AGMA 927- A01AMERICAN 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 ii 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 iii Hertzian deflections. 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. The limitations of the previous AGMA procedures are overcome by the method defined in this information sheet. 1996. Load Distribution Factors -. This version was approved by the AGMA membership on October 22. i. The method provides a significant improvement from the procedures used to define numeric values of face load distribution factor in current AGMA standards. The analytical method described in this information sheet is based on a ”thin slice” model of a gear mesh. 2000. tooth shear.Analytical Methods for Cylindrical Gears.e. They should be sent to the American Gear Manufacturers Association. 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. 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. mesh stiffness (Cγm). in this document are provided for informational purposes only and are not to be construed as a part of AGMA Information Sheet 927--A01. Current AGMA standards utilize either an empirical procedure or a simplified closed form analytical calculation. Alexandria.) by one constant. footnotes and annexes. Suggestions for improvement of this document will be welcome. The method also represents all of the elastic effects of a set of meshing teeth (tooth bending.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION Foreword [The foreword. The first draft of this information sheet was made in February.] This information sheet provides an analytical method to calculate a numeric value for the face load distribution factor for cylindrical gearing.. These parameters include the elastic effects due to deformations under load. if any. Virginia 22314. which provides a description of the analytical procedures that are used in several software programs that have been developed by various gear manufacturing companies. etc. 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. tooth rotation. 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. This is a new document. 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. 1500 King Street. This model treats the distribution of load across the face width of the gear mesh as being independent of the any transverse effects. Without his foresight and contributions this information sheet may not have been possible. Suite 201. Despite these simplifying assumptions. iv . L. . .E. . Neesley . . . Lian . . Charles E. . . . Neesley . . . M. General Electric Company Gear Engineers. . . .A. . Renk AG Milwaukee Gear Company. Luz . . . . R. . Inc. Dalton . F. Berndt . McVittie . Gonzalez Rey M. Pizzichil . . . . A. . . Inc. . . . E.D.G. L.W. M. W. J. Pasquier . . . Amendola . . . Amarillo Gear Company The Falk Corporation Lufkin Industries. Inc. . . . Consultant The Gear Works -. . D. Inc. . Octrue . . . Inc. M.R. Sikorsky Aircraft Division The Timken Company The Horsburgh & Scott Co. . . . R. . . .R. Becker . . . . Pizzichil . . . .J. Lufkin Industries. . . . . Lloyd . Hirt . McCarthy . Henriot . . Hartman . Euclid--Hitachi Heavy Equip.A. Consultant Dudley Technical Group. . . Danieli United. D. . J. LaBath .W. . . Hyde . . Dudley . . Inc. . DeLange . Cardis . . . . Acheson . . The Lubrizol Corporation Philadelphia Gear Corp. The Falk Corporation SUBCOMMITTEE ACTIVE MEMBERS K. M. T. . . . . . . . Ltd. . . . . . . . Dorris Company Vice Chairman: M. M. . .J. .C. . . Inc. . D. Incorporated Consultant GEARTECH Equilon Lubricants The Timken Company The Cincinnati Gear Co. . . . G. The Gear Works -. . . J. . . . . Hinton . . Borden.L. Faure . .B. . G. C. . . O. . Xtek. Nagorny . . . . Bradley . . . . D. . Luz . Consultant General Electric Company Rockwell Automation/Dodge C. General Electric Company Gear Engineers. . . . . . . Rockwell Automation/Dodge M. . Inc. . . . .E. Lisiecki . The Cincinnati Gear Co. Glasener . Incorporated Mobil Technology Company The Horsburgh & Scott Co. S. . . .L. . . Dansdill . G. . Contour Hardening. . . D.A. . . I. . . J. LaBath . Incorporated ISPJAE ITW Rolls--Royce Corporation Consultant Xtek. Dalton . . Lisiecki . .M. Ivers . . . . . . . J. . .Seattle Consultant General Electric Company Prager. . J. . . Inc. Laskin . Inc. . M. . . . .A. . Johnson . . Partridge .F. . . . . . .W. J. .L. . . . .C. .J. . Hoeprich . Beveridge . Mowers . . . . J. . M. . Gonnella . G. . . . . .W. Acheson .V. McVittie . Antosiewicz . . . Inc. .S. . Okamoto .S. R. O. Cragg . . . G. Milburn . . . . . Borden . M. The Timken Company Xtek. . . Bartolomeo . .W. . . . . Inc.H. .J. Hawkins . Flender Corporation G. . . . . Taliaferro . O’Connor . W. Lloyd . . Gay . . Smith . Jackson . .AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 PERSONNEL of the AGMA Helical Rating Committee and Load Distribution SubCommittee Chairman: D. R.A. J. . . Inc.P. . . E. .A. Ciszak . . . ISPJAE Consultant UTC Pratt & Whitney Aircraft CETIM Nippon Gear Company. Cohen . J. . . . Ltd. T. . . L. . Inc. . .P. COMMITTEE ASSOCIATE MEMBERS M. . A. Korossy . Klundt . . . . . .R. Lufkin Industries. .A. . B.S. of Newcastle--Upon--Tyne Rockwell Automation/Dodge Delft University of Technology COMMITTEE ACTIVE MEMBERS K. . Copeland . . . . .W. .E. . . . . . Incorporated v . . Inc. . . DeLange . . . . Bodensieck . . . . R. . . . . . . . Inc.J. . WesTech Gear Corporation Philadelphia Gear Corp. . Holzman . . Broglie . . Gay & Company. . A. . J. .J. Nay . . W. . . W. M.C. . . The Gear Works--Seattle. Errichello . D. A. Nagorny & Associates Philadelphia Gear Corp. K. .R. Escanaverino . . . A. . .Seattle. Bradley . Inc. . V. Nuttall Gear LLC Besco Bodensieck Engineering Co. . . G. . Eberle . . . . Engranes y Maquinaria Arco SA Gear Products. . . . . . . J. F. . .B. . Maddock . . . . . . . . . . . CETIM Univ. . .R. . . Kish . Inc. New Venture Gear. . . Solar Turbines. .G.F. T. . . . .R.M. Chaplin . . . Polder . H.P. . . D. R. . . J. Uherek . T. . . . . . . Gimper . . . . . Phillips . . . . The Falk Corporation SubCommittee Chairman: J. G. .F. . Inc. . . L. . . Mobil Technology Center General Electric Company Prager. . Milburn Engineering.R. A.A. R. . Pennell . . . . . . MAAG Gear AG Caterpillar. . . .A. NASA/Lewis Research Center Dodge Flender Corporation MAAG Gear AG 3E Software & Eng. . . . . . H. . . . Thoma .A. . Ward . Consulting Recovery Systems. . L. . Wasilewski . . . .S. E. . C. . . L. . . Shirley . Schultz . Spiers . . Corp. . . . . A. . . . Sharma . LLC Arrow Gear Company Technische Univ. . F. .C. . . Tellman . . . . B. Scott .W. Smith . . . Sandberg . . . B. R. Muenchen . .A. Y. Uherek .F. Thoma. . . Seireg . . . . . .W. . A. J. C. . Inc. . . Von Graefe . . . . . IIT Research Institute/INFAC Dodge F. . . . . . . Winter .A. . L. . . Wang . . vi 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. .C. A. . . F. . Townsend . . . Swiglo . Tzioumis . . . . . D. . . .D.J.AGMA 927--A01 E. 1. based on combined twisting and bending displacements of a mating gear and pinion. The load distribution factors for use in AGMA parallel axis gear rating standards are defined. Volume 1 1 . The knowledge and judgment required to evaluate the results of this method come from experience in designing. Historically. 1984 Timken Engineering Design Manual. 