Order reduction of forced nonlinear systems using updated LELSM modes with new Ritz vectors

Mohammad A. Al-Shudeifat, Eric A. Butcher

Research output: Contribution to journalArticle

12 Scopus citations


Enhanced modal-based order reduction of forced structural dynamic systems with isolated nonlinearities has been performed using the updated LELSM (local equivalent linear stiffness method) modes and new Ritz vectors. The updated LELSM modes have been found via iteration of the modes of the mass normalized local equivalent linear stiffness matrix of the nonlinear systems. The optimal basis vector of principal orthogonal modes (POMs) is found via simulation and used for POD-based order reduction for comparison. Two new Ritz vectors are defined as static load vectors. One of them gives a static displacement to the mass connected to the periodic forcing load and the other gives a static displacement to the mass connected to the nonlinear element. It is found that the use of these vectors, which are augmented to the updated LELSM modes in the order reduction modal matrix, reduces the number of modes used in order reduction and considerably enhances the accuracy of the order reduction. The combination of the new Ritz vectors with the updated LELSM modes in the order reduction matrix yields more accurate reduced models than POD-based order reduction of the forced nonlinear systems. Hence, the LELSM modal-based order reduction is enhanced via new Ritz vectors when compared with POD-based and linear-based order reductions. In addition, the main advantage of using the updated LELSM modes for order reduction is that, unlike POMs, they do not require a priori simulation and thus they can be combined with new Ritz vectors and applied directly to the system.

Original languageEnglish (US)
Pages (from-to)821-840
Number of pages20
JournalNonlinear Dynamics
Issue number4
StatePublished - Dec 1 2010
Externally publishedYes



  • Local equivalent linear stiffness
  • Model order reduction
  • Nonlinear dynamic systems
  • Proper orthogonal decomposition
  • Smooth nonlinearity

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering

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