In some industrial applications, influence coefficient balancing methods fail to find the optimum vibration reduction due to the limitations of the least-squares optimization methods. Previous min-max balancing methods have not included practical constraints often encountered in industrial balancing. In this paper, the influence coefficient balancing equations, with suitable constraints on the level of the residual vibrations and the magnitude of correction weights, are cast in linear matrix inequality (LMI) forms and solved with the numerical algorithms developed in convex optimization theory. The effectiveness and flexibility of the proposed method have been illustrated by solving two numerical balancing examples with complicated requirements. It is believed that the new methods developed in this work will help in reducing the time and cost of the original equipment manufacturer or field balancing procedures by finding an optimum solution of difficult balancing problems. The resulting method is called the optimum min-max LMI balancing method.

1.
Darlow
,
M.
, 1989, “
Balancing of High-Speed Machinery
,”
Springer-Verlag
, New York, p.
185
.
2.
Goodman
,
T. P.
, 1964, “
A Least Squares Method for Computing Balance Corrections
,”
ASME J. Eng. Ind.
0022-0817,
94
(
1
), pp.
233
242
.
3.
Little
,
R. M.
, and
Pilkey
,
W. D.
, 1976, “
A Linear Programming Approach for Balancing Flexible Rotors
,”
ASME J. Eng. Ind.
0022-0817,
98
(
3
), pp.
1030
1035
.
4.
Woomer
,
E.
, and
Pilkey
,
W. D.
, 1981, “
The Balancing of Rotating Shafts by Quadratic Programming
,”
ASME J. Mech. Des.
1050-0472,
103
, pp.
831
834
.
5.
Kanki
,
H.
,
Kawanishi
,
M.
, and
Ono
,
K.
, 1998, “
A New Balancing Method Applying LMI Optimization Method
,”
Proc. of 5th International Conference on Rotor Dynamics
,
Vieweg
,
Darmstadt, Germany
, IFToMM, pp.
667
678
.
6.
Li
,
G.
,
Lin
,
Z.
,
Untaroiu
,
C. D.
, and
Allaire
,
P. E.
, 2003, “
Balancing of High-Speed Rotating Machinery Using Convex Optimization
,”
Proc. of 42nd Conference on Decision and Control
,
IEEE
,
New York
, pp.
4351
4356
.
7.
El-Shafei
,
A.
,
El-Kabbany
,
A. S.
, and
Younan
,
A. A.
, 2004, “
Rotor Balancing Without Trial Weights
,”
ASME J. Eng. Gas Turbines Power
0742-4795,
126
(
3
), pp.
604
609
.
8.
Gahinet
,
P.
,
Nemirovski
,
A.
, and
Laub
,
A.
, 1995,
LMI Control Toolbox for Use With MATLAB
,
MathWorks
,
Natick, US
.
9.
Ehrich
,
F. F.
, 2004,
Handbook of Rotordynamics
,
Krieger
,
Malabar, FL
.
10.
API
, 1996, “
Tutorial on the API Standard Paragraphs Covering Rotor Dynamics and Balancing: An Introduction to Lateral Critical and Train Torsional Analysis and Rotor Balancing
,” American Petroleum Institute (API), API, 684.
11.
Foiles
,
W. C.
,
Allaire
,
P. E.
, and
Gunter
,
E. J.
, 2000, “
Min-Max Optimum Flexible Rotor Balancing Compared to Weighted Least Squares
,”
Proc. of 7th International Conference on Vibrations in Rotating Machinery
,
Proceedings IMechE
,
Nottingham, UK
, pp.
141
148
.
12.
Vandenberghe
,
L.
, and
Balakrishnan
,
V.
, 1997, “
Algorithms and Software for LMI Problems in Control
,”
IEEE Control Syst. Mag.
0272-1708,
17
, pp.
89
95
.
13.
Nesterov
,
Y. E.
, and
Nemirovski
,
A. S.
, 1994,
Interior Point Polynomial Algorithms in Convex Programming: Theory and Applications
,
SIAM
,
Philadelphia
.
14.
Boyd
,
S. P.
,
El-Ghaoui
,
L.
,
Feron
,
E.
, and
Balakrishnan
,
V.
, 1997,
Linear Matrix Inequalities in System and Control Theory
,
SIAM
,
Philadelphia
,
Studies in Applied Mathematics
, Vol.
15
.
15.
Bazaraa
,
M. S.
,
Jarvis
,
J. J.
, and
Sherali
,
H. D.
, 1990,
Linear Programming and Network Flows
,
Wiley
,
New York
.
You do not currently have access to this content.