Research ArticleAnalytic Formulation for the Sound Absorption of a PanelAbsorber under the Effects of Microperforation Air PumpingLinear Vibration and Nonlinear Vibration

Y Y Lee

Department of Civil and Architectural Engineering City University of Hong Kong Kowloon Tong Kowloon Hong Kong

Correspondence should be addressed to Y Y Lee bcrayleecityueduhk

Received 27 March 2014 Revised 24 June 2014 Accepted 1 July 2014 Published 23 July 2014

Academic Editor Rehana Naz

Copyright copy 2014 Y Y Lee This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This study includes the first work about the absorption of a panel absorber under the effects of microperforation air pumpingand linear and nonlinear vibrations In practice thin perforated panel absorber is backed by a flexible wall to enhance the acousticperformance within the room The panel is easily excited to vibrate nonlinearly and the wall can vibrate linearly However theassumptions of linear panel vibration and nowall vibration are adopted inmany research worksThe development of the absorptionformula is based on the classical approach and the electroacoustic analogy in which the impedances of microperforation airpumping and linear and nonlinear vibrations are in parallel and connected to that of the air cavity in series Unlike those finiteelement numerical integration and multiscale solution methods and so forth the analytic formula to calculate the absorption of apanel absorber does not require heavy computation effort and is suitable for engineering calculation purposeThe theoretical resultobtained from the proposed method shows reasonable agreement with that from a previous numerical integration method It canbe concluded that the overall absorption bandwidth of a panel absorber with an appropriate configuration can be optimized andwidened by making use of the positive effects of microperforation air pumping and panel vibration

1 Introduction

Numerous studies of structural-acoustic interaction havebeen carried out in recent decades (eg [1ndash6]) Particularlythe topics of microperforated panel and panel absorber haveattracted the interest of many researchers It is because (1)microperforated panel and panel absorber can be made ofany durable materials (2) their maintenance is relativelysimple and (3) microperforated panel requires small space toachieve high sound absorption when compared with typicalfoam or porous materials Acoustic theories for these twotreatments were developed by Mulholland and Parbrook [7]Ford and McCormick [8] and Maa [9 10] According to theassumptions of the previous studies the wall behind a panelabsorber was considered as rigid panel (ie no vibration) andthe panel absorber was considered to vibrate linearly (in factthin panels are easy to vibrate nonlinearly) Although manylinear structural-acoustic or nonlinear structural vibrationproblems had been solved (eg [11ndash17]) very limited studies

considered the nonlinear vibration effect on the soundabsorption performance of a panel absorber In fact it ismore appropriate to employ nonlinear approach to analyzethe vibration of a thin panel There are five of those limitednonlinear structural-acoustic research works [18ndash22] whichfocused on the natural frequencies of nonlinear structural-acoustic systems and the sound absorption radiation andtransmission of nonlinear flatcurved panelsThese works arerelevant to the nonlinear structural-acoustic problem han-dled in this paper but the researchers adopted the solutionmethods which required heavy computational efforts (egfinite element method numerical integration method andharmonic balancemethod) Other solutionmethods for largeamplitude structural vibrations and nonlinear oscillations(eg the method of multiple scales the method of normalforms the method of Shaw and Pierre and the method ofKing and Vakakis [23ndash27]) also require relatively tedioussolution implementation and heavy computational efforts In[28] the simple and straightforward electroacoustic analogy

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 906506 12 pageshttpdxdoiorg1011552014906506

2 Abstract and Applied Analysis

was adopted for modelling of the sound absorption of apanel absorber Besides the air pumping effect (or Helmholtzresonance effect) has beenmentioned in many textbooks andarticles (eg [29]) but it has not been incorporated into theresearch problem of nonlinear panel absorber In this paperit focuses on the development of the analytic absorptionformula of a panel absorber which considers the four effects(iemicroperforation air pumping and linear and nonlinearvibrations)

2 Analytic Formulation

21 Nonlinear Panel Vibration and Liner Wall VibrationFigure 1 shows the proposed panel absorber subject to theeffects of microperforation air pumping and linear andnonlinear vibrations Firstly the impedances of the nonlinearpanel and linear wall are derived in this subsection Accord-ing to the formulation of Chu and Herrmann [30] the gov-erning equation for the nonlinear vibration of a rectangularpanel subject to uniformly distributed harmonic excitationcan be expressed in the following form (see Appendix A formore)

1205881198892120576

1198891205912+ 1205881205962

119900120576 + 120573120576

3+ 119865 (120591) = 0 (1)

where

119865(120591) = 119865119900119898119899

sin(120596120591) =modal harmonic excitation

119865119900119898119899

= 119865119900Λ119898119899

= 120582120588119905119892Λ119898119899

= modal excitationmagnitude

119865119900= 120582 120588119905119892 = physical excitation magnitude

119892 = gravity acceleration (981msminus2)

120591 = time

120582 = dimensionless excitation parameter

Λ119898119899=

(int119887

0int119886

0sin (119898120587119909119886) sin (119899120587119910119887) 119889119909 119889119910)

int119887

0int119886

0[sin (119898120587119909119886) sin (119899120587119910119887)]2119889119909 119889119910

= modal coefficient

(2)

120596 = excitation frequency

120596119900= linear resonant frequency of the (119898 119899) mode

119898 119899 = structural mode numbers

120573 =119864119905

12 (1 minus ]2) 119886431205874

2

times [1

21198982(1198982+ ]11989921199032)(minus

3

2+ ]

11989921199032

1198982)

minus3

41198982(1198982+ ]11989921199032)

+1

211989921199032(11989921199032+ ]1198982)(minus

3

2+ ]

1198982

11989921199032)

minus3

411989921199032(11989921199032+ ]1198982)]

= nonlinear stiffness coefficient(3)

119903 = 119886119887 is the aspect ratio 119864 is Youngrsquos modulus 120588 isthe surface density ] is Poissonrsquos ratio and 119905 119886 and 119887are the thickness length and width

Consider the approximation of 120576 = 119860 cos(120596120591) in (1) Then

3

41205731198603+ 120588 (120596

2

119900minus 1205962)119860 + 119865

119900119898119899= 0 (4)

rArr

1198603+4

3120573120588 (1205962

119900minus 1205962)119860 +

4

3120573119865119900119898119899

= 0 (5)

where 119860 =modal vibration amplitudeFor simplicity let119901 = (43120573)120588(1205962

119900minus1205962) 119902 = (43120573)119865

119900119898119899

and 119860 = 119861 minus (1199013119861) (ie Vietarsquos substitution [31]) Equation(5) can be rewritten as

1198613+ 119902 minus

1199013

271198613= 0 (6)

rArr

1198616+ 1199021198613minus1199013

27= 0 (7)

If 1198613 is considered as an independent unknown (7) is aldquomodifiedrdquo quadratic equation Then the analytic formulasfor the cubic solutions are given below

1198601= minus

1

3

3radic1

2(Θ + Ω) minus

1

3

3radic1

2(Θ minus Ω) (8a)

1198602=1 + 119895radic3

6

3radic1

2(Θ + Ω) +

1 minus 119895radic3

6

3radic1

2(Θ minus Ω) (8b)

1198603=1 minus 119895radic3

6

3radic1

2(Θ + Ω) +

1 + 119895radic3

6

3radic1

2(Θ minus Ω) (8c)

where Θ = radic(27119902)2+ 4(3119901)

3Ω = 27119902In the next step the nonlinear stiffness can be defined as

(34)1205731198602 and overall panel stiffness is 1205881205962

119900+ (34)120573119860

2 Thepanel impedance of the (119898 119899) mode is given by

119885119875119898119899

=119865119900119898119899

119881119875119898119899

=

2120588120585120596120596119873+ 119895 [120588 (120596

2minus 1205962

119900) minus (34) 120573119860

2]

120596

(9)

where 119895 = radicminus1 = complex number 120585 is the panel dampingratio 119881

119875119898119899is the modal panel velocity Note that 120596

119873is

Abstract and Applied Analysis 3

D

Linear wall vibration

z

x

Nonlinear perforated panel vibration

Air mass moving along the tube

Uniform sound pressure

Cavity

Rigi

d

Rigi

dFigure 1 A panel absorber subject to the effects of microperforation air pumping and linear and nonlinear vibrations

the nonlinear peak frequency instead of the linear naturalfrequency 120596

119900in [32] (see Appendix B for the derivation of

120596119873)When it is the case of small amplitude vibration or linear

vibration 120596119873is equal to 120596

119900

Consider the overall panel velocity contributed from themodal velocities and find the overall panel impedance

119881119875=

119872

sum

119898

119873

sum

119899

119865119900Λ119898119899Λ1015840

119898119899

119885119875119898119899

(10)

rArr

119885119875=119865119900

119881119875

= (

119872

sum

119898

119873

sum

119899

Λ119898119899Λ1015840

119898119899

119885119875119898119899

)

minus1

(11)

where Λ1015840

119898119899= (int

119887

0int119886

0sin(119898120587119909119886) sin(119899120587119910119887)119889119909 119889119910)119886119887

According to [33] the sound absorption of two linearlyvibrating plates with different boundary conditions was veryclose It was concluded that the boundary condition wouldnot affect the sound absorption significantlyThat is why onlyone boundary condition case is considered here

Consider the linear case (or small amplitude case) inthe model representing the panel vibration and the corre-sponding modal and overall panel impedances (ie (9) (11))It is noted that for the linear vibration the fundamentalharmonic balance can lead to the exact solution (no higherharmonic terms are required) Hence the modal and overallwall impedances are given below

119885119882119898119899

=

2120588119882120585119882120596120596119882+ 119895 [120588 (120596

2minus 1205962

119882)]

120596

(12)

119885119882= (

119872

sum

119898

119873

sum

119899

Λ119898119899Λ1015840

119898119899

119885119882119898119899

)

minus1

(13)

where 120585119882 120588119882 and 120596

119882are the wall damping factor density

and natural frequency

22 Air Pumping (Helmholtz Resonance) In Figure 1 theimpedance of the air pumping along the tubemounted on the

panel absorber is obtained here The differential equation forthe air mass pumping inout along the tube is given by [29]

1205881198861198711015840 1198892Λ

1198891205912+(119877119903+ 119877119908)

119878

119889Λ

119889120591+ 119875 (120591) = 0 (14)

where Λ = air mass displacement 119875(120591) = acoustic pressureexcitation 119877

119903= 120588119886119888119886119896211987824120587 = damping due to radiation

resistance 119877119908= 21205881198861198711015840119878120572119908= damping due to thermoviscous

resistance 1198711015840 = 119871 + 14119886119905= effective tube length 120572

119908=

(1119886119905119888119886)(1205781205962120588

119886)05(1 + (120574radicPr)) = absorption coefficient for

wall losses and 120588119886is the air density 1198711015840 is the effective length

of the tube with unflanged outer end 119896 is the wave number119871 is the tube length 119886

119905is the radius of the tube 119878 is the cross

section area of the tube 120578 is the air viscosity 120574 is the ratio ofspecific heat Pr is Prandtl number

Similar to the overall panel impedance in (9) theimpedance of air pumping along the tube is given below

1198851015840

119867=((119877119903+ 119877119908) 119878) 120596 + 119895120588

11988611987110158401205962

120596 (15)

Using (15) the air mass velocity is given by

1198851015840

119867=119875119900

1198811015840119867

997904rArr 1198811015840

119867=119875119900

1198851015840119867

(16)

where 119875119900= acoustic pressure magnitude

Consider the average velocity over the panel surface

119881119867= 1198811015840

119867120590119867 (17)

where 120590119867= the ratio of tube area to panel area

Then the overall impedance of air pumping is obtained bysubstituting (17) into (16)

119885119867=1198851015840

119867

120590119867

(18)

23 Microperforation In Figure 1 the impedance of themicroholes over the panel surface is obtained here According

4 Abstract and Applied Analysis

to [9 10] the real and imaginary parts of the acousticimpedance of a microhole are given below

1198851015840

119872119877= 120588119886119888119886(0147

119905

1198892)(radic9 +

1001198892119891

32+ 1768radic119891

1198892

119905)

(19a)

1198851015840

119872119868= 1205881198861198881198861847 119891119905(1 +

1

radic9 + 501198892119891

+ 085119889

119905)

(19b)

where 1198851015840119872= 1198851015840

119872119877+ 1198951198851015840

119872119868= the impedance of a microhole

119891 is frequency 119905 is panel thickness and 119889 is the perforationdiameter 120588

119886and 119888119886are air density and sound speed

Then the overall impedance of all microholes over thepanel is given by

119885119872=1198851015840

119872

120590 (20)

where 120590 is the perforation ratio

24 Air Cavity In Figure 1 the impedance of the air cavityis obtained here The acoustic velocity potential within arectangular cavity is given by the following homogeneouswave equation [1]

nabla2120601 minus

1

1198882119886

1205972120601

1205971205912= 0 (21)

where 120601 is the velocity potential functionThe air particle velocities in the 119909 119910 and 119911 direc-

tions and pressures within the air cavity can be given by120597120601120597119909 120597120601120597119910 120597120601120597119911 and minus120588

119886(120597120601120597120591) respectively

The boundary conditions of the rectangular cavity to besatisfied are

120597120601

120597119909

10038161003816100381610038161003816100381610038161003816119909=0=120597120601

120597119909

10038161003816100381610038161003816100381610038161003816119909=119886= 0 (22a)

120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

=120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=119887

= 0 (22b)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=0

= 0 (22c)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=minus119863= 119881119863 (22d)

where 119886 and 119887 are the length and width of the rectangularcavity or the panel and 119881

119863is the average velocity at 119911 =

119863 = cavity depth Note that the cavity impedance of no wallvibration is derived here and used into the sound absorptionformulation in the next subsection which can consider boththe panel and wall vibration effects and is derived using theelectroacoustic analogy

By applying the boundary conditions in (22a) and (22b)the solution of (21) can be expressed as

