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强度计算

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Andrzej Kawalec

e-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 Ceglarek

Department of Industrial and Systems

Engineering,

The University of Wisconsin-Madison,

Madison, WI

e-mail: darek@enger.wisc.edu

Comparative Analysis

of Tooth-Root Strength Using ISO

and AGMA Standards in Spur and

Helical Gears With FEM-based

V eriﬁcation

Current 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 AGMA

standards and verifying them with developed models and simulations using the ﬁnite

element method (FEM). The presented analysis is conducted for (1) wide range of spur

and 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 each

modeled gear. FEM results are compared with stresses calculated based on the ISO and

AGMA 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 current

standards applied in engineering practice as well as provide a basis for future improve-

ments and/or uniﬁcations of gear standards. 关DOI: 10.1115/1.2214735兴

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

1 Introduction

1.1 Motivation. Gears are essential to the global economy

and are used in nearly all applications where power transfer is

required, such as automobiles, industrial equipment, airplanes, he-

licopters, and marine vessels. Frequency of product model change

and the vast amounts of time and cost required to make a change-

over, also called time-based competition, has become a character-

istic feature of modern global manufacturing and new product

development in automotive, aerospace, and other industries 关1兴.

This forces gear manufacturers to respond with improved gear

systems that are designed and manufactured faster than before.

Simultaneously, current trends in engineering globalization re-

quire research to revisit various normalized standards to determine

their common fundamentals and those approaches needed to iden-

tify “best practices” in industries. This can lead to various beneﬁts

including reduction in redundancies, cost containment related to

adjustments between manufacturers for missing part interchange-

ability, and performance due to incompatibility of different stan-

dards. Given the range of differences that exist in engineering

practices today, frequently manufacturers must seek certiﬁcates of

their products in Asia, Europe, and/or/ in the USA. This coincides

with the recently developed “Gear Industry Vision in 2025” stra-

tegic goals. The report recommends as one of the ﬁve strategic

goals to establish common standards with special emphasis on: 共i兲

one global system of design and testing standards by 2015; and

共ii兲 standardized quality and veriﬁcation procedures 关2兴.

Gear transmissions are widely used in various industries and

their efﬁciency and reliability are critical in the ﬁnal product per-

formance evaluation. Gear transmissions affect energy consump-

tion during usage, vibrations, noise, and warranty costs among

others factors 关3–6兴. 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, low

vibrations and noise. Considering their reliability and efﬁciency

are some of the most important factors, problems of distribution

of loads and, consequently, distribution of stresses in the whole

gear transmission, particularly in teeth of mating gears, need to be

thoroughly analyzed.

Gears have been manufactured for a number of years with ex-

tensive ongoing research related to their efﬁciency, operational

quality, and durability. They are relatively complex and there are a

number of design parameters involved in gear design. The design

of 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 complicated

and time consuming. It involves selection of appropriate informa-

tion from a large amount of engineering/standards data available

in engineering catalogues and design handbooks. While the

knowledge in gearing design is vast, however, there is an acute

paucity 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 different

design solutions obtained for the same gear under the same set of

working conditions. Yet, the ﬁeld lacks systematic comparative

analysis of both standards supported and justiﬁed by the detailed

ﬁnite element method 共FEM兲 analysis.

In the ISO standard the ﬁnite element method is treated as the

most accurate method of gear strength determination 共method A兲

and can be used for veriﬁcation purposes of the results obtained

with other methods. This paper attempts to provide a systematic

Contributed by the Power Transmission and Gearing Committee of ASME for

publication in the J

OURNAL OF MECHANICAL DESIGN. Manuscript received July 21,

2004; ﬁnal manuscript received November 18, 2005. Review conducted by Prof.

David Dooner.

comparative analysis between the ISO and AGMA standards, fo-

cused on tooth-root strength, based on the precise FEM modeling

of spur and helical gears.

1.2 Literature Review. When the loads occurring in gear

transmission cross certain thresholds, very often a fatigue break-

age of gear tooth takes place due to repeated high stress exceeding

allowable stress limits usually due to improper structure of gear

transmission, inappropriate tooth contact, stress concentration

around a notch, material defects, and other factors. Tooth break-

age results in destroying gear transmission and breaking links be-

tween driving and driven shafts. The fracture usually starts at the

stretched tooth ﬁllet. Therefore, stress state at tooth root is a rep-

resentative factor to estimate tooth-root strength of gearings for

both involute cylindrical gears with symmetric proﬁles and for

involute spur gears with asymmetric proﬁles 关7兴 in static and dy-

namic conditions 关8,9兴.

