Mechanisms of active control of noise transmission through triple-panel system using single control force on the middle plate

Applied Acoustics 85 (2014) 111–122

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Mechanisms of active control of noise transmission through triple-panel system using single control force on the middle plate Xiyue Ma ⇑, Kean Chen, Shaohu Ding, Haoxin Yu, Jue Chen School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, PR China

a r t i c l e

i n f o

Article history: Received 13 March 2013 Received in revised form 1 April 2014 Accepted 14 April 2014

Keywords: Triple-panel structure Active control Sound energy transmission Pass- and stop-band Physical mechanisms

a b s t r a c t This paper presents an active triple-panel sound insulation structure with an idealized controllable point force acting on the middle plate. A novel analytical approach based on sound energy transmission rule is proposed to achieve the physical mechanism study. The transfer impedance matrix of the incident and middle plate is calculated using numerical approach. And the rule of sound energy transmission through the triple-panel structure is concluded by indirectly analyzing the radiated sound power of the three plates. Finally the physical mechanism of noise insulation is investigated from the point of view of the change in behaviors of energy transmission in controlled and uncontrolled conditions. Results obtained demonstrate that there exist four different energy transmission paths for four panel mode groups. The energy transmission is independent in each path and they are all of band-pass characteristic. The role of the middle plate and two cavities is very similar to the band-pass filter whose pass-band is different for different mode groups. The essence of active noise insulation lies in the fact that the energy transmission in each path is suppressed in its pass-band after control. This greatly improves sound insulation capability of the triple-panel structure and leads to sound propagation being blocked. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Double-panel structure has superior sound insulation capability and has been widely used in various noise control engineering. Typical examples include aircraft or ship fuselage shells, grazing windows, partition walls in buildings, and so on. The sound insulation performance is better in high and middle frequency, but deteriorates rapidly in low frequency due to vibro-acoustic coupling effects. Traditional methods of low-frequency noise reduction require heavy damping material which leads to significant weight penalties and offsets the performance gains. Hence, active control technique is introduced thereby constituting the active doublepanel structure [1–6]. According to type of secondary sources and arrangement locations, the control strategy in existed investigations can be classified into two approaches, i.e., cavity control [1,2] and panel control [2– 6]. The cavity control approach can effectively block the noise transmission path and is useful to control broad-band noise or tonal noise with a variable frequency [7]. But arranging bulky sound sources in the air gap is always difficult, which makes the system unimplementable. Though vibration control of skin panel ⇑ Corresponding author. Tel.: +86 029 88474104. E-mail address: [email protected] (X. Ma). http://dx.doi.org/10.1016/j.apacoust.2014.04.014 0003-682X/Ó 2014 Elsevier Ltd. All rights reserved.

is more efficient than acoustic control for an excitation such as a turbulent boundary layer [8], but there also have some limitations which would impede their practical applications. Incident panel which may be the fuselage shells for the aircraft or ship is usually the heavy structure. Direct force actuation on it requires large amounts of energy and may cause structural fatigue. Also installation and repair of sensors and actuators would be extremely difficult since the fuselage shell is not removable. Direct force actuation on the radiating panel which is usually referred to as the interior trim panel, may lead to the control spillover in the form of significantly increased vibrational energy of this panel. This effect of control on increased sound field close to the panel will go against certain applications such as in aircraft where passengers have to sit close to fuselage trim panels. The active triple-panel structure with controllable point forces acting on the middle plate can effectively eliminate these disadvantages. And the physical characteristics of the middle plate can be designed specially to guarantee it being easily actuated [9]. The physical mechanism study has attracted considerable attention in the field of active control. Exploring the mechanism of noise reduction cannot only provide help for understanding the physical nature inherent in the system, but also offer guidance for system design such as optimal arrangement of secondary sources, implementation of error sensing strategy, and so on. In

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past two decades, much effort has been concentrated on the mechanism study for active control of sound transmission through a double-wall [10,11]. It is well known that there basically exist two mechanisms no matter what type of control strategy is used. For cavity control approach, one mechanism relies on modal suppression of the cavity modes which attenuates cavity sound field. Consequently, this reduces the vibration of the radiating panel. Another involved is the modal rearrangement of cavity modes so as to change vibration pattern of the radiating panel to form a weaker radiator. For panel control approach, there are of similar control mechanisms. The research involved in active structure acoustic control of single plate [12,13] and active control of noise transmission through a panel into a cavity [14,15] also demonstrates these mechanisms. Usually the two phenomenons coexist. Therefore, the interaction of not only the amplitudes but also the phases of these structural or acoustic modes must be considered to achieve an understanding of noise reduction. And if we consider the power radiated by any one structural mode, we must take into account the amplitudes and phases of all other structural or acoustic modes since they are interdependent. Hence, it is very difficult to gain a clear understanding of how the structural or acoustic modes should be altered to achieve noise reduction. Moreover, the coupling relations for the triple-panel structure will become more complicated after the middle plate is introduced. Therefore, it will make the analytical process incomprehensible if the above two phenomenons are still used here for the triple-panel case. A more effective method for single plate is using the concept of radiation modes [16]. Upon applying control, the amplitudes of a limited number of the most efficient radiation modes are reduced, which leads to the weakened radiated sound of the plate [17]. Another method is the wave-number approach. The response of the supersonic region in wave number domain, which represents the plate vibration parts that radiates sound into far field, is significantly reduced with control [18,19]. The two approaches are usually introduced to facilitate interpretation of the relationship between the structure response and the corresponding acoustic response of the plate. It may be also valid for the double- or triple-panel case, but can hardly explore the control nature inherent in the energy transmission process. Hence, it will be of limited usefulness. A novel approach exploring the mechanism from the change in behaviors of energy transmission is proposed to make the analytical process clear and intuitive. Concerned with the research on sound transmission characteristics, the existed literatures have been mainly focused on the sound insulation of double-panel system [20–22], and multiplelayer structure with complicated mechanical links [23,24] or specific boundary conditions [25]. The emphasis was concentrated on the total sound insulation of the system, but the rule of energy transmission occurred in the process is little understanding. However, this will be the necessary preconditions for the mechanism study of the active triple-panel structure and will also be helpful for optimizing various passive control techniques [26,27]. Hence, the emphasis is firstly put on the analysis of the rule of sound energy flow. An indirect approach adopted is to analyze the radiated power of each plate. Compared with the plate radiating sound into an acoustic free field, the calculation of radiated power for the incident and middle plate is relative complicated since they radiate sound into a rectangular enclosed space. One of effective methods is so called discrete elemental approach [28,29]. It separates the plate into a number of elemental radiators whose dimensions should be much less in comparison to acoustic wavelength corresponded to the upper limit frequency of interest. The total power output is the sum of the net power of all elemental radiators. The main issue encountered for the incident and middle plate is to calculate the transfer

impedance matrix. The impedance or mobility matrix functions have been used in [30,31] to express the relation between the forces acting on elements and the corresponding vibration of the plate at center of elements in enclosed space. Referred to this, the numerical approach is introduced to calculate the transfer impedance matrix of the incident and middle plate. The objective of the paper is to investigate the physical mechanism of the active triple-panel system. The main contribution is the analysis of control mechanism from the view point of energy transmission in a clear and intuitive way. The paper is organized as follows: the theoretical model is established in Section 2. After the transfer impedance matrix is calculated in Section 3, the rule of sound energy flow is investigated as an important part in Section 4. The physical mechanism is analyzed in Section 5. The designing of the middle plate in applications are discussed in Section 6. Finally, concluding remarks are provided in Section 7.

