Order reduction of parametrically excited linear and nonlinear structural systems represented by a set of second order equations is considered. First, the system is converted into a second order system with time invariant linear system matrices and (for nonlinear systems) periodically modulated nonlinearities via the Lyapunov-Floquet transformation. Then a master-slave separation of degrees of freedom is used and a relation between the slave coordinates and the master coordinates is constructed. Two possible order reduction techniques are suggested. In the first approach a constant Guyan-like linear kernel which accounts for both stiffness and inertia is employed with a possible periodically modulated nonlinear part for nonlinear systems. The second method for nonlinear systems reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates. In the process, closed form expressions for “true internal” and “true combination” resonances are obtained for various nonlinearities which are generalizations of those previously reported for time-invariant systems. No limits are placed on the size of the time-periodic terms thus making this method extremely general even for strongly excited systems. A four degree-of-freedom mass- spring-damper system with periodic stiffness and damping as well as two and five degree-of-freedom inverted pendula with periodic follower forces are used as illustrative examples. The nonlinear-based reduced models are compared with linear-based reduced models in the presence and absence of nonlinear resonances.
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August 2006
Technical Papers
Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems
Venkatesh Deshmukh,
Venkatesh Deshmukh
Department of Mechanical Engineering,
University of Alaska Fairbanks
, Fairbanks, AK 99775
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Eric A. Butcher,
Eric A. Butcher
Department of Mechanical Engineering,
e-mail: ffeab@uaf.ediu
University of Alaska Fairbanks
, Fairbanks, AK 99775
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S. C. Sinha
S. C. Sinha
Nonlinear Systems Research Laboratory, Department of Mechanical Engineering,
Auburn University
, Auburn, AL 36849
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Venkatesh Deshmukh
Department of Mechanical Engineering,
University of Alaska Fairbanks
, Fairbanks, AK 99775
Eric A. Butcher
Department of Mechanical Engineering,
University of Alaska Fairbanks
, Fairbanks, AK 99775e-mail: ffeab@uaf.ediu
S. C. Sinha
Nonlinear Systems Research Laboratory, Department of Mechanical Engineering,
Auburn University
, Auburn, AL 36849J. Vib. Acoust. Aug 2006, 128(4): 458-468 (11 pages)
Published Online: December 21, 2005
Article history
Received:
January 26, 2004
Revised:
December 21, 2005
Citation
Deshmukh, V., Butcher, E. A., and Sinha, S. C. (December 21, 2005). "Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems." ASME. J. Vib. Acoust. August 2006; 128(4): 458–468. https://doi.org/10.1115/1.2202151
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