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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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ii
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
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--779--7
American
GearManufacturers Association
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AGMA 927--A01AMERICAN GEAR MANUFACTURERS ASSOCIATION
iii
Contents
Page
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
iv
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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AGMA 927--A01AMERICAN GEAR MANUFACTURERS ASSOCIATION
v
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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AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION
vi
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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1
AGMA 927--A01AMERICAN GEAR MANUFACTURERS ASSOCIATION
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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AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION
2
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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AGMA 927--A01AMERICAN GEAR MANUFACTURERS ASSOCIATION
3
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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AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION
4
-- 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
used:
-- 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
*
+BTZ
+BT
+BTN
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-
ment.
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%
where
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:
(2)
F sG !& F tG
&'# A D H R sin % sin "G ( tan Ô cos "G%)
cos %
where
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:
(3)
aG !& F
tG&( A)'# A D H R sin % cos "
G $ tanÔ sin "
G%)
cos %
where
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
--1
0
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
--1
Apex toward reference end (see figure 3), or no apex
Apex away from reference end (see figure 3)
Rotation R +1
--1
Clockwise viewed from reference end
Counterclockwise viewed from reference end
Drive D +1
--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
Hand
Apex
Right handspiral
Left handspiral
Figure 3 -- Hand of cut for gears and explanation of apex for bevel gears
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F s
F s
F t
F t
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
FaP3
Mesh 3FtG3
+BT -- Axis along base tangent plane of target mesh+BTN -- Axis normal to base tangent plane of target mesh
+BT
+BTN
CLGearface
A
A
Reference endand origin of shaft for mesh 3
Mesh 2
FtP3
FsG3
Bearing
DriverRH
DrivenLH
DrivenRH
DriverRH
Shaft 1
Mesh 1
FtP1 FsG1
Shaft 2
Shaft 4Base tangent
plane formesh 3
Base diameter formember typical
FtG2
FaP2
Base tangentplane for mesh 2
Driven
LH
FtG1
+BTZ
FsG1
FaP1FaG1
FaG3
FtP2
DriverLH
Reference endand origin of shaft for mesh 2
Shaft 3
FsP2
FsG2
FaG2
FsP3
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
2
where
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
RH
DriverLH
DrivenLH
BT -- Axis along base tangent plane of target mesh
BTN -- Axis normal to base tangent plane of target mesh
+BT
+BTN
Base tangent coordinate
system for mesh 3
Shaft 4
Base tangentline
TargetMesh #3
Shaft 2
Mesh 2DrivenRH
Shaft 3
&2
FsP33
+BTZ
F tG23
F sG23
F tP33
M G34
M G23
M P33
F tG34
F sG34
Figure 6 -- View A--A from figure 5
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T a
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
22.0
--13500 +9000
+6180 --1680
22.0
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
%)s
where
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
where
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
in i%
64 +++ i ! 1,2,3, +++ n
where
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.
(15)
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.
(16)
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
*n$1
DI i
(20) Ls !i ! 1
*n
xi
(21) IC !$ Sy
Ls
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)
0.0
+0.01 (1/ mm)
25.0
1 2 3 4 5 6 7
22.0
--13500 +9000
+6180 --1680
28.0 28.0 25.0 22.0
+10000 (N)
--10000 (N)
0.0
350000 (Nmm)
--150000 (Nmm)
0.0
Diagram M EI
0.0Slope Curve
0.0
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 !
( IC)
# x
i(1%+++i
! 1,2,3,
+++n
$ 1
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
zero.
(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%-./
i
* j ! 1
L j -01-./
i $ 1
* j ! 1
X j-01
4 d 2
G '#d 4 $ d 4in%
(24)
where
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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X 1
X 2
X 3
X 4
X 5
d
Torqueinput
Torsionaldeflection
Li Load on teeth
L1
L2
L3
L4
L5
L6
d in
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
-23
k
* j ! 1
L j X k-45-0
18 d 2
G '#d 4 $ d 4in%
(25)
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
are:
-- 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
gap.
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-
tion
Torsionaldeflec-
tion
Toothmodifi-
cation
Leadvaria-
tion
Shaft mis-
alignment
Shaft #3
gap
Bendingdeflec-
tion
Torsionaldeflec-
tion
Toothmodifi-
cation
Leadvaria-
tion
Shaft mis-
alignment
Shaft #4
gap
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
--20
--18
--16
--14
--12
--10
--8
--6
--4
--2
24
6
8
10
12
14
16
18
20
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
--20
--18
--16
--14
--12
--10
--8
--6
--4
--2
24
6
8
10
12
14
16
18
20
08 9 10 11 12 13 14 15 16 17
SHAFT #4
M i c r o m e t e r s
Figure 11 -- Shaft number 4 gap
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SHAFT #3
--20
--18
--16
--14
--12
--10
--8
--6
--4
--2
24
6
8
10
12
14
16
18
20
08 9 10 11 12 13 14 15 16 17
SHAFT #4SHAFT #4
M i c r o m e t e r s
Figure 12 -- Total mesh gap
SHAFT #3
--20
--18
--16
--14
--12
--10
--8
--6
--4
--2
24
6
8
10
12
14
16
18
20
010 11 12 13 14 15 16 17
SHAFT #4
SHAFT #4
8 9
M i c r o m e t e r s
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
where
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
$t
C"m
Li meshgap,$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 !