2. Theoretical model 2.1. Vibrational response of the system Fig. 1(a) shows an active triple-panel sound insulation structure. Its profile is shown in Fig. 1(b). Name the panel a, b, and c as an incident plate, a middle plate and a radiating plate, respectively. Assume that the three plates are simply supported and baffled in an infinite wall. Let lx and ly denote the length and wide, respectively. ha, hb and hc are their corresponding thickness. The depths of two cavities are h1 and h2, and the medium in the cavity is air with density q0 and sound speed c0. Except for the three flexible plates, other surrounding walls of the two cavities are all acoustically rigid. The primary excitation is assumed to be an oblique incident plane wave. In general, the total pressure acting on the incident plate can be decomposed into three parts, i.e., the incident pressure, the reflected pressure when the plate is assumed rigid, and the radiated pressure. The incident and reflected pressure magnitudes can be assumed equal since the impedance of the plate approximates a rigid boundary for air loading. And it was also found that the radiated pressure is rather low compared to the

(a)

P0

y

Infinite rigid baffle

x Two cavities

Simply supported plate a

z

Simply supported plate b Simply supported plate c

(b)

x

ha h1 h2

hb

Cavity1 Cavity2

a Fs

Infinite baffle

b c

hc

z Fig. 1. Systemic model of the active triple-panel structure. (a) Systemic sketch; (b) systemic profile.

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other two components for the fluid of air (also called light fluid loaded case). By neglecting its influence, the excitation pressure on the incident plate is twice the magnitude of incident wave, which is known as the blocked pressure [6,22]. In light of modal superposition theory, the displacement of each plate wi(i = a, b, c) can be decomposed over their respective mode shape functions, i.e.,

wi ðx; y; tÞ ¼

X

q ðtÞ/i;m ðx; yÞ m i;m

ði ¼ a; b; cÞ:

ð1Þ

where /i,m(x, y) and qi,m(t) are the mode shape function and instantaneous modal displacement of the mth mode. Using the equation of motion and the orthogonality of mode shape functions, yields the relations that the modal displacement qi,m(t)(i = a, b, c) satisfy

€a;m ðtÞ þ 2na;m xa;m q_ a;m ðtÞ þ x2a;m qa;m ðtÞ M a;m ½q ¼ Q pm ðtÞ  q0 c20 A

N1 X P1n ðtÞ n¼1

M 1;n

L1nm ;

The combination of Eqs. 2, 3, 4, (6), and (7) gives the coupled differential equations which represents the vibro-acoustic behavior of the system. The equations are resolved in frequency domain by performing Fourier transform on both sides of the equations. Substituting the expressions of the modal displacements qi,m(t)(i = a, b, c) derivated from Eqs. (2)–(4) into (6) and (7), yields the relations that modal pressures Pi,n(x)(i = 1, 2) satisfy, M1 N1 X X V1 P1n ðxÞ  B1m ðxÞL1nm L1nm P1n ðxÞ þ M1n H1n ðxÞA m¼1 n¼1 ! M2 N1 X X P1n ðxÞ B2m ðxÞL2nm L2nm þ M1n m¼1 n¼1 ! M2 N2 X X P2n ðxÞ B2m ðxÞL2nm L3nm  M2n m¼1 n¼1

ð2Þ ¼

M1 X

A1m ðxÞL1nm Q pm ðxÞ þ

m¼1

€b;m ðtÞ þ 2nb;m xb;m q_ b;m ðtÞ þ x2b;m qb;m ðtÞ M b;m ½q N1 N2 X X P1n ðtÞ P2n ðtÞ L2nm  q0 c20 A L3nm  Q sm ðtÞ; ¼ q0 c20 A M M 2;n 1;n n¼1 n¼1

ð3Þ

M2 X

B2m ðxÞL3nm

m¼1

 ð4Þ

where Mi,m(i = a, b, c) is the generalized modal mass of the mth plate R mode, Mi;m ¼ qh A ð/i;m Þ2 ds. xi,m(i = a, b, c) and ni,m(i = a, b, c) are its natural frequency and damping ratio, respectively. Mi,n(i = 1, 2) is the generalized modal mass of the nth acoustic mode, R Mi;n ¼ 1=V i V i ðui;n Þ2 dV. L1nm (or L2nm) is the coupling coefficient between the nth acoustic mode of cavity 1 and the mth mode of panel a (or b). L3nm (or L4nm) is the coupling coefficient between the nth acoustic mode of cavity 2 and the mth mode of panel b R (or c), Lnm = 1/V A/munds. Qpm(t) is the generalized primary modal force, and its expression in blocked pressure assumption can be referenced in Ref. [7]. The harmonic control force can be expressed as Fs = fsd(r  rc)ejxt. Then Qsm(t), which is the generalized secondary R modal force, can be expressed as Qsm(t) = A/m(x, y)FsdA. Note that fs is the control force amplitude, and rc is the acting location. Similarly, the sound pressure in two cavities can also be decomposed on the basis of acoustic mode shape functions [32], i.e.,

X pi ðx; y; z; tÞ ¼ Pi;n ðtÞui;n ðx; y; zÞ ði ¼ 1; 2Þ:



L2nm

N2 X P2n ðxÞ

M2n

n¼1

M3 X

N2 X P2n ðxÞ

B3m ðxÞL4nm

M2 X

n¼1

M2n

V2 P2n ðxÞ H2n ðxÞA !

