TY - JOUR
T1 - Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations
AU - Szalai, Robert
AU - Ehrhardt, David
AU - Haller, George
N1 - The work of R.S. and D.E. was partly funded by EPSRC under the Engineering Nonlinearity Programme grant no. EP/K003836/1. Acknowledgements
PY - 2017
Y1 - 2017
N2 - 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.
AB - 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.
KW - invariant manifolds
KW - model identification
KW - nonlinear normal modes
KW - nonlinear vibrations
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U2 - 10.1098/rspa.2016.0759
DO - 10.1098/rspa.2016.0759
M3 - Article
AN - SCOPUS:85119614592
SN - 1364-5021
VL - 473
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
IS - 2202
M1 - 20160759
ER -