Agma 927-a01.pdf

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

A G M A 9 2 7 - A 0 1

AGMA 927- A01

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Load Distribution Factors - Analytical

Methods for Cylindrical Gears

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Load Distribution Factors - Analytical Methods for Cylindrical Gears AGMA 927--A01

CAUTION NOTICE: AGMA technical publications are subject to constant improvement,

revision or withdrawal as dictated by experience. Any person who refers to any AGMA

technical publication should be sure that the publication is the latest available from the As-

sociation on the subject matter.

[Tables or other self--supporting sections may be quoted or extracted. Credit lines should

read: Extracted from AGMA 927--A01, Load Distribution Factors -- Analytical Methods for

Cylindrical Gears, with the permission of the publisher, the American Gear Manufacturers

Association, 1500 King Street, Suite 201, Alexandria, Virginia 22314.]

Approved October 22, 2000

ABSTRACT

This information sheet describes an analytical procedure for the calculation of the face load distribution. The

iterative solution that is described is compatible with the definitions of the term face load distribution ( K H) of

AGMA standards and longitudinal load distribution ( K H! and K F!) of the ISO standards. The procedure is easily

programmable and flow charts of the calculation scheme as well as examples from typical software are

presented.

Published by

American Gear Manufacturers Association1500 King Street, Suite 201, Alexandria, Virginia 22314

Copyright ! 2000 by American Gear Manufacturers Association

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--779--7

American

GearManufacturers Association

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AGMA 927--A01AMERICAN GEAR MANUFACTURERS ASSOCIATION

Contents

Foreword iv. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

2 References 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

4 Iterative analytical method 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Coordinate system, sign convention, gearing forces and moments 4. . . . . . . . .6 Shaft bending deflections 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Shaft torsional deflection 14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 Gap analysis 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 Load distribution 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Future considerations 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Annexes

A Flowcharts for load distribution factor 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B Load distribution examples 26. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figures

1 Base tangent coordinate system for CW driven rotation from reference end 5.2 Base tangent coordinate system for CCW driven rotation from reference end 6

3 Hand of cut for gears and explanation of apex for bevel gears 7. . . . . . . . . . . . .

4 Gearing force sense of direction for positive value from equations 8. . . . . . . . . .

5 Example general case gear arrangement (base tangent coordinate system) 8.

6 View A--A from figure 5 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Example shaft 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 Calculated shaft diagrams 13. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 Torsional increments 15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Shaft number 3 gap 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 Shaft number 4 gap 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 Total mesh gap 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 Relative mesh gap 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 Tooth section with spring constant C"m, load L, and deflection Cd 19. . . . . . . . .

15 Deflection sections 19. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 Mesh gap section grid 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Tables

1 Symbols and definitions 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Values for factors hand, apex, rotation, and drive 7. . . . . . . . . . . . . . . . . . . . . . . .

3 Calculation data and results 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Evaluation of mesh gap for mesh #3, mm 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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AGMA 927--A01 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 927--A01, Load Distribution Factors -- Analytical Methods for Cylindrical Gears.]

This information sheet provides an analytical method to calculate a numeric value for the

face load distribution factor for cylindrical gearing.

This is a new document, which provides a description of the analytical procedures that are

used in several software programs that have been developed by various gear

manufacturing companies. The method provides a significant improvement from the

procedures used to define numeric values of face load distribution factor in current AGMA

standards. Current AGMA standards utilize either an empirical procedure or a simplified

closed form analytical calculation. The empirical procedure which is used in ANSI/AGMA

2101--C95 only allows for a nominal assessment of the influence of many parameters which

effect the numeric value of the face load distribution factor. The closed form analytic

formulations which have been found in AGMA standards suffer from the limitation that the

shape of the load distribution across the face width is limited to a linear form.

The limitations of the previous AGMA procedures are overcome by the method defined in

this information sheet. This method allows for including a sufficiently accurate

representation of many of the parameters that influence the distribution of load along the

face width of cylindrical gears. These parameters include the elastic effects due to

deformations under load, and the inelastic effects of geometric errors as well as the tooth

modifications which are typically utilized to offset the deleterious effects of the deformations

and errors.

The analytical method described in this information sheet is based on a ”thin slice” model of

a gear mesh. This model treats the distribution of load across the face width of the gear

mesh as being independent of the any transverse effects. The method also represents all of

the elastic effects of a set of meshing teeth (tooth bending, tooth shear, tooth rotation,

Hertzian deflections, etc.) by one constant, i.e., mesh stiffness (C"m). Despite these

simplifying assumptions, this method provides numeric values of the face load distribution

factor that are sufficiently accurate for industrial applications of gearing which fall within the

limitations specified.

The first draft of this information sheet was made in February, 1996. This version was

approved by the AGMA membership on October 22, 2000.

Special mention must be made of the devotion of Louis Lloyd of Lufkin for his untiring efforts

from the submittal of the original software code through the prodding for progress during the

long process of writing this information sheet. Without his foresight and contributions this

information sheet may not have been possible.

Suggestions for improvement of this document will be welcome. They should be sent to the

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

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PERSONNEL of the AGMA Helical Rating Committee and Load Distribution SubCommittee

Chairman: D. McCarthy Dorris Company. . . . . . . . . . . . . . . . . . . . . . . . .

Vice Chairman: M. Antosiewicz The Falk Corporation. . . . . . . . . . . . . . . . . .

SubCommittee Chairman: J. Lisiecki The Falk Corporation. . . . . . . . . . . . . .

SUBCOMMITTEE ACTIVE MEMBERS

K.E. Acheson The Gear Works -- Seattle. . .

W.A. Bradley Consultant. . . .

M.F. Dalton General Electric Company. . . . .

G.A. DeLange Prager, Inc.. . .

O. LaBath The Cincinnati Gear Co.. . . . . .

L. Lloyd Lufkin Industries, Inc.. . . . . . . .

J.J. Luz General Electric Company. . . . . . . .

D.R. McVittie Gear Engineers, Inc.. . . .

M.W. Neesley WesTech Gear Corporation. . .

W.P. Pizzichil Philadelphia Gear Corp.. . .

F.C. Uherek Flender Corporation. . . . .

COMMITTEE ACTIVE MEMBERS

K.E. Acheson The Gear Works--Seattle, Inc.. . .

J.B. Amendola MAAG Gear AG. .

T.A. Beveridge Caterpillar, Inc.. .

W.A. Bradley Consultant. . . .M.J. Broglie Dudley Technical Group, Inc.. . . . .

A.B. Cardis Mobil Technology Center. . . . .

M.F. Dalton General Electric Company. . . . .

G.A. DeLange Prager, Incorporated. . .

D.W. Dudley Consultant. . . .

R.L. Errichello GEARTECH. . .

D.R. Gonnella Equilon Lubricants. . .

M.R. Hoeprich The Timken Company. .

O.A. LaBath The Cincinnati Gear Co.. . . .

G. Lian Amarillo Gear Company. . . . . . . . .

J.V. Lisiecki The Falk Corporation. . . . .

L. Lloyd Lufkin Industries, Inc.. . . . . . . .

J.J. Luz General Electric Company. . . . . . . .D.R. McVittie Gear Engineers, Inc.. . . .

A.G. Milburn Milburn Engineering, Inc.. . . .

G.W. Nagorny Nagorny & Associates. . .

M.W. Neesley Philadelphia Gear Corp.. . .

B. O’Connor The Lubrizol Corporation. . . .

W.P. Pizzichil Philadelphia Gear Corp.. . .

D.F. Smith Solar Turbines, Inc.. . . . . .

K. Taliaferro Rockwell Automation/Dodge. . . .

COMMITTEE ASSOCIATE MEMBERS

M. Bartolomeo New Venture Gear, Inc.. .

A.C. Becker Nuttall Gear LLC. . . .

E. Berndt Besco. . . . . . .

E.J. Bodensieck Bodensieck Engineering Co..

D.L. Borden D.L. Borden, Inc.. . . .

M.R. Chaplin Contour Hardening, Inc.. . . .

R.J. Ciszak Euclid--Hitachi Heavy Equip. Inc.. . . . .

A.S. Cohen Engranes y Maquinaria Arco SA. . . . .

S. Copeland Gear Products, Inc.. . . .

R.L. Cragg Consultant. . . . .

T.J. Dansdill General Electric Company. . . .

F. Eberle Rockwell Automation/Dodge. . . . . . .

L. Faure C.M.D.. . . . . . . .C. Gay Charles E. Gay & Company, Ltd.. . . . . . . . .

J. Gimper Danieli United, Inc.. . . . . .

T.C. Glasener Xtek, Incorporated. . .

G. Gonzalez Rey ISPJAE

M.A. Hartman ITW. . .

J.M. Hawkins Rolls--Royce Corporation. . .

G. Henriot Consultant. . . . . .

G. Hinton Xtek, Incorporated. . . . . . .

M. Hirt Renk AG. . . . . . . . .

R.W. Holzman Milwaukee Gear Company, Inc.. .