120601 =

119880

sum

119906=0

119882

sum

119908=0

[119871119906119908

cosh (120583119906119908119911) + 119873

119906119908sinh (120583

119906119908119911)]

times cos(119906120587119909119886) cos(

119908120587119910

119887) 119890119894120596120591

(23)

where 120583119906119908

= radic((119906120587119886)2+ (119908120587119887)

2) minus (120596119888

119886)2 119906 and 119908 are

the acoustic mode numbers 119880 and 119882 are the numbers ofacoustic modes used in the 119909 and 119910 directions and 119871

119906119908and

119873119906119908

are coefficients to be determined in the following stepSubstituting (23) into (22c) and (22d) gives

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=0

= 0 997904rArr 119873119906119908= 0 (24a)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=119863

= 119881119863119890119894120596120591997904rArr 119871

119906119908

= minusint119886

0int119887

0119881119863cos (119906120587119909119886) cos (119908120587119910119887) 119889119909 119889119910120581119906119908120583119906119908

sinh (120583119906119908119863)

(24b)

where 120581119906119908= int119886

0int119887

0cos (119906120587119909119886)2 cos (119908120587119910119887)2119889119909 119889119910

By inputting (24a) and (24b) into (23) the pressure at 119911 =119863 is obtained and given below

119865119863

=

119880

sum

119906=0

119882

sum

119908=0

[119895120588119886120596coth (120583

119906119908119863)

120581119906119908120583119906119908

times int

119886

0

int

119887

0

119881119863cos(119906120587119909

119886) cos(

119908120587119910

119887) 119889119909 119889119910]

times cos(119906120587119909119886) cos(

119908120587119910

119887)

(25)

The average pressure force at 119911 = 119863 is then given by takingintegration over the panel area

119865119863=int119886

0int119887

0119865119863cos (119906120587119909119886) cos (119908120587119910119887) 119889119909 119889119910

119886119887

= 119881119863119895120588119886120596

119880

sum

119906=0

119882

sum

119908=0

[1205811015840

119906119908

120581119906119908

coth (120583119906119908119863)

120583119906119908

]

(26)

where 1205811015840119906119908= (int119886

0int119887

0cos(119906120587119909119886) cos(119908120587119910119887)119889119909 119889119910)

2

119886119887Finally the impedance of the air cavity is given by

119885cav =119865119863

119881119863

= 119895120588119886120596

119880

sum

119906=0

119882

sum

119908=0

[1205811015840

119906119908

120581119906119908

coth (120583119906119908119863)

120583119906119908

] (27)

Note that since even pressure distribution is considered then119906 119908 = 0 2 4 If 119886 and 119887 are less than 03m then the

Abstract and Applied Analysis 5

frequencies of the fundamental resonances parallel to thecavity length or width are higher than 1000Hz (note that1000Hz is not within the frequency range concerned in thisstudy) Then in this case only the mode numbers 119906 = 0 and119908 = 0 are considered The impedance can be approximatedby

119885cav asymp minus119895120588119886120596cot(120596

119888119886

119863) (28)

25 Electroacoustic Analogy According to [34] the acousticimpedance of a double panel absorber can be analogous tothat of the electrical circuit The acoustic impedance is givenbelow

119885dou = 1198851199011 + 119885cav1 +(119885cav1)

2

1198851199012+ 119885cav1 + 119885cav2

(29)

where 1198851199011

and 1198851199012= acoustic impedance of the 1st and 2nd

panels 119885cav1 and 119885cav2 = acoustic impedance of the 1st and2nd cavities

If the impedance of the 2nd cavity is set to zero that isequivalent to the case of a panel absorber backed by a flexiblewall the overall impedance in (29) is rewritten as

119885119900= 119885com + 119885cav +

(119885cav)2

119885119882+ 119885cav

(30)

where 119885com = (119885119872119885119875119885119867)(119885119872119885119875+ 119885119875119885119867+ 119885119867119885119872) =

combined impedance 119885119872= impedance of the microperfo-

ration119885119875= acoustic impedance of the panel absorber119885

119867=

impedance of the air pumping 119885119882= impedance of the wall

If the wall is perfectly rigid (ie no vibration) the naturalfrequency of wall120596

119882in (12)rarr infin Hence119885

119882119898119899119885119882rarr infin

and (119885cav)2(119885119882+ 119885cav) asymp 0 Equation (30) can be rewritten

as

119885119900= 119885com + 119885cav (31)

Then the absorption coefficient of the absorber 120572119900for normal

incidence can be calculated by the well-known formula (see[29])

120572119900=

4Re (119885119900)

(1 + Re(119885119900))2

+ (Im(119885119900))2 (32)

3 Results and Discussions

Using (32) the sound absorption is obtained for the casesconsidered in this section Each panel absorber is madeof aluminum panel The material properties are Youngrsquosmodulus 119864 = 7 times 10

10Nm2 Poissonrsquos ratio ] = 03 andmass density 120588 = 2700 kgm3 The sound speed and airdensity are 340ms and 12 kgm3 respectively Air viscosity120578 = 185 times 10

minus5 Pa sdot s the ratio of specific heat 120574 = 1402Prandtl number Pr = 071 Various excitation levels dampingfactors panel sizes and so forth are considered and theireffects on the sound absorption performances are studied inthe following paragraphs

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [31]

Figure 2 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01 m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 58)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [32]

Figure 3 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 29)

Figures 2 and 3 present comparisons between the fre-quency-absorption curves obtained from the proposedmethod and numerical integration method [32] The resultsshow that there is an absorption peak due to the microperfo-ration around 178Hz The well-known ldquojump phenomenonrdquocan be seen on both curves The results obtained from thetwo methods are generally in reasonably good agreementalthough some deviations are observed at the absorptionpeaks around 422Hz in Figure 2 and 390Hz in Figure 3Besides small differences between the jump frequencies (ie464 and 470Hz in Figure 2 and 410 and 420Hz in Figure 3)are also foundThe deviations are caused by that the dampingterm in the proposed formulation is proportional to thenonlinear peak frequency 120596

119873 instead of the linear natural

frequency 120596119900 which was adopted in [32] The excitation

level in Figure 3 is lower than that in Figure 2 Therefore the

6 Abstract and Applied Analysis

structural vibration in Figure 3 is less nonlinear (ie 120596119873is

closer to 120596119900) and the deviation between the peak values from

the two methods is smaller Note that in [32] the numericalintegration method did not deliver the unstable solutionsTherefore only stable results obtained by the proposedmethod in Figure 3 are compared with the numerical resultsin [32] Besides the unstable results are not obtainable inpractical environments and not very useful for practicalengineering purpose

Figures 4(a)ndash4(c) present the sound absorption plottedagainst the excitation frequency for various panel resonantfrequencies In each of these three figures there are twononlinear solution types The ldquojump uprdquo or ldquojump downrdquosolution is obtained by sweeping the excitation frequencyfrom ldquohigh to lowrdquo or ldquolow to highrdquo In Figures 4(a)ndash4(c) thelinear (11) mode panel resonant frequencies of the three casesare 387Hz 172Hz and 97Hz respectively The absorptionpeak frequency due to the microperforation effect is 178HzIt can be seen that (1) if the panel resonant frequencyis higher than the absorption peak frequency due to themicroperforation effect the panel vibration can enhanceand widen the absorption bandwidth and (2) if the panelresonant frequency is close to or lower than the absorptionpeak frequency the panel vibration degrades the absorptionperformance and the peak absorption is deteriorated

Figures 5(a) and 5(b) present the sound absorption plot-ted against the excitation frequency for various excitation lev-els The ldquojump downrdquo solutions are shown in Figure 5(a) (iesweeping the excitation frequency from ldquohigh to lowrdquo) whilethe ldquojump uprdquo solutions are in Figure 5(b) (ie sweepingthe excitation frequency from ldquolow to highrdquo) The absorptionpeak around 178Hz is due to the microperforation effect andis not significantly affected by the excitation level It is notedthat in the cases of 120581 = 1 there is no jump phenomenon as theexcitation force is quite small (ie linear case) Besides it canbe seen that (1) in Figure 5(a) the higher the excitation forcethe wider the absorption peak due to the panel vibrationand (2) in Figure 5(b) if the panel vibration is linear (or lowexcitation level) the absorption peak is very narrow and notvery useful for widening the absorption bandwidth Figures6(a) and 6(b) present the sound absorption plotted against theexcitation frequency for various damping ratios The ldquojumpdownrdquo solutions are shown in Figure 6(a) while the ldquojumpuprdquo solutions are in Figure 6(b) The absorption peak dueto the microperforation effect is not significantly affected bythe panel damping ratio Besides it can be seen that (1) inFigure 6(a) the higher the damping ratio the narrower theabsorption peak due to the ldquojump downrdquo panel vibration andthe lower the ldquojump downrdquo frequency and (2) in Figure 6(b)the absorption peak due to the ldquojump uprdquo panel vibration isnot significantly affected by the damping ratio

Figures 7(a) and 7(b) present the sound absorption andvibration amplitude plotted against the excitation frequencyNote that this case does not consider any perforation effectand other effects There are two peaks in each of the twofigures which are due to the (11) and (13) mode resonancesrespectively It can be seen that the peaks due to the (11)mode resonance are more nonlinear As the two peaks are farfrom each other no strong mode coupling effect is induced

In Figure 7(b) the vibration amplitude is almost linearlyincreasing from 100Hz to 286Hz while the sound absorptionis almost linearly increasing from 100Hz to 150Hz only andslowly and nonlinearly increasing from 150Hz to 286HzBesides the absorption peak due to the (13) mode resonanceis much more significant than the vibration peak It canbe implied that higher mode responses are important forabsorption but not for vibration

Figures 8(a) and 8(b) present the sound absorptionplotted against the excitation frequency and show the effectsof air pumping (Helmholtz resonance) andmicroperforationIn Figure 8(a) the peak frequencies of air pumping andmicroperforation are around 100Hz It can be seen that theabsorption bandwidth of the case with both effects is muchwider and its peak absorption is much higher than the caseswith air pumping or microperforation only The two effectscan interact with each other and significantly enhance theabsorption performance In Figure 8(b) the peak frequenciesof air pumping and microperforation are quite far from eachother The enhancement of the absorption performance dueto the two effects is not as good as that in Figure 8(a)

Figures 9(a) and 9(b) present the sound absorptionplotted against the excitation frequency for various dampingratios and show the effect of linear wall vibration Thepeak frequencies of wall panel and microperforation arearound 80Hz and 178Hz respectively It can be seen thatthe wall vibration does not affect the absorption performancesignificantly and just induces an abrupt change around 80HzThemagnitude of the abrupt change depends on the dampingratio (ie the higher the damping ratio the smaller the abruptchange) Figures 10(a) and 10(b) present the sound absorptionplotted against the excitation frequency and show the effectsof microperforation and linear and nonlinear vibrations Itcan be seen that the effects ofmicroperforation and nonlinearpanel vibration are positive and significant while the effect oflinear wall vibration is negative but not significant

4 Conclusions

In this study the analytic absorption formulation has beendeveloped for the panel absorber under the effects of microp-erforation air pumping and linear and nonlinear vibrationFrom the predictions it can be concluded that (1) if thepanel resonant frequency is higher than the absorptionpeak frequency due to the microperforation effect the panelvibration can enhance and widen the absorption bandwidth(2) if the panel resonant frequency is close to or lowerthan the absorption peak frequency the panel vibrationdegrades the absorption performance or the peak absorptionis deteriorated (3) the higher the excitation force the widerthe absorption peak due to the panel vibration (4) if the panelvibration is linear (or low excitation level) the absorptionpeak is very narrow and not very useful for widening theabsorption bandwidth (5) the absorption bandwidth of thecase with the air pumping and microperforation effects ismuch wider and its peak absorption is much higher thanthe cases with any one of these two effects If the peakfrequencies of air pumping andmicroperforation are far from

Abstract and Applied Analysis 7

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump up

Jump down

Peak freq = 178HzPeak freq = 452Hz

Jump up freq = 422HzJump dn freq = 522Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump downJump down

Jump dn freq = 218HzJump dn freq = 366Hz

(b)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump upJump down

Jump up freq = 294Hz

Jump dn freq = 144Hz

(c)

Figure 4 (a) The effect of the (11) mode vibration (120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 001 120590 = 05

119889 = 04mm and 120581 = 10) (b) The effect of the (11) mode vibration (120596119900= 17211 times 2120587Hz 119905 = 08mm 119886 = 119887 = 015m 119863 = 250mm 120585 = 001

120590 = 05 119889 = 04mm and 120581 = 10) (c) The effect of the (11) mode vibration (120596119900= 9681 times 2120587Hz 119905 = 08mm 119886 = 119887 = 02m 119863 = 250mm

120585 = 001 120590 = 05 119889 = 04mm and 120581 = 10)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178 Hz

120581 = 10

120581 = 5

120581 = 1

Jump dn freq = 524Hz

Jump dn freq = 454 Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120581 = 10

120581 = 1

120581 = 5

Peak freq = 178Hz

Jump up freq = 410Hz

Jump up freq = 424Hz

(b)

Figure 5 (a) The effect of excitation level (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01 m 119863 = 250mm

120585 = 001 120590 = 05 and 119889 = 04mm) (b) The effect of excitation level (high to low frequency excitation 120596119900= 38725 times 2120587119867119911 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 and 119889 = 04mm)

8 Abstract and Applied Analysis

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump dn freq = 438Hz

Jump dn freq= 524Hz

Jump dn freq= 552Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump up freq = 424Hz

Jump up freq = 424Hz

Jump up freq = 424Hz

(b)

Figure 6 (a) The effect of damping ratio (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm

120590 = 05 119889 = 04mm and 120581 = 10) (b) The effect of damping ratio (high to low frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120590 = 05 119889 = 04mm and 120581 = 10)

00

02

04

06

08

10

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 286Hz

(a)

