In this work, the basic problem of order reduction nonlinear systems subjected to an external periodic excitation is considered. This problem deserves attention because the modes that interact (linearly or nonlinearly) with the external excitation dominate the response. A linear approach like the Guyan reduction does not always guarantee accurate results, particularly when nonlinear interactions are strong. In order to overcome limitations of the linear approach, a nonlinear order reduction methodology through a generalization of the invariant manifold technique is proposed. Traditionally, the invariant manifold techniques for unforced problems are extended to the forced problems by ‘augmenting’ the state space, i.e., forcing is treated as an additional degree of freedom and an invariant manifold is constructed. However, in the approach suggested here a nonlinear time-dependent relationship between the dominant and the non-dominant states is assumed and the dimension of the state space remains the same. This methodology not only yields accurate reduced order models but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. Following this approach, various ‘reducibility conditions’ are obtained that show interactions among the eigenvalues, the nonlinearities and the external excitation. One can also recover all ‘resonance conditions’ commonly obtained via perturbation or averaging techniques. These methodologies are applied to some typical problems and results for large-scale and reduced order models are compared. It is anticipated that these techniques will provide a useful tool in the analysis and control of large-scale externally excited nonlinear systems.

1.
Mahmoud, M. S. and M. G. Singh, 1981, Large Scale Systems Modelling, Pergamon Press, Oxford, UK.
2.
Guyan, R. J., 1965, “Reduction of Stiffness and Mass Matrices.” AIAA Journal 3(380).
3.
Antoulus A. 1999, “Approximations of Linear Dynamical Systems.” Wiley Encyclopedia of Electrical and Electronics Engineering 11, pp. 403–422.
4.
Shaw
S. W.
and
Pierre
C.
,
1991
, “
Nonlinear Normal Modes and Invariant Manifolds
.”
Journal of Sound and Vibration
150
(
1
), pp.
170
173
.
5.
Shaw
S. W.
,
Pierre
C.
and
Pesheck
E.
,
1999
, “
Modal Analysis-Based Reduced-Order Models for Nonlinear Structures - An Invariant Manifold Approach
.”
The Shock and Vibration Digest
,
31
(
1
), pp.
3
16
.
6.
Burton, T.D. and Young M.E., 1994, “Model Reduction and Nonlinear Normal Modes in Structural Dynamics.” ASME Winter Annual Meeting, Chicago, IL, ASME.
7.
Burton
T. D.
and
Rhee
R. W.
,
2000
, “
The Reduction of Nonlinear Structural Dynamic Models
.”
Journal of Vibration and Control
,
6
(
4
), pp.
531
556
.
8.
Newman A. J., 1996, Model Reduction via Karhunen-Loe´ve Expansion Part I: An Exposition, Technical Report, Electrical Engineering Department, University of Maryland, College Park.
9.
Jiang D., Pierre C and Shaw S. W., 2003, “Nonlinear Normal Modes for Vibratory Systems under Periodic Excitation,” The proceedings of 19th Biennial Conference on Mechanical Vibration and Noise, ASME 2003 DETC Chicago, Illinois USA, September 2–6, Paper No. VIB-48443.
10.
Agnes
G. S.
and
Inman
D. J.
,
2001
, “
Performance of Nonlinear Vibration Absorbers for Multi-Degrees-of-Freedom Systems Using Nonlinear Normal Modes
,”
Nonlinear Dynamics
25
(
1
), pp.
275
292
.
11.
Carr J., 1981, Applications of Center Manifold Theory, Springer, New York, USA.
12.
Evan-Iwanowski R. M., 1976, Resonance Oscillations in Mechanical Systems, Elsevier Scientific Publishing Company, Netherlands.
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