Invariant manifold-based order reduction of time-periodic structural dynamic systems

Eric A. Butcher, Venkatesh Deshmukh

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Order reduction of parametrically excited 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 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. The proposed method reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates of the transformed system. 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 two degree-of-freedom inverted pendulum with a periodic follower force is used as an illustrative example. The nonlinear-based reduced models are compared with linearbased reduced models in the presence and absence of nonlinear resonances.

Original languageEnglish (US)
Title of host publicationIMAC-XXV - Celebrating 25 Years of IMAC
StatePublished - 2007
Event25th Conference and Exposition on Structural Dynamics 2007, IMAC-XXV - Orlando, FL, United States
Duration: Feb 19 2007Feb 22 2007

Publication series

NameConference Proceedings of the Society for Experimental Mechanics Series
ISSN (Print)2191-5644
ISSN (Electronic)2191-5652


Other25th Conference and Exposition on Structural Dynamics 2007, IMAC-XXV
Country/TerritoryUnited States
CityOrlando, FL

ASJC Scopus subject areas

  • Engineering(all)
  • Computational Mechanics
  • Mechanical Engineering


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