00

05

10

15

20

25

30

0 100 200 300 400 500 600 700Frequency (Hz)

Vibr

atio

n am

plitu

dep

anel

thic

knes

s

Jump dn freq = 286Hz

(b)

Figure 7 (a) The absorption peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm 119886 = 119887 = 015m119863 = 250mm and 119896 = 5) (b) The vibration peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm119886 = 119887 = 015m119863 = 250mm and 120581 = 5)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Air pumping

Both effects

Microperforation

Peak freq = 102Hz

Peak freq = 100Hz

Peak freq = 100Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Both effectsMicroperforation

Air pumping

Peak freq = 324Hz

Peak freq = 308Hz

Peak freq= 116Hz

(b)

Figure 8 (a) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 150mm 120590 = 05 119889 = 03mm119871 = 10mm and 119886

119905= 5mm) (b) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 100mm 120590 = 028

119889 = 03mm 119871 = 10mm and 119886119905= 5mm)

Abstract and Applied Analysis 9

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 9 (a) The effects of linear wall vibration and microperforation (120596119882= 80 times 2120587Hz 120588

119908= 240m2 119905 = 08mm 119886 = 119887 = 01m

119863 = 250mm 120585119882= 002 120590 = 05 and 119889 = 04mm) (b) The effects of linear wall vibration and microperforation (120596

119882= 80 times 2120587Hz

120588119908= 240m2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585

119882= 001 120590 = 05 and 119889 = 04mm)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 524Hz

Peak freq = 178Hz

Dip freq = 80Hz

Peak freq = 452Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump up freq = 424Hz

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 10 (a) The effects of microperforation and linear and nonlinear vibrations (low to high frequency excitation 120596119900= 38725 times 2120587Hz

120596119882= 80 times 2120587Hz 120588

119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10) (b) The

effects of microperforation and linear and nonlinear vibrations (high to low frequency excitation 120596119900= 38725 times 2120587Hz 120596

119882= 80 times 2120587Hz

120588119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10)

each other the enhancement of the absorption performanceis small and (6) the wall vibration does not affect theabsorption performance significantly and just induces a smallabrupt change on the absorption bandwidth around the wallresonant frequency

Appendices

A Derivation of the NonlinearDifferential Equation

The explanation of how the formulation of a nonlinearpanel is expressed into the form of Duffing equation whichis abstracted from [30] is shown here The flexible panelvibration is governed by the Von-Karman plate theory Thewell-known partial differential equation of the nonlinear

panel can be presented in terms of its displacement and theAiry stress function as follows

1

12(1198892119908

1198891205852+ nabla4119908)

= (1205972119865

1205971205782

1205972119908

1205971205852+1205972119865

1205971205852

1205972119908

1205971205782minus 2

1205972119865

120597120578120597120585

1205972119908

120597120578120597120585)

(A1)

Note that (A1) is abstracted [30] and the notations are thesame as those in [30] but the definitions of some notationsare different from those in the main text (see the followingdefinitions) 119908 = panel displacement 120585 and 120578 are thedimensionless space variables of the 119909 and 119910 directions 120585is the dimensionless time variable ] is Poissonrsquos ratio 119865 isthe Airy stress function According to [30] the derivativesof 119865 can be expressed in terms of the in-plane displacements

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

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Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

was adopted for modelling of the sound absorption of apanel absorber Besides the air pumping effect (or Helmholtzresonance effect) has beenmentioned in many textbooks andarticles (eg [29]) but it has not been incorporated into theresearch problem of nonlinear panel absorber In this paperit focuses on the development of the analytic absorptionformula of a panel absorber which considers the four effects(iemicroperforation air pumping and linear and nonlinearvibrations)

2 Analytic Formulation

21 Nonlinear Panel Vibration and Liner Wall VibrationFigure 1 shows the proposed panel absorber subject to theeffects of microperforation air pumping and linear andnonlinear vibrations Firstly the impedances of the nonlinearpanel and linear wall are derived in this subsection Accord-ing to the formulation of Chu and Herrmann [30] the gov-erning equation for the nonlinear vibration of a rectangularpanel subject to uniformly distributed harmonic excitationcan be expressed in the following form (see Appendix A formore)

1205881198892120576

1198891205912+ 1205881205962

119900120576 + 120573120576

3+ 119865 (120591) = 0 (1)

where

119865(120591) = 119865119900119898119899

sin(120596120591) =modal harmonic excitation

119865119900119898119899

= 119865119900Λ119898119899

= 120582120588119905119892Λ119898119899

= modal excitationmagnitude

119865119900= 120582 120588119905119892 = physical excitation magnitude

119892 = gravity acceleration (981msminus2)

120591 = time

120582 = dimensionless excitation parameter

Λ119898119899=

(int119887

0int119886

0sin (119898120587119909119886) sin (119899120587119910119887) 119889119909 119889119910)

int119887

0int119886

0[sin (119898120587119909119886) sin (119899120587119910119887)]2119889119909 119889119910

= modal coefficient

(2)

120596 = excitation frequency

120596119900= linear resonant frequency of the (119898 119899) mode

119898 119899 = structural mode numbers

120573 =119864119905

12 (1 minus ]2) 119886431205874

2

times [1

21198982(1198982+ ]11989921199032)(minus

3

2+ ]

11989921199032

1198982)

minus3

41198982(1198982+ ]11989921199032)

+1

211989921199032(11989921199032+ ]1198982)(minus

3

2+ ]

1198982

11989921199032)

minus3

411989921199032(11989921199032+ ]1198982)]

= nonlinear stiffness coefficient(3)

119903 = 119886119887 is the aspect ratio 119864 is Youngrsquos modulus 120588 isthe surface density ] is Poissonrsquos ratio and 119905 119886 and 119887are the thickness length and width

Consider the approximation of 120576 = 119860 cos(120596120591) in (1) Then

3

41205731198603+ 120588 (120596

2

119900minus 1205962)119860 + 119865

119900119898119899= 0 (4)

rArr

1198603+4

3120573120588 (1205962

119900minus 1205962)119860 +

4

3120573119865119900119898119899

= 0 (5)

where 119860 =modal vibration amplitudeFor simplicity let119901 = (43120573)120588(1205962

119900minus1205962) 119902 = (43120573)119865

119900119898119899

and 119860 = 119861 minus (1199013119861) (ie Vietarsquos substitution [31]) Equation(5) can be rewritten as

1198613+ 119902 minus

1199013

271198613= 0 (6)

rArr

1198616+ 1199021198613minus1199013

27= 0 (7)

If 1198613 is considered as an independent unknown (7) is aldquomodifiedrdquo quadratic equation Then the analytic formulasfor the cubic solutions are given below

1198601= minus

1

3

3radic1

2(Θ + Ω) minus

1

3

3radic1

2(Θ minus Ω) (8a)

1198602=1 + 119895radic3

6

3radic1

2(Θ + Ω) +

1 minus 119895radic3

6

3radic1

2(Θ minus Ω) (8b)

1198603=1 minus 119895radic3

6

3radic1

2(Θ + Ω) +

1 + 119895radic3

6

3radic1

2(Θ minus Ω) (8c)

where Θ = radic(27119902)2+ 4(3119901)

3Ω = 27119902In the next step the nonlinear stiffness can be defined as

(34)1205731198602 and overall panel stiffness is 1205881205962

119900+ (34)120573119860

2 Thepanel impedance of the (119898 119899) mode is given by

119885119875119898119899

=119865119900119898119899

119881119875119898119899

=

2120588120585120596120596119873+ 119895 [120588 (120596

2minus 1205962

119900) minus (34) 120573119860

2]

120596

(9)

where 119895 = radicminus1 = complex number 120585 is the panel dampingratio 119881

119875119898119899is the modal panel velocity Note that 120596

119873is

Abstract and Applied Analysis 3

D

Linear wall vibration

z

x

Nonlinear perforated panel vibration

Air mass moving along the tube

Uniform sound pressure

Cavity

Rigi

d

Rigi

dFigure 1 A panel absorber subject to the effects of microperforation air pumping and linear and nonlinear vibrations

the nonlinear peak frequency instead of the linear naturalfrequency 120596

119900in [32] (see Appendix B for the derivation of

120596119873)When it is the case of small amplitude vibration or linear

vibration 120596119873is equal to 120596

119900

Consider the overall panel velocity contributed from themodal velocities and find the overall panel impedance

119881119875=

119872

sum

119898

119873

sum

119899

119865119900Λ119898119899Λ1015840

119898119899

119885119875119898119899

(10)

rArr

119885119875=119865119900

119881119875

= (

119872

sum

119898

119873

sum

119899

Λ119898119899Λ1015840

119898119899

119885119875119898119899

)

minus1

(11)

where Λ1015840

119898119899= (int

119887

0int119886

0sin(119898120587119909119886) sin(119899120587119910119887)119889119909 119889119910)119886119887

According to [33] the sound absorption of two linearlyvibrating plates with different boundary conditions was veryclose It was concluded that the boundary condition wouldnot affect the sound absorption significantlyThat is why onlyone boundary condition case is considered here

Consider the linear case (or small amplitude case) inthe model representing the panel vibration and the corre-sponding modal and overall panel impedances (ie (9) (11))It is noted that for the linear vibration the fundamentalharmonic balance can lead to the exact solution (no higherharmonic terms are required) Hence the modal and overallwall impedances are given below

119885119882119898119899

=

2120588119882120585119882120596120596119882+ 119895 [120588 (120596

2minus 1205962

119882)]

120596

(12)

119885119882= (

119872

sum

119898

119873

sum

119899

Λ119898119899Λ1015840

119898119899

119885119882119898119899

)

minus1

(13)

where 120585119882 120588119882 and 120596

119882are the wall damping factor density

and natural frequency

22 Air Pumping (Helmholtz Resonance) In Figure 1 theimpedance of the air pumping along the tubemounted on the

panel absorber is obtained here The differential equation forthe air mass pumping inout along the tube is given by [29]

1205881198861198711015840 1198892Λ

1198891205912+(119877119903+ 119877119908)

119878

119889Λ

119889120591+ 119875 (120591) = 0 (14)

where Λ = air mass displacement 119875(120591) = acoustic pressureexcitation 119877

119903= 120588119886119888119886119896211987824120587 = damping due to radiation

resistance 119877119908= 21205881198861198711015840119878120572119908= damping due to thermoviscous

resistance 1198711015840 = 119871 + 14119886119905= effective tube length 120572

119908=

(1119886119905119888119886)(1205781205962120588

119886)05(1 + (120574radicPr)) = absorption coefficient for

wall losses and 120588119886is the air density 1198711015840 is the effective length

of the tube with unflanged outer end 119896 is the wave number119871 is the tube length 119886

119905is the radius of the tube 119878 is the cross

section area of the tube 120578 is the air viscosity 120574 is the ratio ofspecific heat Pr is Prandtl number

Similar to the overall panel impedance in (9) theimpedance of air pumping along the tube is given below

1198851015840

119867=((119877119903+ 119877119908) 119878) 120596 + 119895120588

11988611987110158401205962

120596 (15)

Using (15) the air mass velocity is given by

1198851015840

119867=119875119900

1198811015840119867

997904rArr 1198811015840

119867=119875119900

1198851015840119867

(16)

where 119875119900= acoustic pressure magnitude

Consider the average velocity over the panel surface

119881119867= 1198811015840

119867120590119867 (17)

where 120590119867= the ratio of tube area to panel area

Then the overall impedance of air pumping is obtained bysubstituting (17) into (16)

119885119867=1198851015840

119867

120590119867

(18)

23 Microperforation In Figure 1 the impedance of themicroholes over the panel surface is obtained here According

4 Abstract and Applied Analysis

to [9 10] the real and imaginary parts of the acousticimpedance of a microhole are given below

1198851015840

119872119877= 120588119886119888119886(0147

119905

1198892)(radic9 +

1001198892119891

32+ 1768radic119891

1198892

119905)

(19a)

1198851015840

119872119868= 1205881198861198881198861847 119891119905(1 +

1

radic9 + 501198892119891

+ 085119889

119905)

(19b)

where 1198851015840119872= 1198851015840

119872119877+ 1198951198851015840

119872119868= the impedance of a microhole

119891 is frequency 119905 is panel thickness and 119889 is the perforationdiameter 120588

119886and 119888119886are air density and sound speed

Then the overall impedance of all microholes over thepanel is given by

119885119872=1198851015840

119872

120590 (20)

where 120590 is the perforation ratio

24 Air Cavity In Figure 1 the impedance of the air cavityis obtained here The acoustic velocity potential within arectangular cavity is given by the following homogeneouswave equation [1]

nabla2120601 minus

1

1198882119886

1205972120601

1205971205912= 0 (21)

where 120601 is the velocity potential functionThe air particle velocities in the 119909 119910 and 119911 direc-

tions and pressures within the air cavity can be given by120597120601120597119909 120597120601120597119910 120597120601120597119911 and minus120588

119886(120597120601120597120591) respectively

The boundary conditions of the rectangular cavity to besatisfied are

120597120601

120597119909

10038161003816100381610038161003816100381610038161003816119909=0=120597120601

120597119909

10038161003816100381610038161003816100381610038161003816119909=119886= 0 (22a)

120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

=120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=119887

= 0 (22b)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=0

= 0 (22c)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=minus119863= 119881119863 (22d)

where 119886 and 119887 are the length and width of the rectangularcavity or the panel and 119881

119863is the average velocity at 119911 =

119863 = cavity depth Note that the cavity impedance of no wallvibration is derived here and used into the sound absorptionformulation in the next subsection which can consider boththe panel and wall vibration effects and is derived using theelectroacoustic analogy

By applying the boundary conditions in (22a) and (22b)the solution of (21) can be expressed as

120601 =

119880

sum

119906=0

119882

sum

119908=0

[119871119906119908

cosh (120583119906119908119911) + 119873

119906119908sinh (120583

119906119908119911)]

times cos(119906120587119909119886) cos(

119908120587119910

119887) 119890119894120596120591

(23)

where 120583119906119908

= radic((119906120587119886)2+ (119908120587119887)