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. D. It is intended to provide a value of load distribution factor and a means by which different gear designs can be analytically compared. This method assumes that the mesh stiffness is a constant through the entire contact roll and across the face. 1965 ANSI/AGMA 1012--F90. New York. 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.2 Limitations of method This method is intended to be used for general gear design and rating purposes.. General guidance for design modifications to improve load distribution is also included.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.Part 1: Basic principles. A general discussion of the design and manufacturing factors which influence load distribution is included. It is not intended for use by the engineering public at large.1 Method A simplified iterative method for calculation of the face load distribution factor. All publications are subject to revision.AMERICAN GEAR MANUFACTURERS ASSOCIATION American Gear Manufacturers Association -- Load Distribution Factors -. introduction and general influence factors Dudley. AGMA 927--A01 1. the editions were valid.W. This method is intended for use by the experienced gear designer.Part 1: Definitions and Allowable Values of Deviations Relevant to Corresponding Flanks of Gear Teeth ISO 6336--1:1996. is presented. Cylindrical Gears -.16. Profile and Longitudinal Corrections on Involute Gears. Calculation of load capacity of spur and helical gears -. to improve communication between users of those standards.ISO System of Accuracy -. Handbook of Practical Gear Design. Gear Nomenclature. Definitions Of Terms With Symbols ANSI/AGMA 2101--C95. manufacturing and operating gear units. The result of this assumption is often overly conservative (high) values of load distribution factors. The method given here is considered more correct. It is not intended for rigorous detailed analysis to calculate the actual distribution of load across the face width of gear sets. The transverse load distribution (in the plane of rotation) is not evaluated in this information sheet. capable of understanding its limitations and purposes. Fundamental Rating Factors And Calculation Methods For Involute Spur And Helical Gear Teeth ANSI/AGMA ISO 1328--1. At the time of publication. McGraw--Hill. 2 References The following documents were used in the development of this information sheet. NOTE: The symbols and definitions used in this standard may differ from other AGMA standards. 3. 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.3 5.-N/mm/mm mm -. KF. The magnitude of KH is affected by two components. effect outside diameter of the teeth Modulus of elasticity Axial thrust force.1 6.--. KFβ and KFα. In past AGMA standards.2 5.3 7. wherever applicable.1 6.3 5.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. pinion member Modulus of elasticity in shear Hand factor Moment of inertia Units -. The symbols and terms. KHα. KH. In current AGMA standards the load distribution factor.-mm4 First referenced 5.1 6. gear Outside effective twist diameter Inside shaft diameter Outside diameter. The face load distribution factor is the focus of this information sheet.2 5. gear member Axial thrust force. for bending strength as found in ISO standards. In ISO standards. KH.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION 3 Definitions and symbols The terms used.3 5.3 5.1 (continued) 2 .1 5.2 6. along with the clause numbers where they are first discussed.--. There is no separate value.--.3 6. the variables Cm and Km have been associated with this factor.1 5.1 Load distribution factor The load distribution factor.2 9. pinion member Tangential force.1 5. The target mesh includes a target pinion and a target gear. Standard procedures to evaluate it have not been established and it is assumed to be unity in this information sheet. are listed in alphabetical order by symbol in table 1.3 5. is used for both pitting resistance and bending strength calculations. 3.3 7. the variables KHβ. Table 1 -. pinion member Total load in the plane of action Gearing or external force at a distance Separating force. for the analysis Helical/bevel gear face width Drive factor Operating pitch diameter.3 5.4 9. modifies the rating equations to reflect the non--uniform distribution of load along the gear tooth lines of contact as they rotate through mesh.1 5. gear member Tangential force. The user should not assume that familiar symbols can be used without a careful study of their definitions.3 5. transverse load distribution factor and face load distribution factor. have been associated with the factor. gear member Separating force.2 Target mesh The target mesh is that mesh for which load distribution is being analyzed. conform to ANSI/AGMA 1012--F90.-mm mm mm mm N/mm2 N N N N N N N N N/mm2 -. 1 7.2 7. The mesh gap is then mathematically closed by compressing the springs until the sum of the spring forces equals the total tooth force.1 9.elastic deflection of a gear body if it is not a solid disk (such as a spoke gear).3 6. resulting from the target mesh loads and loads external to the mesh.--.1 5.-N N rpm -. The iterative method combines the calculated elastic deflection of the pinion and the gear with other misalignments.1 5.1 9.1 5. -.3 -- tooth alignment deviations of pinion and gear.shaft elastic deflections due to twisting and bending.mesh elastic deflections due to Hertzian contact and tooth bending.3 5.1 9. The result defines a “mesh gap” in the base tangent plane which is the net mismatch between the gear and the pinion.3 5.--.1 6. 3 .1 6.-N N/mm N mm N mm -.--.4 6.1 6.1 6. including bearing clearances and housing bore alignment.3 5.1 5.1 6.-kW -.3 7.4 6.-mm N mm mm mm mm mm mm degrees degrees degrees degrees First referenced 6.1 6. The teeth in mesh are modeled by an equally spaced series of independent parallel compression springs which represent the mesh stiffness. -.3 6. -.1 6.alignment of the axes of rotation of the pinion and gear. -- tooth alignment and crowning modification.3 5. This information sheet presents an iterative analytical method for determining a value of load distribution factor.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 -.1 9. 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: -. 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. This is accomplished by compressing the springs until the sum of the spring forces equals the total tooth force. -- variations of stiffness of the gear teeth.1.for double helical gears the net thrust force is zero as the thrust force from each helix cancels each other. Elastic shaft deflections include shaft twist and bending.displacements of the gearing due to bearing deflection. 4 -. 1) Calculate the mesh gap resulting from an initial uniform load distribution. 4. -. Elastic tooth deflections include Hertzian contact and tooth bending. . -- thermal or centrifugal effects. spacing and runout deviations. alignment and deflection.AGMA 927--A01 -. AMERICAN GEAR MANUFACTURERS ASSOCIATION of the housing and -. housing and 4. -- 3) Calculate a new mesh gap resulting from the new load distribution.elastic deflection foundations. Other displacements that are treated by combining them as an equivalent deflection at the target mesh include: -. -.2 Equivalent misalignment inputs -. 