Control of Ultra- and Subharmonic Resonances

M. M. Fyrillas, A. J. Szeri

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


Periodically forced, nonlinear oscillators often show a strong response when driven near a rational multiple of the natural frequency of the linearized system: Such occurrences are known as subharmonic, ultraharmonic, and ultrasubharmonic resonances. In these systems, at these resonances, nonlinearities conspire to shift the response from the driving frequency to (near) the natural frequency of the linearized system. We demonstrate that such resonances are particularly sensitive to a very slight modulation of the forcing to include a component at (near) the natural frequency. This modulation can be constructed - using techniques developed herein - to annihilate or to enhance the subharmonic or ultraharmonic resonance. The ideas are first introduced using a multiple-scales perturbation analysis of the Duffing equation. Later, fully nonlinear techniques are developed that can be implemented using continuation algorithms on a computer. For example, optimal control theory is used to devise a very small modulation of the forcing that annihilates the principal subharmonic of the Duffing equation via a contrived saddle-node bifurcation of periodic orbits. Following this manufactured saddle-node bifurcation, the surviving attractor is of a much smaller amplitude. Similar efforts aimed at the sharp ultraharmonics of the Ray leigh-Plesset equation of nonlinear bubble dynamics are equally successful - although the rich modal structure of the response requires some variations on the control strategy. Finally, these ideas open up the possibility of controlling the shape of broad areas of the response diagrams important in many applications.

Original languageEnglish
Pages (from-to)131-159
Number of pages29
JournalJournal of Nonlinear Science
Issue number2
Publication statusPublished - Jan 1 1998

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering(all)
  • Applied Mathematics

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