Agma 927-a01.pdf

AGMA 927- A01

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927- A01

Load Distribution Factors - Analytical Methods for Cylindrical Gears

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

Load Distribution Factors - Analytical Methods for Cylindrical Gears American AGMA 927--A01 Gear Manufacturers CAUTION NOTICE: AGMA technical publications are subject to constant improvement, revision or withdrawal as dictated by experience. Any person who refers to any AGMA Association technical publication should be sure that the publication is the latest available from the Association on the subject matter.

[Tables or other self--supporting sections may be quoted or extracted. Credit lines should read: Extracted from AGMA 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 (KH) of AGMA standards and longitudinal load distribution (KHβ and KFβ) 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 Association 1500 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

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927--A01

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 B

Flowcharts for load distribution factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Load distribution examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Base tangent coordinate system for CW driven rotation from reference end . 5 Base tangent coordinate system for CCW driven rotation from reference end 6 Hand of cut for gears and explanation of apex for bevel gears . . . . . . . . . . . . . 7 Gearing force sense of direction for positive value from equations . . . . . . . . . . 8 Example general case gear arrangement (base tangent coordinate system) . 8 View A--A from figure 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Example shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Calculated shaft diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Torsional increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Shaft number 3 gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Shaft number 4 gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Total mesh gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Relative mesh gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Tooth section with spring constant Cγm, load L, and deflection Cd . . . . . . . . . 19 Deflection sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Mesh gap section grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Tables 1 2 3 4

Symbols and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Values for factors hand, apex, rotation, and drive . . . . . . . . . . . . . . . . . . . . . . . . 7 Calculation data and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 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 (Cγm). 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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AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927--A01

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 . . . W.A. Bradley . . . . M.F. Dalton . . . . . G.A. DeLange . . . O. LaBath . . . . . . L. Lloyd . . . . . . . .

The Gear Works -- Seattle Consultant General Electric Company Prager, Inc. The Cincinnati Gear Co. Lufkin Industries, Inc.

J.J. Luz . . . . . . . . D.R. McVittie . . . . M.W. Neesley . . . W.P. Pizzichil . . . F.C. Uherek . . . . .

General Electric Company Gear Engineers, Inc. WesTech Gear Corporation Philadelphia Gear Corp. Flender Corporation

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

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

M. Hirt . . . . . . . . . R.W. Holzman . . R.S. Hyde . . . . . . V. Ivers . . . . . . . . A. Jackson . . . . . H.R. Johnson . . . J.G. Kish . . . . . . . R.H. Klundt . . . . . J.S. Korossy . . . . I. Laskin . . . . . . . . J. Maddock . . . . . J. Escanaverino . G.P. Mowers . . . . R.A. Nay . . . . . . . M. Octrue . . . . . . T. Okamoto . . . . . J.R. Partridge . . . M. Pasquier . . . . J.A. Pennell . . . . . A.E. Phillips . . . . . J.W. Polder . . . . .

Renk AG Milwaukee Gear Company, Inc. The Timken Company Xtek, Incorporated Mobil Technology Company The Horsburgh & Scott Co. Sikorsky Aircraft Division The Timken Company The Horsburgh & Scott Co. Consultant The Gear Works -- Seattle, Inc. ISPJAE Consultant UTC Pratt & Whitney Aircraft CETIM Nippon Gear Company, Ltd. Lufkin Industries, Inc. CETIM Univ. of Newcastle--Upon--Tyne Rockwell Automation/Dodge Delft University of Technology

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

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

COMMITTEE ASSOCIATE MEMBERS M. Bartolomeo . . A.C. Becker . . . . E. Berndt . . . . . . . E.J. Bodensieck . D.L. Borden . . . . M.R. Chaplin . . . . R.J. Ciszak . . . . . A.S. Cohen . . . . . S. Copeland . . . . R.L. Cragg . . . . . T.J. Dansdill . . . . F. Eberle . . . . . . . L. Faure . . . . . . . . C. Gay . . . . . . . . . J. Gimper . . . . . . T.C. Glasener . . . G. Gonzalez Rey M.A. Hartman . . . J.M. Hawkins . . . G. Henriot . . . . . . G. Hinton . . . . . . .