R.S. Hyde The Timken Company. . . . . .

V. Ivers Xtek, Incorporated. . . . . . . .

A. Jackson Mobil Technology Company. . . . .

H.R. Johnson The Horsburgh & Scott Co.. . .

J.G. Kish Sikorsky Aircraft Division. . . . . . .

R.H. Klundt The Timken Company. . . . .

J.S. Korossy The Horsburgh & Scott Co.. . . .

I. Laskin Consultant. . . . . . . .

J. Maddock The Gear Works -- Seattle, Inc.. . . . .

J. Escanaverino ISPJAE.

G.P. Mowers Consultant. . . .R.A. Nay UTC Pratt & Whitney Aircraft. . . . . . .

M. Octrue CETIM. . . . . .

T. Okamoto Nippon Gear Company, Ltd.. . . . .

J.R. Partridge Lufkin Industries, Inc.. . .

M. Pasquier CETIM. . . .

J.A. Pennell Univ. of Newcastle--Upon--Tyne. . . . .

A.E. Phillips Rockwell Automation/Dodge. . . . .

J.W. Polder Delft University of Technology. . . . .

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E. Sandberg Det Nordske Veritas. . . .

C.D. Schultz Pittsburgh Gear Company. . . .

E.S. Scott The Alliance Machine Company. . . . . .

A. Seireg University of Wisconsin. . . . . . .

Y. Sharma Philadelphia Gear Corporation. . . . . .

B.W. Shirley Emerson Power Transmission. . . .

L.J. Smith Invincible Gear Company. . . . . .

L. Spiers Emerson Power Trans. Corp.. . . . . . .

A.A. Swiglo IIT Research Institute/INFAC. . . . .J.W. Tellman Dodge. . . .

F.A. Thoma F.A. Thoma, Inc.. . . . .

D. Townsend NASA/Lewis Research Center. . . .

L. Tzioumis Dodge. . . . .

F.C. Uherek Flender Corporation. . . . .

A. Von Graefe MAAG Gear AG. . .

C.C. Wang 3E Software & Eng. Consulting. . . . .

B. Ward Recovery Systems, LLC. . . . . . . .

R.F. Wasilewski Arrow Gear Company.

H. Winter Technische Univ. Muenchen. . . . . . .

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American Gear Manufacturers Association --

Load Distribution Factors

-- Analytical Methods for

Cylindrical Gears

1 Scope

This information sheet covers a method for the

evaluation of load distribution across the teeth of

parallel axis gears. A general discussion of the

design and manufacturing factors which influence

load distribution is included.

The load distribution factors for use in AGMA parallel

axis gear rating standards are defined, to improve

communication between users of those standards.

Historically, analytical methods for evaluating load

distribution in both AGMA and ISO standards have

been limited by the assumption that load is linearly

distributed across the face width of the meshing gear

set. The result of this assumption is often overly

conservative (high) values of load distribution fac-

tors. The method given here is considered more

correct.

1.1 Method

A simplified iterative method for calculation of the

face load distribution factor, based on combinedtwisting and bending displacements of a mating gear

and pinion, is presented. The transverse load

distribution (in the plane of rotation) is not evaluated

in this information sheet. This method assumes that

the mesh stiffness is a constant through the entire

contact roll and across the face. General guidance

for design modifications to improve load distribution

is also included.

1.2 Limitations of method

This method is intended to be used for general gear

design and rating purposes. It is intended to provide

a value of load distribution factor and a means by

which different gear designs can be analytically

compared. It is not intended for rigorous detailed

analysis to calculate the actual distribution of load

across the face width of gear sets.

The knowledge and judgment required to evaluate

the results of this method come from experience in

designing, manufacturing and operating gear units.

This method is intended for use by the experienced

gear designer, capable of understanding its limita-

tions and purposes. It is not intended for use by the

engineering public at large.

2 References

The following documents were used in the develop-

ment of this information sheet. At the time of

publication, the editions were valid. All publications

are subject to revision, and the users of this manual

are encouraged to investigate the possibility of

applying the most recent editions of the publications

listed:

AGMA Technical Paper P109.16, Profile and

Longitudinal Corrections on Involute Gears, 1965

ANSI/AGMA 1012--F90, Gear Nomenclature,

Definitions Of Terms With Symbols

ANSI/AGMA 2101--C95, Fundamental Rating Fac-

tors And Calculation Methods For Involute Spur

And Helical Gear Teeth

ANSI/AGMA ISO 1328--1, Cylindrical Gears -- ISO

System of Accuracy -- Part 1: Definitions and

Allowable Values of Deviations Relevant to Corres- ponding Flanks of Gear Teeth

ISO 6336--1:1996, Calculation of load capacity of

spur and helical gears -- Part 1: Basic principles,

introduction and general influence factors

Dudley, D.W., Handbook of Practical Gear Design,

McGraw--Hill, New York, 1984

Timken Engineering Design Manual, Volume 1

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3 Definitions and symbols

The terms used, wherever applicable, conform to

ANSI/AGMA 1012--F90.

NOTE: The symbols and definitions used in this stan-

dard may differ from other AGMA standards. The user

should not assume that familiar symbols can be used

without a careful study of their definitions.

The symbols and terms, along with the clause

numbers where they are first discussed, are listed in

alphabetical order by symbol in table 1.

3.1 Load distribution factor

The load distribution factor, K H, modifies the rating

equations to reflect the non--uniform distribution of

load along the gear tooth lines of contact as they

rotate through mesh. In past AGMA standards, the

variables Cm and K m have been associated with this

factor. In ISO standards, the variables K H!, K H#, K F!and K F#, have been associated with the factor. In

current AGMA standards the load distribution factor,

K H, is used for both pitting resistance and bending

strength calculations. There is no separate value,

K F, for bending strength as found in ISO standards.

The magnitude of K H is affected by two components,

transverse load distribution factor and face load

distribution factor.

The transverse load distribution factorpertains to theplane of rotation and is affected primarily by the

correctness of the profiles and indexing of themating

teeth. Standard procedures to evaluate it have not

been established and it is assumed to be unity in this

information sheet.

The face load distribution factor is the focus of this

information sheet.

3.2 Target mesh

The target mesh is that mesh for which load

distribution is being analyzed. The target meshincludes a target pinion and a target gear.

Table 1 -- Symbols and definitions

Symbol Definition Units First

referenced

A Apex factor -- -- 5.3

BT Axis in the base tangent plane -- -- 5.2

BTN Axis normal to base tangent plane -- -- 5.2

BTZ Axis in the base tangent plane perpendicular to BT -- -- 5.2

C"m Tooth stiffness constant, for the analysis N/mm/ mm 9.1

b Helical/bevel gear face width mm 5.3

D Drive factor -- -- 5.3

DpG Operating pitch diameter, gear mm 5.3

d Outside effective twist diameter mm 7.1

d in Inside shaft diameter mm 6.1

d sh Outside diameter, effect outside diameter of the teeth mm 6.1

E Modulus of elasticity N/mm2 6.1

F aG Axial thrust force, gear member N 5.3

F aP Axial thrust force, pinion member N 5.4

F g Total load in the plane of action N 9.2

F i Gearing or external force at a distance N 6.1

F sG Separating force, gear member N 5.3

F sP Separating force, pinion member N 5.3

F tG Tangential force, gear member N 5.3

F tP Tangential force, pinion member N 5.3

G Modulus of elasticity in shear N/mm2 7.1

H Hand factor -- -- 5.3

I Moment of inertia mm4 6.1

(continued)

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Table 1 (concluded)

Symbol Definition Units First

referenced

IC Integration constant -- -- 6.1

i Station number -- -- 6.1

K H Load distribution factor -- -- 9.4

Ls Distance between the supports (reactions) -- -- 6.1

L j Load at station N 7.1 L$ Load intensity N/mm 9.1

M Bending moment N mm 6.1

M G Moment due to axial thrust force N mm 5.4

n Station number at end support -- -- 6.1

P Power transmitted through the mesh kW 5.3

R Rotation factor -- -- 5.3

RL Reaction at the left bearing N 6.1

RR Reaction at the right bearing N 6.1

S Speed of shaft rpm 5.3

SLi Station slope value -- -- 6.3

t $i Torsional deflection at a station mm 7.1V Shear N 6.1

xi Length of face where point load applied mm 9.2

X j Distance between adjacent stations mm 7.1

X fi Distance from left support to load location mm 6.1

x Distance between stations mm 6.1

y Deflection along the line of action mm 6.1

$ti Tooth deflection at a load point mm 9.1

"G Bevel pitch angle of gear degrees 5.3

"P Bevel pitch angle of pinion degrees 5.3

% Helix angle/spiral angle degrees 5.3

Ô Normal pressure angle degrees 5.3

4 Iterative analytical method

This information sheet presents an iterative analyti-

cal method for determining a value of load distribu-

tion factor. The iterative method combines the

calculated elastic deflection of the pinion and the

gear with other misalignments. The result defines a

“mesh gap” in the base tangent plane which is the net

mismatch between the gear and the pinion. The

teeth in mesh are modeled by an equally spaced

series of independent parallel compression springs

which representthe mesh stiffness. The mesh gapis

then mathematically closed by compressing the

springs until the sum of the spring forces equals the

total tooth force.

