comparative analysis of tooth-root strength using iso and agma standards

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Andrzej Kawalece-mail: [email protected]

Jerzy Wiktor

Rzeszów University of Technology,Department of Mechanical and Aerospace

Engineering,ul. W. Pola 2,

Rzeszów, PL-35-959,Poland

Dariusz CeglarekDepartment of Industrial and Systems

Engineering,The University of Wisconsin-Madison,

Madison, WIe-mail: [email protected]

Comparative Analysisof Tooth-Root Strength Using ISOand AGMA Standards in Spur andHelical Gears With FEM-basedVerificationCurrent trends in engineering globalization require researchers to revisit various normal-ized standards that determine “best practices” in industries. This paper presents com-parative analysis of tooth-root strength evaluation methods used within ISO and AGMAstandards and verifying them with developed models and simulations using the finiteelement method (FEM). The presented analysis is conducted for (1) wide range of spurand helical gears manufactured using racks or gear tools; and for (2) various combina-tions of key geometrical (gear design), manufacturing (racks and gear tools), and per-formance (load location) parameters. FEM of tooth-root strength is performed for eachmodeled gear. FEM results are compared with stresses calculated based on the ISO andAGMA standards. The comparative analysis for various combinations of design, manu-facturing, and performance parameters are illustrated graphically and discussed briefly.The results will allow for a better understanding of existing limitations in the currentstandards applied in engineering practice as well as provide a basis for future improve-ments and/or unifications of gear standards. �DOI: 10.1115/1.2214735�

Keywords: gear design and manufacturing, gear strength, finite element analysis

Introduction

1.1 Motivation. Gears are essential to the global economynd are used in nearly all applications where power transfer isequired, such as automobiles, industrial equipment, airplanes, he-icopters, and marine vessels. Frequency of product model changend the vast amounts of time and cost required to make a change-ver, also called time-based competition, has become a character-stic feature of modern global manufacturing and new productevelopment in automotive, aerospace, and other industries �1�.his forces gear manufacturers to respond with improved gearystems that are designed and manufactured faster than before.imultaneously, current trends in engineering globalization re-uire research to revisit various normalized standards to determineheir common fundamentals and those approaches needed to iden-ify “best practices” in industries. This can lead to various benefitsncluding reduction in redundancies, cost containment related todjustments between manufacturers for missing part interchange-bility, and performance due to incompatibility of different stan-ards. Given the range of differences that exist in engineeringractices today, frequently manufacturers must seek certificates ofheir products in Asia, Europe, and/or/ in the USA. This coincidesith the recently developed “Gear Industry Vision in 2025” stra-

egic goals. The report recommends as one of the five strategicoals to establish common standards with special emphasis on: �i�ne global system of design and testing standards by 2015; andii� standardized quality and verification procedures �2�.

Gear transmissions are widely used in various industries andheir efficiency and reliability are critical in the final product per-ormance evaluation. Gear transmissions affect energy consump-

Contributed by the Power Transmission and Gearing Committee of ASME forublication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 21,004; final manuscript received November 18, 2005. Review conducted by Prof.
avid Dooner.
ournal of Mechanical Design Copyright © 20

tion during usage, vibrations, noise, and warranty costs amongothers factors �3–6�. These factors are critical in modern competi-tive manufacturing, especially in the aviation industry which de-mands exceptional operational requirements concerning high reli-ability and strength, low weight and energy consumption, lowvibrations and noise. Considering their reliability and efficiencyare some of the most important factors, problems of distributionof loads and, consequently, distribution of stresses in the wholegear transmission, particularly in teeth of mating gears, need to bethoroughly analyzed.

Gears have been manufactured for a number of years with ex-tensive ongoing research related to their efficiency, operationalquality, and durability. They are relatively complex and there are anumber of design parameters involved in gear design. The designof gears requires an iterative approach to optimize design param-eters, which govern both the kinematic as well as strength perfor-mance. Due to the complex combinations of these parameters,conventional design office practice tends to become complicatedand time consuming. It involves selection of appropriate informa-tion from a large amount of engineering/standards data availablein engineering catalogues and design handbooks. While theknowledge in gearing design is vast, however, there is an acutepaucity of research on comparative analysis between various stan-dards and engineering practices. Currently, the most popular stan-dards are ISO and AGMA. These standards vary in selected ap-proaches as well as models and methods resulting in differentdesign solutions obtained for the same gear under the same set ofworking conditions. Yet, the field lacks systematic comparativeanalysis of both standards supported and justified by the detailedfinite element method �FEM� analysis.

In the ISO standard the finite element method is treated as themost accurate method of gear strength determination �method A�and can be used for verification purposes of the results obtained
with other methods. This paper attempts to provide a systematic
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omparative analysis between the ISO and AGMA standards, fo-used on tooth-root strength, based on the precise FEM modelingf spur and helical gears.

1.2 Literature Review. When the loads occurring in gearransmission cross certain thresholds, very often a fatigue break-ge of gear tooth takes place due to repeated high stress exceedingllowable stress limits usually due to improper structure of gearransmission, inappropriate tooth contact, stress concentrationround a notch, material defects, and other factors. Tooth break-ge results in destroying gear transmission and breaking links be-ween driving and driven shafts. The fracture usually starts at thetretched tooth fillet. Therefore, stress state at tooth root is a rep-esentative factor to estimate tooth-root strength of gearings foroth involute cylindrical gears with symmetric profiles and fornvolute spur gears with asymmetric profiles �7� in static and dy-amic conditions �8,9�.

Accurate evaluation of the stress state and distribution oftresses at tooth root is a complex task. The precise determinationf stress distribution requires application of experimental methodssing electric resistance wire strain gauges �9,10� or photo-elasticauges �10,11�, digital photoelastic system involving real timemaging �8�, as well as numerical methods �4,10,12–15�, applyingEM-based simulations �16–18�, bipotential equations, and/or

ig. 1 The finite element model of a segment of a gear and twooad cases: prescribed force applied at the tip and prescribedorce applied at the highest point of single tooth contactHPSTC…

ig. 2 The FEM model of a representative gear „z=45, mn2.75 mm, �n=20 deg, b=32 mm…: segment of three-toothade of 3D isoparametric 20-node brick finite elements;

oundary conditions as in Fig. 1; load uniformly distributed
long the line of contact at the HPSTC
142 / Vol. 128, SEPTEMBER 2006

utilizing Airy function �11�, finite prism method �19�, or thetheory of Muskhelishvili applied to circular elastic rings �20�.

Significant development in analysis of strength properties ofgear transmissions follows the achievements in computational de-sign, simulation of meshing, and tooth contact analysis made byLitvin et al. �21,22�, Lewicki et al. �23�, and Handschuh and Bibel�24�. In particular, Litvin et al. �21� developed an analytical ap-proach for simulation and meshing as well as contact of a face-gear drive with a spur involute pinion, followed by numericalstress analysis of the model of five contacting teeth. In the paperLitvin et al. �22� investigated the contact force and its distributionover contact ellipse, tooth deflection, and load share as well asmaximum bending stress along the width of the gear in loadedgear drive. The results of the performed computations were simi-lar to the known experimental ones. Lewicki et al. �23� studied theeffect of moving gear load on crack propagation predictions. Theyperformed two-dimensional �2D� analysis using finite element andboundary element methods. Comparison of computed results toexperimental ones validated crack simulation based on calculatedstress intensity factors and mixed mode crack angle predictiontechniques in a model of tooth loaded at the highest point of singletooth contact. Handschuh and Bibel �24� investigated anaerospace-quality spiral bevel gear set numerically and experi-mentally. The developed nonlinear finite element model allowed

Fig. 3 The FEM model of a representative gear „z=45, mn=2.75 mm, �n=20 deg, b=32 mm… made of 3D 20-node isopara-metric and eight-node brick finite elements; all degrees of free-dom on the internal cylinder of gear disk fixed; load uniformlydistributed along the line of contact at the HPSTC; model of thewhole gear „left…; enlarged view of the loaded tooth „right…

Fig. 4 Distribution of effective stress in the loaded part of the
whole gear shown in Fig. 3
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hem to simulate three-dimensional �3D� multitooth contact. Theesults were then compared to experimental ones, obtained withtrain gauges. Barkah et al. �25� applied finite element models fortatic stress evaluation of non-circular gears, focusing on the toothllet region. Computed results of stress distributions were in fullgreement with existing analytical solutions. Sainsot and Velex20� developed a new bidimensional analytical model based on

uskhelishvili’s theory applied to elastic rings. The model ap-roximates stress distributions at the root circle and allows toalculate gear body-induced tooth deflections more precisely thanhe Weber’s equation, which is based on the semi-infinite platepproximation. Guingand et al. �19� applied the finite prismethod �FPM� for quasi-static analysis of helical gears to build aodel of cylindrical gears with reduced computational complexity

s compared to conventional FEM models. The developed FPModels allow computing load and tooth-root stresses. Velex andaud �9� experimentally investigated quasi-static and dynamic

ooth-root stresses in spur and helical gears and compared themith numerical predictions obtained by using FEM. They devel-ped a specific FE model which accounts for non-linear time-arying mesh stiffness, tooth geometrical errors, and flexural-

Fig. 5 Distribution of the effective stress i

Fig. 6 Comparison of effective stressfillet of the same gear computed with t„i… the whole spur gear built of 3D 20-finite elements; „ii… segment of threebrick finite elements; and „iii… segmenstress finite elements; distance � des
tooth is shown in Fig. 7 right
ournal of Mechanical Design

torsional-axial couplings. However, they did not consider the gearfabrication process, thus their simulations model does not allow toanalyze the impact of the fillet shape on tooth-root stresses.