þ L3nm

! L4nm

A2m ðxÞL3nm Q sm ðxÞ ðn ¼ 1; 2; . . . N 2 Þ:

ð9Þ

m¼1

where N1 (or N2) and M1(or M2, M3) is the upper limit number of the acoustic modes and plate modes considered in simulation. Define the following variables:

x2

Hin ðxÞ ¼

ði ¼ 1; 2Þ;

x  x þ 2jnin xin x 2 in

Him ðxÞ ¼

Aim ðxÞ ¼

2

1

x2im  x2 þ 2jnim xim x Him ðxÞ M im

ði ¼ a; b; cÞ;

ði ¼ a; b; cÞ;

q0 c20 AHim ðxÞ Mim

ð10Þ

ð11Þ

ð12Þ

ði ¼ a; b; cÞ:

ð13Þ

Then if the following terms are also defined

C 1 ði; jÞ ¼ 

€ 1n ðtÞ þ 2n1n x1n P_ 1n ðtÞ þ x2 P 1n ðtÞ P 1n

þ

M1 X V1 B1m ðxÞL1im L1jm dði  jÞ þ AH1i ðxÞ M 1j m¼1 M2 X B2m ðxÞL2im L2jm m¼1

ð6Þ

M 1j

ði; j ¼ 1; 2; . . . N 1 Þ;

ð14Þ

M2 X B2m ðxÞL2im L3jm D1 ði; jÞ ¼  M2j m¼1

€ 2n ðtÞ þ 2n2n x2n P_ 2n ðtÞ þ x2 P 2n ðtÞ P 2n M2 M3 A X A X €b;m ðtÞL3nm  €c;m ðtÞL4nm : q q V 2 m¼1 V 2 m¼1

M1n

!

m¼1

Bim ðxÞ ¼

where ui,n(x, y, z) and Pi,n(t) are the mode shape function and instantaneous modal pressure of the nth acoustic mode. Using the homogeneous wave equation for the sound field in two cavities, the Green’s function and orthogonality of mode shape functions, yields the relations that the modal pressure Pi,n(t)(i = 1, 2) satisfy

¼

B2m ðxÞL3nm

m¼1

¼

ð8Þ

N1 X P1n ðxÞ n¼1

M2 X

ð5Þ

n

M1 M2 A X A X €a;m ðtÞL1nm  €b;m ðtÞL2nm ; q q ¼ V 1 m¼1 V 1 m¼1

A2m ðxÞL2nm Q sm ðxÞ

m¼1

ðn ¼ 1; 2; . . . N1 Þ;

€c;m ðtÞ þ 2nc;m xc;m q_ c;m ðtÞ þ x2c;m qc;m ðtÞ M c;m ½q N2 X P2n ðtÞ ¼ q0 c20 A L4nm : M2;n n¼1

M2 X

!

ði ¼ 1; 2; . . . ; N1 ; j ¼ 1; 2; . . . ; N2 Þ;

ð7Þ

where x1n (or x2n) and n1n (or n2n) are the natural frequency and damping ratio of the nth acoustic mode, respectively. V1 and V2 are the volumes of the two cavities, and A is the plate’s surface area.

E1 ðiÞ ¼

ð15Þ

M1 M2 X X A1m ðxÞL1im Q pm ðxÞ þ A2m ðxÞL2im Q sm ðxÞ m¼1

ði ¼ 1; 2; . . . ; N1 Þ;

m¼1

ð16Þ

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X. Ma et al. / Applied Acoustics 85 (2014) 111–122

C 2 ði; jÞ ¼

M2 X B2m ðxÞL3im L2jm

M1j

m¼1

ði ¼ 1; 2; . . . ; N2 ; j

¼ 1; 2; . . . ; N1 Þ; D2 ði; jÞ ¼

ð17Þ

M2 X V2 B2m ðxÞL3im L3jm dði  jÞ  AH2i ðxÞ M 2j m¼1



M3 X B3m ðxÞL4im L4jm

M 2j

m¼1

ði; j

M2 X

ð18Þ

A2m ðxÞL3im Q sm ðxÞ ði ¼ 1; 2; . . . ; N2 Þ:

ð19Þ

m¼1

Eqs. (8) and (9) can be expressed in matrix form as



C1 C2

D1 D2



P1 P2



 ¼

0

0

¼ 1; 2; . . . N2 Þ; E2 ðiÞ ¼

1 11 0 L3;1M2 L3;11 B B . C B . CC C CC B B G3 ¼ B @ A21 @ .. A    A2M2 @ .. A A; L3;N 1 L3;N2 M2 0 0 2 1 0 11 A31 L4;N2 1 A31 L4;11 B A B C B CC .. .. A B C    pffiffiffiffiffiffiffiffi B C C; G4 ¼ B . . @ pffiffiffiffiffiffi A AA M 2N @ M 21 @ 2 A3M3 L4;1M3 A3M3 L4;N2 M3 0

 E1 : E2

ð20Þ

where P1 ¼ ½P11 ðxÞ; P12 ðxÞ; .. . ; P1N1 ðxÞT and P2 ¼ ½P21 ðxÞ;P 22 ðxÞ;...; P2N2 ðxÞT are the vectors of the modal pressure of the two cavities, which can be derived using Eq. (20). Then qi,m(t)(i = a, b, c) can be accordingly derived by substituting P1 and P2 into Eqs. (2)–(4). The response of the coupled system without control can also be obtained by resolving Eq. (20). But E1 and E2 should be modified in this case by setting Fs = 0 in Eqs. (16) and (19). 2.2. Optimal secondary control force The optimal cost function in principle is the radiated power of the plate c. The stated objective can achieve maximal improvement of sound transmission loss for the triple-panel structure. According to the discrete elemental approach, with the plate being evenly divided into N elements, its power output can be expressed as [17]:

u1 ðx1 ; y1 Þ    uM3 ðx1 ; y1 Þ

1

C .. C; A .   u1 ðxN ; yN Þ    uM3 ðxN ; yN Þ 1 0 q0 c20 p ffiffiffiffiffiffi C B M21 C B C B . K2 ¼ B C; .. C B @ 2 q0 c 0 A pffiffiffiffiffiffiffiffi

B W ¼ jxB @

M2N

2

U2 ðrc Þ ¼ ½u1 ðrc Þ; u2 ðrc Þ; . . . ; uM2 ðrc ÞT . Substituting Eqs. (22) into (21), the radiated power can be rewritten as H

W c ¼ ða þ bfs Þ Rc ða þ bfs Þ:

ð25Þ

Given that the matrix Rc is real, symmetrical and positive definite, Eq. (25) has a Hermitian quadratic form. It will have a unique minimal value when fs = (bHRcb)1bHRca by using the linear quadratic optimal method. Accordingly the maximum achievable sound attenuation of the triple-panel structure can be achieved. This has only a theoretical significance because it is very difficult to obtain a measure of the radiated power in a real control system. 3. Surface transfer impedance and radiated sound power

where H denotes complex conjugate transpose, Vc is the velocity vector for an array of N elemental radiators. Rc = DSRe(Zc)/2, Zc is the N  N transfer impedance matrix. The velocity vector Vc derived from Eqs. (4) and (20) can be expressed as

An indirect approach adopted for investigating energy transmission rule is to analyze the radiated power of each plate. According to the discrete elemental approach, the transfer impedance matrix should be obtained above all. The numerical approach is proposed for this purpose in the enclosed sound field case. Then the formulation for calculating the radiated power of each plate is given.

