### 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 language | English |
---|---|

Pages (from-to) | 131-159 |

Number of pages | 29 |

Journal | Journal of Nonlinear Science |

Volume | 8 |

Issue number | 2 |

Publication status | Published - Mar 1998 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Nonlinear Science*,

*8*(2), 131-159.

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

Research output: Contribution to journal › Article

*Journal of Nonlinear Science*, vol. 8, no. 2, pp. 131-159.

}

TY - JOUR

T1 - Control of Ultra- and Subharmonic Resonances

AU - Fyrillas, M. M.

AU - Szeri, A. J.

PY - 1998/3

Y1 - 1998/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0011101023&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0011101023&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0011101023

VL - 8

SP - 131

EP - 159

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 2

ER -