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


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 (iiase 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 publication14th Biennial Conference on Mechanical Vibration and Noise
Subtitle of host publicationNonlinear Vibrations
PublisherAmerican Society of Mechanical Engineers (ASME)
Number of pages9
ISBN (Electronic)9780791811719
StatePublished - 1993
EventASME 1993 Design Technical Conferences, DETC 1993 - Albuquerque, United States
Duration: Sep 19 1993Sep 22 1993

Publication series

NameProceedings of the ASME Design Engineering Technical Conference
VolumePart F167972-3


ConferenceASME 1993 Design Technical Conferences, DETC 1993
Country/TerritoryUnited States

ASJC Scopus subject areas

  • Mechanical Engineering
  • Computer Graphics and Computer-Aided Design
  • Computer Science Applications
  • Modeling and Simulation


Dive into the research topics of 'Nonlinear normal modes in a class of nonlinear continuous systems'. Together they form a unique fingerprint.

Cite this