).



η ν = 10 3 Ã

(60)

where: ν

is kinematic viscosity, mm2/s;

ρ

is density, kg/m3.

ASTM D445 [9] is the most widely used method for measuring the kinematic viscosity of lubricants for many different applications. The most commonly

9

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used temperatures are 40°C and 100°C; these measurements are generally made at atmospheric pressure.

then

Under Newton’s law, viscosity is independent of shear rate. Fluids such as these are referred to as Newtonian fluids. Most conventional, single grade lubricants made with a relatively low molecular weight base stock (non--polymeric) are considered Newtonian fluids. However, there are fluids that do not exhibit this ideal behavior because their viscosity is not independent of shear rate. These are usually finished blends containing higher molecular weight polymers (viscosity modifiers or viscosity index improvers, as well as pour point depressants) that are sensitive to shear rate. Some exhibit shear thinning, whereas others result in shear thickening. It is more common in gear lubricant applications to find shear thinning relationships due to the nature of the polymers typically used in these formulations. This shear thinning translates into lower effective viscosities in the contact region under operation than might be expected from a non--polymer blend of similar viscosity.

Another common numeric designation that provides information about the viscosity--temperature relationship of a fluid is the viscosity index or VI. The viscosity index of a fluid can be calculated by ASTM method D2270 [11]. This arbitrary measure gives a relative viscosity--temperature sensitivity for a given oil. The higher the value the less change in viscosity with temperature.

Lubricant viscosity varies inversely with temperature. A truly ideal fluid would have a viscosity that is constant over all temperature. ASTM method D341 [10] can be used to obtain the viscosity--temperature relationship. A simplified form can be used to estimate the kinematic viscosity of a fluid at a given temperature if there is some viscometric information available for the fluid at two other temperatures (see equation 61). (61)

where: T

is absolute temperature, K;

ν

is kinematic viscosity, mm2/s;

A, B are dimensionless constants.

10

log 10 log 10ν 40 + 0.7 − log 10 log 10ν 100 + 0.7 log 10(373.15) − log10(313.15)

5.1.2 Viscosity--pressure relationship Equally important to temperature on the fluid viscosity is the pressure acting on it. This is especially important in highly loaded contacts such as gears and rolling element bearings where pressures can easily exceed 1 GPa. The viscosity of lubricant trapped in a concentrated contact increases exponentially with pressure. In 1893, C. Barus established an empirical equation to describe the isothermal viscosity--pressure relationship for a given liquid as shown in equation 64. (64)

where ηP

is viscosity at pressure, p, mPa•s;

ηatm is viscosity mPa•s; α

at

atmospheric

pressure,

is pressure--viscosity coefficient, mm2/N.

Today the model continues to be refined. So and Klaus [12] provided a comparison of the many models developed since the Barus equation was first introduced. The continued research aided by the development of high pressure rheology techniques to generate empirical information have shown that the viscosity--pressure response of a fluid is also related to its chemical structure [13, 14, 15]. This can have a profound effect on the film forming capabilities of the fluid in question and the overall life of the component involved. 5.2 Film thickness equation

A and B can be determined by solving equation 61 simultaneously with equations 62 and 63, using the kinematic viscosity of the fluid measured at standard temperatures of 40°C and 100°C. B=

(63)

η P = η atm e α p

5.1.1 Viscosity – temperature relationship

log 10 log 10(ν + 0.7) = A − B × log 10 T

A = log 10 log 10ν 40 + 0.7 + B × log 10(313.15)

(62)

Dowson, Higginson and Toyoda have authored various papers on EHL film thickness [16, 17, 18, 19]. The film thickness equations given in these papers account for the exponential increase of lubricant viscosity with pressure, tooth geometry, velocity of the gear teeth, material elastic properties and the transmitted load. The film thickness determines the operating regime of the gearset and

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has been found to be a useful index of wear related distress probability. Wellauer and Holloway [20] also found that specific film thickness could be correlated with the probability of tooth surface distress. The Dowson and Toyoda [19] equation for line contact central EHL film thickness will be used as shown below.

η1

is dynamic viscosity at temperature θ1, mPa•s;

η2

is dynamic viscosity at temperature θ2, mPa•s;

θ1

is temperature at which η1 was determined, °C;

Dimensionless central film thickness:

θ2

is temperature at which η2 was determined, °C.

H c = 3.06

G 0.56U 0.69 

 i

W  i

i 0.10

(65)

where (i)

d= (as a subscript) defines a point on the line of action,

and the dimensionless parameters G, U(i) and W(i) are defined below:

speed parameter, U(i) ηM ve

 i

2E r à n

× 10 −6



log 10η 40 + 0.9

− d log 10 θ 1 + 273.15)

(71)

(72)





c = log 10 log 10η 40 + 0.9 − 2.495752 d (68)

 i

is dynamic viscosity at the gear tooth temperature, mPa•s.

η M = 10 g − 0.9

(69)

where d

is tooth temperature, °C (see 6.3).

The parameters c and d required for calculating ηM can either be taken from table 2 or calculated with equations 70 and 72, respectively. Equations 70 and 72, derived from a modification of the Walther equation [10], will yield the parameters c and d if two dynamic viscosities, η1 and η2, are known at two corresponding temperatures, θ1 and θ2. Since dynamic viscosity is generally available at 40°C and 100°C, equations 70 and 72 are modified in equations 71 and 73 to incorporate terms corresponding to those temperatures.

(73)

is pressure--viscosity coefficient, mm2/N. Values range from 0.725 ¢ 10 --2 mm2/N to 2.9 ¢ 10 --2 mm2/N for typical gear lubricants. Values for pressure--viscosity coefficients vs. dynamic viscosity can be obtained from equation 74.

α

where

g = 10 cθ M + 273.15



log 10η 100 + 0.9

when θ1 = 40°C and θ2 = 100°C,

 i

Er Ãn

(70)

when θ1 = 40°C and θ2 = 100°C,

(67)

i

XΓ wn

θM



θ +273.15 log 10 2 θ 1+273.15

c = log 10 log 10η 1 + 0.9

load parameter, W(i)

ηM



log 10η 1+0.9

(66)

G = α Er

W i =



log 10η 2+0.9

d = 13.13525 log 10

materials parameter, G

U  i =



log 10

α = k η sM

(74)

Table 2 contains viscosity information for mineral oils, MIL--L spec. oils, polyalphaolefin (PAO) based synthetic oils (which contain ester) and polyalkylene glycol (PAG) based synthetic oils, as well as constants c, d, k and s for use in the equations 69 through 74. These values were obtained from the data shown in figures 7 through 11 [22]. It is important that the film thickness is calculated with values of viscosity and pressure--viscosity coefficient for the gear tooth temperature, θM, (see 6.3). The central film thickness at a given point is: h c = H c à n × 10 3  i

i

 i

(75)

(see clause 4 for à n ). i

11

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Table 2 -- Data for determining viscosity and pressure--viscosity coefficient Lubricant Mineral oil

PAO -- based synthetic non-VI improved oil

PAG -- based synthetic2)

MIL--L--7808K Grade 3 MIL--L--7808K Grade 4 MIL--L--23699E

ISO VG1) 32 46 68 100 150 220 320 460 680 1000 1500 2200 3200 150 220 320 460 680 1000 1500 2200 3200 6800 100 150 220 320 460 680 1000

η40 27.17816 39.35879 58.64514 86.91484 131.4335 194.2414 284.6312 412.0824 613.8288 909.4836 1374.931 2031.417 2975.954 128.5772 189.9828 278.3370 402.8943 600.0179 868.1710 1310.350 1933.070 2827.726 6077.362 102.630 153.950 225.790 328.430 472.130 697.920 1026.37

η100 4.294182 5.440514 7.059163 9.251199 12.27588 15.98296 20.60709 26.34104 34.24003 38.56783 49.58728 62.69805 78.56109 16.17971 21.60933 28.66405 37.54020 53.20423 68.60767 91.03300 118.0509 151.2132 244.5559 19.560 27.380 40.090 56.710 77.250 113.43 163.30

c 10.20076 10.07933 9.90355 9.65708 9.42526 9.24059 9.09300 8.96420 8.84572 9.25943 9.19946 9.15646 9.13012 7.99428 7.79927 7.63035 7.49799 7.16434 7.12008 7.07678 7.06113 7.06594 7.11907 6.42534 6.19586 5.76552 5.49394 5.35027 5.06011 4.85075

d --4.02279 --3.95628 --3.86833 --3.75377 --3.64563 --3.55832 --3.48706 --3.42445 --3.36585 --3.52128 --3.48702 --3.46064 --3.44157 --3.07304 --2.98154 --2.90169 --2.83762 --2.69277 --2.66528 --2.63766 --2.62221 --2.61561 --2.62091 --2.45259 --2.34616 --2.16105 --2.04065 --1.97254 --1.84558 --1.75175

k 0.010471 0.010471 0.010471 0.010471 0.010471 0.010471 0.010471 0.010471 0.010471 0.010471 0.010471 0.010471 0.010471 0.010326 0.010326 0.010326 0.010326 0.010326 0.010326 0.010326 0.010326 0.010326 0.010326 0.0047 0.0047 0.0047 0.0047 0.0047 0.0047 0.0047

s 0.1348 0.1348 0.1348 0.1348 0.1348 0.1348 0.1348 0.1348 0.1348 0.1348 0.1348 0.1348 0.1348 0.0507 0.0507 0.0507 0.0507 0.0507 0.0507 0.0507 0.0507 0.0507 0.0507 0.1572 0.1572 0.1572 0.1572 0.1572 0.1572 0.1572

12

11.35364

2.701402

9.58596

--3.82619

0.005492

0.25472

17

16.09154

3.609883

9.08217

--3.60300

0.005492

0.25472

23

22.56448

4.591235

8.91638

--3.51779

0.006515

0.16530

NOTES: 1) ν (mm2/s) 40 2) Copolymer of ethylene oxide and propylene oxide in 50% weight ratio.

The specific film thickness is the ratio of film thickness divided by the composite roughness of the contacting gear teeth and can be used to assess performance. To determine this ratio, the cutoff wavelength for the composite surface roughness measurement (σx) should be comparable to the width of the Hertzian contact, 2b H . This results in σx becoming σ 2b as  i

12

H  i

shown in equation 76. hc λ 2b

H  i

=σ

2b

 i

(76)

H  i

This may not be practical because many surface measuring instruments have a fixed cutoff wavelength (usually 0.8 mm).

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1 000 000 ISO VG 3200 2200 100 000

1500 1000 680 460

Dynamic viscosity (mPa⋅s)

10 000

320 220 150 100

1000

68 46 32

100

10

1 200

250

300

350

400

450

500

Temperature (K)

Figure 7 -- Dynamic viscosity versus temperature for mineral oils

Following the concepts in [21], equation 76 can be approximated by:

λ 2b

H  i

=

0.5

 Lx  σ x 2b   Hi

hc

i

σ x = Ra 21x + Ra 22x

0.5

(77)

(78)

where λ 2b

H  i

is specific film thickness at point i with a filter cutoff wavelength of 2bH;

Lx

is filter cutoff wavelength used in measuring surface roughness, mm. Any cutoff length, Lx, can be used (for example, L0.8 = 0.8 mm cutoff);

13

AGMA 925--A03

σx

AMERICAN GEAR MANUFACTURERS ASSOCIATION

is composite surface roughness for filter cutoff wavelength Lx, mm;

Ra1x is pinion average surface roughness for Lx, mm;

Rq 2x ∝ L x where

Rqx2 is variance or square of the root mean square roughness, mm.

