On the design of external excitations in order to make nonlinear oscillators respond as free oscillators of the same or different type

Zvonko Rakaric, Ivana Kovacic, Matthew Cartmell

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

This study deals with nonlinear oscillators whose restoring force has a polynomial nonlinearity of the cubic or quadratic type. Conservative unforced oscillators with such a restoring force have closed-form exact solutions in terms of Jacobi elliptic functions. This fact can be used to design the form of the external elliptic-type excitation so that the resulting forced oscillators also have closed-form exact steady-state solutions in terms of these functions. It is shown how one can use the amplitude of such excitations to change the way in which oscillators behave, making them respond as free oscillators of the same or different type. Thus, in cubic oscillators, a supercritical or subcritical pitchfork bifurcation can appear, whilst in quadratic oscillators, a transcritical bifurcation can take place.
LanguageEnglish
Number of pages11
JournalInternational Journal of Non-Linear Mechanics
Early online date1 Jul 2016
DOIs
Publication statusE-pub ahead of print - 1 Jul 2016

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Nonlinear Oscillator
Excitation
Polynomials
Closed-form
Transcritical Bifurcation
Jacobi Elliptic Function
Pitchfork Bifurcation
Steady-state Solution
Design
Exact Solution
Nonlinearity
Polynomial

Keywords

  • nonlinear oscillator
  • exact solution
  • Jacobi elliptic function
  • external excitation

Cite this

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abstract = "This study deals with nonlinear oscillators whose restoring force has a polynomial nonlinearity of the cubic or quadratic type. Conservative unforced oscillators with such a restoring force have closed-form exact solutions in terms of Jacobi elliptic functions. This fact can be used to design the form of the external elliptic-type excitation so that the resulting forced oscillators also have closed-form exact steady-state solutions in terms of these functions. It is shown how one can use the amplitude of such excitations to change the way in which oscillators behave, making them respond as free oscillators of the same or different type. Thus, in cubic oscillators, a supercritical or subcritical pitchfork bifurcation can appear, whilst in quadratic oscillators, a transcritical bifurcation can take place.",
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AB - This study deals with nonlinear oscillators whose restoring force has a polynomial nonlinearity of the cubic or quadratic type. Conservative unforced oscillators with such a restoring force have closed-form exact solutions in terms of Jacobi elliptic functions. This fact can be used to design the form of the external elliptic-type excitation so that the resulting forced oscillators also have closed-form exact steady-state solutions in terms of these functions. It is shown how one can use the amplitude of such excitations to change the way in which oscillators behave, making them respond as free oscillators of the same or different type. Thus, in cubic oscillators, a supercritical or subcritical pitchfork bifurcation can appear, whilst in quadratic oscillators, a transcritical bifurcation can take place.

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