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

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AGMA 927--A01

E. Sandberg . . . . C.D. Schultz . . . . E.S. Scott . . . . . . A. Seireg . . . . . . . Y. Sharma . . . . . . B.W. Shirley . . . . L.J. Smith . . . . . . L. Spiers . . . . . . . A.A. Swiglo . . . . . J.W. Tellman . . . .

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Det Nordske Veritas Pittsburgh Gear Company The Alliance Machine Company University of Wisconsin Philadelphia Gear Corporation Emerson Power Transmission Invincible Gear Company Emerson Power Trans. Corp. IIT Research Institute/INFAC Dodge

F.A. Thoma . . . . . D. Townsend . . . . L. Tzioumis . . . . . F.C. Uherek . . . . . A. Von Graefe . . . C.C. Wang . . . . . B. Ward . . . . . . . . R.F. Wasilewski . H. Winter . . . . . . .

F.A. Thoma, Inc. NASA/Lewis Research Center Dodge Flender Corporation MAAG Gear AG 3E Software & Eng. Consulting Recovery Systems, LLC Arrow Gear Company Technische Univ. Muenchen

AMERICAN 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 factors. 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 combined twisting 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.

AGMA 927--A01

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 limitations and purposes. It is not intended for use by the engineering public at large.

2 References The following documents were used in the development 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 Factors 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 Corresponding 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

3 Definitions and symbols The terms used, wherever applicable, conform to ANSI/AGMA 1012--F90. NOTE: The symbols and definitions used in this standard 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, KH, 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 Km have been associated with this factor. In ISO standards, the variables KHβ, KHα, KFβ and KFα, have been associated with the factor. In

current AGMA standards the load distribution factor, KH, is used for both pitting resistance and bending strength calculations. There is no separate value, KF, for bending strength as found in ISO standards. The magnitude of KH is affected by two components, transverse load distribution factor and face load distribution factor. The transverse load distribution factor pertains to the plane of rotation and is affected primarily by the correctness of the profiles and indexing of the mating 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 mesh includes a target pinion and a target gear.

Table 1 -- Symbols and definitions Symbol A BT BTN BTZ Cγm b D DpG d din dsh E FaG FaP Fg Fi FsG FsP FtG FtP G H I

Definition Apex factor Axis in the base tangent plane Axis normal to base tangent plane Axis in the base tangent plane perpendicular to BT Tooth stiffness constant, for the analysis Helical/bevel gear face width Drive factor Operating pitch diameter, gear Outside effective twist diameter Inside shaft diameter Outside diameter, effect outside diameter of the teeth Modulus of elasticity Axial thrust force, gear member Axial thrust force, pinion member Total load in the plane of action Gearing or external force at a distance Separating force, gear member Separating force, pinion member Tangential force, gear member Tangential force, pinion member Modulus of elasticity in shear Hand factor Moment of inertia

Units -- --- --- --- -N/mm/mm mm -- -mm mm mm mm N/mm2 N N N N N N N N N/mm2 -- -mm4

First referenced 5.3 5.2 5.2 5.2 9.1 5.3 5.3 5.3 7.1 6.1 6.1 6.1 5.3 5.4 9.2 6.1 5.3 5.3 5.3 5.3 7.1 5.3 6.1 (continued)

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927--A01

Table 1 (concluded) Symbol IC i KH Ls Lj Lδ M MG n P R RL RR S SLi tδi V xi Xj Xfi x y δti γG γP ψ Ô

Definition

Units

Integration constant Station number Load distribution factor Distance between the supports (reactions) Load at station Load intensity Bending moment Moment due to axial thrust force Station number at end support Power transmitted through the mesh Rotation factor Reaction at the left bearing Reaction at the right bearing Speed of shaft Station slope value Torsional deflection at a station Shear Length of face where point load applied Distance between adjacent stations Distance from left support to load location Distance between stations Deflection along the line of action Tooth deflection at a load point Bevel pitch angle of gear Bevel pitch angle of pinion Helix angle/spiral angle Normal pressure angle