Ra2x is gear average surface roughness for Lx, mm.

also [25]:

Use of the radical term in equation 77 for roughness adjustment is developed below.

Ra x =

From Gaussian statistics [24], it is seen that:

(79)

2π Rqx

(80)

From equations 79 and 80: Ra x ∝ L 0.5 x

(81)

1 000 000 ISO VG 6800 100 000

3200 2200 1500 1000 680

Dynamic viscosity (mPa⋅s)

10 000

460 320 220 150

1000

100

10

1 200

300 350 400 450 500 Temperature (K) Figure 8 -- Dynamic viscosity versus temperature for PAO--based synthetic non--VI--improved oils

14

250

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Hence, for a 0.8 mm cutoff length,

Ra 2b

H  i

= Ra 0.8

σ 0.8

0.5

 i  L0.8    2b H

= Ra



0.5

2 2 10.8 + Ra 2 0.8

yields equation 83

which is equation 77 developed for a 0.8 mm cutoff length.

(82)

Substitute equation 82 into equation 78 once each for Ra1x and for Ra2x to obtain σ 2b . H  i Using this in equation 76, noting that

0.5

 i L λ 2b = σ  0.8  0.8 2b H H  i  i hc

(83)

1010000000 000 000

1000000 1 000 000

Dynamic viscosity (mPa⋅s)

100 000 100000

1010000 000

1000 1000

100 100 ISO VG 1000 680 460 320 220 150 100

10 10

11 200 225 250 275 300 325 350 375 400 425 450 475 500 Temperature (K) Figure 9 -- Dynamic viscosity versus temperature for PAG--based synthetic oils

15

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

1000

MIL--L--23699E MIL--L--7808K Grade 4

100 Dynamic viscosity (mPa⋅s)

MIL--L--7808K Grade 3

10

1

0.1

200

250

300 350 Temperature (K)

400

450

500

Figure 10 -- Dynamic viscosity versus temperature for MIL Spec. oils

Pressure--viscosity coefficient (mm2/N)

1

Mineral oil MIL--L--7808K MIL--L--23699E Synthetic oil (PAO) Synthetic oil (PAG)

0.1

0.01

0.001

0.1

1

10 100 1000 Dynamic viscosity (mPa⋅s)

10 000

100 000 1 000 000

Figure 11 -- Pressure--viscosity coefficient versus dynamic viscosity 16

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6 Scuffing 6.1 General The term scuffing as used in this information sheet is defined as localized damage caused by solid--phase welding between surfaces in relative motion. It is accompanied by transfer of metal from one surface to another due to welding and subsequent tearing, and may occur in any highly loaded contact where the oil film is too thin to adequately separate the surfaces. Scuffing appears as a matte, rough finish due to the microscopic tearing at the surface. It occurs most commonly at extreme end regions of the contact path or near points of single tooth contact. Scuffing is also known generically as severe adhesive wear. Scoring was a term commonly used in the U.S. to describe the same phenomenon now defined as scuffing (welding and tearing of mating surfaces). See ANSI/AGMA 1010--E95 or ISO 10825:1995. 6.1.1 Mechanism of scuffing The basic mechanism of scuffing is caused by intense frictional heat generated by a combination of high sliding velocity and high contact stress. Scuffing occurs under thin film, boundary lubrication conditions and can be affected by physical and chemical properties of the lubricant, nature of the oxide films, and gear material. When gear teeth are separated by a thick lubricant film, contact between surface asperities is minimized and there is usually no scuffing. As lubricant film thickness decreases, asperity contact increases and scuffing becomes more probable. A very thin film, such as in boundary lubrication, together with a high contact temperature suggests a high probability of scuffing is possible in the absence of antiscuff additives in the lubricant. 6.1.2 Probability of scuffing Blok’s [1] contact temperature theory states that scuffing will occur in gear teeth that are sliding under boundary--lubricated conditions, when the maximum contact temperature reaches a critical magnitude. The contact temperature is the sum of two components: the flash temperature and the tooth temperature. See 6.4. Scuffing most commonly occurs at one of the two extreme end regions of the contact path or near the points of single tooth contact.

AGMA 925--A03

Prediction of the probability of scuffing is possible by comparing the calculated contact temperature with limiting scuffing temperature. The limiting scuffing temperature can be calculated from an appropriate gear scuffing test, or can be provided by field investigations. For non--additive mineral oils, each combination of oil and gear materials has a limiting scuffing temperature that is constant regardless of the operating conditions. It is believed that the limiting scuffing temperature is not constant for synthetic and high--additive EP lubricants, and it must be determined from tests that closely simulate the operating condition of the gearset. 6.2 Flash temperature The flash temperature is the calculated increase in gear tooth surface temperature at a given point along the line of action resulting from the combined effects of gear tooth geometry, load, friction, velocity and material properties during operation. 6.2.1 Fundamental temperature, θ fl

formula

for

flash

i

The fundamental formula is based on Blok’s [1] equation. XΓ wn θ fl = 31.62 K m m  i

 i

 i

  bH

×

  

 i

0.5

  

v r1 − v r2  i  i

B M1 v r1  i

0.5

0.5

(84)

+ B M2 v r2  i

where is 0.80, numerical factor valid for a semi-elliptic (Hertzian) distribution of frictional heat over the instantaneous width, 2 bH, of the rectangular contact band;

K

mm XΓ

 i

i

is load sharing factor (see 4.3); is normal unit load, N/mm (see equation 44);

wn v r1

is mean coefficient of friction (see 6.2.2);

 i

is rolling tangential velocity of the pinion, m/s (see equation 36);

17

AGMA 925--A03

v r2

 i

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is rolling tangential velocity of the gear, m/s (see equation 37);

BM1 is thermal contact coefficient of the pinion material, N/[mm s0.5K] (see 6.2.3); BM2 is thermal contact coefficient of the gear material, N/[mm s0.5K] (see 6.2.3); bH

 i

i

is semi--width of Hertzian contact band, mm

The surface roughness is taken as an average of the average values: Ra + Ra 2x (87) R avgx = 1x 2 where

(see equation 57);

Ra1x is pinion average surface roughness for filter cutoff length, Lx, mm;

(as a subscript) defines a point on the line of action.

Ra2x is gear average surface roughness for filter cutoff length, Lx, mm.

In this equation, the coefficient of friction may be approximated by different expressions, for instance as proposed by Kelley [2, 4] and AGMA 217.01 [7]. The influence of surface roughness is incorporated in the approximation of the coefficient of friction. 6.2.2 Mean coefficient of friction, m m

i 

The mean coefficient of friction is an approximation of the actual coefficient of friction on the tooth flank, which is an instantaneous and local value depending on several properties of the oil, surface roughness, lay of the surface irregularities like grinding marks, material properties, tangential velocities, forces and dimensions. Three methods may be used to determine the value of m m to be used in equation 84.  i

-- input a value based upon experience, which is a constant; -- input a value from equation 85, which is also a constant; -- input a value from equation 88, which varies along the line of action. 6.2.2.1 Approximation by a constant A constant coefficient of friction along the line of action has been assumed by AGMA 217.01 [7] and Kelley [2]: m m = m m const = 0.06 × C R  i avg

x

(85)

The surface roughness constant, C R , is limited avgx to a maximum value of 3.0: 1.0 ≤ C R

avgx

=

1.13 ≤ 3.0 1.13 − R avgx

(86)

Equation 85 gives a typical value for gears operating in the partial EHL regime. It may be too low for 18

boundary lubricated gears where mm may be higher than 0.2, or too high for gears operating in the full--film regime where mm may be less than 0.01.

6.2.2.2 Empirical equation An empirical equation for a variable coefficient of friction is the Benedict and Kelley [5] equation, supplemented with the influence of roughness:

29 700 XΓiwn log 10 m m = 0.0127 C R  2  i avgx  ηMvsivei 

(88)

where the surface roughness expression is taken in accordance with equations 86 and 87. Equation 88 is not valid at or near the operating pitch point, as vs goes to zero. where ηM

is dynamic viscosity of the oil at gear tooth temperature, θM, mPa•s;

vs

is sliding velocity, m/s (see equation 38);

i

ve

i

is entraining velocity, m/s (see equation 39).

6.2.3 Thermal contact coefficient, BM The thermal contact coefficient accounts for the influence of the material properties of pinion and gear: B M1 = λ M1 × Ã M1 × c M1

0.5

B M2 = λ M2 × Ã M2 × c M2

0.5

(89) (90)

For martensitic steels the range of heat conductivity, λM , is 41 to 52 N/[s K] and the product of density times the specific heat per unit mass, ρM ¢ cM is about 3.8 N/[mm2K], so that the use of the average value BM = 13.6 N/[mm s0.5 K] for such steels will not introduce a large error when the thermal contact coefficient is unknown. 6.2.4 Maximum flash temperature To locate and determine the maximum flash temperature, the flash temperature should be calculated

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AGMA 925--A03

at a sufficient number of points (for example, 25 to 50) on the line of action. Calculate flash temperatures at points between SAP and LPSTC during double tooth contact, at LPSTC and HPSTC for single tooth contact, and between HPSTC and EAP during double tooth contact.

tion, an accurate value of the gear tooth temperature be used for the analysis.

If the contact temperature (see 6.4) is greater than the mean scuffing temperature (see 6.5) for the lubricant being used, there is a potential risk for scuffing (see 6.5.5).

6.3.3 Thermal network

6.3 Tooth temperature The tooth temperature, θM, is the equilibrium temperature of the surface of the gear teeth before they enter the contact zone. In some cases [26], the tooth temperature may be significantly higher than the temperature of the oil supplied to the gear mesh. 6.3.1 Rough approximation For a very rough approximation, the tooth temperature may be estimated by the sum of the oil temperature, taking into account some impediment in heat transfer for spray lubrication if applicable, and a portion that depends mainly on the flash temperature, for which the maximum value is taken: θ M = k sump θ oil + 0.56 θ fl max where

(91)

The tooth temperature can be measured by testing, or determined according to experience.

The tooth temperature can be calculated from a thermal network analysis [43] (see figure 12). The tooth temperature is determined by the heat flow balance in the gearbox. There are several sources of frictional heat, of which the most important ones are the tooth friction and the bearing friction. Other heat sources, like seals and oil flow, may also contribute. For gear pitchline velocities above 80 m/s, churning loss, expulsion of oil between meshing teeth, and windage loss become important heat sources that should be considered. Heat is conducted and transferred to the environment by conduction, convection and radiation. 6.4 Contact temperature 6.4.1 Contact temperature at any point At any point on the line of action (see figure 13) the contact temperature is: (92)

θ B = θ M + θ fl  i i

ksump = 1.0 if splash lube; 1.2 if spray lube; θoil

6.3.2 Measurement and experience

where

is oil supply or sump temperature, ° C;

θfl max is maximum flash temperature, ° C, see 6.2. However, for a reliable evaluation of the scuffing risk, it is important that instead of the rough approxima-

θM

is tooth temperature, °C (see 6.3);

θ fl

is flash temperature, °C (see 6.2).

i

 i

(as a subscript) defines a point on the line of action.

