Modal interaction refers to the way that the modes of a structure interact when its geometry and material properties are perturbed. The amount of interaction between the neighboring modes depends on the closeness of the natural frequencies, the mode shapes, and the magnitude and distribution of the perturbation. By formulating the structural eigenvalue problem as a normalized modal eigenvalue problem, it is shown that the amount of interaction in two modes can be simply characterized by six normalized modal parameters and the difference between the normalized frequencies. In this paper, the statistical behaviors of the normalized frequencies and modes are investigated based on a perturbation analysis. The results are independently verified by Monte Carlo simulations.

1.
Collins
J. D.
, and
Thomson
W. T.
,
1969
, “
The Eigenvalue Problem for Structural Systems With Statistical Properties
,”
AIAA Journal
, Vol.
7
, No.
4
, pp.
642
648
.
2.
Fox
R. L.
, and
Kapoor
M. P.
,
1968
, “
Rates of Change of Eigenvalues and Eigenvectors
,”
AIAA Journal
, Vol.
6
, No.
12
, pp.
2426
2429
.
3.
Kiefling
L. A.
,
1970
, “
Comment on ‘The Eigenvalue Problem for Structural Systems With Statistical Properties’
,”
AIAA Journal
, Vol.
8
, No.
7
, pp.
1371
1372
.
4.
Schiff
A. J.
, and
Bogdanoff
J. L.
,
1972
a, “
An Estimator for the Standard Deviation of a Natural Frequency—Part 1
,”
ASME Journal of Applied Mechanics
, Vol.
39
, pp.
535
538
.
5.
Schiff
A. J.
, and
Bogdanoff
J. L.
,
1972
b, “
An Estimator for the Standard Deviation of a Natural Frequency—Part 2
,”
ASME Journal of Applied Mechanics
, Vol.
39
, pp.
539
544
.
6.
Shinozuka, M., and Yamazaki, F., 1988, “Stochastic Finite Element Analysis: An Introduction,” Stochastic Structural Dynamics—Progress in Theory and Applications, Elsevier, New York, pp. 241–291.
7.
Sobczyk
K.
,
1972
, “
Free Vibrations of Elastic Plate With Random Properties—The Eigenvalue Problem
,”
Journal of Sound and Vibration
, Vol.
22
, No.
1
, pp.
33
39
.
8.
Ramu
S. A.
, and
Ganesan
R.
,
1993
, “
A Galerkin Finite Element Technique for Stochastic Field Problems
,”
Computer Methods in Applied Mechanics and Engineering
, Vol.
105
, pp.
315
331
.
9.
Vanmarcke
E.
, and
Grigoriu
M.
,
1983
, “
Stochastic Finite Element Analysis of Simple Beams
,”
Journal of Engineering Mechanics
, Vol.
109
, No.
5
, pp.
1203
1214
.
This content is only available via PDF.