Research Article Vibroacoustic Analysis of a Rectangular...

18
Research Article Vibroacoustic Analysis of a Rectangular Enclosure Bounded by a Flexible Panel with Clamped Boundary Condition Yuan Wang, Jianrun Zhang, and Vanquynh Le College of Mechanical Engineering, Southeast University, Nanjing 211189, China Correspondence should be addressed to Jianrun Zhang; [email protected] Received 15 November 2012; Accepted 13 April 2013; Published 9 March 2014 Academic Editor: Reza Jazar Copyright © 2014 Yuan Wang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A lot of research on the properties of a panel-enclosure coupled system which consists of an enclosure with a simply supported flexible wall was conducted. However, an enclosure with clamped flexible walls is commonly encountered. e properties of a panel-enclosure system, with one clamped flexible wall, are different from those of a simply supported one, and there are only a few studies on it. us, this paper is dedicated to the study of the effect of structural-acoustic coupling on the properties of a panel-enclosure system, which consists of an enclosure with a clamped panel. e resonance frequencies and the decay times of the coupled system are obtained using the classical modal coupling theory. e effects of enclosure and its flexible boundary on the resonance frequencies and the modal decay times of the coupled system are then analyzed. It is shown that, when panel thickness is changed, coupling strength is determined by the difference between the resonance frequencies of the panel and the enclosure modes. However, for the variation of enclosure depth, the factor which determines the coupling strength between panel and enclosure modes is the enclosure depth or the difference between their resonance frequencies. 1. Introduction e interaction between a sound field in an enclosure and its flexible boundary is a critical problem, and a good understanding of it is particularly important to the control of sound field in an enclosure. Considerable effort has been devoted to the study of this classical model for many years. e works of Dowell [1, 2] represent some of early research into modeling of vibrations of panel backed by an enclosure. In the works of Pan of and Bies [3], the effect of acoustic- structural coupling on the sound field in an enclosure, whose flexible wall is simply supported along its edges, was studied using the classical modal coupling method. However, the influence of enclosure depth and absorptive walls of enclosure on the coupling degree was not considered in this work. Kim and Brennan [4] proposed an impedance-mobility approach for the analysis of panel-enclosure system, and an enclosure with a simply supported wall was used to validate the proposed approach by experiment. In the efforts of Pan et al. [5] and Xin et al. [6, 7], the weighted residual method was used to calculate the response of a panel-enclosure coupled system. Based on the analysis of acoustic-structural coupling, the forced response of panel-enclosure coupled system with different kinds of excitation [811] and active noise control [1215] were studied. In the effort of Balachandran et al. [15], modal coupling method was used to study the active noise control of panel-enclosure coupled system, in which the flexible panel was clamped along its edges. However, the free vibration characteristics of a panel-enclosure coupled system were not analyzed. e flexible wall of an enclosure with clamped bound- ary is commonly encountered, and the acoustic-structural coupling in panel-enclosure system which consists of an enclosure with a clamped flexible wall is different from that of a simply supported one. us, the boundary condition of panel has significant influence on the coupling with enclosure sound field. e panel with different boundary conditions has attracted many attentions using different methods. Rayleigh- Ritz approach was used to analyze the vibration and acoustic radiation of a baffled panel with arbitrary boundary condi- tions [16, 17], and a baffled and unbaffled panel in light and heavy fluid with arbitrary conditions [18]. In the efforts of Hindawi Publishing Corporation Shock and Vibration Volume 2014, Article ID 872963, 17 pages http://dx.doi.org/10.1155/2014/872963

Transcript of Research Article Vibroacoustic Analysis of a Rectangular...

Research ArticleVibroacoustic Analysis of a Rectangular Enclosure Bounded bya Flexible Panel with Clamped Boundary Condition

Yuan Wang Jianrun Zhang and Vanquynh Le

College of Mechanical Engineering Southeast University Nanjing 211189 China

Correspondence should be addressed to Jianrun Zhang zhangjrseueducn

Received 15 November 2012 Accepted 13 April 2013 Published 9 March 2014

Academic Editor Reza Jazar

Copyright copy 2014 Yuan Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A lot of research on the properties of a panel-enclosure coupled system which consists of an enclosure with a simply supportedflexible wall was conducted However an enclosure with clamped flexible walls is commonly encountered The properties of apanel-enclosure system with one clamped flexible wall are different from those of a simply supported one and there are onlya few studies on it Thus this paper is dedicated to the study of the effect of structural-acoustic coupling on the properties ofa panel-enclosure system which consists of an enclosure with a clamped panel The resonance frequencies and the decay timesof the coupled system are obtained using the classical modal coupling theory The effects of enclosure and its flexible boundaryon the resonance frequencies and the modal decay times of the coupled system are then analyzed It is shown that when panelthickness is changed coupling strength is determined by the difference between the resonance frequencies of the panel and theenclosure modes However for the variation of enclosure depth the factor which determines the coupling strength between paneland enclosure modes is the enclosure depth or the difference between their resonance frequencies

1 Introduction

The interaction between a sound field in an enclosure andits flexible boundary is a critical problem and a goodunderstanding of it is particularly important to the controlof sound field in an enclosure Considerable effort has beendevoted to the study of this classical model for many yearsThe works of Dowell [1 2] represent some of early researchinto modeling of vibrations of panel backed by an enclosureIn the works of Pan of and Bies [3] the effect of acoustic-structural coupling on the sound field in an enclosurewhose flexible wall is simply supported along its edges wasstudied using the classical modal couplingmethod Howeverthe influence of enclosure depth and absorptive walls ofenclosure on the coupling degree was not considered in thiswork Kim andBrennan [4] proposed an impedance-mobilityapproach for the analysis of panel-enclosure system and anenclosure with a simply supported wall was used to validatethe proposed approach by experiment In the efforts of Pan etal [5] and Xin et al [6 7] the weighted residual method wasused to calculate the response of a panel-enclosure coupled

system Based on the analysis of acoustic-structural couplingthe forced response of panel-enclosure coupled system withdifferent kinds of excitation [8ndash11] and active noise control[12ndash15] were studied In the effort of Balachandran et al[15] modal coupling method was used to study the activenoise control of panel-enclosure coupled system inwhich theflexible panel was clamped along its edges However the freevibration characteristics of a panel-enclosure coupled systemwere not analyzed

The flexible wall of an enclosure with clamped bound-ary is commonly encountered and the acoustic-structuralcoupling in panel-enclosure system which consists of anenclosure with a clamped flexible wall is different from thatof a simply supported one Thus the boundary condition ofpanel has significant influence on the couplingwith enclosuresound fieldThe panel with different boundary conditions hasattracted many attentions using different methods Rayleigh-Ritz approach was used to analyze the vibration and acousticradiation of a baffled panel with arbitrary boundary condi-tions [16 17] and a baffled and unbaffled panel in light andheavy fluid with arbitrary conditions [18] In the efforts of

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 872963 17 pageshttpdxdoiorg1011552014872963

2 Shock and Vibration

X

Y

Z

Clamped panel

120572

Lx = 10m

Ly = 08m

Lz

120579

Pi

q

Figure 1 Panel-enclosure geometry and coordinate system

Sung and Jan [19] andArenas [20 21] the transverse vibrationresponse and sound radiation by a clamped panel wereobtained using the virtual work principle Comparedwith theRayleigh-Ritz approach virtual work principle provides aneasier way to calculate the surface response of panel In thecurrent work the mode shape function proposed by Sungand Arenas is used to analyze the characteristics of panel-enclosure coupled systemwhich consists of an enclosure withone clamped flexible wall

In this paper the vibroacoustic analysis of a rectangularenclosure which is bounded by a clamped flexible wall isconducted using the modal coupling method Acousticalmodes are used to describe the coupled system behaviorFirstly the free vibration characteristics of panel-enclosuresystem are solved based on the modal parameters of uncou-pled panel and enclosure and the characteristics of coupledsystem which reflect the intrinsic properties are obtainedSecondly the features of coupled system which includeresonance frequency and modal decay time are analyzedwith different variables including panel thickness panelinternal damping and enclosure depth The energy transferresonance frequencies and decay times of acoustical modesare distinguished from others when the panel mode andenclosure mode are well coupled This work is an extensionof the work carried by Pan and Bies [3] and Sung and Jan [19]where the effect of fluid-structural coupling on the soundfield in the enclosure is studied with a simply supportedflexible wall using the modal coupling method and theclamped boundary panel with new mode shape function isused to analyze the vibration response respectively

2 Analytical Model

A panel-enclosure coupled system is shown in Figure 1 Theenclosure has five small absorptive walls and one flexible walland the dimensions of enclosure are 119871

119909 119871119910 and 119871

119911 The

panel which is clamped along its edges at 119911 = 119871119911 has

the dimensions of 119871119909 119871119910 119875119894119899is the amplitude of plane wave

119875119894 and the 120572 120579 are respectively the elevation and azimuth

angles with respect to the panel center 119902 is volume velocityof source in the enclosure and it is located at origin ofcoordinate (119909 119910 119911) = (0 0 0)mThe sound pressure 119901 in theenclosure is described by an inhomogeneous wave equation

nabla2

119901 minus1

1198882

0

1205972

119901

1205971199052= minus1205880

120597119902

120597119905 (1)

where 1198880 1205880are the speed of sound and air density respec-

tively At the five absorptive walls inside the enclosure andthe flexible enclosure interface the boundary conditions takethe following form

120597119901

120597n= minus119895

1205961205880

119885119860

119901 at absorbent walls

120597119901

120597119911

10038161003816100381610038161003816100381610038161003816119911=119871119911

= minus1205880

1205972

119908

1205971199052at flexible wall

(2)

where n indicates the normal direction of the boundarysurfaces 119885

119860is the specific acoustic impedance on the

absorptive walls inside the enclosure119908(119909 119910 119911) is the normaldisplacement of flexible panel

The governing equation of flexible vibration of thinisotropic panel due to the internal pressure 119901 and plane wave119901119894119899on the outside of the panel is

120588ℎ1205972

119908

1205971199052+ 119863nabla4

119908 = 119901 minus 119901119894119899 (3)

where 119863 = 119864ℎ3

12(1 minus 1205832

) and 120588 119864 120583 ℎ are the densityYoungrsquos modulus Poissonrsquos ration and thickness of the flex-ible panel respectively In this analysis the panel radiationis neglected and the positive direction for displacement ofclamped panel is towards outside of the enclosure

In order to use the modal shape expansion the soundpressure 119901 inside enclosure the flexible panel velocity V =

119895120596119908 the sound pressure 119901119894119899on the outside of the panel and

the monopole acoustic source 119902 inside the enclosure can beexpanded by mode shape functions

119901 (r) = [120601119873]119879

[119901119873] = Φ

TP (4)

V (120590) = [119904119872]119879

[V119872]119879

= STV (5)

Shock and Vibration 3

119901119894119899(120590) =[119904

119872]119879

[119875119894119899119872

]119879

= STP119894119899 (6)

119902 (r) = [120601119873]119879

[119876119873] = Φ

119879Q (7)

In these equations 119901119873 V119872 119875119894119899119872

119876119873are modal amplitude

of sound pressure inside enclosure flexible panel vibrationvelocity the sound pressure on the outside of the clampedpanel surface and the acoustic source inside enclosurerespectively120601

119873 119904119872aremodal shape functions of rigidlywalls

enclosure and uncoupled clamped panel respectively Herethey are the base functions for the analysis of enclosure withmodally reactive flexible panel and locally reactive walls

The mode shape function 120601119873of rigid wall enclosure is

given by

120601119873(r) = 120601

119873(119909 119910 119911) = cos(119897120587119909

119871119909

) cos(119898120587119910

119871119910

) cos(119899120587119911119871119911

)

(8)

Here 119897 119898 119899 are themode indices of the119873th enclosuremodeThemode shape function 119878

119872of uncoupled clamped panel

is given by Sung and Jan [19] and it has been validated byexperiment

119904119872(120590) = 119904

119872(119909 119910) = 120595

119906(119909) 120595V (119910)

120595119906(119909) = 120574(

120582119906119909

119871119909

) minus120574 (120582119906)

119867 (120582119906)119867(

120582119906119909

119871119909

)

120595V (119910) = 120574(120582V119910

119871119910

) minus120574 (120582V)

119867 (120582V)119867(

120582V119910

119871119910

)

(9)

where 120574(119904) = cosh(119904) minus cos(119904)119867(119904) = sinh(119904) minus sin(119904)120582119906 120582V are determined by (10) and 119906 V are mode indices

of clamped panel modes

cosh (120582) cos (120582) minus 1 = 0 (10)

Considering the acoustic boundary condition in (2) andusing the Green function technique the complex acousticpressure 119901 in the enclosure can be expressed as

119901 = 1198951205961205880int119860119891

V119866119860119889119904 + 119895120596120588

0int1198810

119902119866119860119889119881 + 119895

120596

1198880

int119860119871

120573119866119860119901119889119904

(11)

Here 119860119891 119860119897 1198810are the surface of flexible panel the

absorbent wall surface inside enclosure and the volume ofenclosure respectively 120573 = 120588

01198880119885119860is the specific acoustic

admittance ratio on the absorbent wall surface 119866119860is the

sound field Greenrsquos function

119866119860(r r0) = sum

119873

120601119873(r) 120601119873(r0)

1198810Λ119873(1198962 minus 119896

2

119886119873)

Λ119873=1

1198810

int1198810

1206012

119873(r) 119889119903

(12)

Similarly the flexible clamped panel vibration velocitycan be expressed as

V = 119895120596

120588ℎint119860119891

(119901 minus 119901119894119899) 119866119901119889119904 (13)

119866119901is panel Greenrsquos function which can be expressed as

119866119901(1205901205900) = minussum

119872

119904119872(120590) 119904119872(1205900)

119860119891Λ119872(1205962 minus 120596

2

119901119872)

Λ119872=

1

119860119891

int119860119891

1199042

119872(120590) 119889120590

(14)

Substituting (4) (5) (6) and (7) into (11) and (13) we canobtain

(120582I minus A)X = Y (15)

Equation (15) can be expressed as

(120582I minus (A11 A12

A21

A22

))(

PV120582P120582V

) = 120582(

0

0

EQEF

) (16)

where I is unit matrix 120582 = minus119895119896 A11and A

12are (119873 + 119872) times

(119873 +119872) zero and unit matrices respectively

EQ = minus11988801205880(11987611198762sdot sdot sdot 119876

119873)119879

(17)

EF=1

120588ℎ1198880

(1198751198941198991

1198751198941198992

sdot sdot sdot 119875119894119899119872

)119879

(18)

A21= minus

[[[[[[[[

[

1198962

1198861

d 0

1198962

119886119873

1198962

1199011

0 d1198962

119901119872

]]]]]]]]

]

(19)

4 Shock and Vibration

A22= minus

[[[[[[[[[[[[[[[[[[[

[

1205781198861

0

11988801205882

011986011989111986111

119872119860

1

sdot sdot sdot

11988801205882

01198601198911198611119872

119872119860

1

d d

0 120578119886119873

11988801205882

01198601198911198611198731

119872119860

119873

11988801205882

0119860119891119861119873119872

119872119860

119873

minus

11986011989111986111

1198880119872119875

1

sdot sdot sdot minus

1198601198911198611198731

1198880119872119875

1

1205781199011

0

d d

minus

1198601198911198611119872

1198880119872119875

119872

minus

119860119891119861119873119872

1198880119872119875

119872

0 120578119901119872

]]]]]]]]]]]]]]]]]]]

]

(20)

119861119873119872

=1

119860119891

int119860119891

119904119872(120590) 120601119873(r) 119889119904 (21)

120578119901119872

=44120587

1198791199011198721198880

(22)

120578119886119873

=44120587

1198791198861198731198880

(23)

where 119896119886119873

= 1205961198861198731198880 119896119901119872

= 1205961199011198721198880 119896119886119873

120596119886119873

are thewavenumber and the resonance angle frequency of the 119873thenclosure mode respectively 119896

119901119872and 120596

119901119872are respectively

the wavenumber and the resonance angle frequency of the119872th panel mode119872119875119872119860 are the modal mass of panel andenclosure respectively 119872119875

119872= 120588ℎ119860

119891Λ119872 119872119860119873= 12058801198810Λ119873

1198601198911198810are the area of flexible panel and volume of enclosure

respectively 120578119901119872

120578119886119873

are respectively the loss factor of119872thpanel mode and 119873th enclosure mode Particularly 120578

119886119873is

related to the integral of the specific acoustic admittance 120573119879119901119872

119879119886119873

are respectively the 60 dB decay time of119872th panelmode and119873th enclosure mode

3 The Modal Properties ofthe Panel-Enclosure System

31 Modal Coupling Coefficient Modal coupling coefficientis the spatial matching degree between enclosure and panelmodes on the interacting surface of enclosure sound fieldand flexible panel from (21) It is the integral of panelmode and enclosure mode on the surface of panel Modalcoupling coefficient only depends on the geometry shape andboundary condition of panel the mode shape of enclosureacoustic field [3] and it has nothing to do with thicknessmaterial properties damping of the panel and enclosuredepth

Like the panel-enclosure system which consists of anenclosure with a simply-supported flexible wall the modecoupling coefficient between clamped panel and enclosure is

nonzero which must satisfy the following two conditions atthe same time

119906 + 119897 = odd number

V + 119898 = odd number(24)

32 Transfer Factor Coupling coefficient determines thematching degree between an enclosure mode and a panelmode on the interacting surface of enclosure sound field andpanel while the coupling extent between a panelmode and anenclosure mode is decided by transfer factor [3] The transferfactor 119865

119873119872between the119873th enclosure acoustical mode and

119872th panel mode is given by

119865119873119872

=

1 + [

(120596119886119873minus 120596119901119872)

2]

2

[1

119861(119873119872)2]

minus1

(25)

119861 (119873119872) = (12058801198882

0

120588ℎ119871119911Λ119873Λ119872

)

12

119861119873119872

(26)

When the modal coupling coefficient between the 119873thenclosure acoustical mode and119872th panel mode changes thetransfer factor 119865

119873119872has different results from (25) and (26)

There are two different kinds of transfer factors (1) as themodal coupling coefficient 119861

119873119872is equal to zero the transfer

factor 119865119873119872

must be equal to zero In this situation thereis not energy transfer between the 119873th enclosure acousticalmode and 119872th panel mode (2) When the modal couplingcoefficient is nonzero the transfer factor relates to manyfactors such as the difference between resonance frequenciesof uncoupled panel and enclosure modes and the enclosuredepth The larger the transfer factor between a panel modeand an enclosure mode is the bigger the coupling strengthbetween them is obtained When the transfer factor 119865

119873119872

is in the order of 10 the energy transfer between the 119873thenclosure mode and119872th panel mode is important

33 Resonance Frequency and Modal Decay Time of CoupledSystem Thesound pressure in the enclosure and the clampedpanel vibration velocity are used to describe the response

Shock and Vibration 5

of coupled panel-enclosure system Through the surface offlexible panel facing the inside of enclosure the sound fieldin the enclosure is coupling with the vibration of flexiblepanel There are two different kinds of acoustical modes inthe panel-enclosure coupled system one is an enclosure-controlled acoustical mode whose most of energy is stored inthe enclosure sound field and the other is a panel-controlledacoustical mode whose most of energy is stored as panelvibration energy [3]

If the external excitation does not exist in the coupledsystem there will be Y = 0 in (15) and it becomes a 2(119873 +

119872) dimensional system of equations Corresponding to theeigenequation there will be 2(119873 + 119872) eigenvalues 120582

119871and

120582lowast

119871 and 119871 = 1 2 sdot sdot sdot (119873 + 119872) The resonance frequency 119891

119871

and the decay time 119879119871of coupled system are Im(120582

119871)11988802120587

and 691Re(120582119871)1198880 respectively When the Y = 0 in (15)

the solution of the coefficient X is the modal amplitude ofcoupled system Then the panel vibration velocity and thesound pressure in the enclosure which describes the forcedresponse of coupled system can be obtained from (4) and (5)