2) minus (120596119888

119886)2 119906 and 119908 are

the acoustic mode numbers 119880 and 119882 are the numbers ofacoustic modes used in the 119909 and 119910 directions and 119871

119906119908and

119873119906119908

are coefficients to be determined in the following stepSubstituting (23) into (22c) and (22d) gives

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=0

= 0 997904rArr 119873119906119908= 0 (24a)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=119863

= 119881119863119890119894120596120591997904rArr 119871

119906119908

= minusint119886

0int119887

0119881119863cos (119906120587119909119886) cos (119908120587119910119887) 119889119909 119889119910120581119906119908120583119906119908

sinh (120583119906119908119863)

(24b)

where 120581119906119908= int119886

0int119887

0cos (119906120587119909119886)2 cos (119908120587119910119887)2119889119909 119889119910

By inputting (24a) and (24b) into (23) the pressure at 119911 =119863 is obtained and given below

119865119863

=

119880

sum

119906=0

119882

sum

119908=0

[119895120588119886120596coth (120583

119906119908119863)

120581119906119908120583119906119908

times int

119886

0

int

119887

0

119881119863cos(119906120587119909

119886) cos(

119908120587119910

119887) 119889119909 119889119910]

times cos(119906120587119909119886) cos(

119908120587119910

119887)

(25)

The average pressure force at 119911 = 119863 is then given by takingintegration over the panel area

119865119863=int119886

0int119887

0119865119863cos (119906120587119909119886) cos (119908120587119910119887) 119889119909 119889119910

119886119887

= 119881119863119895120588119886120596

119880

sum

119906=0

119882

sum

119908=0

[1205811015840

119906119908

120581119906119908

coth (120583119906119908119863)

120583119906119908

]

(26)

where 1205811015840119906119908= (int119886

0int119887

0cos(119906120587119909119886) cos(119908120587119910119887)119889119909 119889119910)

2

119886119887Finally the impedance of the air cavity is given by

119885cav =119865119863

119881119863

= 119895120588119886120596

119880

sum

119906=0

119882

sum

119908=0

[1205811015840

119906119908

120581119906119908

coth (120583119906119908119863)

120583119906119908

] (27)

Note that since even pressure distribution is considered then119906 119908 = 0 2 4 If 119886 and 119887 are less than 03m then the

Abstract and Applied Analysis 5

frequencies of the fundamental resonances parallel to thecavity length or width are higher than 1000Hz (note that1000Hz is not within the frequency range concerned in thisstudy) Then in this case only the mode numbers 119906 = 0 and119908 = 0 are considered The impedance can be approximatedby

119885cav asymp minus119895120588119886120596cot(120596

119888119886

119863) (28)

25 Electroacoustic Analogy According to [34] the acousticimpedance of a double panel absorber can be analogous tothat of the electrical circuit The acoustic impedance is givenbelow

119885dou = 1198851199011 + 119885cav1 +(119885cav1)

2

1198851199012+ 119885cav1 + 119885cav2

(29)

where 1198851199011

and 1198851199012= acoustic impedance of the 1st and 2nd

panels 119885cav1 and 119885cav2 = acoustic impedance of the 1st and2nd cavities

If the impedance of the 2nd cavity is set to zero that isequivalent to the case of a panel absorber backed by a flexiblewall the overall impedance in (29) is rewritten as

119885119900= 119885com + 119885cav +

(119885cav)2

119885119882+ 119885cav

(30)

where 119885com = (119885119872119885119875119885119867)(119885119872119885119875+ 119885119875119885119867+ 119885119867119885119872) =

combined impedance 119885119872= impedance of the microperfo-

ration119885119875= acoustic impedance of the panel absorber119885

119867=

impedance of the air pumping 119885119882= impedance of the wall

If the wall is perfectly rigid (ie no vibration) the naturalfrequency of wall120596

119882in (12)rarr infin Hence119885

119882119898119899119885119882rarr infin

and (119885cav)2(119885119882+ 119885cav) asymp 0 Equation (30) can be rewritten

as

119885119900= 119885com + 119885cav (31)

Then the absorption coefficient of the absorber 120572119900for normal

incidence can be calculated by the well-known formula (see[29])

120572119900=

4Re (119885119900)

(1 + Re(119885119900))2

+ (Im(119885119900))2 (32)

3 Results and Discussions

Using (32) the sound absorption is obtained for the casesconsidered in this section Each panel absorber is madeof aluminum panel The material properties are Youngrsquosmodulus 119864 = 7 times 10

10Nm2 Poissonrsquos ratio ] = 03 andmass density 120588 = 2700 kgm3 The sound speed and airdensity are 340ms and 12 kgm3 respectively Air viscosity120578 = 185 times 10

minus5 Pa sdot s the ratio of specific heat 120574 = 1402Prandtl number Pr = 071 Various excitation levels dampingfactors panel sizes and so forth are considered and theireffects on the sound absorption performances are studied inthe following paragraphs

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [31]

Figure 2 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01 m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 58)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [32]

Figure 3 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 29)

Figures 2 and 3 present comparisons between the fre-quency-absorption curves obtained from the proposedmethod and numerical integration method [32] The resultsshow that there is an absorption peak due to the microperfo-ration around 178Hz The well-known ldquojump phenomenonrdquocan be seen on both curves The results obtained from thetwo methods are generally in reasonably good agreementalthough some deviations are observed at the absorptionpeaks around 422Hz in Figure 2 and 390Hz in Figure 3Besides small differences between the jump frequencies (ie464 and 470Hz in Figure 2 and 410 and 420Hz in Figure 3)are also foundThe deviations are caused by that the dampingterm in the proposed formulation is proportional to thenonlinear peak frequency 120596

119873 instead of the linear natural

frequency 120596119900 which was adopted in [32] The excitation

level in Figure 3 is lower than that in Figure 2 Therefore the

6 Abstract and Applied Analysis

structural vibration in Figure 3 is less nonlinear (ie 120596119873is

closer to 120596119900) and the deviation between the peak values from

the two methods is smaller Note that in [32] the numericalintegration method did not deliver the unstable solutionsTherefore only stable results obtained by the proposedmethod in Figure 3 are compared with the numerical resultsin [32] Besides the unstable results are not obtainable inpractical environments and not very useful for practicalengineering purpose

Figures 4(a)ndash4(c) present the sound absorption plottedagainst the excitation frequency for various panel resonantfrequencies In each of these three figures there are twononlinear solution types The ldquojump uprdquo or ldquojump downrdquosolution is obtained by sweeping the excitation frequencyfrom ldquohigh to lowrdquo or ldquolow to highrdquo In Figures 4(a)ndash4(c) thelinear (11) mode panel resonant frequencies of the three casesare 387Hz 172Hz and 97Hz respectively The absorptionpeak frequency due to the microperforation effect is 178HzIt can be seen that (1) if the panel resonant frequencyis higher than the absorption peak frequency due to themicroperforation effect the panel vibration can enhanceand widen the absorption bandwidth and (2) if the panelresonant frequency is close to or lower than the absorptionpeak frequency the panel vibration degrades the absorptionperformance and the peak absorption is deteriorated

Figures 5(a) and 5(b) present the sound absorption plot-ted against the excitation frequency for various excitation lev-els The ldquojump downrdquo solutions are shown in Figure 5(a) (iesweeping the excitation frequency from ldquohigh to lowrdquo) whilethe ldquojump uprdquo solutions are in Figure 5(b) (ie sweepingthe excitation frequency from ldquolow to highrdquo) The absorptionpeak around 178Hz is due to the microperforation effect andis not significantly affected by the excitation level It is notedthat in the cases of 120581 = 1 there is no jump phenomenon as theexcitation force is quite small (ie linear case) Besides it canbe seen that (1) in Figure 5(a) the higher the excitation forcethe wider the absorption peak due to the panel vibrationand (2) in Figure 5(b) if the panel vibration is linear (or lowexcitation level) the absorption peak is very narrow and notvery useful for widening the absorption bandwidth Figures6(a) and 6(b) present the sound absorption plotted against theexcitation frequency for various damping ratios The ldquojumpdownrdquo solutions are shown in Figure 6(a) while the ldquojumpuprdquo solutions are in Figure 6(b) The absorption peak dueto the microperforation effect is not significantly affected bythe panel damping ratio Besides it can be seen that (1) inFigure 6(a) the higher the damping ratio the narrower theabsorption peak due to the ldquojump downrdquo panel vibration andthe lower the ldquojump downrdquo frequency and (2) in Figure 6(b)the absorption peak due to the ldquojump uprdquo panel vibration isnot significantly affected by the damping ratio

Figures 7(a) and 7(b) present the sound absorption andvibration amplitude plotted against the excitation frequencyNote that this case does not consider any perforation effectand other effects There are two peaks in each of the twofigures which are due to the (11) and (13) mode resonancesrespectively It can be seen that the peaks due to the (11)mode resonance are more nonlinear As the two peaks are farfrom each other no strong mode coupling effect is induced

In Figure 7(b) the vibration amplitude is almost linearlyincreasing from 100Hz to 286Hz while the sound absorptionis almost linearly increasing from 100Hz to 150Hz only andslowly and nonlinearly increasing from 150Hz to 286HzBesides the absorption peak due to the (13) mode resonanceis much more significant than the vibration peak It canbe implied that higher mode responses are important forabsorption but not for vibration

Figures 8(a) and 8(b) present the sound absorptionplotted against the excitation frequency and show the effectsof air pumping (Helmholtz resonance) andmicroperforationIn Figure 8(a) the peak frequencies of air pumping andmicroperforation are around 100Hz It can be seen that theabsorption bandwidth of the case with both effects is muchwider and its peak absorption is much higher than the caseswith air pumping or microperforation only The two effectscan interact with each other and significantly enhance theabsorption performance In Figure 8(b) the peak frequenciesof air pumping and microperforation are quite far from eachother The enhancement of the absorption performance dueto the two effects is not as good as that in Figure 8(a)

Figures 9(a) and 9(b) present the sound absorptionplotted against the excitation frequency for various dampingratios and show the effect of linear wall vibration Thepeak frequencies of wall panel and microperforation arearound 80Hz and 178Hz respectively It can be seen thatthe wall vibration does not affect the absorption performancesignificantly and just induces an abrupt change around 80HzThemagnitude of the abrupt change depends on the dampingratio (ie the higher the damping ratio the smaller the abruptchange) Figures 10(a) and 10(b) present the sound absorptionplotted against the excitation frequency and show the effectsof microperforation and linear and nonlinear vibrations Itcan be seen that the effects ofmicroperforation and nonlinearpanel vibration are positive and significant while the effect oflinear wall vibration is negative but not significant

4 Conclusions

In this study the analytic absorption formulation has beendeveloped for the panel absorber under the effects of microp-erforation air pumping and linear and nonlinear vibrationFrom the predictions it can be concluded that (1) if thepanel resonant frequency is higher than the absorptionpeak frequency due to the microperforation effect the panelvibration can enhance and widen the absorption bandwidth(2) if the panel resonant frequency is close to or lowerthan the absorption peak frequency the panel vibrationdegrades the absorption performance or the peak absorptionis deteriorated (3) the higher the excitation force the widerthe absorption peak due to the panel vibration (4) if the panelvibration is linear (or low excitation level) the absorptionpeak is very narrow and not very useful for widening theabsorption bandwidth (5) the absorption bandwidth of thecase with the air pumping and microperforation effects ismuch wider and its peak absorption is much higher thanthe cases with any one of these two effects If the peakfrequencies of air pumping andmicroperforation are far from

Abstract and Applied Analysis 7

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump up

Jump down

Peak freq = 178HzPeak freq = 452Hz

Jump up freq = 422HzJump dn freq = 522Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump downJump down

Jump dn freq = 218HzJump dn freq = 366Hz

(b)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump upJump down

Jump up freq = 294Hz

Jump dn freq = 144Hz

(c)

Figure 4 (a) The effect of the (11) mode vibration (120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 001 120590 = 05

119889 = 04mm and 120581 = 10) (b) The effect of the (11) mode vibration (120596119900= 17211 times 2120587Hz 119905 = 08mm 119886 = 119887 = 015m 119863 = 250mm 120585 = 001

120590 = 05 119889 = 04mm and 120581 = 10) (c) The effect of the (11) mode vibration (120596119900= 9681 times 2120587Hz 119905 = 08mm 119886 = 119887 = 02m 119863 = 250mm

120585 = 001 120590 = 05 119889 = 04mm and 120581 = 10)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178 Hz

120581 = 10

120581 = 5

120581 = 1

Jump dn freq = 524Hz

Jump dn freq = 454 Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120581 = 10

120581 = 1

120581 = 5

Peak freq = 178Hz

Jump up freq = 410Hz

Jump up freq = 424Hz

(b)

Figure 5 (a) The effect of excitation level (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01 m 119863 = 250mm

120585 = 001 120590 = 05 and 119889 = 04mm) (b) The effect of excitation level (high to low frequency excitation 120596119900= 38725 times 2120587119867119911 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 and 119889 = 04mm)

8 Abstract and Applied Analysis

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump dn freq = 438Hz

Jump dn freq= 524Hz

Jump dn freq= 552Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump up freq = 424Hz

Jump up freq = 424Hz

Jump up freq = 424Hz

(b)

Figure 6 (a) The effect of damping ratio (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm

120590 = 05 119889 = 04mm and 120581 = 10) (b) The effect of damping ratio (high to low frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120590 = 05 119889 = 04mm and 120581 = 10)

00

02

04

06

08

10

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 286Hz

(a)

00

05

10

15

20

25

30

0 100 200 300 400 500 600 700Frequency (Hz)

Vibr

atio

n am

plitu

dep

anel

thic

knes

s

Jump dn freq = 286Hz

(b)