Vc ¼ a þ fs b;

3.1. Surface transfer impedance

W c ¼ VHc Rc Vc :

ð21Þ

ð22Þ

where

a ¼ WG4 K2 X21 G1 Q p ;

ð23Þ

b ¼ WG4 K2 ðX21 G2 þ X22 G3 ÞU2 ðrc Þ: 

ð24Þ 

C1 D1 in Eq. (20) is prior known, and C2 D2   X11 X12 ; which can be obtained by its inverse is assumed as X21 X22 inverse operation. The variable fs in Eq. (22) is the control force amplitude, which is introduced in the term Qsm(t). In Eqs. (23) and (24), other matrix are defined as follows The coefficient matrix

0

0

B B B B G1 ¼ B A11 B @ @ 0

0

B B B B G2 ¼ B A21 B @ @

L1;11 .. . L1;N1 1 L2;11 .. . L2;N1 1

1

0

C B C B C    A1M1 B A @ 1

Much work [16,32] has been contributed to calculating the transfer impedance in the free field case. For the plate c, according to the physical significant of the transfer impedance matrix Zc, Zc(i, j) can be expressed as [16]:

0

C B C B C    A2M2 B A @

L1;1M1 .. . L1;N1 M1 L2;1M2 .. . L2;N1 M2

11 CC CC C C; AA 11 CC CC C C; AA

Z c ði; jÞ ¼

8 < jq0 c0 kDSejkrij :

2pr ij

q0 c0 ð1  e

pffiffiffiffiffiffiffiffi

jk

DS=p

i–j

:

ð26Þ

Þ i¼j

where DS is the elemental area, k = x/c0 is the acoustic wave number, and rij is the distance between the center of the ith and jth element. Let Za and Zb denote the transfer impedance matrix of the plate a and b, respectively. According to the physical significance of Za (or Zb) [32], Za(i, j) (or Zb(i, j)) is the ratio between the jth elemental complex sound pressure produced by the ith vibrating element, and the ith elemental velocity v(i), i.e., Za(i, j) = P(j)/v(i). The expressions for Za(i, j) and Zb(i, j) as in the case of free field are difficult to obtain since the ith element locates in the enclosed sound field. In order to calculate Za(i, j), the complex sound pressure at the center of the jth element needs to be obtained when only the ith element vibrates with velocity v(i). The plate a with only the ith element vibrating can be equaled as a combination of a rigid wall

X. Ma et al. / Applied Acoustics 85 (2014) 111–122

Cavity1

x

i

3.2. Radiated sound power

Cavity2

Vibrating plate b

As for the discrete elemental approach, the sound radiation from a surface can be approximated by a number of elemental radiators which are oscillating harmonically. The sound power radiated by the ith element will be due to its complex pressure pi and velocity vi at that elemental position [17]:

Vibrating plate c

p(j)

Equivalent Point source

Wi ¼ Z

Rigid plate a

Fig. 2. The equivalent system for calculating the transfer impedance matrix of the plate a.

and a point source with its location in the center of the ith element. Then it translates into calculating the jth elemental sound pressure in a coupled system under the point source excitation. The corresponding coupled system consists of the rigid plate a, the vibrational plate b and c, and the two cavities, as shown in Fig. 2. The instantaneous modal pressure P1n(t) of the cavity 1 satisfies the following relation

DSi Reðv i pi Þ: 2

¼

u1;n ðxi ; yi ; 0Þ _ A QðtÞ 

2 1n P 1n ðtÞ M2 X

€b;m ðtÞL2nm : q V 1 m¼1

V1

ð27Þ

W¼

DS ReðVH PÞ: 2

Cavity2

x

i

Vibrating plate c

Equivalent

p(j)

Point source

Rigid wall b

Z

Fig. 3. The equivalent system for calculating the transfer impedance matrix of the plate b.

ð29Þ

Theoretically there is no restriction on the sound field for calculating the power output of the three plates when using Eq. (29). The sound pressure vector Pi is the function of the normal velocity Vi

ði ¼ a; b; cÞ:

ð30Þ

where i denotes the incident, middle or radiating plate. Substituting Eqs. (30) into (29) produces an expression for the radiated power purely as a function of the velocity

W i ¼ VHi Ri Vi

where Q(t) = DSv(i, t) is the intensity of the equivalent point source. The equation that P2n(t) satisfies is similar with Eq. (7), and the equations that qb,m(t) and qc,m(t) satisfy are also similar in structure to Eqs. (3) and (4). Eqs. (27), (7), (3), and (4) describe the vibroacoustic behavior of the coupled system. Za(i, j) can be obtained when the sound pressure of the jth element is calculated by resolving the coupled equations. The ith row’s elements of Za can be obtained when all elemental sound pressure are calculated. Other rows can also be obtained using the same steps when other elements are orderly chosen as the vibrating source. The approach for calculating Zb is similar with that adopted for Za. The plate b with only ith element vibrating is also equaled as the combination of a rigid wall and a point source. Then the sound pressure at the jth element is also calculated by resolving the response of the coupled system. Such system consists of the rigid wall b, the vibrational plate c and the cavity 2 in this case, as shown in Fig. 3. The elements of Zb can be obtained in a similar way as that used for Za, in which the detailed process is omitted for brevity.