The time-averaged acoustic potential energy 119864119886119873

in theenclosure and the time-averaged vibration kinetic energy119864119901119872

of the flexible panel are given by [14]

119864119886119873

=1

(412058801198882

0) int1198810

1003816100381610038161003816119901 (r 120596)1003816100381610038161003816

2

119889119881

=1198810PHΛ119886119873

P(412058801198882

0)

(27)

119864119901119872

=120588ℎ

4int119860119891

|V (120590 120596)|2

119889119904

=

120588ℎ119860119891VH

Λ119901119872

V4

(28)

HereΛ119886119873

is a119873times119873 diagonalmatrix with each diagonal termconsisting of Λ

119873 and Λ

119901119872is a119872times119872 diagonal matrix with

each diagonal term consisting of Λ119872

4 Results and Discussion

To demonstrate the properties of the panel-enclosure coupledsystemwhich consists of an enclosure with a clamped flexiblewall the resonance frequencies and modal decay times ofacoustical modes are investigated with different panel modaldensity panel internal damping and enclosure depth respec-tively The panel-enclosure coupled system which consists ofan enclosure with a clamped panel on top and five absorptivewalls is shown in Figure 1 The panel material properties aretaken as follows the material of clamped panel is aluminumwith density 120588 = 2770 kgm3 Youngrsquos modulus 119864 = 71Gpaand Poissonrsquos ratio 120583 = 033

41 Effects of Different Panel Physical Parameters onAcoustical Modes

411 Panel Modal Density From (25) as the differencebetween resonance frequencies of uncoupled enclosure andpanel modes is decreased the transfer factors betweenthem become larger when the modal coupling coefficient is

nonzero The resonance frequencies of rigid walls enclosuremode and uncoupled clamped panel mode [21] are given by

119891119897119898119899

=1198880

2[(

119897

119871119909

)

2

+ (119898

119871119910

)

2

+ (119899

119871119911

)

2

] (29)

119891119906V =

1

2120587radic119863

120588

radic(120582119906

119871119909

)

4

+ (120582V

119871119910

)

4

+ 2(120582119906120582V

119871119909119871119910

)

2

120581119906120581V

120585119906120585V

(30)

120581119894=1

4(1 + 119863

2

119894) sinh (2120582

119894) minus

1

2119863119894cosh (2120582

119894)

minus1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) minus 119863119894cos2 (120582

119894)

minus 1198632

119894120582119894+3

2119863119894

(31)

120585119894=1

4(1 + 119863

2

119894) sinh (2120582

119894)

+ sinh (120582119894) [2119863119894sin (120582

119894) minus (1 minus 119863

2

119894) cos (120582

119894)]

minus (1 + 1198632

119894) sin (120582

119894) cosh (120582

119894)

+1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) + 120582119894

minus1

2119863119894[1 + cosh (2120582

119894)] + 119863

119894cos2 (120582

119894)

(32)

119863119894=120574 (120582119894)

119867 (120582119894) (33)

Similar to an enclosure with a simply supported flexiblewall the modal density of uncoupled clamped panel andenclosure will affect energy transfer between them Themodal density of uncoupled clamped panel and enclosuresound field are given by [3 10]

119899119901=

radic3119860119891

119862119871ℎ (34)

119899119886=412058711988101198912

1198883

0

+120587119878119891

21198882

0

+119871

81198880

(35)

Here 119860119891 119862119871are the area and longitudinal wave speed

of clamped flexible wall respectively 119891 is the excitationfrequency 119878 119871 are the total surface area and the total edgelengths inside the enclosure

As mentioned above the transfer factor determines thecoupling strength between a panel mode and an enclosuremode From (34) and (25) by adjusting the panel modaldensity (corresponding to panel thickness) the distributionof the resonance frequencies of panel is changed whichalso leads to changes of the transfer factors between paneland enclosure modes In Figure 2 transfer factors between

6 Shock and Vibration

005 01 015 02 025 030

02

04

06

08

1

Tran

sfer f

acto

r

np (Hzminus1)

(11)

(13)

(33)

(15)

(35)

(51)

(17)

(53)

(37)

(55)

(31)

Figure 2 Transfer factors between the (001) enclosure mode andpanelmodes as a function of panelmodal density119871

119885= 06m119879

119886119873=

15 s 119879119901119872

= 05 s

the (001) enclosure mode and panel modes are plottedagainst panel modal density When the panel modal densityis low only the (31) panel mode participates in the couplingwith the (001) enclosure mode and satisfies the well-coupledcondition As the panel modal density is increased moreand more high-order panel modes in which the couplingcoefficients with the (001) enclosure mode are not equalto zero participate in coupling At these points of transferfactors when their values are in the order of 10 there is largeenergy transfer between enclosure and panel modes Similarto the transfer factors between the (001) enclosure modeand panel modes the distribution of transfer factors betweenother enclosure modes and panel modes are that only fewpanel modes participate in the coupling with enclosuremodes in the low panel modal density region andmore panelmodes participate in the coupling with enclosure mode forhigh panel modal density region

As the coupling extent between enclosure and panelmodes changes with the variation of panel modal densitythe resonance frequency of acoustical mode which is oneimportant characteristic of it is altered also Figure 3(a)shows resonance frequencies of the first few enclosure-controlled acoustical modes as a function of panel modaldensity Forty panel modes and forty enclosure modes areused in this analysis The decay times of all uncoupledenclosure and panel modes are 15 s and 05 s respectivelyThedepth of enclosure is 06m As the panel modal density isincreased the resonance frequencies of enclosure-controlledmodes jump to higher frequencies Compared with theresonance frequencies of acoustical mode in the low panelmodal density region they become larger in the high panelmodal density region The reason is that many high-order

panel modes participate in the coupling with enclosuremodes Similar to the panel-enclosure coupled system with asimply supported flexible panel the energy transfer from theenclosuremode is distributed overmany panel modes and nopanel modes are well coupling with enclosure modes

Figure 3(b) shows plot of the variation of decay times ofthe first few enclosure-controlled modes with panel modaldensity When the panel modal density is small the modaldecay time is longer on the average As the panel modaldensity is increased the modal decay times show someminima It is because of that only few panel modes satisfythe well-coupled condition with enclosure modes in the lowpanel modal density region As the panel modal density isincreased more panel modes participate in the coupling withenclosure modes and decay times of enclosure-controlledmodes become shorter At the same time there are no panelmodes which are well coupled with enclosure mode and theenergy which is stored in the panel is equally distributed overmany panel modes

In order to describe the process of strong couplingbetween panel and enclosure modes when the panel modaldensity is altered we analyze from three points of view ofenergy ratio between the panel vibration and sound field inthe enclosure resonance frequency and modal decay time ofcoupled system respectively In this analysis the plane wave119901119894with amplitude 1 Pa on the panel surface is used to drive the

vibration of panel the excitation frequencies are respectivelythe resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The angles of elevation 120572 andazimuth 120579 of plane wave are 90∘ and 0∘ respectively Thedecay times of all uncoupled panel modes and enclosuremodes are 15 s and 05 s respectively As seen in Figure 2 thetransfer factor between the (31) panel mode and the (001)enclosure mode is approximately equal to 10 in low panelmodal density region and only these two modes satisfy well-coupled condition which means that the energy conversionbetween the sound field and the panel is almost entirelybetween these two modes

The energy ratio between the panel vibration and soundfield in the enclosure is shown in Figure 4(a) which containspanel-controlled and enclosure-controlled acoustical modesThe resonance frequencies of acoustical modes uncoupledpanel and enclosure modes are shown in Figure 4(b) as afunction of panel modal density Figure 4(c) shows the varia-tion of the decay time of acoustical modes with panel modaldensity The effect of truncation numbers on the resonancefrequencies of acoustical modes is shown in Figure 4(d)As the panel modal density tends to the point of transferfactor between the (31) panel mode and the (001) enclosuremode which is in the order of 10 energy ratio resonancefrequencies decay times of the (31) panel-controlled modeand the (001) enclosure-controlled mode tend to be equaltoo

Near the point of panel modal density where the maxi-mum interaction strength between the (31) panel mode andthe (001) enclosure mode the energy which is stored in thepanel vibration and enclosure sound field of two acousticalmodes including the (001) enclosure-controlled mode andthe (31) panel-controlled mode are approximately equal to

Shock and Vibration 7

01 02 03 01 02 03

01 02 03

01 02 0301 02 03

01 02 03

210

220

230

280

290

300

160

170

180

350

360

370

270

280

330

335

340

(001)

(100) (011)

(101)(110)

(010)

fL

(Hz)

fL

(Hz)

fL

(Hz)

np (Hzminus1) np (Hzminus1)

(a)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(101)

(100)

(001)(010)

(011)

(110)

01 02 03

01 02 03 01 02 03

01 02 0301 02 03

01 02 03

TL

(s)

TL

(s)

TL

(s)

np (Hzminus1)np (Hzminus1)

(b)

Figure 3 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal density (a)resonance frequency (b) decay time 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

each other as shown in Figure 4(a) The further the distancefrom that point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlledmodeThe result also shows that energyratio curve of the (001) enclosure-controlled mode is acontinuation of that of the (31) panel-controlled mode as afunction of panel modal density and the energy ratio curveof the (31) panel-controlled mode is a continuation of that ofthe (001) enclosure-controlled mode

It is shown in Figure 4(b) that in the vicinity of thepoint of panel modal density where the transfer factor is

approximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the panel modal density is increased the resonancefrequency of the (001) enclosure-controlled mode jumpsfrom lower than the resonance frequency of the (001) rigidwalls enclosure mode to higher than it when it passes by thepoint ofmaximumenergy transfer Comparedwith the (001)

8 Shock and Vibration

003 00305 0031 00315 0032 00325 0033 00335 0034

0

5

10

15

minus5

minus10

minus15

np (Hzminus1)

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

003 00305 0031 00315 0032 00325 0033 00335 0034260

270

280

290

300

310

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

np (Hzminus1)

fL

(Hz)

(b)

0

1

2

3

4

5

6

7

8

003 00305 0031 00315 0032 00325 0033 00335 0034

(001) enclosure-controlled mode(31) panel-controlled mode

np (Hzminus1)

TL

(s)

(c)

265

270

275

280

285

290

295

300

305

310

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

003 00305 0031 00315 0032 00325 0033 00335 0034

np (Hzminus1)

fL

(Hz)

(d)

Figure 4 (a) Ratio between the sound field energy and panel vibration energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers 119871

119885= 06m

119879119901119872

= 05 s 119879119886119873

= 15 s

enclosure-controlled mode it is opposite to the (31) panel-controlled mode As the panel modal density is away fromthe point of maximum interaction strength the resonancefrequencies of the (31) panel-controlled mode and the (001)enclosure-controlled mode tend to resonance frequencies ofuncoupled them respectively

It is shown in Figure 4(c) that as the panel modal densityis increased decay time of the (001) enclosure-controlledmode firstly reduces When panel modal density arrives atthe point of strong coupling the minimum decay time is gotThen decay time of the (001) enclosure-controlled mode

becomes longer as the panel modal density is increasedCompared with the (001) enclosure-controlled mode decaytime curve of the (31) panel-controlled mode is opposite asthe panel modal density is increased In addition at the pointof panel modal density where the decay times of the (001)enclosure-controlled mode and the (31) panel-controlledmode are approximately equal to each other the energytransfer from the panel vibration to the enclosure acousticfield is relative maximum

Due to the limited number of uncoupled panel and enclo-sure modes used in this analysis some important modes may

Shock and Vibration 9

0 2 4 6 8

Pane

l mod

e

|VM| (ms) times10minus4

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 5 Modal amplitude of the panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

be excluded and truncation errors exist in the calculation ofresonance frequencies and decay times of acoustical modesFigure 4(d) shows the resonance frequencies of two acousti-calmodes using different truncated numbers Comparedwiththe solution for the resonance frequencies using the combi-nation of 64 enclosure modes and 81 panel modes and of 125enclosure modes and 121 panel modes the solution for theresonance frequencies of the (001) enclosure-controlled andthe (31) panel-controlled acoustical modes using 40 panelmodes and 40 enclosuremodesmeets the requirement in thisanalysis and the computation efficiency is also improved

When the panel thickness is 00882m the two subsystemsmodal amplitudes of panel vibration and enclosure acousticfield are shown in Figures 5 and 6 respectively The panelvibration of coupled system is controlled by the (31) panelmode in Figures 5(a) and 5(b) The sound field in the enclo-sure of coupled system is controlled by the (001) enclosuremode in Figures 6(a) and 6(b) Combined with energy ratioshown in Figure 4(a) it is proved that the coupled systemis named the (31) panel-controlled acoustical mode and the(001) enclosure-controlled acoustical mode respectively

412 Panel Internal Damping The decay times of uncoupledclamped panel modes are used to describe the mechanicaldamping of the clamped panel in the analysis The panelvibration energy is dissipated by its internal damping andthen the vibration level of panel will be reduced The dis-turbing degree of the enclosure sound field due to the panelvibration becomes smaller and the energy flow betweenthe panel and enclosure becomes smaller too The decaytimes and the resonance frequencies of enclosure-controlledmodes are related to the decay times of uncoupled panelmodes Figures 7(a) and 7(b) show plot of the variation ofthe resonance frequencies and the decay times of the firstfew enclosure-controlled modes with the decay time of panelmodes respectively In this analysis the decay times of allrigid wall enclosure modes are 15 s The thickness of panel is0007m and the depth of enclosure is 06mThe decay timesof all uncoupled panel modes are equal

In Figure 7(a) the resonance frequencies of the firstfew enclosure-controlled modes approach the uncoupledenclosure resonance frequency as the panelmodal decay timeis reduced (corresponding to the damping increased) Thebigger the damping in the panel is the more the energy is

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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2 Shock and Vibration

X

Y

Z

Clamped panel

120572

Lx = 10m

Ly = 08m

Lz

120579

Pi

q

Figure 1 Panel-enclosure geometry and coordinate system

Sung and Jan [19] andArenas [20 21] the transverse vibrationresponse and sound radiation by a clamped panel wereobtained using the virtual work principle Comparedwith theRayleigh-Ritz approach virtual work principle provides aneasier way to calculate the surface response of panel In thecurrent work the mode shape function proposed by Sungand Arenas is used to analyze the characteristics of panel-enclosure coupled systemwhich consists of an enclosure withone clamped flexible wall

In this paper the vibroacoustic analysis of a rectangularenclosure which is bounded by a clamped flexible wall isconducted using the modal coupling method Acousticalmodes are used to describe the coupled system behaviorFirstly the free vibration characteristics of panel-enclosuresystem are solved based on the modal parameters of uncou-pled panel and enclosure and the characteristics of coupledsystem which reflect the intrinsic properties are obtainedSecondly the features of coupled system which includeresonance frequency and modal decay time are analyzedwith different variables including panel thickness panelinternal damping and enclosure depth The energy transferresonance frequencies and decay times of acoustical modesare distinguished from others when the panel mode andenclosure mode are well coupled This work is an extensionof the work carried by Pan and Bies [3] and Sung and Jan [19]where the effect of fluid-structural coupling on the soundfield in the enclosure is studied with a simply supportedflexible wall using the modal coupling method and theclamped boundary panel with new mode shape function isused to analyze the vibration response respectively

2 Analytical Model

A panel-enclosure coupled system is shown in Figure 1 Theenclosure has five small absorptive walls and one flexible walland the dimensions of enclosure are 119871

119909 119871119910 and 119871

119911 The

panel which is clamped along its edges at 119911 = 119871119911 has

the dimensions of 119871119909 119871119910 119875119894119899is the amplitude of plane wave

119875119894 and the 120572 120579 are respectively the elevation and azimuth

angles with respect to the panel center 119902 is volume velocityof source in the enclosure and it is located at origin ofcoordinate (119909 119910 119911) = (0 0 0)mThe sound pressure 119901 in theenclosure is described by an inhomogeneous wave equation

nabla2

119901 minus1

1198882

0

1205972

119901

1205971199052= minus1205880

120597119902

120597119905 (1)

where 1198880 1205880are the speed of sound and air density respec-

tively At the five absorptive walls inside the enclosure andthe flexible enclosure interface the boundary conditions takethe following form

120597119901

120597n= minus119895

1205961205880

119885119860

119901 at absorbent walls

120597119901

120597119911

10038161003816100381610038161003816100381610038161003816119911=119871119911

= minus1205880

1205972

119908

1205971199052at flexible wall

(2)

where n indicates the normal direction of the boundarysurfaces 119885

119860is the specific acoustic impedance on the

absorptive walls inside the enclosure119908(119909 119910 119911) is the normaldisplacement of flexible panel

The governing equation of flexible vibration of thinisotropic panel due to the internal pressure 119901 and plane wave119901119894119899on the outside of the panel is

120588ℎ1205972

119908

1205971199052+ 119863nabla4

119908 = 119901 minus 119901119894119899 (3)

where 119863 = 119864ℎ3

12(1 minus 1205832

) and 120588 119864 120583 ℎ are the densityYoungrsquos modulus Poissonrsquos ration and thickness of the flex-ible panel respectively In this analysis the panel radiationis neglected and the positive direction for displacement ofclamped panel is towards outside of the enclosure

In order to use the modal shape expansion the soundpressure 119901 inside enclosure the flexible panel velocity V =

119895120596119908 the sound pressure 119901119894119899on the outside of the panel and

the monopole acoustic source 119902 inside the enclosure can beexpanded by mode shape functions

119901 (r) = [120601119873]119879

[119901119873] = Φ

TP (4)

V (120590) = [119904119872]119879

[V119872]119879

= STV (5)

Shock and Vibration 3

119901119894119899(120590) =[119904

119872]119879

[119875119894119899119872

]119879

= STP119894119899 (6)

119902 (r) = [120601119873]119879

[119876119873] = Φ

119879Q (7)

In these equations 119901119873 V119872 119875119894119899119872

119876119873are modal amplitude

of sound pressure inside enclosure flexible panel vibrationvelocity the sound pressure on the outside of the clampedpanel surface and the acoustic source inside enclosurerespectively120601

119873 119904119872aremodal shape functions of rigidlywalls

enclosure and uncoupled clamped panel respectively Herethey are the base functions for the analysis of enclosure withmodally reactive flexible panel and locally reactive walls

The mode shape function 120601119873of rigid wall enclosure is

given by

120601119873(r) = 120601

119873(119909 119910 119911) = cos(119897120587119909

119871119909

) cos(119898120587119910

119871119910

) cos(119899120587119911119871119911

)

(8)

Here 119897 119898 119899 are themode indices of the119873th enclosuremodeThemode shape function 119878

119872of uncoupled clamped panel

is given by Sung and Jan [19] and it has been validated byexperiment

119904119872(120590) = 119904

119872(119909 119910) = 120595

119906(119909) 120595V (119910)

120595119906(119909) = 120574(

120582119906119909

119871119909

) minus120574 (120582119906)

119867 (120582119906)119867(

120582119906119909

119871119909

)

120595V (119910) = 120574(120582V119910

119871119910

) minus120574 (120582V)

119867 (120582V)119867(

120582V119910

119871119910

)

(9)

where 120574(119904) = cosh(119904) minus cos(119904)119867(119904) = sinh(119904) minus sin(119904)120582119906 120582V are determined by (10) and 119906 V are mode indices

of clamped panel modes

cosh (120582) cos (120582) minus 1 = 0 (10)

Considering the acoustic boundary condition in (2) andusing the Green function technique the complex acousticpressure 119901 in the enclosure can be expressed as

119901 = 1198951205961205880int119860119891

V119866119860119889119904 + 119895120596120588

0int1198810

119902119866119860119889119881 + 119895

120596

1198880

int119860119871

120573119866119860119901119889119904

(11)

Here 119860119891 119860119897 1198810are the surface of flexible panel the

absorbent wall surface inside enclosure and the volume ofenclosure respectively 120573 = 120588

01198880119885119860is the specific acoustic

admittance ratio on the absorbent wall surface 119866119860is the

sound field Greenrsquos function

119866119860(r r0) = sum

119873

120601119873(r) 120601119873(r0)

