Multi-degree-of-freedom low-frequency electroacoustic absorbers through coupled resonators

doi:10.1016/j.apacoust.2017.10.019

Applied Acoustics

Volume 132, March 2018, Pages 109-117

https://doi.org/10.1016/j.apacoust.2017.10.019 Get rights and content

Abstract

Electroacoustic absorbers represent an interesting solution for low-frequency sound absorption in rooms. These systems simply consist of closed-box electrodynamic loudspeakers, whose acoustic impedance at the diaphragm is judiciously adjusted by connecting a passive or active electrical control circuit. This paper presents a method for designing different electroacoustic absorber systems constituted of simple electrical and mechanical components that are coupled to a primary loudspeaker, resulting in multi-degree-of-freedom resonators. Each system is optimised to maximise the sound absorption performance with respect to different metrics. Experimental evaluations in an impedance tube confirm the model accuracy and method efficiency for achieving low-frequency sound absorption.

Introduction

Room modes cause uneven distributions in space and frequency of the sound field, thus altering the sound quality [1]. Conventional passive techniques based on foam-based absorbers and diffusers are often used to control the reverberation time and early reflections [2]. Since their size is dictated by the wavelengths in the targeted frequency range, they are not well suited to the low frequencies. Resonant absorbers and Helmholtz resonators are available for low-frequency sound absorption [3], [4], [5]. Nevertheless, even if the design of Helmholtz resonators can be optimised for a given room, their high quality factor causes a narrow frequency band of efficient sound absorption [6]. Thus, Helmholtz resonators with two and three degrees of freedom (DOF), which consist of pairs of cylindrical necks and cavities stacked in series, were designed to improve the sound absorption capabilities [7], [8], [9]. The effect of resonator arrays on the sound field in cavities were evaluated in Refs. [10], [11]. These resonators can also be combined with micro-perforated panels constituted of very thin perforations backed by a cavity, which were firstly introduced in Ref. [12], so as to improve the sound absorption performance at higher frequencies [13], [14]. Recently, an original design constituted of panels arranged with parallel extended tubes, was proposed in Ref. [15], resulting in four peaks of absorption from 150 Hz to 440 Hz.

Another approach is the active sound absorption with electroacoustic absorbers. These active absorbers are closed-box electrodynamic loudspeaker systems, whose acoustic impedance at the diaphragms is judiciously adjusted with [16] or without sensor [17], so as to maximize their sound absorption performance in rooms [18]. When an appropriate electrical resistance is connected to the transducer terminals, an optimal acoustic resistance can be achieved at the diaphragm, but limited to the resonance frequency of the system [19]. With parallel resistance - capacitance (RC) or resistance - inductance (RL) electrical networks, the peak of sound absorption can be tuned below or above the transducer resonance frequency respectively, thanks to the reactive electronic components [20], [21]. Connecting a series resistance - inductance - capacitance (RLC) electrical network to the transducer terminals becomes a two-DOF resonator, resulting in two peaks of sound absorption [21].

Such Helmholtz resonators and shunted transducers can even be combined together. A multi-DOF electromechanical Helmholtz resonator, consisting of a Helmholtz resonator coupled to a shunted piezoelectric at the bottom of the cavity, was developed in Ref. [22]. This way, both resonance frequencies were tuned thanks to the shunted electrical load. The low-frequency sound absorption was also efficiently improved using a thin micro-perforated plate coated with a shunted piezoelectric transducer [23]. These different approaches introduce the idea of using mechanical or mixed electrical/mechanical resonators coupled to the primary closed-box loudspeaker interacting with the sound field, so as to imitate or improve the sound absorption capabilities relative to those obtained with electrical shunts.

The objective of this paper is to design innovative systems of multi-DOF electroacoustic absorber, which are only constituted of conventional electrical and mechanical components. First, the model of the closed-box electrodynamic loudspeaker is introduced, before presenting the sound absorption performance through the definitions of the specific acoustic impedance, as well as the corresponding sound reflection coefficient and sound absorption coefficient. Then, different systems of electrical and mechanical resonators coupled to the primary closed-box loudspeaker are studied, and the sound absorption performance of each system is optimised with respect to specific objective functions. Finally, an experimental evaluation of these systems and a discussion on the measured sound absorption performance are given.

