This paper summarizes the authors’ previous efforts on solving inverse eigenvalue problems for linear vibrating systems described by a vector differential equations with constant coefficient matrices and nonproportional damping. The inverse problem of interest here is that of determining symmetric, real, positive definite coefficient matrices assumed to represent mass normalized velocity and position coefficient matrices, given a set of specified complex eigenvalues and eigenvectors. Two previous solutions to the symmetric inverse eigenvalue problem, presented by Starek and Inman, are reviewed and then extended to the design of underdamped vibrating systems with nonproportional damping.

Issue Section:
Technical Papers
Keywords:
damping, vibrations, eigenvalues and eigenfunctions, differential equations, inverse problems
Topics:
Eigenvalues, Inverse problems, Polynomials, Vibration, Damping, Design
1.
Starek
,
L.
,
Inman
,
D. J.
, and
Kress
,
A.
,
1992
, “
A Symmetric Inverse Vibration Problem
,”
ASME J. Vibr. Acoust.
,
114
, pp.
564
–
568
.
2.
Starek
,
L.
, and
Inman
,
D. J.
,
1997
, “
A Symmetric Inverse Vibration Problem for Non-Proportional, Underdamped System
,”
ASME J. Appl. Mech.
,
64
(
3
), pp.
601
–
605
.
3.
Gohberg, I. et al., 1982, Matrix Polynomials, Academic Press.
4.
Danek, O., 1982, Non conservative Dynamic Systems, Strojnicky Casopis 33, Chap. 6, pp. 667–680 (In Czech).
5.
Gladwell, G. M. L., 1986, Inverse Problem in Vibration, Martinus Nijhoff Publishers, Boston, MA.
6.
Lancaster
,
P.
, and
Maroulas
,
J.
,
1987
, “
Inverse Eigenvalue Problems for Damped Vibrating Systems
,”
J. Math. Anal. Appl.
,
123
(
1
), pp.
238
–
261
.
7.
Starek
,
L.
, and
Inman
,
D. J.
,
1991
, “
On the Inverse Vibration Problem With Rigid Body Modes
,”
ASME J. Appl. Mech.
,
58
, pp.
1101
–
1104
.
8.
Starek
,
L.
, and
Inman
,
D. J.
,
1995
, “
A Symmetric Inverse Vibration Problem With Over-Damped Modes
,”
J. Sound Vib.
,
181
(
5
), pp.
893
–
903
.
9.
Starek, L., Inman, D. J., and Pilkey, D. J., 1995, “A Symmetric Positive Definite Inverse Vibration Problem With Underdamped Modes,” Proceedings of the Design Engineering Technical Conference, Boston, MA, DE-Vol. 84-3, Volume 3-Part C, ASME, pp. 1089–1094.
10.
Caughy
,
T. K.
, and
O’Kelly
,
M. E. J.
,
1965
, “
Classical Normal Modes in Damped Linear Dynamic Systems
,”
ASME J. Appl. Mech.
,
132
, pp.
583
–
588
.
11.
Shapiro, E. Y., and Chung, J. C., 1981, “Application of Eigenvalue/Eigenvector Assignment by Constant Output Feedback to Flight Control Systems Design,” Proceedings 15th Annual Conference on Information Sciences and Systems, Baltimore, MD, pp. 164–169.
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