Control of Ultra- and Subharmonic Resonances

M. M. Fyrillas, A. J. Szeri

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

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
Volume8
Issue number2
Publication statusPublished - Mar 1998
Externally publishedYes

Fingerprint

Subharmonics
Natural Frequency
resonant frequencies
Natural frequencies
Duffing Equation
Saddle-node Bifurcation
Modulation
saddles
modulation
Forcing
Bubble Dynamics
control theory
Optimal Control Theory
Multiple Scales
Perturbation Analysis
Nonlinear Oscillator
Fully Nonlinear
optimal control
Control theory
Rayleigh

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mathematics(all)
  • Applied Mathematics
  • Mathematical Physics
  • Statistical and Nonlinear Physics

Cite this

Fyrillas, M. M., & Szeri, A. J. (1998). Control of Ultra- and Subharmonic Resonances. Journal of Nonlinear Science, 8(2), 131-159.

Control of Ultra- and Subharmonic Resonances. / Fyrillas, M. M.; Szeri, A. J.

In: Journal of Nonlinear Science, Vol. 8, No. 2, 03.1998, p. 131-159.

Research output: Contribution to journalArticle

Fyrillas, MM & Szeri, AJ 1998, 'Control of Ultra- and Subharmonic Resonances', Journal of Nonlinear Science, vol. 8, no. 2, pp. 131-159.
Fyrillas MM, Szeri AJ. Control of Ultra- and Subharmonic Resonances. Journal of Nonlinear Science. 1998 Mar;8(2):131-159.
Fyrillas, M. M. ; Szeri, A. J. / Control of Ultra- and Subharmonic Resonances. In: Journal of Nonlinear Science. 1998 ; Vol. 8, No. 2. pp. 131-159.
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