Bridging the gap between nonlinear normal modes and modal derivatives

Cees Sombroek*, Ludovic Renson, Paolo Tiso, Gaetan Kerschen

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference Contribution (Conference Proceeding)

7 Citations (Scopus)
333 Downloads (Pure)


Nonlinear Normal Modes (NNMs) have a clear conceptual relation to the classical linear normal modes (LNMs), yet they offer a solid theoretical framework for interpreting a wide class of non-linear dynamical phenomena with no linear counterpart. The main difficulty associated with NNMs is that their calculation for large-scale models is expensive, particularly for distributed nonlinearities. Repeated direct time integrations need to be carried out together with extensive sensitivity analysis to reproduce the frequency-energy dependence of the modes of interest. In the present paper, NNMs are computed from a reduced model obtained using a quadratic transformation comprising LNMs and Modal Derivatives (MDs). Previous studies have shown that MDs can capture the essential dynamics of geometrically nonlinear structures and can greatly reduce the computational cost of time integration. A direct comparison with the NNMs computed from another standard reduction technique highlights the capability of the proposed reduction method to capture the essential nonlinear phenomena. The methodology is demonstrated using simple examples with 2 and 4 degrees of freedom.
Original languageEnglish
Title of host publicationDynamic Behavior of Materials
Subtitle of host publicationProceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, 2015
EditorsGaëtan Kerschen
PublisherSpringer, New York, NY
Number of pages13
ISBN (Electronic)9783319152219
ISBN (Print)9783319152202
Publication statusPublished - 2016
Event33rd IMAC Conference and Exposition on Structural Dynamics, 2015 - Orlando, FL, United States
Duration: 2 Feb 20155 Feb 2015

Publication series

NameConference Proceedings of the Society for Experimental Mechanics Series
PublisherSpringer International Publishing
ISSN (Print)2191-5644


Conference33rd IMAC Conference and Exposition on Structural Dynamics, 2015
Country/TerritoryUnited States
CityOrlando, FL


  • Modal derivatives
  • Model reduction
  • Nonlinear normal modes
  • Quadratic manifold transformation


Dive into the research topics of 'Bridging the gap between nonlinear normal modes and modal derivatives'. Together they form a unique fingerprint.

Cite this