Accurate evaluation of the stress state and distribution of

stresses at tooth root is a complex task. The precise determination

of stress distribution requires application of experimental methods

using electric resistance wire strain gauges 关9,10兴 or photo-elastic

gauges 关10,11兴, digital photoelastic system involving real time

imaging 关8兴, as well as numerical methods 关4,10,12–15兴, applying

FEM-based simulations 关16–18兴, bipotential equations, and/or

utilizing Airy function 关11兴, ﬁnite prism method 关19兴,orthe

theory of Muskhelishvili applied to circular elastic rings 关20兴.

Signiﬁcant development in analysis of strength properties of

gear transmissions follows the achievements in computational de-

sign, simulation of meshing, and tooth contact analysis made by

Litvin et al. 关21,22兴, Lewicki et al. 关23兴, and Handschuh and Bibel

关24兴. In particular, Litvin et al. 关21兴 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 numerical

stress analysis of the model of ﬁve contacting teeth. In the paper

Litvin et al. 关22兴 investigated the contact force and its distribution

over contact ellipse, tooth deﬂection, and load share as well as

maximum bending stress along the width of the gear in loaded

gear drive. The results of the performed computations were simi-

lar to the known experimental ones. Lewicki et al. 关23兴 studied the

effect of moving gear load on crack propagation predictions. They

performed two-dimensional 共2D兲 analysis using ﬁnite element and

boundary element methods. Comparison of computed results to

experimental ones validated crack simulation based on calculated

stress intensity factors and mixed mode crack angle prediction

techniques in a model of tooth loaded at the highest point of single

tooth contact. Handschuh and Bibel 关24兴 investigated an

aerospace-quality spiral bevel gear set numerically and experi-

mentally. The developed nonlinear ﬁnite element model allowed

Fig. 1 The ﬁnite element model of a segment of a gear and two

load cases: prescribed force applied at the tip and prescribed

force applied at the highest point of single tooth contact

„HPSTC…

Fig. 2 The FEM model of a representative gear „z =45, m

=2.75 mm,

␣

=20 deg, b =32 mm…: segment of three-tooth

made of 3D isoparametric 20-node brick ﬁnite elements;

boundary conditions as in Fig. 1; load uniformly distributed

along the line of contact at the HPSTC

Fig. 3 The FEM model of a representative gear „z =45, m

=2.75 mm,

␣

=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 uniformly

distributed along the line of contact at the HPSTC; model of the

whole 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

1142 / Vol. 128, SEPTEMBER 2006 Transactions of the ASME

them to simulate three-dimensional 共3D兲 multitooth contact. The

results were then compared to experimental ones, obtained with

strain gauges. Barkah et al. 关25兴 applied ﬁnite element models for

static stress evaluation of non-circular gears, focusing on the tooth

ﬁllet region. Computed results of stress distributions were in full

agreement with existing analytical solutions. Sainsot and Velex

关20兴 developed a new bidimensional analytical model based on

Muskhelishvili’s theory applied to elastic rings. The model ap-

proximates stress distributions at the root circle and allows to

calculate gear body-induced tooth deﬂections more precisely than

the Weber’s equation, which is based on the semi-inﬁnite plate

approximation. Guingand et al. 关19兴 applied the ﬁnite prism

method 共FPM兲 for quasi-static analysis of helical gears to build a

model of cylindrical gears with reduced computational complexity

as compared to conventional FEM models. The developed FPM

models allow computing load and tooth-root stresses. Velex and

Baud 关9兴 experimentally investigated quasi-static and dynamic

tooth-root stresses in spur and helical gears and compared them

with numerical predictions obtained by using FEM. They devel-

oped a speciﬁc FE model which accounts for non-linear time-

varying mesh stiffness, tooth geometrical errors, and ﬂexural-

torsional-axial couplings. However, they did not consider the gear

fabrication process, thus their simulations model does not allow to

analyze the impact of the ﬁllet shape on tooth-root stresses.

The development in analysis of gear strength shown in papers

关21–25兴 validates the use of correctly deﬁned ﬁnite element mod-

els for analysis of not only spur or helical gears, but also other

types of gears, like hypoid or spiral-bevel gears, etc. Experimental

methods, 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 transmission

design. They enable estimation of stresses at tooth root with ac-

curacy acceptable for engineering design.

