Modal energy exchanges in an impulsively loaded beam with a geometrically nonlinear boundary condition: computation and experiment

Alireza Mojahed, Yang Liu, Lawrence A. Bergman, Alexander F. Vakakis

Research output: Contribution to journalArticlepeer-review


The capability of a geometrically nonlinear boundary condition, i.e., a strong local nonlinearity, in “redistributing” a broadband input energy (generated by an impulsive load) among the vibration modes of a cantilever Euler–Bernoulli beam is investigated. It is shown that this modal energy redistribution increases the inherent capacity of the cantilever for passive energy dissipation. The nonlinear boundary condition is realized by grounding the free end of the cantilever through an inclined linear spring–damper pair with initial angle of inclination ϕ relative to the neutral axis of the beam while at rest. For ϕ0<90∘, the inclined spring–damper pair is geometrically nonlinear, whereas in the limiting case ϕ0=90∘ the boundary condition becomes linear. To study the nonlinear modal energy redistribution in the cantilever, a multi-step system identification method to identify the unknown parameters of the experimental fixture is employed; this informs a computational reduced-order finite element (FE) model of the fixture. First, the Multi-input Multi-output Frequency Domain Identification (MFDID) technique to analyze the experimental frequency response functions of the “base” linear cantilever without the boundary condition is employed and its modal parameters are identified. Next, the boundary condition for the limiting angle ϕ0=90∘ is imposed, so that again a linear fixture is obtained. Through reconciliation of computational and experimental measurements, the (linear) stiffness and damping coefficients of the boundary are identified, as well. Finally, by varying the angle of inclination in the range 0∘≤ϕ0<90∘, the nonlinear transient responses of the identified FE model with the nonlinear boundary condition are computed and projected onto the linearized modal basis of the system in the limit of zero energy. The computational FE results favorably compare with experimental measurements. Following this, the time-averaged modal energies of the system are computed and used to estimate the portion of the total energy of the beam allocated to each mode. Additionally, by employing these modal energies one may study and track the nonlinear energy exchanges between subsets of modes for different angles of initial inclination ϕ of the nonlinear boundary attachment. The computational results are validated by experimental measurements, thus highlighting the predictive capacity of the computational FE model. In the last step, a scalar measure for modal energy exchange is defined by computing the maximum fluctuation in the percentage of each of the instantaneous modal energies that is the maximum percentage of energy being exchanged by the modes. This measure proves to be dependent on both the initial energy and the initial angle of inclination ϕ. Again, experimental measurements favorably compare to computational FE simulations.

Original languageEnglish (US)
Pages (from-to)3443-3463
Number of pages21
JournalNonlinear Dynamics
Issue number4
StatePublished - Mar 2021


  • Cantilever beam
  • Geometrically nonlinear boundary condition
  • Hardening, softening nonlinearity
  • Modal energy exchange
  • System identification

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Electrical and Electronic Engineering
  • Applied Mathematics


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