Section snippets

Passive loudspeaker

In the low-frequency approximation, an electrodynamic loudspeaker can be modeled as a one-DOF oscillator mechanically driven by a voice coil within a magnetic field. All forces acting on the transducer, especially those resulting from the sound pressures $P_{f}$ and $P_{r}$ at the front and rear faces of the diaphragm, are assumed small enough so that the governing equations should remain linear. The mechanical part is modeled as a simple mass - spring - damper system in the low-frequency range, that is

Strategy

In Section 2, the specific acoustic impedance expressions of systems A, B, and C given in Eqs. (8), (17), (19) respectively show the complexity to analyse the effect of each system model parameter. To maximise the sound absorption performance, parametric optimisations are proposed through two objective functions using the simplex search method developed in [24]. First, a bandwidth BW of efficient sound absorption is defined as the frequency range over which the total sound intensity in front of

Experimental setup

To experimentally validate the results found in Section 3, a waveguide was designed (length $L = 1.97$  m and internal diameter $\emptyset = 150$  mm) as depicted in Fig. 12. The duct was closed by each system one after another at one end and by a closed-box electrodynamic loudspeaker, delivering a band-limited pink noise of bandwidth [2 Hz to 2 kHz] at the other end. The specific acoustic impedance and corresponding sound absorption coefficient were evaluated according to ISO 10534-2 standard [25]. Three 1/2”

Conclusion

The method presented in this paper aims at designing efficient low-frequency multi-degree-of-freedom electroacoustic absorbers through coupled resonators. First, an appropriate electrical shunt, such as the series resistance - inductance - capacitance network, can be connected to the loudspeaker terminals, resulting in a two-DOF resonator. Stacking loudspeakers in closed boxes points out that the order of the system increases in the same manner with an equivalent sound absorption performance

Acknowledgment

This research was supported by the Swiss Commission for Technology and Innovation (CTI) under grant agreement n^o 14220.1 PFNM-NM.

References (25)

W. Frommhold et al.
Acoustic performance of membrane absorbers
J Sound Vib
(1994)
F. Fahy et al.
A note on the interaction between a helmholtz resonator and an acoustic mode of an enclosure
J Sound Vib
(1980)
A. Doria
Control of acoustic vibrations of an enclosure by means of multiple resonators
J Sound Vib
(1995)
M. Xu et al.
Dual helmholtz resonator
Appl Acoust
(2010)
C. Guan et al.
Modeling and optimal design of 3 degrees of freedom helmholtz resonator in hydraulic system
Chin J Aeronaut
(2012)
A. Cummings
The effects of a resonator array on the sound field in a cavity
J Sound Vib
(1992)
D. Li et al.
Acoustically coupled model of an enclosure and a helmholtz resonator array
J Sound Vib
(2007)
X.-L. Gai et al.
Sound absorption of microperforated panel mounted with helmholtz resonators
Appl Acoust
(2016)
X.-D. Zhao et al.
Improving low-frequency sound absorption of micro-perforated panel absorbers by using mechanical impedance plate combined with helmholtz resonators
Appl Acoust
(2016)
D. Li et al.
Enhancing the low frequency sound absorption of a perforated panel by parallel-arranged extended tubes
Appl Acoust
(2016)

R. Boulandet et al.

Optimization of electroacoustic absorbers by means of designed experiments

Appl Acoust

(2010)

H. Kuttruff

Room acoustics

(2009)

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Multi-degree-of-freedom low-frequency electroacoustic absorbers through coupled resonators

Abstract

Introduction

Section snippets

Passive loudspeaker

Strategy

Experimental setup

Conclusion

Acknowledgment

J Sound Vib

J Sound Vib

J Sound Vib

Appl Acoust

Chin J Aeronaut

J Sound Vib

J Sound Vib

Appl Acoust

Appl Acoust

Appl Acoust

Appl Acoust

Room acoustics