4. gearing forces and moments 5.total tooth load including increases due to application influences and tooth dynamics. -. sign convention. gearing forces and moments are: -- the target mesh shafts are mutually parallel. gears with one helix 4. The iterative analytical method consists of the following basic steps: -. -- running--in or lapping effects. -.the coordinate system for all calculations lies in the base tangent plane. -. The method does not consider the following influences: -- tooth profile.shear coupling between the mesh gap compression springs representing the mesh stiffness is ignored.2 Assumptions and simplifications The following assumptions and simplifications are used: -- the weight of components is ignored. 4) Repeat steps 2 and 3 until the change in load distribution from the previous iteration is negligible. sign convention.1 Methodology of the -- thermal or centrifugal effects.the base tangent plane is a plane tangent to the base circles of the target mesh.1 Rules The rules that govern the coordinate system.equivalent elastic deflection of non--solid body gears (such as a spoke gear). load on these meshes is treated as concentrated in the center of the mesh.displacements due to bearing clearance. 2) Calculate a new load distribution by mathematically closing the mesh gap.elastic deflection foundations. this is generally true as long as one member can float with respect to the other with no external axial load applied. plus the mesh. 5) The load distribution factor is then calculated from this final load distribution.1. all shafts are supported on two bearings. -- running--in or lapping effects. -. -. -.alignment deviations and modifications of pinion and gear teeth.1 Calculated elastic deflections Deflections which are calculated within the iterative method include the elastic deflections of the pinion and gear shafts.mesh stiffness is a constant across the full width of tooth. -.effects of uneven distribution of load on meshes other than the target mesh are ignored.double helical overloaded. 5 Coordinate system.for double helical gears the tangential and separating force is distributed equally on each hand helix. -. + BT: obtained by right hand rule. they are assigned negative values. of the target mesh and is defined as the base tangent coordinate system. Base tangent plane . Using this definition of the refence end. and BTZ. the reference end is the end of the driving element shaft opposite the torque input end. The BTZ axis is parallel to the axes of the target mesh shafts. -. 5. BTN and BTZ are assigned positive values. moment and deflection along the positive direction of BT. the input torque is counterclockwise when viewed from the reference end. For consistency in defining the positive direction of the BTCS axes and in calculating the mesh loads.a modified Timken sign convention is followed. In figure 2. the positive directions of the BTCS axes are determined as follows: + BTZ: away from the reference end. a “reference end” needs to be identified. Along the negative direction of BT.Base tangent coordinate system for CW driven rotation from reference end 5 . plane and the edge of the target mesh face closest to the reference end (see figures 1 and 2). -. The BT axis lies in the BTP and is perpendicular to the BTZ axis. The force. BTN and BTZ. 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 -. -. 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. For purposes of this information sheet. BTN (base tangent normal).2). The BTCS is comprised of three orthogonal axes: BT. BTP. BTN to BTZ.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.each analysis includes only the two shafts under consideration.AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 -. The coordinate system is aligned with the base tangent plane.2 Coordinate system and sign convention + BTN: toward the driven element. The BTN axis is perpendicular to both the BT and the BTZ axes (normal to the base tangent plane). In figure 1.the input torque to the driving element enters the shaft from one side only and is fully balanced by torque in the target mesh. the input torque is clockwise when viewed from the reference end. BTCS. mm. these factors will ensure that the proper direction of the forces are determined. H is hand factor (see table 2). the values of factors H. The directions obtained will be consistent with the BTCS definition presented in 5. S is speed of gear shaft. mm. degrees. degrees. degrees. 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. kW. gear. D is drive factor (see table 2). N. R. N. rpm. The separating force is calculated as: F sG = cos ψ FsG is separating force. 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. The tangential force is calculated as: F tG = 1. gear member.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION Target shaft -driver Driver Base diameter -driving element Input torque +BTZ * Target mesh Base diameter -driven element . G − tan Ô sin γ G cos ψ FtG is tangential force.Base tangent coordinate system for CCW driven rotation from reference end 5.3 Gearing forces and signs R is rotation factor (see table 2). ψ is helix angle/spiral angle. The forces on the gear member are given by equations 1 through 3. These elastic deflections need to be combined with all other sources of potential misalignment. gear member. Ô is normal pressure angle. and D are obtained using table 2. gear. Base tangent plane +BT Reference end +BTN Driven Target shaft -driven Figure 2 -. Meshing gear members transmitting torque will cause forces and moments to develop on the shafts that carry these gear members. In these equations. b is helical/bevel gear face width.91 × 10 7 P (D R ) S D pG − b sin γ G (1) DpG is operating pitch diameter. A. When properly applied. γG is bevel pitch angle. A is apex (bevel) factor (see table 2).2. (3) . gear member. 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. N. spiral. or pitch angles. For this example. rotation. set the values of these angles equal to zero in equations 1 to 3. replace the gear values in equations 1 through 3 with the corresponding pinion values. 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. or herringbone Apex toward reference end (see figure 3). the tangential mesh load on the driving element will introduce positive mesh displacement in the base tangent plane. straight bevel. To obtain the force for the pinion member.AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 Table 2 – Values for factors hand. The forces must be determined for each mesh on each of the target mesh shafts. Hand Right hand helix Left hand helix Right hand spiral Left hand spiral Apex Away from reference Toward reference Figure 3 -. With the sign convention of figure 3 and the definition of the BT axis. Figure 4 shows the sign convention to use for the direction of the gear forces.Hand of cut for gears and explanation of apex for bevel gears 7 . Shafts 3 and 4 are the target shafts. 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. mesh 3 is the target mesh. The direction shown is for the positive value of forces evaluated by the Figure 5 shows a general arrangement. For gears having no helix. apex. Figure 5 -.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.Axis along base tangent plane of target mesh +BTN -.