Oil

Pinion

Case Friction power

Gear

Air

Shafts

Bearings

Friction power Figure 12 -- Example of thermal network

19

AGMA 925--A03

θB

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machines. The mean scuffing temperature was derived from data published by Blok [27].

θB max  i

θ fl

Equation 94 gives the scuffing temperature for non--antiscuff mineral oils (R&O in accordance with ANSI/AGMA 9005--E02 [28]).

θfl max  i

θ S = 63 + 33 ln ν 40 where ν40

θM

(94)

is kinematic viscosity at 40° C, mm2/s (table 2).

Equation 95 gives the scuffing temperature for antiscuff mineral oils (EP gear oil in accordance with ANSI/AGMA 9005--E02). A

B

C

D

E

Figure 13 -- Contact temperature along the line of action

6.4.2 Maximum contact temperature

(93)

where θfl max is maximum flash temperature, °C (see 6.2). 6.5 Scuffing temperature The scuffing temperature is the temperature in the tooth contact zone at which scuffing is likely to occur with the chosen combination of lubricant and gear materials. The scuffing temperature is assumed to be a characteristic value for the material--lubricant system of a gear pair, to be determined by gear tests with the same material--lubricant system. When θB max (see figure 13) reaches the scuffing temperature of the system, scuffing is likely. The mean scuffing temperature is the temperature at which there is a 50% chance of scuffing. 6.5.1 Mean scuffing temperature for mineral oils Scuffing temperatures for mineral oils with low concentrations of antiscuff additives are independent of operating conditions. Viscosity grade is a convenient index of oil composition, and thus of scuffing temperature. Equations 94 and 95 are approximate guides for mineral oils and steels typical of IAE and FZG test 20

Table 3 gives the mean scuffing temperature for oils with steels typical of the aerospace industry. Table 3 -- Mean scuffing temperatures for oils and steels typical of the aerospace industry

The maximum contact temperature is: θ B max = θ M + θ fl max

(95) θ S = 118 + 33 ln ν 40 6.5.2 Mean scuffing temperature for oils and steels typical of aerospace industry

Lubricant MIL--L--7808 MIL--L--23699 DERD2487 DERD2497 DOD--L--85734 ISO VG 32 PAO DexronR II1)

Mean scuffing temperature, ° C 205 220 225 240 260 280 290

NOTE: 1) DexronR is a registered trademark of General Motors Corporation.

6.5.3 Extension of test gear scuffing temperature for one steel to other steels The scuffing temperature determined from test gears with low--additive mineral oils may be extended to different gear steels, heat treatments or surface treatments by introducing an empirical welding factor. θ S = X Wθ fl max, test + θ M, test

(96)

where XW

is welding factor (see table 4);

θfl max, test is maximum flash temperature of test gears, °C;

AMERICAN GEAR MANUFACTURERS ASSOCIATION

is tooth temperature of test gears, °C.

θM, test

6.5.4 Scuffing temperature for oils used in hypoid gear application Scuffing temperature for high--additive oils (hypoid gear oil) may be dependent on operating conditions. Therefore, the scuffing temperature should be obtained from tests that closely simulate operating conditions of the gears. Table 4 -- Welding factors, XW Material Through hardened steel Phosphated steel Copper--plated steel Bath or gas nitrided steel Hardened carburized steel -- Less than 20% retained austenite -- 20 to 30% retained austenite -- Greater than 30% retained austenite Austenite steel (stainless steel)

XW 1.00 1.25 1.50 1.50 1.15 1.00 0.85 0.45

6.5.5 Scuffing risk Scuffing risk can be calculated from a Gaussian distribution of scuffing temperature about the mean value. Typically, the coefficient of variation is at least 15%. Therefore, use the procedure of annex B to calculate the probability of scuffing: where

proposed which may support the gear geometry and rotor dimensions most suitable to the gear application. Gear drives cover a wide field of operating conditions from relatively low pitch line velocities with high specific tooth loads, to very high pitch line velocities and moderate specific tooth loads. Lubricants vary, as well, between mineral oils with little or no additives to antiscuff lubricants with substantial additives. The flash temperature method described in 6.2 through 6.5 is based on Blok’s contact temperature theory. The flash temperature, θfl, must be added to the steady gear tooth temperature, θM, to give the total contact temperature, θB. The value of the contact temperature for every point in the contact zone must be less than the mean scuffing temperature of the material--lubricant system or scuffing may occur. 6.6.1 Integral temperature method The integral temperature method [29] has been proposed as an alternative to the flash temperature method by which the influence of the gear geometry imposes a critical energy level based on the integrated temperature distribution (for example, numerically integrating using Simpson’s rule) along a path of contact and adopting a steady gear tooth temperature. This method involves the calculation of a scuffing load basically independent of speed, but controlled by gear geometry. Application requires comparison of the proposed gearset based on a test rig result to a known test rig gearset and tested oil. A comparison of the flash temperature method and integral temperature method has shown the following:

y = θB max my = θs σy = 0.15 θs Table 5 gives the evaluation of scuffing risk based on the probability of scuffing [7]. Table 5 -- Scuffing risk Probability of scuffing <10% 10 to 30% >30%

AGMA 925--A03

Scuffing risk Low Moderate High

6.6 Alternative scuffing risk evaluation The calculation of the scuffing load capacity is a very complex problem. Several alternative methods are

-- Blok’s method and the integral temperature method give essentially the same assessment of scuffing risk for most gearsets; -- Blok’s method and the integral temperature method give different assessments of scuffing risk for those cases where there are local temperature peaks. These cases usually occur in gearsets that have low contact ratio, contact near the base circle, or other sensitive geometries; -- Blok’s method is sensitive to local temperature peaks because it is concerned with the maximum instantaneous temperature, whereas the integral temperature method is insensitive to these peaks because it averages the temperature distribution.

21

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

7 Surface fatigue (micro-- and macropitting)

6.6.2 Other scuffing methods 6.6.2.1 PVT Method

7.1 General information

Almen [30] popularized the PVT method for predicting scuffing where: P

is Hertzian pressure;

V

is sliding velocity;

T

is distance along line of action.

PVT was used during World War II by designers in automotive and aircraft industries. It worked well for a narrow range of gear designs, but was unreliable when extrapolated to other gear applications. 6.6.2.2 Borsoff scoring factor method Borsoff [31, 32, 33, 34] conducted many scuffing tests during the 1950’s and found scuffing resistance increased when test gears were run at high speeds. Borsoff introduced a scoring factor, Sf: 2b Sf = v H s

(97)

where Sf

is contact time, ms (sec ¢ 10 --3);

bH

is semi--width of Hertzian contact band, mm;

vs

is sliding velocity, m/s.

Sf is the time required for a point on one tooth to traverse the Hertzian band of the mating tooth. Borsoff’s test data showed a linear relationship between scuffing load and scoring factor, Sf. Borsoff recommended that a number of considerations should be made before using his method for specific applications. 6.6.2.3 Simplified scuffing criteria for high speed gears Annex B of ANSI/AGMA 6011--H98 [35] has been used to evaluate scuffing risk of high speed gear applications. There are other methods for evaluating scuffing of gear teeth not mentioned here. Other methods may also have application merit. Most importantly, the gear designer should recognize scuffing as a gear design criteria. 22

Surface fatigue, commonly referred to as pitting or spalling, is a wear mode that results in loss of material as a result of repeated stress cycles acting on the surface. There are two major sub--groups under surface fatigue known as micro-- and macropitting. As their names imply, the type of pitting is related to the size of the pit. Macropits usually can be seen with the naked eye as irregular shaped cavities in the surface of the tooth. Damage beginning on the order of 0.5 to 1.0 mm in diameter is considered to be a macropit. The number of stress cycles occurring before failure is referred to as the fatigue life of the component. The surface fatigue life of a gear is inversely proportional to the contact stress applied. Although contact stress is probably the major factor governing life, there are many others that influence life. These include design factors such as tip relief and crowning, surface roughness, physical and chemical properties of the lubricant and its additive system, and external contaminants such as water and hard particulate matter. 7.2 Micropitting Micropitting is a fatigue phenomenon that occurs in Hertzian contacts that operate in elastohydrodynamic or boundary lubrication regimes and have combined rolling and sliding. Besides operating conditions such as load, speed, sliding, temperature and specific film thickness, the chemical composition of a lubricant strongly influences micropitting. Damage can start during the first 105 to 106 stress cycles with generation of numerous surface cracks. The cracks grow at a shallow angle to the surface forming micropits that are about 10 – 20 mm deep by about 25 -- 100 mm long and 10 – 20 mm wide. The micropits coalesce to produce a continuous fractured surface which appears as a dull, matte surface to the observer. Micropitting is the preferred name for this mode of damage, but it has also been referred to as grey staining, grey flecking, frosting, and peeling. Although micropitting generally occurs with heavily loaded, carburized gears, it also occurs with nitrided, induction hardened and through--hardened gears. Micropitting may arrest after running--in. If micropitting continues to progress, however, it may result in

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reduced gear tooth accuracy, increased dynamic loads and noise. Eventually, it can progress to macropitting and gear failure. 7.2.1 Micropitting risk evaluation Factors that influence micropitting are gear tooth geometry, surface roughness, lubricant viscosity, coefficient of friction, load, tangential speed, oil temperature and lubricant additives. Common methods suggested for reducing the probability of micropitting include:

AGMA 925--A03

turning or branching back to the surface. Eventually, material will dislodge from the surface forming a pit, an irregular shaped cavity in the surface of the material. With gears the origin of the crack is more likely surface initiated because lubricant film thickness is low resulting in a high amount of asperity or metal--to--metal contact. For high--speed gears with smooth surface finishes, film thickness is larger and sub--surface initiated crack formation may dominate. In these cases an inclusion or small void in the material is a source for stress concentration.

--

reduce surface roughness;

--

increase film thickness;

--

use higher viscosity oil;

--

reduce coefficient of friction;

--

run at higher speeds if possible;

Laboratory testing commonly uses a 1% limit on tooth surface area damage as a criteria to stop a test. However, for field service applications one should always abide by the equipment manufacturer’s recommendations or guidelines for acceptable limits of damage to any gear or supporting component.

--

reduce oil temperature;

7.4 Regimes of lubrication

-- use additives with demonstrated micropitting resistance; -- protect gear teeth during run--in with suitable coatings, such as manganese phosphate, copper or silver plating. CAUTION: Silver or copper plating of carburized gear elements will cause hydrogen embrittlement, which could result in a reduction in bending strength and fatigue life. Thermal treatment shortly after plating may reduce this effect.

