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Research Papers

Targeted Energy Transfer From One Acoustical Mode to an Helmholtz Resonator With Nonlinear Behavior

[+] Author and Article Information
Emmanuel Gourdon

LTDS UMR CNRS 5513,
Univ Lyon, ENTPE,
rue Maurice Audin,
Vaulx-en-Velin F-69518, France
e-mail: emmanuel.gourdon@entpe.fr

Alireza Ture Savadkoohi

Mem. ASME
LTDS UMR CNRS 5513,
Univ Lyon, ENTPE,
rue Maurice Audin,
Vaulx-en-Velin F-69518, France
e-mail: alireza.turesavadkoohi@entpe.fr

Valentin Alamo Vargas

LTDS UMR CNRS 5513,
Univ Lyon, ENTPE,
rue Maurice Audin,
Vaulx-en-Velin F-69518, France
e-mail: valentin.alamovargas@entpe.fr

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 21, 2017; final manuscript received April 5, 2018; published online May 7, 2018. Assoc. Editor: Maurizio Porfiri.

J. Vib. Acoust 140(6), 061005 (May 07, 2018) (8 pages) Paper No: VIB-17-1550; doi: 10.1115/1.4039960 History: Received December 21, 2017; Revised April 05, 2018

Targeted energy transfer from one acoustical mode to a Helmholtz resonator (HR) with nonlinear behaviors is studied. For the HR, nonlinear restoring forces and nonlinear damping are taken into account. A time multiple scale method around a 1:1 resonance is used to detect slow invariant manifold (SIM) of the system, its equilibrium and singular points. Analytical predictions are compared with those which are obtained by direct numerical integration of system equations. Experimental verifications are performed and presented for free and forced vibrating system.