4 Iterative analytical method

-- --- --- --- -N N/mm N mm N mm -- -kW -- -N N rpm -- -mm N mm mm mm mm mm mm degrees degrees degrees degrees

First referenced 6.1 6.1 9.4 6.1 7.1 9.1 6.1 5.4 6.1 5.3 5.3 6.1 6.1 5.3 6.3 7.1 6.1 9.2 7.1 6.1 6.1 6.1 9.1 5.3 5.3 5.3 5.3

--

tooth alignment deviations of pinion and gear;

--

tooth alignment and crowning modification;

This information sheet presents an iterative analytical method for determining a value of load distribution 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 represent the mesh stiffness. The mesh gap is then mathematically closed by compressing the springs until the sum of the spring forces equals the total tooth force.

Influences that may be accounted for by estimating values and including them as equivalent misalignments of the target shaft axes are:

The method has the ability to consider the following influences:

-- elastic deflection of a gear body if it is not a solid disk (such as a spoke gear);

-- 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 and bending, resulting from the target mesh loads and loads external to the mesh.

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AGMA 927--A01

-- elastic deflection foundations;

AMERICAN GEAR MANUFACTURERS ASSOCIATION

of

the

housing

and

-- equivalent elastic deflection of non--solid body gears (such as a spoke gear);

-- displacements of the gearing due to bearing deflection;

-- elastic deflection foundations;

--

thermal or centrifugal effects;

--

running--in or lapping effects.

-- displacements due to bearing clearance, alignment and deflection;

The method does not consider the following influences: --

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 overloaded.

gears

with

one

helix

4.1 Methodology

of

the

--

thermal or centrifugal effects;

--

running--in or lapping effects.

housing

and

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;

The iterative analytical method consists of the following basic steps:

-- shear coupling between the mesh gap compression springs representing the mesh stiffness is ignored;

1) Calculate the mesh gap resulting from an initial uniform load distribution;

-- mesh stiffness is a constant across the full width of tooth;

2) Calculate a new load distribution by mathematically closing the mesh gap. This is accomplished 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 load distribution 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.

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 one member 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;

4.1.2 Equivalent misalignment inputs

-- the coordinate system for all calculations lies in the base tangent plane;

Other displacements that are treated by combining them as an equivalent deflection at the target mesh include:

-- the base tangent plane is a plane tangent to the base circles of the target mesh;

-- alignment deviations and modifications of pinion and gear teeth; 4

-- 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;

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927--A01

-- a modified Timken sign convention is followed;

plane and the edge of the target mesh face closest to the reference end (see figures 1 and 2).

-- each analysis includes only the two shafts under consideration;

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 the end of the driving element shaft opposite the torque input end.

-- 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 (see 5.2); -- the input torque to the driving element enters the shaft from one side only and is fully balanced by torque in the target mesh.

Using this definition of the refence end, the positive directions of the BTCS axes are determined as follows: + BTZ: away from the reference end;

5.2 Coordinate system and sign convention

+ BTN: toward the driven element;

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.

+ BT: obtained by right hand rule; BTN to 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

Figures 1 and 2 illustrate the base tangent plane and the base tangent coordinate system for a typical target mesh. In figure 1, the input torque is clockwise when viewed from the reference end. In figure 2, the input 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.

Base tangent plane

.

Driver Base diameter -driving element

Input torque Target shaft -driver Target mesh +BTZ

+BT

Base diameter -driven element Target shaft -driven

* Reference end

+BTN

Driven

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

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AGMA 927--A01

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Target shaft -driver

Driver Base diameter -driving element

Input torque +BTZ

*

Target mesh Base diameter -driven element

.

Base tangent plane

+BT

Reference end

+BTN Driven

Target shaft -driven

Figure 2 -- Base tangent coordinate system for CCW driven rotation from reference end 5.3 Gearing forces and signs

R

is rotation factor (see table 2);

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 combined with all other sources of potential misalignment.