Surface roughness strongly influences the tendency to micropit. Gears finished to a mirrorlike finish have been reported to eliminate micropitting [36, 37, 38]. Gear teeth have maximum micropitting resistance when the teeth of the high speed member are harder than the mating teeth and are as smooth as possible [39]. Currently there is no standard test for determining micropitting resistance of lubricants. However, FVA Information Sheet 54/IV describes a test that uses the FZG C--GF type gears to rank micropitting performance of oils [40]. At present, the influence of lubricant additives is unresolved. Therefore, the micropitting resistance of a lubricant should be determined by field testing on actual gears or by laboratory tests. 7.3 Macropitting Macropitting is also a fatigue phenomenon. Cracks can initiate either at or near the surface of a gear tooth. The crack usually propagates for a short distance at a shallow angle to the surface before

7.4.1 Introduction to regimes of lubrication Gear rating standards have progressed and been refined to take into account many of the major variables that affect gear life. With respect to calculated stress numbers, variables such as load distribution, internally induced dynamic loading and externally induced dynamic loading are accounted for by derating factors. Variables such as material quality, cycle life and reliability are accounted for by allowable stress numbers, stress cycle factors and reliability factors. Along with these influences, it has been recognized that adequate lubrication is necessary for gears to realize their calculated capacity. Indeed, AGMA gearing standards have acknowledged this fact by stating this need as a requirement in order to apply the various rating methods. Much of the groundwork for lubrication theory came about in the 1960’s and 1970’s. This period saw the advent and proliferation of jet travel, space travel, advanced manufacturing processes and advanced power needs. These technological and industrial developments led to the need for better gear rating methods which, in turn, resulted in rapid progress in industrial, vehicle and aerospace gearing standards. High speed gearing was coming into greater use, but it was not as well understood as the industry would have liked. To compensate, designs tended to focus on making higher speed stages of gearing more successful, sometimes to the detriment of slower speed stages. This is how the gearing industry

23

AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

started to get its first glimpses into the importance of lubrication on the life of gearing.

defined the three regimes of lubrication at the operating pitch diameter as follows:

It was not uncommon to see a three (3) stage industrial gear drive with problems as follows: a high speed set of gears that looked relatively undamaged, an intermediate speed set of gears that was experiencing initial pitting, and a slow speed set of gears that was experiencing advanced pitting and tooth breakage. In the event that all three stages were designed to have similar load intensity factors (K--factors and unit loads) the problem could be particularly puzzling. Rating theory at the time indicated that with all other things equal, the higher speed stages of gearing should have been failing sooner than the lower speed stages, due to greater stress cycles.

-- Regime III: Full EHL oil film is developed and separates the asperities of gear flanks in motion relative to one another;

At issue was the tribological condition between surfaces of two mating teeth. Elastohydrodynamic lubrication (EHL) theory showed that factors like relative surface velocity and local oil viscosity at the contact area directly affected thickness of the EHL oil film that separated asperities on surfaces of two mating gear teeth. For a multiple stage gear reducer, higher speed stages of gearing, with higher surface velocities, tended to produce thicker EHL oil films, better capable of separating asperities on mating teeth. Lower speed stages, with lower surface velocities, tended to produce thinner EHL oil films, less capable of separating asperities on mating teeth. Through the years, a great many researchers and companies inside and outside of the gear industry have sought to quantify the effects of EHL oil film theory on the life of gearing. There are many ways in which one could hypothesize the effects of inadequate oil films on degradation of gear tooth surfaces and its results on the life of gearing. Indeed, a comprehensive treatment of this subject could fill many volumes. Added to this is the fact that this is still a very active area of gear research. With this in mind, it is still useful to put forth a simplified description of how inadequate oil films can lead to decreased life of gears. So, very simply put, thinner oil films lead to a greater chance of more frequent and more detrimental degree of contact between asperities on mating gear teeth. The more severe this is, the more likely it will lead to pitting, a recognized form of surface fatigue in gearing. The effects of this phenomenon on the fatigue life of gearing were introduced by Bowen [41]. Dudley [42] 24

-- Regime II: Partial EHL oil film is developed and there is occasional contact of the asperities of gear flanks in motion relative to one another; -- Regime I: Only boundary lubrication exists with essentially no EHL film and contact of the asperities of gear flanks in motion relative to one another is pronounced. The implementation of this theory involves what is currently referred to as the stress cycle factor for the surface durability of gears, ZN, (this used to be called the life factor for surface durability). Keeping in mind that regime of lubrication depends ultimately on the degree of separation between asperities, Dudley proposed that the effect could be quantified by making proper adjustments to the curves that determine the stress cycle factor. Thus, we have as follows: 7.4.2 Regime III This regime of lubrication, characterized by full EHL oil film development, occurs mainly when gears have relatively high pitch line velocity, good care is taken to ensure that an adequate supply of clean, cool oil is available (of adequate viscosity and formulation), and good surface finishes are achieved on the gearing. As such, aerospace gearing, high speed marine gearing, and good quality industrial gear drives tend to have gears that operate within regime III. Thus, stress cycle factor curves that appear in standards for these gears are the basis for rating gears that operate within regime III. 7.4.3 Regime II This regime of lubrication, characterized by partial EHL oil film development, occurs mainly when gears have moderate pitch line velocities, moderate care is taken to ensure that an adequate supply of clean, cool oil is available (of adequate viscosity and formulation), and moderately good surface finishes are achieved on the gearing. As such, vehicle gearing is very characteristic of gears that operate within regime II. Dudley uses information from the stress cycle factor curves in vehicle standards to create a branch from the regime III curve for cycles greater than 100 000. It is felt that effects of operation within regime II on fatigue life will not begin to be realized until this point in the life of a gear.

AMERICAN GEAR MANUFACTURERS ASSOCIATION

7.4.4 Regime I This regime of lubrication, characterized by boundary lubrication, occurs mainly when gears have low pitch line velocities, little care is taken to ensure that an adequate supply of clean, cool oil is available (of adequate viscosity and formulation), and relatively rough surface finishes are achieved on the gearing. Many types of gearing can fall into this range of operation, including all types mentioned above. Dudley used fatigue curves generated for ball and roller bearings as a basis for regime I stress cycle factor curves. These curves, first developed in the 1940’s, indicated that with a ten--fold increase in

AGMA 925--A03

cycles, load capacity of a bearing drops off by a factor of 2.0. Thus, a stress curve for Hertzian contact would drop off by about a factor of 1.41 (square root of 2.0). Bearings back in the 1940’s commonly had surface finishes and oil films very analogous to gears operating in regime I. This information is used to create a branch from the regime III curve at cycles greater than 100 000. Figure 14 shows the curves that result from Dudley’s method of regimes of lubrication. Below, the method is described in fuller detail and calculations are given to show how one assesses which regime of lubrication should be applied to a given set of gears.

4.00 3.00 2.00

Stress cycle factor, ZN

1.50 1.00 0.90 0.80 0.70 0.60 0.50

Regime III Regime II

0.40 0.30

Regime I

0.20 0.15 0.10 0.09 0.08 0.07 0.06 0.05

102

103

104

105

106

107

108

Number of load cycles, N

109

1010

1011

1012

Figure 14 -- Plot of regimes of lubrication versus stress cycle factor

Table 6 -- Stress cycle factor equations for regimes I, II and III Regime of lubrication Regime III Regime II Regime I

Stress cycle factor for surface durability Z N = 1.47 for N < 10 000 cycles Z N = 2.46604 × N −0.056

for N ≥ 10 000 cycles

ZN =

3.83441 × N −0.094

for N ≥ 100 000 cycles

ZN =

7.82078 × N −0.156

for N ≥ 100 000 cycles

25

AGMA 925--A03

7.5 Estimating life with respect to surface durability After calculating the minimum EHL film thickness based on 5.2, one must calculate the specific film thickness. In figure 14, specific film thicknesses greater than or equal to 1.0 indicate the beginning of regime III and the end of regime II lubrication. Specific film thicknesses between 0.4 and 1.0 indicate operation within regime II and specific film thicknesses less than or equal to 0.4 indicate regime I. Once the regime of lubrication is determined, one can calculate the stress cycle factor, ZN, shown in figure 14. ZN is used to calculate gear rating in ANSI/AGMA 2101--C95.

8 Wear Wear is a term describing change to a gear tooth surface involving removal or displacement of material, due to mechanical, chemical or electrical action. In the boundary lubrication regime, some wear is inevitable. Many gears, because of practical limits on lubricant viscosity, speed and temperature, must operate under boundary lubricated conditions. Mild wear occurs during running--in and usually subsides with time, resulting in a tolerable wear rate and a satisfactory lifetime for the gearset. Wear that occurs during running--in may be beneficial if it smoothes tooth surfaces (increasing specific film thickness) and increases the area of contact by removing minor imperfections through local wear. The amount of wear that is tolerable depends on the expected lifetime for the gearset, and on requirements for noise and vibration. Wear rate may become excessive if tooth profiles are worn to the extent that high dynamic loads are encountered. Excessive wear may also be caused by contamination of the lubricant by abrasive particles. When wear becomes aggressive and is not preempted by scuffing or bending fatigue, wear and pitting will likely compete for the predominate failure mode. 8.1 Abrasive wear Abrasive wear is removal or displacement of material due to the presence of hard particles suspended in 26

AMERICAN GEAR MANUFACTURERS ASSOCIATION

the lubricant or embedded in flanks of mating teeth. The choice of lubricant usually does not have any direct effect on abrasive wear. Abrasive particles can be present, however, as debris from other forms of wear such as fatigue pitting and adhesion. The lubricant should not react with any systemic materials or with any contaminants. Products of these reactions can be abrasive. In large open gears, the film thickness of highly viscous lubricants may prevent three--body abrasion from small particles. 8.2 Wear risk evaluation The boundary lubrication regime consists of exceedingly complex interactions between additives in the lubricant, metal, and atmosphere making it impossible to assess accurately the chance of wear or scuffing from a single parameter such as specific film thickness. However, empirical data of figure 15 have been used as an approximate guide to the probability of wear related distress. Figure 15 is based on data published by Wellauer and Holloway [20] that were obtained from several hundred laboratory tests and field applications. The curves of figure 15 apply to through--hardened steel gears ranging in size from 25 mm to 4600 mm in diameter that were lubricated with mineral--based, non--EP gear lubricants. The authors [20] defined tooth flank surface distress as surface pitting or wear that might be destructive or could shorten the gear life. Most of the data of figure 15 pertain to gears that experienced lives in excess of 10 million cycles. 8.2.1 Adjustments to the surface distress and specific film thickness curves The surface distress and specific film thickness curves (figure 15) were derived from the Wellauer and Holloway curves. The curves are adjusted to account for different definitions of composite surface roughness and specific film thickness. 8.2.1.1 Average surface roughness adjustment Reference [20] used root mean square surface roughness. This information sheet uses average surface roughness. The relationship between root mean square and average surface roughness varies with the machining process. Typically, Rq x ≅ 1.11 Ra x

(98)

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Specific film thickness, λ

10

5% 40% 80%

1

0.1

0.01

0.1

1

10 Pitch line velocity (m/s)

100

1000

Figure 15 -- Probability of wear related distress 8.2.1.2 Composite adjustment

roughness

h c , of equation 75, provides film thickness values

Reference [20] used an arithmetic average for the composite surface roughness:

1.316 times the Dowson and Higginson [17] minimum film thickness, hmin, used by the Wellauer and Holloway paper [20].

