Order reduction of parametrically excited nonlinear systems: Techniques and applications

Nonlinear Dynamics

Volumn 41, Issue 1-3, 2005, Pages 237-273

  Sinha, S C ^a   Redkar, Sangram ^a   Deshmukh, Vankatesh ^b   Butcher, Eric A ^b  

^a AUBURN UNIVERSITY   (United States)

^b University of Alaska Fairbanks   (United States)

Author keywords

Lyapunov Floquet transformation; Nonlinear time periodic systems; Order reduction

Indexed keywords

APPROXIMATION THEORY; EQUATIONS OF MOTION; GEOMETRY; LARGE SCALE SYSTEMS; LYAPUNOV METHODS; MATHEMATICAL TRANSFORMATIONS; PROBLEM SOLVING; TIME VARYING SYSTEMS;

LYAPUNOV-FLOQUET TRANSFORMATION; NONLINEAR TIME PERIODIC SYSTEMS; ORDER REDUCTION; SECOND ORDER NONLINEAR SYSTEMS;

NONLINEAR SYSTEMS;

EID: 21844433471     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-005-2822-z     Document Type: Article

References (29)

1. Mahmoud, M. S. and Singh, M. G., Large Scale Systems Modelling, Pergamon Press, Oxford, UK, 1981.
2. Guyan, R. J., 'Reduction of stiffness and mass matrices', AIAA Journal 3, 1965, 380.
3. Shokoohi, S., Silverman, L., and Van Dooren, P., 'Linear time-variable systems: Balancing and model reduction', IEEE Transactions on Automatic Control AC-28, 1983, 810-822.
4. Shaw, S. W. and Pierre, C., 'Nonlinear normal modes and invariant manifolds', Journal of Sound and Vibration 150(1), 1991, 170-173.
5. Vakakis, A. F., 'Non-linear normal modes (NNMs) and their applications in vibration theory: An overview', Mechanical Systems and Signal Processing 11(1), 1997, 3-22.
6. Shaw, S. W., Pierre, C., and Pesheck, E., 'Modal analysis-based reduced-order models for nonlinear structures - an invariant manifold approach', The Shock and Vibration Digest 31(1), 1999, 3-16.
7. Burton, T. D. and Young, M. E., 'Model reduction and nonlinear normal modes in structural dynamics', in Proceedings of ASME Winter Annual Meeting, Chicago, IL, 1994.
8. Burton, T. D. and Rhee, R. W., 'The reduction of nonlinear structural dynamic models,' Journal of Vibration and Control 6(4), 2000, 531-556.
9. Gallivan, K., Grimme, E., and Van Dooren, P., 'Multi-point Padé approximants of large scale system via two-sided rational Krylov algorithm', in Proceedings of the IEEE Conference on Decision and Control, Lake Buena Vista, FL, 1994.
10. Guilhen, P. M., Bertheier, P., Ferraris, G., and Lalanne, M., 'Dynamic behavior of dissymmetric rotor bearings modelled periodic coefficient large system', in Proceedings of 58th Shock and Vibration Symposium, Huntsville, AL, 1987.
11. Sinha, S. C. and Pandiyan, R., 'Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov-Floquet Transformation', International Journal of Non-Linear Mechanics 29, 1994, 687-702.
12. Sinha, S. C., Pandiyan, R., and Bibb, J. S., 'Liapunov-Floquet transformation: Computation and applications to periodic systems', ASME Journal of Vibration and Acoustics 118, 1996, 209-219.
13. Deshmukh, V., Sinha, S. C., and Joseph, P., 'Order reduction and control of parametrically excited dynamical systems', Journal of Vibration and Control 6, 2000, 1017-1028.
14. Sinha, S. C., Redkar, S., and Butcher, E., 'Order reduction of nonlinear time periodic coefficients using invariant manifolds', Journal of Sound and Vibration, 2005 (in press).
15. Deshmukh, V., Butcher, E., and Sinha, S. C., 'Order reduction of parametrically excited nonlinear structural systems via invariant manifolds', in review Journal of Vibration and Acoustics, 2004.
16. Arnold, V. I., Ordinary Differential Equations, MIT Press, Boston, MA, 1978.
17. Wooden, S., Application of Lyapunov-Floquet Transformation and Quasiperiodic Normal Forms to Dynamical Systems, MS thesis, Auburn University, Auburn, AL, 2002.
18. Yakubovich, V. A. and Starzhinski, V. M., Linear Differential Equations with Periodic Coefficients, Vols. 1 and 2, Wiley, New York, 1975.
19. Kokotović, P., Khalil, H., and O'Reilly, J., Singular Perturbation Methods in Control Analysis and Design, SIAM Press, Philadelphia, PA, 1999.
20. Kan, D., Nonlinear Galerkin Based Model Reduction, Technical Report No. (98-49), Department of Mathematics, UCLA, Los Angeles, CA, 1998.
21. Redkar, S., Sinha, S. C., and Butcher, E., 'Some techniques for order reduction of nonlinear time periodic systems', in Proceedings of IMECE 2003, Washington DC, November 15-21, 2003, Paper No. 42552.
22. Steindl, A. and Troger, H., 'Methods for dimension reduction and their application in nonlinear dynamics', International Journal of Solids and Structures 38, 2001, 2131-2147.
23. Sinha, S. C., Redkar, S., and Butcher, E., 'On macromodeling of nonlinear systems with time periodic coefficients', Communications in Nonlinear Science and Numerical Simulation, 2005 (in press).
24. Garcia-Archilla B., Novo, J., and Titi, E., 'Postprocessing the Galerkin method: A novel approach to approximate inertial manifolds', SIAM Journal on Numerical Analysis 35(3), 1998, 941-958.
25. Evan-Iwanowski, R. M., Resonance Oscillations in Mechanical Systems, Elsevier, Amsterdam, The Netherlands, 1976.
26. Hsu, C. S., 'On the parametric excitation of a dynamic system having multiple degree of freedom', Journal of Applied Mechanics 30, 1963, 367-372.
27. Malkin, I. G., Some Problems in the Theory of Nonlinear Oscillations (Translated from Russian), State Publishing House of Technical and Theoretical Literature, Moscow, 1956.
28. Sinha, S. C., Redkar, S., Butcher, E., and Deshmukh, V., 'Order reduction of nonlinear time periodic systems using invariant manifolds', Paper no. VIB-48445, in Proceedings of 19th Biennial Conference on Mechanical Vibration and Noise, Chicago, IL, September 2-6, 2003.
29. Sinha, S. C., Butcher, E. A., and David, A., 'Construction of dynamically equivalent time-invariant forms for time-periodic systems', Nonlinear Dynamics 16, 1998, 203-221.