Figure 7 (a) The absorption peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm 119886 = 119887 = 015m119863 = 250mm and 119896 = 5) (b) The vibration peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm119886 = 119887 = 015m119863 = 250mm and 120581 = 5)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Air pumping

Both effects

Microperforation

Peak freq = 102Hz

Peak freq = 100Hz

Peak freq = 100Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Both effectsMicroperforation

Air pumping

Peak freq = 324Hz

Peak freq = 308Hz

Peak freq= 116Hz

(b)

Figure 8 (a) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 150mm 120590 = 05 119889 = 03mm119871 = 10mm and 119886

119905= 5mm) (b) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 100mm 120590 = 028

119889 = 03mm 119871 = 10mm and 119886119905= 5mm)

Abstract and Applied Analysis 9

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 9 (a) The effects of linear wall vibration and microperforation (120596119882= 80 times 2120587Hz 120588

119908= 240m2 119905 = 08mm 119886 = 119887 = 01m

119863 = 250mm 120585119882= 002 120590 = 05 and 119889 = 04mm) (b) The effects of linear wall vibration and microperforation (120596

119882= 80 times 2120587Hz

120588119908= 240m2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585

119882= 001 120590 = 05 and 119889 = 04mm)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 524Hz

Peak freq = 178Hz

Dip freq = 80Hz

Peak freq = 452Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump up freq = 424Hz

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 10 (a) The effects of microperforation and linear and nonlinear vibrations (low to high frequency excitation 120596119900= 38725 times 2120587Hz

120596119882= 80 times 2120587Hz 120588

119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10) (b) The

effects of microperforation and linear and nonlinear vibrations (high to low frequency excitation 120596119900= 38725 times 2120587Hz 120596

119882= 80 times 2120587Hz

120588119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10)

each other the enhancement of the absorption performanceis small and (6) the wall vibration does not affect theabsorption performance significantly and just induces a smallabrupt change on the absorption bandwidth around the wallresonant frequency

Appendices

A Derivation of the NonlinearDifferential Equation

The explanation of how the formulation of a nonlinearpanel is expressed into the form of Duffing equation whichis abstracted from [30] is shown here The flexible panelvibration is governed by the Von-Karman plate theory Thewell-known partial differential equation of the nonlinear

panel can be presented in terms of its displacement and theAiry stress function as follows

1

12(1198892119908

1198891205852+ nabla4119908)

= (1205972119865

1205971205782

1205972119908

1205971205852+1205972119865

1205971205852

1205972119908

1205971205782minus 2

1205972119865

120597120578120597120585

1205972119908

120597120578120597120585)

(A1)

Note that (A1) is abstracted [30] and the notations are thesame as those in [30] but the definitions of some notationsare different from those in the main text (see the followingdefinitions) 119908 = panel displacement 120585 and 120578 are thedimensionless space variables of the 119909 and 119910 directions 120585is the dimensionless time variable ] is Poissonrsquos ratio 119865 isthe Airy stress function According to [30] the derivativesof 119865 can be expressed in terms of the in-plane displacements

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

D

Linear wall vibration

z

x

Nonlinear perforated panel vibration

Air mass moving along the tube

Uniform sound pressure

Cavity

Rigi

d

Rigi

dFigure 1 A panel absorber subject to the effects of microperforation air pumping and linear and nonlinear vibrations

the nonlinear peak frequency instead of the linear naturalfrequency 120596

119900in [32] (see Appendix B for the derivation of

120596119873)When it is the case of small amplitude vibration or linear

vibration 120596119873is equal to 120596

119900

Consider the overall panel velocity contributed from themodal velocities and find the overall panel impedance

119881119875=

119872

sum

119898

119873

sum

119899

119865119900Λ119898119899Λ1015840

119898119899

119885119875119898119899

(10)

rArr

119885119875=119865119900

119881119875

= (

119872

sum

119898

119873

sum

119899

Λ119898119899Λ1015840

119898119899

119885119875119898119899

)

minus1

(11)

where Λ1015840

119898119899= (int

119887

0int119886

0sin(119898120587119909119886) sin(119899120587119910119887)119889119909 119889119910)119886119887

According to [33] the sound absorption of two linearlyvibrating plates with different boundary conditions was veryclose It was concluded that the boundary condition wouldnot affect the sound absorption significantlyThat is why onlyone boundary condition case is considered here

Consider the linear case (or small amplitude case) inthe model representing the panel vibration and the corre-sponding modal and overall panel impedances (ie (9) (11))It is noted that for the linear vibration the fundamentalharmonic balance can lead to the exact solution (no higherharmonic terms are required) Hence the modal and overallwall impedances are given below

119885119882119898119899

=

2120588119882120585119882120596120596119882+ 119895 [120588 (120596

2minus 1205962

119882)]

120596

(12)

119885119882= (

119872

sum

119898

119873

sum

119899

Λ119898119899Λ1015840

119898119899

119885119882119898119899

)

minus1

(13)

where 120585119882 120588119882 and 120596

119882are the wall damping factor density

and natural frequency

22 Air Pumping (Helmholtz Resonance) In Figure 1 theimpedance of the air pumping along the tubemounted on the

panel absorber is obtained here The differential equation forthe air mass pumping inout along the tube is given by [29]

1205881198861198711015840 1198892Λ

1198891205912+(119877119903+ 119877119908)

119878

119889Λ

119889120591+ 119875 (120591) = 0 (14)

where Λ = air mass displacement 119875(120591) = acoustic pressureexcitation 119877

119903= 120588119886119888119886119896211987824120587 = damping due to radiation

resistance 119877119908= 21205881198861198711015840119878120572119908= damping due to thermoviscous

resistance 1198711015840 = 119871 + 14119886119905= effective tube length 120572

119908=

(1119886119905119888119886)(1205781205962120588

119886)05(1 + (120574radicPr)) = absorption coefficient for

wall losses and 120588119886is the air density 1198711015840 is the effective length

of the tube with unflanged outer end 119896 is the wave number119871 is the tube length 119886

119905is the radius of the tube 119878 is the cross

section area of the tube 120578 is the air viscosity 120574 is the ratio ofspecific heat Pr is Prandtl number

Similar to the overall panel impedance in (9) theimpedance of air pumping along the tube is given below

1198851015840

119867=((119877119903+ 119877119908) 119878) 120596 + 119895120588

11988611987110158401205962

120596 (15)

Using (15) the air mass velocity is given by

1198851015840

119867=119875119900

1198811015840119867

997904rArr 1198811015840

119867=119875119900

1198851015840119867

(16)

where 119875119900= acoustic pressure magnitude

Consider the average velocity over the panel surface

119881119867= 1198811015840

119867120590119867 (17)

where 120590119867= the ratio of tube area to panel area

Then the overall impedance of air pumping is obtained bysubstituting (17) into (16)

119885119867=1198851015840

119867

120590119867

(18)

23 Microperforation In Figure 1 the impedance of themicroholes over the panel surface is obtained here According

4 Abstract and Applied Analysis

to [9 10] the real and imaginary parts of the acousticimpedance of a microhole are given below

1198851015840

119872119877= 120588119886119888119886(0147

119905

1198892)(radic9 +

1001198892119891

32+ 1768radic119891

1198892

119905)

(19a)

1198851015840

119872119868= 1205881198861198881198861847 119891119905(1 +

1

radic9 + 501198892119891

+ 085119889

119905)

(19b)

where 1198851015840119872= 1198851015840

119872119877+ 1198951198851015840

119872119868= the impedance of a microhole

119891 is frequency 119905 is panel thickness and 119889 is the perforationdiameter 120588

119886and 119888119886are air density and sound speed

Then the overall impedance of all microholes over thepanel is given by

119885119872=1198851015840

119872

120590 (20)

where 120590 is the perforation ratio

24 Air Cavity In Figure 1 the impedance of the air cavityis obtained here The acoustic velocity potential within arectangular cavity is given by the following homogeneouswave equation [1]

nabla2120601 minus

1

1198882119886

1205972120601

1205971205912= 0 (21)

where 120601 is the velocity potential functionThe air particle velocities in the 119909 119910 and 119911 direc-

tions and pressures within the air cavity can be given by120597120601120597119909 120597120601120597119910 120597120601120597119911 and minus120588

119886(120597120601120597120591) respectively

The boundary conditions of the rectangular cavity to besatisfied are

120597120601

120597119909

10038161003816100381610038161003816100381610038161003816119909=0=120597120601

120597119909

10038161003816100381610038161003816100381610038161003816119909=119886= 0 (22a)

120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

=120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=119887

= 0 (22b)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=0

= 0 (22c)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=minus119863= 119881119863 (22d)

where 119886 and 119887 are the length and width of the rectangularcavity or the panel and 119881

119863is the average velocity at 119911 =

119863 = cavity depth Note that the cavity impedance of no wallvibration is derived here and used into the sound absorptionformulation in the next subsection which can consider boththe panel and wall vibration effects and is derived using theelectroacoustic analogy

By applying the boundary conditions in (22a) and (22b)the solution of (21) can be expressed as

120601 =

119880

sum

119906=0

119882

sum

119908=0

[119871119906119908

cosh (120583119906119908119911) + 119873

119906119908sinh (120583

119906119908119911)]

times cos(119906120587119909119886) cos(

119908120587119910

119887) 119890119894120596120591

(23)

where 120583119906119908

= radic((119906120587119886)2+ (119908120587119887)

2) minus (120596119888

119886)2 119906 and 119908 are

the acoustic mode numbers 119880 and 119882 are the numbers ofacoustic modes used in the 119909 and 119910 directions and 119871

119906119908and

119873119906119908

are coefficients to be determined in the following stepSubstituting (23) into (22c) and (22d) gives

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=0

= 0 997904rArr 119873119906119908= 0 (24a)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=119863

= 119881119863119890119894120596120591997904rArr 119871

119906119908

= minusint119886

0int119887

0119881119863cos (119906120587119909119886) cos (119908120587119910119887) 119889119909 119889119910120581119906119908120583119906119908

sinh (120583119906119908119863)

(24b)

where 120581119906119908= int119886

0int119887

0cos (119906120587119909119886)2 cos (119908120587119910119887)2119889119909 119889119910

By inputting (24a) and (24b) into (23) the pressure at 119911 =119863 is obtained and given below

119865119863

=

119880

sum

119906=0

119882

sum

119908=0

[119895120588119886120596coth (120583

119906119908119863)

120581119906119908120583119906119908

times int

119886

0

int

119887

0

119881119863cos(119906120587119909

119886) cos(

119908120587119910

119887) 119889119909 119889119910]

times cos(119906120587119909119886) cos(

119908120587119910

119887)

(25)

The average pressure force at 119911 = 119863 is then given by takingintegration over the panel area

119865119863=int119886

0int119887

0119865119863cos (119906120587119909119886) cos (119908120587119910119887) 119889119909 119889119910

119886119887

= 119881119863119895120588119886120596

119880

sum

119906=0

119882

sum

119908=0

[1205811015840

119906119908

120581119906119908

coth (120583119906119908119863)

120583119906119908

]

(26)

where 1205811015840119906119908= (int119886

0int119887

0cos(119906120587119909119886) cos(119908120587119910119887)119889119909 119889119910)

2

119886119887Finally the impedance of the air cavity is given by

119885cav =119865119863

119881119863

= 119895120588119886120596

119880

sum

119906=0

119882

sum

119908=0

[1205811015840

119906119908

120581119906119908

coth (120583119906119908119863)

120583119906119908

] (27)

Note that since even pressure distribution is considered then119906 119908 = 0 2 4 If 119886 and 119887 are less than 03m then the

Abstract and Applied Analysis 5

frequencies of the fundamental resonances parallel to thecavity length or width are higher than 1000Hz (note that1000Hz is not within the frequency range concerned in thisstudy) Then in this case only the mode numbers 119906 = 0 and119908 = 0 are considered The impedance can be approximatedby

119885cav asymp minus119895120588119886120596cot(120596

119888119886

119863) (28)

25 Electroacoustic Analogy According to [34] the acousticimpedance of a double panel absorber can be analogous tothat of the electrical circuit The acoustic impedance is givenbelow

119885dou = 1198851199011 + 119885cav1 +(119885cav1)

2

1198851199012+ 119885cav1 + 119885cav2

(29)

where 1198851199011

and 1198851199012= acoustic impedance of the 1st and 2nd

panels 119885cav1 and 119885cav2 = acoustic impedance of the 1st and2nd cavities

If the impedance of the 2nd cavity is set to zero that isequivalent to the case of a panel absorber backed by a flexiblewall the overall impedance in (29) is rewritten as

119885119900= 119885com + 119885cav +

(119885cav)2

119885119882+ 119885cav

(30)

where 119885com = (119885119872119885119875119885119867)(119885119872119885119875+ 119885119875119885119867+ 119885119867119885119872) =

combined impedance 119885119872= impedance of the microperfo-

ration119885119875= acoustic impedance of the panel absorber119885

119867=

impedance of the air pumping 119885119882= impedance of the wall

If the wall is perfectly rigid (ie no vibration) the naturalfrequency of wall120596

119882in (12)rarr infin Hence119885

119882119898119899119885119882rarr infin

and (119885cav)2(119885119882+ 119885cav) asymp 0 Equation (30) can be rewritten

as

119885119900= 119885com + 119885cav (31)

Then the absorption coefficient of the absorber 120572119900for normal

incidence can be calculated by the well-known formula (see[29])

120572119900=

4Re (119885119900)

(1 + Re(119885119900))2

+ (Im(119885119900))2 (32)

3 Results and Discussions

Using (32) the sound absorption is obtained for the casesconsidered in this section Each panel absorber is madeof aluminum panel The material properties are Youngrsquosmodulus 119864 = 7 times 10

10Nm2 Poissonrsquos ratio ] = 03 andmass density 120588 = 2700 kgm3 The sound speed and airdensity are 340ms and 12 kgm3 respectively Air viscosity120578 = 185 times 10

minus5 Pa sdot s the ratio of specific heat 120574 = 1402Prandtl number Pr = 071 Various excitation levels dampingfactors panel sizes and so forth are considered and theireffects on the sound absorption performances are studied inthe following paragraphs