S

is speed of gear shaft, rpm;

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: F tG =

1.91 × 10 7 P (D R )





S D pG − b sin γ G

(1)

DpG is operating pitch diameter, gear, mm; b

is helical/bevel gear face width, mm;

γG

is bevel pitch angle, gear, degrees.

The separating force is calculated as: F sG =

cos ψ

FsG is separating force, gear member, N; A

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

H

is hand factor (see table 2);

ψ

is helix angle/spiral angle, degrees;

Ô

is normal pressure angle, degrees.

The thrust (axial) force is calculated as: F aG =

F (A )A D H R sin ψ cos γ tG

is power transmitted through the mesh, kW;

D

is drive factor (see table 2);



G − tan Ô sin γ G

cos ψ

FtG is tangential force, gear member, N; P

(2)

where

where

6

F A D H R sin ψ sin γ + tan Ô cos γ  tG G G

where FaG is axial thrust force, gear member, N.

(3)

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927--A01

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

Factor H

Apex (bevel)

A

Rotation

R

Drive

D

Value +1 --1 0 +1 --1 +1 --1 +1 --1

Condition Right hand helix or spiral (see figure 3) Left hand helix or spiral (see figure 3) Spur, straight bevel, or herringbone Apex toward reference end (see figure 3), or no apex Apex away from reference end (see figure 3) Clockwise viewed from reference end Counterclockwise viewed from reference end Driving element Driven element above equations. The forces must be determined for each mesh on each of the target mesh shafts.

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.

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 displacement in the base tangent plane.

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

Figure 5 shows a general arrangement. For this example, mesh 3 is the target mesh. Shafts 3 and 4 are the target shafts.

Hand

Right hand helix

Left hand helix

Right hand spiral

Left hand spiral

Apex

Away from reference

Toward reference

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

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

Mating target shaft

One target shaft

Mate shaft

If mate to target shaft is on the left, use these positive force directions

Ft Fa

Fa

Fs

Fs Ft

If mate to target shaft is on the right, use these positive force directions

View direction from reference end

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

Mesh 1 Shaft 1

FtG1

FaP1

FaG1

A Driver RH FsG1

FtP1

Driver LH Shaft 2 Reference end and origin of shaft for mesh 2

FsP2

FsG1

FtP2

Driven LH

Driver RH

FaG2

Shaft 3 Reference end and origin of shaft for mesh 3

FaP2

Base tangent FtG2 plane for mesh 2 Mesh 2

FsG2

FsP3

+BT

FtG3 Mesh 3 FaG3

FaP3 FtP3 +BTZ

Driven RH Bearing

+BTN Base tangent coordinate system for mesh 2

Base tangent plane for mesh 3

Base diameter for member typical

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

Shaft 4

FsG3

Driven LH

CL Gear face

A

Example showing actual direction of the forces as determined from the sign of the values calculated in the force equations. Figure 5 -- Example general case gear arrangement (base tangent coordinate system)

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AGMA 927--A01

5.4 Gearing moments

6 Shaft bending deflections

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.

Gears transmitting power will impose forces and moments on their shafts, which will cause elastic deflections. These deflections can affect the alignment of the gear teeth and therefore affect the load distribution across the gear face width.

The moment due to an axial thrust force on the gear member is given by equation 4.

MG =

F aG D pG

(4)

2

This section presents a simplified computer programmable 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 calculating load distribution factor are also presented.

where MG 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 on shafts 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.

