Microgeometry optimization has become an important phase of gear design that can remarkably enhance gear performance. For spiral bevel and hypoid gears, microgeometry is typically represented by ease-off topography. The optimal ease-off shape can be defined as the outcome of a process where generally conflicting objective functions are simultaneously minimized (or maximized), in the presence of constraints. This matter naturally lends itself to be framed as a multi-objective optimization problem. This paper proposes a general algorithmic framework for ease-off multi-objective optimization, with special attention given to computational efficiency. Its implementation is fully detailed. A simulation model for loaded tooth contact analysis is assumed to be available. The proposed method is demonstrated on a face-hobbed hypoid gear set. Three objectives are defined: maximization of gear mesh mechanical efficiency, minimization of loaded transmission error, minimization of maximum contact pressure. Bound constraints on the design variables are imposed, as well as a nonlinear constraint aimed at keeping the loaded contact pattern inside a predefined allowable contact region. The results show that the proposed method can obtain optimal ease-off topographies that significantly improve the basic design performances. It is also evident that the method is general enough to handle geometry optimization of any gear type.

References

References
1.
Litvin
,
F. L.
, 1989,
Theory of Gearing
,
NASA, Reference Publication
1212
.
2.
Stadtfeld
,
H. J.
, 1993,
Handbook of Bevel and Hypoid Gears
,
Rochester Institute of Technology
,
Rochester, NY
.
3.
Achtmann
,
J.
, and
Bär
,
G.
, 2003, “
Optimized Bearing Ellipses of Hypoid Gears
,”
J. Mech. Des.
,
125
, pp.
739
745
.
4.
Stadtfeld
,
H. J.
, and
Gaiser
,
U.
, 2000, “
The Ultimate Motion Graph
,”
J. Mech. Des.
,
122
, pp.
317
322
.
5.
Wang
,
P.-Y.
, and
Fong
,
Z.-H.
, 2006, “
Fourth-Order Kinematic Synthesis for Face-Milling Spiral Bevel Gears With Modified Radial Motion (MRM) Correction
,”
J. Mech. Des.
,
128
, pp.
457
467
.
6.
Artoni
,
A.
,
Bracci
,
A.
,
Gabiccini
,
M.
, and
Guiggiani
,
M.
, 2009, “
Optimization of the Loaded Contact Pattern in Hypoid Gears by Automatic Topography Modification
,”
J. Mech. Des.
,
131
,
011008
.
7.
Artoni
,
A.
,
Kolivand
,
M.
, and
Kahraman
,
A.
, 2010. “
An Ease-Off Based Optimization of the Loaded Transmission Error of hypoid Gears
,”
J. Mech. Des.
,
132
,
011010
.
8.
Gabiccini
,
M.
,
Bracci
,
A.
, and
Guiggiani
,
M.
, 2010. “
Robust Optimization of the Loaded Contact Pattern in Hypoid Gears With Uncertain Misalignments
,”
J. Mech. Des.
,
132
,
041010
.
9.
Kolivand
,
M.
, and
Kahraman
,
A.
, 2009. “
A Load Distribution Model for Hypoid Gears Using Ease-Off Topography and Shell Theory
,”
Mech. Mach. Theory
,
44
(
10
), pp.
1848
1865
.
10.
Artoni
,
A.
,
Gabiccini
,
M.
, and
Guiggiani
,
M.
, 2008, “
Nonlinear Identification of Machine Settings for Flank Form Modifications in Hypoid Fears
,”
J. Mech. Des.
,
130
,
112602
.
11.
Miettinen
,
K. M.
, 1999,
Nonlinear Multiobjective Optimization
,
Kluwer Academic Publishers
,
Norwell, MA
.
12.
Deb
,
K.
, 2001,
Multi-Objective Optimization Using Evolutionary Algorithms
,
John Wiley & Sons
,
Chichester, West Sussex, England
.
13.
Branke
,
J.
,
Deb
,
K.
,
Miettinen
,
K.
, and
Słowiński
,
R.
, eds., 2008, “
Multiobjective Optimization—Interactive and Evolutionary Approaches
,”
Lecture Notes in Computer Science
,
Springer-Verlag
,
New York
, Vol.
5252
.
14.
Wierzbicki
,
A. P.
, 1979, “
The Use of Reference Objectives in Multiobjective Optimization—Theoretical Implications and Practical Experience, WP-79-66
,” International Institute for Applied Systems Analysis, Laxenburg, Austria.
15.
Gill
,
P. E.
,
Murray
,
W.
, and
Wright
,
M. H.
, 1982,
Practical Optimization
,
Academic Press
,
London and New York
.
16.
Nocedal
,
J.
, and
Wright
,
S. J.
, 2006, “
Numerical Optimization
,”
Springer Series in Operations Research and Financial Engineering
,
Springer
,
New York
.
17.
Neumaier
,
A.
, Global Optimization Software, retrieved date December 16, 2010, www.mat.univie.ac.at/~neum/glopt/software_g.htmlwww.mat.univie.ac.at/~neum/glopt/software_g.html
18.
Jones
,
D. R.
, 2001, “
Direct Global Optimization Algorithm
,” in
Encyclopedia of Optimization
,
C. A.
Floudas
and
P. M.
Pardalos
, eds.,
Kluwer Academic Publishers
,
Dordrecht, The Netherlands
, pp.
431
440
.
19.
Kelley
,
C. T.
, Implicit Filtering, retrieved date December 16, 2010, www4.ncsu.edu/~ctk/iffco.htmlwww4.ncsu.edu/~ctk/iffco.html
20.
TOMLAB, TOMLAB Base Module Solvers, retrieved date December 22, 2010, http://tomopt.com/tomlab/products/base/solvers/http://tomopt.com/tomlab/products/base/solvers/
21.
Kolivand
,
M.
,
Li
,
S.
, and
Kahraman
,
A.
, 2010, “
Prediction of Mechanical Gear Mesh Efficiency of Hypoid Gear Pairs
,”
Mech. Mach. Theory
,
45
(
11
), pp.
1568
1582
.
22.
ANSI/AGMA 2005–D03, 2003, Design Manual for Bevel Gears, American Gear Manufacturers Association, 500 Montgomery Street, Suite 350, Alexandria, VA, http://www.agma.orghttp://www.agma.org.
23.
Finkel
,
D. E.
, 2005, “
Global Optimization With the DIRECT Algorithm
,” Ph.D. thesis, North Carolina State University, Raleigh, NC.
You do not currently have access to this content.