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References

Housner, G. W. , Bergman, L. A. , Caughey, T. K. , Chassiakos, A. G. , Claus, R. O. , Masri, S. F. , Skelton, R. E. , Soong, T. T. , Spencer, B. F. , and Yao, J. T. P. , 1997, “Structural Control: Past, Present and Future,” J. Eng. Mech., 123(9), pp. 897–971. [CrossRef]
Korkmaz, S. , 2011, “A Review of Active Structural Control: Challenges for Engineering Informatics,” Comput. Struct., 89(23–24), pp. 2113–2132. [CrossRef]
Mead, D. J. , 1999, Passive Vibration Control, Wiley, West Sussex, UK.
Frahm, H. , 1911, “Device for Damping Vibrations of Bodies,” U.S. Patent No. 989,958A. https://patents.google.com/patent/US989958
Hartog, D. J. , 1956, Mechanical Vibrations, McGraw-Hill Book, New York.
Roberson, R. E. , 1952, “Synthesis of a Nonlinear Dynamic Vibration Absorber,” J. Franklin Inst., 254(3), pp. 205–220. [CrossRef]
Vakakis, A. , Gendelman, O. , Bergman, L. , McFarland, D. , Kerschen, G. , and Lee, Y. , 2009, Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems, Springer, Dordrecht, The Netherlands.
Vakakis, A. F. , 2001, “Inducing Passive Nonlinear Energy Sinks in Vibrating Systems,” ASME J. Vib. Acoust., 123(3), pp. 324–332. [CrossRef]
Gendelman, O. V. , 2001, “Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators,” Nonlinear Dyn., 25(1–3), pp. 237–253. [CrossRef]
Starosvetsky, Y. , and Gendelman, O. , 2008, “Strongly Modulated Response in Forced 2D of Oscillatory System With Essential Mass and Potential Asymmetry,” Phys. D, 237(13), pp. 1719–1733. [CrossRef]
Gendelman, O. V. , 2008, “Targeted Energy Transfer in Systems With Non-Polynomial Nonlinearity,” J. Sound Vib., 315(3), pp. 732–745. [CrossRef]
Mohammad, A. , and AL-Shudeifat, J. , 2017, “Nonlinear Energy Sinks With Nontraditional Kinds of Nonlinear Restoring Forces,” ASME J. Vib. Acoust., 139(2), p. 024503. [CrossRef]
Nucera, F. , Vakakis, A. F. , McFarland, M. D. , Bergman, L. A. , and Kerschen, G. , 2007, “Targeted Energy Transfers in Vibro-Impact Oscillators for Seismic Mitigation,” Nonlinear Dyn., 50(3), pp. 651–677. [CrossRef]
Gendelman, O. V. , 2012, “Analytic Treatment of a System With a Vibro-Impact Nonlinear Energy Sink,” J. Sound Vib., 331(21), pp. 4599–4608. [CrossRef]
Pennisi, G. , Stephan, C. , Gourc, E. , and Michon, G. , 2017, “Experimental Investigation and Analytical Description of a Vibro-Impact NES Coupled to a Single-Degree-of-Freedom Linear Oscillator Harmonically Forced,” Nonlinear Dyn., 88(3), pp. 1769–1784. [CrossRef]
Gourc, E. , Michon, G. , Seguy, S. , and Berlioz, A. , 2015, “Targeted Energy Transfer Under Harmonic Forcing With a Vibro-Impact Nonlinear Energy Sink: Analytical and Experimental Developments,” ASME J. Vib. Acoust., 137(3), p. 031008. [CrossRef]
Ture Savadkoohi, A. , Lamarque, C.-H. , and Dimitrijevic, Z. , 2012, “Vibratory Energy Exchange Between a Linear and a Nonsmooth System in the Presence of the Gravity,” Nonlinear Dyn., 70(2), pp. 1473–1483. [CrossRef]
Ture Savadkoohi, A. , Lamarque, C.-H. , and Contessa, M. V. , 2016, “Trapping Vibratory Energy of Main Linear Structures by Coupling Light Systems With Geometrical and Material Non-Linearities,” Int. J. Non-Linear Mech., 80, pp. 3–13. [CrossRef]
Gourdon, E. , Alexander, N. , Taylor, C. , Lamarque, C. , and Pernot, S. , 2007, “Nonlinear Energy Pumping Under Transient Forcing With Strongly Nonlinear Coupling: Theoretical and Experimental Results,” J. Sound Vib., 300(3–5), pp. 522–551. [CrossRef]
Wierschem, N. E. , Luo, J. , AL-Shudeifat, M. , Hubbard, S. , Ott, R. , Fahnestock, A. , Dane Quinn, D. , McFarland, D. M. , Spencer, B. F. , Spencer, A. , and Bergman, L. A. , 2014, “Experimental Testing and Numerical Simulation of a Six-Story Structure Incorporating Two-Degree-of-Freedom Nonlinear Energy Sink,” J. Struct. Eng., 14(6), p. 04014027. [CrossRef]
Lee, Y. S. , Vakakis, A. F. , Bergman, L. A. , McFarland, D. M. , and Kerschen, G. , 2007, “Suppression of Aeroelastic Instability by Means of Broadband Passive Tet—Part I: Theory,” AIAA J., 45(3), pp. 693–711. [CrossRef]
Gendelman, O. V. , Vakakis, A. F. , Bergman, L. A. , and McFarland, D. M. , 2010, “Asymptotic Analysis of Passive Nonlinear Suppression of Aeroelastic Instabilities of a Rigid Wing in Subsonic Flow,” SIAM J. Appl. Math., 70(5), pp. 1655–1677. [CrossRef]
Cochelin, B. , Herzog, P. , and Mattei, P.-O. , 2006, “Experimental Evidence of Energy Pumping in Acoustics,” C. R. Méc., 334(11), pp. 639–644. [CrossRef]
Bellet, R. , Cochelin, B. , Herzog, P. , and Mattei, P.-O. , 2010, “Experimental Study of Targeted Energy Transfer From an Acoustic System to a Nonlinear Membrane Absorber,” J. Sound Vib., 329(14), pp. 2768–2791. [CrossRef]
Wu, X. , Shao, J. , and Cochelin, B. , 2016, “Study of Targeted Energy Transfer Inside Three-Dimensional Acoustic Cavity by Two Nonlinear Membrane Absorbers and an Acoustic Mode,” ASME J. Vib. Acoust., 138(3), p. 031017. [CrossRef]
Helmholtz, H. V. , 1863, Die lehre von den tonempfindungen als physiologische grundlagefur die theorie der music, Friedrich Vieweg Und Sons, Braunschweig, Germany.
Sivian, L. J. , 1935, “Acoustic Impedance of Small Orifices,” J. Acoust. Soc. Am., 7(2), pp. 94–101. [CrossRef]
Bolt, R. H. , Labate, S. , and Ingard, U. , 1949, “The Acoustic Reactance of Small Circular Orifices,” J. Acoust. Soc. Am., 21(2), pp. 94–97. [CrossRef]
Ingard, U. , and Ising, H. , 1967, “Acoustic Nonlinearity of an Orifice,” J. Acoust. Soc. Am., 42(1), p. 6. [CrossRef]
Singh, D. , and Rienstra, S. , 2013, “A Systematic Impedance Model for Non-Linear Helmholtz Resonator Liner,” AIAA Paper No. 2013-2223.
Achilleos, V. , Richoux, O. , Theocharis, G. , and Frantzeskakis, D. J. , 2015, “Acoustic Solitons in Waveguides With Helmholtz Resonators: Transmission Line Approach,” Phys. Rev. E - Stat., Nonlinear, Soft Matter Phys., 91(2), p. 023204. [CrossRef]
Gourdon, E. , and Ture Savadkoohi, A. , 2015, “Nonlinear Structuring of Helmholtz Resonators for Increasing the Range of Sound Absorption,” The 10th European Congress and Exposition on Noise Control Engineering (EuroNoise), Maastricht, The Netherlands, May 31–June 3, pp. 973–976. http://www.conforg.fr/euronoise2015/proceedings/data/articles/000034.pdf
Vargas, V. A. , Gourdon, E. , and Ture Savadkoohi, A. , 2018, “Nonlinear Softening and Hardening Behavior in Helmholtz Resonators for Nonlinear Regimes,” Nonlinear Dyn., 91(1), pp. 217–231. [CrossRef]
Yu, G. K. , Zhang, Y. D. , and Shen, Y. Y. , 2011, “Nonlinear Amplitude-Frequency Response of a Helmholtz Resonator,” ASME J. Vib. Acoust., 133(2), p. 024502.
Boullosa, R. , and Orduña-Bustamante, F. , 1992, “The Reaction Force on a Helmholtz Resonator Driven at High Sound Pressure Amplitudes,” Am. J. Phys., 60(8), pp. 722–726.
Manevitch, L. I. , 2001, “The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables,” Nonlinear Dyn., 25(1), pp. 95–109. [CrossRef]
Nayfeh, A. H. , 1979, Nonlinear Oscillation, Wiley, New York.

