### Abstract

An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ω_{3}≈2ω_{2} and ω_{2}≈2ω_{1} to a harmonic excitation of the third mode, where the ω_{m} 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 language | English (US) |
---|---|

Pages (from-to) | 385-410 |

Number of pages | 26 |

Journal | Nonlinear Dynamics |

Volume | 3 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1992 |

Externally published | Yes |

### Fingerprint

### Keywords

- Autoparametric resonance
- chaos
- Hopf bifurcation
- torus

### ASJC Scopus subject areas

- Mechanics of Materials
- Mechanical Engineering
- Computational Mechanics

### Cite this

*Nonlinear Dynamics*,

*3*(5), 385-410. https://doi.org/10.1007/BF00045074

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

Research output: Contribution to journal › Article

*Nonlinear Dynamics*, vol. 3, no. 5, pp. 385-410. https://doi.org/10.1007/BF00045074

}

TY - JOUR

T1 - Three-mode interactions in harmonically excited systems with quadratic nonlinearities

AU - Nayfeh, T. A.

AU - Asrar, W.

AU - Nayfeh, A. H.

PY - 1992/9

Y1 - 1992/9

N2 - 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.

AB - 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.

KW - Autoparametric resonance

KW - chaos

KW - Hopf bifurcation

KW - torus

UR - http://www.scopus.com/inward/record.url?scp=0012131896&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012131896&partnerID=8YFLogxK

U2 - 10.1007/BF00045074

DO - 10.1007/BF00045074

M3 - Article

AN - SCOPUS:0012131896

VL - 3

SP - 385

EP - 410

JO - Nonlinear Dynamics

JF - Nonlinear Dynamics

SN - 0924-090X

IS - 5

ER -