The method has the ability to consider the following

influences:

-- tooth alignment deviations of pinion and gear;

-- tooth alignment and crowning modification;

-- alignment of the axes of rotation of the pinion

and gear, including bearing clearances and

housing bore alignment;

-- mesh elastic deflections due to Hertzian

contact and tooth bending;

-- shaft elastic deflections due to twisting andbending, resulting from the target mesh loads and

loads external to the mesh.

Influences that may be accounted for by estimating

values and including them as equivalent misalign-

ments of the target shaft axes are:

-- elastic deflection of a gear body if it is not a

solid disk (such as a spoke gear);

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-- elastic deflection of the housing and

foundations;

-- displacements of the gearing due to bearing

deflection;

-- thermal or centrifugal effects;

-- running--in or lapping effects.

The method does not consider the followinginfluences:

-- tooth profile, spacing and runout deviations;

-- total tooth load including increases due to

application influences and tooth dynamics;

-- variations of stiffness of the gear teeth;

-- double helical gears with one helix

overloaded.

4.1 Methodology

The iterative analytical method consists of thefollowing basic steps:

1) Calculate the mesh gap resulting from an

initial uniform load distribution;

2) Calculate a new load distribution by mathe-

matically closing the mesh gap. This is accom-

plished by compressing the springs until the sum

of the spring forces equals the total tooth force;

3) Calculate a new mesh gap resulting from the

new load distribution;

4) Repeat steps 2 and 3 until the change in loaddistribution from the previous iteration is

negligible;

5) The load distribution factor is then calculated

from this final load distribution.

4.1.1 Calculated elastic deflections

Deflections which are calculated within the iterative

method include the elastic deflections of the pinion

and gear shafts, plus the mesh. Elastic shaft

deflections include shaft twist and bending. Elastic

tooth deflections include Hertzian contact and tooth

bending.

4.1.2 Equivalent misalignment inputs

Other displacements that are treated by combining

them as an equivalent deflection at the target mesh

include:

-- alignment deviations and modifications of

pinion and gear teeth;

-- equivalent elastic deflection of non--solid

body gears (such as a spoke gear);

-- elastic deflection of the housing and

foundations;

-- displacements due to bearing clearance,

alignment and deflection;

-- thermal or centrifugal effects;

-- running--in or lapping effects.

4.2 Assumptions and simplifications

The following assumptions and simplifications are

-- the weight of components is ignored;

-- effects of uneven distribution of load on

meshes other than the target mesh are ignored;

load on these meshes is treated as concentrated

in the center of the mesh;

-- shear coupling between the mesh gap com-

pression springs representing the mesh stiffness

is ignored;

-- mesh stiffness is a constant across the full

width of tooth;

-- all shafts are supported on two bearings;

-- for double helical gears the net thrust force is

zero as the thrust force from each helix cancels

each other;

-- for double helical gears the tangential and

separating force is distributed equally on each

hand helix; this is generally true as long as onemember can float with respect to the other with no

external axial load applied.

5 Coordinate system, sign convention,

gearing forces and moments

5.1 Rules

The rules that govern the coordinate system, sign

convention, gearing forces and moments are:

-- the target mesh shafts are mutually parallel;

-- the coordinate system for all calculations lies

in the base tangent plane;

-- the base tangent plane is a plane tangent to

the base circles of the target mesh;

-- the driving element is the element for which

contact first occurs in the root of the tooth and

traverses to the tip of the tooth;

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-- a modified Timken sign convention is

followed;

-- each analysis includes only the two shafts

under consideration;

-- the origin of the shaft is the bearing or point of

application of a force or moment on the target

pinion shaft which is most remote from the target

mesh toward the reference end of the shaft (see5.2);

-- the input torque to the driving element enters

the shaft from one side only and is fully balanced

by torque in the target mesh.

5.2 Coordinate system and sign convention

The coordinate system is aligned with the base

tangent plane, BTP, of the target mesh and is defined

as the base tangent coordinate system, BTCS. The

BTCS is comprised of three orthogonal axes: BT,

BTN (base tangent normal), and BTZ.

The BTZ axis is parallel to the axes of the target

mesh shafts. The BT axis lies in the BTP and is

perpendicular to the BTZ axis. The BTN axis is

perpendicular to both the BT and the BTZ axes

(normal to the base tangent plane). The origin of the

BTCS lies at the intersection of the base tangent

plane and the edge of the target mesh face closest to

the reference end (see figures 1 and 2).

For consistency in defining the positive direction of

the BTCS axes and in calculating the mesh loads, a

“reference end” needs to be identified. For purposes

of this information sheet,the reference end is theend

of the driving element shaft opposite the torque input

end.Using this definition of the refence end, the positive

directions of the BTCS axes are determined as

follows:

+ BTZ: away from the reference end;

+ BTN: toward the driven element;

+ BT: obtained by right hand rule; BTN to BTZ.

Figures 1 and 2 illustrate the base tangent plane and

the base tangent coordinate system for a typical

targetmesh. In figure 1, the input torque is clockwise

when viewed from the reference end. In figure 2, theinput torque is counterclockwise when viewed from

the reference end.

The force, moment and deflection along the positive

direction of BT, BTN and BTZ are assigned positive

values. Along the negative direction of BT, BTN and

BTZ, they are assigned negative values.

+BTZ+BT

+BTNReferenceend

Driver

Driven

Base tangent plane

Base diameter --driven element

Target shaft --driven

Target shaft --driver

Target mesh

Base diameter --driving element

Inputtorque

Figure 1 -- Base tangent coordinate system for CW driven rotation from reference end

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Referenceend

Driver

Driven

Base diameter --driven element

Target shaft --

driven

Target shaft --driver

Target meshBase diameter --driving element

Inputtorque

Base tangentplane.

Figure 2 -- Base tangent coordinate system for CCW driven rotation from reference end

5.3 Gearing forces and signs

Meshing gear members transmitting torque will

cause forces and moments to develop on the shafts

that carry these gear members. These forces and

moments will cause deflections of the shafts that will

tend to affect the alignment and ultimately the

distribution of the load across the face width of the

mesh. These elastic deflections need to be com-bined with all other sources of potential misalign-

The forces on the gear member are given by

equations 1 through 3. In these equations, the

values of factors H , A, R, and D are obtained using

table 2. When properly applied, these factors will

ensure that the proper direction of the forces are

determined. The directions obtained will be

consistent with the BTCS definition presented in 5.2.

The tangential force is calculated as:

(1) F tG ! 1.91 " 107 P ( D R)

S # DpG $ b sin "G%

F tG is tangential force, gear member, N;

P is power transmitted through the mesh, kW;

D is drive factor (see table 2);

R is rotation factor (see table 2);

S is speed of gear shaft, rpm;

DpG is operating pitch diameter, gear, mm;

b is helical/bevel gear face width, mm;

"G is bevel pitch angle, gear, degrees.

The separating force is calculated as:

F sG !& F tG

&'# A D H R sin % sin "G ( tan Ô cos "G%)

F sG is separating force, gear member, N;

A is apex (bevel) factor (see table 2);

H is hand factor (see table 2);

% is helix angle/spiral angle, degrees;

Ô is normal pressure angle, degrees.

The thrust (axial) force is calculated as:

aG !& F

tG&( A)'# A D H R sin % cos "

G $ tanÔ sin "

F aG is axial thrust force, gear member, N.

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Table 2 – Values for factors hand, apex, rotation, and drive

Factor description Factor Value Condition

Hand H +1

Right hand helix or spiral (see figure 3)

Left hand helix or spiral (see figure 3)

Spur, straight bevel, or herringbone

Apex (bevel) A +1

Apex toward reference end (see figure 3), or no apex

Apex away from reference end (see figure 3)

Rotation R +1

Clockwise viewed from reference end

Counterclockwise viewed from reference end

Drive D +1

Driving element

Driven element

For gears having no helix, spiral, or pitch angles, set

the values of these angles equal to zero in equations

1 to 3.

To obtain the force for the pinion member, replace

the gear values in equations 1 through 3 with the

corresponding pinion values.

Figure 4 shows the sign convention to use for the

direction of the gear forces. The direction shown is

for the positive value of forces evaluated by the

above equations. The forces must be determined for

each mesh on each of the target mesh shafts.

With the sign convention of figure 3 and the definition

of the BT axis, the tangential mesh load on the

driving element will introduce positive mesh dis-

placement in the base tangent plane.

Figure 5 shows a general arrangement. For this

example, mesh 3 is the target mesh. Shafts 3 and 4

are the target shafts.