Figures

Grahic Jump Location
Fig. 2

Scheme of the coupled acoustic system with a HR

Grahic Jump Location
Fig. 1

Scheme of the straight neck HR

Grahic Jump Location
Fig. 7

Pressure amplitude in the middle of the small diameter tube versus frequency in forced regime for system with HR (with a cavity of Lcav = 26.5 mm) (-) and without HR (-) for three cases: (a) 138 dB, (b) 150 dB, and (c) 157.5 dB

Grahic Jump Location
Fig. 8

Pressure amplitude in the middle of the small diameter tube in forced regime for system without HR (-) and with HR (-) at three cases: (a) 138 dB, (b) 150 dB, and (c) 157.5 dB

Grahic Jump Location
Fig. 3

(a) N1 versus N2 analytical (-) and numerical (-) for initial conditions (τ = 0) U1=U2=0,(dU1/dτ)=(dU2/dτ)=1×10−5 and (b) numerical N1 with HR (-), N1 without HR (-)

Grahic Jump Location
Fig. 4

(a) N1 versus N2 analytical (-) and numerical (-) for initials conditions (τ = 0) U1=U2=0, (dU1/dτ)=1×10−5, (dU2/dτ)=5×10−5 and (b) Numerical N1 with HR (-) and N1 without HR (-)

Grahic Jump Location
Fig. 5

Experimental setup

Grahic Jump Location
Fig. 6

N1 versus τ with HR (-) and N1 without HR versus τ (-) for three cases: (a) 144 dB, (b) 150 dB, and (c) 153.5 dB

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