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [31]

Figure 2 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01 m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 58)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [32]

Figure 3 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 29)

Figures 2 and 3 present comparisons between the fre-quency-absorption curves obtained from the proposedmethod and numerical integration method [32] The resultsshow that there is an absorption peak due to the microperfo-ration around 178Hz The well-known ldquojump phenomenonrdquocan be seen on both curves The results obtained from thetwo methods are generally in reasonably good agreementalthough some deviations are observed at the absorptionpeaks around 422Hz in Figure 2 and 390Hz in Figure 3Besides small differences between the jump frequencies (ie464 and 470Hz in Figure 2 and 410 and 420Hz in Figure 3)are also foundThe deviations are caused by that the dampingterm in the proposed formulation is proportional to thenonlinear peak frequency 120596

119873 instead of the linear natural

frequency 120596119900 which was adopted in [32] The excitation

level in Figure 3 is lower than that in Figure 2 Therefore the

6 Abstract and Applied Analysis

structural vibration in Figure 3 is less nonlinear (ie 120596119873is

closer to 120596119900) and the deviation between the peak values from

the two methods is smaller Note that in [32] the numericalintegration method did not deliver the unstable solutionsTherefore only stable results obtained by the proposedmethod in Figure 3 are compared with the numerical resultsin [32] Besides the unstable results are not obtainable inpractical environments and not very useful for practicalengineering purpose

Figures 4(a)ndash4(c) present the sound absorption plottedagainst the excitation frequency for various panel resonantfrequencies In each of these three figures there are twononlinear solution types The ldquojump uprdquo or ldquojump downrdquosolution is obtained by sweeping the excitation frequencyfrom ldquohigh to lowrdquo or ldquolow to highrdquo In Figures 4(a)ndash4(c) thelinear (11) mode panel resonant frequencies of the three casesare 387Hz 172Hz and 97Hz respectively The absorptionpeak frequency due to the microperforation effect is 178HzIt can be seen that (1) if the panel resonant frequencyis higher than the absorption peak frequency due to themicroperforation effect the panel vibration can enhanceand widen the absorption bandwidth and (2) if the panelresonant frequency is close to or lower than the absorptionpeak frequency the panel vibration degrades the absorptionperformance and the peak absorption is deteriorated

Figures 5(a) and 5(b) present the sound absorption plot-ted against the excitation frequency for various excitation lev-els The ldquojump downrdquo solutions are shown in Figure 5(a) (iesweeping the excitation frequency from ldquohigh to lowrdquo) whilethe ldquojump uprdquo solutions are in Figure 5(b) (ie sweepingthe excitation frequency from ldquolow to highrdquo) The absorptionpeak around 178Hz is due to the microperforation effect andis not significantly affected by the excitation level It is notedthat in the cases of 120581 = 1 there is no jump phenomenon as theexcitation force is quite small (ie linear case) Besides it canbe seen that (1) in Figure 5(a) the higher the excitation forcethe wider the absorption peak due to the panel vibrationand (2) in Figure 5(b) if the panel vibration is linear (or lowexcitation level) the absorption peak is very narrow and notvery useful for widening the absorption bandwidth Figures6(a) and 6(b) present the sound absorption plotted against theexcitation frequency for various damping ratios The ldquojumpdownrdquo solutions are shown in Figure 6(a) while the ldquojumpuprdquo solutions are in Figure 6(b) The absorption peak dueto the microperforation effect is not significantly affected bythe panel damping ratio Besides it can be seen that (1) inFigure 6(a) the higher the damping ratio the narrower theabsorption peak due to the ldquojump downrdquo panel vibration andthe lower the ldquojump downrdquo frequency and (2) in Figure 6(b)the absorption peak due to the ldquojump uprdquo panel vibration isnot significantly affected by the damping ratio

Figures 7(a) and 7(b) present the sound absorption andvibration amplitude plotted against the excitation frequencyNote that this case does not consider any perforation effectand other effects There are two peaks in each of the twofigures which are due to the (11) and (13) mode resonancesrespectively It can be seen that the peaks due to the (11)mode resonance are more nonlinear As the two peaks are farfrom each other no strong mode coupling effect is induced

In Figure 7(b) the vibration amplitude is almost linearlyincreasing from 100Hz to 286Hz while the sound absorptionis almost linearly increasing from 100Hz to 150Hz only andslowly and nonlinearly increasing from 150Hz to 286HzBesides the absorption peak due to the (13) mode resonanceis much more significant than the vibration peak It canbe implied that higher mode responses are important forabsorption but not for vibration

Figures 8(a) and 8(b) present the sound absorptionplotted against the excitation frequency and show the effectsof air pumping (Helmholtz resonance) andmicroperforationIn Figure 8(a) the peak frequencies of air pumping andmicroperforation are around 100Hz It can be seen that theabsorption bandwidth of the case with both effects is muchwider and its peak absorption is much higher than the caseswith air pumping or microperforation only The two effectscan interact with each other and significantly enhance theabsorption performance In Figure 8(b) the peak frequenciesof air pumping and microperforation are quite far from eachother The enhancement of the absorption performance dueto the two effects is not as good as that in Figure 8(a)

Figures 9(a) and 9(b) present the sound absorptionplotted against the excitation frequency for various dampingratios and show the effect of linear wall vibration Thepeak frequencies of wall panel and microperforation arearound 80Hz and 178Hz respectively It can be seen thatthe wall vibration does not affect the absorption performancesignificantly and just induces an abrupt change around 80HzThemagnitude of the abrupt change depends on the dampingratio (ie the higher the damping ratio the smaller the abruptchange) Figures 10(a) and 10(b) present the sound absorptionplotted against the excitation frequency and show the effectsof microperforation and linear and nonlinear vibrations Itcan be seen that the effects ofmicroperforation and nonlinearpanel vibration are positive and significant while the effect oflinear wall vibration is negative but not significant

4 Conclusions

In this study the analytic absorption formulation has beendeveloped for the panel absorber under the effects of microp-erforation air pumping and linear and nonlinear vibrationFrom the predictions it can be concluded that (1) if thepanel resonant frequency is higher than the absorptionpeak frequency due to the microperforation effect the panelvibration can enhance and widen the absorption bandwidth(2) if the panel resonant frequency is close to or lowerthan the absorption peak frequency the panel vibrationdegrades the absorption performance or the peak absorptionis deteriorated (3) the higher the excitation force the widerthe absorption peak due to the panel vibration (4) if the panelvibration is linear (or low excitation level) the absorptionpeak is very narrow and not very useful for widening theabsorption bandwidth (5) the absorption bandwidth of thecase with the air pumping and microperforation effects ismuch wider and its peak absorption is much higher thanthe cases with any one of these two effects If the peakfrequencies of air pumping andmicroperforation are far from

Abstract and Applied Analysis 7

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump up

Jump down

Peak freq = 178HzPeak freq = 452Hz

Jump up freq = 422HzJump dn freq = 522Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump downJump down

Jump dn freq = 218HzJump dn freq = 366Hz

(b)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump upJump down

Jump up freq = 294Hz

Jump dn freq = 144Hz

(c)

Figure 4 (a) The effect of the (11) mode vibration (120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 001 120590 = 05

119889 = 04mm and 120581 = 10) (b) The effect of the (11) mode vibration (120596119900= 17211 times 2120587Hz 119905 = 08mm 119886 = 119887 = 015m 119863 = 250mm 120585 = 001

120590 = 05 119889 = 04mm and 120581 = 10) (c) The effect of the (11) mode vibration (120596119900= 9681 times 2120587Hz 119905 = 08mm 119886 = 119887 = 02m 119863 = 250mm

120585 = 001 120590 = 05 119889 = 04mm and 120581 = 10)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178 Hz

120581 = 10

120581 = 5

120581 = 1

Jump dn freq = 524Hz

Jump dn freq = 454 Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120581 = 10

120581 = 1

120581 = 5

Peak freq = 178Hz

Jump up freq = 410Hz

Jump up freq = 424Hz

(b)

Figure 5 (a) The effect of excitation level (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01 m 119863 = 250mm

120585 = 001 120590 = 05 and 119889 = 04mm) (b) The effect of excitation level (high to low frequency excitation 120596119900= 38725 times 2120587119867119911 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 and 119889 = 04mm)

8 Abstract and Applied Analysis

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump dn freq = 438Hz

Jump dn freq= 524Hz

Jump dn freq= 552Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump up freq = 424Hz

Jump up freq = 424Hz

Jump up freq = 424Hz

(b)

Figure 6 (a) The effect of damping ratio (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm

120590 = 05 119889 = 04mm and 120581 = 10) (b) The effect of damping ratio (high to low frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120590 = 05 119889 = 04mm and 120581 = 10)

00

02

04

06

08

10

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 286Hz

(a)

00

05

10

15

20

25

30

0 100 200 300 400 500 600 700Frequency (Hz)

Vibr

atio

n am

plitu

dep

anel

thic

knes

s

Jump dn freq = 286Hz

(b)

Figure 7 (a) The absorption peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm 119886 = 119887 = 015m119863 = 250mm and 119896 = 5) (b) The vibration peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm119886 = 119887 = 015m119863 = 250mm and 120581 = 5)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Air pumping

Both effects

Microperforation

Peak freq = 102Hz

Peak freq = 100Hz

Peak freq = 100Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Both effectsMicroperforation

Air pumping

Peak freq = 324Hz

Peak freq = 308Hz

Peak freq= 116Hz

(b)

Figure 8 (a) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 150mm 120590 = 05 119889 = 03mm119871 = 10mm and 119886

119905= 5mm) (b) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 100mm 120590 = 028

119889 = 03mm 119871 = 10mm and 119886119905= 5mm)

Abstract and Applied Analysis 9

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 9 (a) The effects of linear wall vibration and microperforation (120596119882= 80 times 2120587Hz 120588

119908= 240m2 119905 = 08mm 119886 = 119887 = 01m

119863 = 250mm 120585119882= 002 120590 = 05 and 119889 = 04mm) (b) The effects of linear wall vibration and microperforation (120596

119882= 80 times 2120587Hz

120588119908= 240m2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585

119882= 001 120590 = 05 and 119889 = 04mm)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 524Hz

Peak freq = 178Hz

Dip freq = 80Hz

Peak freq = 452Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump up freq = 424Hz

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 10 (a) The effects of microperforation and linear and nonlinear vibrations (low to high frequency excitation 120596119900= 38725 times 2120587Hz

120596119882= 80 times 2120587Hz 120588

119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10) (b) The

effects of microperforation and linear and nonlinear vibrations (high to low frequency excitation 120596119900= 38725 times 2120587Hz 120596

119882= 80 times 2120587Hz

120588119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10)

each other the enhancement of the absorption performanceis small and (6) the wall vibration does not affect theabsorption performance significantly and just induces a smallabrupt change on the absorption bandwidth around the wallresonant frequency

Appendices

A Derivation of the NonlinearDifferential Equation

The explanation of how the formulation of a nonlinearpanel is expressed into the form of Duffing equation whichis abstracted from [30] is shown here The flexible panelvibration is governed by the Von-Karman plate theory Thewell-known partial differential equation of the nonlinear

panel can be presented in terms of its displacement and theAiry stress function as follows

1

12(1198892119908

1198891205852+ nabla4119908)

= (1205972119865

1205971205782

1205972119908

1205971205852+1205972119865

1205971205852

1205972119908

1205971205782minus 2

1205972119865

120597120578120597120585

1205972119908

120597120578120597120585)

(A1)

Note that (A1) is abstracted [30] and the notations are thesame as those in [30] but the definitions of some notationsare different from those in the main text (see the followingdefinitions) 119908 = panel displacement 120585 and 120578 are thedimensionless space variables of the 119909 and 119910 directions 120585is the dimensionless time variable ] is Poissonrsquos ratio 119865 isthe Airy stress function According to [30] the derivativesof 119865 can be expressed in terms of the in-plane displacements

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

to [9 10] the real and imaginary parts of the acousticimpedance of a microhole are given below

1198851015840

119872119877= 120588119886119888119886(0147

119905

1198892)(radic9 +

1001198892119891

32+ 1768radic119891

1198892

119905)

(19a)

1198851015840

119872119868= 1205881198861198881198861847 119891119905(1 +

1

radic9 + 501198892119891

+ 085119889

119905)

(19b)

where 1198851015840119872= 1198851015840

119872119877+ 1198951198851015840

119872119868= the impedance of a microhole

119891 is frequency 119905 is panel thickness and 119889 is the perforationdiameter 120588

119886and 119888119886are air density and sound speed

Then the overall impedance of all microholes over thepanel is given by

119885119872=1198851015840

119872

120590 (20)

where 120590 is the perforation ratio

24 Air Cavity In Figure 1 the impedance of the air cavityis obtained here The acoustic velocity potential within arectangular cavity is given by the following homogeneouswave equation [1]

nabla2120601 minus

1

1198882119886

1205972120601

1205971205912= 0 (21)

where 120601 is the velocity potential functionThe air particle velocities in the 119909 119910 and 119911 direc-

tions and pressures within the air cavity can be given by120597120601120597119909 120597120601120597119910 120597120601120597119911 and minus120588

119886(120597120601120597120591) respectively

The boundary conditions of the rectangular cavity to besatisfied are

120597120601

120597119909

10038161003816100381610038161003816100381610038161003816119909=0=120597120601

120597119909

10038161003816100381610038161003816100381610038161003816119909=119886= 0 (22a)

120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

=120597120601

120597119910

10038161003816100381610038161003816100381610038161003816119910=119887

= 0 (22b)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=0

= 0 (22c)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=minus119863= 119881119863 (22d)

where 119886 and 119887 are the length and width of the rectangularcavity or the panel and 119881