Rq x avg =

Rq1x + Rq 2x 2

surface

(99)

 i

Specific film thickness adjustment factor is derived as follows: Wellauer and Holloway [20] defined λ as:

where Rq1x, Rq2x is root mean square surface roughness, pinion and gear respectively, for filter cutoff length, Lx, mm. Composite surface roughness used in this information sheet is root mean square average of average surface roughness, see equation 78. If Rq1x = Rq2x and Ra1x = Ra2x (similar surface roughnesses), σ x = 2 Ra1x = 2 Ra 2x

(100)

Rq x avg = Rq 1x = Rq 2x

(101)

8.2.1.3 Specific film thickness adjustment The curves of figure 15 were also adjusted for different definitions of film thickness. The Dowson and Toyoda equation for central film thickness [19],

λ W&H =

h min Rqx avg

(102)

This information sheet uses h c and σx defined by  i equations 75 and 100: hc  i (103) λ  i = σ x Substituting adjustment factors into the equation for λ gives: λ min =

1.316 (1.11)h min 2 Rq

(104)

x avg

(105) λ min = 1.033 λ W&H and is used to adjust the specific film thickness provided by Wellauer and Holloway. This vertical axis adjustment is now reflected in figure 15.

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Finally, the units of pitch line velocity, vt, were adjusted from feet per minute to meters per second. Note that specific film thickness is dimensionless. 8.2.2 Wear risk probability The curves of figure 15 can be fitted with the following equations:





+ 0.47767 λ 5% = 2.68863 vt



−1



−1



−1

+ 0.64585 λ 40% = 4.90179 vt



λ 80% = 9.29210 + 0.95507 vt

(106) (107) (108)

Using the following definition, the mean minimum specific film thickness, mλ min, and the standard deviation, σλ min, can be calculated by simultaneous solution (two equations in two unknowns) using any two of the adjusted Wellauer and Holloway curves (5% and 40%, 40% and 80%, or 5% and 80%): x=

λ min − m λ min σ λ min

(ref [24])

(109)

where x

is value of the standard normal variable determined by probability;

λmin

is specific film thickness (equation 105);

mλ min is mean minimum specific film thickness; σλ min is standard deviation of the minimum specific film thickness. Figure 15 and equations 106 through 108 are listed in the percent failure mode, Q(x). This must first be converted to a percent survival mode, P(x), by the equation P (x) = 1 − Q (x). With P(x) known, the value “x” may be determined from the table “Normal Probability Function and Derivatives” of reference [24]. λ5%:

P (x) = 20% x 80% = − 0.84163389 Use several film thickness values from figure 15 to find how mean minimum specific film thickness, mλ min, and standard deviation of the minimum specific film thickness, σλ min, vary with pitch line velocity. An example is shown below: v t = 5 m∕s λ 5% = 0.9849 λ 40% = 0.6149 This gives the following equations that are solved for σλ min: 1.6449 = 0.2534 =

 

 

0.9849 − m λ min σ λ min 0.6149 − m λ min σ λ min

1.6449 σ λ min = 0.9849 − m λ min 0.2534 σ λ min = 0.6149 − m λ min Subtracting the bottom equation from the upper equation yields: 1.3915 σ λ min = 0.3700 σ λ min = 0.3700 = 0.2659 1.3915 Using σλ min in the first equation, mλ min is found: 1.6449 =

0.9849 − m λ min

0.2659 m λ min = 0.9849 − 1.6449 (0.2659) m λ min = 0.5475 This process was repeated for all data points along the curves in the following combinations: 5%--40%, 40%--80% and 5%--80%. Results of these calculations were averaged and the values are shown in table 7.

Q (x) = 5%

Curve--fitting the inverse of the mean, m 1

P (x) = 95%

inverse of the standard deviation, σ 1 , versus the λ min 1 inverse of the pitch line velocity, v , results in the t following:

x 5% = 1.64491438 λ40%: Q (x) = 40% P (x) = 60% x 40% = 0.25335825 λ80%: 28

Q (x) = 80%

λ min

, and the

for vt ≤ 5 m/s





m λ min = 5.43389 + 0.71012 vt

−1

(110)

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σ λ min =



0.01525 + 9.43942 + 2.06085 vt v 2t



−1

(111)

for vt > 5 m/s:





m λ min = 5.47432 + 0.70153 vt

σ λ min =



−1

(112)



9.7849 + 6.19681 + 2.34174 vt v 2t

−1

(113) Association of a mean and standard deviation with each pitch line velocity allows the probability of wear distress to be assigned given specific EHL operating conditions using the procedure of annex B and using: y = λ min m y = m λ min σ y = σ λ min

AGMA 925--A03

Table 7 -- Calculation results vt (m/s) 0.25 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 100.00 150.00 200.00 250.00

mλ min 0.04455408 0.08636353 0.16271966 0.23073618 0.29172511 0.34673387 0.39660952 0.44204486 0.48361240 0.52178951 0.55697759 0.80016431 0.93691698 1.02464932 1.08573704 1.13072662 1.16524421 1.19256659 1.21473204 1.23307514 1.32309469 1.35614631 1.37331023 1.38382249

σλ min 0.02496302 0.04757665 0.08689583 0.11982298 0.14771523 0.17158123 0.19218459 0.21011292 0.22582491 0.23968331 0.25197825 0.32484801 0.35693985 0.37431229 0.38496185 0.39205782 0.39707727 0.40079104 0.40363655 0.40587858 0.41541491 0.41831071 0.41968741 0.42048785

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Annex A (informative) Flow chart for evaluating scuffing risk and oil film thickness [The foreword, footnotes and annexes, if any, are provided for informational purposes only and should not be construed as a part of AGMA 925--A03, Effect of Lubrication on Gear Surface Distress.]

START

z1, z2, mn, β, aw, αn, ra1, ra2, b, n1, P, Ko, Km, Kv, E1, E2, ν1, ν2, Ra1x, Ra2x, nop, Tip, Driver, mmet, θM, BM1, BM2, θoil, ksump, ηM, α, Lx, θM,test, XW, ν40, θfl max, test, θ S met



Get Input Data

P1 Tip

profile modification 0 = none 1 = modified for high load capacity 2 = modified for smooth meshing

mmet

method for approximating mean coefficient of friction 1 = Kelley and AGMA 217.01 method (constant) 2 = Benedict and Kelley method (variable) Other = enter own value for mm (constant)

θM

gear tooth temperature (°C) 0 = program calculates with equation 91 ¸ 0 ! input own value

Driver driving member 1 = pinion 2 = gear nop

number of calculation points along the line of action (25 recommended)

ηM

dynamic viscosity (mPa⋅s) at gear tooth temperature, θM 0 = calculate using table 2 and equation 69 ¸ 0 ! input own value (must also input α)

α

pressure viscosity coefficient (mm2/N) 0 = calculate using table 2 and equation 74 ¸ 0 ! input own value (must also input ηM)

ksump = 1.0 if splash lube = 1.2 if spray lube

of calculating scuffing θS met method temperature, θs 0 = from test gears (need to also input θfl max, test, θM, test and XW from table 4 1 = R&O mineral oil 2 = EP mineral oil Other = enter own value of θs (°C), (see table 3)

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P1

u r1 r2 rw1 αt rb1 rb2 αwt pbt pbn px βb βw αwn

(Eq 1) (Eq 2) (Eq 3) (Eq 4) (Eq 5) (Eq 6) (Eq 7) (Eq 8) (Eq 9) (Eq 10) (Eq 11) (Eq 12) (Eq 13) (Eq 14)

CF CA CC CD CE CB Z

(Eq 15) (Eq 16) (Eq 17) (Eq 18) (Eq 19) (Eq 20) (Eq 21)

εβ (Eq 23) na = fractional part of εβ

(1 − n r) ≥ n a yes

εα (Eq 22) nr = fractional part of εα

Lmin

(Eq 25)

ω1 ω2 vt (Ft)nom KD Ft Fwn wn Er

(Eq 33) (Eq 34) (Eq 35) (Eq 40) (Eq 41) (Eq 42) (Eq 43) (Eq 44) (Eq 58)

R avg

(Eq 87)

CR

no helical gear

yes spur gear

32

εβ

(Eq 24)

Lmin

(Eq 27)

x

avgx m m const

σx

β=0

no

(Eq 86) (Eq 85) (Eq 78)

P2

Lmin

(Eq 26)

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

P2

C1, C2, C3, C4, C5 = CA, CB, CC, CD, CE

ξ1, ξ2, ξ3, ξ4, ξ5

(Eq 28)

ξA, ξB, ξC, ξD, ξE = ξ1, ξ2, ξ3, ξ4, ξ5

i=1

i>5

XΓi XΓi XΓi

(Eq 45) (Eq 46) (Eq 47)

XΓi XΓi XΓi

(Eq 54) (Eq 55) (Eq 56)

XΓi XΓi XΓi

(Eq 48) (Eq 49) (Eq 50)

XΓi XΓi XΓi

(Eq 51) (Eq 52) (Eq 53)

yes no

ξ i = ξ A + (i − 6 )

ξE − ξA (nop − 1)

(Eq 29) (Eq 30) (Eq 31) (Eq 32) (Eq 36) (Eq 37) (Eq 38) (Eq 39)

Ã1i Ã2i Ãri Ãni vr1i vr2i vsi vei

yes

yes

no

(eq 57)

bH1 yes

Tip = 0

i = i+ 1

no Tip = 2 no

i = nop + 6 yes

Driver = 1 no

P3

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

P3

K = 0.8

yes

θM = 0

yes

no (θM input)

mmet = 2 no yes

(Eq 85)

mm const

mm const = mmet

ηM = 0

ηM = 0

mmet = 1 no

no (ηM & α input)

yes

yes no (ηM & α input)

mm const

ηM*

(Eq 69)

α*

(Eq 74)

(Eq 85)

Call subroutine Max_Flash_Temp θfl max Call subroutine Max_Flash_Temp θfl max

Call subroutine Max_Flash_Temp θfl max

(Eq 91)

θM

P3A

θM

yes

θM = 0

(Eq 91)

no (θM input)

yes

θM1 = θM

Call subroutine Max_Flash_Temp θfl max

ηM*

(Eq 69)

α*

(Eq 74)

mm const = 0

ηM = 0 P3A

ηM

(Eq 69)

α

(Eq 74)

no (εo & α input)

same page

Call subroutine Max_Flash_Temp θfl max

θM

P4

yes

* See table 2 for constants in these equations calculated per 71 and 73.