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION Mating target shaft One target shaft Mate shaft If mate to target shaft is on the left.Example general case gear arrangement (base tangent coordinate system) 8 . use these positive force directions View direction from reference end Figure 4 -. use these positive force directions Ft Fa Fa Fs Fs Ft If mate to target shaft is on the right.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 -. However. N mm. For a double helical mesh the net moment will be zero. To obtain the moment due to an axial thrust force on the pinion member.Axis along base tangent plane of target mesh BTN -. These forces affect the load distribution of mesh 3.AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 5. where MG moment due to axial thrust force. which will cause elastic deflections. Figure 6 demonstrates the resolution of the shaft 3 and 4 forces and moments into the base tangent coordinate system for mesh 3. For each additional mesh on the target shafts.View A--A from figure 5 9 . when calculating shaft deflections.1 Simplified bending calculation routine As explained in other sections. Figure 6 shows the tangential and separating forces and the axial thrust moments acting on shafts 3 and 4 of figure 5. 6. These deflections can affect the alignment of the gear teeth and therefore affect the load distribution across the gear face width. three changes in diameter. Gears transmitting power will impose forces and moments on their shafts. This is as shown in figure 7 and table 3.4 Gearing moments 6 Shaft bending deflections The axial thrust forces acting on the pinion and gear cause moments. Rules for calculating bending deflection when calculating load distribution factor are also presented. the area of the gear teeth is broken into eighteen separate load application sections. 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. For the target mesh.Axis normal to base tangent plane of target mesh Figure 6 -. and two point loads. 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 -. the moments can be determined for each mesh section. replace the gear values by the corresponding pinion values. to simplify the explanation of the deflection calculation method the following model and explanation will be of a stepped shaft with two supports. the resulting moment is assumed to act at the center of the face width. The moment due to an axial thrust force on the gear member is given by equation 4. AMERICAN GEAR MANUFACTURERS ASSOCIATION Table 3 -.Calculation data and results AGMA 927--A01 10 . The basic equation for small deflection of a stepped shaft is: d2 y =M EI dx 2 x is the distance between stations.0 6 7 38. N/mm2. RR.0 4 50. Vi. 2. mm.AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 --13500 22. y is the deflection.Example shaft All modeling will be from the left--hand support moving toward the right--hand support. Integrating equation 8 twice gives deflection. one station below the station for which it is calculated. mm.0 25. Nmm. where X fi = x i + X fi−1 i = 1. Step 1: Divide the shaft into lengths with intervals beginning at each force and at each change in section (see figure 7). Using standard static force analyses calculate the reaction.0 +6180 --1680 Figure 7 -. (6) Then calculate the reaction at the left using the total sum of the loads. Fi.0 1 28.0 +9000 28. The last shear value 11 . The gearing forces and any other external forces are used to obtain the free body force diagram. RR = Fi Xfi Ls (5) F is the force applied at a distance. A tabulated form as shown in table 3 lends itself to the process. 3. mm4. are specified. at each station by summing the values in column 4. Deflection at supports is zero. M is the bending moment. N. Step 4: List free body forces in column 4 on the same lines as the station numbers at which they occur.0 5 44. Step 5: Calculate the shear. I is the moment of inertia. Fi.0 22. The following step by step procedure applied to the stepped shaft as shown in figure 7 will illustrate the procedure evaluating shaft deflection. RL = Fi − RR (7) It is critical that sign convention be maintained during the calculations with the preceding formulas. In the force diagram the forces. i. Care should be taken to designate proper signs to forces (upward forces are considered positive in this example). on alternate lines in column 1 of calculation sheet (see table 3). and the distances they act from the left support. Xfi is the distance from left support to load location.0 3 25.0 2 35. Ls is the distance between the two supports. at the right side support by summing the moments about the left support. 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. Tabulate each shear value in column 5. Step 3: List station numbers. E is the modulus of elasticity. Xfi. Step 9: Multiply each Ii value by modulus of elasticity. n (10) Step 8: Calculate the moment of inertia. These values. 3. AMEI i = Step 6: In column 6. in column 10. for each interval by averaging the values on the lines on which the station is listed and the following line.0 and Mn = 0. 2. SL i+1 = SL i + AMEI ix i+1 i = 1. 12 Step 11: Obtain the average MEI values. to obtain the integration constant per mm of length. 3. i = 1 and i = n. xi (column 6).e. the constant is obtained by summing the deflection increment values in column 14 to obtain Sy. n − 1 (17) 2 Step 14: Obtain the deflection increment values. Mi. The moment at the first and last station. 2. and insert the EIi value in column 9 on the same lines as corresponding Ii values. 2. i is the station number. 2. 2. 3. Value at the first station is zero. 2. n is the station number at end support. 2. din is the inside shaft diameter. Ls. For the simply supported shaft with no load outside of the supports as shown in figure 7. 3. 3. DIi. List the average values on the lines between stations in column 11. Ii. in bending for each interval. mm. Dividing the EIi values by 103 before tabulating them in column 9 results in units of µm for the rest of the tabulation. N. SL1=0). AMEIi. ASLi. M1=0. n − 1 EI i (14) MEI ui + MEI li i = 1. n − 1 (9) where V is the shear. n (11) where dsh is the outside shaft diameter (see 6.0). 3. 2. (21) will change the . Place the I value in column 8 on the line between the two stations at which the interval begins and ends. should be zero (i. DI i = ASL ix i+1 i = 1. These values are listed on the same lines as the stations. EI i = ( E )I i i = 1. Values at succeeding stations are obtained by summing the products of shear force. 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. For steel use E = 206 000 N/mm2. on the same line as the station number. Step 7: Calculate bending moment. List these two values. MEIui and MEIli. in column 14 by multiplying the average slope value in column 13 and the xi value from the next lower line in column 6. M i+1 = M i + V i+1x i+1 i = 1. 3. list the distance to the preceding station. 2. n−1 Sy = i=1 DI i (19) xi (20) n Ls = IC = i=1 − Sy Ls Other shaft configurations integration constant. E. 3. 3.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION should be numerically equal to but opposite in sign to the last force listed in column 4. The sign of Sy is changed and the sum divided by the distance between the reaction. n − 1 (16) Step 13: Average the slope values in column 12 at the beginning and end of each interval. in column 12 starting with zero at station 1 (i. n − 1 2 (15) Step 12: Calculate the slope value. MEI ui = Mi i = 1. V i+1 = V i + F i⋅⋅⋅ i = 1. n − 1 (18) Step 15: The next step is to evaluate the integration constant which depends on type of shaft. mm. Ii = π d4 sh i − d4 64 in i i = 1. 