Right handhelix

Left handhelix

Away fromreference

Towardreference

Right handspiral

Left handspiral

Figure 3 -- Hand of cut for gears and explanation of apex for bevel gears

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F a F a

If mate to target shaft is onthe right, use these positiveforce directions

View direction fromreference end

Mateshaft

One targetshaft

Mating targetshaft

If mate to targetshaft is on theleft, use thesepositive forcedirections

Figure 4 -- Gearing force sense of direction for positive value from equations

Mesh 3FtG3

+BT -- Axis along base tangent plane of target mesh+BTN -- Axis normal to base tangent plane of target mesh

CLGearface

Reference endand origin of shaft for mesh 3

Mesh 2

Bearing

DriverRH

DrivenLH

DrivenRH

DriverRH

Shaft 1

Mesh 1

FtP1 FsG1

Shaft 2

Shaft 4Base tangent

plane formesh 3

Base diameter formember typical

Base tangentplane for mesh 2

Driven

FaP1FaG1

DriverLH

Reference endand origin of shaft for mesh 2

Shaft 3

Base tangent coordinatesystem for mesh 2

Example showing actual direction of the forces as determined from the sign of the values calculated in theforce equations.

Figure 5 -- Example general case gear arrangement (base tangent coordinate system)

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5.4 Gearing moments

The axial thrust forces acting on the pinion and gear

cause moments. For the target mesh, the moments

can be determined for each mesh section. For each

additional mesh on the target shafts, the resulting

moment is assumed to act at the center of the face

width. For a double helical mesh the net moment will

be zero.The moment due to an axial thrust force on the gear

member is given by equation 4.

(4) M G ! aG pG

M G moment due to axial thrust force, N mm.

To obtain the moment due to an axial thrust force on

the pinion member, replace the gear values by the

corresponding pinion values.

Figure 6 shows the tangential and separating forces

and the axial thrust moments acting onshafts 3 and 4

of figure 5. These forces affect the load distribution

of mesh 3. Figure 6 demonstrates the resolution of

the shaft 3 and 4 forces and moments into the base

tangent coordinate system for mesh 3.

6 Shaft bending deflections

Gears transmitting power will impose forces and

moments on their shafts, which will cause elastic

deflections. These deflections can affect the align-

ment of the gear teeth and therefore affect the load

distribution across the gear face width.

This section presents a simplified computer pro-

grammable integration method for calculating the

bending deflection of a stepped shaft with radial

loads imposed and two bearing supports.

Rules for calculating bending deflection when calcu-

lating load distribution factor are also presented.

6.1 Simplified bending calculation routine

As explained in other sections, when calculating

shaft deflections, the area of the gear teeth is broken

into eighteen separate load application sections.

However, to simplify the explanation of the deflection

calculation method the following model and explana-

tion will be of a stepped shaft with two supports,three

changes in diameter, and two point loads. This is as

shown in figure 7 and table 3.

Driver

DriverLH

DrivenLH

BT -- Axis along base tangent plane of target mesh

BTN -- Axis normal to base tangent plane of target mesh

Base tangent coordinate

system for mesh 3

Shaft 4

Base tangentline

TargetMesh #3

Shaft 2

Mesh 2DrivenRH

Shaft 3

F tG23

F sG23

F tP33

F tG34

F sG34

Figure 6 -- View A--A from figure 5

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b l e 3

- - C a l c u l a t i o n

d a t a a n d

r e s u l t s

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1 2 3 4 5 6 7

--13500 +9000

+6180 --1680

50.0 38.044.035.0

28.0 28.0 25.0 25.0

Figure 7 -- Example shaft

All modeling will be from the left--hand supportmoving toward the right--hand support. Deflection at

supports is zero. The gearing forces and any other

external forces are used to obtain the free body force

diagram. In the force diagram the forces, F i, and the

distances they act from the left support, X fi, are

specified.

Using standard static force analyses calculate the

reaction, RR, at the right side support by summing

the moments about the left support.

(5) RR !

*'# F i% # X fi

F is the force applied at a distance, N;

Ls is the distance between the two supports;

X fi is the distance from left support to loadlocation, F i.

(6) X fi ! x i ( X f #i$1%+++ i ! 1,2,3, +++ n

Then calculate the reaction at the left using the total

sum of the loads.

(7) RL ! * F i $ RR

It is critical that sign convention be maintained during

the calculations with the preceding formulas.

The basic equation for small deflection of a stepped

shaft is:

(8)d 2 y

dx2 ! M

where x is the distance between stations, mm;

M is the bending moment, Nmm;

I is the moment of inertia, mm4;

E is the modulus of elasticity, N/mm2;

y is the deflection, mm.

Integrating equation 8 twice gives deflection. The

following step by step procedure applied to the

stepped shaft as shown in figure 7 will illustrate the

procedure evaluating shaft deflection. A tabulated

form as shown in table 3 lends itself to the process.

Step 1: Divide the shaft into lengths with intervals

beginning at each force and at each change in

section (see figure 7).

Step 2: Label the ends of intervals with station

numbers beginning at the left support with station i=1

and ending at the right support with station i = n.

Step 3: List station numbers, i, on alternate lines in

column 1 of calculation sheet (see table 3).

Step 4: List free body forces in column 4 on the

same lines as the station numbers at which they

occur. Care should be taken to designate proper

signs to forces (upward forces are considered

positive in this example).

Step 5: Calculate the shear, V i, at each station by

summing the values in column 4. Tabulate each

shear value in column 5, one station below the

station for which it is calculated. The last shear value

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should be numericallyequal to but opposite in sign to

the last force listed in column 4.

(9)V i(1 ! V i ( i,,, i ! 1,2,3, +++ n $ 1

V is the shear, N;

i is the station number;

n is the station number at end support.

Step 6: In column 6, on the same line as the station

number, list the distance to the preceding station.

Step 7: Calculate bending moment, M i, at each

station and list the value in column 7. Value at the

first station is zero. Values at succeeding stations

are obtained by summing the products of shear

force, V i (column 5), and distance between stations,

xi (column 6). The moment at the first and last

station, i =1and i = n, should be zero(i.e. M 1=0.0 and

M n = 0.0).

(10) M i(1 ! M i ( #V i(1

%# xi(1%+++ i ! 1,2,3, +++ n

Step 8: Calculate the moment of inertia, I i, in

bending for each interval. Place the I value in

column 8 on the line between the two stations at

which the interval begins and ends.

(11) I i !'#d 4

sh i $ d 4

64 +++ i ! 1,2,3, +++ n

d sh is the outside shaft diameter (see 6.2), mm;

d in is the inside shaft diameter, mm;

Step 9: Multiply each I i value by modulus of

elasticity, E, and insert the EI i value in column 9 on

the same lines as corresponding I i values. For steel

use E = 206 000 N/mm2. Dividing the EI i values by

103 before tabulating them in column 9 results in

units of "m for the rest of the tabulation.

(12) EI i ! ( E)# I i%+++ i ! 1,2, 3, +++ n $ 1

Step 10: Divide each bending moment M i value incolumn7 by the EI i value in column9 which precedes

and follows it. List these two values, MEI ui and MEI li,

in column 10.

(13) MEI ui ! M i EI i

+++ i ! 1,2,3, +++ n $ 1

(14) MEI li ! M i(1

EI i+++ i ! 1,2,3, +++ n $ 1

Step 11: Obtain the average MEI values, AMEI i, for

each interval by averaging the values on the lines on

which the station is listed and the following line. List

the average values on the lines between stations in

column 11.

MEI i ! M ui ( M li

2 +++ i ! 1,2,3, +++ n $ 1

Step 12: Calculatethe slope value, SLi, incolumn 12

starting with zero at station 1 (i.e., SL1=0). Succeed-

ing values are obtained by summing the products of

AMEI i from column 11 and the xi value on the next

lower line of column6. These values arelisted on the

same lines as the stations.

SLi(1 ! SL i ( # AMEI i%# xi(1

%+++ i ! 1,2,3, +++ n $ 1

Step 13: Average the slope values in column 12 at

the beginning and end of each interval. These

values, ASLi, are listed on the lines between stationsin column 13.

(17) SLi ! S i ( S i(1

2 +++ i ! 1,2,3, +++ n $ 1

Step 14: Obtainthe deflection increment values, DI i,

in column 14 by multiplying the average slope value

in column 13 and the xi value from the next lower line

in column 6.

(18) DI i ! # ASLi%# xi(1

%+++ i ! 1,2, 3, +++ n $ 1

Step 15: The next step is to evaluate the integration

constant which depends on type of shaft. For thesimply supported shaft with no load outside of the

supports as shown in figure 7, the constant is

obtained by summing the deflection increment

values in column 14 to obtain Sy. The sign of Sy is

changed and the sum divided by the distance

between the reaction, Ls, to obtain the integration

constant per mm of length.

(19) Sy !i ! 1

(20) Ls !i ! 1

(21) IC !$ Sy

Other shaft configurations will change the

integration constant.