The development in analysis of gear strength shown in papers�21–25� validates the use of correctly defined finite element mod-els for analysis of not only spur or helical gears, but also othertypes of gears, like hypoid or spiral-bevel gears, etc. Experimentalmethods, though necessary for validation of theoretical and nu-merical analyses, are especially complicated, expensive, and labo-rious. Therefore, they are applied mainly in special cases. In prac-tice, simplified formulas are usually used in gear transmissiondesign. They enable estimation of stresses at tooth root with ac-curacy acceptable for engineering design.

In every case, strength properties of gear transmissions arestrongly influenced by gear geometry, applied manufacturing pro-cesses, and dimensional accuracy of manufactured gears�4,12,13,18�. For example, investigations described in Liu andPines’ paper �26� illustrate that some fundamental gear designparameters such as diametral pitch �proportional to the normalmodule, used in Europe as the characteristic parameter in geardesign�, pressure angle, and number of teeth, may have a signifi-

e loaded tooth of the gear shown in Fig. 3

stributions along tooth line and alonguse of different finite element models:e isoparametric and eight-node brickth built of 3D 20-node isoparametricf three teeth built of eight-node planeing location of transversal section of

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ant influence on damage detection sensitivity. In the aforemen-ioned investigations, the three main types of gear damage wereonsidered: pitting and wear of tooth flanks, concentration oftress, and cracks at tooth root.

The final geometry of every gear depends on the assumed geartructural parameters and the applied manufacturing processes,pplied tools, tool machines, etc. �3,12,13,18�. Therefore, the es-ablished standards consider models of both gear structures and

anufacturing techniques, and take into account certain structuralactors, mostly geometric and operational, as well as informationoncerning applied manufacturing processes, applied tools, andaterials. In particular, suitable international ISO �27� standards

eveloped over the years in Europe �or fully compatible Germantandard DIN �28��, and American standard AGMA �29–37� de-cribe procedures for approximate calculations of stresses at toothoot of the involute gearings. In all these standards evaluation ofooth-root strength is performed based on maximum stress at toothoot on the stretched side of the tooth. Moreover, in all of thesetandards it is assumed that the maximum stress occurs in theritical section of the tooth. It is clear that proper evaluation ofaximum bending stress at tooth root is one of the most critical

arameters determining whether or not a particular gear transmis-ion will function properly. Tooth breakage due to operating stresshat significantly exceeds the maximum allowable stress in gearransmission can be extremely dangerous in automotive, aero-pace, or space industries �3,4,38,39�. Therefore, an accurate

Fig. 7 Absolute values of obtained differecomputed according to Eq. „1… for the 2D pmodel of the whole gear built of 3D 20-noelements in sections associated with �=2lines where the effective stresses are com

Fig. 8 Comparative analysis of the efor three magnitudes of rim thicknesrelative differences in the effective staccording to Eq. „2… for the model of
20-node isoparametric, for load applied a

estimation of critical strength parameters is of great importance insuch cases.

The models considered in ISO and AGMA standards are similarfrom both theoretical and computational points of view, thoughthey are not identical. They have been developed using state of theart industrial practice as well as theoretical and experimental re-search in this field. However, there are a number of discrepancieswithin the current standards. These discrepancies can be classifiedas: �1� each standard defines different geometrical parameters ascritical for gear performance; �2� the calculations undertaken toestimate the thresholds for these critical parameters are based ondifferent assumptions and on different mathematical relations fol-lowing them; and �3� results of computations differ for each stan-dard under the same conditions.

Currently, there is a need to unify both standards �35�. Initialstudies, which intended to compare both standards, were done byHösel �36,37�. However, Hösel only considered the older versionsof the standards, thus limiting his analysis to simple numericalcomparison of both standards without any other verification ofcomputed results.

In this paper corresponding models of gears, computationalmethods used for their analysis, and results obtained according tothe ISO and AGMA standards are briefly described and comparedwith each other. In the next step we verify these with the use ofthe developed FEM models and simulations. This approach allows

s in the effective stresses along fillet ��effe stress model with three teeth and for theisoparametric and eight-node brick finite„left…; explanation of the sections and the

ed „right…

ctive stress distribution at tooth rootf gear „left…; absolute values of the

ses along fillet ��eff„*1 , *i… computed

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or better understanding of existing limitations in the current stan-ards as applied in engineering practice and will provide a basisor future improvements of the standards.

1.3 Approach. In both the ISO and AGMA standards, stressest tooth root are calculated for load applied at the highest point ofingle tooth contact, i.e., at the HPSTC �in the case of spur gear-ngs� or at the tooth tip �in the case of helical gearings�. In thease of helical gears, calculations are performed for the virtualear with straight teeth and virtual number of teeth denoted zn.he analysis is conducted in three steps to calculate: �i� transmit-

ed load; �ii� nominal tooth-root stress �FO; and, �iii� local stress athe tooth root, i.e., to the tooth-root stress �F.

Transmitted load in the plane of action, which is normal to theooth flank, is converted to the nominal tangential load at theeference diameter Ft �according to the ISO standard� or to theransmitted tangential load at the operating pitch diameter Fwtaccording to the AGMA standard�. For a given load, the nominal

Table 1

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traight tooth gearings:ccording to method B �accurate computational method�aximum stress at tooth root is calculated for the load

pplied at the HPSTC �for gear transmission with theransverse contact ratio 1��n�2� or at theighest point of double tooth contact �for 2��n�3�.ccording to method C �simplified computationalethod� maximum stress at the tooth root is calculated

or the load applied at the tooth tip. Then, with the usef the coefficient Y� it is converted to the stressenerated with the load applied at the HPSTC �27�.elical gearings:aximum stress at tooth root is calculated for the virtual

earings with straight tooth �with virtual number of teeth

n� and for load applied at the tip.

ransmitted load in the plane of action normal to theooth flank is converted to the tangential load at theperating pitch diameter �27�:

Ft=2000T1,2

d1,2

here: Ft - tangential load at the operating pitchiameter �N�; T1,2 - pinion torque �1�, wheel torque �2�Nm�; d1,2 - operating pitch diameter of pinion �1�,heel �2� �mm�.ext, unit tooth load is calculated:

Ft

bmn

here: b - facewidth �mm�; mn - normal module �mm�.

Fig. 9 Determination of the critical sof the critical section sFn, hFe, and �
standards „for comparison see Table 7 in

tooth-root stress �FO is calculated with consideration to the toothfillet shape and stress concentration caused by the notch. Then, thenominal tooth-root stress is converted using load factors to thelocal stress at the tooth root, i.e., to the tooth-root stress �F. Theload factors take into account increase of stresses caused by mis-alignments in manufacturing of gear transmission, external andinternal dynamic loads acting on gear transmission, as well asnon-uniform distribution of load.

The FEM simulation models of tooth-root strength use amethod for computations of tooth profiles as presented in Ref.�40�. This method allows for accurate computation of real toothprofile, including fillet for both tools rack or gear tool. Addition-ally, the results are automatically interpreted by the software de-scribed in Ref. �40�. The development of the precise FEM modelof gear tooth requires accurate determination of tooth profile withconsideration of the selected machining process of gear fabrica-tion. Such methods are described in Refs. �40,41�.

oth load

AGMA

Straight tooth gearings:Maximum stress at tooth root is calculated fortransmitted load applied at the HPSTC �the standardallows calculation of stresses for the load applied at thetooth tip� �29�.

Helical gearings:Maximum stress at tooth root is calculated for the virtualgearings with straight tooth �with virtual number of teethzn�and for the load applied at the tooth tip �29�.Transmitted load in the plane of action normal to thetooth flank is converted to the tangential load at thereference diameter �33�

Fwt=2000 T1,2

dw1,2

where: Ftw - tangential load at the reference diameter�N�; T1,2 - pinion torque �1�, wheel torque �2� �Nm�; d1,2- reference diameter of pinion �1�, wheel �2� �mm�.

Next, unit tooth load is calculated �33�:Fwt

bmn

where: b - facewidth �mm�; mn - normal module �mm�.

on location, angle �F and parametersccording to the ISO and the AGMA

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It should be noted that the ISO and AGMA standards do notrovide procedures for calculation of the whole tooth profile. Thetandards describe ways to calculate the main parameters of theasic rack profile. In addition, the formulas given in each of thetandards do not cover all range of gear manufacturing tools usedn industrial practice.