1198810Λ119873(1198962 minus 119896

2

119886119873)

Λ119873=1

1198810

int1198810

1206012

119873(r) 119889119903

(12)

Similarly the flexible clamped panel vibration velocitycan be expressed as

V = 119895120596

120588ℎint119860119891

(119901 minus 119901119894119899) 119866119901119889119904 (13)

119866119901is panel Greenrsquos function which can be expressed as

119866119901(1205901205900) = minussum

119872

119904119872(120590) 119904119872(1205900)

119860119891Λ119872(1205962 minus 120596

2

119901119872)

Λ119872=

1

119860119891

int119860119891

1199042

119872(120590) 119889120590

(14)

Substituting (4) (5) (6) and (7) into (11) and (13) we canobtain

(120582I minus A)X = Y (15)

Equation (15) can be expressed as

(120582I minus (A11 A12

A21

A22

))(

PV120582P120582V

) = 120582(

0

0

EQEF

) (16)

where I is unit matrix 120582 = minus119895119896 A11and A

12are (119873 + 119872) times

(119873 +119872) zero and unit matrices respectively

EQ = minus11988801205880(11987611198762sdot sdot sdot 119876

119873)119879

(17)

EF=1

120588ℎ1198880

(1198751198941198991

1198751198941198992

sdot sdot sdot 119875119894119899119872

)119879

(18)

A21= minus

[[[[[[[[

[

1198962

1198861

d 0

1198962

119886119873

1198962

1199011

0 d1198962

119901119872

]]]]]]]]

]

(19)

4 Shock and Vibration

A22= minus

[[[[[[[[[[[[[[[[[[[

[

1205781198861

0

11988801205882

011986011989111986111

119872119860

1

sdot sdot sdot

11988801205882

01198601198911198611119872

119872119860

1

d d

0 120578119886119873

11988801205882

01198601198911198611198731

119872119860

119873

11988801205882

0119860119891119861119873119872

119872119860

119873

minus

11986011989111986111

1198880119872119875

1

sdot sdot sdot minus

1198601198911198611198731

1198880119872119875

1

1205781199011

0

d d

minus

1198601198911198611119872

1198880119872119875

119872

minus

119860119891119861119873119872

1198880119872119875

119872

0 120578119901119872

]]]]]]]]]]]]]]]]]]]

]

(20)

119861119873119872

=1

119860119891

int119860119891

119904119872(120590) 120601119873(r) 119889119904 (21)

120578119901119872

=44120587

1198791199011198721198880

(22)

120578119886119873

=44120587

1198791198861198731198880

(23)

where 119896119886119873

= 1205961198861198731198880 119896119901119872

= 1205961199011198721198880 119896119886119873

120596119886119873

are thewavenumber and the resonance angle frequency of the 119873thenclosure mode respectively 119896

119901119872and 120596

119901119872are respectively

the wavenumber and the resonance angle frequency of the119872th panel mode119872119875119872119860 are the modal mass of panel andenclosure respectively 119872119875

119872= 120588ℎ119860

119891Λ119872 119872119860119873= 12058801198810Λ119873

1198601198911198810are the area of flexible panel and volume of enclosure

respectively 120578119901119872

120578119886119873

are respectively the loss factor of119872thpanel mode and 119873th enclosure mode Particularly 120578

119886119873is

related to the integral of the specific acoustic admittance 120573119879119901119872

119879119886119873

are respectively the 60 dB decay time of119872th panelmode and119873th enclosure mode

3 The Modal Properties ofthe Panel-Enclosure System

31 Modal Coupling Coefficient Modal coupling coefficientis the spatial matching degree between enclosure and panelmodes on the interacting surface of enclosure sound fieldand flexible panel from (21) It is the integral of panelmode and enclosure mode on the surface of panel Modalcoupling coefficient only depends on the geometry shape andboundary condition of panel the mode shape of enclosureacoustic field [3] and it has nothing to do with thicknessmaterial properties damping of the panel and enclosuredepth

Like the panel-enclosure system which consists of anenclosure with a simply-supported flexible wall the modecoupling coefficient between clamped panel and enclosure is

nonzero which must satisfy the following two conditions atthe same time

119906 + 119897 = odd number

V + 119898 = odd number(24)

32 Transfer Factor Coupling coefficient determines thematching degree between an enclosure mode and a panelmode on the interacting surface of enclosure sound field andpanel while the coupling extent between a panelmode and anenclosure mode is decided by transfer factor [3] The transferfactor 119865

119873119872between the119873th enclosure acoustical mode and

119872th panel mode is given by

119865119873119872

=

1 + [

(120596119886119873minus 120596119901119872)

2]

2

[1

119861(119873119872)2]

minus1

(25)

119861 (119873119872) = (12058801198882

0

120588ℎ119871119911Λ119873Λ119872

)

12

119861119873119872

(26)

When the modal coupling coefficient between the 119873thenclosure acoustical mode and119872th panel mode changes thetransfer factor 119865

119873119872has different results from (25) and (26)

There are two different kinds of transfer factors (1) as themodal coupling coefficient 119861

119873119872is equal to zero the transfer

factor 119865119873119872

must be equal to zero In this situation thereis not energy transfer between the 119873th enclosure acousticalmode and 119872th panel mode (2) When the modal couplingcoefficient is nonzero the transfer factor relates to manyfactors such as the difference between resonance frequenciesof uncoupled panel and enclosure modes and the enclosuredepth The larger the transfer factor between a panel modeand an enclosure mode is the bigger the coupling strengthbetween them is obtained When the transfer factor 119865

119873119872

is in the order of 10 the energy transfer between the 119873thenclosure mode and119872th panel mode is important

33 Resonance Frequency and Modal Decay Time of CoupledSystem Thesound pressure in the enclosure and the clampedpanel vibration velocity are used to describe the response

Shock and Vibration 5

of coupled panel-enclosure system Through the surface offlexible panel facing the inside of enclosure the sound fieldin the enclosure is coupling with the vibration of flexiblepanel There are two different kinds of acoustical modes inthe panel-enclosure coupled system one is an enclosure-controlled acoustical mode whose most of energy is stored inthe enclosure sound field and the other is a panel-controlledacoustical mode whose most of energy is stored as panelvibration energy [3]

If the external excitation does not exist in the coupledsystem there will be Y = 0 in (15) and it becomes a 2(119873 +

119872) dimensional system of equations Corresponding to theeigenequation there will be 2(119873 + 119872) eigenvalues 120582

119871and

120582lowast

119871 and 119871 = 1 2 sdot sdot sdot (119873 + 119872) The resonance frequency 119891

119871

and the decay time 119879119871of coupled system are Im(120582

119871)11988802120587

and 691Re(120582119871)1198880 respectively When the Y = 0 in (15)

the solution of the coefficient X is the modal amplitude ofcoupled system Then the panel vibration velocity and thesound pressure in the enclosure which describes the forcedresponse of coupled system can be obtained from (4) and (5)

The time-averaged acoustic potential energy 119864119886119873

in theenclosure and the time-averaged vibration kinetic energy119864119901119872

of the flexible panel are given by [14]

119864119886119873

=1

(412058801198882

0) int1198810

1003816100381610038161003816119901 (r 120596)1003816100381610038161003816

2

119889119881

=1198810PHΛ119886119873

P(412058801198882

0)

(27)

119864119901119872

=120588ℎ

4int119860119891

|V (120590 120596)|2

119889119904

=

120588ℎ119860119891VH

Λ119901119872

V4

(28)

HereΛ119886119873

is a119873times119873 diagonalmatrix with each diagonal termconsisting of Λ

119873 and Λ

119901119872is a119872times119872 diagonal matrix with

each diagonal term consisting of Λ119872

4 Results and Discussion

To demonstrate the properties of the panel-enclosure coupledsystemwhich consists of an enclosure with a clamped flexiblewall the resonance frequencies and modal decay times ofacoustical modes are investigated with different panel modaldensity panel internal damping and enclosure depth respec-tively The panel-enclosure coupled system which consists ofan enclosure with a clamped panel on top and five absorptivewalls is shown in Figure 1 The panel material properties aretaken as follows the material of clamped panel is aluminumwith density 120588 = 2770 kgm3 Youngrsquos modulus 119864 = 71Gpaand Poissonrsquos ratio 120583 = 033

41 Effects of Different Panel Physical Parameters onAcoustical Modes

411 Panel Modal Density From (25) as the differencebetween resonance frequencies of uncoupled enclosure andpanel modes is decreased the transfer factors betweenthem become larger when the modal coupling coefficient is

nonzero The resonance frequencies of rigid walls enclosuremode and uncoupled clamped panel mode [21] are given by

119891119897119898119899

=1198880

2[(

119897

119871119909

)

2

+ (119898

119871119910

)

2

+ (119899

119871119911

)

2

] (29)

119891119906V =

1

2120587radic119863

120588

radic(120582119906

119871119909

)

4

+ (120582V

119871119910

)

4

+ 2(120582119906120582V

119871119909119871119910

)

2

120581119906120581V

120585119906120585V

(30)

120581119894=1

4(1 + 119863

2

119894) sinh (2120582

119894) minus

1

2119863119894cosh (2120582

119894)

minus1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) minus 119863119894cos2 (120582

119894)

minus 1198632

119894120582119894+3

2119863119894

(31)

120585119894=1

4(1 + 119863

2

119894) sinh (2120582

119894)

+ sinh (120582119894) [2119863119894sin (120582

119894) minus (1 minus 119863

2

119894) cos (120582

119894)]

minus (1 + 1198632

119894) sin (120582

119894) cosh (120582

119894)

+1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) + 120582119894

minus1

2119863119894[1 + cosh (2120582

119894)] + 119863

119894cos2 (120582

119894)

(32)

119863119894=120574 (120582119894)

119867 (120582119894) (33)

Similar to an enclosure with a simply supported flexiblewall the modal density of uncoupled clamped panel andenclosure will affect energy transfer between them Themodal density of uncoupled clamped panel and enclosuresound field are given by [3 10]

119899119901=

radic3119860119891

119862119871ℎ (34)

119899119886=412058711988101198912

1198883

0

+120587119878119891

21198882

0

+119871

81198880

(35)

Here 119860119891 119862119871are the area and longitudinal wave speed

of clamped flexible wall respectively 119891 is the excitationfrequency 119878 119871 are the total surface area and the total edgelengths inside the enclosure

As mentioned above the transfer factor determines thecoupling strength between a panel mode and an enclosuremode From (34) and (25) by adjusting the panel modaldensity (corresponding to panel thickness) the distributionof the resonance frequencies of panel is changed whichalso leads to changes of the transfer factors between paneland enclosure modes In Figure 2 transfer factors between

6 Shock and Vibration

005 01 015 02 025 030

02

04

06

08

1

Tran

sfer f

acto

r

np (Hzminus1)

(11)

(13)

(33)

(15)

(35)

(51)

(17)

(53)

(37)

(55)

(31)

Figure 2 Transfer factors between the (001) enclosure mode andpanelmodes as a function of panelmodal density119871

119885= 06m119879

119886119873=

15 s 119879119901119872

= 05 s

the (001) enclosure mode and panel modes are plottedagainst panel modal density When the panel modal densityis low only the (31) panel mode participates in the couplingwith the (001) enclosure mode and satisfies the well-coupledcondition As the panel modal density is increased moreand more high-order panel modes in which the couplingcoefficients with the (001) enclosure mode are not equalto zero participate in coupling At these points of transferfactors when their values are in the order of 10 there is largeenergy transfer between enclosure and panel modes Similarto the transfer factors between the (001) enclosure modeand panel modes the distribution of transfer factors betweenother enclosure modes and panel modes are that only fewpanel modes participate in the coupling with enclosuremodes in the low panel modal density region andmore panelmodes participate in the coupling with enclosure mode forhigh panel modal density region

As the coupling extent between enclosure and panelmodes changes with the variation of panel modal densitythe resonance frequency of acoustical mode which is oneimportant characteristic of it is altered also Figure 3(a)shows resonance frequencies of the first few enclosure-controlled acoustical modes as a function of panel modaldensity Forty panel modes and forty enclosure modes areused in this analysis The decay times of all uncoupledenclosure and panel modes are 15 s and 05 s respectivelyThedepth of enclosure is 06m As the panel modal density isincreased the resonance frequencies of enclosure-controlledmodes jump to higher frequencies Compared with theresonance frequencies of acoustical mode in the low panelmodal density region they become larger in the high panelmodal density region The reason is that many high-order

panel modes participate in the coupling with enclosuremodes Similar to the panel-enclosure coupled system with asimply supported flexible panel the energy transfer from theenclosuremode is distributed overmany panel modes and nopanel modes are well coupling with enclosure modes

Figure 3(b) shows plot of the variation of decay times ofthe first few enclosure-controlled modes with panel modaldensity When the panel modal density is small the modaldecay time is longer on the average As the panel modaldensity is increased the modal decay times show someminima It is because of that only few panel modes satisfythe well-coupled condition with enclosure modes in the lowpanel modal density region As the panel modal density isincreased more panel modes participate in the coupling withenclosure modes and decay times of enclosure-controlledmodes become shorter At the same time there are no panelmodes which are well coupled with enclosure mode and theenergy which is stored in the panel is equally distributed overmany panel modes

In order to describe the process of strong couplingbetween panel and enclosure modes when the panel modaldensity is altered we analyze from three points of view ofenergy ratio between the panel vibration and sound field inthe enclosure resonance frequency and modal decay time ofcoupled system respectively In this analysis the plane wave119901119894with amplitude 1 Pa on the panel surface is used to drive the

vibration of panel the excitation frequencies are respectivelythe resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The angles of elevation 120572 andazimuth 120579 of plane wave are 90∘ and 0∘ respectively Thedecay times of all uncoupled panel modes and enclosuremodes are 15 s and 05 s respectively As seen in Figure 2 thetransfer factor between the (31) panel mode and the (001)enclosure mode is approximately equal to 10 in low panelmodal density region and only these two modes satisfy well-coupled condition which means that the energy conversionbetween the sound field and the panel is almost entirelybetween these two modes

The energy ratio between the panel vibration and soundfield in the enclosure is shown in Figure 4(a) which containspanel-controlled and enclosure-controlled acoustical modesThe resonance frequencies of acoustical modes uncoupledpanel and enclosure modes are shown in Figure 4(b) as afunction of panel modal density Figure 4(c) shows the varia-tion of the decay time of acoustical modes with panel modaldensity The effect of truncation numbers on the resonancefrequencies of acoustical modes is shown in Figure 4(d)As the panel modal density tends to the point of transferfactor between the (31) panel mode and the (001) enclosuremode which is in the order of 10 energy ratio resonancefrequencies decay times of the (31) panel-controlled modeand the (001) enclosure-controlled mode tend to be equaltoo

Near the point of panel modal density where the maxi-mum interaction strength between the (31) panel mode andthe (001) enclosure mode the energy which is stored in thepanel vibration and enclosure sound field of two acousticalmodes including the (001) enclosure-controlled mode andthe (31) panel-controlled mode are approximately equal to

Shock and Vibration 7

01 02 03 01 02 03

01 02 03

01 02 0301 02 03

01 02 03

210

220

230

280

290

300

160

170

180

350

360

370

270

280

330

335

340

(001)

(100) (011)

(101)(110)

(010)

fL

(Hz)

fL

(Hz)

fL

(Hz)

np (Hzminus1) np (Hzminus1)

(a)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(101)

(100)

(001)(010)

(011)

(110)

01 02 03

01 02 03 01 02 03

01 02 0301 02 03

01 02 03

TL

(s)

TL

(s)

TL

(s)

np (Hzminus1)np (Hzminus1)

(b)

Figure 3 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal density (a)resonance frequency (b) decay time 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

each other as shown in Figure 4(a) The further the distancefrom that point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlledmodeThe result also shows that energyratio curve of the (001) enclosure-controlled mode is acontinuation of that of the (31) panel-controlled mode as afunction of panel modal density and the energy ratio curveof the (31) panel-controlled mode is a continuation of that ofthe (001) enclosure-controlled mode

It is shown in Figure 4(b) that in the vicinity of thepoint of panel modal density where the transfer factor is

approximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the panel modal density is increased the resonancefrequency of the (001) enclosure-controlled mode jumpsfrom lower than the resonance frequency of the (001) rigidwalls enclosure mode to higher than it when it passes by thepoint ofmaximumenergy transfer Comparedwith the (001)

8 Shock and Vibration

003 00305 0031 00315 0032 00325 0033 00335 0034

0

5

10

15

minus5

minus10

minus15

np (Hzminus1)

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

003 00305 0031 00315 0032 00325 0033 00335 0034260

270

280

290

300

310

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

np (Hzminus1)

fL

(Hz)

(b)

0

1

2

3

4

5

6

7

8

003 00305 0031 00315 0032 00325 0033 00335 0034

(001) enclosure-controlled mode(31) panel-controlled mode

np (Hzminus1)

TL

(s)

(c)

265

270

275

280

285

290

295

300

305

310

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

003 00305 0031 00315 0032 00325 0033 00335 0034

np (Hzminus1)

fL

(Hz)

(d)

Figure 4 (a) Ratio between the sound field energy and panel vibration energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers 119871

119885= 06m

119879119901119872

= 05 s 119879119886119873

= 15 s

enclosure-controlled mode it is opposite to the (31) panel-controlled mode As the panel modal density is away fromthe point of maximum interaction strength the resonancefrequencies of the (31) panel-controlled mode and the (001)enclosure-controlled mode tend to resonance frequencies ofuncoupled them respectively

It is shown in Figure 4(c) that as the panel modal densityis increased decay time of the (001) enclosure-controlledmode firstly reduces When panel modal density arrives atthe point of strong coupling the minimum decay time is gotThen decay time of the (001) enclosure-controlled mode

becomes longer as the panel modal density is increasedCompared with the (001) enclosure-controlled mode decaytime curve of the (31) panel-controlled mode is opposite asthe panel modal density is increased In addition at the pointof panel modal density where the decay times of the (001)enclosure-controlled mode and the (31) panel-controlledmode are approximately equal to each other the energytransfer from the panel vibration to the enclosure acousticfield is relative maximum

Due to the limited number of uncoupled panel and enclo-sure modes used in this analysis some important modes may

Shock and Vibration 9

0 2 4 6 8

Pane

l mod

e

|VM| (ms) times10minus4

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 5 Modal amplitude of the panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

be excluded and truncation errors exist in the calculation ofresonance frequencies and decay times of acoustical modesFigure 4(d) shows the resonance frequencies of two acousti-calmodes using different truncated numbers Comparedwiththe solution for the resonance frequencies using the combi-nation of 64 enclosure modes and 81 panel modes and of 125enclosure modes and 121 panel modes the solution for theresonance frequencies of the (001) enclosure-controlled andthe (31) panel-controlled acoustical modes using 40 panelmodes and 40 enclosuremodesmeets the requirement in thisanalysis and the computation efficiency is also improved

When the panel thickness is 00882m the two subsystemsmodal amplitudes of panel vibration and enclosure acousticfield are shown in Figures 5 and 6 respectively The panelvibration of coupled system is controlled by the (31) panelmode in Figures 5(a) and 5(b) The sound field in the enclo-sure of coupled system is controlled by the (001) enclosuremode in Figures 6(a) and 6(b) Combined with energy ratioshown in Figure 4(a) it is proved that the coupled systemis named the (31) panel-controlled acoustical mode and the(001) enclosure-controlled acoustical mode respectively

412 Panel Internal Damping The decay times of uncoupledclamped panel modes are used to describe the mechanicaldamping of the clamped panel in the analysis The panelvibration energy is dissipated by its internal damping andthen the vibration level of panel will be reduced The dis-turbing degree of the enclosure sound field due to the panelvibration becomes smaller and the energy flow betweenthe panel and enclosure becomes smaller too The decaytimes and the resonance frequencies of enclosure-controlledmodes are related to the decay times of uncoupled panelmodes Figures 7(a) and 7(b) show plot of the variation ofthe resonance frequencies and the decay times of the firstfew enclosure-controlled modes with the decay time of panelmodes respectively In this analysis the decay times of allrigid wall enclosure modes are 15 s The thickness of panel is0007m and the depth of enclosure is 06mThe decay timesof all uncoupled panel modes are equal

