Force appropriation of nonlinear structures

Ludovic Renson*, Tom Hill, David Ehrhardt, David Barton, Simon Neild

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

15 Citations (Scopus)
312 Downloads (Pure)


Nonlinear normal modes (NNMs) are widely used as a tool for developing mathematical models of nonlinear structures and understanding their dynamics. NNMs can be identified experimentally through a phase quadrature condition between the system response and the applied excitation. This paper demonstrates that this commonly used quadrature condition can give results that are significantly different from the true NNM, in particular, when the excitation applied to the system is limited to one input force, as is frequently used in practice. The system studied is a clamped-clamped cross-beam with two closely spaced modes. This paper shows that the regions where the quadrature condition is (in)accurate can be qualitatively captured by analysing transfer of energy between the modes of the system, leading to a discussion of the appropriate number of input forces and their locations across the structure.

Original languageEnglish
Article number20170880
Number of pages23
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2214
Early online date13 Jun 2018
Publication statusPublished - 13 Jun 2018

Structured keywords

  • Engineering Mathematics Research Group


  • Energy transfer
  • Force appropriation
  • Nonlinear dynamics
  • Nonlinear normal modes
  • Phase quadrature


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