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. n − 1 EI i (13) MEI li = M i+1 i = 1. at each station and list the value in column 7. and distance between stations. SLi. 2. 3.e. are listed on the lines between stations in column 13.. ASL i = SL i + SL i+1 i = 1. Vi (column 5).2). 0 +6180 22.0 3 25.0 1 35.0 5 44.0 --150000 (Nmm) Moment Diagram.0 50.0 --0.0 Slope Curve 0.0 28.0 --1680 +10000 (N) 0. M +0.Calculated shaft diagrams 13 .AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 --13500 22.0 6 7 38. V 350000 (Nmm) 0.01 (1/mm) 0.0 2 +9000 28.0 4 25.0 --10000 (N) Shear Diagram.01 (1/mm) M Diagram EI 0.4 (mrad) 0.0 --10 (mm) Deflection Curve Figure 8 -. 1 Torsional deflection The torque input end is subjected to full torque. yn. i is station number. the following rules apply: -- This is a two dimensional deflection analysis. The torque value decreases along the face until it becomes zero at the other end. din.1 to calculate load distribution. mm. is the distance between adjacent stations. Consider a cylindrical shaft with circular cross section with outside effective twist diameter. no longer than 30 mm. The twist must be converted from radians to a deflection in the base tangent plane. inside diameter. if adding more does not significantly change the calculation results. y i+1 = y i + DI i + ICS i i = 1. The twist will cause deflection at the teeth that will affect the load distribution across their face width. ICS i = ( IC )x i+1 i = 1. i.When calculating shaft deflections. IC.Only forces acting in the base tangent plane are considered. List these values in column 15 on the same line as the average slope and deflection increments. the following rules also apply: Lj is load at a station. no longer than 3 times the shortest section of the non--gear tooth portion of the shaft. is now calculated. d. and incremental length. should be very close to zero. 3. Hence the direction of torque path is of importance. the number of original stations is adequate. n − 1 (23) 6. n − 1 (22) Step 17: Column 16 is the calculated deflection. ICSi. Rules for station length are: no longer than 1/2 diameter of the station. 7 Shaft torsional deflection Meshing gear sets transmitting torque will also twist the shafts that carry the gear elements. mm. The equation for torsional twist can be found in machinery design text.The effective bending outside diameter of the teeth is the (tip diameter minus root diameter)/2 plus the root diameter. Multiply integration constant. -.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. d is effective twist diameter (see 7. din is inside diameter.0. the area of the gear teeth is broken into eighteen equal sections.2). Place zero at left support location. -.The length between any two stations is critical to the accuracy of this calculation. calculated in step 15 by xi value on the next lower line from column 6 to obtain the constant for each section. N. Xj -. 14 . 7. This results in the equation: −1 2 i i X L j j4 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. These deflection values are inserted on the same line as the station. as shown in figure 9. mm. -.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION Step 16: The integration constant for each section. because support locations must have zero deflection. The torsional deflection must be calculated over the length of the tooth face. Xj. For all other stations the deflection values are obtained by summing together the deflection increment and integration constant values from columns 14 and 15. 2. y1=0.2 Rules When using the shaft bending deflection routine explained in 6. When in doubt about the number of stations. When calculating bending deflection for load distribution factor. 3. G is shear modulus (83 000 N/mm2 for steel). As a math check when summing the values of yi the calculated value at the right support location.e. -. Equation 24 is in a form that allows summation using the discreet stations used in this document. -- Shear deflections are not included. 2. the outside effective twist diameter of tooth section is the root diameter plus 0. The theoretically correct equation would be an integration. Retain the positive or negative sign of the bending deflection. 8 Gap analysis Elastic bending and torsional deflections. 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.Torsional increments At the first point of interest on the tooth where j = 1. it is assumed that the deflection in the base tangent plane is proportional to the twist angle. tooth modifications. CAUTION: Equations 24 and 25 only cover torques in the target mesh that arise from gear tooth loading. The distance between non contacting points along the face width of the mating teeth is defined as the gap. the summation of Xj will be zero and the torsional deflection is zero. 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 15 . see figure 9.the twist of all elements except the target mesh being analyzed is ignored. Equation 24 is an approximation which yields reasonable results for gearing.AMERICAN GEAR MANUFACTURERS ASSOCIATION 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 -.2 Rules -. t δi = (i − 1) k 10 3 L j X k8 d 2 k = 1j = 1 (25) G π d 4 − d 4in 7.4 times the normal module. Continued calculation of the torsional twist toward the end of the tooth face where torque is being applied results in a maximum torsional deflection. A slightly more accurate approximation is found in equation 25. Since the angle is small. -. Bending deflection: Use the values obtained from the bending analysis for each shaft increment of the target mesh. Other torques may require additional modeling. lead variations and shaft misalignments cause the gear teeth to not be in contact across the entire face width. AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION on the shaft.3 10 11.1 --0. Table 4 -.0 2.8 --0.3 10.4 --12.4 0. At final verification stage use actual lead variation measured for the gear set.7 --1.1 --11.9 3.5 --8.3 1.0 0.4 --11.3 --2.3 --0. The lead variation must be incorporated so as to increase the total mesh gap (check both directions).2 11.8 18.1 0.0 5.0 3.9 18. Lead variation: The actual lead variation of the gear set is not available at the design stage.6 0.4 11.0 0.9 14.1 2. At design stage.0 --1.0 --1. tooth modifications.8 0. The last column in table 4 reflects the relative mesh gap. The sign convention for tooth modification as illustrated in table 4 is the following: if the load direction on the teeth is positive.3 7. lead variation and misalignment.0 0.6 --0.Evaluation of mesh gap for mesh #3. The difference between the individual shaft gap positions is the total mesh gap.6 15.0 0.8 --7.5 --2.3 --6.7 --8.5 --10. 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 5.0 1.0 10. housing bore non--parallelism.5 3. 