In Figure 7(a) the resonance frequencies of the firstfew enclosure-controlled modes approach the uncoupledenclosure resonance frequency as the panelmodal decay timeis reduced (corresponding to the damping increased) Thebigger the damping in the panel is the more the energy is

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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Shock and Vibration

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Shock and Vibration 3

119901119894119899(120590) =[119904

119872]119879

[119875119894119899119872

]119879

= STP119894119899 (6)

119902 (r) = [120601119873]119879

[119876119873] = Φ

119879Q (7)

In these equations 119901119873 V119872 119875119894119899119872

119876119873are modal amplitude

of sound pressure inside enclosure flexible panel vibrationvelocity the sound pressure on the outside of the clampedpanel surface and the acoustic source inside enclosurerespectively120601

119873 119904119872aremodal shape functions of rigidlywalls

enclosure and uncoupled clamped panel respectively Herethey are the base functions for the analysis of enclosure withmodally reactive flexible panel and locally reactive walls

The mode shape function 120601119873of rigid wall enclosure is

given by

120601119873(r) = 120601

119873(119909 119910 119911) = cos(119897120587119909

119871119909

) cos(119898120587119910

119871119910

) cos(119899120587119911119871119911

)

(8)

Here 119897 119898 119899 are themode indices of the119873th enclosuremodeThemode shape function 119878

119872of uncoupled clamped panel

is given by Sung and Jan [19] and it has been validated byexperiment

119904119872(120590) = 119904

119872(119909 119910) = 120595

119906(119909) 120595V (119910)

120595119906(119909) = 120574(

120582119906119909

119871119909

) minus120574 (120582119906)

119867 (120582119906)119867(

120582119906119909

119871119909

)

120595V (119910) = 120574(120582V119910

119871119910

) minus120574 (120582V)

119867 (120582V)119867(

120582V119910

119871119910

)

(9)

where 120574(119904) = cosh(119904) minus cos(119904)119867(119904) = sinh(119904) minus sin(119904)120582119906 120582V are determined by (10) and 119906 V are mode indices

of clamped panel modes

cosh (120582) cos (120582) minus 1 = 0 (10)

Considering the acoustic boundary condition in (2) andusing the Green function technique the complex acousticpressure 119901 in the enclosure can be expressed as

119901 = 1198951205961205880int119860119891

V119866119860119889119904 + 119895120596120588

0int1198810

119902119866119860119889119881 + 119895

120596

1198880

int119860119871

120573119866119860119901119889119904

(11)

Here 119860119891 119860119897 1198810are the surface of flexible panel the

absorbent wall surface inside enclosure and the volume ofenclosure respectively 120573 = 120588

01198880119885119860is the specific acoustic

admittance ratio on the absorbent wall surface 119866119860is the

sound field Greenrsquos function

119866119860(r r0) = sum

119873

120601119873(r) 120601119873(r0)

1198810Λ119873(1198962 minus 119896

2

119886119873)

Λ119873=1

1198810

int1198810

1206012

119873(r) 119889119903

(12)

Similarly the flexible clamped panel vibration velocitycan be expressed as

V = 119895120596

120588ℎint119860119891

(119901 minus 119901119894119899) 119866119901119889119904 (13)

119866119901is panel Greenrsquos function which can be expressed as

119866119901(1205901205900) = minussum

119872

119904119872(120590) 119904119872(1205900)

119860119891Λ119872(1205962 minus 120596

2

119901119872)

Λ119872=

1

119860119891

int119860119891

1199042

119872(120590) 119889120590

(14)

Substituting (4) (5) (6) and (7) into (11) and (13) we canobtain

(120582I minus A)X = Y (15)

Equation (15) can be expressed as

(120582I minus (A11 A12

A21

A22

))(

PV120582P120582V

) = 120582(

0

0

EQEF

) (16)

where I is unit matrix 120582 = minus119895119896 A11and A

12are (119873 + 119872) times

(119873 +119872) zero and unit matrices respectively

EQ = minus11988801205880(11987611198762sdot sdot sdot 119876

119873)119879

(17)

EF=1

120588ℎ1198880

(1198751198941198991

1198751198941198992

sdot sdot sdot 119875119894119899119872

)119879

(18)

A21= minus

[[[[[[[[

[

1198962

1198861

d 0

1198962

119886119873

1198962

1199011

0 d1198962

119901119872

]]]]]]]]

]

(19)

4 Shock and Vibration

A22= minus

[[[[[[[[[[[[[[[[[[[

[

1205781198861

0

11988801205882

011986011989111986111

119872119860

1

sdot sdot sdot

11988801205882

01198601198911198611119872

119872119860

1

d d

0 120578119886119873

11988801205882

01198601198911198611198731

119872119860

119873

11988801205882

0119860119891119861119873119872

119872119860

119873

minus

11986011989111986111

1198880119872119875

1

sdot sdot sdot minus

1198601198911198611198731

1198880119872119875

1

1205781199011

0

d d

minus

1198601198911198611119872

1198880119872119875

119872

minus

119860119891119861119873119872

1198880119872119875

119872

0 120578119901119872

]]]]]]]]]]]]]]]]]]]

]

(20)

119861119873119872

=1

119860119891

int119860119891

119904119872(120590) 120601119873(r) 119889119904 (21)

120578119901119872

=44120587

1198791199011198721198880

(22)

120578119886119873

=44120587

1198791198861198731198880

(23)

where 119896119886119873

= 1205961198861198731198880 119896119901119872

= 1205961199011198721198880 119896119886119873

120596119886119873

are thewavenumber and the resonance angle frequency of the 119873thenclosure mode respectively 119896

119901119872and 120596

119901119872are respectively

the wavenumber and the resonance angle frequency of the119872th panel mode119872119875119872119860 are the modal mass of panel andenclosure respectively 119872119875

119872= 120588ℎ119860

119891Λ119872 119872119860119873= 12058801198810Λ119873

1198601198911198810are the area of flexible panel and volume of enclosure

respectively 120578119901119872

120578119886119873

are respectively the loss factor of119872thpanel mode and 119873th enclosure mode Particularly 120578

119886119873is

related to the integral of the specific acoustic admittance 120573119879119901119872

119879119886119873

are respectively the 60 dB decay time of119872th panelmode and119873th enclosure mode

3 The Modal Properties ofthe Panel-Enclosure System

31 Modal Coupling Coefficient Modal coupling coefficientis the spatial matching degree between enclosure and panelmodes on the interacting surface of enclosure sound fieldand flexible panel from (21) It is the integral of panelmode and enclosure mode on the surface of panel Modalcoupling coefficient only depends on the geometry shape andboundary condition of panel the mode shape of enclosureacoustic field [3] and it has nothing to do with thicknessmaterial properties damping of the panel and enclosuredepth

Like the panel-enclosure system which consists of anenclosure with a simply-supported flexible wall the modecoupling coefficient between clamped panel and enclosure is

nonzero which must satisfy the following two conditions atthe same time

119906 + 119897 = odd number

V + 119898 = odd number(24)

32 Transfer Factor Coupling coefficient determines thematching degree between an enclosure mode and a panelmode on the interacting surface of enclosure sound field andpanel while the coupling extent between a panelmode and anenclosure mode is decided by transfer factor [3] The transferfactor 119865

119873119872between the119873th enclosure acoustical mode and

119872th panel mode is given by

119865119873119872

=

1 + [

(120596119886119873minus 120596119901119872)

2]

2

[1

119861(119873119872)2]

minus1

(25)

119861 (119873119872) = (12058801198882

0

120588ℎ119871119911Λ119873Λ119872

)

12

119861119873119872

(26)

When the modal coupling coefficient between the 119873thenclosure acoustical mode and119872th panel mode changes thetransfer factor 119865

119873119872has different results from (25) and (26)

There are two different kinds of transfer factors (1) as themodal coupling coefficient 119861

119873119872is equal to zero the transfer

factor 119865119873119872

must be equal to zero In this situation thereis not energy transfer between the 119873th enclosure acousticalmode and 119872th panel mode (2) When the modal couplingcoefficient is nonzero the transfer factor relates to manyfactors such as the difference between resonance frequenciesof uncoupled panel and enclosure modes and the enclosuredepth The larger the transfer factor between a panel modeand an enclosure mode is the bigger the coupling strengthbetween them is obtained When the transfer factor 119865

119873119872

is in the order of 10 the energy transfer between the 119873thenclosure mode and119872th panel mode is important

33 Resonance Frequency and Modal Decay Time of CoupledSystem Thesound pressure in the enclosure and the clampedpanel vibration velocity are used to describe the response

Shock and Vibration 5

of coupled panel-enclosure system Through the surface offlexible panel facing the inside of enclosure the sound fieldin the enclosure is coupling with the vibration of flexiblepanel There are two different kinds of acoustical modes inthe panel-enclosure coupled system one is an enclosure-controlled acoustical mode whose most of energy is stored inthe enclosure sound field and the other is a panel-controlledacoustical mode whose most of energy is stored as panelvibration energy [3]

If the external excitation does not exist in the coupledsystem there will be Y = 0 in (15) and it becomes a 2(119873 +

119872) dimensional system of equations Corresponding to theeigenequation there will be 2(119873 + 119872) eigenvalues 120582

119871and

120582lowast

119871 and 119871 = 1 2 sdot sdot sdot (119873 + 119872) The resonance frequency 119891

119871

and the decay time 119879119871of coupled system are Im(120582

119871)11988802120587

and 691Re(120582119871)1198880 respectively When the Y = 0 in (15)

the solution of the coefficient X is the modal amplitude ofcoupled system Then the panel vibration velocity and thesound pressure in the enclosure which describes the forcedresponse of coupled system can be obtained from (4) and (5)

The time-averaged acoustic potential energy 119864119886119873

in theenclosure and the time-averaged vibration kinetic energy119864119901119872

of the flexible panel are given by [14]

119864119886119873

=1

(412058801198882

0) int1198810

1003816100381610038161003816119901 (r 120596)1003816100381610038161003816

2

119889119881

=1198810PHΛ119886119873

P(412058801198882

0)

(27)

119864119901119872

=120588ℎ

4int119860119891

|V (120590 120596)|2

119889119904

=

120588ℎ119860119891VH

Λ119901119872

V4

(28)

HereΛ119886119873

is a119873times119873 diagonalmatrix with each diagonal termconsisting of Λ

119873 and Λ

119901119872is a119872times119872 diagonal matrix with

each diagonal term consisting of Λ119872

4 Results and Discussion

To demonstrate the properties of the panel-enclosure coupledsystemwhich consists of an enclosure with a clamped flexiblewall the resonance frequencies and modal decay times ofacoustical modes are investigated with different panel modaldensity panel internal damping and enclosure depth respec-tively The panel-enclosure coupled system which consists ofan enclosure with a clamped panel on top and five absorptivewalls is shown in Figure 1 The panel material properties aretaken as follows the material of clamped panel is aluminumwith density 120588 = 2770 kgm3 Youngrsquos modulus 119864 = 71Gpaand Poissonrsquos ratio 120583 = 033

41 Effects of Different Panel Physical Parameters onAcoustical Modes

411 Panel Modal Density From (25) as the differencebetween resonance frequencies of uncoupled enclosure andpanel modes is decreased the transfer factors betweenthem become larger when the modal coupling coefficient is

nonzero The resonance frequencies of rigid walls enclosuremode and uncoupled clamped panel mode [21] are given by

119891119897119898119899

=1198880

2[(

119897

119871119909

)

2

+ (119898

119871119910

)

2

+ (119899

119871119911

)

2

] (29)

119891119906V =

1

2120587radic119863

120588

radic(120582119906

119871119909

)

4

+ (120582V

119871119910

)

4

+ 2(120582119906120582V

119871119909119871119910

)

2

120581119906120581V

120585119906120585V

(30)

120581119894=1

4(1 + 119863

2

119894) sinh (2120582

119894) minus

1

2119863119894cosh (2120582

119894)

minus1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) minus 119863119894cos2 (120582

119894)

minus 1198632

119894120582119894+3

2119863119894

(31)

120585119894=1

4(1 + 119863

2

119894) sinh (2120582

119894)

+ sinh (120582119894) [2119863119894sin (120582

119894) minus (1 minus 119863

2

119894) cos (120582

119894)]

minus (1 + 1198632

119894) sin (120582

119894) cosh (120582

119894)

+1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) + 120582119894

minus1

2119863119894[1 + cosh (2120582

119894)] + 119863

119894cos2 (120582

119894)

(32)

119863119894=120574 (120582119894)

119867 (120582119894) (33)

Similar to an enclosure with a simply supported flexiblewall the modal density of uncoupled clamped panel andenclosure will affect energy transfer between them Themodal density of uncoupled clamped panel and enclosuresound field are given by [3 10]

119899119901=

radic3119860119891

119862119871ℎ (34)

119899119886=412058711988101198912

1198883

0

+120587119878119891

21198882

0

+119871

81198880

(35)

Here 119860119891 119862119871are the area and longitudinal wave speed

of clamped flexible wall respectively 119891 is the excitationfrequency 119878 119871 are the total surface area and the total edgelengths inside the enclosure

As mentioned above the transfer factor determines thecoupling strength between a panel mode and an enclosuremode From (34) and (25) by adjusting the panel modaldensity (corresponding to panel thickness) the distributionof the resonance frequencies of panel is changed whichalso leads to changes of the transfer factors between paneland enclosure modes In Figure 2 transfer factors between

6 Shock and Vibration

005 01 015 02 025 030

02

04

06

08

1

Tran

sfer f

acto

r

np (Hzminus1)

(11)

(13)

(33)

(15)

(35)

(51)

(17)

(53)

(37)

(55)

(31)

Figure 2 Transfer factors between the (001) enclosure mode andpanelmodes as a function of panelmodal density119871

119885= 06m119879

119886119873=

15 s 119879119901119872

= 05 s

the (001) enclosure mode and panel modes are plottedagainst panel modal density When the panel modal densityis low only the (31) panel mode participates in the couplingwith the (001) enclosure mode and satisfies the well-coupledcondition As the panel modal density is increased moreand more high-order panel modes in which the couplingcoefficients with the (001) enclosure mode are not equalto zero participate in coupling At these points of transferfactors when their values are in the order of 10 there is largeenergy transfer between enclosure and panel modes Similarto the transfer factors between the (001) enclosure modeand panel modes the distribution of transfer factors betweenother enclosure modes and panel modes are that only fewpanel modes participate in the coupling with enclosuremodes in the low panel modal density region andmore panelmodes participate in the coupling with enclosure mode forhigh panel modal density region

As the coupling extent between enclosure and panelmodes changes with the variation of panel modal densitythe resonance frequency of acoustical mode which is oneimportant characteristic of it is altered also Figure 3(a)shows resonance frequencies of the first few enclosure-controlled acoustical modes as a function of panel modaldensity Forty panel modes and forty enclosure modes areused in this analysis The decay times of all uncoupledenclosure and panel modes are 15 s and 05 s respectivelyThedepth of enclosure is 06m As the panel modal density isincreased the resonance frequencies of enclosure-controlledmodes jump to higher frequencies Compared with theresonance frequencies of acoustical mode in the low panelmodal density region they become larger in the high panelmodal density region The reason is that many high-order

panel modes participate in the coupling with enclosuremodes Similar to the panel-enclosure coupled system with asimply supported flexible panel the energy transfer from theenclosuremode is distributed overmany panel modes and nopanel modes are well coupling with enclosure modes

Figure 3(b) shows plot of the variation of decay times ofthe first few enclosure-controlled modes with panel modaldensity When the panel modal density is small the modaldecay time is longer on the average As the panel modaldensity is increased the modal decay times show someminima It is because of that only few panel modes satisfythe well-coupled condition with enclosure modes in the lowpanel modal density region As the panel modal density isincreased more panel modes participate in the coupling withenclosure modes and decay times of enclosure-controlledmodes become shorter At the same time there are no panelmodes which are well coupled with enclosure mode and theenergy which is stored in the panel is equally distributed overmany panel modes

In order to describe the process of strong couplingbetween panel and enclosure modes when the panel modaldensity is altered we analyze from three points of view ofenergy ratio between the panel vibration and sound field inthe enclosure resonance frequency and modal decay time ofcoupled system respectively In this analysis the plane wave119901119894with amplitude 1 Pa on the panel surface is used to drive the

vibration of panel the excitation frequencies are respectivelythe resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The angles of elevation 120572 andazimuth 120579 of plane wave are 90∘ and 0∘ respectively Thedecay times of all uncoupled panel modes and enclosuremodes are 15 s and 05 s respectively As seen in Figure 2 thetransfer factor between the (31) panel mode and the (001)enclosure mode is approximately equal to 10 in low panelmodal density region and only these two modes satisfy well-coupled condition which means that the energy conversionbetween the sound field and the panel is almost entirelybetween these two modes

The energy ratio between the panel vibration and soundfield in the enclosure is shown in Figure 4(a) which containspanel-controlled and enclosure-controlled acoustical modesThe resonance frequencies of acoustical modes uncoupledpanel and enclosure modes are shown in Figure 4(b) as afunction of panel modal density Figure 4(c) shows the varia-tion of the decay time of acoustical modes with panel modaldensity The effect of truncation numbers on the resonancefrequencies of acoustical modes is shown in Figure 4(d)As the panel modal density tends to the point of transferfactor between the (31) panel mode and the (001) enclosuremode which is in the order of 10 energy ratio resonancefrequencies decay times of the (31) panel-controlled modeand the (001) enclosure-controlled mode tend to be equaltoo

Near the point of panel modal density where the maxi-mum interaction strength between the (31) panel mode andthe (001) enclosure mode the energy which is stored in thepanel vibration and enclosure sound field of two acousticalmodes including the (001) enclosure-controlled mode andthe (31) panel-controlled mode are approximately equal to

Shock and Vibration 7

01 02 03 01 02 03

01 02 03

01 02 0301 02 03

01 02 03

210

220

230

280

290

300

160

170

180

350

360

370

270

280

330

335

340

(001)

(100) (011)

(101)(110)

(010)

fL

(Hz)

fL

(Hz)

fL

(Hz)

np (Hzminus1) np (Hzminus1)

(a)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(101)

(100)

(001)(010)

(011)

(110)

01 02 03

01 02 03 01 02 03

01 02 0301 02 03

01 02 03

TL

(s)

TL

(s)

TL

(s)

np (Hzminus1)np (Hzminus1)

(b)

Figure 3 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal density (a)resonance frequency (b) decay time 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

each other as shown in Figure 4(a) The further the distancefrom that point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlledmodeThe result also shows that energyratio curve of the (001) enclosure-controlled mode is acontinuation of that of the (31) panel-controlled mode as afunction of panel modal density and the energy ratio curveof the (31) panel-controlled mode is a continuation of that ofthe (001) enclosure-controlled mode

It is shown in Figure 4(b) that in the vicinity of thepoint of panel modal density where the transfer factor is

approximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the panel modal density is increased the resonancefrequency of the (001) enclosure-controlled mode jumpsfrom lower than the resonance frequency of the (001) rigidwalls enclosure mode to higher than it when it passes by thepoint ofmaximumenergy transfer Comparedwith the (001)

8 Shock and Vibration

003 00305 0031 00315 0032 00325 0033 00335 0034

0

5

10

15

minus5

minus10

minus15

np (Hzminus1)

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

003 00305 0031 00315 0032 00325 0033 00335 0034260

270

280

290

300

310

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

np (Hzminus1)

fL

(Hz)

(b)

0

1

2

3

4

5

6

7

8

003 00305 0031 00315 0032 00325 0033 00335 0034

(001) enclosure-controlled mode(31) panel-controlled mode

np (Hzminus1)

TL

(s)

(c)

265

270

275

280

285

290

295

300

305

310

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

003 00305 0031 00315 0032 00325 0033 00335 0034

np (Hzminus1)

fL

(Hz)

(d)

Figure 4 (a) Ratio between the sound field energy and panel vibration energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers 119871