ð28Þ

where DSi is the elemental area, Re(  ) represents the real part of the bracketed quantity, and denotes conjugation. Then the total radiated power should be the sum of the net power of all the elements. Assume that P and V are the N-length sound pressure and normal velocity vectors for an array of N elements, then the net power output to one side can be written as

Pi ¼ Z i V i € 1n ðtÞ þ 2n1n x1n P_ 1n ðtÞ þ x P

115

ði ¼ a; b; cÞ:

ð31Þ

where Ri = (DS/2)Re(Zi) is the radiation matrix, which is proportional to the real part of Zi. The physical significance for Eq. (31) is that it can be viewed that the plate a radiates sound into an equivalent medium which consists of the right side parts of the plate a. The quantity of radiated power represents the capability of transmitting sound energy into the equivalent medium. For the plate b, it radiates sound into another equivalent medium, which consists of the cavity 2, the plate c and the free field on right side of the plate c. The following analysis for energy transmission rule is just carried out based on this physical significance. 4. Rule of sound energy transmission 4.1. Model parameters Assume that three plates are made of aluminum with density

q = 2790 kg/m3, Young’s modulus E = 7.2  1010 N/m2, Poisson’s ratio r = 0.34, and structural damping ratio ni,m = 0.005. Their length and wide are lx = 0.6 m and ly = 0.42 m, and their thickness are ha = 0.002 m, hb = 0.003 and hc = 0.004 m. The depths of the two cavities are h1 = 0.2 m and h2 = 0.2 m. The medium in the cavity is air with density q0 = 1.21 kg/m3, sound speed c0 = 340 m/s, and acoustical damping ratio ni,n = 0.001. N = 10  10 is the number of elements. The primary excitation is a plane wave with amplitude P0 = 1 Pa, incident at h = p/4 and a = p/4. The secondary point force locates at (0.1lx, 0.1ly) so as to excite most of the plate modes contained in a wide frequency range. In general, the accuracy of the solution can be satisfied by increasing the number of modes until the convergence is achieved in the frequency range of interest. It is admissible that once the solution is convergent at a given frequency, it is also convergent for all frequencies lower than that [33]. After a careful convergence study the necessary number of modes is determined by the highest frequency of interest. Fifty plate and cavity modes are sufficient. A table of the first few uncoupled natural frequencies of the three plates and two cavities is listed in Table 1, which would help

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X. Ma et al. / Applied Acoustics 85 (2014) 111–122

Table 1 Uncoupled natural frequencies of the three plates and two cavities. Plate modes

(1, 1) (2, 1) (1, 2) (3, 1) (2, 2) (3, 2) (4, 1) (1, 3) (2, 3) (4, 2) (5, 1) (3, 3) (5, 2) (1, 4) (4, 3) (2, 4) (6, 1)

Uncoupled natural frequencies (Hz) Plate a

Plate b

Plate c

41.0 81.5 123.6 149.0 164.1 231.6 243.4 261.3 301.8 326.0 364.9 369.3 447.5 454.1 463.7 494.5 513.3

61.5 122.2 185.5 223.5 246.2 347.4 365.1 392.0 452.7 489.0 547.3 553.9 671.2 681.1 695.6 741.8 770.0

82.1 163.0 247.3 297.9 328.2 463.2 486.8 522.6 603.6 652.1 729.7 738.5 894.9 908.1 927.4 989.1 1026.5

Fig. 4. The radiated sound power of the three plates. —— Radiated by the plate a; radiated by the plate b; ———— radiated by the plate c.

navigating the plots of spectrum and giving an idea of the strength of coupling effects for the following analysis. 4.2. Sound energy transmission The curves of radiated sound power of the three plates calculated by Eq. (31) are plotted in Fig. 4 as a function of frequency. It is concluded that the sound energy flow decreases gradually when the sound propagates onwards and there has energy transmission loss in the transmitting process. A small part of the transmission loss is due to the damping of the three plates and the medium of air in two cavities which dissipates the sound energy. The majority of the transmission loss may store in the three plates and two cavities because of the low energy transfer efficiency between the plate and the cavity which is correlated with the coupling strength of the coupled mode pairs. The root causes for the sound energy transmission between the plate and the cavity is due to the coupling effect in the coupled mode pairs. The occurrence of energy transmission only exists in the coupled panel and cavity mode. According to Pan’s work [34], the coupling of the panel modes and the cavity modes is very selective. One panel mode can only couple with these cavity modes that have the opposite parity of mode index. As far as the triple-panel structure is concerned, it can be concluded that the sound energy transmits among the same type of mode groups in the three plates. Consequently there form four equivalent energy transmission paths for four types of panel mode group. According to different

Cavity modes (cavity 1 or 2)

Uncoupled natural frequencies (Hz)

(0, 0, 0) (1, 0, 0) (0, 1, 0) (1, 1, 0) (2, 0, 0) (2, 1, 0) (0, 2, 0) (0, 0, 1) (3, 0, 0) (1, 2, 0) (1, 0, 1) (0, 1, 1)

0 286.7 409.5 499.9 573.3 704.6 819.0 860.0 860.0 867.8 906.5 952.5

parity of mode index, the panel mode can be categorized into four groups, i.e., (odd, odd), (even, odd), (odd, even) and (even, even) mode group. In the low frequency range of 0–500 Hz, the four mode groups in the three plates couples mainly with the following four acoustic modes of the two cavities, i.e., (0, 0, 0), (1, 0, 0), (0, 1, 0) and (1, 1, 0), respectively. That is to say, the energy contained in the (odd, odd) mode group in the plate a transmits into the (odd, odd) mode group in the plate b and finally into the (odd, odd) mode group in the plate c, duo to the coupling effects between this mode group and the (0, 0, 0) acoustic mode. Similarly, the energy contained in other three mode groups in the plate a also transmits into the same mode groups in the plate b and subsequently into the same mode groups in the plate c. Only the first four acoustic modes are considered in the analysis because of their well couplings with the plate modes in the low frequency range of interest. And other high order acoustic modes couple weakly with these plate modes contained below the upper frequency limit, thereby ignoring their influence. The four equivalent energy transmission paths are summarized as: (1) Coupled with the (0, 0, 0) acoustic mode, energy contained in (odd, odd) mode group in the plate a transmits into this mode group in the plate b and finally into this mode group in the plate c. (2) Coupled with the (1, 0, 0) acoustic mode, energy contained in (even, odd) mode group in the plate a transmits into this mode group in the plate b and finally into this mode group in the plate c. (3) Coupled with the (0, 1, 0) acoustic mode, energy contained in (odd, even) mode group in the plate a transmits into this mode group in the plate b and finally into this mode group in the plate c. (4) Coupled with the (1, 1, 0) acoustic mode, energy contained in (even, even) mode group in the plate a transmits into this mode group in the plate b and finally into this mode group in the plate c. Moreover, the sound energy should transmit independently in each path because of the selectivity of the mode coupling. This will be demonstrated in the following section. 4.3. Band-pass characteristic of each transmitting path An indirect approach to explore the specific energy transmission rule in each path is to analyze the radiated sound power of