119863is the average velocity at 119911 =

119863 = cavity depth Note that the cavity impedance of no wallvibration is derived here and used into the sound absorptionformulation in the next subsection which can consider boththe panel and wall vibration effects and is derived using theelectroacoustic analogy

By applying the boundary conditions in (22a) and (22b)the solution of (21) can be expressed as

120601 =

119880

sum

119906=0

119882

sum

119908=0

[119871119906119908

cosh (120583119906119908119911) + 119873

119906119908sinh (120583

119906119908119911)]

times cos(119906120587119909119886) cos(

119908120587119910

119887) 119890119894120596120591

(23)

where 120583119906119908

= radic((119906120587119886)2+ (119908120587119887)

2) minus (120596119888

119886)2 119906 and 119908 are

the acoustic mode numbers 119880 and 119882 are the numbers ofacoustic modes used in the 119909 and 119910 directions and 119871

119906119908and

119873119906119908

are coefficients to be determined in the following stepSubstituting (23) into (22c) and (22d) gives

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=0

= 0 997904rArr 119873119906119908= 0 (24a)

120597120601

120597119911

10038161003816100381610038161003816100381610038161003816119911=119863

= 119881119863119890119894120596120591997904rArr 119871

119906119908

= minusint119886

0int119887

0119881119863cos (119906120587119909119886) cos (119908120587119910119887) 119889119909 119889119910120581119906119908120583119906119908

sinh (120583119906119908119863)

(24b)

where 120581119906119908= int119886

0int119887

0cos (119906120587119909119886)2 cos (119908120587119910119887)2119889119909 119889119910

By inputting (24a) and (24b) into (23) the pressure at 119911 =119863 is obtained and given below

119865119863

=

119880

sum

119906=0

119882

sum

119908=0

[119895120588119886120596coth (120583

119906119908119863)

120581119906119908120583119906119908

times int

119886

0

int

119887

0

119881119863cos(119906120587119909

119886) cos(

119908120587119910

119887) 119889119909 119889119910]

times cos(119906120587119909119886) cos(

119908120587119910

119887)

(25)

The average pressure force at 119911 = 119863 is then given by takingintegration over the panel area

119865119863=int119886

0int119887

0119865119863cos (119906120587119909119886) cos (119908120587119910119887) 119889119909 119889119910

119886119887

= 119881119863119895120588119886120596

119880

sum

119906=0

119882

sum

119908=0

[1205811015840

119906119908

120581119906119908

coth (120583119906119908119863)

120583119906119908

]

(26)

where 1205811015840119906119908= (int119886

0int119887

0cos(119906120587119909119886) cos(119908120587119910119887)119889119909 119889119910)

2

119886119887Finally the impedance of the air cavity is given by

119885cav =119865119863

119881119863

= 119895120588119886120596

119880

sum

119906=0

119882

sum

119908=0

[1205811015840

119906119908

120581119906119908

coth (120583119906119908119863)

120583119906119908

] (27)

Note that since even pressure distribution is considered then119906 119908 = 0 2 4 If 119886 and 119887 are less than 03m then the

Abstract and Applied Analysis 5

frequencies of the fundamental resonances parallel to thecavity length or width are higher than 1000Hz (note that1000Hz is not within the frequency range concerned in thisstudy) Then in this case only the mode numbers 119906 = 0 and119908 = 0 are considered The impedance can be approximatedby

119885cav asymp minus119895120588119886120596cot(120596

119888119886

119863) (28)

25 Electroacoustic Analogy According to [34] the acousticimpedance of a double panel absorber can be analogous tothat of the electrical circuit The acoustic impedance is givenbelow

119885dou = 1198851199011 + 119885cav1 +(119885cav1)

2

1198851199012+ 119885cav1 + 119885cav2

(29)

where 1198851199011

and 1198851199012= acoustic impedance of the 1st and 2nd

panels 119885cav1 and 119885cav2 = acoustic impedance of the 1st and2nd cavities

If the impedance of the 2nd cavity is set to zero that isequivalent to the case of a panel absorber backed by a flexiblewall the overall impedance in (29) is rewritten as

119885119900= 119885com + 119885cav +

(119885cav)2

119885119882+ 119885cav

(30)

where 119885com = (119885119872119885119875119885119867)(119885119872119885119875+ 119885119875119885119867+ 119885119867119885119872) =

combined impedance 119885119872= impedance of the microperfo-

ration119885119875= acoustic impedance of the panel absorber119885

119867=

impedance of the air pumping 119885119882= impedance of the wall

If the wall is perfectly rigid (ie no vibration) the naturalfrequency of wall120596

119882in (12)rarr infin Hence119885

119882119898119899119885119882rarr infin

and (119885cav)2(119885119882+ 119885cav) asymp 0 Equation (30) can be rewritten

as

119885119900= 119885com + 119885cav (31)

Then the absorption coefficient of the absorber 120572119900for normal

incidence can be calculated by the well-known formula (see[29])

120572119900=

4Re (119885119900)

(1 + Re(119885119900))2

+ (Im(119885119900))2 (32)

3 Results and Discussions

Using (32) the sound absorption is obtained for the casesconsidered in this section Each panel absorber is madeof aluminum panel The material properties are Youngrsquosmodulus 119864 = 7 times 10

10Nm2 Poissonrsquos ratio ] = 03 andmass density 120588 = 2700 kgm3 The sound speed and airdensity are 340ms and 12 kgm3 respectively Air viscosity120578 = 185 times 10

minus5 Pa sdot s the ratio of specific heat 120574 = 1402Prandtl number Pr = 071 Various excitation levels dampingfactors panel sizes and so forth are considered and theireffects on the sound absorption performances are studied inthe following paragraphs

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [31]

Figure 2 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01 m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 58)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [32]

Figure 3 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 29)

Figures 2 and 3 present comparisons between the fre-quency-absorption curves obtained from the proposedmethod and numerical integration method [32] The resultsshow that there is an absorption peak due to the microperfo-ration around 178Hz The well-known ldquojump phenomenonrdquocan be seen on both curves The results obtained from thetwo methods are generally in reasonably good agreementalthough some deviations are observed at the absorptionpeaks around 422Hz in Figure 2 and 390Hz in Figure 3Besides small differences between the jump frequencies (ie464 and 470Hz in Figure 2 and 410 and 420Hz in Figure 3)are also foundThe deviations are caused by that the dampingterm in the proposed formulation is proportional to thenonlinear peak frequency 120596

119873 instead of the linear natural

frequency 120596119900 which was adopted in [32] The excitation

level in Figure 3 is lower than that in Figure 2 Therefore the

6 Abstract and Applied Analysis

structural vibration in Figure 3 is less nonlinear (ie 120596119873is

closer to 120596119900) and the deviation between the peak values from

the two methods is smaller Note that in [32] the numericalintegration method did not deliver the unstable solutionsTherefore only stable results obtained by the proposedmethod in Figure 3 are compared with the numerical resultsin [32] Besides the unstable results are not obtainable inpractical environments and not very useful for practicalengineering purpose

Figures 4(a)ndash4(c) present the sound absorption plottedagainst the excitation frequency for various panel resonantfrequencies In each of these three figures there are twononlinear solution types The ldquojump uprdquo or ldquojump downrdquosolution is obtained by sweeping the excitation frequencyfrom ldquohigh to lowrdquo or ldquolow to highrdquo In Figures 4(a)ndash4(c) thelinear (11) mode panel resonant frequencies of the three casesare 387Hz 172Hz and 97Hz respectively The absorptionpeak frequency due to the microperforation effect is 178HzIt can be seen that (1) if the panel resonant frequencyis higher than the absorption peak frequency due to themicroperforation effect the panel vibration can enhanceand widen the absorption bandwidth and (2) if the panelresonant frequency is close to or lower than the absorptionpeak frequency the panel vibration degrades the absorptionperformance and the peak absorption is deteriorated

Figures 5(a) and 5(b) present the sound absorption plot-ted against the excitation frequency for various excitation lev-els The ldquojump downrdquo solutions are shown in Figure 5(a) (iesweeping the excitation frequency from ldquohigh to lowrdquo) whilethe ldquojump uprdquo solutions are in Figure 5(b) (ie sweepingthe excitation frequency from ldquolow to highrdquo) The absorptionpeak around 178Hz is due to the microperforation effect andis not significantly affected by the excitation level It is notedthat in the cases of 120581 = 1 there is no jump phenomenon as theexcitation force is quite small (ie linear case) Besides it canbe seen that (1) in Figure 5(a) the higher the excitation forcethe wider the absorption peak due to the panel vibrationand (2) in Figure 5(b) if the panel vibration is linear (or lowexcitation level) the absorption peak is very narrow and notvery useful for widening the absorption bandwidth Figures6(a) and 6(b) present the sound absorption plotted against theexcitation frequency for various damping ratios The ldquojumpdownrdquo solutions are shown in Figure 6(a) while the ldquojumpuprdquo solutions are in Figure 6(b) The absorption peak dueto the microperforation effect is not significantly affected bythe panel damping ratio Besides it can be seen that (1) inFigure 6(a) the higher the damping ratio the narrower theabsorption peak due to the ldquojump downrdquo panel vibration andthe lower the ldquojump downrdquo frequency and (2) in Figure 6(b)the absorption peak due to the ldquojump uprdquo panel vibration isnot significantly affected by the damping ratio

Figures 7(a) and 7(b) present the sound absorption andvibration amplitude plotted against the excitation frequencyNote that this case does not consider any perforation effectand other effects There are two peaks in each of the twofigures which are due to the (11) and (13) mode resonancesrespectively It can be seen that the peaks due to the (11)mode resonance are more nonlinear As the two peaks are farfrom each other no strong mode coupling effect is induced

In Figure 7(b) the vibration amplitude is almost linearlyincreasing from 100Hz to 286Hz while the sound absorptionis almost linearly increasing from 100Hz to 150Hz only andslowly and nonlinearly increasing from 150Hz to 286HzBesides the absorption peak due to the (13) mode resonanceis much more significant than the vibration peak It canbe implied that higher mode responses are important forabsorption but not for vibration

Figures 8(a) and 8(b) present the sound absorptionplotted against the excitation frequency and show the effectsof air pumping (Helmholtz resonance) andmicroperforationIn Figure 8(a) the peak frequencies of air pumping andmicroperforation are around 100Hz It can be seen that theabsorption bandwidth of the case with both effects is muchwider and its peak absorption is much higher than the caseswith air pumping or microperforation only The two effectscan interact with each other and significantly enhance theabsorption performance In Figure 8(b) the peak frequenciesof air pumping and microperforation are quite far from eachother The enhancement of the absorption performance dueto the two effects is not as good as that in Figure 8(a)

Figures 9(a) and 9(b) present the sound absorptionplotted against the excitation frequency for various dampingratios and show the effect of linear wall vibration Thepeak frequencies of wall panel and microperforation arearound 80Hz and 178Hz respectively It can be seen thatthe wall vibration does not affect the absorption performancesignificantly and just induces an abrupt change around 80HzThemagnitude of the abrupt change depends on the dampingratio (ie the higher the damping ratio the smaller the abruptchange) Figures 10(a) and 10(b) present the sound absorptionplotted against the excitation frequency and show the effectsof microperforation and linear and nonlinear vibrations Itcan be seen that the effects ofmicroperforation and nonlinearpanel vibration are positive and significant while the effect oflinear wall vibration is negative but not significant

4 Conclusions

In this study the analytic absorption formulation has beendeveloped for the panel absorber under the effects of microp-erforation air pumping and linear and nonlinear vibrationFrom the predictions it can be concluded that (1) if thepanel resonant frequency is higher than the absorptionpeak frequency due to the microperforation effect the panelvibration can enhance and widen the absorption bandwidth(2) if the panel resonant frequency is close to or lowerthan the absorption peak frequency the panel vibrationdegrades the absorption performance or the peak absorptionis deteriorated (3) the higher the excitation force the widerthe absorption peak due to the panel vibration (4) if the panelvibration is linear (or low excitation level) the absorptionpeak is very narrow and not very useful for widening theabsorption bandwidth (5) the absorption bandwidth of thecase with the air pumping and microperforation effects ismuch wider and its peak absorption is much higher thanthe cases with any one of these two effects If the peakfrequencies of air pumping andmicroperforation are far from

Abstract and Applied Analysis 7

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump up

Jump down

Peak freq = 178HzPeak freq = 452Hz

Jump up freq = 422HzJump dn freq = 522Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump downJump down

Jump dn freq = 218HzJump dn freq = 366Hz

(b)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump upJump down

Jump up freq = 294Hz

Jump dn freq = 144Hz

(c)

Figure 4 (a) The effect of the (11) mode vibration (120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 001 120590 = 05

119889 = 04mm and 120581 = 10) (b) The effect of the (11) mode vibration (120596119900= 17211 times 2120587Hz 119905 = 08mm 119886 = 119887 = 015m 119863 = 250mm 120585 = 001

120590 = 05 119889 = 04mm and 120581 = 10) (c) The effect of the (11) mode vibration (120596119900= 9681 times 2120587Hz 119905 = 08mm 119886 = 119887 = 02m 119863 = 250mm

120585 = 001 120590 = 05 119889 = 04mm and 120581 = 10)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178 Hz

120581 = 10

120581 = 5

120581 = 1

Jump dn freq = 524Hz

Jump dn freq = 454 Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120581 = 10

120581 = 1

120581 = 5

Peak freq = 178Hz

Jump up freq = 410Hz

Jump up freq = 424Hz

(b)

Figure 5 (a) The effect of excitation level (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01 m 119863 = 250mm

120585 = 001 120590 = 05 and 119889 = 04mm) (b) The effect of excitation level (high to low frequency excitation 120596119900= 38725 times 2120587119867119911 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 and 119889 = 04mm)

8 Abstract and Applied Analysis

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump dn freq = 438Hz

Jump dn freq= 524Hz

Jump dn freq= 552Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump up freq = 424Hz