34

(Eq 91)

|θM1 -- θM| < 0.01

no

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

P4

(Eq 66)

G

hmin = 100

λmin > λ2bH(i) λmin = 100 yes

no i=1

U(i)

(Eq 67)

W(i)

(Eq 68)

λmin = λ2bH(i)

i=i+1 no

i = nop + 6 yes

Hc(i)

(Eq 65)

hc(i)

(Eq 75)

(eq 93)

θ B max

P5

hmin > hc(i) yes

no

hmin = hc(i)

λ2bH(i)

(Eq 77)

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AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

P5 yes

P scuff < 0.10

θ S met = 0

Srisk = low

no

yes

θS (eq 96) no

R&O Mineral Oil θ S met = 2

yes

θS (Eq 95)

no θ s = θ S met

S risk = high

yes

θS (eq 94)

no

Srisk = moderate

no

test gears (need θfl max, test, θM test & XW input)

θ S met = 1

yes

P scuff ≤ 0.30

Rq1x

(Eq 98)

Rq2x

(Eq 98)

Rqx avg

(Eq 99)

λmin

(Eq 105)

no

v t ≤ 5 m∕s

EP Mineral Oil

yes mλ min

(Eq 110)

σλ min

(Eq 111)

y = θ B max

mλ min

(Eq 112)

my = θs

σλ min

(Eq 113)

Enter own value of θs

σ y = 0.15 θ s

Call subroutine “Probability”

y = λ min m y = m λ min σ y = σ λ min

Return POF

Call subroutine “Probability” Return POF

Pscuff = POF

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Subroutine “Probability” y, m y, σ y input

x (eq B.1)

|x| > 1.6448

yes Q = 0.05

no t (eq B.4) ZQ (eq B.3) Q (eq B.2)

x>0 no

yes POF = 1.0 -- Q

POF = Q

Return POF

POF = Probability of failure

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AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Subroutine “Max_Flash_Temp i=1

θfl max = 0

mm const = 0

no

yes

mm is a (given) constant or calculated by equation 85 (AGMA 217.01 and Kelley)

νs(i) or XΓ(i) < εmach**

yes

no mm(i) (Eq 88) (Benedict and Kelley)

mm(i) = mm const

mm(i) = 0

yes mm(i) or bH(i) < εmach**

yes

no

mm(i) = 0

no

yes θfl max = θfl(i)

(Eq 84)

θfl(i)

θfl(i) = 0

θfl(i) > θfl max no

i=i+ 1

i = nop + 6 yes Return

**Eq 88 is not valid at vs(i) = 0 or XΓ(i) = 0 or near zero, and Eq 84 is not valid at bH(i) = 0 or near zero. εmach is a small finite number (e.g., 10 --10). In case the calculated mm(i) < 0, set mm(i) = 0.

38

no

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Annex B (informative) Normal or Gaussian probability [The foreword, footnotes and annexes, if any, are provided for informational purposes only and should not be construed as a part of AGMA 925--A03, Effect of Lubrication on Gear Surface Distress.]

where

B.1 Normal or Gaussian probability For random variables that follow normal (Gaussian) distributions, the following procedure [24] can be used to calculate probabilities of failure in the range of 5% to 95%: x=

y − m y  σy

(B.1)

Q

is the tail area of the normal probability function;

ZQ

is the normal probability density function.

Probability of failure: if x > 0, then: probability of failure = 1 -- Q;

where

else probability of failure = Q

x

is the standard normal variable;

y

is the random variable;

my

is the mean value of random variable y;

σy

is the standard deviation of random variable y.

where

Evaluation of Q:

ZQ

2  −0.5(x )  = 0.3989422804 e

(B.3)

b 1 = 0.319381530 b 2 = − 0.356563782 b 3 = 1.781477937

if x > 1.6448, then:

b 4 = − 1.821255978

Q = 0.05;

b 5 = 1.330274429

else Q = Z Q b 1t + b 2t 2 + b 3t 3 + b 4t 4 + b 5t 5 (B.2)

p = 0.2316419 1 t= 1 + p|x|

(B.4)

are constants given in reference [24].

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40

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Annex C (informative) Test rig gear data [The foreword, footnotes and annexes, if any, are provided for informational purposes only and should not be construed as a part of AGMA 925--A03, Effect of Lubrication on Gear Surface Distress.]

C.1 Test rig gear data Table C.1 provides a summary of gear data for several back to back test rigs that have been used for gear lubrication rating and research.

41

42

Pinion torque range

Primary wear assessment a mn αn β αwt z1 z2 b ra1 ra2 x1 x2 Quality number Quality standard Ra1 Ra2 n1 θoil Ref document

Symbol

AGMA 925--A03

Nm

mm mm rpm deg C -- --

mm mm deg deg deg -- --- -mm mm mm -- --- --- --

Units

91.5 4.5 20 0 22.44 16 24 20 44.385 56.25 0.8635 -- 0.5103 5 ISO 1328 0.3 -- 0.7 0.3 -- 0.7 2170 90--140 ISO 14635--1 ASTM D5182--97 CEC L--07--A--95 3.3 -- 534.5

Scuffing

FZG “A”

3.3--534.5

91.5 4.5 20 0 22.44 16 24 10 44.385 56.25 0.8635 --0.5103 5 ISO 1328 0.3 -- 0.7 0.3 -- 0.7 2170 90--120 ISO/WD 14635--2

Scuffing

FZG “A10”

135 -- 376

Pitting (micro & macro) 91.5 4.5 20 0 22.44 16 24 14 41.23 59.18 0.1817 0.1715 5 DIN 3962 0.3 -- 0.5 0.3 -- 0.5 2250 90--120 FVA Info Sheet 54/7

FZG “C”

28 -- 265

91.5 4.5 20 0 22.44 16 24 14 41.23 59.18 0.1817 0.1715 5 DIN 3962 0.4 -- 0.6 0.4 -- 0.6 2250 90 FVA Info Sheet 54/I--IV

FZG “C -- GF” Micropitting

0 -- 100

88.9 3.175 20 0 20 28 28 6.35/2.8 47.625 47.625 0 0 13 AGMA 2000 0.3 -- 0.4 0.3 -- 0.4 10000 49 -- 77 NASA TP -- 2047 (1982)

Pitting

NASA

Table C.1 -- Summary of gear data for lubricant testing

0 -- 270

88.9 3.175 22.5 0 22.5 28 28 6.35 47.22 47.22 0 0 13 AGMA 2000 0.46 -- 0.64 0.46 -- 0.64 10000 74 ASTM D1947--83 (1984)

Scuffing

Ryder

250 -- 400

Pitting (micro & macro) 91.5 3.629 20 0 21.31 20 30 14 40.82 58.18 0.2231 0.0006 12--13 AGMA 2000 0.5 -- 0.8 0.5 -- 0.8 2250 80 -- --

AGMA

20 -- 407

82.55 5.08 20 0 26.25 15 16 4.76 45.02 47.69 0.3625 0.3875 5 ISO 1328 0.3 -- 0.8 0.3 -- 0.8 4K -- 6K 70 -- 110 IP166/77 (1992)

Scuffing

IAE

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Annex D (informative) Example calculations [The foreword, footnotes and annexes, if any, are provided for informational purposes only and should not be construed as a part of AGMA 925--A03, Effect of Lubrication on Gear Surface Distress.]

****************************************************************************** SCUFFING AND WEAR RISK ANALYSIS ver 1.0.9 -- AGMA925--A03 SCORING+ EX.#1 DATE:2002/04/18 TIME:08:08:23 ****************************************************************************** ***** GENERAL AND GEOMETRY INPUT DATA ***** SCORING+ EX.#1 Input unit (=1 SI, =2 Inch) (iInputUnit) 1.000000 Output unit (=1 SI, =2 Inch) (iOutputUnit) 1.000000 Gear type (=1 external, =2 internal) (iType) 1.000000 Driving member (=1 pinion, =2 gear) (iDriver) 2.000000 Number of pinion teeth (z1) 21.000000 Number of gear teeth (z2) 26.000000 Normal module (mn) 4.000000 mm Helix angle (Beta) 0.000000 deg Operating center distance (aw) 96.000000 mm Normal generating pressure angle (Alphan) 20.000000 deg Standard outside radius, pinion (ra1) 46.570900 mm Standard outside radius, gear (ra2) 57.277000 mm Face width (b) 66.040000 mm Profile mod (=0 none, =1 hi load, =2 smooth) (iTip) 1.000000 ***** Material input data ***** Modulus of elasticity, pinion (E1) 206842.718795 N/mm^2 Modulus of elasticity, gear (E2) 206842.718795 N/mm^2 Poisson’s ratio, pinion (Nu1) 0.300000 Poisson’s ratio, gear (Nu2) 0.300000 Average surface roughness at Lx, pinion (Ra1x) 0.508000 mu m Average surface roughness at Lx, gear (Ra2x) 0.508000 mu m Filter cutoff of wavelength x (Lx) 0.800000 mm Method for approximate mean coef. friction (Mumet) 1.000000 Welding factor (Xw) 1.000000 ***** Load data ***** Pinion speed (n1) 308.570000 rpm Transmitted power (P) 20.619440 kW Overload factor (Ko) 1.000000 Load distribution factor (Km) 1.400000 Dynamic factor (Kv) 1.063830 ***** Lubrication data ***** Lubricant type (=1 Mineral, =2 Synthetic, =3 MIL--L--7808K, =4 MIL--L--23699E) (iLubeType) 1.000000 ISO viscosity grade number (nIsoVG) 460.000000 Kinematic viscosity at 40 deg C (Nu40) 407.000000 mm^2/s ***** Input temperature data ***** Tooth temperature (ThetaM) 82.222222 deg C Thermal contact coefficient, pinion (BM1) 16.533725 N/[mm s^.5K] Thermal contact coefficient, gear (BM2) 16.533725 N/[mm s^.5K] Oil inlet or sump temperature (Thetaoil) 71.111111 deg C Parameter for calculating tooth temperature (ksump) 1.000000 Dynamic viscosity at gear tooth temperature (EtaM) 43.000000 mPa⋅s Pressure--viscosity coefficient (Alpha) 0.022045 mm^2/N Method of calculating scuffing temperature (Thetasmet) 2.000000 Maximum flash temperatrue of test gears (Thetaflmaxtest) 0.000000 Tooth temperature of test gear (ThetaMtest) 0.000000 Number of calculation points (nNop) 25.000000

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AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

****************************************************************************** SCUFFING AND WEAR RISK ANALYSIS ver 1.0.9 -- AGMA925--A03 SCORING+ EX.#1 DATE:2002/04/18 TIME:08:08:23 ****************************************************************************** ***** GEOMETRY CALCULATION ***** Gear ratio (u) 1.238095 Standard pitch radius, pinion (r1) 42.000000 mm Standard pitch radius, gear (r2) 52.000000 mm Pinion operating pitch radius (rw1) 42.893617 mm Transverse generating pressure angle (Alphat) 20.000000 deg Base radius, pinion (rb1) 39.467090 mm Base radius, gear (rb2) 48.864016 mm Transverse operating pressure angle (Alphawt) 23.056999 deg Transverse base pitch (pbt) 11.808526 mm Normal base pitch (pbn) 11.808526 mm Axial pitch (px) ---------------Base helix angle (Betab) 0.000000 deg Operating helix angle (Betaw) 0.000000 deg Normal operating pressure angle (Alphawn) 23.056999 deg Distance along line of action -- Point A (CA) 7.715600 mm Distance along line of action -- Point B (CB) 12.913884 mm Distance along line of action -- Point C (CC) 16.799142 mm Distance along line of action -- Point D (CD) 19.524126 mm Distance along line of action -- Point E (CE) 24.722409 mm Distance along line of action -- Point F (CF) 37.598080 mm Active length of line of action (Z) 17.006810 mm Transverse contact ratio (EpsAlpha) 1.440214 Fractional part of EpsAlpha (nr) 0.440214 Axial contact ratio (EpsBeta) 0.000000 Fractional part of EpsBeta (na) 0.000000 Minimum contact length (Lmin) 66.040000 mm ***** GEAR TOOTH VELOCITY AND LOADS ***** Rotational (angular) velocity, pinion (Omega1) 32.313375 Rotational (angular) velocity, gear (Omega2) 26.099264 Operating pitch line velocity (vt) 1.386038 Nominal tangential load (Ftnom) 14876.538066 Combined derating factor (KD) 1.489362 Actual tangential load (Ft) 22156.550486 Normal operating load (Fwn) 24080.178937 Normal unit load (wn) 364.630208

rad/s rad/s m/s N N N N/mm

***** MATERIAL PROPERTY AND TOOTH SURFACE FINISH ***** Reduced modulus of elasticity (Er) 227299.690984 N/mm^2 Average of pinion and gear average roughness (Ravgx) 0.508000 mu m Surface roughness constant (CRavgx) 1.816720 Composite surface roughness at filter cuttoff (Sigmax) 0.718420 mu m