For the sake of comparative analysis the following conditionsnd assumptions are made:

�1� Loads applied in the FEM, ISO, and AGMA standards arethe same.

�2� There are no misalignments in gear transmission, and loadsare static and uniformly distributed.

�3� Location of the critical tooth section and maximum tooth-root stresses were calculated following the procedures ofthe ISO �27� and AGMA �29–35� standards. Both standardsprovide formulas for calculation of tooth-root stress at thecritical section while the conducted FEM simulations allowobtaining the whole distributions of stresses in the modeledgears. From those distributions of stresses the maximumtooth-root stresses with their locations were found by scan-ning the FEM results.

�4� The notation follows both ISO �27� and AGMA standards�29–35�. Regarding the analysis and computations focusedon the ISO procedures, the whole notation is based on the

Table 2 Methods of determ

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ased on calculations and analysis of experimentalesults it is assumed that the critical section isetermined by the points of tangency of the fillet withhe straight lines inclined at 30 deg to the tooth centerine �Fig. 9, left�

herefore, the angle �F between tooth center line andangent to fillet at the critical point F is constant andndependent from shape of the fillet and from location ofpplied load:

�F=30 deg

Fig. 10 Influence of the number of gcenterline and tangent to fillet at the cdard, AGMA standard, and FEM: load aHPSTC „right…; main parameters of th�1=0 deg, main parameters of the rac
=0.18mn

ISO standard �27�. Concerning analysis and computationsfocused on the ISO procedures in cases where the presentednotation is the same for both standards, only the ISO nota-tion is used in order to emphasize similarities between thestandards. Other parameters used in formulas are denotedaccording to the ANSI/AGMA 2101-C95 �33� and AGMA908-B89 �29� standards. The nomenclature list for bothstandards is presented in Table 7 in the Appendix.

The details of the related approaches, computational models,methods of calculations, and results are presented in the tables andfigures included in the following sections. The outline of the paperis as follows. Section 2 presents the FEM-based approach used forcalculation and analysis of gear strength at tooth root. Section 3describes the loads, critical section definition, and parameters, co-efficients, and load factors used in computations as well as recal-culations of nominal tooth-root stresses to local tooth root stress.In Sec. 4, tooth-root stress analysis following the ISO, AGMA,and FEM procedures is presented. Finally, the results are summa-rized and conclusions are drawn in Sec. 5.

2 FEM ModelingSpur and helical gear models were analyzed based on formulas

given in the ISO and AGMA standards as well as based on the

ation of the critical section

AGMA

Location of the critical section is determined on the basisof the modified Lewis method �Fig. 9, right�. The toothis considered as a beam fixed at its root and loaded withthe transmitted load. The critical section is determinedby the tangency points of a parabola inscribed into thetooth profile. This parabola represents profile of thebeam with uniform strength along its axis.The angle �F between tooth center line and tangent tofillet at the critical point F is not constant - its valuedepends on profile of the fillet and on location of thepoint where load is applied. In the examples consideredin this paper the load is applied at the tip and the angle�F which varies �depending on parameters of the gearand machining tool� within the range:

�F=13–16 degFor load applied at the HPSTC the angle �F varieswithin the range:

�F=24–30 deg

teeth on the angle �F between toothcal point F according to the ISO stan-lied at the tip „left…, load applied at theear: mn=2.75, �n=20 deg, s1=4.265,

�0=20 deg, s0=4.320, hfP=1.22mn, �fP

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onducted FEM modeling �16–18�. FEM computations of stressistribution, as presented in this paper, require precise modelingf tooth, especially accurate representation of its fillet, which con-titutes a typical geometric notch. Therefore, a specialized meshenerating program integrated with the ADINA �42� finite elementystem was developed. The developed program allows for para-etric and automatic generation of well shaped finite elementeshes for two-dimensional �2D� and three-dimensional �3D�

roblems with consideration to proper node numbering and re-uired tooth fillet profile �40�. A precise representation of thehole tooth profiles in their various sections was based on cubicermite splines and on the least squares approximations �43,44�.he three-dimensional geometric models of tooth profiles were

ransformed to the ADINA system �42� for further finite elementnalyses.

Considering the geometric symmetry of the gear only certainear segments were defined and used for FEM computations.hey represented plane stress models made of the eight-node iso-arametric plane stress finite elements for spur gears and 3D mod-ls made of isoparametric 20-node and eight-node brick finite el-ments for spur and helical gears. All nodes at the internal circlef gear rim in models considered in this work had all degrees ofreedom �DOE� fixed. In addition, the nodes at both radial bound-ry sections had all DOF fixed, except the ones which are relatedo radial movements. All nodes at both radial boundary conditions

Fig. 11 Influence of the number of tbetween tooth centerline and tangening to the ISO standard, AGMA stan„left…, load applied at the HPSTC „ri=2.75, z1=45, �n=20 deg, s1=4.265,tool: �0=20 deg, s0=4.320, hfP=1.22m

ig. 12 Influence of the normal module mn on the angle �Fetween tooth centerline and tangent to fillet at the criticaloint F according to the ISO standard, AGMA standard for loadpplied at the tip „Tip…, and at the HPSTC; main parameters ofhe gear: z1=45, �n=20 deg, �1=0 deg; main parameters of the
ack: �0=20 deg, hfP=1.25mn, �fP=0.5

were defined in their local coordinate systems �Fig. 1�. Accordingto the ISO and AGMA standards the models were loaded at toothtip or at the highest point of single tooth contact with correspond-ing prescribed load �Fig. 1�. Considering load distribution alongthe path of contact associated with �i� contact of a single pair ofmating teeth, or �ii� simultaneous contact of two pairs of matingteeth, the load applied at tooth tip was half of the load applied atthe HPSTC.

The selected plane stress model �Fig. 1� is the most suitable forstrength analysis of spur gears presented in this paper. The selec-tion of the plane stress model for strength analysis of spur gears isbased on fulfilment of necessary conditions for replacement of 3Dstrength analysis with plane stress strength analysis of structures�45� like spur gears �much longer face width than whole depth,same shape in each transversal section, uniform load distributionalong tooth line� as well as on conducting a large number of two-and three-dimensional finite element gears simulations. The meshis chosen to satisfy: �i� accuracy requirements for modeling oftooth shape; and �ii� to properly represent necessary boundaryconditions, which are located far from the fillets of loaded toothand region of load application. Following recommendations ofKawalec and Wiktor �44�, the optimum mesh density was gener-ated and selected using the criterion of change in the maximumeffective stress, i.e., the mesh density in the volume of tooth rootis increased until the changes in the maximum effective stress attooth root �eff,max are less than 0.4%. In order to compare resultsof plane stress models with three-dimensional ones several 3DFEM models were developed and analyzed: �1� models built ofeight-node 2D isoparametric plane stress elements; �2� modelsbuilt of 20-node 3D isoparametric brick elements; and �3� modelsbuilt of 20-node isoparametric and eight-node brick 3D finite el-ements �FEs�. Two of the models are shown in Figs. 2 and 3.Figures 2 and 3 also show the key parameters of gears used in thispaper.

The simulations conducted in the paper for 3D models use load,which is uniformly distributed along tooth line and corresponds tothe point load applied in plane stress models. The application ofload uniformly distributed along tooth line as well as point load isjustified, according to the Saint-Venant’s principle �45�. That prin-ciple states that self-equilibrated systems of tractions result instresses that decay rapidly with distance from the region wheretractions are applied. In other words, in the cases of simulations ofbending stresses at tooth root which are considered in this paper,the tooth root is located far from the area of load application, andthus, the overall load forces can be replaced with the equivalent

h z0 of the gear tool on the angle �Ffillet at the critical point F accord-

d, and FEM: load applied at the tip…; main parameters of the gear: mn0 deg, main parameters of the gear�fP=0.18mn

eett todarght�1=

static forces having the same total moments and the same resultant

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orces. The results of FEM stress simulations are shown in Figs.–6 and 8. From analysis of tooth stress distribution, the follow-

ng conclusions can be drawn �see also Figs. 4 and 5�: largetresses in the area of load application and at the fillet are sepa-ated from each other by the area of very low stresses. Therefore,onsidering obtained distributions of stresses and the Saint-enant’s principle applied to bending of tooth caused by load

ransmitted between teeth in mesh, the propagation of errors fromoint or distributed loadings of tooth to the area/volume at thellet can be neglected.Figure 6 contains comparison of distributions of von Mises ef-

ective stress along fillet of the same gear computed with the usef different finite element models: �1� the whole gear built of 3D0-node isoparametric and eight-node brick finite elements; �2�egment of three teeth built of 3D 20-node isoparametric bricknite elements; and �3� segment of three teeth built of eight-nodelane stress finite elements.

