Three-mode interactions in harmonically excited systems with quadratic nonlinearities

T. A. Nayfeh, W. Asrar, A. H. Nayfeh

Research output: Contribution to journalArticle

Abstract

An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ω3≈2ω2 and ω2≈2ω1 to a harmonic excitation of the third mode, where the ωm are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitude F of the excitation as a control parameter. As the excitation amplitude F is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. As F is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. As F is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.

Original languageEnglish (US)
Pages (from-to)385-410
Number of pages26
JournalNonlinear Dynamics
Volume3
Issue number5
DOIs
StatePublished - Sep 1992
Externally publishedYes

Fingerprint

Mode Interaction
Modulation Equations
Hopf bifurcation
Modulation
Nonlinearity
Hopf Bifurcation
Excitation
Fixed point
Limit Cycle
Motion
Bifurcation (mathematics)
Method of multiple Scales
Ordinary differential equations
Period-doubling Bifurcation
Natural frequencies
Nonlinear Ordinary Differential Equations
Natural Frequency
Control Parameter
Breakdown
Torus

Keywords

  • Autoparametric resonance
  • chaos
  • Hopf bifurcation
  • torus

ASJC Scopus subject areas

  • Mechanics of Materials
  • Mechanical Engineering
  • Computational Mechanics

Cite this

Three-mode interactions in harmonically excited systems with quadratic nonlinearities. / Nayfeh, T. A.; Asrar, W.; Nayfeh, A. H.

In: Nonlinear Dynamics, Vol. 3, No. 5, 09.1992, p. 385-410.

Research output: Contribution to journalArticle

Nayfeh, T. A. ; Asrar, W. ; Nayfeh, A. H. / Three-mode interactions in harmonically excited systems with quadratic nonlinearities. In: Nonlinear Dynamics. 1992 ; Vol. 3, No. 5. pp. 385-410.
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