Characterization of a nonlinear sound absorber at low frequencies and high sound levels

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Submitted on 29 Nov 2019

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Characterization of a nonlinear sound absorber at low frequencies and high sound levels

M Volpe, S Bellizzi, Renaud Côte

To cite this version:

M Volpe, S Bellizzi, Renaud Côte. Characterization of a nonlinear sound absorber at low frequencies and high sound levels. Seventh Conference on Nonlinear Vibrations, Localization and Energy Transfer (NNM2019), Jul 2019, Marseille, France. �hal-02387235�

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M. Volpe, S. Bellizzi, R. Côte

Aix-Marseille Univ, CNRS, Centrale Marseille, LMA, Marseille, France

Characterization of a nonlinear sound absorber at low frequencies and high sound levels

Contact : volpe@lma.cnrs-mrs.fr

Context and Aims

Issues:

• Noise reduction at low frequency and high sound level

• Requirement of a characterization method for nonlinear sound absorbers

ex: micro-perforated panels, dynamic absorbers like Nonlinear Energy Sink (NES)

Aims:

• Development of a set-up for the characterization of non-linear acoustic absorbers

 Short Kundt Tube (SKT)

[1] R. Mariani et al, JSV, 330:5245–5258 (2011) [2] P-Y. Bryk et al, JSV, 424:224-237 (2018)

[3] A.Chauvin et al, JSV, 416:244-257 (2018) [4] H. Bodén, JSV, 331:3050-3067 (2012)

[5] Novak et al, JAES, 63:786-798 (2015)

Perspectives

• Identification of nonlinear systems (Hammerstein model, Voltera series)

• Adaptation of the methods for a Short Kundt tube with 2 microphones

• Test on other NESs (multiple membrane, microperforated panel)

• Applications to nonlinear absorbers:

 At low level: linear behaviour

 Beyond: resonance frequency shift towards high frequencies

Linearization method [3]

Set-up (top) and Electoacoustic equivalent model in the frequency domain (bottom)

Aim: Characterization of acoustic loads in term of apparent impedance (𝑍_𝑇) or apparent reflexion coefficient (𝑅_𝑇)

• From measurement of transfer function 𝐻_𝑚(𝑓) between 𝑢(𝑡) and 𝑝(𝑡)

• Excitation with synchronized sweep sine, for different excitation levels

• Equivalent linear loads characteristics from electroacoustic model:

𝑍_𝑇 = ^𝐻^𝑚^𝑍^𝑆

𝐻_𝑎𝑒𝑍_𝑆−ℎ 𝑅_𝑇 = ^𝐻^𝑚^𝑍^𝑆^−𝑍^𝑐^(𝐻^𝑎𝑒^−𝐻^𝑚⁾

𝐻_𝑚𝑍_𝑆−𝑍_𝑐(𝐻_𝑎𝑒−𝐻_𝑚

Method by nonlinear signature

Excitation mode 1: stepped sines [4]

𝑢 𝑡 = 𝐴 sin 2𝜋𝑓𝑡 + Ф

• 𝑓: signal excitation frequency

• Acquisition of pressure harmonics by demodulation (for each excitation frequency 𝑓)

Excitation mode 2: synchronized swept sine [5]

𝑢 𝑡 = 𝐴 sin 𝜑(𝑡) with 𝜑 𝑡 = 2𝜋𝑓₁𝐿𝑒^𝑡/𝐿

• [𝑓₁, 𝑓₂]: signal frequency band

• Reconstruction of pressure harmonics: 𝑝 𝑡 = 𝑢 𝑡 ∗ ℎ(𝑡)

• Impulse response: ℎ 𝑡 = σ_𝑛=1^𝑁 ℎ_𝑛 𝑡 + ∆𝑡_𝑛

with ℎ_𝑛(t) the n^th harmonics impulse response on [𝑛𝑓₁, 𝑛𝑓₂]

• Separation of 𝑁 harmonics with

sin 𝑛𝜑(𝑡) = sin 𝜑 𝑡 + ∆𝑡 𝑃_1𝑓⁻ (𝑓)

⋮

𝑃_𝑛𝑓⁻ (𝑓)

⋮

𝑃_𝑁𝑓⁻ (𝑓)

=

𝑆₁₁(𝑓) ⋯ 𝑆_1𝑛(𝑓) ⋯ 𝑆_1𝑁(𝑓)

⋮ ⋱ ⋮

𝑆_𝑛1(𝑓) 𝑆_𝑛𝑛(𝑓) 𝑆_𝑛𝑁(𝑓)

⋮ ⋱ ⋮

𝑆_𝑁1(𝑓) ⋯ 𝑆_𝑁𝑛(𝑓) ⋯ 𝑆_𝑁𝑁(𝑓)

𝑃_1𝑓⁺ (𝑓)

⋮

𝑃_𝑛𝑓⁺ (𝑓)

⋮

𝑃_𝑁𝑓⁺ (𝑓)

Aim: Taking nonlinearity signature into account in the characterization of acoustic absorbers

• Extension of the notion of reflexion coefficient to nonlinear systems in terms of scattering matrix

 Relation between 𝑁 harmonics of incident pressure 𝑝⁺ 𝑡 and backward pressure 𝑝⁻ 𝑡

 Note: for a linear load, scattering matrix reduces to 𝑆₁₁ 𝑓 = 𝑅_𝑇 𝑓

Notation: 𝑃_𝑛𝑓⁻ 𝑓 is the 𝑛^𝑡ℎ harmonic of the fundamental frequency 𝑓 of the backward pressure wave 𝑝⁻(𝑡) emitted by the load (respectively 𝑃_𝑛𝑓⁺ (𝑓) for the incident pressure 𝑝⁺ 𝑡 )

• Estimation of the scattering matrix using stepped sines (mode 1) and synchronized sewpt sine (mode 2)

• For the motorless loudspeaker NES:

Absorber Source

Estimated scattering matrix coefficient by excitation mode 1 Comparison of the frequency responses 𝐻_𝑛(𝑓) obtained

with excitation modes 1 and 2 for the first 4 harmonics Estimated impulse responses ℎ_𝑛(𝑡) and frequency responses

𝐻_𝑛 𝑓 for the first 4 harmonics

Estimated impedance of hybrid absorber [2]

𝑢(𝑡)

𝑝(𝑡)

𝑢(𝑡)

𝑝(𝑡)

SKT set-up

_• Frequency domain:

[10 ; 200] Hz

• High levels: typ. 300Pa (≈ 144 dB SPL)

• Source: 4 synchronous

loudspeakers, controlled in voltage 𝑢(𝑡), linear functioning

• Studied NES:

Motorless loudspeaker NES [1], hybrid NES [2]

Experimental set-up and schematic diagram

Load/Absorber SKT

Loudspeaker 𝑢(𝑡)

𝑝(𝑡)