6.1 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 explanation 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 LH Base tangent coordinate system for mesh 3

Shaft 2

+BTZ Driven RH

Mesh 2 FsG23

Shaft 3

Driven LH

FtG34 FsP33

MG23 Driver RH

+BTN

Base tangent line

FtG23

θ2

+BT

MP33

FsG34

Shaft 4

FtP33

Target Mesh #3

MG34

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

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

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Table 3 -- Calculation data and results

AGMA 927--A01

10

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927--A01

--13500 22.0 1

28.0 2

35.0

+9000 28.0

3

25.0 4

50.0

25.0 5

44.0

22.0 6

7

38.0

+6180

--1680

Figure 7 -- Example shaft All modeling will be from the left--hand support moving 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, Fi, and the distances they act from the left support, Xfi, are specified. Using standard static force analyses calculate the reaction, RR, at the right side support by summing the moments about the left support. RR =

Fi Xfi Ls

(5)

F

is the force applied at a distance, N;

Ls

is the distance between the two supports;

Xfi

is the distance from left support to load location, Fi. (6)

Then calculate the reaction at the left using the total sum of the loads. RL =

 Fi − RR

(7)

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: d2 y =M EI dx 2

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).

where

X fi = x i + X fi−1 i = 1, 2, 3,  n

where

(8)

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, Vi, 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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AMERICAN GEAR MANUFACTURERS ASSOCIATION

should be numerically equal to but opposite in sign to the last force listed in column 4. V i+1 = V i + F i⋅⋅⋅ i = 1, 2, 3,  n − 1

(9)

where V

is the shear, N;

i

is the station number;

n

is the station number at end support.

AMEI i =

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, Mi, 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, Vi (column 5), and distance between stations, xi (column 6). The moment at the first and last station, i = 1 and i = n, should be zero (i.e. M1=0.0 and Mn = 0.0). M i+1 = M i + V i+1x i+1 i = 1, 2, 3,  n (10) Step 8: Calculate the moment of inertia, Ii, 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. Ii =



π d4

sh i

− d4

64



in i

 i = 1, 2, 3,  n

(11)

where dsh

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

din

is the inside shaft diameter, mm;

Step 9: Multiply each Ii value by modulus of elasticity, E, and insert the EIi value in column 9 on the same lines as corresponding Ii values. For steel use E = 206 000 N/mm2. Dividing the EIi values by 103 before tabulating them in column 9 results in units of µm for the rest of the tabulation. EI i = ( E )I i i = 1, 2, 3,  n − 1

(12)

Step 10: Divide each bending moment Mi value in column 7 by the EIi value in column 9 which precedes and follows it. List these two values, MEIui and MEIli, in column 10.

12

Step 11: Obtain the average MEI values, AMEIi, 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 ui =

Mi  i = 1, 2, 3,  n − 1 EI i

(13)

MEI li =

M i+1  i = 1, 2, 3,  n − 1 EI i

(14)

MEI ui + MEI li  i = 1, 2, 3,  n − 1 2 (15)

Step 12: Calculate the slope value, SLi, in column 12 starting with zero at station 1 (i.e., SL1=0). Succeeding values are obtained by summing the products of AMEIi from column 11 and the xi value on the next lower line of column 6. These values are listed on the same lines as the stations. SL i+1 = SL i + AMEI ix i+1 i = 1, 2, 3,  n − 1

(16)

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 stations in column 13. ASL i =

SL i + SL i+1  i = 1, 2, 3,  n − 1 (17) 2

Step 14: Obtain the deflection increment values, DIi, in column 14 by multiplying the average slope value in column 13 and the xi value from the next lower line in column 6. DI i = ASL ix i+1 i = 1, 2, 3,  n − 1

(18)

Step 15: The next step is to evaluate the integration constant which depends on type of shaft. For the simply 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. n−1 Sy =



i=1

DI i

(19)

xi

(20)

n

Ls =

IC =



i=1 − Sy Ls

Other shaft configurations integration constant.

(21) will

change

the

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927--A01

--13500 22.0 1

35.0

28.0 2

+9000 28.0

3

25.0 4

25.0 5

44.0

50.0

+6180

22.0 6

7

38.0

--1680

+10000 (N) 0.0 --10000 (N)

Shear Diagram, V

350000 (Nmm)

0.0 --150000 (Nmm)

Moment Diagram, M

+0.01 (1/mm)

0.0

--0.01 (1/mm)

M Diagram EI

0.4 (mrad)

0.0

Slope Curve

0.0 --10 (mm)

Deflection Curve

Figure 8 -- Calculated shaft diagrams

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

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. ICS i = ( IC )x i+1 i = 1, 2, 3,  n − 1

(22)

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 deflection. For all other stations the deflection values are obtained by summing together the deflection increment 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. y i+1 = y i + DI i + ICS i i = 1, 2, 3,  n − 1 (23) 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.

