Nonlinear Dynamics

, Volume 96, Issue 4, pp 2547–2566 | Cite as

Modal interactions in the nonlinear dynamics of a beam–cable–beam

  • Vincenzo Gattulli
  • Marco LepidiEmail author
  • Francesco Potenza
  • Umberto Di Sabatino
Original Paper


Quadratic and cubic modal interactions characterize the geometrically nonlinear dynamics of a parametric analytical model composed by two cantilever beams connected by a suspended shallow cable. The natural frequencies and modes of the linearized model are determined exactly, by solving the integral–differential eigenproblem governing the undamped free oscillations. Interesting phenomena of linear cable–beam interaction (frequency veering and modal hybridization) are recognized in the spectrum. Global and local modes are distinguished by virtue of the two localization factors measuring the modal kinetic energy stored in the beams and cable, respectively. The localization level is also put in relation to the magnitude of the quadratic and cubic nonlinearities. Therefore, the exact linear eigensolution is employed to formulate a nonlinearly coupled two-degrees-of-freedom model, defined in the reduced space of the modal amplitudes corresponding to a global and a local mode. The modal interactions between the two modes are analyzed, with focus on the autoparametric excitation mechanisms that can be favored by the occurrence of integer frequency ratios (1:2 and 2:1). Such internal resonance conditions enable significant transfers of mechanical energy—essentially governed by the quadratic coupling terms—from the small amplitudes of the externally excited global mode to the high amplitudes of the autoparametrically excited local mode. Different regimes of periodic and quasi-periodic oscillations are identified.


Dynamic interaction Cable vibration Nonlinear oscillations Autoparametric excitation Energy transfer 



The research leading to these results has received funding from the Italian Government under Cipe resolution no. 135 (Dec. 21, 2012), project INnovating City Planning through Information and Communication Technologies (authors V.G., F.P. and U.D.S.). The author M.L. acknowledges the financial support of the (MURST) Italian Department for University and Scientific and Technological Research in the framework of the research MIUR Prin15 project 2015LYYXA8, “Multi-scale mechanical models for the design and optimization of micro-structured smart materials and metamaterials.”

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflicts of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Structural and Geotechnical EngineeringSapienza – University of RomeRomeItaly
  2. 2.Department of Civil Architectural and Environmental EngineeringUniversity of L’AquilaL’AquilaItaly
  3. 3.Department of Civil, Chemical and Environmental EngineeringUniversity of GenoaGenoaItaly

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