Relative differences in distributions of the effective stresseslong fillet ��eff for the 2D plane stress model with three teetheff,�2D� and for the model of the whole gear built of 3D 20-node

soparametric and eight-node brick finite elements �eff,�3D� were

Table 3 Parameters

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he ISO standard describes how to calculate thearameters of the critical section for gearings generatedith rack. In the case of gearings generated with gear

ool �e.g., with gear-shaper cutter� the real tool is in the.n. procedure of calculations replaced with a virtualne—a rack with the same reference tooth profile. Itignificantly simplifies calculations, but can lead torroneous results. The tooth profile generated with theear tool differs from the one generated with the rack.herefore, parameters calculated for the virtual rack canignificantly differ from the real ones. This may lead torrors in estimation of tooth-root strength, due to wrongesults of stress calculations.ormulas determining parameters of the critical sectionre derived from the geometric relations given in Fig. 13up� which shows both machined gear and generatingack in a position in which the rack generates point Felonging to the fillet �the point which determinesocation of the critical section�.ooth-root chord at the critical section sFn:

sFn

mn=zn sin��

3−��+�3� G

cos �−

fP

mn�

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F

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mn+

2G2

cos ��zn cos2 �−2G�

ending moment arm application at the tip hFe:

hFe

mn=

1

2 ��cos e−sin etg�Fen�den

mn�

−1

2 �zn cos ��

3−��+

G

cos �−

fP

mn�

nalysis of the geometric relations shown in Figs. 9 and3 leads to the following equation from which anuxiliary angle � can be calculated in an iterativerocedure:

omputed according to the formula:


��eff =�eff,�2D� − �eff,�3D�

�eff,�2D�100 �1�

Absolute values of the computed differences ��eff are shown inFig. 7. From the performed analysis it follows that the maximumdifference in tooth-root stresses between the 2D model �Fig. 1�and the 3D model of the whole gear �Fig. 3� are less than 5%.

Strength computations of gears require analysis of the rimthickness �46,47�. Therefore, in order to investigate the influenceof rim thickness sR on the maximum tooth-root strength, various3D models of three teeth segments of gear with different magni-tudes of rim thickness sR made of the 20-node isoparametric finiteelements were computed. Relative differences in distributions ofthe effective stresses along fillet ��eff�

*1 , *i� for the 3D modelsof three-tooth segments of gear with various magnitudes of rimthickness i*sR were computed according to the formula:

��eff�*1, *i� =

��eff,max�*1� − �eff,max�

*i��eff,max�

*1�100 �2�

where i� �0.5,0.6,0.75,2.0,4.0�, and �eff,max�*1�, �eff,max�*i� de-

the critical section

AGMA

The AGMA standard contains procedure of calculation ofthe parameters of the critical section for gearingsgenerated with gear tool. In the case of gearingsgenerated with rake the calculations are performed thesame way, but the rack is represented by the gear toolwitha very large number of teeth, i.e. z0=10,000. Theparameter of the critical section calculated in such a wayare comparable with the real ones. The differences canbe omitted as far tooth-root strength of gears generatedwith racks is concerned.

Formulas determining parameters of the critical sectionare derived from the geometric relations given in Fig. 13�bottom� which shows both machined gear andgenerating gear tool in a position in which the toolgenerates point F belonging to the fillet �the point whichdetermines location of the critical section�.Coordinates of the critical point F �29�:

nF=rwn1 sin �n1+KF cos �F

�nF=rwn1 cos �n1+KF sin �F

Tooth-root chord at the critical section sFn:sFn=2nF

Radius of fillet at the section F:

F= fP+KS

2

rwn1rwn0 cos �

rwn1+rwn0−KS

Bending moment arm application at the tip hFe

hFe=rbn1

cos �Fen−�nF

where:KS=rwn0 cos �−rRn0 cos ��Rn0−��

KF=KS− fP

of

note the maximum tooth-root stress computed in the model with


bcsFdr1

smtdreemst=

Fsc

J

asic rim thickness sR=2.0*mn, the maximum tooth-root stressomputed in the model with rim thickness of magnitude i*sR, re-pectively. The basic finite element model used for all furtherEM strength computations, i.e., the model with rim thicknessefined as i*sR=1.0*sR=2.0*mn, is shown in Fig. 2, and the cor-esponding 2D plane stress finite element model is shown in Fig..

Absolute values of the computed differences ��eff�*1 , *i� are

hown in Fig. 8. From the simulations results it follows that theaximum difference in the maximum tooth-root stresses between

he investigated models for i�0.75, which is equivalent with con-ition sR�1.5*mn, is less than 2% �Fig. 8�. That differenceeaches appox. 20% at the end of fillet for i=4.0*sR. The differ-nce of 20% occurs in that case, however, in the areas where theffective stresses are more than three times smaller than the maxi-um effective tooth-root stress. As far as the maximum tooth-root

tresses at the critical section are concerned, the absolute values ofhe computed differences ��eff�

*1 , *i� for i=0.6*sR and i*

ig. 13 Determination of parameters of the critical section:Fn, hFe, and �F, according to the ISO standard „top… and ac-ording to the AGMA standard „bottom…

0.5 sR reach approx. 8% and 16%, respectively �Fig. 8�. From


the conducted analysis it follows, that the rim effect takes placefor i�0.75. Therefore, influence of rim thickness in developedFEM models used for further computations, i.e., models with i=1.0, on the maximum tooth-root stress is negligible.

Distributions and magnitudes of effective stress at tooth root in2D and 3D isoparametric models are almost the same �Figs. 6 and7�. In all models the critical sections were located very close toeach other �abscissas of local maxima in Fig. 6�. The models builtof 3D 20-node isoparametric and eight-node brick finite elementsare a more realistic representation of real gears than the 2D finiteelement models. However, the effective stresses at tooth root com-puted for the 2D plane stress model differ not more than 5% fromthe effective tooth-root stresses computed for the 3D models ofthe whole gear. Moreover, increasing two times and four times thebasic thickness of gear rim changes the maximum tooth-rootstresses less than 2%.

In all 2D and 3D finite element models generally isoparametricquadrilateral or brick finite elements with very regular rule-basedmesh were used. They give more flexibility in precise modeling ofthe gear shape, at least in the case of rule-based meshes, which isvery important in precise modeling of tooth flank and fillet. Theyalso give more flexibility in modeling of strength properties, be-cause they use higher order shape functions than simple tetrahe-dral elements �16,17�. The triangle and tetrahedral elements wereomitted, as their linear shape functions are too primitive to cor-rectly model curvilinear geometry and stresses as compared to thequadrilateral and brick elements, respectively.

From the comparison of results obtained for the various finiteelement models of gears it follows that application of the precise2D plane stress finite element models for the analysis of tooth rootstrength, especially its maximum magnitudes and location of criti-cal section in spur gears is justified. Analysis of tooth root stressof helical gears requires 3D models. However, as it follows fromresults obtained for the 3D finite element models of the wholegear and three-tooth segment �Figs. 6 and 7�, simulations of thethree-tooth segment are sufficiently accurate for computations ofthe local tooth-root stresses at the root of loaded tooth. The de-tailed results of the 3D finite element simulations of helical gearsare given in Sec. 4.

Generally, it is important to mention that both analyzed stan-dards, i.e., ISO and AGMA, clearly indicate the finite elementmethod as one of the most precise methods for computations oftooth strength. Therefore, considering the approach given in theISO and AGMA standards, and considering the obtained simula-tions results, it can be concluded that: �1� the developed 2D finiteelement plane stress models represent tooth-root strength of spurgears with good accuracy, and, thus they will be used in thispaper; and �2� in the case of helical gears, the 3D models will beused.

3 Determination of Geometric and Strength Param-eters and Factors

3.1 Loads. In both standards tooth-root stresses are calculatedfor the unit nominal load. The load in the plane of action normalto the tooth flank is converted to the nominal tangential load at thereference diameter Ft �ISO standard� or to the transmitted tangen-tial load at the operating pitch diameter Fwt �AGMA standard�—Table 1.

3.2 Critical Section. In both standards it is assumed that
maximum stress at tooth root occurs at the critical section, which
SEPTEMBER 2006, Vol. 128 / 1149

i�

sml

stmpstsats

irina�t=an

fztn

1

s determined in two different ways specific for each standardFig. 9 and Table 2�.

The magnitudes of the angle �F based on the ISO and AGMAtandards, and calculated for different parameter values of theanufactured gears and generating tools as well as for different

oad cases, are shown in Figs. 10–12.The angles �F calculated on the basis of the ISO and AGMA

tandards are compared with the angles representing location ofhe critical section determined with the use of the finite element

ethod �16,17� in ADINA �42�. For this purpose tooth forms wererecisely modeled with fine mesh consisting of eight-node planetress isoparametric finite elements. Suitable automatic analysis ofhe results, focused on searching for location of the maximumtresses at tooth root, allowed determination of magnitudes of thengle �F �i.e., slope of the line tangent to the fillet at the point ofhe computed maximum stress� for a wide range of consideredpur gears models.

Figure 10 presents results of computations performed for gear-ngs with different number of teeth z1 generated with the sameack. According to the AGMA standard the angle �F grows withncreasing number of gear teeth z1 �Fig. 10�. In the case when theumber of teeth of generated gear z1 varies in the range 17, 101nd load is applied at the HPSTC the angle �F increases fromF=24 deg to �F=29.2 deg. When the same load is applied at theooth tip the angle �F increases from �F=13.2 deg to �F16.2 deg. Also in these cases, according to the ISO standard, thengle �F is constant and equals 30 deg independently from theumber of gear teeth.

Figure 11 contains results of computations of the angle �F madeor gears generated with gear tools with different number of teeth0. Similarly to the case of racks, according to the ISO standardhe angle �F is constant and equals 30 deg independently from the

Fig. 14 Influence of the number of gsection: sFn, hFe, �F according to the ISand at the HPSTC; main parameters othe same as in Fig. 10

umber of teeth of the tool and from the location of applied load.


According to the AGMA standard, this angle decreases slightlywith the growing number z0 and in the case of load applied at theHPSTC it equals approximately 29 deg which is very close to thevalue assumed in the ISO standard. On the other hand, in the caseof load applied at the tip, the angle �F is almost twice less andequals approximately 16 deg. Results obtained with the use of thefinite element method significantly differ from the ones calculatedwith consideration of both ISO and AGMA standards �see Table3�. In the cases of load applied at the HPSTC the angles �F variedin the range 44.25 deg,47.25 deg and in the cases of load ap-plied at the tip the angles �F varied in the range36.0 deg,38.8 deg. Different magnitudes of the angle �F be-tween tooth center line and tangent to fillet at the critical point Fcalculated in accordance with the ISO and AGMA standards fordifferent values of the normal module mn are shown in Fig. 12.

3.3 Parameters of the Critical Section. The following pa-rameters of the critical section �Figs. 9 and 13� are necessary inorder to calculate stress at tooth root: tooth-root chord at the criti-cal section sFn; tooth-root fillet radius of the critical section F andbending moment arm application at the tooth tip hFe.

The differences between the ISO and AGMA standards in de-termination of location of the critical section causes that for thesame parameters of gearings and generating tools the parametersof the critical section can significantly differ from each other. Thisfurther results in differences in calculated stresses. The selectedparameters are calculated in accordance with the ISO and theAGMA standards and are compared in Figs. 14–17.

Figure 14 shows the parameters of the critical section for gear-ings with various number of teeth z1 generated with rack and Fig.15 shows the parameters for gearings generated with gear tools

teeth z1 on parameters of the criticalAGMA for load applied at the tip „Tip…,e rack „generating tool… and gear are

earO,f th

with various number of teeth z0. In all examples the parameters


wc

Sfr�

atttf

Fra„

e0

J

ere computed for load applied at the highest point of single toothontact �HPSTC� and for load applied at the tip �Tip�.

In the case of gears generated with rack and loaded at the HP-TC, results of calculations of the parameters sFn, hFe, F per-ormed in accordance with the ISO standard are close to the cor-esponding results obtained on the basis of the AGMA standardFigs. 14 and 15�.

Slightly bigger differences occur between the results for loadpplied at the tip. In the case of gears generated with gear tools,he differences are more significant, especially, for load applied athe tip. It follows from the fact that according to the ISO standardhe calculations are made not for real gear tool used in gear manu-acturing but for a virtual rack with the same reference tooth form.

Fig. 15 Influence of the number ofof the critical section: sFn, hFe, �F acplied at the tip „Tip… and at the HPS„generating tool… and gear are the sa

ig. 16 Influence of the range of sharpening of gear tool „pa-ameter u… on parameters of the critical section: sFn, hFe, �Fccording to the AGMA standard for load applied at the HPSTCleft…, the range of sharpening of gear tool „right…; main param-ters of the gear tool „generating tool…, and the gear in section
-0 are the same as in Fig. 11

Therefore, calculations made in accordance with the ISO standarddo not consider either influence of the number of teeth of the toolz0 or a sharpening range of the tool �in the case of gear tool eachsharpening of the tool induces changes in parameters of the tooland changes in some parameters of the generated gear—Fig. 16�.

Calculations made for gears with different magnitudes of thenormal module mn show that all parameters of the critical sectionare nearly proportional to the normal module �Fig. 17�.

3.4 Coefficients Used in Computational Procedures. In or-der to calculate nominal tooth-root stresses in accordance withboth the ISO and AGMA standards, it is necessary to calculatecertain coefficients, which take into account profile of the fillet�parameters of the critical section�, complex stress state in toothroot and concentration of stresses caused by a geometric notch�fillet at tooth root�. Methods of calculation of these coefficientsare given in Table 4.

In both standards, calculations for the helical gearings are per-formed the same way as in the case of spur straight tooth gearingswith the virtual number of teeth zn. An influence of the helicaltooth trace on tooth-root strength is additionally taken into ac-count by related coefficients. In the ISO standard it is a helixangle factor Y� �Table 4�. In the AGMA standard the formula forthe bending strength geometry factor YJ considers the load sharingratio mN �in the case of spur gears mN=1.0� and the formula forthe tooth form factor Y considers � and �w, i.e., helix angle at thereference cylinder and helix angle at the operating pitch cylinder,respectively �Table 4�.

3.5 Load Factors. The stresses at tooth root result from realloads transmitted in gear transmission and not from nominal ones.Therefore, both ISO and AGMA standards consider the corre-sponding load factors �Table 5�. These factors take into account

th z0 of the gear tool on parametersding to the ISO, AGMA for load ap-; main parameters of the gear toolas in Fig. 11

teecorTCme

increase of stresses due to errors arising in gear transmission

SEPTEMBER 2006, Vol. 128 / 1151

mndd

btraT�

gfttti

4I

wmif1cwo

w

1

anufacturing, due to internal and external dynamic loads andon-uniform distribution of loads in gear transmission. The stan-ards contain suitable formulas or tables based on which one canetermine values of the load factors.

3.6 Stress at Tooth Root. Stress at tooth root is considered inoth ISO and AGMA standards as a representative parameter inhe evaluation of tooth-root strength of gear. First-nominal tooth-oot stress �FO is calculated for the nominal load and for thessumed form of the fillet and parameters of the critical section.hen, the stress is recalculated with the use of the load factors

Table 5� to get local tooth-root stress �F �Table 6�.The objective of this work is to compare influence of tooth

eometry �resulting from applied method of gear generation androm parameters of the tools� on the tooth-root strength, accordingo both the ISO and AGMA standards. Therefore, in all computa-ions the load factors �Table 5� were not taken into account, ashey do not introduce any new information as far as the analysis ofnfluence of tooth geometry on tooth-root strength is considered.

Analysis of Tooth-Root Stresses According to theSO and AGMA Standards and FEM

Stresses calculated for various models of gears in accordanceith the ISO and AGMA standards as well as based on the FEModels developed in the ADINA �42� environment, are compared

n Figs. 18–25. Results of computations made for gears with dif-erent number of teeth z1 generated with racks are shown in Fig.8. Similarly, as in the case of parameters calculations of theritical section given in Sec. 3.3, computations of tooth-root stressere made for two load cases: for load applied at the highest pointf single tooth contact and for load applied at the tip, respectively.

Figure 19 presents stresses calculated for gears manufactured

Fig. 17 Influence of the normal modution: sFn, hFe, �F according to the ISOthe tip „Tip…, and at the HPSTC; mainand gear are the same as in Fig. 12

ith gear tools for various number of teeth z0, Figure 20 shows


stresses calculated for gear manufactured with one exemplary geartool, but with different magnitudes of parameter u related to thegeometric effect of tool sharpening. In the ISO standard any realgear tool is replaced with a related virtual rack. Therefore, tooth-root stresses calculated in accordance with this standard are con-stant, i.e., they do not depend on the parameters z0 and u.

Influence of some other important geometric parameters ofgears, like normal pressure angle �n, addendum modification co-efficient x1, helix angle at the reference cylinder �1, and normalmodule mn on magnitude of tooth-root stress are given in Figs.21–25, respectively.

The models of helical gears were built of 3D 20-node isopara-metric and eight-node brick finite elements. They representedthree teeth segments of gears with load applied to the central toothin the most unfavorable case �assuming no misalignments in gearaxes� of load distributed along the diagonal line of contact passingthrough the tip corner of tooth flank. The boundary conditionsreflected the ones shown in Fig. 1, and were applied on corre-sponding boundary surfaces of the 3D models of gear segments.Distributions of the effective stresses along tooth line passingthrough critical sections in various FEM models of helical gearswith different helix angles � and along fillet in various transversalsections of helical gear with helix angle �=17 deg are shown inFig. 24.