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the four mode groups in the three plates. No matter what type of sound field the plate radiates into, the radiated power of one single mode should be the sum of the self and mutual radiated power. The mutual radiation is induced by its coupling with other plate modes with the same type. The total power output of one mode group will not be affected by other mode groups because of there being no coupling effects. Hence, it should be logical to make the assumption that the four mode groups radiate sound independently, and the total sound radiation is the sum of the power output of the four mode groups. According to the modal superposition theory, the total velocity distribution of the four mode groups should be the sum of the velocity of all modes with the same type. They are denoted as Vi,oo, Vi,eo, Vi,oe and Vi,ee, which are N-length column vectors for an array of N elements and can be the velocity distribution either before or after control. These velocities can be expressed as

Vi;oo ¼ v i;ð1;1Þ þ v i;ð1;3Þ þ    þ v i;oo þ    ;

ð32Þ

Vi;eo ¼ v i;ð2;1Þ þ v i;ð2;3Þ þ    þ v i;eo þ    ;

ð33Þ

Vi;oe ¼ v i;ð1;2Þ þ v i;ð3;2Þ þ    þ v i;oe þ    ;

ð34Þ

Vi;ee ¼ v i;ð2;2Þ þ v i;ð2;4Þ þ    þ v i;ee þ    :

ð35Þ

where vi,oo, vi,eo, vi,oe and vi,ee are also N-length column vectors which denote the velocity distribution of single mode. With the assumption of the four mode groups being the orthogonal contributors, their respective total radiated power can be expressed as

W i;oo ¼ VHi;oo Ri Vi;oo ;

ð36Þ

W i;eo ¼ VHi;eo Ri Vi;eo ;

ð37Þ

W i;oe ¼ VHi;oe Ri Vi;oe ;

ð38Þ

W i;ee ¼ VHi;ee Ri Vi;ee :

ð39Þ

For the plate c, it is evident that one mode group uncouples with other groups because the plate radiates sound into free fields. The four mode groups are just the four radiation clusters mentioned in Ref. [35]. It has been demonstrated that each of the radiation clusters uncouples with the other cluster and therefore the clusters are orthogonal contributors, i.e., independent contributors. As for the case of plate radiating sound into enclosed space, Kaizuka’s work [36] also shows that the four radiation clusters which represent the four mode groups are also the orthogonal contributors with respect to the global potential energy in the enclosed sound field. But for the plate a (or b), the model is slight different from the above literature since the enclosure has two elastic plates. Here, a simplified approach for the demonstration of the orthogonality of the four mode groups in the plate a (or b) is proposed to further validate the above conclusions. Take the (odd, odd) mode group in the plate a for example, we define the following expression

W 0a;oo ¼

DS ReðVHa;oo Pa Þ: 2

ð40Þ

where Va,oo is the velocity distribution of this mode group, Pa is the sound pressures in vicinity of the N element in cavity 1. Eq. (40) represents the radiated power of the (odd, odd) mode group, which takes the other mode groups’ coupling effect, if it has, into consideration. If the radiated power calculated by Eq. (36) which does not consider the coupling effect is in good agreement with that calculated by Eq. (40), it will illustrate that this mode group is an independent contributor with respect to the other three mode groups. A lot of simulations have been accomplished, and only the result for

117

Fig. 5. The radiated sound power of the (odd, odd) mode group in the plate a. —— calculated by Eq. (41); calculated by Eq. (37).

(odd, odd) mode group is listed here as an example to demonstrate the orthogonality of the four mode groups, as shown in Fig. 5. The curves of radiated sound power of the four mode groups in the three plates are plotted in Fig. 6 as a function of frequency. The radiated power of the four mode groups decreases gradually in the low frequency range as the sound energy transmits orderly from the plate a to the plate c. It illustrates that the sound energy contained in the four mode groups in the plate a attenuates gradually in the transmitting process. However, the attenuation is different for different mode groups. As for the (odd, odd) mode group, the level of the energy attenuation is smaller in the frequency range of 0–230 Hz than that in the residual frequency band, when the energy transmits into the plate b and orderly into the plate c. The level of attenuation is almost up to 60 dB in the residual frequency band, as shown in Fig. 6(a). Herein, the frequency band for 0–230 Hz is referred to as an equivalent pass-band for the (odd, odd) mode group. And the residual frequency range in which the attenuation is relative large is referred to as a stop-band. For the (even, odd) mode group, it is shown in Fig. 6(b) that the sound energy attenuation is relatively small in the frequency band from 230 Hz to 380 Hz. Therefore this frequency band is referred to as the pass-band. The attenuation is very large in the neighboring frequency bands which are therefore referred to as the stop-band. For the (odd, even) mode group, the pass-band is the frequency range from 380 Hz to 470 Hz, and the residual neighboring frequency bands are the stop-band, as shown in Fig. 6(c). Finally for the (even, even) mode group, only the small band of 470–500 Hz is the pass-band and the rest frequency band is the stop-band, as shown in Fig. 6(d). Note that the division of the pass- and stop-band is not rigorous in principle, and the purpose here is to qualitatively illuminate the energy transmission rule. The pass-bands for these mode groups are shown in Fig. 6(a–d), which are the parts with a grey background color. The sound energy contained in each mode group does not transmit in the whole frequency range of interest. The corresponding pass-bands for different transmission paths occupy different frequency range. Besides, the pass-band mentioned does not represent the sound energy all transmitting through the system without any loss, and merely represents that the level of attenuation is relative small. On the contrary, the energy transmitting in the stop-bands is attenuated by up to 60 dB, and hence, the energy transmitting into the plate c is much small. This transmission characteristic illustrates that the three plates and two cavities behave analogously with the band-pass filter whose pass-band is different for different mode groups. 4.4. Discussion The main reason for the band-pass characteristics is due to the different coupling strength in different coupled mode pairs. It is

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Fig. 6. (a–d) Are the radiated sound power of the four mode groups in the three plates, (a) the (odd, odd) mode group; (b) the (even, odd) mode group; (c) the (odd, even) mode group; (d) the (even, even) mode group. —— In the plate a; in the plate b; ————— in the plate c. The filed with a grey background color represents the passband, and the rest fields are the stop-band.

well known that the energy transfer is larger in the strongly coupled mode pairs than that in the weakly coupled. In order to quantitatively describe the coupling strength and energy transfer efficiency, Pan’s work [34] introduced the following energy transfer factor

"

F m;n

ðxc;n  xp;m Þ2 ¼ 1þ Bðm; nÞ2 4

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 c20 Lp;c : Bðm; nÞ ¼ qcV c Mp;m Mc;n nm

#1 ;

ð41Þ

ð42Þ

where xp,m and xc,n are the natural frequencies of the plate and cavity mode, Mp,m and Mc,n are the generalized modal mass, and Lp;c nm is the coupling coefficient. The energy transfer is highly correlated with the difference in the natural frequencies for the plate and cavity mode. The smaller the difference is, the stronger the coupling strength and the more energy transfer will be. For the sound energy contained in the (odd, odd) mode group, its transmission is mainly due to the coupling effect between this mode group in the three plates and the (0, 0, 0) acoustic mode in the two cavities. The natural frequency of the (0, 0, 0) acoustic mode is 0 Hz. Hence, these (odd, odd) modes whose natural frequencies are close to 0 Hz couple well with the (0, 0, 0) acoustic mode in the two cavities and there has large energy transfer between them. Consequently, the amount of the sound energy transmitting through the triple-layer structure is large in the frequency range of 0–230 Hz, which is referred to as the pass-band. As the frequency increases, the (odd, odd) modes contained in the higher frequency range couple weakly with the (0, 0, 0) acoustic mode because of the large difference in their natural frequencies.