119885= 06m

119879119901119872

= 05 s 119879119886119873

= 15 s

enclosure-controlled mode it is opposite to the (31) panel-controlled mode As the panel modal density is away fromthe point of maximum interaction strength the resonancefrequencies of the (31) panel-controlled mode and the (001)enclosure-controlled mode tend to resonance frequencies ofuncoupled them respectively

It is shown in Figure 4(c) that as the panel modal densityis increased decay time of the (001) enclosure-controlledmode firstly reduces When panel modal density arrives atthe point of strong coupling the minimum decay time is gotThen decay time of the (001) enclosure-controlled mode

becomes longer as the panel modal density is increasedCompared with the (001) enclosure-controlled mode decaytime curve of the (31) panel-controlled mode is opposite asthe panel modal density is increased In addition at the pointof panel modal density where the decay times of the (001)enclosure-controlled mode and the (31) panel-controlledmode are approximately equal to each other the energytransfer from the panel vibration to the enclosure acousticfield is relative maximum

Due to the limited number of uncoupled panel and enclo-sure modes used in this analysis some important modes may

Shock and Vibration 9

0 2 4 6 8

Pane

l mod

e

|VM| (ms) times10minus4

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 5 Modal amplitude of the panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

be excluded and truncation errors exist in the calculation ofresonance frequencies and decay times of acoustical modesFigure 4(d) shows the resonance frequencies of two acousti-calmodes using different truncated numbers Comparedwiththe solution for the resonance frequencies using the combi-nation of 64 enclosure modes and 81 panel modes and of 125enclosure modes and 121 panel modes the solution for theresonance frequencies of the (001) enclosure-controlled andthe (31) panel-controlled acoustical modes using 40 panelmodes and 40 enclosuremodesmeets the requirement in thisanalysis and the computation efficiency is also improved

When the panel thickness is 00882m the two subsystemsmodal amplitudes of panel vibration and enclosure acousticfield are shown in Figures 5 and 6 respectively The panelvibration of coupled system is controlled by the (31) panelmode in Figures 5(a) and 5(b) The sound field in the enclo-sure of coupled system is controlled by the (001) enclosuremode in Figures 6(a) and 6(b) Combined with energy ratioshown in Figure 4(a) it is proved that the coupled systemis named the (31) panel-controlled acoustical mode and the(001) enclosure-controlled acoustical mode respectively

412 Panel Internal Damping The decay times of uncoupledclamped panel modes are used to describe the mechanicaldamping of the clamped panel in the analysis The panelvibration energy is dissipated by its internal damping andthen the vibration level of panel will be reduced The dis-turbing degree of the enclosure sound field due to the panelvibration becomes smaller and the energy flow betweenthe panel and enclosure becomes smaller too The decaytimes and the resonance frequencies of enclosure-controlledmodes are related to the decay times of uncoupled panelmodes Figures 7(a) and 7(b) show plot of the variation ofthe resonance frequencies and the decay times of the firstfew enclosure-controlled modes with the decay time of panelmodes respectively In this analysis the decay times of allrigid wall enclosure modes are 15 s The thickness of panel is0007m and the depth of enclosure is 06mThe decay timesof all uncoupled panel modes are equal

In Figure 7(a) the resonance frequencies of the firstfew enclosure-controlled modes approach the uncoupledenclosure resonance frequency as the panelmodal decay timeis reduced (corresponding to the damping increased) Thebigger the damping in the panel is the more the energy is

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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International Journal of

4 Shock and Vibration

A22= minus

[[[[[[[[[[[[[[[[[[[

[

1205781198861

0

11988801205882

011986011989111986111

119872119860

1

sdot sdot sdot

11988801205882

01198601198911198611119872

119872119860

1

d d

0 120578119886119873

11988801205882

01198601198911198611198731

119872119860

119873

11988801205882

0119860119891119861119873119872

119872119860

119873

minus

11986011989111986111

1198880119872119875

1

sdot sdot sdot minus

1198601198911198611198731

1198880119872119875

1

1205781199011

0

d d

minus

1198601198911198611119872

1198880119872119875

119872

minus

119860119891119861119873119872

1198880119872119875

119872

0 120578119901119872

]]]]]]]]]]]]]]]]]]]

]

(20)

119861119873119872

=1

119860119891

int119860119891

119904119872(120590) 120601119873(r) 119889119904 (21)

120578119901119872

=44120587

1198791199011198721198880

(22)

120578119886119873

=44120587

1198791198861198731198880

(23)

where 119896119886119873

= 1205961198861198731198880 119896119901119872

= 1205961199011198721198880 119896119886119873

120596119886119873

are thewavenumber and the resonance angle frequency of the 119873thenclosure mode respectively 119896

119901119872and 120596

119901119872are respectively

the wavenumber and the resonance angle frequency of the119872th panel mode119872119875119872119860 are the modal mass of panel andenclosure respectively 119872119875

119872= 120588ℎ119860

119891Λ119872 119872119860119873= 12058801198810Λ119873

1198601198911198810are the area of flexible panel and volume of enclosure

respectively 120578119901119872

120578119886119873

are respectively the loss factor of119872thpanel mode and 119873th enclosure mode Particularly 120578

119886119873is

related to the integral of the specific acoustic admittance 120573119879119901119872

119879119886119873

are respectively the 60 dB decay time of119872th panelmode and119873th enclosure mode

3 The Modal Properties ofthe Panel-Enclosure System

31 Modal Coupling Coefficient Modal coupling coefficientis the spatial matching degree between enclosure and panelmodes on the interacting surface of enclosure sound fieldand flexible panel from (21) It is the integral of panelmode and enclosure mode on the surface of panel Modalcoupling coefficient only depends on the geometry shape andboundary condition of panel the mode shape of enclosureacoustic field [3] and it has nothing to do with thicknessmaterial properties damping of the panel and enclosuredepth

Like the panel-enclosure system which consists of anenclosure with a simply-supported flexible wall the modecoupling coefficient between clamped panel and enclosure is

nonzero which must satisfy the following two conditions atthe same time

119906 + 119897 = odd number

V + 119898 = odd number(24)

32 Transfer Factor Coupling coefficient determines thematching degree between an enclosure mode and a panelmode on the interacting surface of enclosure sound field andpanel while the coupling extent between a panelmode and anenclosure mode is decided by transfer factor [3] The transferfactor 119865

119873119872between the119873th enclosure acoustical mode and

119872th panel mode is given by

119865119873119872

=

1 + [

(120596119886119873minus 120596119901119872)

2]

2

[1

119861(119873119872)2]

minus1

(25)

119861 (119873119872) = (12058801198882

0

120588ℎ119871119911Λ119873Λ119872

)

12

119861119873119872

(26)

When the modal coupling coefficient between the 119873thenclosure acoustical mode and119872th panel mode changes thetransfer factor 119865

119873119872has different results from (25) and (26)

There are two different kinds of transfer factors (1) as themodal coupling coefficient 119861

119873119872is equal to zero the transfer

factor 119865119873119872

must be equal to zero In this situation thereis not energy transfer between the 119873th enclosure acousticalmode and 119872th panel mode (2) When the modal couplingcoefficient is nonzero the transfer factor relates to manyfactors such as the difference between resonance frequenciesof uncoupled panel and enclosure modes and the enclosuredepth The larger the transfer factor between a panel modeand an enclosure mode is the bigger the coupling strengthbetween them is obtained When the transfer factor 119865

119873119872

is in the order of 10 the energy transfer between the 119873thenclosure mode and119872th panel mode is important

33 Resonance Frequency and Modal Decay Time of CoupledSystem Thesound pressure in the enclosure and the clampedpanel vibration velocity are used to describe the response

Shock and Vibration 5

of coupled panel-enclosure system Through the surface offlexible panel facing the inside of enclosure the sound fieldin the enclosure is coupling with the vibration of flexiblepanel There are two different kinds of acoustical modes inthe panel-enclosure coupled system one is an enclosure-controlled acoustical mode whose most of energy is stored inthe enclosure sound field and the other is a panel-controlledacoustical mode whose most of energy is stored as panelvibration energy [3]

If the external excitation does not exist in the coupledsystem there will be Y = 0 in (15) and it becomes a 2(119873 +

119872) dimensional system of equations Corresponding to theeigenequation there will be 2(119873 + 119872) eigenvalues 120582

119871and

120582lowast

119871 and 119871 = 1 2 sdot sdot sdot (119873 + 119872) The resonance frequency 119891

119871

and the decay time 119879119871of coupled system are Im(120582

119871)11988802120587

and 691Re(120582119871)1198880 respectively When the Y = 0 in (15)

the solution of the coefficient X is the modal amplitude ofcoupled system Then the panel vibration velocity and thesound pressure in the enclosure which describes the forcedresponse of coupled system can be obtained from (4) and (5)

The time-averaged acoustic potential energy 119864119886119873

in theenclosure and the time-averaged vibration kinetic energy119864119901119872

of the flexible panel are given by [14]

119864119886119873

=1

(412058801198882

0) int1198810

1003816100381610038161003816119901 (r 120596)1003816100381610038161003816

2

119889119881

=1198810PHΛ119886119873

P(412058801198882

0)

(27)

119864119901119872

=120588ℎ

4int119860119891

|V (120590 120596)|2

119889119904

=

120588ℎ119860119891VH

Λ119901119872

V4

(28)

HereΛ119886119873

is a119873times119873 diagonalmatrix with each diagonal termconsisting of Λ

119873 and Λ

119901119872is a119872times119872 diagonal matrix with

each diagonal term consisting of Λ119872

4 Results and Discussion

To demonstrate the properties of the panel-enclosure coupledsystemwhich consists of an enclosure with a clamped flexiblewall the resonance frequencies and modal decay times ofacoustical modes are investigated with different panel modaldensity panel internal damping and enclosure depth respec-tively The panel-enclosure coupled system which consists ofan enclosure with a clamped panel on top and five absorptivewalls is shown in Figure 1 The panel material properties aretaken as follows the material of clamped panel is aluminumwith density 120588 = 2770 kgm3 Youngrsquos modulus 119864 = 71Gpaand Poissonrsquos ratio 120583 = 033

41 Effects of Different Panel Physical Parameters onAcoustical Modes

411 Panel Modal Density From (25) as the differencebetween resonance frequencies of uncoupled enclosure andpanel modes is decreased the transfer factors betweenthem become larger when the modal coupling coefficient is

nonzero The resonance frequencies of rigid walls enclosuremode and uncoupled clamped panel mode [21] are given by

119891119897119898119899

=1198880

2[(

119897

119871119909

)

2

+ (119898

119871119910

)

2

+ (119899

119871119911

)

2

] (29)

119891119906V =

1

2120587radic119863

120588

radic(120582119906

119871119909

)

4

+ (120582V

119871119910

)

4

+ 2(120582119906120582V

119871119909119871119910

)

2

120581119906120581V

120585119906120585V

(30)

120581119894=1

4(1 + 119863

2

119894) sinh (2120582

119894) minus

1

2119863119894cosh (2120582

119894)

minus1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) minus 119863119894cos2 (120582

119894)

minus 1198632

119894120582119894+3

2119863119894

(31)

120585119894=1

4(1 + 119863

2

119894) sinh (2120582

119894)

+ sinh (120582119894) [2119863119894sin (120582

119894) minus (1 minus 119863

2

119894) cos (120582

119894)]

minus (1 + 1198632

119894) sin (120582

119894) cosh (120582

119894)

+1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) + 120582119894

minus1

2119863119894[1 + cosh (2120582

119894)] + 119863

119894cos2 (120582

119894)

(32)

119863119894=120574 (120582119894)

119867 (120582119894) (33)

Similar to an enclosure with a simply supported flexiblewall the modal density of uncoupled clamped panel andenclosure will affect energy transfer between them Themodal density of uncoupled clamped panel and enclosuresound field are given by [3 10]

119899119901=

radic3119860119891

119862119871ℎ (34)

119899119886=412058711988101198912

1198883

0

+120587119878119891

21198882

0

+119871

81198880

(35)

Here 119860119891 119862119871are the area and longitudinal wave speed

of clamped flexible wall respectively 119891 is the excitationfrequency 119878 119871 are the total surface area and the total edgelengths inside the enclosure

As mentioned above the transfer factor determines thecoupling strength between a panel mode and an enclosuremode From (34) and (25) by adjusting the panel modaldensity (corresponding to panel thickness) the distributionof the resonance frequencies of panel is changed whichalso leads to changes of the transfer factors between paneland enclosure modes In Figure 2 transfer factors between

6 Shock and Vibration

005 01 015 02 025 030

02

04

06

08

1

Tran

sfer f

acto

r

np (Hzminus1)

(11)

(13)

(33)

(15)

(35)

(51)

(17)

(53)

(37)

(55)

(31)

Figure 2 Transfer factors between the (001) enclosure mode andpanelmodes as a function of panelmodal density119871

119885= 06m119879

119886119873=

15 s 119879119901119872

= 05 s

the (001) enclosure mode and panel modes are plottedagainst panel modal density When the panel modal densityis low only the (31) panel mode participates in the couplingwith the (001) enclosure mode and satisfies the well-coupledcondition As the panel modal density is increased moreand more high-order panel modes in which the couplingcoefficients with the (001) enclosure mode are not equalto zero participate in coupling At these points of transferfactors when their values are in the order of 10 there is largeenergy transfer between enclosure and panel modes Similarto the transfer factors between the (001) enclosure modeand panel modes the distribution of transfer factors betweenother enclosure modes and panel modes are that only fewpanel modes participate in the coupling with enclosuremodes in the low panel modal density region andmore panelmodes participate in the coupling with enclosure mode forhigh panel modal density region

As the coupling extent between enclosure and panelmodes changes with the variation of panel modal densitythe resonance frequency of acoustical mode which is oneimportant characteristic of it is altered also Figure 3(a)shows resonance frequencies of the first few enclosure-controlled acoustical modes as a function of panel modaldensity Forty panel modes and forty enclosure modes areused in this analysis The decay times of all uncoupledenclosure and panel modes are 15 s and 05 s respectivelyThedepth of enclosure is 06m As the panel modal density isincreased the resonance frequencies of enclosure-controlledmodes jump to higher frequencies Compared with theresonance frequencies of acoustical mode in the low panelmodal density region they become larger in the high panelmodal density region The reason is that many high-order

panel modes participate in the coupling with enclosuremodes Similar to the panel-enclosure coupled system with asimply supported flexible panel the energy transfer from theenclosuremode is distributed overmany panel modes and nopanel modes are well coupling with enclosure modes

Figure 3(b) shows plot of the variation of decay times ofthe first few enclosure-controlled modes with panel modaldensity When the panel modal density is small the modaldecay time is longer on the average As the panel modaldensity is increased the modal decay times show someminima It is because of that only few panel modes satisfythe well-coupled condition with enclosure modes in the lowpanel modal density region As the panel modal density isincreased more panel modes participate in the coupling withenclosure modes and decay times of enclosure-controlledmodes become shorter At the same time there are no panelmodes which are well coupled with enclosure mode and theenergy which is stored in the panel is equally distributed overmany panel modes

In order to describe the process of strong couplingbetween panel and enclosure modes when the panel modaldensity is altered we analyze from three points of view ofenergy ratio between the panel vibration and sound field inthe enclosure resonance frequency and modal decay time ofcoupled system respectively In this analysis the plane wave119901119894with amplitude 1 Pa on the panel surface is used to drive the

vibration of panel the excitation frequencies are respectivelythe resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The angles of elevation 120572 andazimuth 120579 of plane wave are 90∘ and 0∘ respectively Thedecay times of all uncoupled panel modes and enclosuremodes are 15 s and 05 s respectively As seen in Figure 2 thetransfer factor between the (31) panel mode and the (001)enclosure mode is approximately equal to 10 in low panelmodal density region and only these two modes satisfy well-coupled condition which means that the energy conversionbetween the sound field and the panel is almost entirelybetween these two modes

The energy ratio between the panel vibration and soundfield in the enclosure is shown in Figure 4(a) which containspanel-controlled and enclosure-controlled acoustical modesThe resonance frequencies of acoustical modes uncoupledpanel and enclosure modes are shown in Figure 4(b) as afunction of panel modal density Figure 4(c) shows the varia-tion of the decay time of acoustical modes with panel modaldensity The effect of truncation numbers on the resonancefrequencies of acoustical modes is shown in Figure 4(d)As the panel modal density tends to the point of transferfactor between the (31) panel mode and the (001) enclosuremode which is in the order of 10 energy ratio resonancefrequencies decay times of the (31) panel-controlled modeand the (001) enclosure-controlled mode tend to be equaltoo

Near the point of panel modal density where the maxi-mum interaction strength between the (31) panel mode andthe (001) enclosure mode the energy which is stored in thepanel vibration and enclosure sound field of two acousticalmodes including the (001) enclosure-controlled mode andthe (31) panel-controlled mode are approximately equal to

Shock and Vibration 7

01 02 03 01 02 03

01 02 03

01 02 0301 02 03

01 02 03

210

220

230

280

290

300

160

170

180

350

360

370

270

280

330

335

340

(001)

(100) (011)

(101)(110)

(010)

fL

(Hz)

fL

(Hz)

fL

(Hz)

np (Hzminus1) np (Hzminus1)

(a)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(101)

(100)

(001)(010)

(011)

(110)

01 02 03

01 02 03 01 02 03

01 02 0301 02 03

01 02 03

TL

(s)

TL

(s)

TL

(s)

np (Hzminus1)np (Hzminus1)

(b)

Figure 3 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal density (a)resonance frequency (b) decay time 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

each other as shown in Figure 4(a) The further the distancefrom that point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlledmodeThe result also shows that energyratio curve of the (001) enclosure-controlled mode is acontinuation of that of the (31) panel-controlled mode as afunction of panel modal density and the energy ratio curveof the (31) panel-controlled mode is a continuation of that ofthe (001) enclosure-controlled mode

It is shown in Figure 4(b) that in the vicinity of thepoint of panel modal density where the transfer factor is

approximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the panel modal density is increased the resonancefrequency of the (001) enclosure-controlled mode jumpsfrom lower than the resonance frequency of the (001) rigidwalls enclosure mode to higher than it when it passes by thepoint ofmaximumenergy transfer Comparedwith the (001)

8 Shock and Vibration

003 00305 0031 00315 0032 00325 0033 00335 0034

0

5

10

15

minus5

minus10

minus15

np (Hzminus1)

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

003 00305 0031 00315 0032 00325 0033 00335 0034260

270

280

290

300

310

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

np (Hzminus1)

fL

(Hz)

(b)

0

1

2

3

4

5

6

7

8

003 00305 0031 00315 0032 00325 0033 00335 0034

(001) enclosure-controlled mode(31) panel-controlled mode

np (Hzminus1)

TL

(s)

(c)

265

270

275

280

285

290

295

300

305

310

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

003 00305 0031 00315 0032 00325 0033 00335 0034

np (Hzminus1)

fL

(Hz)

(d)

Figure 4 (a) Ratio between the sound field energy and panel vibration energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers 119871

119885= 06m

119879119901119872

= 05 s 119879119886119873

= 15 s

enclosure-controlled mode it is opposite to the (31) panel-controlled mode As the panel modal density is away fromthe point of maximum interaction strength the resonancefrequencies of the (31) panel-controlled mode and the (001)enclosure-controlled mode tend to resonance frequencies ofuncoupled them respectively

It is shown in Figure 4(c) that as the panel modal densityis increased decay time of the (001) enclosure-controlledmode firstly reduces When panel modal density arrives atthe point of strong coupling the minimum decay time is gotThen decay time of the (001) enclosure-controlled mode

becomes longer as the panel modal density is increasedCompared with the (001) enclosure-controlled mode decaytime curve of the (31) panel-controlled mode is opposite asthe panel modal density is increased In addition at the pointof panel modal density where the decay times of the (001)enclosure-controlled mode and the (31) panel-controlledmode are approximately equal to each other the energytransfer from the panel vibration to the enclosure acousticfield is relative maximum

