Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations

Robert Szalai, David Ehrhardt, George Haller

Research output: Contribution to journalArticlepeer-review


In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude-frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here, a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. This model identification utilizes Taken's delay-embedding theorem, as well as a least square fit to the Taylor expansion of the sampling map associated with that embedding. The SSMs are then constructed for the sampling map using the parametrization method for invariant manifolds, which assumes that the manifold is an embedding of, rather than a graph over, a spectral subspace. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.

Original languageEnglish (US)
Article number20160759
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2202
StatePublished - 2017
Externally publishedYes


  • invariant manifolds
  • model identification
  • nonlinear normal modes
  • nonlinear vibrations

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)


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