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Shear Diagram, V

Moment Diagram, M

--10 (mm)

0.4 (mrad)

--0.01 (1/ mm)

+0.01 (1/ mm)

1 2 3 4 5 6 7

--13500 +9000

+6180 --1680

28.0 28.0 25.0 22.0

+10000 (N)

--10000 (N)

350000 (Nmm)

--150000 (Nmm)

Diagram M EI

0.0Slope Curve

Deflection Curve

35.0 50.0 44.0 38.0

Figure 8 -- Calculated shaft diagrams

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Step 16: The integration constant for each section,

ICSi, is now calculated. Multiply integration constant,

IC, calculated in step 15 by xi value on the next lower

line from column 6 to obtain the constant for each

section. List these values in column 15 on the same

line as the average slope and deflection increments.

(22) ICSi !

i(1%+++i

! 1,2,3,

Step 17: Column 16 is the calculated deflection.

Place zero at left support location, i.e. y1=0.0,

because support locations must have zero deflec-

tion. For all other stations the deflection values are

obtained by summing together the deflection incre-

ment and integration constant values from columns

14 and 15. These deflection values are inserted on

the same line as the station. As a math check when

summing the values of yi the calculated value at the

right support location, yn, should be very close to

(23)i(1 ! y i ( I i ( ICSi+++ i ! 1,2, 3, +++ n $ 1

6.2 Rules

When using the shaft bending deflection routine

explained in 6.1 to calculate load distribution, the

following rules apply:

-- This is a two dimensional deflection analysis;

-- Shear deflections are not included;

-- The length between any two stations is critical

to the accuracy of this calculation. Rules for

station length are: no longer than 1/2 diameter of

the station; no longer than 3 times the shortest

section of the non--gear tooth portion of the shaft;

no longer than 30 mm.

When in doubt about the number of stations, if

adding more does not significantly change the

calculation results, the number of original stations

is adequate.When calculating bending deflection for load dis-

tribution factor, the following rules also apply:

-- Only forces acting in the base tangent plane

are considered;

-- When calculating shaft deflections, the area

of the gear teeth is broken into eighteen equal

sections;

-- The effective bending outside diameter of the

teeth is the (tip diameter minus root diameter)/2

plus the root diameter;

-- The moment couple applied to single helical

gears due to the thrust component of tooth

loading can be modeled as equal positive and

negative forces at a location just to the left and

right of the gear tooth area.

7 Shaft torsional deflection

Meshing gear sets transmitting torque will also twist

the shafts that carry the gear elements. The twist will

cause deflection at the teeth that will affect the load

distribution across their face width.

7.1 Torsional deflection

The torque input end is subjected to full torque. The

torque value decreases along the face until itbecomes zero at the other end. Hence the direction

of torque path is of importance.

Consider a cylindrical shaft with circular cross

section with outside effective twist diameter, d , inside

diameter, d in, and incremental length, X j, as shown in

figure 9.

The equation for torsional twist can be found in

machinery design text. The torsional deflection must

be calculated over the length of the tooth face. The

twist must be converted from radians to a deflection

in the base tangent plane. Equation 24 is in a formthat allows summation using the discreet stations

used in this document. This results in the equation:

t $i !

#103%-./

* j ! 1

L j -01-./

* j ! 1

X j-01

G '#d 4 $ d 4in%

t $i is torsional deflection at a station, mm;

L j is load at a station, N;

X j is the distance between adjacent stations,mm;

d is effective twist diameter (see 7.2), mm;

d in is inside diameter, mm;

i is station number;

G is shear modulus (83 000 N/mm2 for steel).

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Torqueinput

Torsionaldeflection

Li Load on teeth

Torqueinput

FacewidthUndeformed

position

Figure 9 -- Torsional increments

At the first point of interest on the tooth where j = 1,

the summation of X j will be zero and the torsional

deflection is zero. Continued calculation of the

torsional twist toward the end of the tooth face where

torque is being applied results in a maximum

torsional deflection, see figure 9.

Equation 24 is an approximation which yields

reasonable results for gearing. The theoreticallycorrect equation would be an integration.

A slightly more accurate approximation is found in

equation 25.

t $i !

#103%-.

(i $ 1)

*k ! 1

* j ! 1

L j X k-45-0

18 d 2

G '#d 4 $ d 4in%

7.2 Rules

Since the angle is small, it is assumed that the

deflection in the base tangent plane is proportional to

the twist angle.

The rules that apply to this shaft torsional deflection

-- the outside effective twist diameter of tooth

section is the root diameter plus 0.4 times the nor-

mal module;

-- the twist of all elements except the target

mesh being analyzed is ignored;

CAUTION: Equations 24 and 25 only cover torques in

thetargetmesh that arise from gear tooth loading. Oth-

er torques may require additional modeling.

8 Gap analysis

Elastic bending and torsional deflections, tooth

modifications, lead variations and shaft misalign-

ments cause the gear teeth to not be in contact

across the entire face width. The distance between

non contacting points along the face width of the

mating teethis defined as thegap. This gap is closed

to some degree when the gear set is loaded due to

the compliance of the gear teeth along the face widthof the target mesh.

Bending deflection: Use the values obtained from

the bending analysis for each shaft increment of the

target mesh. Retain the positive or negative sign of

the bending deflection.

Torsional deflection: Use the values from the

torsional analysis for each shaft increment of the

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target mesh. Retain the positive or negative sign of

torsional deflection.

Tooth modification: Tooth modification accounts

for lead modification and crowning. The sign

convention for tooth modification as illustrated in

table 4 is the following: if the load direction on the

teeth is positive, removal of metal at an individual

station is entered as a positive value; if the directionof load on the teeth is negative, the removal of metal

at an individual station is entered as a negative

value.

Lead variation: The actuallead variation of thegear

set is not available at the design stage. At this stage

lead variation based on the expected ANSI/AGMA

ISO 1328--1 tolerance of the gear set may be used.

The lead variation must be incorporated so as to

increase the total mesh gap (check both directions).

At final verification stage use actual lead variation

measured for the gear set. The lead variationcorresponding to material removal from the tooth

flank has the same sign as the load on the tooth flank

when it is entered in table 4.

Shaft misalignment: Shaft misalignment accounts

for the error in concentricity of the bearing diameters

on the shaft, bearing clearance, housing bore

non--parallelism, etc. At design stage, values should

be based on expected manufacturing accuracy.

Incorporate expected shaft misalignment so as to

increase mesh gap (check both directions).

At final verification stage use actual shaft misalign-

ment. The shaft misalignment that corresponds to

material removal on the tooth flank has the samesign as the load on the tooth flank when entered in

table 4.

Use the deflections, modifications, variations and

misalignment values with proper positive or negative

signs for each shaft of the target mesh to form table

4. In table 4, the shaft gap is the algebraic sum of all

deflections, tooth modifications, lead variation and

misalignment. The difference between the individual

shaft gap positions is the total mesh gap. To

evaluate load distribution by the iterative method the

relative gap is used. Relative mesh gap at each

station of interest is obtained by subtracting the least

total mesh gap from the total mesh gap at the station.

The last column in table 4 reflects the relative mesh

Table 4 is an example of the mesh gap evaluated for

mesh #3 of general arrangement shown in figure 5.

Table 4 -- Evaluation of mesh gap for mesh #3, mm

Shaft #3 Shaft #4

Stationnumber

Bendingdeflec-

Torsionaldeflec-

Toothmodifi-

cation

Leadvaria-

Shaft mis-

alignment

Shaft #3

Bendingdeflec-

Torsionaldeflec-

Toothmodifi-

cation

Leadvaria-

Shaft mis-

alignment

Shaft #4

otameshgap

e at vmeshgap

8 11.8 --9.1 5.0 0.0 0.0 7.7 --12.8 8.6 0.0 0.0 0.0 --4.2 11.9 0.0

9 11.7 --8.9 3.5 0.3 0.8 7.4 --12.7 8.4 0.0 --0.3 --0.8 --5.4 12.8 0.9

10 11.5 --8.5 2.7 0.6 1.3 7.6 --12.6 8.0 0.0 --0.6 --1.3 --6.5 14.1 2.2

11 11.3 --7.9 2.0 0.8 1.8 8.0 --12.4 7.4 0.0 --0.8 --1.8 --7.6 15.6 3.7

12 11.0 --7.1 1.3 1.0 2.3 8.5 --12.1 6.6 0.0 --1.0 --2.3 --8.8 17.3 5.4

13 10.7 --6.1 0.7 1.3 2.8 9.4 --11.8 5.6 0.0 --1.3 --2.8 --10.3 19.7 7.8

14 10.3 --4.9 0.0 1.5 3.3 10.2 --11.4 4.4 0.0 --1.5 --3.3 --11.8 22.0 10.1

15 9.9 --3.5 0.0 1.7 3.8 11.9 --11.0 3.0 0.0 --1.7 --3.8 --13.5 25.4 13.5

16 9.5 --2.1 1.0 2.0 4.3 14.7 --10.5 1.6 0.0 --2.0 --4.3 --15.2 29.9 18.0

17 9.1 --0.8 3.5 2.2 4.8 18.8 --9.9 0.8 0.0 --2.2 --4.8 --16.1 34.9 23.0

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SHAFT #3

09 10 11 12 13 14 15 16 178

M i c r o m e t e r s

Figure 10 -- Shaft number 3 gap

08 9 10 11 12 13 14 15 16 17

SHAFT #4

Figure 11 -- Shaft number 4 gap

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SHAFT #3

08 9 10 11 12 13 14 15 16 17

SHAFT #4SHAFT #4

Figure 12 -- Total mesh gap

SHAFT #3

010 11 12 13 14 15 16 17

SHAFT #4

Figure 13 -- Relative mesh gap

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9 Load Distribution

9.1 Tooth deflection

This method uses the concept of a tooth mesh

stiffness constant, C"m, to compare the tooth load

intensity and tooth deflection with the total load and

overall mesh gap. For simplicity, the base tangent

plane along the line of action is used and multiple

teeth in contact are ignored. Effectively the mesh is

analyzed as if it were a spur set. For the purpose of

illustrating this concept, this clause will use only 6

sections in the mesh area. Hertzian contact and

tooth bending deflections are combined to produce a

single mesh stiffness constant, C"m, and themeshis

assumed to be a set of independent springs (as

shown in figure 14).