The energy transfer efficiency is low (Fm,n is small), which results in the higher frequency range becoming the stop-band. Similarly, the transmission of sound energy contained in the (even, odd), (odd, even) and (even, even) mode groups is mainly due to these mode groups coupling with the (1, 0, 0), (0, 1, 0) and (1, 1, 0) modes, respectively. The natural frequencies of the three acoustic modes are 287 Hz, 410 Hz and 500 Hz. Therefore, the above three types of panel modes contained in their respective pass-bands couple well with the three acoustic modes, respectively. This results in the large energy transfer in their respective pass-bands. Also the three types of modes contained in their respective stop-bands couple weakly with the three acoustic modes, respectively, due to the large difference in their natural frequencies. This finally results in the small energy transfer (Fm,n is extreme small) in these frequency bands. 5. Physical mechanisms for active control 5.1. Control results Fig. 7 shows the radiated sound power of the plate c with and without control. The power output of the plate c decreases obviously in the entire frequency range of interest when applying control. The level of noise reduction achieved is nearly up to 60 dB especially in the extreme low frequency range. It is reasonable for the large noise reduction below 100 Hz because the first three peaks in the radiated power curve without control are the (1, 1) mode of the three plates, respectively. It has a volumetric vibrational pattern that can be easily controlled. There are also some resonant frequencies on which the radiated power did not reduce, which may be mainly due to the location in which the point force acts and the limited control authority with single control force. The

X. Ma et al. / Applied Acoustics 85 (2014) 111–122

Fig. 7. The radiated sound power of the plate c before and after control. —— Before control; after control.

high level of noise reduction produced by applying active control in this type of control strategy is mainly due to the special control mechanisms involved in such system. The current understanding of the mechanisms involved in the use of point forces to control structurally transmitted noise is that, noise control is achieved either by suppressing the structural modes to reduce the level of vibration, or by rearranging the structural modes to change the structure into a weaker radiator. However, these two mechanisms are of limited usefulness for the mechanism study of the active triple-panel structure because of its complicated coupling relations. The rule of sound energy transmission has been clearly interpreted in Section 4. Such proposed method simplifies the analysis of the energy transmission effectively, and should be correspondingly offer some possibility to facilitate the physical mechanisms study when applying control. 5.2. Physical mechanisms In the following, the physical mechanism is investigated by comparing the change of the energy transmission rule in controlled and uncontrolled conditions. Fig. 8 shows the radiated sound power of the four mode groups in the plate b and c before and after control. Note that the optimal control force used here is the same obtained in the minimization of the power output of the plate c in Section 5.1. The radiated sound power of the (odd, odd) mode group in the plate b has been reduced obviously in the pass-band upon applying control, as shown in Fig. 8(a1). This illustrates that the sound transmission of this mode group is blocked in pass-band in controlled condition so that the energy transmitting into this mode group in the plate c is highly reduced. This consequently results in the reduction of radiated sound power of this mode group in the plate c, as shown in Fig. 8(a2). In the stop-band, the radiated power of this mode group in the plate b has been increased after control. But the energy transmission efficiency is low so that only a small amount of the increased sound energy can transmit into the plate c. Moreover, the sound energy contained in the (odd, odd) mode group in the plate c is not dominant in the stop-band. Hence, the little increased sound energy in this mode group has little influences on the total energy. The total effect is that the energy transmission in the pass-band is suppressed upon applying control so that the pass-band also becomes the stop-band. For the (even, odd) mode group, the radiated sound power of this mode group in the plate b is significantly decreased in the pass-band after control, as shown in Fig. 8(b1), which effectively suppress the energy transmitting into the plate c, as shown in Fig. 8(b2). The little increased sound energy in the stop-bands has little influence on the total energy in the plate c.

119

The situation is very similar for the (odd, even) and (even, even) mode groups. The energy transmission in their respective passbands is suppressed in controlled condition. The sound energy contained in these mode groups in the plate b can hardly transmit into the plate c, which results in the reduced radiated power of these mode groups in the plate c, as shown in Fig. 8(c1), (c2), (d1) and (d2). It is concluded that the sound energy transmission in the four pass-bands being suppressed after control and the corresponding four pass-bands becoming the stop-bands are the mechanisms involved in such control system. The introduction of point force effectively improves the sound insulation capability of the system so as to achieve the minimization of the objective function. It is interesting to note that thought the noise reduction is achieved only in the pass-bands for the four mode groups in the plate c and there is always some increment in the stop-bands, but the noise reduction for the plate c is achieved in the entire low frequency range. This is mainly due to the interesting phenomenon that the stop-band of one mode group is just constituted by the pass-bands of the other three mode groups. The amount of reduction of the radiated power in their respective pass-bands for these mode groups is much higher than the increment of that mode group in its stop-band. Therefore, the total effect is the reduced radiated power after control. The sum of the radiated powers of the four mode groups with control in Fig. 8(a2), (b2), (c2) and (d2) would give the total power output of the plate c with control, which is the grey curve in Fig. 7. Fig. 9 shows the flow chart of the physical mechanisms stated above, which only take the (odd, odd) mode group as an example to clarify the interpretations. Fig. 10 shows the total kinetic energy of the plate c before and after control. The kinetic energy decreases significantly after control, which demonstrates the feasibility of the stated mechanisms concluded from the energy transmission point of view, and also illustrates that the control spillover on the interior trim panel can be avoided. This analytical approach is much clearer than that in terms of modal analysis because it decouples the system into four independent parts. In this research, the analysis of the energy transmission rule in terms of pass- and stop-band is accomplished only for a single example. And there exist some limitations. For example, the dimensions of the three plates and air cavities should be small. The large panel and cavity will have a relative high modal density. More than one acoustic mode will participate in certain mode group’s energy transmission, which would make it difficult to distinctly separate the pass- or stop band. And if the structural response of the three plates are very different and the mode groups fall in different frequency ranges, the energy transfer factor defined in Eq. (41) is likely to be small for all the four mode groups. In this case, it would also be difficult to separate the spectrum of the radiated power in distinct pass- and stop-band. The method cannot work in these cases, and therefore, its the ongoing research to overcome these limitations.