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 4.3 8.0 --4.8 8. the shaft gap is the algebraic sum of all deflections.6 8.1 34. target mesh. In table 4.2 4.9 2.0 1. removal of metal at an individual station is entered as a positive value.5 --12.3 5. modifications.0 --5.8 --9.6 --11. variations and misalignment values with proper positive or negative signs for each shaft of the target mesh to form table 4.7 --9.5 0.8 9.8 3. Retain the positive or negative sign of torsional deflection.8 1. At this stage lead variation based on the expected ANSI/AGMA ISO 1328--1 tolerance of the gear set may be used.8 7.9 .4 7.7 11 11.4 13.0 7.0 0.3 --8.0 --12.2 --4.9 --1.1 6.5 0.7 3.7 1.6 --12.7 7. the removal of metal at an individual station is entered as a negative value.0 9 11.9 0.8 0.0 --2.7 --12.6 1.0 12 11.7 8. if the direction of load on the teeth is negative. At final verification stage use actual shaft misalignment. Use the deflections. Tooth modification: Tooth modification accounts for lead modification and crowning.9 23.5 25. etc.0 --0.3 19.2 29.0 --2.8 10.2 15 9.0 2.6 0.8 0.2 --1. mm Shaft #3 Shaft #4 Station number Bending deflection Torsional deflection Tooth modification Lead variation Shaft misalignment 8 11. To evaluate load distribution by the iterative method the relative gap is used. values should be based on expected manufacturing accuracy. 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.8 --10.7 14 Total mesh gap Relative mesh gap --4.0 0.0 0. 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. bearing clearance.0 --7.0 0.4 0.3 2.1 13 10.1 5.5 2.5 14.3 --11.4 0.8 4.4 12.3 17 9.8 22.9 --3.6 3.8 17.8 16 Lead variation Shaft misalignment Shaft #4 gap 0.6 --6.0 --2. Incorporate expected shaft misalignment so as to increase mesh gap (check both directions).0 --1.3 --4.5 2.8 8.0 1.3 --15.8 16 9.8 --13.3 --7.8 --16.4 0.1 1.7 --3.5 --3.0 4.5 1.9 0.0 Shaft #3 gap Bending deflection Torsional deflection Tooth modification 0. 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 . L L δi = i Xi (27) where Xi is length of face where point load is applied. Also note that the tooth is divided into equal length sections such that all values of Xi are equal. N/mm. 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. δt. Since the method for calculating mesh gap uses point loads. the point loads must be converted to load intensity.Tooth section with spring constant Cγm. This is shown in equation 27. to compare the tooth load intensity and tooth deflection with the total load and overall mesh gap. Cγm. and the mesh is assumed to be a set of independent springs (as shown in figure 14).AMERICAN GEAR MANUFACTURERS ASSOCIATION 9 Load Distribution Clause 8 explains the methods used to calculate the mesh gap. mm. BTP. Note that load is not applied directly on the ends of the tooth. In addition. mm. the base tangent plane along the line of action is used and multiple teeth in contact are ignored. For simplicity. N. with point loads. is proportional to the difference in 19 . Xi. L2 L3 L4 L5 L6 X1 X2 X3 X4 X5 X6 Xi Cγm δt Face width Bearing Figure 15 -. F g = L 1 + L 2 + L 3 + + L n (28) where Fg is total load in plane of action. For double helical.2 Mesh gap analysis The mesh gap analysis divides the target mesh into discreet equal length sections. analyze each helix separately. applied in the center of each of these sections (see figure 15). load L. δi Face width Figure 14 -. (26) L δi = δ ti C γm AGMA 927--A01 9. Effectively the mesh is analyzed as if it were a spur set. i and j. δti is tooth deflection at a load point “i”. N. while the tooth deflections per equation 26 are based on load intensity. this clause will use only 6 sections in the mesh area. The difference in load intensity between any two points. Cγm. N/mm/mm (~11 N/mm/mm for steel gears). Hertzian contact and tooth bending deflections are combined to produce a single mesh stiffness constant. 9. as shown in figure 14 and equation 26. C) analytical methods where the load distribution was assumed as a straight line over the whole face width.1 Tooth deflection This method uses the concept of a tooth mesh stiffness constant. This should improve accuracy as mesh stiffness is generally lower at the ends of the teeth. L1 Cγm is tooth stiffness constant for the analysis.Deflection sections Li mesh gap. but it is assumed constant in this analysis. For the purpose of illustrating this concept. This gap in the mesh must be accommodated by deflection of the teeth. the sum of the individual loads must equal the total load on the gearset as shown in equation 28. Li is load at a specific point “i”. where Lδi is load intensity. Li. and deflection δ This assumed linearity differs from previous AGMA (AGMA 218) and ISO (ISO 6336--1. only one value of tooth stiffness. (29) L δi − L δj = δ i − δ j C γm In terms of the point loads used in the mesh gap analysis. 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 -. Remember. Areas with greater mesh gap have lower tooth load and areas with lower mesh gap have higher tooth load. rather the change in mesh gap which is equal to the change in tooth deflection is used. This iteration process is continued until the newly calculated gaps differ from the previous ones by only a small amount.δi 1 2 n 2 n 1 1 = δ 1 − δ 1 + δ 2 − δ 1 +⋅⋅⋅ δ n − δ 1 C γm (35) 0.Mesh gap section grid (37) Using equation 33 the rest of the values for loads can be calculated. Cγm. note that in equation 30 as mesh gap. must get smaller.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION mesh gap between these two points multiplied by the tooth stiffness constant. Therefore. gets larger. and all values of Xi are equal. iterations are required to get an acceptable error (less than 3. Using figure 16 as a guide. δi. Li. Usually only a few. A sum of the values for all locations referenced to location “1” can then be created.0 mm change in gaps calculated). One location is selected as a reference.0 δ1 δ2 δ3 δ4 δ5 δ6 The sum of all loads always equals the base tangent plane load. equation 29 below can be derived from equation 26 (see figure 16).4 KH evaluation from loads Sign convention is very important and is explained further in clause 5. This new load distribution is then used to calculate a new set of gaps. 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. Notice the switch in terms. an uneven load distribution is calculated. 2 or 3. 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. 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 . The absolute tooth deflection is not used. Fg. the load. a uniform load distribution across the mesh is assumed and gaps are calculated. From these initial gaps.3 Summation and load solution 9. in this example it is location “1” (see figure 16). For the first iteration. 9. e.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. the load must be zero at that location. KH.The face width shall be divided into eighteen sections for the actual gap analysis and load distribution factor calculations.. Set the value of load to zero at all stations that had a change in sign. 9.1 Differential thermal conditions Temperature differences are developed between the pinion and mating gear elements and they may vary along the face width. 21 . -.e. and therefore. between those locations. even if a change in sign is calculated. have the same sign. -. Li. Lead correction may be used to compensate for this. This thermal differential will cause pinion base pitch increases that exceed those of the cooler mating gear.