In every case, strength properties of gear transmissions are

strongly inﬂuenced by gear geometry, applied manufacturing pro-

cesses, and dimensional accuracy of manufactured gears

关4,12,13,18兴. For example, investigations described in Liu and

Pines’ paper 关26兴 illustrate that some fundamental gear design

parameters such as diametral pitch 共proportional to the normal

module, used in Europe as the characteristic parameter in gear

design兲, pressure angle, and number of teeth, may have a signiﬁ-

Fig. 5 Distribution of the effective stress in the loaded tooth of the gear shown in Fig. 3

Fig. 6 Comparison of effective stress distributions along tooth line and along

ﬁllet of the same gear computed with the use of different ﬁnite element models:

„i… the whole spur gear built of 3D 20-node isoparametric and eight-node brick

ﬁnite elements; „ii… segment of three teeth built of 3D 20-node isoparametric

brick ﬁnite elements; and „iii… segment of three teeth built of eight-node plane

stress ﬁnite elements; distance ⌬ describing location of transversal section of

tooth is shown in Fig. 7 right

Journal of Mechanical Design SEPTEMBER 2006, Vol. 128 / 1143

cant inﬂuence on damage detection sensitivity. In the aforemen-

tioned investigations, the three main types of gear damage were

considered: pitting and wear of tooth ﬂanks, concentration of

stress, and cracks at tooth root.

The ﬁnal geometry of every gear depends on the assumed gear

structural parameters and the applied manufacturing processes,

applied tools, tool machines, etc. 关3,12,13,18兴. Therefore, the es-

tablished standards consider models of both gear structures and

manufacturing techniques, and take into account certain structural

factors, mostly geometric and operational, as well as information

concerning applied manufacturing processes, applied tools, and

materials. In particular, suitable international ISO 关27兴 standards

developed over the years in Europe 共or fully compatible German

standard DIN 关28兴兲, and American standard AGMA 关29–37兴 de-

scribe procedures for approximate calculations of stresses at tooth

root of the involute gearings. In all these standards evaluation of

tooth-root strength is performed based on maximum stress at tooth

root on the stretched side of the tooth. Moreover, in all of these

standards it is assumed that the maximum stress occurs in the

critical section of the tooth. It is clear that proper evaluation of

maximum bending stress at tooth root is one of the most critical

parameters determining whether or not a particular gear transmis-

sion will function properly. Tooth breakage due to operating stress

that signiﬁcantly exceeds the maximum allowable stress in gear

transmission can be extremely dangerous in automotive, aero-

space, or space industries 关3,4,38,39兴. Therefore, an accurate

estimation of critical strength parameters is of great importance in

such cases.

The models considered in ISO and AGMA standards are similar

from both theoretical and computational points of view, though

they are not identical. They have been developed using state of the

art industrial practice as well as theoretical and experimental re-

search in this ﬁeld. However, there are a number of discrepancies

within the current standards. These discrepancies can be classiﬁed

as: 共1兲 each standard deﬁnes different geometrical parameters as

critical for gear performance; 共2兲 the calculations undertaken to

estimate the thresholds for these critical parameters are based on

different assumptions and on different mathematical relations fol-

lowing them; and 共3兲 results of computations differ for each stan-

dard under the same conditions.

Currently, there is a need to unify both standards 关35兴. Initial

studies, which intended to compare both standards, were done by

Hösel 关36,37兴. However, Hösel only considered the older versions

of the standards, thus limiting his analysis to simple numerical

comparison of both standards without any other veriﬁcation of

computed results.

In this paper corresponding models of gears, computational

methods used for their analysis, and results obtained according to

the ISO and AGMA standards are brieﬂy described and compared

with each other. In the next step we verify these with the use of

the developed FEM models and simulations. This approach allows

Fig. 7 Absolute values of obtained differences in the effective stresses along ﬁllet

␦

␴

eff

computed according to Eq. „1… for the 2D plane stress model with three teeth and for the

model of the whole gear built of 3D 20-node isoparametric and eight-node brick ﬁnite

elements in sections associated with ⌬=2 mm „left…; explanation of the sections and the

lines where the effective stresses are compared „right…

Fig. 8 Comparative analysis of the effective stress distribution at tooth root

for three magnitudes of rim thickness of gear „left…; absolute values of the

relative differences in the effective stresses along ﬁllet ⌬

␴

eff

„

i… computed

according to Eq. „2… for the model of three-teeth segment of gear built of 3D

20-node isoparametric, for load applied at the HPSTC F

/b =500 N/mm „right…

1144 / Vol. 128, SEPTEMBER 2006 Transactions of the ASME

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