6. Applications According to the energy transmission rule and its fundamental causations, different type of materials of the three plates should be used in practice so as to effectively weaken the couplings and improve the energy transmission loss. In typical practical applications such as the aircraft or ship, the materials of the incident (fuselage shells) and radiating plate (interior trim panel) may be special to meet the special requirements and therefore they cannot be replaced or removed. Hence the middle plate should be designed in such a way that it should weaken couplings of the coupled system as much as possible.

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Fig. 8. (a1), (b1), (c1) and (d1) are the radiated sound power for the four mode groups in the plate b before and after control. (a2), (b2), (c2) and (d2) are the radiated sound power of the four mode groups in the plate c before and after control. (a1) and (a2) the (odd, odd) mode group, (b1) and (b2) the (even, odd) mode group, (c1) and (c2) the (odd, even) mode group, (d1) and (d2) the (even, even) mode group. —— Before control; after control; The filled with a grey background color represents the passband, and the rest fields are the stop-band.

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Frequency band of 0~230Hz

(odd,odd)mode (even,odd)mode

Total incident energy

(odd,even)mode (even,even)mode a b c

(b)

(odd,odd)mode (even,odd)mode

Total incident energy

Fs

(odd,even)mode (even,even)mode a b

c

(odd,odd)mode group

The transmitted energy

The other three mode groups

Frequency band of 0~230Hz

(a)

(odd,odd)mode group The other three mode groups

The transmitted energy

Fig. 9. The flow chart of the physical mechanisms. The four arrowheads represent the sound energy transmission in the four transmitting paths, respectively. The relative thickness of the arrowheads represents the level of sound energy. (a) Before control, (b) after control.

Fig. 10. The total kinetic energy of the plate c before and after control. —— Before control; after control.

For the passive noise reduction effects, the fundamental designing principle is that the frequencies of the structural modes in each mode group should keep away from the frequency of the cavity mode with which the mode group couples. This can effectively weaken the couplings of the coupled mode pairs and suppress the sound energy transmission in each transmitting path. Here three cases of the middle plate b with different materials and thickness are used to demonstrate the stated conclusions. The three cases are, case1 (Aluminum, 3 mm), case2 (Steel, 3 mm) and case3 (Steel, 6 mm), respectively. Fig. 11 presents the radiated sound powers of the radiating plate c with the different cases of the middle plate. Compared the case1 with case2, the radiated power of the plate c is decreased obviously when the material of the middle plate changes to steel. This is just due to the fact that the differences of the resonant frequencies of some coupled mode pairs increase when the middle plate is the case2 (The resonant frequencies of the modes of the middle plate in case2 are not listed here for briefly). The resonant frequency of (1, 1) mode of the middle plate in case2 nearly does not change compared with the case1, therefore the coupling strength between this mode and (0, 0, 0) acoustic mode of two cavities nearly does not weaken. And the sound energy transmission for the (odd, odd) mode group is slightly suppressed in the extreme low frequency (below 100 Hz), which results in the little decreased radiated power of

Fig. 11. The radiated sound power of the plate c without control when the middle plate b is the case1, case2 and case3, respectively. —— Case1; case2; case3.

the plate c. When the middle plate changes from the case2 to case3, the number of the resonant modes contained below 500 Hz reduces to only four, i.e., the (1, 1), (2, 1), (1, 2) and (3, 1) modes. A superficial anticipation is that the radiated power of the radiating plate should be decreased significantly since the couplings between the middle plate and two cavities may become weak when structural modes are highly reduced. However, the situations are not all what was expected. With the middle plate as the case3, the resonant frequencies of the (2, 1), (1, 2) and (2, 2) modes are much closer to the resonant frequencies of the (1, 0, 0), (0, 1, 0) and (1, 1, 0) acoustic modes, respectively. Therefore, these mode pairs are well coupled and the level of sound energy transmission in the latter three paths are not suppressed (or even reinforced at certain frequencies) through only single structural mode in each mode group may participate in the energy transmission in these paths. The resonant frequency of the (1, 1) mode in case3 is increased obviously, therefore its coupling effects with (0, 0, 0) acoustic mode weakens significantly so that the energy transmission is suppressed in the low frequency range of below 100 Hz. This results in the highly reduced radiated power of the plate c. For active control effects, the fundamental designing principle is that the number of modes in each mode group in middle plate should be small so that these modes can be easily and significantly

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Fig. 12. The level of active noise reduction of the triple-panel system when the middle plate b is the case1, case2 and case3, respectively. —— Case1; case2; case3.

suppressed by controllable point forces to guarantee the pass-band being maximally blocked in each path. Fig. 12 presents the active noise reduction level (ARL, the differences of the radiated power of the radiating plate with and without control) for the different cases of the middle plate. The ARL in case3 is obviously larger than that obtained in case1 or case2 in the entire low frequency, which mainly dues to the smaller number of the structural modes in case3. Therefore, the optimal middle plate used in practical applications should be designed in such a way that the number of resonant modes should be small, and moreover their resonant frequencies should keep away from the frequency of the coupled acoustic mode as much as possible. 7. Conclusions The rule of sound energy transmission through the triple-panel structure is firstly investigated as an important part. Then based on this, the physical mechanism is analyzed from the change in behaviors of energy transmission in controlled and uncontrolled conditions. The energy contained in the four mode groups in the incident plate transmits independently, which results in forming four transmission paths. They all have the band-pass characteristics. The mechanism involved lies in the fact that the energy transmission in each transmitting path is suppressed in the pass-band after control. Consequently the energy transmitting into the radiating plate is significantly reduced, which illustrates the pass-bands also becoming the stop-bands. This is an encouraging approach to interpret the physical mechanism in a simple and effective way and also to make the analytical process easier for the reader. Moreover, these rules offer some principles for optimally designing the middle plate to guarantee high transmission loss with either passive or active concern. Acknowledgements This work was financially supported by Doctorate Foundation of Northwestern Polytechnical University (Grant No. CX201004), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20096102110007) and Aeronautical Science Foundation of China (Grant No. 2011ZA53004). References [1] Sas P, Bao C, Augusztinovicz F, Desmet W. Active control of sound transmission through a double panel partition. J Sound Vib 1995;180:609–25. [2] Pietrzko SJ, Mao Q. New results in active and passive control of sound transmission through double wall structures. Aerosp Sci Technol 2008;12:42–53.

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