Tunable-with-energy intense modal interactions induced by geometric nonlinearity

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

Research output: Contribution to journalArticlepeer-review


Modal interactions are distinct features of nonlinear systems that can be exploited in applications such as vibration and shock mitigation, targeted (irreversible) energy transfers (TET), and acoustic/stress wave tailoring. For such applications, different types of nonlinearities, e.g. hardening, softening, smooth, non-smooth, material or geometric, have been considered. In this work, we examine the geometric nonlinearity resulting from an initially inclined element consisting of a linear spring and a viscous damper connected in parallel, having an initial angle of inclination, ɸ0. Because of its inclined configuration, this element possesses strong (and doubly tunable with respect to ɸ0 and energy) geometrically nonlinear stiffness and damping effects, despite the linear constitutive laws governing its constituent components. First, we consider a single-degree-of-freedom linearly grounded oscillator attached to the nonlinear inclined element. Omitting dissipative effects, we investigate the frequency–energy relation of this system by employing the canonical action-angle transformation and show that, depending on the initial angle of inclination and the energy-level, the resulting nonlinearity can be tuned to be softening, hardening or a combination of both. Next, we explore the efficacy of the geometric nonlinearity to induce strong modal interactions by considering a three-degree-of-freedom lightly damped primary system that is weakly coupled to a single-degree-of-freedom lightly damped attachment with the inclined nonlinear element, subjected to impulsive excitation. Varying ɸ0 and the input energy, we demonstrate strong modal energy-exchanges between the modes of the primary system and the nonlinear attachment over broad energy-dependent spans of ɸ0. We show that the passive self-adaptiveness of the nonlinear damping and the hardening–softening geometric nonlinearity can induce narrowband or broadband frequency TET, including high-to-low frequency energy transfers. Interestingly, over a definitive range of ɸ0, these modal interactions may be limited only between the nonlinear mode of the attachment and the highest-frequency linear mode of the primary system, inducing strong high-frequency targeted energy transfer to the primary system.


  • geometric nonlinearity
  • hardening
  • modal energy exchange
  • Nonlinear systems
  • softening nonlinearity
  • targeted energy transfer

ASJC Scopus subject areas

  • Mechanical Engineering

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