-- 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 it becomes 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, din, and incremental length, Xj, 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 form that allows summation using the discreet stations used in this document. This results in the equation: −1  2  i i  X L   j  j4 d j = 1 j = 1   

10 3 t δi =



(24)



G π d 4 − d 4in

where tδi

is torsional deflection at a station, mm;

When calculating bending deflection for load distribution factor, the following rules also apply:

Lj

is load at a station, N;

Xj

-- Only forces acting in the base tangent plane are considered;

is the distance between adjacent stations, mm;

d

is effective twist diameter (see 7.2), mm;

-- When calculating shaft deflections, the area of the gear teeth is broken into eighteen equal sections;

din

is inside diameter, mm;

i

is station number;

G

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

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AGMA 927--A01

Undeformed position

Facewidth

X1 X2

L1

X3

L2

X4

L3

X5

L4 din Torsional deflection

Torque input

L5 L6

d

Torque input

Li Load on teeth

Figure 9 -- Torsional increments At the first point of interest on the tooth where j = 1, the summation of Xj 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 theoretically correct equation would be an integration. A slightly more accurate approximation is found in equation 25.

t δi =

(i − 1) k  10 3    L j X k8 d 2  k = 1j = 1  

(25)



G π d 4 − d 4in

7.2 Rules

-- the outside effective twist diameter of tooth section is the root diameter plus 0.4 times the normal module; -- the twist of all elements except the target mesh being analyzed is ignored; CAUTION: Equations 24 and 25 only cover torques in the target mesh that arise from gear tooth loading. Other torques may require additional modeling.

8 Gap analysis Elastic bending and torsional deflections, tooth modifications, lead variations and shaft misalignments 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 teeth is defined as the gap. This gap is closed to some degree when the gear set is loaded due to the compliance of the gear teeth along the face width of the target mesh.

Since the angle is small, it is assumed that the deflection in the base tangent plane is proportional to the twist angle.

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.

The rules that apply to this shaft torsional deflection are:

Torsional deflection: Use the values from the torsional analysis for each shaft increment of the

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AMERICAN GEAR MANUFACTURERS ASSOCIATION

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).

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 direction of load on the teeth is negative, the removal of metal at an individual station is entered as a negative value.

At final verification stage use actual shaft misalignment. The shaft misalignment that corresponds to material removal on the tooth flank has the same sign as the load on the tooth flank when entered in table 4.

At final verification stage use actual lead variation measured for the gear set. The lead variation corresponding 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.

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.

Shaft misalignment: Shaft misalignment accounts for the error in concentricity of the bearing diameters

Table 4 is an example of the mesh gap evaluated for mesh #3 of general arrangement shown in figure 5.

Lead variation: The actual lead variation of the gear 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).