44

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

********************************************************************************** SCUFFING AND WEAR RISK ANALYSIS ver 1.0.9 -- AGMA925--A03 SCORING+ EX.#1 DATE:2002/04/18 TIME:08:08:23 ********************************************************************************** ***** LOAD SHARING RATIO AND bH ***** Index (A) (B) (C) (D) (E)

Roll Ang(rad) 0.19549 0.32721 0.42565 0.49469 0.62641

XGamma 0.14286 1.00000 1.00000 1.00000 0.00000

Rhon(mm) 6.13226 8.47833 9.29314 9.38554 8.46633

bH 0.05982 0.18610 0.19484 0.19581 0.00000

Index (A) (B) (C) (D) (E)

( ( ( ( (

1) 2) 3) 4) 5)

0.19549 0.21345 0.23140 0.24936 0.26731

0.14286 0.25970 0.37654 0.49339 0.61023

6.13226 6.53669 6.91441 7.26541 7.58971

0.05982 0.08327 0.10313 0.12101 0.13755

( ( ( ( (

( 6) ( 7) ( 8) ( 9) ( 10) ( 11) ( 12) ( 13) ( 14) ( 15)

0.28527 0.30322 0.32118 0.33913 0.35709 0.37504 0.39300 0.41095 0.42890 0.44686

0.72708 0.84392 0.96076 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000

7.88729 8.15816 8.40233 8.61978 8.81052 8.97455 9.11187 9.22247 9.30637 9.36356

0.15306 0.16770 0.18160 0.18765 0.18971 0.19147 0.19293 0.19410 0.19498 0.19558

( 6) ( 7) ( 8) ( 9) ( 10) ( 11) ( 12) ( 13) ( 14) ( 15)

( 16) ( 17) ( 18) ( 19) ( 20)

0.46481 0.48277 0.50072 0.51868 0.53663

1.00000 1.00000 0.81791 0.70106 0.58422

9.39403 9.39780 9.37485 9.32520 9.24883

0.19589 0.19593 0.17698 0.16342 0.14857

( 16) ( 17) ( 18) ( 19) ( 20)

( 21) ( 22) ( 23) ( 24) ( 25)

0.55459 0.57254 0.59050 0.60845 0.62641

0.46737 0.35053 0.23369 0.11684 0.00000

9.14575 9.01596 8.85946 8.67625 8.46633

0.13214 0.11362 0.09196 0.06435 0.00000

( 21) ( 22) ( 23) ( 24) ( 25)

1) 2) 3) 4) 5)

**** P3 -- Calculate flash temperature **** Dynamic viscosity at 40 deg C Dynamic viscosity at 100 deg C Factor c Factor d Factor k Factor s Mumet -- use Kelley and AGMA 217.01 Surface roughness constant Mean coef. of friction, const. (Eq 85)

(Eta40C) (Eta100C) (c_coef) (d_coef) (k_coef) (s_coef) (Mumet) (CRavgx) (Mumconst)

412.082400 26.341040 8.964201 --3.424449 0.010471 0.134800 1.000000 1.816720 0.109003

mPa⋅s mPa⋅s

45

AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

********************************************************************************** SCUFFING AND WEAR RISK ANALYSIS ver 1.0.9 -- AGMA925--A03 SCORING+ EX.#1 DATE:2002/04/18 TIME:08:08:23 ********************************************************************************** **** Calculate flash temperature **** Index (A) (B) (C) (D) (E)

K 0.80 0.80 0.80 0.80 0.80

Mum 0.1090 0.1090 0.1090 0.1090 0.0000

XGamma 0.1429 1.0000 1.0000 1.0000 0.0000

bH (mm) 0.059822 0.186102 0.194840 0.195806 0.000000

vs (m/s) 0.5306 0.2269 0.0000 0.1592 0.4628

vr1 (m/s) 0.2493 0.4173 0.5428 0.6309 0.7989

vr2 (m/s) 0.7799 0.6442 0.5428 0.4717 0.3360

1) 2) 3) 4) 5)

0.80 0.80 0.80 0.80 0.80

0.1090 0.1090 0.1090 0.1090 0.1090

0.1429 0.2597 0.3765 0.4934 0.6102

0.059822 0.083275 0.103129 0.121010 0.137549

0.5306 0.4892 0.4478 0.4064 0.3650

0.2493 0.2722 0.2951 0.3180 0.3409

0.7799 0.7614 0.7429 0.7244 0.7059

13.6320 19.2004 22.7228 24.7713 25.6466

( ( ( ( (

( 6) ( 7) ( 8) ( 9) ( 10)

0.80 0.80 0.80 0.80 0.80

0.1090 0.1090 0.1090 0.1090 0.1090

0.7271 0.8439 0.9608 1.0000 1.0000

0.153056 0.167704 0.181595 0.187648 0.189713

0.3236 0.2822 0.2408 0.1995 0.1581

0.3638 0.3867 0.4096 0.4325 0.4554

0.6874 0.6689 0.6505 0.6320 0.6135

25.5359 24.5661 22.8276 19.2753 15.1349

( 6) ( 7) ( 8) ( 9) ( 10)

( 11) ( 12) ( 13) ( 14) ( 15)

0.80 0.80 0.80 0.80 0.80

0.1090 0.1090 0.1090 0.1090 0.1090

1.0000 1.0000 1.0000 1.0000 1.0000

0.191471 0.192930 0.194098 0.194979 0.195577

0.1167 0.0753 0.0339 0.0075 0.0489

0.4783 0.5012 0.5241 0.5470 0.5699

0.5950 0.5765 0.5580 0.5395 0.5210

11.0832 7.1033 3.1799 0.7011 4.5531

( 11) ( 12) ( 13) ( 14) ( 15)

( 16) ( 17) ( 18) ( 19) ( 20)

0.80 0.80 0.80 0.80 0.80

0.1090 0.1090 0.1090 0.1090 0.1090

1.0000 1.0000 0.8179 0.7011 0.5842

0.195895 0.195934 0.176983 0.163420 0.148569

0.0903 0.1317 0.1731 0.2145 0.2559

0.5928 0.6157 0.6386 0.6615 0.6844

0.5025 0.4840 0.4655 0.4470 0.4285

8.3886 12.2201 13.8125 15.2621 15.9136

( 16) ( 17) ( 18) ( 19) ( 20)

( 21) ( 22) ( 23) ( 24) ( 25)

0.80 0.80 0.80 0.80 0.80

0.1090 0.1090 0.1090 0.1090 0.0000

0.4674 0.3505 0.2337 0.1168 0.0000

0.132141 0.113623 0.091964 0.064352 0.000000

0.2973 0.3386 0.3800 0.4214 0.4628

0.7073 0.7302 0.7531 0.7760 0.7989

0.4100 0.3915 0.3730 0.3545 0.3360

15.6888 14.4671 12.0443 7.9953 0.0000

( 21) ( 22) ( 23) ( 24) ( 25)

( ( ( ( (

Thetafl (C) 13.6320 22.0835 0.0000 14.7688 0.0000

Index (A) (B) (C) (D) (E)

-------------------------------------------------------------------------------------------------------------------------------------------------------------------The max. flash temp. occurs at point (10) (Thetaflmax) 25.646608 deg C -------------------------------------------------------------------------------------------------------------------------------------------------------------------Dynamic viscosity at the gear tooth temperature Pressure--viscosity coefficient

46

(EtaM) (Alpha)

43.000000 0.022045

mPa⋅s mm^2/N

1) 2) 3) 4) 5)

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

********************************************************************************** SCUFFING AND WEAR RISK ANALYSIS ver 1.0.9 -- AGMA925--A03 SCORING+ EX.#1 DATE:2002/04/18 TIME:08:08:23 ********************************************************************************** ********** P4 -- Specific film thickness ********** Material parameter (eq 66)

(G)

5010.821688

Index (A) (B) (C) (D) (E)

U 1.587561e--11 1.184300e--11 1.105036e--11 1.111223e--11 1.267961e--11

W 0.000037 0.000189 0.000173 0.000171 0.000000

Hc 3.539329e--05 2.458469e--05 2.365326e--05 2.376807e--05 0.000000e+00

hc (mu m) 0.217041 0.208437 0.219813 0.223076 0.000000

Lambda2bH 0.781203 0.425354 0.438395 0.443804 0.000000

Index (A) (B) (C) (D) (E)

( ( ( ( (

1) 2) 3) 4) 5)

1.587561e--11 1.495710e--11 1.420027e--11 1.357156e--11 1.304655e--11

0.000037 0.000064 0.000087 0.000109 0.000129

3.539329e--05 3.220166e--05 3.010396e--05 2.854084e--05 2.730927e--05

0.217041 0.210492 0.208151 0.207361 0.207269

0.781203 0.642142 0.570609 0.524768 0.491992

( ( ( ( (

( 6) ( 7) ( 8) ( 9) ( 10)

1.260712e--11 1.223958e--11 1.193348e--11 1.168076e--11 1.147515e--11

0.000148 0.000166 0.000183 0.000186 0.000182

2.630903e--05 2.548198e--05 2.479093e--05 2.439213e--05 2.414785e--05

0.207507 0.207886 0.208301 0.210255 0.212755

0.466937 0.446895 0.430320 0.427292 0.430014

( 6) ( 7) ( 8) ( 9) ( 10)

( 11) ( 12) ( 13) ( 14) ( 15)

1.131183e--11 1.118707e--11 1.109806e--11 1.104277e--11 1.101981e--11

0.000179 0.000176 0.000174 0.000172 0.000171

2.395433e--05 2.380784e--05 2.370556e--05 2.364541e--05 2.362595e--05

0.214979 0.216934 0.218624 0.220053 0.221223

0.432510 0.434789 0.436856 0.438717 0.440375

( 11) ( 12) ( 13) ( 14) ( 15)

( 16) ( 17) ( 18) ( 19) ( 20)

1.102840e--11 1.106830e--11 1.113982e--11 1.124380e--11 1.138168e--11

0.000171 0.000171 0.000140 0.000121 0.000101

2.364633e--05 2.370628e--05 2.428942e--05 2.481221e--05 2.546118e--05

0.222134 0.222787 0.227710 0.231379 0.235486

0.441830 0.443084 0.476505 0.503875 0.537840

( 16) ( 17) ( 18) ( 19) ( 20)

1) 2) 3) 4) 5)