The tooth deflection at a given point is a linear

function of the load intensity at that point and the

tooth mesh stiffness as shown in equation 26 below.

(26) L$i ! $ ti C"m

L$i is load intensity, N/mm;

$ti is tooth deflection at a load point “i”, mm;

C"m is tooth stiffness constant for the analysis,

N/mm/ mm (611 N/mm/ mm for steel gears).

Face width

Li meshgap,$i

Figure 14 -- Tooth section with spring constant

C !m, load L, and deflection "

This assumed linearity differs from previous AGMA

(AGMA 218) and ISO (ISO 6336--1, C) analytical

methods where the load distribution was assumed

as a straight line over the whole face width.

Clause 8 explains the methods used to calculate the

mesh gap. This gap in the mesh must be accommo-

dated by deflection of the teeth,$t, as shown in figure

14 and equation 26.

9.2 Mesh gap analysis

The mesh gap analysis divides the target mesh into

discreet equal length sections, X i, with point loads,

Li, applied in the center of each of these sections(see figure 15). For double helical, analyze each

helix separately. Since the method for calculating

mesh gap uses point loads, while the tooth deflec-

tions per equation 26 are based on load intensity, the

point loads must be converted to load intensity. This

is shown in equation 27.

(27)$i !

X i is length of face where point load is applied,

Li is load at a specific point “i”, N.

Face width

X 1 X 2 X 3 X 4 X 5 X 6

Bearing

Figure 15 -- Deflection sections

Note that load is not applied directly on the ends of

the tooth. This should improve accuracy as mesh

stiffness is generally lower at the ends of the teeth,

but it is assumed constant in this analysis. Also note

that the tooth is divided into equal length sections

such that all values of X i are equal. In addition, the

sum of the individual loads must equal the total load

on the gearset as shown in equation 28.

(28) F g ! L1 ( L2 ( L3 (+++( Ln

F g is total load in plane of action, BTP , N.

The difference in load intensity between any two

points, i and j, is proportional to the difference in

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mesh gap between these two points multiplied by the

tooth stiffness constant. Notice the switch in terms.

The absolute tooth deflection is not used, rather the

change in mesh gap which is equal to the change in

tooth deflection is used. Therefore, equation 29

below can be derived from equation 26 (see figure

(29) L$i $ L$ j ! #$i $ $ j%C"m

In terms of the point loads used in the mesh gap

analysis, equation 29 may be rewritten as:

(30) Li

X j! #$ i $ $ j

0.0$2 $5$4$3 $6

Face width

M e s h g a p , # i

Total gear deflection

Total pinion deflection

Figure 16 -- Mesh gap section grid

9.3 Summation and load solution

Sign convention is very important and is explained

further in clause 5. Areas with greater mesh gap

have lower tooth load andareas with lower mesh gap

have higher tooth load. Using figure 16 as a guide,

note that in equation 30 as mesh gap, $i, gets larger,

the load, L i, must get smaller.

One location is selected as a reference, in this

example itis location“1” (seefigure 16). A sum of the

values for all locations referenced to location “1” can

then be created. This is done by setting term “j” in

equation 30 to location “1” and rearranging the

equation as shown below:

(31) L i

X 1! #$i $ $1

(32)1 ! X 1' Li

X i$ #$i $ $1

% C"m) And:

(33)i ! X i' L1

X 1$ #$i $ $1

% C"m)Sum up the values for all locations using equation 31

and get equation 34 below. Remember, only one

value of tooth stiffness, C"m, is used and the tooth

face width is broken into equally spaced segments:

X 1% ( # L2

X 1% (,,,# Ln

! '#$1 $ $1% ( #$2 $ $1

% (,,, #$n $ $1%) C"m

Simplifying equation 34 gives:

X 2(,,,

X n% $

! '#$1 $ $1% ( #$2 $ $1

% (,,, #$n $ $1%) C"m

The sum of all loads always equals the base tangent

plane load, F g, and all values of X i are equal, so:

X 2(,,,

X n% !

X n(36)

Solving the equations for the value of L1 gives:

1 ! F g

C"m X ii

'#$1 $ $1% ( #$2 $ $1

(,,, #$n $ $1%)

Using equation 33 the rest of the values for loads can

be calculated.

9.4 K H evaluation from loads

For the first iteration, a uniform load distribution

across the mesh is assumed and gaps are calcu-

lated. From these initial gaps, an uneven load

distribution is calculated. This new load distribution

is then used to calculate a new set of gaps. This

iteration process is continued until the newly calcu-

lated gaps differ from the previous ones by only a

small amount. Usually only a few, 2 or 3, iterations

are required to get an acceptable error (less than 3.0

mm change in gaps calculated).

The loads that correspond to the final iteration that

results in negligible change in gaps calculated are

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then used to calculate the load distribution factor, K H.

This is defined as the highest or peak load divided by

the average load.

(38) K H ! L i peak

Li ave

where:

(39) Li ave ! gn

9.5 Partial face contact

Initiallyall loads on the face width are assumed in the

same direction, i.e., have the same sign. If there is

not full contact across the face width some stations

will have their load value change sign. This indicates

tooth separation and there is no tooth contact at that

location, and therefore, the load must be zero at that

location. The method used to correct this condition

relies on the difference in load between stations

being a function of the change in deflection betweenstations. Therefore, even if a change in sign is

calculated, the difference in load between stations

with tooth contact will be correct.

To find the actual loads at these stations do the

following. Sum all the loads thathad a change in sign

and divide by the total number of loads that had a

change in sign. Subtract this value from each load

that did not have a change in sign. Set the value of

load to zero at all stations that had a change in sign.

The sum of loads at all stations that have contact will

now equal the total load on the face width and the

difference in load between these stations has not

changed.

9.6 Restatement of rules

The rules that govern the loads on the face width are:

-- The sum of the individual loads on the face

width, Li, must equal the total load on the gearset,

-- The change in load intensity, Li -- L j, between

any two locations on the face width must equal the

change in tooth deflection, $ti -- $ tj, or change in

mesh gap, $i -- $ j, between those locations;

-- Areas on the face width with more mesh gap

(mesh misalignment) have lower tooth load and

areas with lower mesh gap (mesh misalignment)

have higher tooth load;

-- Areas where load changes sign represent

areas where the teeth are not in contact and their

sum must be included in the loads that did not

change sign, i.e., ( Li = F g;

-- The face width shall be divided into eighteen

sections for the actual gap analysis and load dis-

tribution factor calculations.

10 Future considerations

10.1 Differential thermal conditions

Temperature differences are developed between the

pinion and mating gear elements and they may vary

along the face width. Both of these phenomena

produce distortions that may require lead com-

pensations to achieve acceptable load distribution.

Under running conditions the pinion element of a

gear set operates at a higher temperature than its

mating gear. This thermal differential will cause

pinion base pitch increases that exceed those of the

cooler mating gear.

In speed reducers the base pitch differential in-

crease is partially offset by elastic tooth deforma-

tions (refer to 5.1). Profile modification is often used

to compensate for this.In helical gear meshes there is also a temperature

differential along the face width due to the heat

generated as lubricant is displaced in wave--like

fashion from leading end to trailing end of the helix.

Lead correction may be used to compensate for this.

10.2 Mesh stiffness variations

The stiffness of a gear tooth at any given location

along its length is buttressed by adjacent tooth

length. A tooth portion at mid--face width is but-

tressed on both sides and has greater stiffness than

a similar tooth portion at the tooth end.

7/17/2019 Agma 927-a01.pdf

Annex A

Flowcharts for load distribution factor

Bending

Torsional

isP&GDone

Gap Analysis

Load Distribution

NewGap

DifferenceSmall

Output

ElasticData

Non--elasticData

Figure A.1 -- Overall flow chart

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Case ID

Units?

