### Abstract

A general methodology is developed for computing the nonlinear normal modes of a class of undamped vibratory systems governed by nonlinear partial differential equations of motion. A nonlinear normal mode is defined as free motion during which all points of the system vibrate equiperiodically, reaching their extremum positions at the same instants of time. The analytical methodology is based on a previous work by Shaw and Pierre (1992b), where the displacements and velocities at any point of a structure were expressed as functions of the displacement and velocity of a single reference point. The dynamics of the continuous system were then restricted to invariant manifolds of the phase space. Motivated by the methodology presented by Shaw and Pierre, we express the displacement of an arbitrary point of the structure as a function of the displacement of a single reference point. Assuming undamped oscillations (and thus conservation of energy), a singular partial differential equation for the function relating the displacements is derived, and is subsequently solved using an asymptotic, power series methodology. Applications of the general theory are then given by computing the nonlinear normal modes of a simply supported beam resting on a nonlinear elastic foundation, and of a cantilever beam having geometric nonlinearities. The stability of the detected modes is then investigated by a linearized stability analysis.

Original language | English (US) |
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Title of host publication | Nonlinear Vibrations |

Editors | Mo Shahinpoor, H.S. Tzou |

Publisher | Publ by ASME |

Pages | 33-41 |

Number of pages | 9 |

ISBN (Print) | 0791811719 |

State | Published - Dec 1 1993 |

Event | 14th Biennial Conference on Mechanical Vibration and Noise - Albuquerque, NM, USA Duration: Sep 19 1993 → Sep 22 1993 |

### Publication series

Name | American Society of Mechanical Engineers, Design Engineering Division (Publication) DE |
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Volume | 54 |

### Other

Other | 14th Biennial Conference on Mechanical Vibration and Noise |
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City | Albuquerque, NM, USA |

Period | 9/19/93 → 9/22/93 |

### ASJC Scopus subject areas

- Engineering(all)

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## Cite this

*Nonlinear Vibrations*(pp. 33-41). (American Society of Mechanical Engineers, Design Engineering Division (Publication) DE; Vol. 54). Publ by ASME.