In all performed computations, tooth-root stresses obtained ac-cording to the ISO standard were greater than the tooth-root stresscalculated according to the AGMA standard. The difference wasbigger in the case of load applied at the tip than in the case of loadapplied at the HPSTC. Stresses computed with the use of the finiteelement method were in between, i.e., were greater than stressesaccording to the AGMA and smaller than stresses according to theISO standard.

mn on parameters of the critical sec-AGMA standards for load applied at

ameters of the rack „generating tool…

leandpar

Comparison of locations of the critical section and the angle


TTtp

wp

HTbt

STan

w

Kln

KcKnKdic

N

L

J

Table 4 Computational coefficients

ISO AGMA

ooth form factor YF: Tooth form factor Y:his factor considers influence of shape of tooth at the

ooth-root stress. This shape is defined in computationalrocedure by the parameters of the critical section

This factor considers influence of shape of tooth at thetooth-root stress. This shape is defined in computationalprocedure by the parameters of the critical section

YF=6�hFe /mn�cos �Fen

�sFn /mn�2 cos �n

Y =cos �w cos �

cos �Fen

cos �wn� 6hFe

sFn2 Ch

−tg�Fen

sFn�

here: �Fen-load angle; �n-normal pressure angle at theitch cylinder.

where: �Fen-load angle; �wn-operating normal pressureangle; �-helix angle at the reference cylinder;�w-operating helix angle; Ch-helix factor.

elix angle factor Y�: Helix factor Ch:his factor considers difference in stresses at tooth rootetween helical gear and virtual spur gear with straighteeth used for calculations

The helix factor Ch is calculated in the followingway:-for spur gears Ch=1.0

-for conventional helical gears

Y�=1−��

�

120Ch=

1

1− � �

100�1−�

100��0.5

where: �=arctg�tg� sin �n�tress correction factor YS: Stress correction factor Kf:his factor considers complex stress state at tooth rootnd stress concentration caused by the fillet �geometricotch�

This factor considers complex stress state at tooth rootand stress concentration caused by the fillet �geometricnotch�

YS= �1.2+0.13L�qsa

Kf =H+ �sFn

F�L� sFn

hFe�M

here: L=sFn /hFe; where: H=0.331−0.436�n;qs=sFn / �2F�; L=0.324−0.492�n; M =0.261+0.54�n

a= �1.21+2.3/L�−1 Bending strength geometry factor YJ:This factor is calculated from the equation

YJ=YC�

KfmN

where: C�-helical overlap factor �for spur andconventional helical transmissions C�=1.0�; mN- loadsharing ratio �for spur gears mN=1.0; for helical gearsmN=b /Lmin�; b-effective face width; Lmin-minimumlength of contact lines.

Table 5 Load factors

ISO AGMA

A - application factor: considers variations of externaloads acting on gear transmission in relation to theominal load �Table 1�;

Ko - overload factor;Kv - dynamic factor;KH - load distribution factor;

v - dynamic factor: considers internal dynamic loadsaused by vibrations of pinion and gear;

KS - size factor: considers influence of size of gearing ontooth-root stress;

F� - face load factor �root stress�: considers possibleon-uniform distribution of load along contact line;

KB - rim thickness factor �33�.

F�-transverse load factor �root stress�: considersistribution of load between the pairs of teeth in contact,ncluding non-uniform distribution of load along path ofontact �27�.

Table 6 Stress at tooth root

ISO AGMA

ominal tooth-root stress �F0 �27�: Local tooth-root stress �F �33,34�

�F0=Ft

bmnYFYSY� �F=FtKoKvKS

1

bmt

KHKB

YJ

ocal tooth-root stress �F: where: mt-transverse module�F=�F0KAKvKF�KF�

ournal of Mechanical Design SEPTEMBER 2006, Vol. 128 / 1153

btc

tdcbmd1

wotweA2

5

cbt

1

etween tangent to fillet and tooth center line computed accordingo the ISO and AGMA standards as well as according to the pre-ise finite element analysis is shown in Fig. 26.

Different locations of critical section at tooth root is caused byhe fact that ISO and AGMA standards use different methods toetermine critical section. This leads to different magnitudes ofritical section parameters, such as: hFe, sFn, F,. Additionally,oth standards determine these parameters by using different for-ulas. Varying locations of critical section are the main reason of

ifferences in tooth-root stresses, which can be seen in Figs.8–23.

In the case of gears manufactured with racks, stresses computedith the use of the finite element method were closer to the resultsf the ISO standard. In the cases of gears manufactured with gearools, stresses computed with the use of the finite element methodere closer to the results of the AGMA standard. Absolute differ-

nces in stresses calculated in accordance with the ISO andGMA standards diminish with growing normal module mn �Fig.5�.

Summary and ConclusionsThis paper conducts a comprehensive analysis of the geometri-

al parameters and their impact on gear performance as describedy the ISO �27� and AGMA �29–37� standards with FEM compu-ations conducted by the authors. Furthermore, the presented

Fig. 18 Influence of the number of gcording to the ISO standard, AGMA sttip Fbn /b=250 N/mm „left… and fo=500 N/mm „right…; main parameters othe same as in Fig. 10

Fig. 19 Influence of the number ofstress �F according to the ISO standapplied at the tip „left… and for loadrameters of the gear tool „generating
11; magnitude of load is the same as in

analysis leads to suggestions and recommendations which can beused in future preparation of gear standards. The presented com-parative analysis is important for modern design and manufactur-ing of gears conducted globally at various locations. Currenttrends in engineering globalization necessitate revisiting variousnormalized standards to determine their common fundamentalsand best approaches needed to develop “best practices” in auto-motive, aerospace and other industries �1,2,48,49�. This can leadto both reduction in redundancies and also cost containment re-lated to needed adjustments between manufacturers for missingpart interchangeability and performance due to incompatibility ofdifferent standards. Also, currently many manufacturers are facedwith the necessity of obtaining certificates of their products inEurope, USA, or in Asia.

Therefore, in this paper methods of evaluation of tooth-rootstrength according to the ISO and AGMA standards are comparedby matching: �i� formulas used in both standards; �ii� estimation ofcritical section location �determined by the angle �F between toothcenterline and tangent to fillet at the critical point F�, and �iii�parameters of the critical section and tooth-root stresses. Calcula-tions were performed for gears generated with racks and geartools for various combinations of main parameters of gear geom-etry and tools used for manufacturing as well as for differentlocations of load application. Stresses determined in accordancewith the ISO and AGMA standards are compared with results of

teeth z1 on tooth-root stress �F ac-ard, and FEM: for load applied at the

oad applied at the HPSTC Fbn /be rack „generating tool… and gear are

th z0 of the gear tool on tooth-root, AGMA standard, and FEM: for loadlied at the HPSTC „right…; main pa-ol… and gear are the same as in Fig.

earandr lf th

teeardapp

to
Fig. 18

rf

Fra=as

Fof=eF

J

elated finite element analysis of the gears. Conclusions followingrom the performed analysis can be grouped in these categories.

1. Strength of helical gears. In both standards tooth-rootstrength analysis is performed for virtual straight tooth spurgearing with virtual number of teeth zn.

Fig. 20 Influence of the range of shtooth-root stress �F according to thethe tip „left… and for load applied at thgear tool „generating tool… and the ge11; magnitude of load is the same as

ig. 21 Influence of the normal pressure angle �n on tooth-oot stress �F according to the ISO, AGMA, and FEM: for loadpplied at the tip; main parameters of the gear: mn=2.75, z145, s1=4.265, �1=0 deg; main parameters of the rack „gener-ting tool…: s0=4.320, �fP=0.18mn; magnitude of load is theame as in Fig. 18

ig. 22 Influence of the addendum modification coefficient x1n tooth-root stress �F according to the ISO, AGMA, and FEM:

or load applied at the tip; main parameters of the gear: mn2.75, �n=20 deg, �1=0 deg; main parameter of the rack „gen-rating tool…: �fP=0.18mn; magnitude of load is the same as in
ig. 18

2. Strength of gears manufactured with racks. In the ISO,standard calculations are performed for rack with assumedstandard basic rack tooth profile. In the AGMA, standardcalculations are made for virtual gear tool with the samestandard basic rack tooth profile and very large number ofteeth �z0=10.000�.

3. Strength of gears manufactured with gear tools. In theISO standard real gear tool is replaced with virtual rack withthe same standard basic rack tooth profile. This significantlysimplifies calculations though it may lead to incorrect resultsin parameters of the critical section and in tooth-root stress�tooth profiles generated with gear tools and racks can sig-nificantly differ from each other�. In the AGMA standardcalculations are done for real gear tool.

4. Location of the critical section. Methods of determinationof the critical section are different in the ISO standard and inthe AGMA standard. This leads to magnitudes of the angle�F between tooth centerline and tangent to fillet at the criti-cal point F and the parameters of the critical section: sFn,hFe, and F. In the case of load applied at the HPSTC thedifferences are relatively small. However, in the case of loadapplied at the tip the differences are meaningful.