U.S. SI

UnitsLabels

Manual Adjustment

in BTCS

Target meshdata

External forces,moments, torques

(Timken convention)

Convert toBTCS

Analysis

Output K H

Yes No

Figure A.2 -- Data flow

7/17/2019 Agma 927-a01.pdf

does any station have a loadreversal (i.e., teeth are not

contacting) or[X6/abs (X6)] * W(j) < 0

INPUT Values

The gear mesh is divided into sections of equal length with loads placed in the center of each section.The sign convention is critical, positive loads and deflections are in same direction.

C"m = tooth stiffness constant N = total number of sections$i (j) = gap at each section

Li (j) = initial load at each sectionXi (j) = length of each section

k = number of sections across the face width

X (j) = Z (1) -- Z (j) relative gap from section 1 to section jX3 = sum [W (j) / Y (j) -- X (j) * e] for j = 1 to k sum of deflection and load

X6 = sum [W (j)] for j = 1 to k total load, this must remain constantM3 = X6/k average load on each section

W4 = Y(1)*X3/k new load on first section [new W(1)]

W (j) = Y (j) * [W4/Y(1) + X(j) * e] new load on each section

sum all loads with a reversalXTOT = sum {[X6/abs (X6)] * W(j) <0}

KTOT = sum number of stations wherethere is load reversal

add XTOT/KTOT to all stationswithout a load reversal

set all stations with a loadreversal to zero (0.0)

CALL SUBROUTINEcalculate deflections and performgap analysis based on new load

distribution

does new gap analysis differfrom last gap analysis by a

significant amount

Find maximum value of W(j) Y5 = max [abs W(i): abs W(k)]

calculate misalignment factorC5 = Y5/abs (M3)

OUTPUTC5 = misalignment factor Km

Z(j) = final gap analysisW(j) = final load distribution

Figure A.3 -- Overall flow chart detail of program CmSolve

7/17/2019 Agma 927-a01.pdf

INPUT Values

The helix is divided into sections of equal length with loads placed in the center of each section.The sign convention is critical.

G = shear modulusm = total number of sections

D(j) = major diameter at section ‘j’ (outside diameter minus 4 standard addendums) A(j) = inside diameter at section ‘j’

W(j) = load at each section ‘j’ (in base tangent plane) Y(j) = length of each section

A = sign multiplier to correct for direction of torsional deflection

for j = 1 to mL(j) = L(j--1) + W(j) sum of load to station ‘j’

U(j) = U(j--1) + Y(j--1) sum of length to station ‘j’T(j) = A * L(j) * U(j) * 4D(j) 2 /[G * 3.1416 * (D(j)4 -- A(j)4)] torsional deflection

OUTPUTT(j) = torsional deflection across mesh

Figure A.4 -- Torsional flow chart of program CmSolve

7/17/2019 Agma 927-a01.pdf

Annex B

Load distribution examples

B.1 K H example calculation

In this example a pinion shaft with dimensions as

shown in figure B.1, and with a total load, F g, of 104 090 N upward is analyzed for mesh gap. This

load is broken into six even loads of 17 348 N each

and gives the shaft deflection shown in table B.1. In

this gap analysis the deflection of the gear is very

small and is assumed to be a straight line. The

values are carried to the significant digits shown to

keep round--off error to a minimum and should notbe

confused with the precision of the deflection analy-

sis. A miscellaneous misalignment of 5.08mm in the

direction to increase mesh gap was included to

account for manufacturing and assembly errors.

Refer to figure B.2 for gap analysis information.

F g = ( Li = 104 090 N

L6 L5 L4

Face width

X 1 X 2 X 3 X 4 X 5 X 6

Bearing support

115.6 115.6

135.6135.6

Rotation

Torque path

Figure B.1 -- Example sections

Solve for L1 using equation 37 and then all other

values of “ L” using equation 33 The values for

deflection are micrometers (1¢ 10--6 meters) and a

value of C$m = 11 N/mm/ mm is used.

X 2(+++

X 6% ! 4560

(B.1)( 19.2) " 11] 22.836

! 16 320

L1 ! [4560 $ (0 $ 2.73 $ 2.23 ( 1.61 ( 8.78

4522.8 45.7 68.5 91.3 1140

Bending and shear deflection

Miscellaneousmismatch

Mesh gap

Torsional deflection

Figure B.2 -- Gap analysis

Table B.1

Deflections, micrometers LoadSta. No., i Bending Torsional Misc. Total, "i "i -- "1 Li, N

1 67.35 0.00 0.00 67.35 0.00 17 348

2 72.05 --0.94 --1.03 70.08 --2.73 17 348

3 74.45 --2.82 --2.06 69.58 --2.23 17 348

4 74.45 --5.64 --3.07 65.74 1.61 17 348

5 72.05 --9.40 --4.08 58.57 8.78 17 348

6 67.35 --14.1 --5.08 48.18 19.20 17 348

7/17/2019 Agma 927-a01.pdf

Solve for other values of “ L” using equation 33:

L2 ! 22.83#16 32022.83

$ 2.73 7 11% ! 15 630

L3 ! 22.83#16 32022.83

$ 2.23 7 11% ! 15 760

(B.4) L4 ! 22.83#16 32022.83 ( 1.61 7 11% ! 16 720

L5 ! 22.83#16 32022.83

( 8.78 7 11% ! 18 520

L6 ! 22.83#16 32022.83

( 19.2 7 11% ! 21 140

Using the non--uniform loads calculated, re--calcu-

late the deflections and new loads in an iteration until

sufficient accuracy has been attained. In this

example, further analysis gives values shown intable B.2. Therefore:

(B.7) K H ! 21 14017 350

! 1.22

Sufficient accuracy was achieved in this example on

the first calculation, and although further iterations

did change the values, they did not change the

overall accuracy of the K H calculation. Only six

stations across the face width were used, and this

may not insure sufficient accuracy. However this

example was also run with 20 load stations across

the face width and it only changed the K H value by4% to 1.27. So within the accuracy of the procedure,

it is not necessary to have large numbers of load

stations. Use of computers make this a moot

question, as more stations and iterations are not

hard to process.

It is necessary to investigate the effects of miscella-

neous misalignment in the other direction, and in

varying amounts, as this can have a big impact on

the K H for a gearset. For this example a miscella-

neous misalignment of 5.08 mm in a direction to

reduce mesh gap gave a K H = 1.18.In this example the deflection of the gear was not

considered. In some cases the deflection of the

mating element could make a major impact, espe-

cially in overhung designs or multiple reductionunits.

This procedure is dependent only on the total

mismatch between the gear teeth and can be used

with equal ease when deflections of both parts are

considered.

B.2 CmSolve example calculation

In this example the load distribution factor for a low

speed mesh of a double reduction parallel shaft gear

drive is shown. The dimensions, loading and

deflections are as shown in Table B.3 with a figure.

This data is also presented as it appears in the form

of the input and output data files to the computer

program CmSolve. The computer software program

CmSolve was developed to do an analysis as

described in this document. It was used to do aninternational comparative analysis in an effort to

improve the calculation of load distribution for load

capacity determinations.

Table B.2

Deflections, micrometers LoadSta. no., i Bending Torsional Misc. Total, "i "i -- "1 Li, N

1 66.98 0.00 0.00 66.98 0.00 16 410

2 71.72 --0.87 --1.03 69.82 --2.85 15 690

3 74.21 --2.58 --2.06 69.57 --2.59 15 760

4 74.31 --5.23 --3.07 66.01 0.97 16 660

5 72.02 --8.98 --4.08 58.95 8.02 18 430

6 67.41 --14.1 --5.08 48.24 18.7 21 140

7/17/2019 Agma 927-a01.pdf

Table B.3 -- CmSolve example

CmSolve Version 4.2.1 01/15/00 AGMA 07:01:00 AM

ISO Double -- LS Pinion -- CW -- fma=(fHB1**2+fHB2**2)**0.5 Crowned

**********************DEFLECTIONS***********************

LENGTH STATION LOAD BENDING TORSIONAL MISC. TOTAL RELATIVE

(MM) NUMBER (N*100) (MU--M) (MU--M) (MU--M) (MU--M) (MU--M)

0 7 36.223 15.4 0 0 15.4 0

6.7 8 44.706 18.0 --0.1 --9.5 8.5 7.0

13.41 9 52.231 20.5 --0.3 --17.9 2.3 13.2

20.11 10 58.850 22.8 --0.7 --25.3 --3.2 18.6

26.81 11 64.605 25.0 --1.2 --31.7 --7.9 23.4

33.52 12 69.535 26.9 --1.9 --37.0 --12.0 27.4

40.22 13 73.668 28.7 --2.8 --41.2 --15.4 30.8

46.92 14 77.030 30.3 --4.0 --44.5 --18.1 33.6

53.62 15 79.640 31.7 --5.3 --46.7 --20.3 35.7

60.33 16 81.508 32.8 --6.8 --47.8 --21.8 37.3

67.03 17 82.642 33.7 --8.5 --48.0 --22.8 38.2

73.73 18 83.040 34.4 --10.5 --47.0 --23.1 38.5

80.44 19 82.694 34.9 --12.6 --45.1 --22.8 38.2

87.14 20 81.590 35.1 --14.9 --42.1 --21.9 37.3

93.84 21 79.708 35.1 --17.4 --38.0 --20.3 35.8

100.55 22 77.017 34.8 --20.0 --33.0 --18.1 33.6

107.25 23 73.483 34.4 --22.7 --26.9 --15.2 30.7

113.95 24 69.064 33.7 --25.5 --19.7 --11.6 27.0

LOAD DISTRIBUTION FACTOR CM= 1.179508MISC MISALIGNMENT VALUE = 19.707 MICRO--METER PER HELIX

CROWN AMOUNT VALUE = 37.533 MICRO--METER PER HELIX

10 15 20 25

LOAD (N*100)