δtj.AMERICAN GEAR MANUFACTURERS ASSOCIATION then used to calculate the load distribution factor. To find the actual loads at these stations do the following.. ΣLi = Fg.1). Sum all the loads that had a change in sign and divide by the total number of loads that had a change in sign. Under running conditions the pinion element of a gear set operates at a higher temperature than its mating gear. 10 Future considerations 10. This is defined as the highest or peak load divided by the average load.6 Restatement of rules The rules that govern the loads on the face width are: -. -. Subtract this value from each load that did not have a change in sign. 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. 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. δi -. the difference in load between stations with tooth contact will be correct. 10.δj. i.The sum of the individual loads on the face width. KH = L i peak L i ave (38) where: Fg L i ave = n (39) 9. between any two locations on the face width must equal the change in tooth deflection. δti -. 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. i. This indicates tooth separation and there is no tooth contact at that location.Lj. Therefore. 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. Fg.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.2 Mesh stiffness variations The stiffness of a gear tooth at any given location along its length is buttressed by adjacent tooth length. must equal the total load on the gearset. If there is not full contact across the face width some stations will have their load value change sign.5 Partial face contact Initially all loads on the face width are assumed in the same direction. Li -. or change in mesh gap. In speed reducers the base pitch differential increase is partially offset by elastic tooth deformations (refer to 5. Profile modification is often used to compensate for this. AGMA 927--A01 -. Both of these phenomena produce distortions that may require lead compensations to achieve acceptable load distribution.The change in load intensity. 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.Overall flow chart 22 .1 -. AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 Case ID U.Data flow 23 . 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.S.2 -. 3 -.X (j) * e] for j = 1 to k sum of deflection and load X6 = sum [W (j)] for j = 1 to k total load.Z (j) relative gap from section 1 to section j X3 = sum [W (j) / Y (j) -. positive loads and deflections are in same direction. 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) <0} KTOT = sum number of stations where there is load reversal add XTOT/KTOT to all stations without a load reversal No CALL SUBROUTINE calculate deflections and perform gap analysis based on new load distribution No set all stations with a load reversal to zero (0.Overall flow chart detail of program CmSolve 24 . 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. The sign convention is critical.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.e.0) Find maximum value of W(j) Y5 = max [abs W(i): abs W(k)] calculate misalignment factor C5 = Y5/abs (M3) does new gap analysis differ from last gap analysis by a significant amount Yes OUTPUT C5 = misalignment factor Km Z(j) = final gap analysis W(j) = final load distribution Figure A. 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) -.. A(j)4)] torsional deflection OUTPUT T(j) = torsional deflection across mesh Figure A. G = shear modulus m = 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 m L(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.AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 INPUT Values The helix is divided into sections of equal length with loads placed in the center of each section.Torsional flow chart of program CmSolve 25 .4 -. The sign convention is critical.1416 * (D(j)4 -. 1 Sta.1 -.08 69.35 70.δ1 0.1.1 KH example calculation In this example a pinion shaft with dimensions as shown in figure B.83 = 16 320 (B. The values are carried to the significant digits shown to keep round--off error to a minimum and should not be confused with the precision of the deflection analysis.00 0.08 mm in the direction to increase mesh gap was included to account for manufacturing and assembly errors.94 --1.00 --0.6 Mesh gap Torque path 45 Bearing support Figure B.64 --3.1.03 --2. Fg.73 --2.18 δi -.08 Total.61 + 8.1 --5. δi 67.2) × 11] 22. of 104 090 N upward is analyzed for mesh gap.3 114 Figure B.78 + 19.40 --4.08 --14. micrometers Torsional Misc.07 --9.05 74.6 22. and with a total load.57 48.20 Load Li.35 Deflections.6 137 Face width Rotation 135.23 1. XL + XL + XL = 4560 1 2 6 1 2 6 L 1 = [4560 − (0 − 2.Example sections 0 22. A miscellaneous misalignment of 5.83 135. Refer to figure B.2 for gap analysis information.45 74.78 19.7 68.06 --5.Gap analysis Table B.73 − 2.6 X1 X2 X3 X4 X5 X6 55 115. In this gap analysis the deflection of the gear is very small and is assumed to be a straight line.2 -. No.45 72.1) 6 Bending and shear deflection 75 Fg = Σ Li = 104 090 N L1 L4 L2 L3 Miscellaneous mismatch 65 L L5 6 Torsional deflection 115. i 1 2 3 4 5 6 26 Bending 67.23 + 1.00 --2.35 72.05 67. 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 Cγm = 11 N/mm/mm is used.74 58. 0.5 91. This load is broken into six even loads of 17 348 N each and gives the shaft deflection shown in table B..61 8.58 65.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION Annex B Load distribution examples B.82 --2.8 45. N 17 348 17 348 17 348 17 348 17 348 17 348 . However this example was also run with 20 load stations across the face width and it only changed the KH value by 4% to 1.AMERICAN GEAR MANUFACTURERS ASSOCIATION Solve for other values of “L” using equation 33: L 2 = 22.83 16 320 + 8. It was used to do an international comparative analysis in an effort to improve the calculation of load distribution for load capacity determinations.98 69.78 ∗ 11 = 18 520 22.83 16 320 + 19.83 16 320 − 2.23 --3.08 mm in a direction to reduce mesh gap gave a KH = 1. and this may not insure sufficient accuracy.23 ∗ 11 = 15 760 22. as more stations and iterations are not hard to process.27. and although further iterations did change the values.18. it is not necessary to have large numbers of load stations. especially in overhung designs or multiple reduction units.61 ∗ 11 = 16 720 22.6) Using the non--uniform loads calculated.22 17 350 (B.83 (B. 0.83 (B.00 --0.2.58 --2.06 --5. and in varying amounts.. loading and deflections are as shown in Table B. It is necessary to investigate the effects of miscellaneous misalignment in the other direction. they did not change the overall accuracy of the KH calculation.02 67. B.1 --5.4) L 5 = 22. The dimensions. The computer software program CmSolve was developed to do an analysis as described in this document.83 16 320 − 2. In this example the deflection of the gear was not considered.00 0. as this can have a big impact on the KH for a gearset.02 18.87 --1.08 Total. Use of computers make this a moot AGMA 927--A01 question.24 δi -. re--calculate the deflections and new loads in an iteration until sufficient accuracy has been attained.95 48.82 69. δi 66.2 ∗ 11 = 21 140 22.57 66.07 --8.31 72.83 16 320 + 1. further analysis gives values shown in table B. i 1 2 3 4 5 6 Bending 66.2 Sta.3) L 4 = 22.59 0.72 74.01 58. N 16 410 15 690 15 760 16 660 18 430 21 140 27 .