( 21) 1.155550e--11 0.000082 2.627996e--05 0.240350 0.582071 ( 21) ( 22) 1.176804e--11 0.000062 2.735014e--05 0.246588 0.644006 ( 22) ( 23) 1.202294e--11 0.000042 2.885557e--05 0.255645 0.742128 ( 23) ( 24) 1.232482e--11 0.000022 3.139471e--05 0.272388 0.945273 ( 24) ( 25) 1.267961e--11 0.000000 0.000000e+00 0.000000 0.000000 ( 25) -------------------------------------------------------------------------------------------------------------------------------------------------------------------Minimum film thickness found at point(5) (hmin) 0.207269 mu m Min. specific film thk. found at point (B) (LambdaMin) 0.425354 Tooth temperature (ThetaM) 82.222222 deg C Max. flash temperature (Thetaflmax) 25.646608 deg C Minimum film thickness (hmin) 0.207269 mu m Maximum contact temperature (ThetaBmax) 107.868830 deg C --------------------------------------------------------------------------------------------------------------------------------------------------------------------

47

AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

********************************************************************************** SCUFFING AND WEAR RISK ANALYSIS ver 1.0.9 -- AGMA925--A03 SCORING+ EX.#1 DATE:2002/04/18 TIME:08:08:23 ********************************************************************************** **** P5 -- Calculate risk of scuffing and wear **** ***** Risk of scuffing ***** Method of calculating scuffing temperature Mean scuffing temperature

(Thetasmet) (Thetas)

2.000000 316.290835

deg C

***** Probability of scuffing ***** Maximum contact temperature (y) 107.868830 deg C Mean scuffing temperature (Muy) 316.290835 deg C Approx. standard deviation of scuffing temp. (Sigmay) 47.443625 deg C Standard normal variable, x = ((y--muy)/Sigmay) --4.393046 -------------------------------------------------------------------------------------------------------------------------------------------------------------------Probability of scuffing Pscuff = 5% or lower Based on AGMA925--A03 Table 5, scuffing risk is low -------------------------------------------------------------------------------------------------------------------------------------------------------------------Average surface roughness, pinion Average surface roughness, gear Average surface roughness (rms), pinion Average surface roughness (rms), gear Arithmetic average of rms roughness Minimum specific film thickness Pitchline velocity is less than 5 m/s Mean min. specific film thk. (eq. 110) Std. dev. of min. spec. film thk. (eq. 111)

**** Risk of wear **** (Ra1x) (Ra2x) (Rq1x) (Rq2x) (Rqxavg) (Lambdamin) (vt) (MuLambdaMin) (SigmaLambdaMin)

0.508000 0.508000 0.563880 0.563880 0.563880 0.425354 1.386038 0.215956 0.112623

mu m mu m mu m mu m mu m m/s

***** Probability of wear ***** Minimum specific film thickness (y) 0.425354 Mean minimum specific film thickness (muy) 0.215956 Standard deviation of the min. specific film (Sigmay) 0.112623 Standard normal variable, x = ((y--muy)/Sigmay) 1.859273 -------------------------------------------------------------------------------------------------------------------------------------------------------------------Probability of wear Pwear = 5% or lower --------------------------------------------------------------------------------------------------------------------------------------------------------------------

48

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

Bibliography

The following documents are either referenced in the text of AGMA 925--A03, Effect of Lubrication on Gear Surface Distress, or indicated for additional information. 1. Blok, H., Les Températures de Surface dans les Conditions de Graissage sans Pression Extrême, Second World Petroleum Congress, Paris, June, 1937. 2. Kelley, B.W., A New Look at the Scoring Phenomena of Gears, SAE transactions, Vol. 61, 1953, pp. 175--188. 3.

Dudley, D.W., Practical Gear Design, McGraw--Hill, New York, 1954.

4. Kelley, B.W., The Importance of Surface Temperature to Surface Damage, Chapter in Engineering Approach to Surface Damage, Univ. of Michigan Press, Ann Arbor, 1958. 5. Benedict, G. H. and Kelley, B. W., Instantaneous Coefficients of Gear Tooth Friction, ASLE transactions, Vol. 4, 1961, pp. 59--70. 6.

Lemanski, A.J., “AGMA Aerospace Gear Committee Gear Scoring Project”, March 1962.

7. AGMA 217.01, AGMA Information Sheet -- Gear Scoring Design for Aerospace Spur and Helical Power Gears, October, 1965. 8.

SCORING+, computer program, GEARTECH Software, Inc., 1985.

9. ASTM D445--97, Standard Test Method for Kinematic Viscosity of Transparent and Opaque Liquids (the Calculation of Dynamic Viscosity). 10.

ASTM D341--93(1998), Standard Viscosity -- Temperature Charts for Liquid Petroleum Products.

11. ASTM D2270--93(1998), Standard Practice for Calculating Viscosity Index From Kinematic Viscosity at 40 and 100°C. 12. So, B. Y. C. and Klaus, E. E., Viscosity--Pressure Correlation of Liquids, ASLE Transactions, Vol. 23, 4, 409--421, 1979. 13. Novak, J. D. and Winer, W. O., Some Measurements of High Pressure Lubricant Rheology, Journal of Lubricant Technology, Transactions of the ASME, Series F, Vol. 90, No. 3, July 1968, pp. 580 – 591. 14. Jones, W. R., Johnson, R. L., Winer, W. O. and Sanborn, D. M., Pressure--Viscosity Measurements for Several Lubricants to 5.5x10 8 N/m 2 (8x10 4 psi) and 149°C (300°F), ASLE Transactions, 18, pp. 249 – 262, 1975. 15. Brooks, F. C. and Hopkins, V., Viscosity and Density Characteristics of Five Lubricant Base Stocks at Elevated Pressures and Temperatures, Preprint number 75--LC--3D--1, presented at the ASLE/ASME Lubrication Conference, Miami Beach, FL, October 21 – 23, 1975. 16. Dowson, D. and Higginson, G. R., New Roller -- Bearing Lubrication Formula, Engineering, (London), Vol. 192, 1961, pp. 158--159. 17. Dowson, D. and Higginson, G.R., Elastohydrodynamic Lubrication -- The Fundamentals of Roller and Gear Lubrication, Pergamon Press (London), 1966. 18. Dowson, D., Elastohydrodynamics, Paper No. 10, Proc. Inst. Mech. Engrs., Vol. 182, Pt. 3A, 1967, pp. 151--167. 19. Dowson, D. and Toyoda, S., A Central Film Thickness Formula for Elastohydrodynamic Line Contacts, 5th Leeds--Lyon Symposium Proceedings, Paper 11 (VII), 1978, pp. 60--65.

49

AGMA 925--A03

AMERICAN GEAR MANUFACTURERS ASSOCIATION

20. Wellauer, E. J. and Holloway, G.A., Application of EHD Oil Film Theory to Industrial Gear Drives, Transactions of ASME, J. Eng., Ind., Vol. 98., series B, No 2, May 1976, pp. 626--634. 21. Moyer, C. A. and Bahney, L.L., Modifying the Lambda Ratio to Functional Line Contacts, STLE Trib. Trans. Vol. 33 (No. 4), 1990, pp. 535--542. 22.

Viscosity and pressure -- viscosity data supplied by Mobil Technology Company and Kluber Lubrication.

23. Sayles, R.S. and Thomas, T.R., Surface Topography as a Nonstationary Random Process, Nature, 271, pp. 431--434, February 1978. 24. Handbook of Mathematical Functions, National Bureau of Standards (NIST), U.S. Government Printing Office, Washington, D.C., 1964. 25.

Rough Surfaces, edited by Thomas, T.R., Longman, Inc., New York, 1982, p. 92.

26.

Errichello, R., Friction, Lubrication and Wear of Gears, ASM Handbook, Vol. 18, Oct. 1992, pp. 535--545.

27. Blok, H., The Postulate About the Constancy of Scoring Temperature, Interdisciplinary Approach to the Lubrication of Concentrated Contacts, NASA SP--237, 1970, pp. 153--248. 28.

ANSI/AGMA 9005--E02, Industrial Gear Lubrication.

29. Winter, H. and Michaelis, K., Scoring Load Capacity of Gears Lubricated with EP--Oils, AGMA Paper No. P219.17, October, 1983. 30.

Almen, J.O., Dimensional Value of Lubricants in Gear Design, SAE Journal, Sept. 1942, pp. 373--380.

31. Borsoff, V.N., Fundamentals of Gear Lubrication, Summary Report for Period March 1953 to May 1954, Bureau of Aeronautics, Shell Development Company, Contract No. 53--356c, p. 12. 32. Borsoff, V.N., On the Mechanism of Gear Lubrication, ASME Journal of Basic Engineering, Vol. 81, pp. 79--93, 1959. 33. Borsoff, V.N. and Godet, M.R., A Scoring Factor for Gears, ASLE Transactions, Vol. 6, No. 2, 1963, pp. 147--153. 34.

Borsoff, V.N., Predicting the Scoring of Gears, Machine Design, January 7, 1965, pp. 132--136.

35.

ANSI/AGMA 6011--H98, Specification for High Speed Helical Gear Units.

36. Nakanishi, T. and Ariura, Y., Effect of Surface--Finishing on Surface Durability of Surface--Hardened Gears, MPT ‘91, JSME International Conference on Motion and Power Transmissions, 1991, pp. 828--833. 37. Tanaka, S., et al, Appreciable Increases in Surface Durability of Gear Pairs with Mirror--Like Finish, ASME Paper No. 84--DET--223, 1984, pp. 1--8. 38. Ueno, T., et al, Surface Durability of Case--Carburized Gears on a Phenomenon of ‘Gray Staining’ of Tooth Surface, ASME Paper No. 80--C2/DET--27, 1980, pp. 1--8. 39. Olver, A.V., Micropitting of Gear Teeth -- Design Solutions, presented at Aerotech 1995, NEC Birmingham, October 1995, published by I. Mech. E., 1995. 40. FVA Information Sheet “Micropitting”, No. 54/7 (July, 1993) Forschungsvereinigung Antriebstechnik e.V., Lyoner Strasse 18, D--60528, Frankfurt/Main. 41. Bowen, C. W., The Practical Significance of Designing to Gear Pitting Fatigue Life Criteria, ASME Paper 77--DET--122, September 1977. 42. Dudley, D.W., Characteristics of Regimes of Gear Lubrication, International Symposium on Gearing and Power Transmissions, Tokyo, Japan, 1981. 43. Blok, H., The Thermal--Network Method for Predicting Bulk Temperatures in Gear Transmissions, Proc. 7th Round Table Discussion on Marine Reduction Gears held in Finspong, Sweden, 9--10 September 1969. 44. Blok, H., Thermo--Tribology -- Fifty Years On, keynote address to the Int. Conf. Tribology; Friction, Lubrication and Wear -- 50 Years On, Inst. Mech. Engrs., London, 1--3 July 1987, Paper No. C 248/87. 50

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 925--A03

45. Ku, P.M. and Baber, B.B., The Effect of Lubricants on Gear Tooth Scuffing, ASLE Transactions, Vol. 2, No. 2, 1960, pp. 184--194. 46. Winter, H., Michaelis, K. and Collenberg, H.F., Investigations on the Scuffing Resistance of High--Speed Gears, AGMA Fall Technical Meeting Paper 90FTM8, 1990. 47.

ANSI/AGMA 6002--B93, Design Guide for Vehicle Spur and Helical Gears.

48.

Barish, T., How Sliding Affects Life of Rolling Surfaces, Machine Design, 1960.

49. Massey, C., Reeves, C. and Shipley, E.E., The Influence of Lubrication on the Onset of Surface Pitting in Machinable Hardness Gear Teeth, AGMA Technical Paper 91FTM17, 1991.

51

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