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.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Pierce, A. D., 1981, Acoustics: An Introduction to its Physical Principles and Applications, McGraw-Hill, New York.
Temkin, S., 1981, Elements of Acoustics, John Wiley and Sons, New York.
Griffin, S., Lane, S. A., and Huybrechts, S., 2001, “Coupled Helmholtz Resonators for Acoustic Attenuation,” ASME J. Vibr. Acoust., 123(1), pp. 11–17. [CrossRef]
Selamat, A. and Lee, I., 2003, “Helmholtz Resonator With Extended Neck,” J. Acoust. Soc. Am.,113(4), pp. 1975–1985. [CrossRef]
Tang, S., 2005, “On Helmholtz Resonators With Tapered Necks,” J. Sound Vib., 279(3–5), pp. 1085–1096. [CrossRef]
de Bedout, J. M., Franchek, M. A., Bernhard, R. J., and Mongeau, L., 1997, “Adaptive-Passive Noise Control With Self-Tuning Helmholtz Resonators,” J. Sound Vib., 202(1), pp. 109–123. [CrossRef]
Esteve, S. J., and Johnson, M. E., 2005, “Adaptive Helmholtz Resonators and Passive Vibration Absorbers for Cylinder Interior Noise Control,” J. Sound Vib., 288(4–5), pp. 1105–1130. [CrossRef]
Yuan, J., 2007, “Active Helmholtz Resonator With Positive Real Impedance,” ASME J. Vibr. Acoust., 129(1), pp. 94–100. [CrossRef]
Maas, L. R. M., 1997, “On the Nonlinear Helmholtz Response of Almost-Enclosed Tidal Basins With Sloping Bottoms,” J. Fluid Mech., 349, pp. 361–380. [CrossRef]
Miles, J. W., 1981, “Nonlinear Helmholtz Oscillations in Harbours and Coupled Basins,” J. Fluid Mech., 104, pp. 407–418. [CrossRef]
Cao, H., 2005, “Primary Resonant Optimal Control for Homoclinic Bifurcations in Single-Degree-of-Freedom Nonlinear Oscillators,” Chaos, Solitons Fractals, 24(5), pp. 1387–1398. [CrossRef]
Farooq, U., and Nudehi, S. S., 2007, “A Nonlinear Acoustic Resonator,” The 2007 ASME International Design Engineering Technical Conferences, Paper No. DETC2007-34700, 21st Biennial Conference on Mechanical Vibration and Noise, Las Vegas, Nevada (on DVD-ROM).
Timoshenko, S., 1959, Theory of Plates and Shells,2nd ed., McGraw-Hill, New York.
Chobotov, V. A., and Binder, R. C., 1964, “Nonlinear Response of a Circular Membrane to Sinusoidal Excitations,” J. Acoust. Soc. Am., 36(1), pp. 59–73. [CrossRef]
Khalil, H. K., 2002, Nonlinear Systems,3rd ed., Prentice-Hall, Upper Saddle River, New Jersey.
Verhulst, F., 2005, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, 1st ed., Springer, New York.
Ferri, A., and Heck, B., 1998, “Vibration Analysis of Dry Friction Damped Turbine Blades Using Singular Perturbation Theory,” ASME J. Vibr. Acoust., 120(2), pp. 588–595. [CrossRef]
Heck, B., and Ferri, A., 1996, “Model Reduction of Coulomb Friction Damped Systems Using Singular Perturbation Theory,” ASME J. Dyn. Syst., Meas., Control, 118(1), pp. 85–91. [CrossRef]
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, John Wiley and Sons, New York.


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



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In