δ1 0. no. In this example.03 --2.83 (B.5) L 6 = 22.98 --4.83 (B. Only six stations across the face width were used.98 71.73 ∗ 11 = 15 630 22.08 --14.3 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.7) Sufficient accuracy was achieved in this example on the first calculation.97 8.2) L 3 = 22.2 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. Table B.85 --2. 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.00 --2.41 Deflections. In some cases the deflection of the mating element could make a major impact. micrometers Torsional Misc.83 (B.7 Load Li. So within the accuracy of the procedure. For this example a miscellaneous misalignment of 5.21 74. Therefore: K H = 21 140 = 1. 017 34.605 25.3 -.3 --4.9 --12.11 10 58.706 18.2.1 38.5 --48.3 35.064 33.81 11 64.9 37.7 60.3 --17.62 15 79.CW -.5 80.8 --21.7 --8.3 93.9 2.3 --3.22 13 73.4 33.8 --0.533 MICRO--METER PER HELIX 100 Double Reduction Low Speed 80 Load / Deflection 60 40 20 0 5 10 15 20 --20 --40 --60 Location Across Face -.5 --18.8 --47.8 --20.0 --12.41 9 52.3 35.0 --22.7 8 44.5 7.1 --9.6 --45.55 22 77.9 --1.14 20 81.2 20.8 --41.7 --5.5 --19.9 --15.2 73.6 26.0 LOAD DISTRIBUTION FACTOR CM= 1.6 27.707 MICRO--METER PER HELIX CROWN AMOUNT VALUE = 37.84 21 79.6 53.4 30.2 87.CmSolve example CmSolve Version 4.0 --18.7 --25.7 --11.2 18.6 107. TOTAL RELATIVE (MM) NUMBER (N*100) (MU--M) (MU--M) (MU--M) (MU--M) (MU--M) 0 7 36.73 18 83. (MU--M) TOTAL (MU--M) RELATIVE (MU--M) .8 37.7 --26.LS Pinion -.642 33.179508 MISC MISALIGNMENT VALUE = 19.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION Table B.590 35.25 23 73.8 46.92 14 77.030 30.694 34.0 --33.708 35.1 --17.0 27.33 16 81.1 --14.4 40.0 13.640 31.1 --21.0 --1.535 26.4 0 6.850 22.0 --20.44 19 82.483 34.7 --20.8 100.7 --7.223 15.9 23.8 38.231 20.7 --25.5 --0.3 13.508 32.5 8.9 --42.fma=(fHB1**2+fHB2**2)**0.0 --0.1 01/15/00 AGMA 07:01:00 AM ISO Double -.7 --2.3 67.8 38.52 12 69.2 30.1 --22.1 33.4 --10.95 24 69.7 113.9 --37.8 --6.040 34.4 --38.668 28.Station Number Saved File Image of Input Data 28 25 LOAD (N*100) BENDING (MU--M) TORSIONAL (MU--M) MISC.3 --46.5 Crowned **********************DEFLECTIONS*********************** LENGTH STATION LOAD BENDING TORSIONAL MISC.03 17 82.2 --15.4 --22.4 0 0 15.0 --23.5 --47.2 --31.0 --44.1 33. 0 2621..22.22.0...0435.0.0..0 2.22.0 -1763.3..0 1584.0.22.0435.22.4..0 0.2639.20 0.0.0.2685.0.0.0..3.0435.22.fma=(fHB1**2+fHHB2**2)**0.0.0.4.2639.46.0.4.2.4.2.2.0.0 1584.22.6.4.451.0..2639.0.37..5.4..0.46.0 1584.4..22.0.0.0.22.77.2.22.0 2.364.0 18.0.2675.0.2639.0435.0435.22..537.4.22.2639.11.0 0.22.22.0.0 1584.0.0.0 1584.0 0.0 0.0435.0507.22..0 1584.0 1584.0.0 0.4.1.0.2639.0 0.364.5” 0 29 .76..2639.0 1763.0.0.48.0..0.0.11.1319.16.0 0.3.22.0.0.0435.35.0.5.4.85.0507.0435.0.0 1584.1477.9..0.451.0435.0 2.3..2639.5.1.4.2639.2639.0.22.528.-775.4.3.8031 300.4.0435.4.1574.707.22.001.4.0 1584..0.0.4.28512.0 1584.0435.001.3.5.0 0.-99.0 3.0.2639.2639.3 ”AGMA” ”ISO Double .0.0.2..0.5.0 0.0.1.22.340.0 -2621.20.0.4.8483.825.0 0.2639.0 0.0.0.22.3.2.4.0435..11.4.4.0 2.2639.11.0 1584..4.0.0.0 1584.0..0435.0435.0435.0.4115.22..22.0.4125.75.0 0.1 10.091.4.5.3..0.0435.0 1 7.92.4..2639.604..-3.0 1584.1.25 2.0 0.0 1584.0435.632.0 1584..0 2.0.4..7834.0 1584.4.728.LS Pinion .22.451..3..3.-3.0 7735...0.22 2.0.0.AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 0.4.11.0 0.6.CW .4.4.0..35.3.-696.4.0 1584.1319.6644 1.2639...0 1584.4.0435..22.4.2639. 19 7040.0 -22.246 6.19 7040.0 SHAFT DIMENSIONS USED FOR TORSIONAL DEFLECTION CALCULATION OUTSIDE DIAMETER 95.78 0.5 40.LS Pinion .188 107.5 -9.000 0.188 107.6 31.000 0.703 6.19 7040.28 *SHAFT DIAMETER* OUTSIDE INSIDE -76.19 7040.00 7040.19 7040.19 7040.200 76.0 -7.2 23.703 6.5 15.684 11.000 0.8 33.2.44 0.7 13.2 -11.19 7040.19 7040.680 290.703 6.00 FREE BODY FORCE N -95459.7 80.000 0.000 0.000 0.19 7040.19 7040.000 0.000 0.0 6.5 23.8 29.100 38.200 -76.00 -11650.703 3.19 7040.703 6.000 0.6 24.00 7040.000 0.44 0.188 107.188 107.188 107.6 16.000 0.0000 TOOTH STIFFNESS CONSTANT = 2.19 7040.000 0.7 -2.703 6.2 19.1 01-15-2000 ISO Double .7 -6.000 0.188 107.188 107.7 15.19 7040.0 73.00 -11650.000 0.2 8.00 -65645.703 6.188 107.000 0.7580 INSIDE DIAMETER 0.0 -13. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 EXTERNAL FORCE N 0.0 13.000 0.452 10.620 11.188 107.00 -7835.684 0.2 20.00 7835.000 0.00 0.5 10.2 3.2 7.0 17.0 -1.78 0.fma=(fHB1**2+fHB2**2)**0.703 6.19 7040.703 6.0 17.1 24.820 6.000 0.16 11650.200 107.000 0.19 7040.632 30 X10^6 07:10:04 0.19 7040.1 22.8 0.78 0.16 11650.188 107.2 46.19 7040.5 33.478 38.19 7040.00 0.00 0.5 107.188 107.1 27.4 87.44 34381.200 76.3 67.000 0.246 9.4 21.19 7040.19 7040.703 6.100 7.6 18.188 107.19 7040.025 3.000 0.19 7040.19 7040.200 0.2 33.7 -25.000 0.680 106.0 -15.00 7835.000 0.3 30.00 0.188 107.1 32.3 28.00 0.6 24.19 7040.000 0.000 LENGTH ****** DEFLECTION ***** FACE BENDING TORS.4 22.00 0.9 .9 22.19 0.2 -0.8 33.6 33.703 6.7 32.703 6.703 6.188 107.19 7040.3 13.CW .188 107.000 0.2 114.1 9.4 6.000 0.2 11.5 19.7 11.AGMA 927--A01 AMERICAN GEAR MANUFACTURERS ASSOCIATION Printed Image of Program Output – Page 1 CmSolve Version 4.0 30.19 7040.2 -17.7 13.188 107.5 -4.0 0.000 0.703 6.78 0.350 0.0 32.9 53.350 6.5 AGMA STA.620 7.795 0.6 60.188 107.000 0.19 7040.680 106.188 107.000 0.000 0.19 7040.188 107.855 88.4 20.00 0.5 -20.5 -3.4 32.188 107.900 76.703 6.8 33.000 0. TOTAL 0.19 0.1 26.1 93.44 34381.6 24.4 19.025 10.6 33.2 26.00 0.19 7040.19 7040.188 107.000 SHAFT LENGTH 9.900 88.5 -1.703 6.8 100.3 24.19 7040.188 106.00 -7835.19 7040.0 -0.19 7040.855 290.000 0.703 6.0 4.35 0.000 0.00 0.19 7040.00 0. 1 -42.1 -0. LOAD (N) 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 3622.8 7703.0 -33.1 34.9 -19.AMERICAN GEAR MANUFACTURERS ASSOCIATION AGMA 927--A01 Printed Image of Program Output – Page 2 CmSolve Version 4.CW .2 -44.8 7701.62 60.2725 MAX LD= 8303.95 24 24 24 CM= 1.8 -4.22 46.0 -26.4 8159.7 30.8 33.2 8304.0 -41. NO.1 -22.5 22.325 SUM 247301.0 0.6 30.0 0.92 53.0 0.4 27.6 23.533 MICRO-METER PER HELIX 31 .8 34.0 0.3 -6.3 4470.8 -21.0 0.8 25.0 -15.0 0. TOTAL RELATIVE CORR (MU-M) (MU-M) (MU-M) (MU-M) (MU-M) (MU-M) 15.8 33.LS Pinion .5 7366.0 0.4 -20.0 0.5 -12.1 5885.5 -10.3 35.70 13.0 6460.4 34.00 6.0 LOAD DIST FACTOR CM= 0.fma=(fHB1**2+fHB2**2)**0.7 34.9 -2.8 33.4 8.0 7.9 -12.0 0.84 100.179403 CM= 1.3 -18.5 38.6 0.6 -14.0 8269.7 -47.0 0.52 40.4 33.7 37.0 7964.186654 CM= 1.8 8264.03 73.4 18.0 0.7 -1.1 -15.73 80.3 -3.5 6953.5 0.5 -46.14 93.7 7348.0 0.7 0.2 18.5 -17.8 -23.0 -45.0 20.179508 LOAD DISTRIBUTION FACTOR = 1.41 20.7 -37.0 0.11 26.3 31.0 0.55 107.25 113.81 33.4 30.179508 MISC MISALIGNMENT VALUE =-19.4 -18.5 2.3 -0.179508 STA.1 35.9 -25.0 0.0 -5.4 MAX LD= 8354.9 35.5 AGMA STA 7 STA 7 STA 7 LENGTH (MM) 0.6 5223.9 -20.0 7970.7 15.0 0.0 -0.6 35.0 0.0 26.9630 TOT LD= 126723 TOT LD= 126723 TOT LD= 126723 07:10:04 AVE LD= 7040 AVE LD= 7040 AVE LD= 7040 SUM 239200.7 -25.2 37.066 SUM 247066.1 -20.3 38.1 01-15-2000 ISO Double .3 6906.9 -17.2282 MAX LD= 8303.2 -11.0 -22.2 -7.0 13.1 -38.9 28.3 -21.229 ****************** DEFLECTIONS ****************** BENDING TORSIONAL MISC.000000 LOAD DIST FACTOR CM= 1.7 32.0 -9.2.3 -31.2 38.0 0.2 -1.0 8150.707 MICRO-METER PER HELIX CROWN AMOUNT VALUE =37.8 -22.8 -48.7 27.33 67.0 -47.8 -8.44 87. PUBLISHED BY AMERICAN GEAR MANUFACTURERS ASSOCIATION 1500 KING STREET. VIRGINIA 22314 . ALEXANDRIA.