TY - GEN

T1 - Nonlinear normal modes in a class of nonlinear continuous systems

AU - King, Melvin E.

AU - Vakakis, Alexander F.

N1 - Funding Information:
to the 3uthors preprints of some of their recent public3tions. This work W3S supported by an NSF Graduate Student Fellowship, and by NSF Grant No MSS 92-07318. Dr. Devendra Garg is the Grant monitor.
Publisher Copyright:
© 1993 American Society of Mechanical Engineers (ASME). All rights reserved.

PY - 1993

Y1 - 1993

N2 - 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.

AB - 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.

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U2 - 10.1115/detc1993-0030

DO - 10.1115/detc1993-0030

M3 - Conference contribution

AN - SCOPUS:85104182037

T3 - Proceedings of the ASME Design Engineering Technical Conference

SP - 33

EP - 41

BT - 14th Biennial Conference on Mechanical Vibration and Noise

PB - American Society of Mechanical Engineers (ASME)

T2 - ASME 1993 Design Technical Conferences, DETC 1993

Y2 - 19 September 1993 through 22 September 1993

ER -