Due to the limited number of uncoupled panel and enclo-sure modes used in this analysis some important modes may

Shock and Vibration 9

0 2 4 6 8

Pane

l mod

e

|VM| (ms) times10minus4

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 5 Modal amplitude of the panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

be excluded and truncation errors exist in the calculation ofresonance frequencies and decay times of acoustical modesFigure 4(d) shows the resonance frequencies of two acousti-calmodes using different truncated numbers Comparedwiththe solution for the resonance frequencies using the combi-nation of 64 enclosure modes and 81 panel modes and of 125enclosure modes and 121 panel modes the solution for theresonance frequencies of the (001) enclosure-controlled andthe (31) panel-controlled acoustical modes using 40 panelmodes and 40 enclosuremodesmeets the requirement in thisanalysis and the computation efficiency is also improved

When the panel thickness is 00882m the two subsystemsmodal amplitudes of panel vibration and enclosure acousticfield are shown in Figures 5 and 6 respectively The panelvibration of coupled system is controlled by the (31) panelmode in Figures 5(a) and 5(b) The sound field in the enclo-sure of coupled system is controlled by the (001) enclosuremode in Figures 6(a) and 6(b) Combined with energy ratioshown in Figure 4(a) it is proved that the coupled systemis named the (31) panel-controlled acoustical mode and the(001) enclosure-controlled acoustical mode respectively

412 Panel Internal Damping The decay times of uncoupledclamped panel modes are used to describe the mechanicaldamping of the clamped panel in the analysis The panelvibration energy is dissipated by its internal damping andthen the vibration level of panel will be reduced The dis-turbing degree of the enclosure sound field due to the panelvibration becomes smaller and the energy flow betweenthe panel and enclosure becomes smaller too The decaytimes and the resonance frequencies of enclosure-controlledmodes are related to the decay times of uncoupled panelmodes Figures 7(a) and 7(b) show plot of the variation ofthe resonance frequencies and the decay times of the firstfew enclosure-controlled modes with the decay time of panelmodes respectively In this analysis the decay times of allrigid wall enclosure modes are 15 s The thickness of panel is0007m and the depth of enclosure is 06mThe decay timesof all uncoupled panel modes are equal

In Figure 7(a) the resonance frequencies of the firstfew enclosure-controlled modes approach the uncoupledenclosure resonance frequency as the panelmodal decay timeis reduced (corresponding to the damping increased) Thebigger the damping in the panel is the more the energy is

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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Shock and Vibration 5

of coupled panel-enclosure system Through the surface offlexible panel facing the inside of enclosure the sound fieldin the enclosure is coupling with the vibration of flexiblepanel There are two different kinds of acoustical modes inthe panel-enclosure coupled system one is an enclosure-controlled acoustical mode whose most of energy is stored inthe enclosure sound field and the other is a panel-controlledacoustical mode whose most of energy is stored as panelvibration energy [3]

If the external excitation does not exist in the coupledsystem there will be Y = 0 in (15) and it becomes a 2(119873 +

119872) dimensional system of equations Corresponding to theeigenequation there will be 2(119873 + 119872) eigenvalues 120582

119871and

120582lowast

119871 and 119871 = 1 2 sdot sdot sdot (119873 + 119872) The resonance frequency 119891

119871

and the decay time 119879119871of coupled system are Im(120582

119871)11988802120587

and 691Re(120582119871)1198880 respectively When the Y = 0 in (15)

the solution of the coefficient X is the modal amplitude ofcoupled system Then the panel vibration velocity and thesound pressure in the enclosure which describes the forcedresponse of coupled system can be obtained from (4) and (5)

The time-averaged acoustic potential energy 119864119886119873

in theenclosure and the time-averaged vibration kinetic energy119864119901119872

of the flexible panel are given by [14]

119864119886119873

=1

(412058801198882

0) int1198810

1003816100381610038161003816119901 (r 120596)1003816100381610038161003816

2

119889119881

=1198810PHΛ119886119873

P(412058801198882

0)

(27)

119864119901119872

=120588ℎ

4int119860119891

|V (120590 120596)|2

119889119904

=

120588ℎ119860119891VH

Λ119901119872

V4

(28)

HereΛ119886119873

is a119873times119873 diagonalmatrix with each diagonal termconsisting of Λ

119873 and Λ

119901119872is a119872times119872 diagonal matrix with

each diagonal term consisting of Λ119872

4 Results and Discussion

To demonstrate the properties of the panel-enclosure coupledsystemwhich consists of an enclosure with a clamped flexiblewall the resonance frequencies and modal decay times ofacoustical modes are investigated with different panel modaldensity panel internal damping and enclosure depth respec-tively The panel-enclosure coupled system which consists ofan enclosure with a clamped panel on top and five absorptivewalls is shown in Figure 1 The panel material properties aretaken as follows the material of clamped panel is aluminumwith density 120588 = 2770 kgm3 Youngrsquos modulus 119864 = 71Gpaand Poissonrsquos ratio 120583 = 033

41 Effects of Different Panel Physical Parameters onAcoustical Modes

411 Panel Modal Density From (25) as the differencebetween resonance frequencies of uncoupled enclosure andpanel modes is decreased the transfer factors betweenthem become larger when the modal coupling coefficient is

nonzero The resonance frequencies of rigid walls enclosuremode and uncoupled clamped panel mode [21] are given by

119891119897119898119899

=1198880

2[(

119897

119871119909

)

2

+ (119898

119871119910

)

2

+ (119899

119871119911

)

2

] (29)

119891119906V =

1

2120587radic119863

120588

radic(120582119906

119871119909

)

4

+ (120582V

119871119910

)

4

+ 2(120582119906120582V

119871119909119871119910

)

2

120581119906120581V

120585119906120585V

(30)

120581119894=1

4(1 + 119863

2

119894) sinh (2120582

119894) minus

1

2119863119894cosh (2120582

119894)

minus1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) minus 119863119894cos2 (120582

119894)

minus 1198632

119894120582119894+3

2119863119894

(31)

120585119894=1

4(1 + 119863

2

119894) sinh (2120582

119894)

+ sinh (120582119894) [2119863119894sin (120582

119894) minus (1 minus 119863

2

119894) cos (120582

119894)]

minus (1 + 1198632

119894) sin (120582

119894) cosh (120582

119894)

+1

2(1 minus 119863

2

119894) sin (120582

119894) cos (120582

119894) + 120582119894

minus1

2119863119894[1 + cosh (2120582

119894)] + 119863

119894cos2 (120582

119894)

(32)

119863119894=120574 (120582119894)

119867 (120582119894) (33)

Similar to an enclosure with a simply supported flexiblewall the modal density of uncoupled clamped panel andenclosure will affect energy transfer between them Themodal density of uncoupled clamped panel and enclosuresound field are given by [3 10]

119899119901=

radic3119860119891

119862119871ℎ (34)

119899119886=412058711988101198912

1198883

0

+120587119878119891

21198882

0

+119871

81198880

(35)

Here 119860119891 119862119871are the area and longitudinal wave speed

of clamped flexible wall respectively 119891 is the excitationfrequency 119878 119871 are the total surface area and the total edgelengths inside the enclosure

As mentioned above the transfer factor determines thecoupling strength between a panel mode and an enclosuremode From (34) and (25) by adjusting the panel modaldensity (corresponding to panel thickness) the distributionof the resonance frequencies of panel is changed whichalso leads to changes of the transfer factors between paneland enclosure modes In Figure 2 transfer factors between

6 Shock and Vibration

005 01 015 02 025 030

02

04

06

08

1

Tran

sfer f

acto

r

np (Hzminus1)

(11)

(13)

(33)

(15)

(35)

(51)

(17)

(53)

(37)

(55)

(31)

Figure 2 Transfer factors between the (001) enclosure mode andpanelmodes as a function of panelmodal density119871

119885= 06m119879

119886119873=

15 s 119879119901119872

= 05 s

the (001) enclosure mode and panel modes are plottedagainst panel modal density When the panel modal densityis low only the (31) panel mode participates in the couplingwith the (001) enclosure mode and satisfies the well-coupledcondition As the panel modal density is increased moreand more high-order panel modes in which the couplingcoefficients with the (001) enclosure mode are not equalto zero participate in coupling At these points of transferfactors when their values are in the order of 10 there is largeenergy transfer between enclosure and panel modes Similarto the transfer factors between the (001) enclosure modeand panel modes the distribution of transfer factors betweenother enclosure modes and panel modes are that only fewpanel modes participate in the coupling with enclosuremodes in the low panel modal density region andmore panelmodes participate in the coupling with enclosure mode forhigh panel modal density region

As the coupling extent between enclosure and panelmodes changes with the variation of panel modal densitythe resonance frequency of acoustical mode which is oneimportant characteristic of it is altered also Figure 3(a)shows resonance frequencies of the first few enclosure-controlled acoustical modes as a function of panel modaldensity Forty panel modes and forty enclosure modes areused in this analysis The decay times of all uncoupledenclosure and panel modes are 15 s and 05 s respectivelyThedepth of enclosure is 06m As the panel modal density isincreased the resonance frequencies of enclosure-controlledmodes jump to higher frequencies Compared with theresonance frequencies of acoustical mode in the low panelmodal density region they become larger in the high panelmodal density region The reason is that many high-order

panel modes participate in the coupling with enclosuremodes Similar to the panel-enclosure coupled system with asimply supported flexible panel the energy transfer from theenclosuremode is distributed overmany panel modes and nopanel modes are well coupling with enclosure modes

Figure 3(b) shows plot of the variation of decay times ofthe first few enclosure-controlled modes with panel modaldensity When the panel modal density is small the modaldecay time is longer on the average As the panel modaldensity is increased the modal decay times show someminima It is because of that only few panel modes satisfythe well-coupled condition with enclosure modes in the lowpanel modal density region As the panel modal density isincreased more panel modes participate in the coupling withenclosure modes and decay times of enclosure-controlledmodes become shorter At the same time there are no panelmodes which are well coupled with enclosure mode and theenergy which is stored in the panel is equally distributed overmany panel modes

In order to describe the process of strong couplingbetween panel and enclosure modes when the panel modaldensity is altered we analyze from three points of view ofenergy ratio between the panel vibration and sound field inthe enclosure resonance frequency and modal decay time ofcoupled system respectively In this analysis the plane wave119901119894with amplitude 1 Pa on the panel surface is used to drive the

vibration of panel the excitation frequencies are respectivelythe resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The angles of elevation 120572 andazimuth 120579 of plane wave are 90∘ and 0∘ respectively Thedecay times of all uncoupled panel modes and enclosuremodes are 15 s and 05 s respectively As seen in Figure 2 thetransfer factor between the (31) panel mode and the (001)enclosure mode is approximately equal to 10 in low panelmodal density region and only these two modes satisfy well-coupled condition which means that the energy conversionbetween the sound field and the panel is almost entirelybetween these two modes

The energy ratio between the panel vibration and soundfield in the enclosure is shown in Figure 4(a) which containspanel-controlled and enclosure-controlled acoustical modesThe resonance frequencies of acoustical modes uncoupledpanel and enclosure modes are shown in Figure 4(b) as afunction of panel modal density Figure 4(c) shows the varia-tion of the decay time of acoustical modes with panel modaldensity The effect of truncation numbers on the resonancefrequencies of acoustical modes is shown in Figure 4(d)As the panel modal density tends to the point of transferfactor between the (31) panel mode and the (001) enclosuremode which is in the order of 10 energy ratio resonancefrequencies decay times of the (31) panel-controlled modeand the (001) enclosure-controlled mode tend to be equaltoo

Near the point of panel modal density where the maxi-mum interaction strength between the (31) panel mode andthe (001) enclosure mode the energy which is stored in thepanel vibration and enclosure sound field of two acousticalmodes including the (001) enclosure-controlled mode andthe (31) panel-controlled mode are approximately equal to

Shock and Vibration 7

01 02 03 01 02 03

01 02 03

01 02 0301 02 03

01 02 03

210

220

230

280

290

300

160

170

180

350

360

370

270

280

330

335

340

(001)

(100) (011)

(101)(110)

(010)

fL

(Hz)

fL

(Hz)

fL

(Hz)

np (Hzminus1) np (Hzminus1)

(a)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(101)

(100)

(001)(010)

(011)

(110)

01 02 03

01 02 03 01 02 03

01 02 0301 02 03

01 02 03

TL

(s)

TL

(s)

TL

(s)

np (Hzminus1)np (Hzminus1)

(b)

Figure 3 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal density (a)resonance frequency (b) decay time 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

each other as shown in Figure 4(a) The further the distancefrom that point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlledmodeThe result also shows that energyratio curve of the (001) enclosure-controlled mode is acontinuation of that of the (31) panel-controlled mode as afunction of panel modal density and the energy ratio curveof the (31) panel-controlled mode is a continuation of that ofthe (001) enclosure-controlled mode

It is shown in Figure 4(b) that in the vicinity of thepoint of panel modal density where the transfer factor is

approximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the panel modal density is increased the resonancefrequency of the (001) enclosure-controlled mode jumpsfrom lower than the resonance frequency of the (001) rigidwalls enclosure mode to higher than it when it passes by thepoint ofmaximumenergy transfer Comparedwith the (001)

8 Shock and Vibration

003 00305 0031 00315 0032 00325 0033 00335 0034

0

5

10

15

minus5

minus10

minus15

np (Hzminus1)

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

003 00305 0031 00315 0032 00325 0033 00335 0034260

270

280

290

300

310

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

np (Hzminus1)

fL

(Hz)

(b)

0

1

2

3

4

5

6

7

8

003 00305 0031 00315 0032 00325 0033 00335 0034

(001) enclosure-controlled mode(31) panel-controlled mode

np (Hzminus1)

TL

(s)

(c)

265

270

275

280

285

290

295

300

305

310

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

003 00305 0031 00315 0032 00325 0033 00335 0034

np (Hzminus1)

fL

(Hz)

(d)

Figure 4 (a) Ratio between the sound field energy and panel vibration energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers 119871

119885= 06m

119879119901119872

= 05 s 119879119886119873

= 15 s

enclosure-controlled mode it is opposite to the (31) panel-controlled mode As the panel modal density is away fromthe point of maximum interaction strength the resonancefrequencies of the (31) panel-controlled mode and the (001)enclosure-controlled mode tend to resonance frequencies ofuncoupled them respectively

It is shown in Figure 4(c) that as the panel modal densityis increased decay time of the (001) enclosure-controlledmode firstly reduces When panel modal density arrives atthe point of strong coupling the minimum decay time is gotThen decay time of the (001) enclosure-controlled mode

becomes longer as the panel modal density is increasedCompared with the (001) enclosure-controlled mode decaytime curve of the (31) panel-controlled mode is opposite asthe panel modal density is increased In addition at the pointof panel modal density where the decay times of the (001)enclosure-controlled mode and the (31) panel-controlledmode are approximately equal to each other the energytransfer from the panel vibration to the enclosure acousticfield is relative maximum

Due to the limited number of uncoupled panel and enclo-sure modes used in this analysis some important modes may

Shock and Vibration 9

0 2 4 6 8

Pane

l mod

e

|VM| (ms) times10minus4

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 5 Modal amplitude of the panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

be excluded and truncation errors exist in the calculation ofresonance frequencies and decay times of acoustical modesFigure 4(d) shows the resonance frequencies of two acousti-calmodes using different truncated numbers Comparedwiththe solution for the resonance frequencies using the combi-nation of 64 enclosure modes and 81 panel modes and of 125enclosure modes and 121 panel modes the solution for theresonance frequencies of the (001) enclosure-controlled andthe (31) panel-controlled acoustical modes using 40 panelmodes and 40 enclosuremodesmeets the requirement in thisanalysis and the computation efficiency is also improved

When the panel thickness is 00882m the two subsystemsmodal amplitudes of panel vibration and enclosure acousticfield are shown in Figures 5 and 6 respectively The panelvibration of coupled system is controlled by the (31) panelmode in Figures 5(a) and 5(b) The sound field in the enclo-sure of coupled system is controlled by the (001) enclosuremode in Figures 6(a) and 6(b) Combined with energy ratioshown in Figure 4(a) it is proved that the coupled systemis named the (31) panel-controlled acoustical mode and the(001) enclosure-controlled acoustical mode respectively

412 Panel Internal Damping The decay times of uncoupledclamped panel modes are used to describe the mechanicaldamping of the clamped panel in the analysis The panelvibration energy is dissipated by its internal damping andthen the vibration level of panel will be reduced The dis-turbing degree of the enclosure sound field due to the panelvibration becomes smaller and the energy flow betweenthe panel and enclosure becomes smaller too The decaytimes and the resonance frequencies of enclosure-controlledmodes are related to the decay times of uncoupled panelmodes Figures 7(a) and 7(b) show plot of the variation ofthe resonance frequencies and the decay times of the firstfew enclosure-controlled modes with the decay time of panelmodes respectively In this analysis the decay times of allrigid wall enclosure modes are 15 s The thickness of panel is0007m and the depth of enclosure is 06mThe decay timesof all uncoupled panel modes are equal

In Figure 7(a) the resonance frequencies of the firstfew enclosure-controlled modes approach the uncoupledenclosure resonance frequency as the panelmodal decay timeis reduced (corresponding to the damping increased) Thebigger the damping in the panel is the more the energy is

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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Shock and Vibration

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DistributedSensor Networks

International Journal of

6 Shock and Vibration

005 01 015 02 025 030

02

04

06

08

1

Tran

sfer f

acto

r

np (Hzminus1)

(11)

(13)

(33)

(15)

(35)

(51)

(17)

(53)

(37)

(55)

(31)

Figure 2 Transfer factors between the (001) enclosure mode andpanelmodes as a function of panelmodal density119871

119885= 06m119879

119886119873=

15 s 119879119901119872

= 05 s

the (001) enclosure mode and panel modes are plottedagainst panel modal density When the panel modal densityis low only the (31) panel mode participates in the couplingwith the (001) enclosure mode and satisfies the well-coupledcondition As the panel modal density is increased moreand more high-order panel modes in which the couplingcoefficients with the (001) enclosure mode are not equalto zero participate in coupling At these points of transferfactors when their values are in the order of 10 there is largeenergy transfer between enclosure and panel modes Similarto the transfer factors between the (001) enclosure modeand panel modes the distribution of transfer factors betweenother enclosure modes and panel modes are that only fewpanel modes participate in the coupling with enclosuremodes in the low panel modal density region andmore panelmodes participate in the coupling with enclosure mode forhigh panel modal density region

As the coupling extent between enclosure and panelmodes changes with the variation of panel modal densitythe resonance frequency of acoustical mode which is oneimportant characteristic of it is altered also Figure 3(a)shows resonance frequencies of the first few enclosure-controlled acoustical modes as a function of panel modaldensity Forty panel modes and forty enclosure modes areused in this analysis The decay times of all uncoupledenclosure and panel modes are 15 s and 05 s respectivelyThedepth of enclosure is 06m As the panel modal density isincreased the resonance frequencies of enclosure-controlledmodes jump to higher frequencies Compared with theresonance frequencies of acoustical mode in the low panelmodal density region they become larger in the high panelmodal density region The reason is that many high-order

panel modes participate in the coupling with enclosuremodes Similar to the panel-enclosure coupled system with asimply supported flexible panel the energy transfer from theenclosuremode is distributed overmany panel modes and nopanel modes are well coupling with enclosure modes

Figure 3(b) shows plot of the variation of decay times ofthe first few enclosure-controlled modes with panel modaldensity When the panel modal density is small the modaldecay time is longer on the average As the panel modaldensity is increased the modal decay times show someminima It is because of that only few panel modes satisfythe well-coupled condition with enclosure modes in the lowpanel modal density region As the panel modal density isincreased more panel modes participate in the coupling withenclosure modes and decay times of enclosure-controlledmodes become shorter At the same time there are no panelmodes which are well coupled with enclosure mode and theenergy which is stored in the panel is equally distributed overmany panel modes