5. Tooth-root stress. In both standards, evaluation of tooth-root strength is performed based on maximum stress at toothroot on the stretched side of the tooth. It is assumed that the

ening of gear tool „parameter u… on, AGMA, and FEM: for load applied atPSTC „right…; main parameters of thein section 0-0 are the same as in Fig.ig. 18

Fig. 23 Influence of the helix angle �1 on tooth-root stress �Faccording to the ISO, AGMA, and FEM for load applied at thetip; main parameters of the gear: mn=5.0, z1=24, �n=20 deg,magnitude of distributed load passing through the tip corner of

arpISOe Harin F

helical tooth flank Fbn /b=274 N/mm

SEPTEMBER 2006, Vol. 128 / 1155

stra

Fstal

1

maximum stress occurs in the critical section of the tooth.Differences in tooth-root stresses calculated according to theISO and AGMA standards �Figs. 18–25� are caused by ap-plication of different formulas for tooth form factor andstress correction factor �Table 4�. Those coefficients arecomputed in both standards using the same parameters ofcritical section. However, in the formula defining tooth formfactor, Y in AGMA, there exists an additional componenttg�Fen /sFn, which does not exist in the corresponding for-mula for tooth form factor YF in the ISO standard. Formulasfor stress correction factor �YS according to ISO and Kf ac-cording to AGMA� differ, first of all, in applied exponents�Table 4�. As a result of that, even in the case when compu-tations are made for load applied at the HPSTC �for whichcase differences in both location and parameters of criticalsection determined according to ISO and AGMA standardsare insignificant - Figs. 10–12, 14, 15, and 17�, computedtooth-root stresses differ from each other. Those differencesin tooth-root stresses calculations according to the ISO andAGMA standards are bigger in the case when computationsare conducted for load applied to tooth tip than to HPSTC.This is caused by the fact that considered ISO and AGMAstandards differently determine both location and parametersof critical section �Fig. 9�. In all considered cases tooth-rootstresses calculated in accordance with the ISO were bigger

Fig. 24 Distribution of the effective scritical sections in various finite elemehelix angles � „left… and along fillet ingear with helix angle �=17 deg „rightz1=24; magnitude of distributed loadtooth flank Fbn /b=274 N/mm; parameof transversal sections of tooth is illu

ig. 25 Influence of the normal module mn on tooth-roottress �F according to the ISO and AGMA for load applied athe tip „Tip… and at the HPSTC; main parameters of the gener-ting rack and gear are the same as in Fig. 12; magnitude of

oad is the same as in Fig. 18


than corresponding results according to the AGMA. At thesame time, the difference in tooth-root stress was much big-ger in the case of load applied at the tip than in the case ofload applied at the HPSTC. Results of finite element com-putations were bigger than AGMA and smaller than ISOresults. In the case of gears generated with racks stressescomputed with finite elements were closer to the ISO resultsand in the case of gears generated with gear tools they werecloser to the AGMA results.

6. Influence of geometric parameters of gearing on tooth-root strength. Both standards differ in evaluation of an in-fluence of the helix angle �1, pressure angle �1, and adden-dum modification coefficient x1 on tooth-root strength �Figs.20–25�.

Detailed conclusions following comparison of related formulasand results of calculations are given in corresponding sections ofthis paper.

AppendixIn order to enable comparison of corresponding relations be-

tween considered parameters used in the ISO and AGMA stan-dards, one of them should be presented and analyzed in relation tothe other. Nomenclature used in this paper follows the ISO stan-dard �27�. Original AGMA nomenclature is given in Table 7. The

ses along tooth line passing throughmodels of helical gears with differentrious transversal sections of helicalain parameters of the gear: mn=5.0,

sing through the tip corner of helical� used for determination of locationted in Fig. 7 left

Fig. 26 Comparison of locations of the critical section and theangle between tangent to fillet and tooth centerline computedaccording to the ISO „left… and AGMA „middle… standards aswell as according to the precise finite element analysis „right…of the gears with the same tooth profiles; magnitude of load is

tresntva

…; mpaster

the same as in Fig. 18


Tsc

R

r

J

able 7 contains symbols, descriptions, and units used for corre-ponding parameters in the ISO �27� and AGMA �29,33� standardsonsidered in this paper.

eferences�1� Ceglarek, D., Huang, W., Zhou, S., Ding, Y., Kumar, R., and Zhou, Y., 2004,

“Time-Based Competition in Manufacturing: Stream-of-Variation Analysis�SOVA� Methodology—Review,” Int. J. of Flexible Manuf. Syst., 16�1�, pp.

Table 7 Li

ISO AGMA

�Fen �nLLoad angle

�n �nStandard normal pr

�wn �wnOperating normal p

� � Helix angle at the r�w �w

Operating helix ang�� mp

Transverse contact�� mF

Overlap ratio, axial�Rn0 �n0

Auxiliary angle locF F� Radius of curvature

fP, a0 a0Root fillet radius of

�F �FTooth-root stress, b

�F0¯ Nominal tooth-root

�F �n Angle between tangcenterline

� ¯ Auxiliary angle usesection �Fig. 13�

�nF, �nF Abscissa and ordinab b Effective face width

�AGMA�d1, d2 d1, d2

Standard pitch diamr1, r2 r1, r2

Standard pitch radiidbn1 dbn0 dnb1, dbn0

Virtual base diametrbn1, rbn0 rnb1, rnb0

Virtual base diametden �ren� dnL �rnL� Virtual load diametdw1, dw2 dw1, dw2

Operating diameter,rw1, dw2 rw1, dw2

Operating radius, piE ¯ Distance from the c

Fbn¯ Transmitted load in

Ft FtTransmitted tangenttangential load at th

hFe hFBending moment ar

hfP, ha0 ha0Tooth dedendum of

KA¯ Application factor f

KF� KF�Transverse load dis

KF� KF�Face load distributi

Kv KvDynamic factor for

mn mnNormal module

mt mtTransverse module

rRn0 rSnoRadius to center of

wn1, rwn0 rn�, rn0� Generating pitch ra

s1, s0 sn, sn0Reference normal c

sFn sFTooth thickness at c

spr �a0Amount of effective

T1, T2 T1, T2Torque on the pinio

Y� ChHelix angle factor

YF Y Tooth form factorYS Kf

Stress correction faz0

¯ Number of tooth ofz1 z1

Number of tooth of¯ KB

Rim thickness factoKF Distance from pitch

¯ KHLoad distribution fa

¯ KoOverload factor

KS Distance from pitch¯ Ks

Size factorLmin

Minimum length ofmN

Load sharing ratioYJ

Bending strength ge

11–44.


�2� Anderson, N. et al., 2004, “Gear Industry Vision. A Vision of Gear Industry in2025,” Technical Report developed at the Gear Industry Vision Workshop,March 10, 2004, Detroit. The report was prepared by Energetics, Inc., andsponsored by U.S. Army, AGMA Foundation, ASME CRTD, Ben FranklinTechnology Center, Boeing, Gleason Foundation, GM, and John Deere.

�3� Dudley, D. W., 2002, Handbook of Practical Gear Design, CRC, Boca Raton,FL.

�4� Townsend, D. P., 1992, Dudley’s Gear Handbook, McGraw–Hill, New York.�5� Smith, J. D., 1999, Gear Noise and Vibration, Dekker, New York.�6� Badgley, R. H., and Hartman, R. H., 1974, “Gearbox Noise Reduction: Pre-

f symbols

Name of parameter Unit

deg, rdre angle deg, rdure angle deg, rdence cylinder deg, rd

deg, rddeg, rd

tact ratio ¯

g the center of tool tip radius deg, rdfillet curve �at the critical point F� mmbasic rack, tool tip radius mm

ng stress number MPass MPato fillet at the critical point F and tooth deg, rd

r calculation of parameters of the critical deg, rd

f critical point FO�, net face width of narrowed member mm

r, pinion �1� and gear �2� mmnion �1� and gear �2�adius� of pinion �1�, mmadius� of gear type tool �0�adius� mmion �1� and gear �2� mmn �1� and gear �2�r of rounded tip of tool to the axis of tooth mmplane of action �normal to the tooth flank� N

load at the pitch diameter �ISO�, transmittedperating pitch diameter �AGMA�

N

mmvirtual basic rack, nominal tool addendum mmending strength ¯

tion factor �root stress� ¯

actor �root stress� ¯

ding strength ¯

mmmm

l tip radius mmof virtual spur gear �1�, virtual tool �0� mm

lar tooth thickness of gear �1�, of tool �0� mmal section mm

otuberance mm�, gear �2� Nm

¯

¯

¯

r-type toolion ¯

¯

int C to point F �Fig. 13�¯

¯

int C to point OR �Fig. 13�¯

tact lines mm¯

etry factor ¯

st o

essuresseferleratiocon

atinofthe

endistreent

d fo

te o�IS

ete, pier �rer �rer �rpinnioentetheiale omthe

or btribuon fben

toodiusircuriticpr

n �1

ctorgeapinrpo

ctor

po

con

om

diction and Measurement of Mesh-Frequency Vibrations Within an Operating

SEPTEMBER 2006, Vol. 128 / 1157

1

Helicopter Rotor-Drive Gearbox,” Trans. ASME J. Eng. Ind., 96�2�, pp. 567–577.

