数理解析研究所講究録
1129 巻 2000 年 1-8

1Modelling of sound fields inside acoustically-structurally
coupled fields by a combined method

A.Gonenc and l.Hagiwara (TIT)

Abstrac$t$, the sound $fie/ds$ inside
thiscoupled enclosed spaces
excited by simple harmonic point sound sources and sfructural forcing under the presence ofresonators $inc/udingtl?e$ a coustic absorption is modeled a special modal $anal\gamma sis$technique. Two case studies are, in the first one, a rectangularhavingboundaries STexible is solved
and the proposed method.
, thethe second caseefficiency $oti\prec elmholtz|$ resonators and a coustic absorptive treatment is compared.
$ln$$\epsilon tud\gamma$$\mathrm{a}coustical/\gamma- strucfur\mathrm{a}//\gamma$$He/mhol\mathrm{f}z$$b\gamma$$\rho \mathrm{e}rformed$$a//$$ca\mathrm{v}i\nu$$/n$$b\gamma\ulcorner\Xi M$$stud\gamma$

Introduction

Study of sound fields inside acoustically-structurally coupled enclosed spaces

alwaysan interesting but difflcult subject for those who are interested in shaping
controlling suchfields by active
passive means. There are several different techniques in modeling these
coupled fields such as FEM, Modal Analysis, BEM, Statistical Energy Analysis etc. In this study,$|\mathrm{s}$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$it is aimed to model the low frequency response ofspaces under the presence of simple harmonic int sound sources

coupled enclosed
structuralforshaping these. Acoustical absorption is also included in the model. Model also includes theresonators which are well-known passive systems tuned to shape sound fields [1].
Efficiency of resonators and absorptive treatment is compared in the coupled field response in
low frequency range inside acoustoelastically coupled enclosed spaces. Besides, as a specialcase, a rectangular cavity having all boundaries flexible is solved with the proposed method andthe results are compared with FEM results obtained in a previous study [2].
$\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{l}\mathrm{c}\mathrm{a}\mathfrak{l}\mathrm{l}\mathrm{y}- \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\dot{\mathrm{a}}\mathrm{I}\mathrm{l}\mathrm{y}$$\mathrm{p}o$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}|\mathrm{n}\mathrm{g}$$\mathrm{f}_{\dot{1}}\mathrm{e}\mathrm{l}\mathrm{d}\mathrm{s}$$\mathrm{H}\mathrm{e}\mathrm{I}\mathrm{m}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{z}$TheorySince itinterested in the low frequency response of coupled fields, matMmaticalmodeI
on a special modal analysis technique, namely, Method of Acoustoelasticity [3].
The
technique is developed and improved to include all the possible excitations andresonators are also included in the mathematica} model for single and multi-cavity systems.
Throughout the formulations of acoustoetastically coupled enclosed spaces the following basic
assumptions are done:. fluid inside the cavity is at rest prior to the motion of the flexible wall.
. sound sources are simple harmonic point sound sources.
. all the sound sources and Helmholtz resonators are stationary. Helmholtz resonators are cornpact in when compared to the size of the enclosed$i\dot{\mathrm{s}}$$\mathrm{i}.\mathrm{s}.\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}6$$\mathrm{o}\mathrm{r}\dagger \mathrm{n}\mathrm{a}\mathrm{l}$$\mathrm{i}$$\mathrm{i}i$$\mathrm{i}\mathrm{i}\mathrm{i}$$\mathrm{i}\mathrm{v}$$\mathrm{s}|\mathrm{z}\mathrm{e}$space.

$\mathrm{v}$. harmonic$\mathrm{v}\mathrm{i}$$\mathrm{v}\mathrm{a}\mathrm{r}\dagger \mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$in time are assumed for all inputs.

. absorption material is to be modeled as a locally reacting material for which a force ata point leads to motion of that point only

Under the presence of Helmholtz resonators there are three different coupled systems,
i.e. acoustical system, structural system and Helmholtz resonators

2$\ddot{P}_{n}+\sum_{l}\frac{A_{A}^{i}}{V}\rho_{\mathit{0}}c_{o}^{2}\frac{\alpha_{1n}^{i}}{M_{n,A}},\dot{P}_{n}+P_{n}=-\frac{A_{F}}{V}\ddot{w}_{n}rightarrow$(t)$\sum\frac{S_{k}}{V}\ddot{\zeta}_{k}F_{n}(r_{Rk})\sim-\sum\frac{\dot{Q}_{l}}{V}F_{n}(r_{Sl})arrow$term on the rightsecond term in Equation (1) is the acoustic absorption, and theresonators
Helmholtz
therepresents
term
secondcoupling,
hand side represents the structuralpointsoundsimple
harmonicrepresents
source-like,thelastterm
and
theas
which are treated$\mathrm{f}|\mathrm{r}\mathrm{s}\mathrm{t}$sources(2)$M_{n},( \ddot{q}_{m}+\omega_{m}^{2}q_{m})=\rho_{\mathit{0}}c_{o}^{2}A_{F}\sum_{m}\frac{P_{n}L_{nm}}{M_{n,A}}+Q_{m}^{E}$

Equation (2) represents the structural system and the first term on the right handthe acoustical loading on it and the second term is the structural external loading$\mathrm{s}|\mathrm{d}\mathrm{e}$is(3)$\rho_{\mathit{0}}l_{k}’S_{k}\ddot{\zeta}_{k}+S_{k}R_{ik}\dot{\zeta}_{k}+\frac{\rho_{\mathit{0}}c_{O}^{2}S^{\frac{9}{k}}}{V_{Rk}}\zeta_{k}=-p(\overline{r}_{Rk})S_{k}$the last equation, Equation (3) is the equation representing the Helmholtz resonator andas it is seen it couples with total acoustic pressure at the mouth in acoustically-structurallycoupled cavities. These three system equations can be reduced into two by defining two newresonator parameters as [1],(4)$H_{M}( \omega_{E})=\frac{S_{k}c_{o}^{2}\omega_{E}^{2}(-\omega_{E}^{2}+\omega_{M}^{2})}{l_{\mathrm{A}}’VR_{k}^{2}\omega_{E}^{2}+(-\omega_{E}^{-}+\omega_{Rk}^{2})\underline{9}}$(5)$H_{d}( \omega_{E})=\frac{S_{k}c_{o}^{2}}{l_{k}’V}\frac{R_{k}\omega_{E}^{8}}{R_{k}^{2}\omega_{E}^{2}+(-\omega_{E}^{2}+\omega_{Rk}^{2})^{2}}$Harmonic inputs are assumed for all the acoustical and structural parameters and somefor the equations above [1]. The coupled system equations then obtained inmatrices arematrix form as
the$\mathrm{d}\mathrm{e}\mathrm{f}\dot{\iota}\mathrm{n}\mathrm{e}\mathrm{d}$$\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}!\mathrm{e}$$[[\mathrm{T}\eta NA]+2^{H_{Rk}}(\omega_{E})[FRMNA]_{k}]\{\overline{P_{nr}}\}+$(6.a)$[ \sum^{H_{d}(\omega_{E})[FRMNA]_{k}}-\sum_{i}[\alpha m^{\langle i\rangle}]]\{\overline{P_{ni}}\}=\omega_{E}^{9}\sim\frac{A_{F}}{V}[LNM]\{qmr\}$$[[7mA]+2^{H_{Rk}}(\omega_{E})[FRMNA]_{k}]\{\overline{P_{ni}}\}-$$[ \sum^{H_{aR}}(\omega_{E})[FRMN\mathrm{A}]_{k}+\sum_{\iota}[\alpha nn^{(i\rangle}]]\{\overline{P_{nr}}\}=\omega_{E}^{2}\frac{A_{F}}{V}[LNM]\{qmi\}$(6.b)$- \sum\frac{\omega_{E}}{V}Q_{l}\{FNRS\}_{l}$$[\mathrm{T}\mathrm{t}MS]\{qmr\}=\rho_{\mathit{0}}c^{\frac{?}{o}}A_{F}[LMNA]\{Pnr\}+\{Q_{m}^{E}\}$(7.a)$[WMS]\{qmi\}=\rho_{\mathit{0}}c^{\frac{9}{o}}A_{F}[LMNA]\{Pni\}$(7.b)

Equations (6.a) to (7.b) are the general equations representing the acoustica} structural
system and Helmholtz resonators. As it is seen that, sound fields should contain phase
information as well. Solution of these equations requires only simple matrix operations.
Calculation of the eigenvalues are performed by bisection method with a dynamic step length,
obtaining the eigenvalues is proponionaI with the step size used in thethus the error
calculations. Besides, the computational time is very short compared with other methods.
$|\mathrm{n}$3Moreover, since the theoretical solutions arein the solution of system equations, it iseasy to determrne the cost function for the optimization procedures for shaping purposes.
$\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{I}\mathrm{o}\mathrm{y}\mathrm{e}\mathrm{d}$Case Study
Within the scope of the present study, two case study is performed.
the first casestudy, a rectangular cavity having all boundaries flexible ofof lm lm $ excitedby unit harmonic structuralsolved by the proposed mode} and the results arecompared with the results obtained from solving the same system by FEM by Ma and Hagiwara[2]. In their study, only lowest 12 odes were held, but the present model considers all themodes, for which the mode numbers changing from 1 to 8 for each structural flexible plate within
the frequency range of interest. The results obtained from the present analysis are tabulated inTable 1.
Table 2, modes calculated by FEM, and the present model are compared andpercentage error is given in the last column. Study of the Table 1, yields that, the coupled system
at low frequency range is highly dominated by structural modes with a very high modaland parallel plates can act
a pistonin the system response. The frequencies which arevery close to zero on the other hand can be considered as rigid body motion of the cavity. Table2illustrates that for the modes retained in the previous study, the results obtained from FEM
and the resultsfrom the proposed method are quite close to each other. The reason ofth percentage difference between the two methods on the other hand, is due to truncation ofand higher modes in the FEM analysis which affects the frequency response.
In the second case study, the efficiency of Helmholtz resonators and absorptive
treatment are compared in coupled field response. Although in the mathematical model,
Helmholtz resonators are treated as source-like, they also introduce an absorptive mechanismto the acoustical system which is very similar to acoustic absorption provided by absorptive
materials and are coupled with total acoustic pressure
the cavity. Therefore, evenHelmholtz resonators are tuned to afrequency, they affect the frequency range of interest.
This can be seen in Figure 2. Figure 3, the response of the cavity having a sound source andresonator is, the coupled acoustic pressure distribution is glven.
. In Figu
Comparison of the figures yield that, in the absence of sound sources resonators arevery promising in shaping the cavity response. When there is sound source, although theefficiency of resonators seems to be decreased compared with the absorptive treatment, thiscan be improved by optimizing the resonator equivalent resistance and. Besides, forthe efficiency, it is necessary to
the resonator-source-receive combinations. When theapplication of these two passive methods are compared, depending on the purpose, Helmholtz
resonators can be still more attractive with proper combinations, since they are much less bulky
and cheap compared with absorptive treatment. Therefore, in shaping the sound fields
passenger cabins such as automobiles or airplane fuselages for which weight hasimportance, the possibility of employing the resonators as a promising passrve method should
also be considered having an appropriate mathematical modeling of the system.
$\ln$$\mathrm{d}\dot{|}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}|\mathrm{o}\mathrm{n}\mathrm{s}$$\mathrm{x}$$\mathrm{x}$$\mathrm{m}$$\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}|\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{S}$$.\mathrm{m}$$\ln$$\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}|\mathrm{t}\mathrm{y}$$i\mathrm{B}\Re$$|\mathrm{l}\mathrm{k}\mathrm{e}$$\mathrm{o}\mathrm{b}\mathrm{t}\mathrm{a}|\mathrm{n}\mathrm{e}\mathrm{d}$$e$$\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{e}\}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}$$\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\ln$$\mathrm{s}|\mathrm{n}\mathrm{g}\mathrm{I}\mathrm{e}$$\ln$$\mathrm{g}\mathrm{I}\mathrm{v}\mathrm{e}\mathrm{n}$$r\mathrm{e}4$$\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}|\mathrm{o}\mathrm{n}\mathrm{s}$$\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{m}|\mathrm{z}\mathrm{e}$$r$$\mathrm{i}\mathrm{n}\mathrm{s}|\mathrm{d}\mathrm{e}$$\mathrm{v}\mathrm{e}\mathrm{h}|\mathrm{c}\mathrm{I}\mathrm{e}$Summary
this study, sound fields inside coupled enclosed spaces excited by acousticalstructural excitations such as sound sources, structural loadings were modeled. Acoustical
was also included in the model. Helmholtz resonators were also introduced to thecoupled cavity as an additional system, but defining two new parameters, system was reduced
into coupling problem between acoustical system and structural system only.
Two case studies were performed. thecase study, the FEM model and the modelobtained from the proposed study were compared. Acoustical pressure distribution inside thecavity and all the coupled modes were calculated. Comparison was realized for the modes
retained in FEM. It was seen that there was a percentage difierence of maximum 4.8% becauseof the truncation of lower andmodes in FEM analysis as well as some numericalmodeling errors introduced by FEM.
the second case study, the efficiency of Helmholtz
resonators was compared with absorptive treatment, and it was illustrated that, Helmholtz
resonators were also promising in shaping the sound { in}enclosed spaces.
$\ln$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{t}|\mathrm{o}\mathrm{n}$$\ln$$i\dot{\mathrm{i}}\mathrm{r}\mathrm{s}\mathrm{t}$$\mathrm{h}\mathrm{I}\mathrm{g}\mathrm{h}\mathrm{e}\mathrm{r}$$\ln$$\mathrm{f}\mathrm{i}\mathrm{e}$$\mathrm{d}\mathrm{s}$$\mathrm{e}\mathrm{d}$$\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{p}$4Reierences
1. Gonenc, A., 1997, ’Shaping of Sound Fields lnside Acoustoelastically Coupled Enclosed$\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\mathrm{s}^{\mathrm{I}},$P.hD. Thesis, M.E.T.

$\cup$

2. Ma, Z., Hagiwara, 1., 1994, ’ Development oi New Mode-Superposition Technique for, Transactions of the JSME, $\vee 01(37)(1)$ , pp.7Higher Order
Truncating the Lower
13 ( Japanese).
3. Dowe}|, E.H., Gorman, G.F., and Smith, D.A., 1977, ’Acoustoelasticity: General Theory and$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{M}\mathrm{o}\mathrm{d}e\mathrm{s}^{1}$$\ln$

Force Response to Sinusoidal Excitation, Including Comparison with Experimentst, Journal ofSound and Vibration, Vol.52(4), pp. 519-542.

5Figure 1. Acoustic Pressure Distribution Inside the Rectangular Cavity

6

7Table 2. Comparison of Coupled Frequencies Calculated by the Proposed Method and FEM

Comparison of Frequencies$\mathrm{u}$6
Number of Retained Modes$\tau \mathit{1}$48Figure 2. Percentage Difference Between Proposed Method and FEM

8Figure 2. Effect of helmholtz Resonator without Sound Source

Figure 3. Eftect of Helmholtz Rcsonator with Sound Sourc. Effect of Absorblive Trcalmenl$\Gamma\prec \mathrm{i}\mathrm{g}\mathrm{u}\mathrm{r}\mathrm{e}4$$e$