Alfonso Fuentes
Faydor L. Litvin
Gear Research Center,
Department of Mechanical Engineering,
University of Illinois at Chicago,
Chicago, IL 60607-7022
Baxter R. Mullins
Ron Woods
Bell Helicopter Textron Inc.,
Forth Worth, TX 76101
Robert F. Handschuh
U.S. Army Research Laboratory,
NASA Glenn Research Center,
Cleveland, OH 44135
Design and Stress Analysis of
Low-Noise Adjusted Bearing
Contact Spiral Bevel Gears
An integrated computerized approach for design and stress analysis of low-noise spiral
bevel gear drives with adjusted bearing contact has been developed. The computational
procedure is an iterative process requiring four separate steps that provide: (a) a para-
bolic function of transmission errors that is able to reduce the effect of errors of align-
ment, and (b) reduction of the shift of bearing contact caused by misalignment. Applica-
tion of finite element analysis permits the contact and bending stresses to be determined
and the formation of the bearing contact to be investigated. The design of finite element
models and boundary conditions is automated and does not require intermediate CAD
computer programs. A commercially available finite element analysis computer program
with contact capability is used to conduct the stress analysis. The theory developed is
illustrated with numerical examples. 关DOI: 10.1115/1.1481364兴
1 Introduction
Spiral bevel gears have found broad application in helicopter
and truck transmissions. Design and stress analysis of such gear
drives is a recent topic of research that has been performed by
many scientists 关1–10兴. Reduction of noise and stabilization of
bearing contact of misaligned spiral bevel gear drives is still a
very challenging topic of research although machine tool manu-
facturers have developed analysis tools and outstanding equip-
ment for manufacture of such gear drives. The phenomenon of
design of spiral bevel gears is that the machine-tool settings of
spiral bevel gears are not standardized and have to be specially
determined for each set of parameters of design to guarantee the
required quality of the gear drives.
A new approach is proposed for the solution of this problem
that is based on the following considerations:
共1兲 The gear machine-tool settings are considered as given
共adapted, for instance, from the Gleason summary兲. The to-be-
determined pinion machine-tool settings have to provide the ob-
servation of assigned conditions of meshing and contact of the
gear drive.
共2兲 Low noise of the gear drive is achieved by application of a
predesigned parabolic function of transmission errors of a limited
value of maximum transmission error of 6–8 arcsec. A prede-
signed parabolic function of transmission errors is able to absorb
almost linear discontinuous functions of transmission errors
caused by errors of alignment that are the source of high noise and
vibration 关4,11兴. The authors’ concept of reduction of noise has
been confirmed by test of prototypes of existing design and pro-
posed design accomplished at NASA Glenn Research Center. The
test showed that the level of noise was reduced by 18 decibels of
the whole amount of 90 decibels and the vibration was reduced by
50% at the spiral bevel gear meshing frequencies and its harmon-
ics 关4兴.
共3兲 The provided orientation of the bearing contact has to re-
duce its shift caused by the errors of alignment of the gear drive.
共4兲 The developed procedure of design is an iterative process
based on simultaneous application of local synthesis and TCA
共Tooth Contact Analysis兲. The local synthesis provides assigned
conditions of meshing and contact at the mean contact point of
tangency of pinion-gear tooth surfaces. The TCA computer pro-
gram can simulate the conditions of meshing and contact for the
entire meshing process.
Finite element method is used for stress analysis and the inves-
tigation of the bearing contact. A model of three contacting teeth
complemented with the boundaries conditions is applied for finite
element analysis 共FEA兲. A general purpose finite element analysis
computer program 关12兴 has been used to conduct the stress analy-
sis. The design of contact models is automated and does not re-
quire application of CAD computer programs.
Computer programs for synthesis, analysis, and automation of
FEA are based on application of the same programming language.
Graphic representation of results of computation is obtained by
application of a commercially available graphical program.
2 Basic Ideas of Developed Approach
Local Synthesis. The mean contact point M is chosen on gear
tooth surface ⌺
2
共Fig. 1兲. Parameters 2a,
2
, and m
12
⬘
are taken at
M and represent the mayor axis of the instantaneous contact el-
lipse, the tangent to the contact path on gear tooth surface, and the
derivative of the gear ratio function m
12
⫽
(1)
/
(2)
where
(1)
and
(2)
are the angular velocities of the pinion and gear rota-
tions.
The program of local synthesis enables the pinion machine-tool
settings to be determined considering as known the gear machine-
tool settings and parameters a,
2
, and m
12
⬘
关11兴. The program
requires solution of ten equations for ten unknowns but six of the
ten equations are represented in echelon form. The algorithm of
local synthesis includes relations between principal curvatures and
directions proposed in 关11,13–15兴.
TCA „Tooth Contact Analysis…. The computer program al-
gorithm is based on conditions of continuous tangency of pinion-
gear tooth surfaces and is illustrated in Fig. 2. The TCA program
enables the function of transmission errors ⌬
2
(
1
) to be deter-
mined and the bearing contact to be obtained for each iteration
whereas the input variable parameters a,
2
, and m
12
⬘
of the re-
spective iteration are applied.
The computational procedure is divided into four separately
applied procedures performed as follows:
Contributed by the Mechanisms and Robotics Committee for publication in the
J
OURNAL OF MECHANICAL DESIGN. Manuscript received December 2000. Associ-
ate Editor: M. Raghavan.
524 Õ Vol. 124, SEPTEMBER 2002 Copyright © 2002 by ASME Transactions of the ASME
