Determining Frequency-Energy Dependency of Nonlinear Normal Modes and Internal Resonances Using a Numerical Independent Approach

  • Arash Ghariblou Shahid Rajaee Teacher Training University, Tehran, Iran
  • Mussa Mahmoudi Shahid Rajaee Teacher Training University, Tehran, Iran


Application of linear normal modes to the nonlinear area provides an in-depth investigation of structures. In this paper, a straightforward approach is proposed to investigate nonlinear normal modes (NNMs) thoroughly and focus on all possible solutions and bifurcations, independent of all initial assumptions and prior solutions. In this context, after discretization of the response domain over an appropriate resolution, a periodicity algorithm is suggested to capture the solutions that meet the NNMs criteria. Afterward, the frequency and energy of the system during accepted responses and degrees of freedom (DOFs)’ relations are derived. Finally, after verifying the proposed approach and acquiring new internal resonances, the frequency-energy plots and NNMs of a nonlinear elastic system with more substantial nonlinearities and a two-story steel structure with nonlinear material are studied. It is worth noting that the periodicity algorithm and capturing all possible solutions and bifurcations are among the apparent novelties of the current paper.


Nonlinear Normal Modes, Frequency-Energy dependency, Independent Approach, Internal Resonances, Bifurcations, Nonlinear dynamic analysis,


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Sep 12, 2022
How to Cite
GHARIBLOU, Arash; MAHMOUDI, Mussa. Determining Frequency-Energy Dependency of Nonlinear Normal Modes and Internal Resonances Using a Numerical Independent Approach. Computer Assisted Methods in Engineering and Science, [S.l.], v. 29, n. 3, p. 261–292, sep. 2022. ISSN 2299-3649. Available at: <>. Date accessed: 08 dec. 2022. doi: