PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Acoustic radiation modes and active structural acoustic control in
coupled enclosure
Haichao ZHU1, Rongfu MAO1, Jinlong LIAO1, Changwei SU1
1

National Key Laboratory on Ship Vibration and Noise, Naval University of Engineering, China

ABSTRACT
There are two problems in the active structural acoustic control (ASAC) of coupled enclosure. Firstly, thestructural mode information is required in order to obtain the acoustic radiation mode (ARM) of coupledenclosure. Secondly, only the radiation efficiency is included in conventional ASAC strategy of coupledenclosure, which has not enough good precision. To overcome these problems, by analogy to ARM theory of
free space, the acoustic potential energy is expressed in quadratic form of normal velocity on coupling
surface and ARM of coupled enclosure is redefined. So the proposed ARM theory of coupled enclosure isconsistent with that of free space and is convenient for application. The control strategy based on thedominant acoustic radiation mode (dARM) is proposed, and corresponding ASAC model of the coupledenclosure is also established. Simulation analysis and experimental research on elastic plate-rectangularcoupled enclosure are carried out. Both simulation and experimental results show that better control effectcan be obtained by controlling APE contributed by dARM. Therefore, the feasibility and effectiveness of the
proposed dARM control strategies are proved.
Keywords: ASAC, Acoustic radiation mode, Coupled enclosure

1. INTRODUCTION
Acoustic Radiation Mode (ARM) theory, which was put forward in 1990s (1,2), is an effective
structural acoustic analytical method. The ARM is a specific radiation pattern of a radiator surface,and all normal velocity or surface pressure on the vibrating surface can be expressed by a linear
combination of ARMs. An important feature of ARM applied in structure radiation research is that
coupling terms of structural modes may be avoided and acoustic radiation power can be reduced bycontrolling an arbitrary order of ARM coefficient amplitude. Therefore, the calculation and control of
acoustic radiation can be performed more effectively (3-5).
The problems of noise control in coupled enclosure exist widely in industry and life environments,
such as cabins of plane, ship and vehicle, rooms for working and living. A lot of researches at home
and aboard have been conducted on this topic. Among them, the applications of Active Structural
Acoustic Control (ASAC) attracted extensive attention, due to avoidance of secondary sound sourcesby actuators applied on structure directly instead and good performance with a few actuators. Point
force control was applied on a cylindrical shell to reduce the transmitted sound power by Simpson (6),and internal noise control in DC-9 cabin was achieved successfully. The mechanism of ASAC incoupled enclosure was studied in detail by Pan (7-9) via both theoretic analysis and experimentresearch. Snyder (10) firstly proposed a formulation to derive the eigenvectors of the error weighting
matrix of coupled enclosure, and corresponding ASAC model was built. Similar researches on ARM of
weak coupled enclosure were also conducted by Cazzolato (11), Hill (12) and Bagha (13), and great
achievements were made.
However, it’s worthy to note that the “ARMs of weak coupled enclosure” in previous studies areincompatible with ARMs in free space in physical significance, namely, the former are a set of basis
functions of structural mode amplitude whereas the latter are a set of basis functions of structural
surface normal vibration or pressure. On the other hand, structural mode is still necessary incalculation and control process. However, accurate structural modes cannot be obtained conveniently
due to complexity of structures in practice and dependence on the boundary conditions. In addition,
only radiation efficiency is considered in the ASAC, which may fail to achieve the expected control
1

haiczhu@163.com

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effect or lead to low control efficiency.
To overcome these problems, by analogy to ARM theory of free space, the acoustic potential energy
was expressed in quadratic form of normal velocity on coupling surface and ARM of coupledenclosure was redefined. Then, the active control strategy based on the dominant acoustic radiation
mode (dARM) is proposed, and the corresponding ASAC model is established.

2. AOUSTIC RADIATION MODE THEORY

Figure 1 –The coupled enclosure
Considering the minimization problem of sound transmission into a coupled enclosure. The
Acoustic Potential Energy (APE) of an enclosure is chosen as the global error criterion, which is
written as
𝐸𝑝 =

2
1
∫ |𝑝(𝒓)| 𝑑𝒓，
4𝜌0 𝑐02 𝑉(1)

where 𝜌0 is the density and 𝑐0 the speed of sound within the acoustic fluid enclosed by the enclosure,𝑝(𝑟) is the acoustic pressure and the integral is evaluated over the enclosure volume 𝑉, and 𝑟 = (𝑥, 𝑦, 𝑧).
With the assumption of weak coupling between the flexible structure and enclosure, the acousticpressure at an any location in the enclosure can be expressed as a linear combination of the ri gid wallacoustic mode shape functions of the enclosure𝑝(𝒓) = ∑∞(2)
𝑖=1 𝑃𝑖 𝝋𝑖 (𝒓),where 𝑃𝑖 is the i-th acoustic mode amplitude and 𝜑𝑖 (𝒓) is the associated mode shape function at thelocation 𝒓 in the enclosure.
Assuming that there is no fluctuating sound source in the closed cavity, the sound pressure 𝑝(𝐫) at
any point 𝐫 in the enclosure generated by the elastic structure vibration can be defined as the Green's
function response equation𝑝(𝒓) = 𝑗𝜌0 𝜔 ∫𝑆 𝐺𝑎 (𝒙|𝒓)𝒗(𝒙)𝑑𝒙，(3)
where 𝒗(𝒙) is the normal velocity on the structural boundary 𝑆 at location 𝒙, and 𝐺𝑎 is the Green’s
function for the enclosure satisfying rigid walled boundary conditions
𝐺𝑎 (𝒙|𝒓) = ∑∞𝑖=1𝝋𝑖 (𝒙)𝝋𝑖 (𝒓)，𝛬𝑖 (𝜅𝑖2 −𝑘 2 +2𝑗𝜅𝑖 𝑘𝜉𝑖 )(4)where 𝜅𝑖 and 𝑘 are acoustic wave number of the 𝑖-th order acoustic mode frequency𝜔𝑖 and theanalysis frequency 𝜔 , 𝜉𝑖 is the modal damping ratio of i-th enclosure mode, Λ𝑖 is the volume
normalization factor of the 𝑖-th acoustic mode with Λ𝑛 = ∫𝑉 𝛗𝑛 (𝐫)∗ 𝛗𝑛 (𝐫) 𝑑𝐫, and the superscript “*”denotes the conjugate operator.
Suppose that the surface of the elastic structure is discretized into 𝑀 elements. Considering thefirst 𝑁-th order acoustic modes, and using the orthogonality between the acoustic modes, the APE of
enclosure can be expressed as the form of the normal velocity of the elastic structure surface (14)𝐸𝑝 = 𝒗𝑯 𝜴𝒗，(5)H(𝑀
where the superscript “ ” denotes the Hermite operator, 𝒗 is an× 𝑀) vector of normal velocity on
the discrete elastic surface, 𝜴 is an (𝑀 × 𝑀) error weighting matrix, and the (𝑚, 𝑛) -th element
expression is𝜴(𝑚, 𝑛) = ∑𝑁
𝑖=1

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𝜌0 𝑘 2 𝑠𝑚 𝑠𝑛
2

4𝛬𝑖 |𝜅𝑖2 −𝑘 2 +2𝑗𝜅𝑖 𝑘𝜉𝑖 |𝝋𝑖 (𝑥𝑚 )∗ 𝝋𝑖 (𝑥𝑛 )，(6)where 𝑠𝑚 and 𝑠𝑛 are the areas of the 𝑚-th and 𝑛-th discrete elements of the elastic structure surface
respectively.
The definition of error weighting matrix 𝛀 shows that 𝛀 is Hermitian. Additionally, from thephysic meaning of the APE 𝐸𝑝 in Equation 5, it can be deduced that 𝐸𝑝 > 0 will be always true
except the normal velocity vector equals 0. Therefore, 𝛀 is positive definite and Hermitian, and it canbe expressed by the eigenvalue decomposition as(7)
𝜴 = 𝑫𝑨𝑫𝐻 ，where 𝐀 is a diagonal matrix with eigenvalues 𝜆𝑖 (𝑖 = 1,2,⋅⋅⋅ 𝐼) decreasing monotonically along the
diagonal, 𝐃 represents an (𝑀 × 𝑀) matrix whose columns are eigenvectors d𝑖 (𝑖 = 1,2,⋅⋅⋅ 𝑀). Since𝛀 is positive definite and Hermitian, all of its eigenvalues 𝜆𝑖 are positive real and its eigenvectors d𝑖
are orthogonal to each other. Thus, it can be seen that eigenvectors d𝑖 are a set of basis functions in the
𝑀 dimensional space, so they’re termed as the Acoustic Radiation Modes of coupled enclosure
specifically. It can be found that no structural modal information of the elastic structure is required in
the calculation of ARM.
Similar to the ARM in free space, the ARM of coupled enclosure represents a particular velocity
pattern on the coupling surface. Both of them are dependent on the exciting frequency an d thegeometry of the flexible surface, but independent of the surface material and its boundary condition.
However, different from the ARM in free space, the ARM of coupled enclosure is dependent on thegeometry, natural frequencies and damp of the enclosure in addition. On this basis, the ARM theory of
coupled enclosure which is consistent with that of free space and convenient for application wasformed.

3. ACTIVE CONTROL STRATEGY
From Equation (7), it can be seen that eigenvectors d𝑖 (𝑖 = 1,2,⋅⋅⋅ 𝑀) are a set of basis functions in
𝑀 dimensional space, and then all arbitrary normal velocity 𝐯 on the elastic surface can be expanded
uniquely by the eigenvectors 𝐝𝐢 as(8)𝐯 = 𝐃𝐲 = ∑𝑀
𝑖=1 𝑦𝑖 𝐝𝐢 ，where 𝐲 is the ARM amplitude vector.
Substituting Equations (7,8) into Equation (5), the APE of coupled enclosure can be further written
as follows
𝑀
2
𝐸𝑝 = 𝐲 𝐻 𝐀𝐲 = ∑𝑀
(9)𝑖=1 𝜆𝑖 |𝑦𝑖 | = ∑𝑖=1 𝐸𝑝𝑖 ，It can be seen from Equation (9) that the APE of coupled enclosure can be decoupled by the ARMs,
that is, the APE can be reduced by control of an arbitrary order of ARM amplitude.
In ASAC, the primary problem is the determination of the dominant acoustic radiation modes(dARM). In most previous studies, dARMs are obtained only based on the radiation efficiency 𝜆𝑖 . As
can be seen from Equation (9), the APE is not only related to the radiation efficiency 𝜆𝑖 , but also
affected by the ARM amplitude |𝑦𝑖 |. Therefore, the conventional control strategy of ASAC is notcomprehensive enough. Combining radiation efficiency and the ARM amplitude, a new dARM
determination method is proposed in the Reference (15). Therefore, once the dARM contributed APE
is cancelled, ideal ASAC effect can be obtained. When the APE contributed by dARM is used as thecontrol target, an active control strategy based on the dARM may be designed.
Suppose there are 𝐺 dARMs 𝑔1 , 𝑔2 ⋯ , 𝑔𝐺 , and the dARM contributed APE can be written as2(10)
𝐸𝑝𝐺 = 𝐲𝐺𝐻 𝐀 𝐺 𝐲𝐺 = ∑𝐺𝑖=1 𝜆𝑔𝑖 |𝑦𝑔𝑖 | ，where 𝐀𝐺 = 𝑑𝑖𝑎𝑔(𝜆𝑔1 , 𝜆𝑔2 , ⋯ , 𝜆𝑔𝐺 ) is a diagonal matrix consisting of the radiation efficiency ofdARM, and 𝐲𝐺 is a ARM amplitude vector of 𝐺 dARMs, according to Equation (8),(11)
𝐲𝐺 = 𝐃𝐺 𝐯𝑛 ，
𝐝
𝐝
⋯
𝐝
(𝑀
[
]
where 𝐃𝐺 = 𝑔1
× 𝐺) dimensional matrix, with 𝐺 dARMs vectors in its
𝑔2
𝑔𝐺 is acolumns.
Considering the coupling effect of the acoustic enclosure, the normal vibration velocity of theelastic structure surface can be expressed as(12)𝐯𝑛 = 𝐓(𝐅 − 𝐩(𝑠))，where 𝐓 is the admittance matrix of the elastic structure without considering the coupling of theenclosure, 𝐅 is the excitation force vector applied to the elastic structure surface, and 𝐩(𝑠) is the4305surface acoustic pressure vector of the elastic structure in the enclosure. According to Equation (3),𝐩(𝑠) is expressed as matrix form is(13)𝐩(𝑠) = 𝐆𝐯𝑛 ，Inserting Equation (13) into Equation (12), the relationship between the normal vibration velocityof the elastic structure surface and the external force can be expressed as(14)
𝐯𝑛 = (𝐈 + 𝐓𝐆𝑠 )−1 𝐓𝐅 = 𝐓 ′ 𝐅，′−1where 𝐓 = (𝐈 + 𝐓𝐆𝑠 ) 𝐓 is the admittance matrix of elastic structure considering coupling effect.
Assuming that m primary excitation forces and n secondary control forces act together on theelastic structure of the coupled enclosure. According to the superposition principle, the normal
vibration velocity of the elastic structure surface can be written as
𝐯𝑛 = 𝐯𝑛𝑝 + 𝐯𝑛𝑠 = 𝐓𝑝′ 𝐅𝑝 + 𝐓𝑠′ 𝐅𝑠 ，
(15)
By substituting Equations (11, 15) into Equation (10), 𝐸𝑝 𝐺 can be written as a quadratic form of
the secondary control force complex amplitude
𝐸𝑝 𝐺 = 𝐅𝑠𝐻 𝐚𝐴 𝐅𝑠 + 𝐅𝑠𝐻 𝒃𝐴 + 𝒃𝐴𝐻 𝐅𝑠 + 𝐜𝐴 ，
(16)
where
′
(17)
𝐚𝐴 = (𝐓𝐬′ )𝐇 𝐃𝐆 𝐀𝐆 𝐃𝐇
𝐆 𝐓𝐬 ，′
𝐛𝐴 = (𝐓𝐬′ )𝐇 𝐃𝐆 𝐀𝐆 𝐃𝐇
𝐆 𝐓𝑝 𝐅𝑝 ，

(18)

𝐇
′
𝐜𝐴 = 𝐅𝑝𝐻 (𝐓𝑝′ ) 𝐃𝐆 𝐀𝐆 𝐃𝐇
𝐆 𝐓𝑝 𝐅𝑝 ，(19)
According to the quadratic optimal theory, the optimal secondary control force can be obtained byminimizing the dARM contributed APE
𝐅𝑠,𝑜𝑝𝑚 = −𝐚𝐴−1 𝐛𝐴 ，
(20)
The minimum value of the dARM contributed APE is
𝐸𝑝𝐺,𝑚𝑖𝑛 = 𝒄𝐴 − 𝒃𝐴𝐻 𝒂𝐴−1 𝒃𝐴 ，
(21)

4. SIMUALTION ANALYSIS
In the following, the effectiveness of the dARM contributed APE control strategy is analyzed. The
same numerical calculation model in Reference (7) is used here. The dimension of the coupledenclosure is given by 𝐿𝑥 × 𝐿𝑦 × 𝐿𝑧 = 0.868 𝑚 × 1.150 𝑚 × 1.000 𝑚, and its top surface consists of a
6mm-thick simply supported aluminum panel and other five surfaces are perfectly rigid walls. The
constants used in the calculation can also be found in reference 7. The primary excitation force is asingle simple harmonic force with a amplitude 𝐹𝑝 = 1𝑁, applied to the lower left corner (𝐿𝑥 ⁄8 , 𝐿𝑦 ⁄8)of the elastic plate to ensure that all the modes of the elastic panel in the analysis band can be excited;Three simple harmonic forces are arranged as secondary control forces, their amplitudes are denoted
as 𝐹𝑠1 , 𝐹𝑠2 and 𝐹𝑠3 , the excitation positions are (0.434𝑚, 0.575𝑚, 1.000𝑚) ,(0.434𝑚, 0.863𝑚, 1.000𝑚), and (0.651𝑚, 0.575𝑚, 1.000𝑚) respectively.

Figure 2 – First five order ARM contributed APE curves

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The first five order ARM contributed APE curves by the primary excitation force are given Figure
2, in which the dARM at a given frequency can be observed clearly.
Figures 3-5 show the APE curves of coupled enclosure before and after control when using one
secondary control force to cancel the APE contributed by the 1 st ARM (i.e. 𝐺 = 1and𝑔1 = 1) , using
two secondary control forces to cancel the APE contributed by the 1 st and 2 nd ARMs (i.e. 𝐺 = 2 and𝑔1 = 1, 𝑔2 = 2), and using two secondary control forces to cancel the APE contributed by the 2 nd and3 rd ARMs (i.e. 𝐺 = 2 and 𝑔1 = 2, 𝑔2 = 3) respectively.
As illustrated by Figure 2 and Figure 3, the APE is significantly reduced only in the frequency band
where the 1st ARM is the dARM, and there is almost no control effect or even control overflow inother frequency bands, such as 132 Hz (the dARM at this frequency is the fourth -order). Similarly,
from Figure 2 and Figure 4, when the 1st and 2nd ARMs are controlled, the APE is greatly reduced in thefrequency band where the 1 st and 2nd ARMs are the dARMs and there is no control effect in otherfrequency bands. Also, from Figure 2 and Figure 5, when 2 nd and 3 rd ARMs are controlled, significant
control effect is obtained in the frequency band where the 2 nd and 3 rd ARMs are the dARMs. But there
is no longer reduction effect or even a large control overflow in the frequency band where the 1 st ARM
is the dARM.
Through the above analysis, it can be concluded that by controlling the dARM contributed APE, asatisfactory control effect can be obtained. Conversely, if any dARM is not cancel led, it is usually
impossible to obtain the desired control effect. This shows that the proposed dARM based ASAC
strategy is effective.

Figure 3 – Results of controlling the 1st ARM using one secondary control force

Figure 4 – Results of controlling the 1st and 2nd ARMs using two secondary control forces4307

Figure 5 – Results of controlling the 2 nd and 3 rd ARMs using two secondary control forces.

5. EXPERIMENTAL RESEARCH
A fixed elastic plate-rectangular coupled enclosure model was designed to carry out experimental
research. The inner wall dimension of the rectangular acoustic enclosure is 𝐿𝑥 × 𝐿𝑦 × 𝐿𝑧 = 0.65𝑚 ×0.50𝑚 × 0.45𝑚.The upper surface is an elastic steel plate with thickness ℎ𝑝 = 0.0026𝑚, and the otherfive surfaces are steel plates with thickness ℎ𝑐 = 0.03𝑚 (equivalent to a rigid wall). The elastic steel
plate is fastened the rectangular enclosure by bolts, and is equivalent to the fixed boundary condition.
The experimental system is shown in Figure 6.
In the plane of upper surface, with the lower left corner of the surface as the coordinate origin, aHEV-50 vibration exciter is installed at the position (0.155𝑚 , 0.05m)as the primary excitation, andanother HEV-50 vibration exciter is at the position (0.325𝑚, 0.250𝑚) as the secondary force source.
The elastic plate is evenly meshed into 4 × 3 small area elements, 12 acceleration sensors are evenly
arranged in the center of the 4 × 3 elements to measure the normal vibration velocity, which is used todetermine the dARM and its amplitude (15). Four MPA416 microphones are arranged in the four
corners of the bottom of the rectangular cavity to measure the APE of coupled enclosure before andafter control (16). The adaptive controller is installed at the NI Compact RIO embedded measurement
and control platform, and the Filtered-x LMS algorithm is used as the adaptive control algorithm.

Figure 6 – Experimental system
The elastic plate (1,1) structure modal frequency 72 Hz was selected as the primary excitation
frequency. The 1 st ARM is determined as the dARM under the primary excitation by the method
proposed in the Reference (15). Then the control system is turned on to cancel the determined thedARM contributed APE. Figure 7 shows the time history of the dARM amplitude during control
process. The APE curves before and after control are compared in Figure.8. It can be seen that bycancelling the dARM contributed APE, the dARM amplitude is effectively reduced, and APE of4308

coupled enclosure also achieves a satisfactory control effect, the APE reduction is as high as 14.7 dB.
This further indicates that the ASAC strategy based on the dARM is effective.

Figure 7 – Time history of dARM amplitude during control

Figure 8 – The APE before and after control

6. CONCLUSIONS(1) By analogy to ARM theory of free space, the acoustic potential energy is expressed in quadratic
form of normal velocity on coupling surface and ARM of coupled enclosure is redefined, no structuralmodal information of the elastic structure is required in the calculation of ARM.
(2) Combining radiation efficiency and the ARM amplitude, a dARM based ASAC strategy is
proposed, which can guarantee satisfactory control effect. The simulation and experimental results
show that the control strategy is effective.

ACKNOWLEDGEMENTS
This article was supported by the National Natural Science Foundation of China (Grant No.
51675529).

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