Nonlinear normal modes in a class of nonlinear continuous systems

Melvin E. King, Alexander F Vakakis

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationNonlinear Vibrations
EditorsMo Shahinpoor, H.S. Tzou
PublisherPubl by ASME
Pages33-41
Number of pages9
ISBN (Print)0791811719
StatePublished - Dec 1 1993
Event14th Biennial Conference on Mechanical Vibration and Noise - Albuquerque, NM, USA
Duration: Sep 19 1993Sep 22 1993

Publication series

NameAmerican Society of Mechanical Engineers, Design Engineering Division (Publication) DE
Volume54

Other

Other14th Biennial Conference on Mechanical Vibration and Noise
CityAlbuquerque, NM, USA
Period9/19/939/22/93

ASJC Scopus subject areas

  • Engineering(all)

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  • Cite this

    King, M. E., & Vakakis, A. F. (1993). Nonlinear normal modes in a class of nonlinear continuous systems. In M. Shahinpoor, & H. S. Tzou (Eds.), Nonlinear Vibrations (pp. 33-41). (American Society of Mechanical Engineers, Design Engineering Division (Publication) DE; Vol. 54). Publ by ASME.