�7� Cavdar, K., Karpat, F., and Babalik, F. C., 2005, “Computer Aided Analysis ofBending Strength of Involute Spur Gears with Asymmetric Profile,” ASME J.Mech. Des., 127�3�, pp. 477–484.

�8� Wang, M.-J., 2003, “A New Photoelastic Investigation of the Dynamic Bend-ing Stress of Spur Gears,” ASME J. Mech. Des., 125�2�, pp. 365–372.

�9� Velex, P., and Baud, S., 2002, “Static and Dynamic Tooth Loading in Spur andHelical Geared Systems—Experiments and Model Validation,” ASME J.Mech. Des., 124�2�, pp. 334–346.

�10� Winter, H., and Hirt, M., 1974, “Zahnfußtragfähigkeit auf der Grundlage derWirklichen Spannungen. Spannungskorrekturfaktor, Kerbempfindlichkeitszahlund Relativer Kerbfaktor in ISO-Ansatz,” VDI-Z, 116�2�, pp. 119–126.

�11� Linke, H., and Sporbert, K., 1985, “Einfluß des Schleifabsatzes auf dieSpannungs-Konzentration bei Verzahnungen,” Maschinenbautechnik, 34�4�,pp. 251–257.

�12� Jaskiewicz, Z., and Wasiewski, A., 1992, Spur Gear Transmissions. Geometry,Strength, Accuracy of Manufacturing �in Polish�, 1, Transport and Communi-cation, Warsaw.

�13� Jaskiewicz, Z., and Wasiewski, A., 1995, Spur Gear Transmissions. Design �inPolish�, 2, Transport and Communication, Warsaw.

�14� Litvin, F. L., Egelja, A., Tan, J., and Heath, G., 1998, “Computerized Design,Generation and Simulation of Meshing of Orthogonal Offset Face-Gear DriveWith a Spur Involute Pinion With Localized Bearing Contact,” Mech. Mach.Theory, 33�1/2�, pp. 87–102.

�15� Pedrero, J. I., Rueda, A., and Fuentes, A., 1999, “Determination of the ISOTooth Form Factor for Involute Spur and Helical Gears,” Mech. Mach. Theory,34�1�, pp. 89–103.

�16� Bathe, K. J., 1996, Finite Element Procedures, Prentice–Hall, EnglewoodCliffs, NJ.

�17� Kleiber, M., ed., 1998, Handbook of Computational Solid Mechanics,Springer, Berlin.

�18� Weck, M., 1992, Moderne Leistungsgetriebe. Verzahnungsauslegung undBetriebsverhalten, Springer-Verlag, Berlin.

�19� Guingand, M., de Vaujany, J. P., and Icard, Y., 2004, “Fast Three-DimensionalQuasi-Static Analysis of Helical Gears Using the Finite Prism Method,”ASME J. Mech. Des., 126�6�, pp. 1082–1088.

�20� Sainsot, P., and Velex, P., 2004, “Contribution of Gear Body to ToothDeflections—A New Bidimensional Analytical Formula,” ASME J. Mech.Des., 126�4�, pp. 748–752.

�21� Litvin, F. L., Fuentes, A., Zani, C., Pontiggia, M., and Handschuh, R. F., 2002,“Face-Gear Drive With Spur Involute Pinion: Geometry, Generation by aWorm, Stress Analysis,” Comput. Methods Appl. Mech. Eng., 191�25–26�, pp.2785–2813.

�22� Litvin, F. L., Chen, J. S., Lu, J., Handschuh, R. F., 1996, “Application of FiniteElement Analysis for Determination of Load Share, Real Contact Ratio, Pre-cision of Motion, and Stress Analysis,” ASME J. Mech. Des., 118�4�, pp.561–567.

�23� Lewicki, D. G., Handschuh, R. F., Spievak, L. E., Wawrzynek, P. A., andIngraffea, A. R., 2001, “Consideration of Moving Tooth Load in Gear CrackPropagation Predictions,” ASME J. Mech. Des., 123�1�, pp. 118–124.

�24� Handschuh, R. F., and Bibel, G. D., 1999, “Experimental and Analytical Studyof Aerospace Spiral Bevel Gear Tooth Fillet Stresses,” ASME J. Mech. Des.,121�4�, pp. 565–572.

�25� Barkah, D., Shafiq, B., and Dooner, D., 2002, “3D Mesh Generation for StaticStress Determination in Spiral Noncircular Gears Used for Torque Balancing,”ASME J. Mech. Des., 124�2�, pp. 313–319.


�26� Liu, L., and Pines, D. J., 2002, “The Influence of Gear Design Parameters onGear Tooth Damage Detection Sensitivity,” ASME J. Mech. Des., 124�4�, pp.794–804.

�27� ISO 6336 Standard, Calculation of Load Capacity of Spur and Helical Gears.�28� DIN 3990, Tragfähigkeitsberechnung von Stirnrädern, Dezember 1987.�29� AGMA 908-B89, April 1989, Geometry Factors for Determining the Pitting

Resistance and Bending Strength of Spur, Helical and Herringbone Gear Teeth.�30� AGMA 918-A93. AGMA Information Sheet, January 1993, A Summary of

Numerical Examples Demonstrating the Procedures for Calculating GeometryFactors for Spur and Helical Gears.

�31� ANSI/AGMA 6002-B93, February 1993, Design Guide for Vehicle Spur andHelical Gears.

�32� ANSI/AGMA 2001-C95, January 1995, Fundamental Rating Factors and Cal-culation Methods for Involute Spur and Helical Gear Teeth.

�33� ANSI/AGMA 2101-C95, January 1995, Metric Edition of ANSI/AGMA 2001-C95. Fundamental Rating Factors and Calculation Methods for Involute Spurand Helical Gear Teeth.

�34� ANSI/AGMA 6110-F97, September 1997, Standard for Spur, Helical, Herring-bone and Bevel Enclosed Drives. Annex C, Illustrative examples.

�35� AGMA Information Sheet 913-A98, Method for Specifying the Geometry ofSpur and Helical Gears.

�36� Hösel, T., 1988, “Einfluß der Zahnform auf die Flanken—undZahnfu�tragfähigkeit nach DIN 3990 und AGMA 218.01—Grenzen für Opti-mierungsrechnungen,” Antriebstechnik, 27�9�, pp. 65–68.

�37� Hösel, T., 1989, “Vergleich der Tragfähigkeitsberechnung für Stirnräder nachANSI/AGMA—ISO/DIN—und RGW-Normen,” Antriebstechnik, 28�11�, pp.77–84.

�38� Niemann, G., and Winter, H., 1985, Maschinenelemente, Band II, Springer-Verlag, Berlin.

�39� Lewicki, D. G., and Ballarini, R., 1997, “Gear Crack Propagation Investiga-tions,” Gear Technol., 14�6�, pp. 18–24.

�40� Kawalec, A., and Wiktor, J., 1999, “Analytical and Numerical Method ofDetermination of Spur Gear Tooth Profile Machined by Gear Tools,” Adv.Technol. Mach. Equipment, 23�2�, pp. 5–28.

�41� Math, V. B., and Chand, S., 2004, “An Approach to the Determination of SpurGear Tooth Root Fillet,” ASME J. Mech. Des., 126�2�, pp. 336–340.

�42� ADINA Theory and Modeling Guide, ADINA R & D, Inc., Watertown, MA,2000.

�43� Kawalec, A., 1997, “Modelling of Tooth Flanks Based on Distorted Measure-ments,” Adv. Technol. Mach. Mech. Equipment, 21�3�, pp. 5–28.

�44� Kawalec, A., and Wiktor, J., 2001, “Analysis of Strength of Tooth Root WithNotch After Finishing of Involute Gears,” Arch. Mech. Eng., 48�3�, pp. 217–248.

�45� Timoshenko, S. P., and Goodyear, J. N., 1970, Theory of Elasticity, McGraw–Hill, New York.

�46� Kramberger, J., Šraml, M., Potrč, I., and Flašker, J., 2004, “Numerical Calcu-lation of Bending Fatigue Life of Thin-Rim Spur Gears,” Eng. Fract. Mech.,71�4–6�, pp. 647–656.

�47� Li, S., 2002, “Gear Contact Model and Loaded Tooth Contact Analysis of aThree-Dimensional Thin-Rimmed Gear,” ASME J. Mech. Des., 124�3�, pp.511–517.

�48� Ceglarek, DS., Shi, J., and Wu, S. M., 1994, “A Knowledge-based DiagnosisApproach for the Launch of the Auto-body Assembly Process,” ASME J. Eng.Ind., 116�4�, pp. 491–499.

�49� Rybak, J., Kawalec, A., and Wiktor, J., 1999, “Analysis in Involute CylindricalGear Transmissions with Modified Tooth Trace,” Proceedings of the 4th WorldCongress on Gearing and Power Transmissions, Paris, M.C.I., 1, pp. 169–181.


comparative analysis of tooth-root strength using iso and agma standards

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