BENDING (MU--M)

TORSIONAL (MU--M)

MISC. (MU--M)

TOTAL (MU --M)

RELATIVE (MU--M)

Location Across Face -- Station Number

Double Reduction Low Speed

L o a d

f l e c t i o n

Saved File Image of Input Data

7/17/2019 Agma 927-a01.pdf

0,-3,0,.364,0

0,3,0,.364,0

0,4.22,0,.2685,0

0,4.22,0,.2675,0

1763.0507,4.22,0,.001,0

0,4.22,0,.1319,0

1584.0435,4.22,0,.2639,0

1584.0435,4.22,0,.2639,01584.0435,4.22,0,.2639,0

1584.0435,4.22,0,.2639,0

1584.0435,4.22,0,.2639,01584.0435,4.22,0,.2639,0

1584.0435,4.22,0,.1319,0

0,4.2,0,.001,0

-1763.0507,4.2,0,.4115,0

0,4.2,0,.4125,0

-2621.35,11.451,0,1.5,0

7735.76,11.451,0,1.5,0

2621.35,3.5,0,.3,0

0,3.5,0,.3,0

0,3,0,.46,0

0,-3,0,0,0

3.77,0,6,0,0,25

2.632,-775.8483,1477.6644

10.5,4.091,16.604,9,20

0,0,-696.528,85,-99,0,1574.8031

300,340.48,1,4.22

2,.728,0,3,0

2,.537,0,4.22,0

0,0,0,0,0

18,4.75,28512.7834,4.22,0

0,0,0,0,02,.825,0,4.2,0

2,3,0,11.451,0

2,.6,0,3.5,0

2,.92,0,3,0

7.2,2.707,11.37,20,11,1,3

”AGMA”

”ISO Double - LS Pinion - CW - fma=(fHB1**2+fHHB2**2)**0.5”

7/17/2019 Agma 927-a01.pdf

Printed Image of Program Output – Page 1

CmSolve Version 4.2.1 01-15-2000 07:10:04

ISO Double - LS Pinion - CW - fma=(fHB1**2+fHB2**2)**0.5

STA. EXTERNAL FREE BODY *SHAFT DIAMETER* SHAFT LENGTH ****** DEFLECTION *****

FORCE N FORCE N OUTSIDE INSIDE LENGTH FACE BENDING TORS. TOTAL

1 0.00 -95459.35 -76.200 0.000 9.246 0.0

2 0.00 0.00 76.200 0.000 9.246 4.2

3 0.00 0.00 107.188 0.000 6.820 8.2

4 0.00 0.00 107.188 0.000 6.795 11.0

5 7835.78 7835.78 107.188 0.000 0.025 13.7

6 0.00 0.00 107.188 0.000 3.350 13.7

7 7040.19 7040.19 107.188 0.000 6.703 0.0 15.0 0.0 15.0

8 7040.19 7040.19 107.188 0.000 6.703 6.7 17.5 -0.2 17.4

9 7040.19 7040.19 107.188 0.000 6.703 13.4 19.9 -0.5 19.4

10 7040.19 7040.19 107.188 0.000 6.703 20.1 22.1 -1.0 21.1

11 7040.19 7040.19 107.188 0.000 6.703 26.8 24.2 -1.7 22.5

12 7040.19 7040.19 107.188 0.000 6.703 33.5 26.1 -2.5 23.6

13 7040.19 7040.19 107.188 0.000 6.703 40.2 27.8 -3.5 24.3

14 7040.19 7040.19 107.188 0.000 6.703 46.9 29.3 -4.7 24.615 7040.19 7040.19 107.188 0.000 6.703 53.6 30.6 -6.0 24.6

16 7040.19 7040.19 107.188 0.000 6.703 60.3 31.7 -7.5 24.2

17 7040.19 7040.19 107.188 0.000 6.703 67.0 32.6 -9.2 23.4

18 7040.19 7040.19 107.188 0.000 6.703 73.7 33.2 -11.0 22.2

19 7040.19 7040.19 107.188 0.000 6.703 80.4 33.6 -13.0 20.6

20 7040.19 7040.19 107.188 0.000 6.703 87.1 33.8 -15.2 18.6

21 7040.19 7040.19 107.188 0.000 6.703 93.8 33.8 -17.5 16.3

22 7040.19 7040.19 107.188 0.000 6.703 100.5 33.5 -20.0 13.5

23 7040.19 7040.19 107.188 0.000 6.703 107.2 33.1 -22.7 10.4

24 7040.19 7040.19 107.188 0.000 3.350 114.0 32.4 -25.5 6.9

25 0.00 0.00 106.680 0.000 0.025 32.0

26 -7835.78 -7835.78 106.680 0.000 10.452 32.0

27 0.00 0.00 106.680 0.000 10.478 30.328 -11650.44 -11650.44 290.855 0.000 38.100 28.2

29 34381.16 34381.16 290.855 0.000 38.100 19.7

30 11650.44 11650.44 88.900 0.000 7.620 11.1

31 0.00 0.00 88.900 0.000 7.620 9.2

32 0.00 0.00 76.200 0.000 11.684 7.2

33 0.00 0.00 76.200 0.000 11.684 3.8

34 0.00 -65645.28 -76.200 0.000 0.000 0.0

SHAFT DIMENSIONS USED FOR TORSIONAL DEFLECTION CALCULATION

OUTSIDE DIAMETER 95.7580 INSIDE DIAMETER 0.0000

TOOTH STIFFNESS CONSTANT = 2.632 X10^6

7/17/2019 Agma 927-a01.pdf

Printed Image of Program Output – Page 2

CmSolve Version 4.2.1 01-15-2000 07:10:04

ISO Double - LS Pinion - CW - fma=(fHB1**2+fHB2**2)**0.5

STA 7 24 CM= 1.186654 MAX LD= 8354.2725 TOT LD= 126723 AVE LD= 7040 SUM 239200.325

****************** DEFLECTIONS ******************

LENGTH STA. LOAD BENDING TORSIONAL MISC. TOTAL RELATIVE CORR

(MM) NO. (N) (MU-M) (MU-M) (MU-M) (MU-M) (MU-M) (MU-M)

0.00 7 3622.3 15.4 0.0 0.0 15.4 0.0 0.0

6.70 8 4470.6 18.0 -0.1 -9.5 8.5 7.0 0.0

13.41 9 5223.1 20.5 -0.3 -17.9 2.3 13.2 0.0

20.11 10 5885.0 22.8 -0.7 -25.3 -3.2 18.6 0.0

26.81 11 6460.5 25.0 -1.2 -31.7 -7.9 23.4 0.0

33.52 12 6953.5 26.9 -1.9 -37.0 -12.0 27.4 0.0

40.22 13 7366.8 28.7 -2.8 -41.2 -15.4 30.8 0.0

46.92 14 7703.0 30.3 -4.0 -44.5 -18.1 33.6 0.0

53.62 15 7964.0 31.7 -5.3 -46.7 -20.3 35.7 0.060.33 16 8150.8 32.8 -6.8 -47.8 -21.8 37.3 0.0

67.03 17 8264.2 33.7 -8.5 -48.0 -22.8 38.2 0.0

73.73 18 8304.0 34.4 -10.5 -47.0 -23.1 38.5 0.0

80.44 19 8269.4 34.9 -12.6 -45.1 -22.8 38.2 0.0

87.14 20 8159.0 35.1 -14.9 -42.1 -21.9 37.3 0.0

93.84 21 7970.8 35.1 -17.4 -38.0 -20.3 35.8 0.0

100.55 22 7701.7 34.8 -20.0 -33.0 -18.1 33.6 0.0

107.25 23 7348.3 34.4 -22.7 -26.9 -15.2 30.7 0.0

113.95 24 6906.4 33.7 -25.5 -19.7 -11.6 27.0 0.0

LOAD DIST FACTOR CM= 0.000000

LOAD DIST FACTOR CM= 1.179508

LOAD DISTRIBUTION FACTOR = 1.179508

MISC MISALIGNMENT VALUE =-19.707 MICRO-METER PER HELIX

CROWN AMOUNT VALUE =37.533 MICRO-METER PER HELIX

7/17/2019 Agma 927-a01.pdf

PUBLISHED BY AMERICAN GEAR MANUFACTURERS ASSOCIATION

Agma 927-a01.pdf

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