Li
X i
where
X i is length of face where point load is applied,
mm;
Li is load at a specific point “i”, N.
L2
L6 L5
L4 L3
L1
Face width
X 1 X 2 X 3 X 4 X 5 X 6
Bearing
X i
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
where
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
16).
(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 i$
L j
X j! #$ i $ $ j
% C"m
0.0$2 $5$4$3 $6
Face width
M e s h g a p , # i
$1
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, $i, gets larger,
the load, L i, must get smaller.
One location is selected as a reference, in this
example itis location“1” (seefigure 16). A sum of the
values for all locations referenced to location “1” can
then be created. This is done by setting term “j” in
equation 30 to location “1” and rearranging the
equation as shown below:
(31) L i
i$
L1
X 1! #$i $ $1
% C"
Or:
(32)1 ! X 1' Li
X i$ #$i $ $1
% C"m) And:
(33)i ! X i' L1
X 1$ #$i $ $1
% C"m)Sum up the values for all locations using equation 31
and get equation 34 below. Remember, only one
value of tooth stiffness, C"m, is used and the tooth
face width is broken into equally spaced segments:
# L1
X 1$
L1
X 1% ( # L2
X 2$
L1
X 1% (,,,# Ln
X n$
L1
X 1%
(34)
! '#$1 $ $1% ( #$2 $ $1
% (,,, #$n $ $1%) C"m
Simplifying equation 34 gives:
# L1
X 1(
L2
X 2(,,,
Ln
X n% $
n L1
X 1
! '#$1 $ $1% ( #$2 $ $1
% (,,, #$n $ $1%) C"m
(35)
The sum of all loads always equals the base tangent
plane load, F g, and all values of X i are equal, so:
# L1
X 1(
L2
X 2(,,,
Ln
X n% !
F g
X n(36)
Solving the equations for the value of L1 gives:
(37)
1 ! F g
i $
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,
F g;
-- 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.
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Annex A
Flowcharts for load distribution factor
Input
Bending
Torsional
isP&GDone
isP&GDone
Yes
No
No
Yes
Gap Analysis
Load Distribution
NewGap
DifferenceSmall
No
Yes
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
Test
Output K H
Yes No
Figure A.2 -- Data flow
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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
Yes
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
No
No
Yes
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
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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
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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
L2
L6 L5 L4
L3 L1
137
Face width
X 1 X 2 X 3 X 4 X 5 X 6
22.83
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.
# L1
X 1(
L2
X 2(+++
L6
X 6% ! 4560
(B.1)( 19.2) " 11] 22.836
! 16 320
L1 ! [4560 $ (0 $ 2.73 $ 2.23 ( 1.61 ( 8.78
65
75
55
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
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Solve for other values of “ L” using equation 33:
(B.2)
L2 ! 22.83#16 32022.83
$ 2.73 7 11% ! 15 630
(B.3)
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
(B.5)
L5 ! 22.83#16 32022.83
( 8.78 7 11% ! 18 520
(B.6)
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
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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
--60
--40
--20
0
20
40
60
80
100
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
/ D e
f l e c t i o n
5
Saved File Image of Input Data
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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,.2639,0
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,.2639,0
1584.0435,4.22,0,.2639,0
1584.0435,4.22,0,.2639,0
1584.0435,4.22,0,.2639,0
1584.0435,4.22,0,.2639,0
1584.0435,4.22,0,.2639,0
1584.0435,4.22,0,.2639,0
1584.0435,4.22,0,.2639,0
1584.0435,4.22,0,.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,.46,0
0,-3,0,0,0
3.77,0,6,0,0,25
2.632,-775.8483,1477.6644
1,1
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
0,0,0,0,0
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
1
7.2,2.707,11.37,20,11,1,3
”AGMA”
”ISO Double - LS Pinion - CW - fma=(fHB1**2+fHHB2**2)**0.5”
0
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AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION
30
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
AGMA
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
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AGMA 927--A01AMERICAN GEAR MANUFACTURERS ASSOCIATION
31
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
AGMA
STA 7 24 CM= 1.186654 MAX LD= 8354.2725 TOT LD= 126723 AVE LD= 7040 SUM 239200.325
STA 7 24 CM= 1.179403 MAX LD= 8303.2282 TOT LD= 126723 AVE LD= 7040 SUM 247301.066
STA 7 24 CM= 1.179508 MAX LD= 8303.9630 TOT LD= 126723 AVE LD= 7040 SUM 247066.229
****************** 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
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PUBLISHED BY AMERICAN GEAR MANUFACTURERS ASSOCIATION