In order to describe the process of strong couplingbetween panel and enclosure modes when the panel modaldensity is altered we analyze from three points of view ofenergy ratio between the panel vibration and sound field inthe enclosure resonance frequency and modal decay time ofcoupled system respectively In this analysis the plane wave119901119894with amplitude 1 Pa on the panel surface is used to drive the

vibration of panel the excitation frequencies are respectivelythe resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The angles of elevation 120572 andazimuth 120579 of plane wave are 90∘ and 0∘ respectively Thedecay times of all uncoupled panel modes and enclosuremodes are 15 s and 05 s respectively As seen in Figure 2 thetransfer factor between the (31) panel mode and the (001)enclosure mode is approximately equal to 10 in low panelmodal density region and only these two modes satisfy well-coupled condition which means that the energy conversionbetween the sound field and the panel is almost entirelybetween these two modes

The energy ratio between the panel vibration and soundfield in the enclosure is shown in Figure 4(a) which containspanel-controlled and enclosure-controlled acoustical modesThe resonance frequencies of acoustical modes uncoupledpanel and enclosure modes are shown in Figure 4(b) as afunction of panel modal density Figure 4(c) shows the varia-tion of the decay time of acoustical modes with panel modaldensity The effect of truncation numbers on the resonancefrequencies of acoustical modes is shown in Figure 4(d)As the panel modal density tends to the point of transferfactor between the (31) panel mode and the (001) enclosuremode which is in the order of 10 energy ratio resonancefrequencies decay times of the (31) panel-controlled modeand the (001) enclosure-controlled mode tend to be equaltoo

Near the point of panel modal density where the maxi-mum interaction strength between the (31) panel mode andthe (001) enclosure mode the energy which is stored in thepanel vibration and enclosure sound field of two acousticalmodes including the (001) enclosure-controlled mode andthe (31) panel-controlled mode are approximately equal to

Shock and Vibration 7

01 02 03 01 02 03

01 02 03

01 02 0301 02 03

01 02 03

210

220

230

280

290

300

160

170

180

350

360

370

270

280

330

335

340

(001)

(100) (011)

(101)(110)

(010)

fL

(Hz)

fL

(Hz)

fL

(Hz)

np (Hzminus1) np (Hzminus1)

(a)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(101)

(100)

(001)(010)

(011)

(110)

01 02 03

01 02 03 01 02 03

01 02 0301 02 03

01 02 03

TL

(s)

TL

(s)

TL

(s)

np (Hzminus1)np (Hzminus1)

(b)

Figure 3 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal density (a)resonance frequency (b) decay time 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

each other as shown in Figure 4(a) The further the distancefrom that point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlledmodeThe result also shows that energyratio curve of the (001) enclosure-controlled mode is acontinuation of that of the (31) panel-controlled mode as afunction of panel modal density and the energy ratio curveof the (31) panel-controlled mode is a continuation of that ofthe (001) enclosure-controlled mode

It is shown in Figure 4(b) that in the vicinity of thepoint of panel modal density where the transfer factor is

approximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the panel modal density is increased the resonancefrequency of the (001) enclosure-controlled mode jumpsfrom lower than the resonance frequency of the (001) rigidwalls enclosure mode to higher than it when it passes by thepoint ofmaximumenergy transfer Comparedwith the (001)

8 Shock and Vibration

003 00305 0031 00315 0032 00325 0033 00335 0034

0

5

10

15

minus5

minus10

minus15

np (Hzminus1)

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

003 00305 0031 00315 0032 00325 0033 00335 0034260

270

280

290

300

310

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

np (Hzminus1)

fL

(Hz)

(b)

0

1

2

3

4

5

6

7

8

003 00305 0031 00315 0032 00325 0033 00335 0034

(001) enclosure-controlled mode(31) panel-controlled mode

np (Hzminus1)

TL

(s)

(c)

265

270

275

280

285

290

295

300

305

310

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

003 00305 0031 00315 0032 00325 0033 00335 0034

np (Hzminus1)

fL

(Hz)

(d)

Figure 4 (a) Ratio between the sound field energy and panel vibration energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers 119871

119885= 06m

119879119901119872

= 05 s 119879119886119873

= 15 s

enclosure-controlled mode it is opposite to the (31) panel-controlled mode As the panel modal density is away fromthe point of maximum interaction strength the resonancefrequencies of the (31) panel-controlled mode and the (001)enclosure-controlled mode tend to resonance frequencies ofuncoupled them respectively

It is shown in Figure 4(c) that as the panel modal densityis increased decay time of the (001) enclosure-controlledmode firstly reduces When panel modal density arrives atthe point of strong coupling the minimum decay time is gotThen decay time of the (001) enclosure-controlled mode

becomes longer as the panel modal density is increasedCompared with the (001) enclosure-controlled mode decaytime curve of the (31) panel-controlled mode is opposite asthe panel modal density is increased In addition at the pointof panel modal density where the decay times of the (001)enclosure-controlled mode and the (31) panel-controlledmode are approximately equal to each other the energytransfer from the panel vibration to the enclosure acousticfield is relative maximum

Due to the limited number of uncoupled panel and enclo-sure modes used in this analysis some important modes may

Shock and Vibration 9

0 2 4 6 8

Pane

l mod

e

|VM| (ms) times10minus4

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 5 Modal amplitude of the panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

be excluded and truncation errors exist in the calculation ofresonance frequencies and decay times of acoustical modesFigure 4(d) shows the resonance frequencies of two acousti-calmodes using different truncated numbers Comparedwiththe solution for the resonance frequencies using the combi-nation of 64 enclosure modes and 81 panel modes and of 125enclosure modes and 121 panel modes the solution for theresonance frequencies of the (001) enclosure-controlled andthe (31) panel-controlled acoustical modes using 40 panelmodes and 40 enclosuremodesmeets the requirement in thisanalysis and the computation efficiency is also improved

When the panel thickness is 00882m the two subsystemsmodal amplitudes of panel vibration and enclosure acousticfield are shown in Figures 5 and 6 respectively The panelvibration of coupled system is controlled by the (31) panelmode in Figures 5(a) and 5(b) The sound field in the enclo-sure of coupled system is controlled by the (001) enclosuremode in Figures 6(a) and 6(b) Combined with energy ratioshown in Figure 4(a) it is proved that the coupled systemis named the (31) panel-controlled acoustical mode and the(001) enclosure-controlled acoustical mode respectively

412 Panel Internal Damping The decay times of uncoupledclamped panel modes are used to describe the mechanicaldamping of the clamped panel in the analysis The panelvibration energy is dissipated by its internal damping andthen the vibration level of panel will be reduced The dis-turbing degree of the enclosure sound field due to the panelvibration becomes smaller and the energy flow betweenthe panel and enclosure becomes smaller too The decaytimes and the resonance frequencies of enclosure-controlledmodes are related to the decay times of uncoupled panelmodes Figures 7(a) and 7(b) show plot of the variation ofthe resonance frequencies and the decay times of the firstfew enclosure-controlled modes with the decay time of panelmodes respectively In this analysis the decay times of allrigid wall enclosure modes are 15 s The thickness of panel is0007m and the depth of enclosure is 06mThe decay timesof all uncoupled panel modes are equal

In Figure 7(a) the resonance frequencies of the firstfew enclosure-controlled modes approach the uncoupledenclosure resonance frequency as the panelmodal decay timeis reduced (corresponding to the damping increased) Thebigger the damping in the panel is the more the energy is

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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Shock and Vibration

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Shock and Vibration 7

01 02 03 01 02 03

01 02 03

01 02 0301 02 03

01 02 03

210

220

230

280

290

300

160

170

180

350

360

370

270

280

330

335

340

(001)

(100) (011)

(101)(110)

(010)

fL

(Hz)

fL

(Hz)

fL

(Hz)

np (Hzminus1) np (Hzminus1)

(a)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(101)

(100)

(001)(010)

(011)

(110)

01 02 03

01 02 03 01 02 03

01 02 0301 02 03

01 02 03

TL

(s)

TL

(s)

TL

(s)

np (Hzminus1)np (Hzminus1)

(b)

Figure 3 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal density (a)resonance frequency (b) decay time 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

each other as shown in Figure 4(a) The further the distancefrom that point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlledmodeThe result also shows that energyratio curve of the (001) enclosure-controlled mode is acontinuation of that of the (31) panel-controlled mode as afunction of panel modal density and the energy ratio curveof the (31) panel-controlled mode is a continuation of that ofthe (001) enclosure-controlled mode

It is shown in Figure 4(b) that in the vicinity of thepoint of panel modal density where the transfer factor is

approximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the panel modal density is increased the resonancefrequency of the (001) enclosure-controlled mode jumpsfrom lower than the resonance frequency of the (001) rigidwalls enclosure mode to higher than it when it passes by thepoint ofmaximumenergy transfer Comparedwith the (001)

8 Shock and Vibration

003 00305 0031 00315 0032 00325 0033 00335 0034

0

5

10

15

minus5

minus10

minus15

np (Hzminus1)

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

003 00305 0031 00315 0032 00325 0033 00335 0034260

270

280

290

300

310

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

np (Hzminus1)

fL

(Hz)

(b)

0

1

2

3

4

5

6

7

8

003 00305 0031 00315 0032 00325 0033 00335 0034

(001) enclosure-controlled mode(31) panel-controlled mode

np (Hzminus1)

TL

(s)

(c)

265

270

275

280

285

290

295

300

305

310

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

003 00305 0031 00315 0032 00325 0033 00335 0034

np (Hzminus1)

fL

(Hz)

(d)

Figure 4 (a) Ratio between the sound field energy and panel vibration energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers 119871

119885= 06m

119879119901119872

= 05 s 119879119886119873

= 15 s

enclosure-controlled mode it is opposite to the (31) panel-controlled mode As the panel modal density is away fromthe point of maximum interaction strength the resonancefrequencies of the (31) panel-controlled mode and the (001)enclosure-controlled mode tend to resonance frequencies ofuncoupled them respectively

It is shown in Figure 4(c) that as the panel modal densityis increased decay time of the (001) enclosure-controlledmode firstly reduces When panel modal density arrives atthe point of strong coupling the minimum decay time is gotThen decay time of the (001) enclosure-controlled mode

becomes longer as the panel modal density is increasedCompared with the (001) enclosure-controlled mode decaytime curve of the (31) panel-controlled mode is opposite asthe panel modal density is increased In addition at the pointof panel modal density where the decay times of the (001)enclosure-controlled mode and the (31) panel-controlledmode are approximately equal to each other the energytransfer from the panel vibration to the enclosure acousticfield is relative maximum

Due to the limited number of uncoupled panel and enclo-sure modes used in this analysis some important modes may

Shock and Vibration 9

0 2 4 6 8

Pane

l mod

e

|VM| (ms) times10minus4

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 5 Modal amplitude of the panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

be excluded and truncation errors exist in the calculation ofresonance frequencies and decay times of acoustical modesFigure 4(d) shows the resonance frequencies of two acousti-calmodes using different truncated numbers Comparedwiththe solution for the resonance frequencies using the combi-nation of 64 enclosure modes and 81 panel modes and of 125enclosure modes and 121 panel modes the solution for theresonance frequencies of the (001) enclosure-controlled andthe (31) panel-controlled acoustical modes using 40 panelmodes and 40 enclosuremodesmeets the requirement in thisanalysis and the computation efficiency is also improved

When the panel thickness is 00882m the two subsystemsmodal amplitudes of panel vibration and enclosure acousticfield are shown in Figures 5 and 6 respectively The panelvibration of coupled system is controlled by the (31) panelmode in Figures 5(a) and 5(b) The sound field in the enclo-sure of coupled system is controlled by the (001) enclosuremode in Figures 6(a) and 6(b) Combined with energy ratioshown in Figure 4(a) it is proved that the coupled systemis named the (31) panel-controlled acoustical mode and the(001) enclosure-controlled acoustical mode respectively

412 Panel Internal Damping The decay times of uncoupledclamped panel modes are used to describe the mechanicaldamping of the clamped panel in the analysis The panelvibration energy is dissipated by its internal damping andthen the vibration level of panel will be reduced The dis-turbing degree of the enclosure sound field due to the panelvibration becomes smaller and the energy flow betweenthe panel and enclosure becomes smaller too The decaytimes and the resonance frequencies of enclosure-controlledmodes are related to the decay times of uncoupled panelmodes Figures 7(a) and 7(b) show plot of the variation ofthe resonance frequencies and the decay times of the firstfew enclosure-controlled modes with the decay time of panelmodes respectively In this analysis the decay times of allrigid wall enclosure modes are 15 s The thickness of panel is0007m and the depth of enclosure is 06mThe decay timesof all uncoupled panel modes are equal

In Figure 7(a) the resonance frequencies of the firstfew enclosure-controlled modes approach the uncoupledenclosure resonance frequency as the panelmodal decay timeis reduced (corresponding to the damping increased) Thebigger the damping in the panel is the more the energy is

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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8 Shock and Vibration

003 00305 0031 00315 0032 00325 0033 00335 0034

0

5

10

15

minus5

minus10

minus15

np (Hzminus1)

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

003 00305 0031 00315 0032 00325 0033 00335 0034260

270

280

290

300

310

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

np (Hzminus1)

fL

(Hz)

(b)

0

1

2

3

4

5

6

7

8

003 00305 0031 00315 0032 00325 0033 00335 0034

(001) enclosure-controlled mode(31) panel-controlled mode

np (Hzminus1)

TL

(s)

(c)

265

270

275

280

285

290

295

300

305

310

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

003 00305 0031 00315 0032 00325 0033 00335 0034

np (Hzminus1)

fL

(Hz)

(d)

Figure 4 (a) Ratio between the sound field energy and panel vibration energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers 119871

119885= 06m

119879119901119872

= 05 s 119879119886119873

= 15 s

enclosure-controlled mode it is opposite to the (31) panel-controlled mode As the panel modal density is away fromthe point of maximum interaction strength the resonancefrequencies of the (31) panel-controlled mode and the (001)enclosure-controlled mode tend to resonance frequencies ofuncoupled them respectively

It is shown in Figure 4(c) that as the panel modal densityis increased decay time of the (001) enclosure-controlledmode firstly reduces When panel modal density arrives atthe point of strong coupling the minimum decay time is gotThen decay time of the (001) enclosure-controlled mode

becomes longer as the panel modal density is increasedCompared with the (001) enclosure-controlled mode decaytime curve of the (31) panel-controlled mode is opposite asthe panel modal density is increased In addition at the pointof panel modal density where the decay times of the (001)enclosure-controlled mode and the (31) panel-controlledmode are approximately equal to each other the energytransfer from the panel vibration to the enclosure acousticfield is relative maximum

Due to the limited number of uncoupled panel and enclo-sure modes used in this analysis some important modes may

Shock and Vibration 9

0 2 4 6 8

Pane

l mod

e

|VM| (ms) times10minus4

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 5 Modal amplitude of the panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

be excluded and truncation errors exist in the calculation ofresonance frequencies and decay times of acoustical modesFigure 4(d) shows the resonance frequencies of two acousti-calmodes using different truncated numbers Comparedwiththe solution for the resonance frequencies using the combi-nation of 64 enclosure modes and 81 panel modes and of 125enclosure modes and 121 panel modes the solution for theresonance frequencies of the (001) enclosure-controlled andthe (31) panel-controlled acoustical modes using 40 panelmodes and 40 enclosuremodesmeets the requirement in thisanalysis and the computation efficiency is also improved

When the panel thickness is 00882m the two subsystemsmodal amplitudes of panel vibration and enclosure acousticfield are shown in Figures 5 and 6 respectively The panelvibration of coupled system is controlled by the (31) panelmode in Figures 5(a) and 5(b) The sound field in the enclo-sure of coupled system is controlled by the (001) enclosuremode in Figures 6(a) and 6(b) Combined with energy ratioshown in Figure 4(a) it is proved that the coupled systemis named the (31) panel-controlled acoustical mode and the(001) enclosure-controlled acoustical mode respectively

412 Panel Internal Damping The decay times of uncoupledclamped panel modes are used to describe the mechanicaldamping of the clamped panel in the analysis The panelvibration energy is dissipated by its internal damping andthen the vibration level of panel will be reduced The dis-turbing degree of the enclosure sound field due to the panelvibration becomes smaller and the energy flow betweenthe panel and enclosure becomes smaller too The decaytimes and the resonance frequencies of enclosure-controlledmodes are related to the decay times of uncoupled panelmodes Figures 7(a) and 7(b) show plot of the variation ofthe resonance frequencies and the decay times of the firstfew enclosure-controlled modes with the decay time of panelmodes respectively In this analysis the decay times of allrigid wall enclosure modes are 15 s The thickness of panel is0007m and the depth of enclosure is 06mThe decay timesof all uncoupled panel modes are equal

In Figure 7(a) the resonance frequencies of the firstfew enclosure-controlled modes approach the uncoupledenclosure resonance frequency as the panelmodal decay timeis reduced (corresponding to the damping increased) Thebigger the damping in the panel is the more the energy is

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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Shock and Vibration

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Shock and Vibration 9

0 2 4 6 8

Pane

l mod

e

|VM| (ms) times10minus4

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 5 Modal amplitude of the panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

be excluded and truncation errors exist in the calculation ofresonance frequencies and decay times of acoustical modesFigure 4(d) shows the resonance frequencies of two acousti-calmodes using different truncated numbers Comparedwiththe solution for the resonance frequencies using the combi-nation of 64 enclosure modes and 81 panel modes and of 125enclosure modes and 121 panel modes the solution for theresonance frequencies of the (001) enclosure-controlled andthe (31) panel-controlled acoustical modes using 40 panelmodes and 40 enclosuremodesmeets the requirement in thisanalysis and the computation efficiency is also improved

When the panel thickness is 00882m the two subsystemsmodal amplitudes of panel vibration and enclosure acousticfield are shown in Figures 5 and 6 respectively The panelvibration of coupled system is controlled by the (31) panelmode in Figures 5(a) and 5(b) The sound field in the enclo-sure of coupled system is controlled by the (001) enclosuremode in Figures 6(a) and 6(b) Combined with energy ratioshown in Figure 4(a) it is proved that the coupled systemis named the (31) panel-controlled acoustical mode and the(001) enclosure-controlled acoustical mode respectively

412 Panel Internal Damping The decay times of uncoupledclamped panel modes are used to describe the mechanicaldamping of the clamped panel in the analysis The panelvibration energy is dissipated by its internal damping andthen the vibration level of panel will be reduced The dis-turbing degree of the enclosure sound field due to the panelvibration becomes smaller and the energy flow betweenthe panel and enclosure becomes smaller too The decaytimes and the resonance frequencies of enclosure-controlledmodes are related to the decay times of uncoupled panelmodes Figures 7(a) and 7(b) show plot of the variation ofthe resonance frequencies and the decay times of the firstfew enclosure-controlled modes with the decay time of panelmodes respectively In this analysis the decay times of allrigid wall enclosure modes are 15 s The thickness of panel is0007m and the depth of enclosure is 06mThe decay timesof all uncoupled panel modes are equal

In Figure 7(a) the resonance frequencies of the firstfew enclosure-controlled modes approach the uncoupledenclosure resonance frequency as the panelmodal decay timeis reduced (corresponding to the damping increased) Thebigger the damping in the panel is the more the energy is

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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Shock and Vibration

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International Journal of

10 Shock and Vibration

0 1 2 3

Enclo

sure

mod

e

|PN| (Pa)

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(a)

0 1 2 3 4|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(300)(023)(040)(230)(132)

(b)

Figure 6 Modal amplitude of sound filed in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 000844m 119871

119885= 06m 119879

119886119873= 15 s 119879

119901119872= 05 s

dissipated in the panel As the damping of panel is increasedthe flexible panel tends to rigid wall gradually and the soundfield in the enclosure will not be affected by the panel

As the panel damping is increased the decay times ofthe first few enclosure-controlled modes decrease and arriveat the minimum and then they increase to the uncoupledenclosure mode as shown in Figure 7(b)

42 Effects of Enclosure Depth on Acoustical Modes Similarto the panel modal density enclosure depth has significanteffect on the panel-enclosure coupled system Due to themodal density of enclosure sound field which relates tomany factors from (35) such as excitation frequency 119891enclosure volume 119881