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

Shaft #4

Station number

Bending deflection

Torsional deflection

Tooth modification

Lead variation

Shaft misalignment

8

11.8

--9.1

5.0

0.0

9

11.7

--8.9

3.5

0.3

10

11.5

--8.5

2.7

11

11.3

--7.9

2.0

12

11.0

--7.1

13

10.7

14

Total mesh gap

Relative mesh gap

--4.2

11.9

0.0

--5.4

12.8

0.9

--1.3

--6.5

14.1

2.2

--1.8

--7.6

15.6

3.7

--1.0

--2.3

--8.8

17.3

5.4

0.0

--1.3

--2.8

--10.3

19.7

7.8

4.4

0.0

--1.5

--3.3

--11.8

22.0

10.1

--11.0

3.0

0.0

--1.7

--3.8

--13.5

25.4

13.5

--10.5

1.6

0.0

--2.0

--4.3

--15.2

29.9

18.0

0.8

0.0

--2.2

--4.8

--16.1

34.9

23.0

Shaft #3 gap

Bending deflection

Torsional deflection

Tooth modification

0.0

7.7

--12.8

8.6

0.0

0.8

7.4

--12.7

8.4

0.0

0.6

1.3

7.6

--12.6

8.0

0.0

0.8

1.8

8.0

--12.4

7.4

0.0

1.3

1.0

2.3

8.5

--12.1

6.6

--6.1

0.7

1.3

2.8

9.4

--11.8

10.3

--4.9

0.0

1.5

3.3

10.2

15

9.9

--3.5

0.0

1.7

3.8

16

9.5

--2.1

1.0

2.0

4.3

17

9.1

--0.8

3.5

2.2

4.8

18.8

16

Lead variation

Shaft misalignment

Shaft #4 gap

0.0

0.0

--0.3

--0.8

--0.6 --0.8

0.0

5.6

--11.4

11.9 14.7

--9.9

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927--A01

20 18 16

SHAFT #3

14 12

Micrometers

10 8 6 4 2 0 8 --2

9

10

11

12

13

14

15

16

17

15

16

17

--4 --6 --8 --10 --12 --14 --16 --18 --20

Figure 10 -- Shaft number 3 gap

20 18 16 14 12

Micrometers

10 8 6 4 2 0 8 --2

9

10

11

12

13

14

--4 --6 --8 --10 --12 --14 --16

SHAFT #4

--18 --20

Figure 11 -- Shaft number 4 gap

17

AGMA 927--A01

AMERICAN GEAR MANUFACTURERS ASSOCIATION 20 18 16

SHAFT #3

14 12

Micrometers

10 8 6 4 2 0 8 --2

9

10

11

12

13

14

15

16

17

14

15

16

17

--4 --6 --8 --10 --12

SHAFT #4

--14 --16 --18 --20

Figure 12 -- Total mesh gap

20 18 16 14 12

Micrometers

10 8

SHAFT #3

6 4 2 0 8 --2

9

10

11

12

13

--4 --6 --8 --10 --12 --14 --16

SHAFT #4 SHAFT #4

--18 --20

Figure 13 -- Relative mesh gap

18

AMERICAN GEAR MANUFACTURERS ASSOCIATION

9 Load Distribution

Clause 8 explains the methods used to calculate the mesh gap. This gap in the mesh must be accommodated by deflection of the teeth, δt, as shown in figure 14 and equation 26.

9.1 Tooth deflection This method uses the concept of a tooth mesh stiffness constant, Cγm, 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, Cγm, and the mesh is 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 δi = δ ti C γm

AGMA 927--A01

9.2 Mesh gap analysis The mesh gap analysis divides the target mesh into discreet equal length sections, Xi, 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 deflections per equation 26 are based on load intensity, the point loads must be converted to load intensity. This is shown in equation 27. L L δi = i Xi

(27)

where Xi

is length of face where point load is applied, mm;

Li

is load at a specific point “i”, N.

where Lδi

is load intensity, N/mm;

δti

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

L1

Cγm is tooth stiffness constant for the analysis, N/mm/mm (~11 N/mm/mm for steel gears).

L2

L3

L4

L5

L6

X1 X2 X3 X4 X5 X6 Xi

Cγm

δt

Face width Bearing Figure 15 -- Deflection sections

Li

mesh gap, δi

Face width Figure 14 -- Tooth section with spring constant Cγm, load L, and deflection δ 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.

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 Xi are equal. In addition, the sum of the individual loads must equal the total load on the gearset as shown in equation 28. F g = L 1 + L 2 + L 3 + + L n

(28)

where Fg

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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AMERICAN GEAR MANUFACTURERS ASSOCIATION

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 δi − L δj = δ i − δ j C γm

In terms of the point loads used in the mesh gap analysis, equation 29 may be rewritten as: Li Lj − = δ i − δ j C γm Xi Xj





(30)

Li L1 − = δ i − δ 1 Cγ Xi X1

(31)

Or:

XL − δ − δ  C 

(32)

