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Research Papers

Perturbation Analysis of a Nonlinear Resonator

[+] Author and Article Information
Shahin S. Nudehi

Mechanical Engineering Department,
Valparaiso University,
Valparaiso, IN 46383
e-mail: shahin.nudehi@valpo.edu

Umar Farooq

Eaton Corporation,
2425 W. Michigan Avenue,
Jackson, MI 49202
e-mail: umarfarooq@eaton.com

1Address all correspondence to this author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 20, 2009; final manuscript received January 23, 2012; published online June 14, 2012. Assoc. Editor: Al Ferri.

J. Comput. Nonlinear Dynam 8(1), 011001 (Jun 14, 2012) (3 pages) Paper No: CND-09-1179; doi: 10.1115/1.4005998 History: Received December 20, 2009; Revised January 23, 2012

A perturbation analysis of a Helmholtz-type resonator with one of the resonator ends replaced by a membrane is studied in this work. A membrane is known to exhibit nonlinear behavior under certain conditions; thus, when attached to a resonator system, it modifies the dynamic characteristics of the original system. This modified resonator system is modeled by coupled nonlinear differential equations and investigated by using the singular perturbation theory. The resonant frequency of the nonlinear resonator in the primary resonance case is analytically obtained using first-order approximate solutions. A good agreement is seen when the frequency response of the first-order approximate system is compared with the numerically simulated results.

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Figures

Grahic Jump Location
Fig. 1

A Helmholtz resonator with a membrane replacing one of its ends

Grahic Jump Location
Fig. 2

Frequency response of the original nonlinear resonator shown by a solid (—) line, the approximate (perturbation) solution of the same system shown by a dotted (…) line and the linear resonator system shown by a dashed (- - -) line

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