0 total surface area of enclosure 119878 and

total edge lengths inside enclosure 119871 the enclosure depth isused as a variable which affects the dimensions of enclosurein subsequent analysis From (25) (26) and (29) when thecoupling coefficient between enclosure and panel modes isnot equal to zero the reason for the variation of transferfactor between them with the change of enclosure depthcan be divided into two different kinds one is the changeof difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0

and the other is the change of enclosure depth when theenclosure modal index 119899 = 0 The effect of enclosure depthon the resonance frequencies of panel-controlled acousticalmode was studied by Dowell et al [2] but only resonancefrequencies of the first two panel-controlled modes werestudied

Asmentioned above the coupling strength between paneland enclosure modes will be altered by adjusting enclosuredepth when the modal coupling coefficient between themis nonzero In Figures 8(a) 8(b) and 8(c) transfer factorsbetween enclosure modes and panel modes (11) (12) and(31) are plotted against enclosure depth respectively Com-pared with the effect of panel modal density upon transferfactors enclosure depth has smaller influence upon transferfactors in the zone of analysis

In Figures 8(a) and 8(b) the transfer factors betweenenclosure modes and panel modes (11) and (12) are all lessthan 10 and it means that no enclosure modes satisfy thewell-coupled condition with panel modes (11) and (12) Butwhen the enclosure depth is shallow the enclosure modes(000) and (010) have relative large coupling strength withclamped panel modes (11) and (12) respectively As theenclosure depth is increased the coupling strength between

Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

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Shock and Vibration

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Shock and Vibration 11

fL

(Hz)

fL

(Hz)

fL

(Hz)

TpM (s) TpM (s)

(010)

(101)

(110)

(011)(100)

(001)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

214

215

216

2865

287

2875

171

172

173

358

359

360

275

2755

276

334

3345

(a)

TpM (s) TpM (s)

0009 01 1 410 0009 01 1 410

0009 01 1 4100009 01 1 410

0009 01 1 410 0009 01 1 410

TL

(s)

TL

(s)

TL

(s)

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

0

10

20

(010)

(101)(110)

(011)(100)

(001)

(b)

Figure 7 Resonance frequencies and decay times of the first few enclosure-controlled modes as a function of panel modal decay time (a)resonance frequency (b) decay time ℎ = 0007m 119871

119885= 06m 119879

119886119873= 15 s

the (000) enclosuremode and the (11) panelmode decreasesgradually and it is similar to transfer factor between the(010) enclosuremode and the (12) panelmodeThe couplingdegree between the (001) enclosure mode and the (11) panelmode is inverse when the enclosure depth is increased andthe influence of it upon transfer factor is less than enclosuremode (000) The reason is that when the mode index 119899 isequal to zero the factor which determines the transfer factorbetween the enclosure and panelmodes is for enclosure depth

rather than for the difference between resonance frequenciesof them from (25) and (26)

The transfer factor between the (001) enclosure modeand the (31) panel mode is in the order of 10 at the enclosuredepth 07182m as shown in Figure 8(c) and there will belarge energy transfer between the (001) enclosure mode andthe (31) panelmodeThe reason is that the difference betweenresonance frequencies of enclosure and panel modes is in theorder of 0 which is different from transfer factor between

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

International Journal of

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Shock and Vibration

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DistributedSensor Networks

International Journal of

12 Shock and Vibration

02 04 06 08 10

005

01

015

02

025

03

035

04

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)

(200)

(201)

(a)

02 04 06 08 10

01

02

03

04

05

06

07

08

Enclosure depth (m)

Tran

sfer f

acto

r

(010)

(011)

(012)

(210)

(030)

(b)

02 04 06 08 10

02

04

06

08

1

Enclosure depth (m)

Tran

sfer f

acto

r

(000)

(001)

(020)

(002)

(021)(200)(201)

(c)

Figure 8 Transfer factors between enclosure modes and panel modes as a function of enclosure depth (a) (11) panel mode (b) (12) panelmode (c) (31) panel mode ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

the (000) enclosure mode and the (11) panel mode aboveWhen the enclosure depth is shallow the (200) enclosuremode has small coupling strength with the (31) panel modeand the coupling strength is reduced with the increase inenclosure depth

Figure 9(a) shows resonance frequencies of the first eightpanel-controlled acoustical modes as a function of enclosuredepth As the enclosure depth is increased the resonancefrequencies of panel-controlled acoustical modes (11) (12)

(21) and (22) approach to those of uncoupled ones Thereason is that transfer factors between panel modes aboveand enclosure modes are less than 10 and do not satisfywell-coupled condition between them in the frequency zoneof analysis and the variation of transfer factors betweenpanel modes (21) (22) and enclosure modes with enclosuredepth is the same as panel modes (11) and (12) as shownin Figures 8(a) and 8(b) The resonance frequencies of panel-controlled acoustical modes (12) (21) and (22) increase

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

fL

(Hz)

fL

(Hz)

fL

(Hz)

fL

(Hz)

(11)(12)

(14)

(21)(13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

75

80

85

175

180

185

136

138

140

330

340

350

236

238

240

390

395

400

235

240

245

550

560

570

(a)

(11) (12)

(14)

(21) (13)

(22)

(31)

(23)

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

02 04 06 08 1 02 04 06 08 1

Enclosure depth (m) Enclosure depth (m)

TL

(s)

TL

(s)

TL

(s)

TL

(s)

05

051

052

055

05

06

05

055

06

05

1

15

05

055

06

06

08

1

06

08

1

0608

1

(b)

Figure 9 Resonance frequencies and decay times of the first few panel-controlled modes as a function of enclosure depth (a) resonancefrequency (b) decay time ℎ = 0007m 119879

119886119873= 15 s 119879

119901119872= 05 s

gradually as the enclosure depth is increasedOn the contrarythe resonance frequency of the (11) panel-controlled modeis reduced This is because of that the resonance frequenciesof the (11) uncoupled panel mode are larger than the (000)rigid wall enclosure mode while the resonance frequenciesof panel modes (12) (21) (22) is less than those of enclosuremodesThe resonance frequencies of panel-controlledmodes(13) (23) (31) and (14) appear jump phenomenon in theprocess of the change of enclosure depth which correspondsto the point of enclosure depth where the transfer factorbetween the (31) panel mode and enclosure mode is in theorder of 10 The variation of transfer factor between panel

modes (13) (23) and (14) and enclosure modes is thesame as the (31) panel mode Simultaneously the interactionbetween panel and enclosure modes becomes stronger andthe larger energy transfer between them is conducted

In Figure 9(b) the decay times of the first eight panel-controlled modes are plotted against enclosure depth Corre-sponding to the points of enclosure depth where resonancefrequencies of panel-controlled modes jump to higher fre-quencies in Figure 9(a) the decay times of them appearpeaks Meanwhile there will be large energy transfer betweenenclosure and panel modes As the enclosure depth isincreased less enclosure modes participate in the coupling

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

066 068 07 072 074 076 078

0

5

10

15

20

Enclosure depth (m)

minus5

minus10

minus15

minus20

(001) enclosure-controlled mode(31) panel-controlled mode

Ener

gy ra

tio (1

0log10

(EpME

aN

))

(a)

066 068 07 072 074 076 078210

220

230

240

250

260

270

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode(001) uncoupled enclosure mode(31) uncoupled panel mode

fL

(Hz)

(b)

066 068 07 072 074 076 0780

2

4

6

8

10

Enclosure depth (m)

(001) enclosure-controlled mode(31) panel-controlled mode

TL

(s)

(c)

066 068 07 072 074 076 078215

220

225

230

235

240

245

250

255

260

265

Enclosure depth (m)

40 enclosure modes and 40 panel modes64 enclosure modes and 81 panel modes125 enclosure modes and 121 panel modes

fL

(Hz)

(d)

Figure 10 (a) Ratio between the panel vibration energy and sound field energy for two acoustical modes (001) enclosure-controlled modeand (31) panel-controlledmode (b) Resonance frequencies of two acousticalmodes the (001) uncoupled enclosuremode and the (31) panelmode (c) 60 dB modal decay time 119879

119871 (d) Resonance frequencies of two acoustical modes with different truncation numbers ℎ = 0007m

119879119886119873

= 15 s 119879119901119872

= 05 s

with panel modes and the decay times of panel-controlledmodes tend to those of uncoupled ones

In order to describe the process of strong couplingbetween panelmode and enclosuremode when the enclosuredepth is changed we analyze the coupled system from threepoints of view of energy ratio between the panel vibration andsound field in the enclosure the resonance frequencies andmodal decay times of coupled system Similar to the analysisof forced response of the coupled system with differentpanel modal density the plane wave 119875

119894with amplitude 1 Pa

on the panel surface is also used to drive the vibration ofpanel and the excitation frequencies are also respectively

the resonance frequencies of panel-controlled and enclosure-controlled acoustical modes The elevation angle 120572 andazimuth angle 120579 of planewave are 90∘ and 0∘ respectivelyThedecay times of all uncoupled panel and enclosure modes are15 s and 05 s respectively As seen in Figure 8(c) the transferfactor between panel mode (31) and enclosure mode (001)is approximately equal to 10 at large enclosure depth whichmeans that the energy transfer between the sound field andthe panel is almost entirely between these two modes

The effect of enclosure depth on energy ratio between thesound field in the enclosure and panel vibration is shownin Figure 10(a) which contains a panel-controlled mode and

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 15

0 05 1 15

Pane

l mod

e

|VM| (ms) times10minus3

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(a)

0 05 1|VM| (ms) times10minus3

Pane

l mod

e

(11)(12)(21)(13)(22)(23)(31)(14)(32)(24)(33)(15)(41)(34)(25)(42)(43)(16)(35)(26)(44)(51)(52)(17)(36)(53)(45)(27)(54)(46)(61)(37)(18)(62)(55)(28)(63)(47)(64)(38)

(b)

Figure 11 Modal amplitude of panel vibration velocity (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled mode ℎ =0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

an enclosure-controlled mode The resonance frequenciesand the decay times of coupled system are plotted againstenclosure depth in Figures 10(b) and 10(c) respectivelyFigure 10(d) shows natural frequencies of two acousticalmodes using three kinds of panel and enclosure modes num-bers In addition the resonance frequencies of uncoupledpanel and enclosure are included in Figure 10(b) As theenclosure depth tends to the point where transfer factorbetween the (31) panel mode and the (001) enclosure modeis in the order of 10 energy ratio resonance frequencies anddecay times of the (31) panel-controlledmode and the (001)enclosure-controlled mode tend to be equal respectively

Similar to the effect of panel modal density on the energyratio near the point of the maximum interaction strengthbetween the (31) panel mode and the (001) enclosuremode the energy which is stored in each part (panel andenclosure) of the (31) panel-controlled mode and of the(001) enclosure-controlled mode approximately is equalas shown in Figure 10(a) The further the distance fromthat point the greater the difference between the energyratio between the (001) enclosure-controlled mode and the(31) panel-controlled mode And the energy ratio curves of

the (31) panel-controlled mode and the (001) enclosure-controlled mode are continuation of those of the (001)enclosure-controlledmode one and the (31) panel-controlledmode one as the enclosure depth is increased respectively

It is shown in Figure 10(b) that in the vicinity ofthe point of enclosure depth where the transfer factor isapproximately equal to 10 the resonance frequencies ofthe (001) enclosure-controlled mode and the (31) panel-controlled mode which deviate from uncoupled resonancefrequencies are greater than others respectively The reasonis that the strength of interaction between panel vibrationand sound field in the enclosure becomes bigger than othersAs the enclosure depth is away from the point of maximuminteraction strength the resonance frequencies of the (31)panel-controlled mode and the (001) enclosure-controlledmode tend to those of uncoupled ones respectively

In Figure 10(c) as the enclosure depth is increased thedecay time of the (001) enclosure-controlledmode decreasesgradually When arriving at the point of strong coupling thedecay time becomes relative minimumThen the decay timeof the (001) enclosure-controlled mode become to increaseas the enclosure depth is increased Comparedwith the (001)

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

16 Shock and Vibration

0 2 4 6

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

|PN| (Pa)

(a)

0 1 2 3|PN| (Pa)

Enclo

sure

mod

e

(000)(010)(001)(100)(011)(110)(101)(020)(111)(002)(021)(120)(012)(200)(102)(121)(210)(112)(201)(030)(022)(211)(031)(130)(220)(122)(003)(131)(202)(221)(013)(212)(103)(032)(113)(3 00)(023)(040)(230)(132)

(b)

Figure 12 Modal amplitude of sound field in the enclosure (a) the (001) enclosure-controlled mode (b) the (31) panel-controlled modeℎ = 0007m 119871

119885= 0722m 119879

119886119873= 15 s 119879

119901119872= 05 s

enclosure-controlled mode the decay time curve of the (31)panel-controlled mode is opposite as the enclosure depth isincreased In addition at the point of panel modal densitywhere the decay times of the (001) enclosure-controlledmode and the (31) panel-controlled mode tend to equal eachother the energy transfer between the (001) enclosure modeand the (31) panel mode is relative maximum for these twoacoustical modes

Similarly with the results shown in Figure 4(d) thetruncation error of the solution for the resonance frequenciesof (001) enclosure-controlled and (31) panel-controlledacoustical modes using 40 panel modes and 40 enclosuremodes can be neglected as shown in Figure 10(d)

When enclosure depth is 0722m modal amplitude ofeach part (panel and enclosure) of two acoustical modesis shown in Figures 11 and 12 respectively Panel vibrationof coupled system is controlled by the (31) panel mode inFigures 11(a) and 11(b) Soundfield in the enclosure of coupledsystem is controlled by the (001) enclosure mode in Figures12(a) and 12(b) Combined with the energy ratio shown inFigure 10(a) it is proved that the coupled system is namedthe (31) panel-controlled mode and the (001) enclosure-controlled mode respectively

5 Conclusions

Thispaper presents a theoretical investigation into the vibroa-coustic analysis of a rectangular enclosure with clampedflexible wall using the classical modal coupling method

The coupling between clamped panel and enclosuremodes is very selective and it is the same as the couplingbetween enclosure and simply supported panel modes Themodal coupling coefficient determines the degree of matchbetween panel and enclosure modes and the couplingstrength between panel and enclosure is determined by thetransfer factorWhen the panel modal density is changed thecoupling strength between panel and enclosure modes onlydepends on the difference between the resonance frequenciesof them But for the enclosure depth the factors whichdetermine the coupling strength between enclosure andpanel modes can be divided into two different kinds one isthe difference between resonance frequencies of panel andenclosure modes when the enclosure modal index 119899 = 0 andthe other is the enclosure depth when the enclosure modalindex 119899 = 0

The transfer factor tends to 10 by adjusting the panelthickness or enclosure depth and the interaction between

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 17

the sound field in the enclosure and the panel vibrationthen becomes stronger gradually In the vicinity of themaximum coupling point the resonance frequencies ofenclosure-controlled or panel-controlledmodes appear jumpphenomenon Simultaneously the resonance frequency andthe decay time of acoustical mode which deviate from thoseof uncoupled ones are more than others and the energyof interaction between panel vibration and sound field inenclosure becomes bigger than others

The vibration of clamped panel which acts on the soundfield in the enclosure can be changed by varying damping ofthe panel and then the energy between the vibration paneland enclosure sound field is altered consequently

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the Science and TechnologySupport Plan of Jiangsu China (Grant no BE2D1047) andcombination of product study and investigation in theprospective Research Program of Jiangsu China (Grant noBY2011151)

References

[1] E H Dowell and H Voss ldquoThe effect of a cavity on panelvibrationrdquo AIAA Journal vol 1 pp 476ndash477 1963

[2] E H Dowell G F Gorman III and D A Smith ldquoAcous-toelasticity general theory acoustic natural modes and forcedresponse to sinusoidal excitation including comparisons withexperimentrdquo Journal of Sound and Vibration vol 52 no 4 pp519ndash542 1977

[3] J Pan and D A Bies ldquoThe effect of fluid-structural coupling onsound waves in an enclosuremdashtheoretical partrdquo Journal of theAcoustical Society of America vol 87 no 2 pp 691ndash707 1990

[4] S M Kim and M J Brennan ldquoA compact matrix formulationusing the impedance and mobility approach for the analysisof structural-acoustic systemsrdquo Journal of Sound and Vibrationvol 223 no 1 pp 97ndash112 1999

[5] J Pan S J Elliott and K-H Baek ldquoAnalysis of low frequencyacoustic response in a damped rectangular enclosurerdquo Journalof Sound and Vibration vol 223 no 4 pp 543ndash566 1999

[6] F X Xin T J Lu and C Q Chen ldquoVibroacoustic behaviorof clamp mounted double-panel partition with enclosure aircavityrdquo Journal of the Acoustical Society of America vol 124 no6 pp 3604ndash3612 2009

[7] F X Xin and T J Lu ldquoAnalytical and experimental investigationon transmission loss of clamped double panels implication ofboundary effectsrdquo Journal of the Acoustical Society of Americavol 125 no 3 pp 1506ndash1517 2009

[8] J Pan ldquoThe forced response of an acoustic-structural coupledsystemrdquo Journal of the Acoustical Society of America vol 91 no2 pp 949ndash956 1992

[9] K S Sum and J Pan ldquoAn analytical model for bandlimitedresponse of acoustic-structural coupled systems I Direct sound

field excitationrdquo Journal of the Acoustical Society of America vol103 no 2 pp 911ndash923 1998

[10] K S Sum and J Pan ldquoA study of the medium frequencyresponse of sound field in a panel-cavity systemrdquo Journal of theAcoustical Society of America vol 103 no 3 pp 1510ndash1519 1998

[11] B Venkatesham M Tiwari and M L Munjal ldquoAnalyticalprediction of the breakout noise from a rectangular cavity withone compliant wallrdquo Journal of the Acoustical Society of Americavol 124 no 5 pp 2952ndash2962 2008

[12] J Pan C H Hansen and D A Bies ldquoActive control of noisetransmission through a panel into a cavity I Analytical StudyrdquoJournal of the Acoustical Society of America vol 87 no 5 pp2098ndash2108 1990

[13] J Pan and C H Hansen ldquoActive control of noise transmissionthrough a panel into a cavity III effect of a actuator locationrdquoJournal of the Acoustical Society of America vol 90 no 3 pp1493ndash1501 1991

[14] S-M Kim and M J Brennan ldquoActive control of harmonicsound transmission into an acoustic enclosure using bothstructural and acoustic actuatorsrdquo Journal of the AcousticalSociety of America vol 107 no 5 pp 2523ndash2534 2000

[15] B Balachandran A Sampath and J Park ldquoActive controlof interior noise in a three-dimensional enclosurerdquo SmartMaterials and Structures vol 5 no 1 pp 89ndash97 1996

[16] A Berry J-L Guyader and J Nicolas ldquoA general formulationfor the sound radiation from rectangular baffled plates witharbitrary boundary conditionsrdquo Journal of the Acoustical Societyof America vol 88 no 6 pp 2792ndash2802 1990

[17] X Zhang and W L Li ldquoA unified approach for predictingsound radiation from baffled rectangular plates with arbitraryboundary conditionsrdquo Journal of Sound and Vibration vol 329no 25 pp 5307ndash5320 2010

[18] H Nelisse O Beslin and J Nicolas ldquoA generalized approachfor the acoustic radiation from a baffled or unbaffled plate witharbitrary boundary conditions immersed in a light or heavyfluidrdquo Journal of Sound and Vibration vol 211 no 2 pp 207ndash225 1998

[19] C-C Sung and J T Jan ldquoThe response of and sound powerradiated by a clamped rectangular platerdquo Journal of Sound andVibration vol 207 no 3 pp 301ndash317 1997

[20] J P Arenas ldquoOn the vibration analysis of rectangular clampedplates using the virtual work principlerdquo Journal of Sound andVibration vol 266 no 4 pp 912ndash918 2003

[21] J P Arenas Analysis of the acoustic radiation resistance matrixand its applications to vibro-acoustic problems [PhD thesis]University of Auburn 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of