XL − δ − δ  C 

(33)

i

L1 = X 1

γm

1

i

i

And: Li = Xi

1

1

γm

1

i

Sum up the values for all locations using equation 31 and get equation 34 below. Remember, only one value of tooth stiffness, Cγm, is used and the tooth face width is broken into equally spaced segments:

XL − XL  + XL − XL  +⋅⋅⋅ XL − XL 

Face width Total pinion deflection

1

1

2

1

n

1

1

1

2

1

n

1

= δ 1 − δ 1 + δ 2 − δ 1 +⋅⋅⋅ δ n − δ 1 C γm (34) Simplifying equation 34 gives:

XL + XL +⋅⋅⋅ XL  − nXL 1

Mesh gap,δi

1

2

n

2

n

1

1

= δ 1 − δ 1 + δ 2 − δ 1 +⋅⋅⋅ δ n − δ 1 C γm (35) 0.0

δ1

δ2

δ3

δ4

δ5

δ6

The sum of all loads always equals the base tangent plane load, Fg, and all values of Xi are equal, so:

XL + XL +⋅⋅⋅ XL  = XF 1

2

n

g

1

2

n

n

(36)

Solving the equations for the value of L1 gives: Total gear deflection

L1 =

F g C γm X i − i i

δ 1 − δ 1 + δ 2 − δ 1

+⋅⋅⋅ δ n − δ 1 Figure 16 -- Mesh gap section grid

(37)

Using equation 33 the rest of the values for loads can be calculated.

9.3 Summation and load solution

9.4 KH evaluation from loads

Sign convention is very important and is explained further in clause 5. Areas with greater mesh gap have lower tooth load and areas with lower mesh gap have higher tooth load. Using figure 16 as a guide, note that in equation 30 as mesh gap, δi, gets larger, the load, Li, must get smaller.

For the first iteration, a uniform load distribution across the mesh is assumed and gaps are calculated. 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 calculated 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).

One location is selected as a reference, in this example it is location “1” (see figure 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: 20

The loads that correspond to the final iteration that results in negligible change in gaps calculated are

AMERICAN GEAR MANUFACTURERS ASSOCIATION

then used to calculate the load distribution factor, KH. This is defined as the highest or peak load divided by the average load. KH =

L i peak L i ave

(38)

where: Fg L i ave = n

(39)

9.5 Partial face contact Initially all 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 between stations. 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 that had 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, Fg; -- The change in load intensity, Li -- Lj, between any two locations on the face width must equal the change in tooth deflection, δti -- δtj, or change in mesh gap, δi -- δj, between those locations;

AGMA 927--A01

-- 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., ΣLi = Fg; -- The face width shall be divided into eighteen sections for the actual gap analysis and load distribution 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 compensations 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 increase is partially offset by elastic tooth deformations (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 buttressed on both sides and has greater stiffness than a similar tooth portion at the tooth end.

21

AGMA 927--A01

AMERICAN GEAR MANUFACTURERS ASSOCIATION

Annex A Flowcharts for load distribution factor

Input Elastic Data

Non--elastic Data Bending

No

is P&G Done Yes Torsional

No

is P&G Done Yes Gap Analysis

Load Distribution

No

New Gap Difference Small

Yes Output

Figure A.1 -- Overall flow chart 22

AMERICAN GEAR MANUFACTURERS ASSOCIATION

AGMA 927--A01

Case ID

U.S.

SI

Units ?

Units Labels Manual Adjustment in BTCS Target mesh data External forces, moments, torques (Timken convention)

Convert to BTCS

Analysis

Yes

Test

No

Output KH Figure A.2 -- Data flow

23

AGMA 927--A01

AMERICAN GEAR MANUFACTURERS ASSOCIATION

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. Cγm = tooth stiffness constant N = total number of sections δi (j) = gap at each section Li (j) = initial load at each section Xi (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 j X3 = 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 constant M3 = 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

does any station have a load reversal (i.e., teeth are not contacting) or [X6/abs (X6)] * W(j) < 0

Yes

sum all loads with a reversal XTOT = sum {[X6/abs (X6)] * W(j)