Jump up freq = 424Hz

Jump up freq = 424Hz

(b)

Figure 6 (a) The effect of damping ratio (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm

120590 = 05 119889 = 04mm and 120581 = 10) (b) The effect of damping ratio (high to low frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120590 = 05 119889 = 04mm and 120581 = 10)

00

02

04

06

08

10

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 286Hz

(a)

00

05

10

15

20

25

30

0 100 200 300 400 500 600 700Frequency (Hz)

Vibr

atio

n am

plitu

dep

anel

thic

knes

s

Jump dn freq = 286Hz

(b)

Figure 7 (a) The absorption peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm 119886 = 119887 = 015m119863 = 250mm and 119896 = 5) (b) The vibration peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm119886 = 119887 = 015m119863 = 250mm and 120581 = 5)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Air pumping

Both effects

Microperforation

Peak freq = 102Hz

Peak freq = 100Hz

Peak freq = 100Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Both effectsMicroperforation

Air pumping

Peak freq = 324Hz

Peak freq = 308Hz

Peak freq= 116Hz

(b)

Figure 8 (a) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 150mm 120590 = 05 119889 = 03mm119871 = 10mm and 119886

119905= 5mm) (b) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 100mm 120590 = 028

119889 = 03mm 119871 = 10mm and 119886119905= 5mm)

Abstract and Applied Analysis 9

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 9 (a) The effects of linear wall vibration and microperforation (120596119882= 80 times 2120587Hz 120588

119908= 240m2 119905 = 08mm 119886 = 119887 = 01m

119863 = 250mm 120585119882= 002 120590 = 05 and 119889 = 04mm) (b) The effects of linear wall vibration and microperforation (120596

119882= 80 times 2120587Hz

120588119908= 240m2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585

119882= 001 120590 = 05 and 119889 = 04mm)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 524Hz

Peak freq = 178Hz

Dip freq = 80Hz

Peak freq = 452Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump up freq = 424Hz

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 10 (a) The effects of microperforation and linear and nonlinear vibrations (low to high frequency excitation 120596119900= 38725 times 2120587Hz

120596119882= 80 times 2120587Hz 120588

119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10) (b) The

effects of microperforation and linear and nonlinear vibrations (high to low frequency excitation 120596119900= 38725 times 2120587Hz 120596

119882= 80 times 2120587Hz

120588119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10)

each other the enhancement of the absorption performanceis small and (6) the wall vibration does not affect theabsorption performance significantly and just induces a smallabrupt change on the absorption bandwidth around the wallresonant frequency

Appendices

A Derivation of the NonlinearDifferential Equation

The explanation of how the formulation of a nonlinearpanel is expressed into the form of Duffing equation whichis abstracted from [30] is shown here The flexible panelvibration is governed by the Von-Karman plate theory Thewell-known partial differential equation of the nonlinear

panel can be presented in terms of its displacement and theAiry stress function as follows

1

12(1198892119908

1198891205852+ nabla4119908)

= (1205972119865

1205971205782

1205972119908

1205971205852+1205972119865

1205971205852

1205972119908

1205971205782minus 2

1205972119865

120597120578120597120585

1205972119908

120597120578120597120585)

(A1)

Note that (A1) is abstracted [30] and the notations are thesame as those in [30] but the definitions of some notationsare different from those in the main text (see the followingdefinitions) 119908 = panel displacement 120585 and 120578 are thedimensionless space variables of the 119909 and 119910 directions 120585is the dimensionless time variable ] is Poissonrsquos ratio 119865 isthe Airy stress function According to [30] the derivativesof 119865 can be expressed in terms of the in-plane displacements

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

frequencies of the fundamental resonances parallel to thecavity length or width are higher than 1000Hz (note that1000Hz is not within the frequency range concerned in thisstudy) Then in this case only the mode numbers 119906 = 0 and119908 = 0 are considered The impedance can be approximatedby

119885cav asymp minus119895120588119886120596cot(120596

119888119886

119863) (28)

25 Electroacoustic Analogy According to [34] the acousticimpedance of a double panel absorber can be analogous tothat of the electrical circuit The acoustic impedance is givenbelow

119885dou = 1198851199011 + 119885cav1 +(119885cav1)

2

1198851199012+ 119885cav1 + 119885cav2

(29)

where 1198851199011

and 1198851199012= acoustic impedance of the 1st and 2nd

panels 119885cav1 and 119885cav2 = acoustic impedance of the 1st and2nd cavities

If the impedance of the 2nd cavity is set to zero that isequivalent to the case of a panel absorber backed by a flexiblewall the overall impedance in (29) is rewritten as

119885119900= 119885com + 119885cav +

(119885cav)2

119885119882+ 119885cav

(30)

where 119885com = (119885119872119885119875119885119867)(119885119872119885119875+ 119885119875119885119867+ 119885119867119885119872) =

combined impedance 119885119872= impedance of the microperfo-

ration119885119875= acoustic impedance of the panel absorber119885

119867=

impedance of the air pumping 119885119882= impedance of the wall

If the wall is perfectly rigid (ie no vibration) the naturalfrequency of wall120596

119882in (12)rarr infin Hence119885

119882119898119899119885119882rarr infin

and (119885cav)2(119885119882+ 119885cav) asymp 0 Equation (30) can be rewritten

as

119885119900= 119885com + 119885cav (31)

Then the absorption coefficient of the absorber 120572119900for normal

incidence can be calculated by the well-known formula (see[29])

120572119900=

4Re (119885119900)

(1 + Re(119885119900))2

+ (Im(119885119900))2 (32)

3 Results and Discussions

Using (32) the sound absorption is obtained for the casesconsidered in this section Each panel absorber is madeof aluminum panel The material properties are Youngrsquosmodulus 119864 = 7 times 10

10Nm2 Poissonrsquos ratio ] = 03 andmass density 120588 = 2700 kgm3 The sound speed and airdensity are 340ms and 12 kgm3 respectively Air viscosity120578 = 185 times 10

minus5 Pa sdot s the ratio of specific heat 120574 = 1402Prandtl number Pr = 071 Various excitation levels dampingfactors panel sizes and so forth are considered and theireffects on the sound absorption performances are studied inthe following paragraphs

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [31]

Figure 2 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01 m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 58)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Classical method Numerical method [32]

Figure 3 Comparison between the results from the proposedclassical and numerical methods (119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 119889 = 04mm and 120581 = 29)

Figures 2 and 3 present comparisons between the fre-quency-absorption curves obtained from the proposedmethod and numerical integration method [32] The resultsshow that there is an absorption peak due to the microperfo-ration around 178Hz The well-known ldquojump phenomenonrdquocan be seen on both curves The results obtained from thetwo methods are generally in reasonably good agreementalthough some deviations are observed at the absorptionpeaks around 422Hz in Figure 2 and 390Hz in Figure 3Besides small differences between the jump frequencies (ie464 and 470Hz in Figure 2 and 410 and 420Hz in Figure 3)are also foundThe deviations are caused by that the dampingterm in the proposed formulation is proportional to thenonlinear peak frequency 120596

119873 instead of the linear natural

frequency 120596119900 which was adopted in [32] The excitation

level in Figure 3 is lower than that in Figure 2 Therefore the

6 Abstract and Applied Analysis

structural vibration in Figure 3 is less nonlinear (ie 120596119873is

closer to 120596119900) and the deviation between the peak values from

the two methods is smaller Note that in [32] the numericalintegration method did not deliver the unstable solutionsTherefore only stable results obtained by the proposedmethod in Figure 3 are compared with the numerical resultsin [32] Besides the unstable results are not obtainable inpractical environments and not very useful for practicalengineering purpose

Figures 4(a)ndash4(c) present the sound absorption plottedagainst the excitation frequency for various panel resonantfrequencies In each of these three figures there are twononlinear solution types The ldquojump uprdquo or ldquojump downrdquosolution is obtained by sweeping the excitation frequencyfrom ldquohigh to lowrdquo or ldquolow to highrdquo In Figures 4(a)ndash4(c) thelinear (11) mode panel resonant frequencies of the three casesare 387Hz 172Hz and 97Hz respectively The absorptionpeak frequency due to the microperforation effect is 178HzIt can be seen that (1) if the panel resonant frequencyis higher than the absorption peak frequency due to themicroperforation effect the panel vibration can enhanceand widen the absorption bandwidth and (2) if the panelresonant frequency is close to or lower than the absorptionpeak frequency the panel vibration degrades the absorptionperformance and the peak absorption is deteriorated

Figures 5(a) and 5(b) present the sound absorption plot-ted against the excitation frequency for various excitation lev-els The ldquojump downrdquo solutions are shown in Figure 5(a) (iesweeping the excitation frequency from ldquohigh to lowrdquo) whilethe ldquojump uprdquo solutions are in Figure 5(b) (ie sweepingthe excitation frequency from ldquolow to highrdquo) The absorptionpeak around 178Hz is due to the microperforation effect andis not significantly affected by the excitation level It is notedthat in the cases of 120581 = 1 there is no jump phenomenon as theexcitation force is quite small (ie linear case) Besides it canbe seen that (1) in Figure 5(a) the higher the excitation forcethe wider the absorption peak due to the panel vibrationand (2) in Figure 5(b) if the panel vibration is linear (or lowexcitation level) the absorption peak is very narrow and notvery useful for widening the absorption bandwidth Figures6(a) and 6(b) present the sound absorption plotted against theexcitation frequency for various damping ratios The ldquojumpdownrdquo solutions are shown in Figure 6(a) while the ldquojumpuprdquo solutions are in Figure 6(b) The absorption peak dueto the microperforation effect is not significantly affected bythe panel damping ratio Besides it can be seen that (1) inFigure 6(a) the higher the damping ratio the narrower theabsorption peak due to the ldquojump downrdquo panel vibration andthe lower the ldquojump downrdquo frequency and (2) in Figure 6(b)the absorption peak due to the ldquojump uprdquo panel vibration isnot significantly affected by the damping ratio

Figures 7(a) and 7(b) present the sound absorption andvibration amplitude plotted against the excitation frequencyNote that this case does not consider any perforation effectand other effects There are two peaks in each of the twofigures which are due to the (11) and (13) mode resonancesrespectively It can be seen that the peaks due to the (11)mode resonance are more nonlinear As the two peaks are farfrom each other no strong mode coupling effect is induced

In Figure 7(b) the vibration amplitude is almost linearlyincreasing from 100Hz to 286Hz while the sound absorptionis almost linearly increasing from 100Hz to 150Hz only andslowly and nonlinearly increasing from 150Hz to 286HzBesides the absorption peak due to the (13) mode resonanceis much more significant than the vibration peak It canbe implied that higher mode responses are important forabsorption but not for vibration

Figures 8(a) and 8(b) present the sound absorptionplotted against the excitation frequency and show the effectsof air pumping (Helmholtz resonance) andmicroperforationIn Figure 8(a) the peak frequencies of air pumping andmicroperforation are around 100Hz It can be seen that theabsorption bandwidth of the case with both effects is muchwider and its peak absorption is much higher than the caseswith air pumping or microperforation only The two effectscan interact with each other and significantly enhance theabsorption performance In Figure 8(b) the peak frequenciesof air pumping and microperforation are quite far from eachother The enhancement of the absorption performance dueto the two effects is not as good as that in Figure 8(a)

Figures 9(a) and 9(b) present the sound absorptionplotted against the excitation frequency for various dampingratios and show the effect of linear wall vibration Thepeak frequencies of wall panel and microperforation arearound 80Hz and 178Hz respectively It can be seen thatthe wall vibration does not affect the absorption performancesignificantly and just induces an abrupt change around 80HzThemagnitude of the abrupt change depends on the dampingratio (ie the higher the damping ratio the smaller the abruptchange) Figures 10(a) and 10(b) present the sound absorptionplotted against the excitation frequency and show the effectsof microperforation and linear and nonlinear vibrations Itcan be seen that the effects ofmicroperforation and nonlinearpanel vibration are positive and significant while the effect oflinear wall vibration is negative but not significant

4 Conclusions

In this study the analytic absorption formulation has beendeveloped for the panel absorber under the effects of microp-erforation air pumping and linear and nonlinear vibrationFrom the predictions it can be concluded that (1) if thepanel resonant frequency is higher than the absorptionpeak frequency due to the microperforation effect the panelvibration can enhance and widen the absorption bandwidth(2) if the panel resonant frequency is close to or lowerthan the absorption peak frequency the panel vibrationdegrades the absorption performance or the peak absorptionis deteriorated (3) the higher the excitation force the widerthe absorption peak due to the panel vibration (4) if the panelvibration is linear (or low excitation level) the absorptionpeak is very narrow and not very useful for widening theabsorption bandwidth (5) the absorption bandwidth of thecase with the air pumping and microperforation effects ismuch wider and its peak absorption is much higher thanthe cases with any one of these two effects If the peakfrequencies of air pumping andmicroperforation are far from

Abstract and Applied Analysis 7

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump up

Jump down

Peak freq = 178HzPeak freq = 452Hz

Jump up freq = 422HzJump dn freq = 522Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump downJump down

Jump dn freq = 218HzJump dn freq = 366Hz

(b)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump upJump down

Jump up freq = 294Hz

Jump dn freq = 144Hz

(c)

Figure 4 (a) The effect of the (11) mode vibration (120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 001 120590 = 05

119889 = 04mm and 120581 = 10) (b) The effect of the (11) mode vibration (120596119900= 17211 times 2120587Hz 119905 = 08mm 119886 = 119887 = 015m 119863 = 250mm 120585 = 001

120590 = 05 119889 = 04mm and 120581 = 10) (c) The effect of the (11) mode vibration (120596119900= 9681 times 2120587Hz 119905 = 08mm 119886 = 119887 = 02m 119863 = 250mm

120585 = 001 120590 = 05 119889 = 04mm and 120581 = 10)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178 Hz

120581 = 10

120581 = 5

120581 = 1

Jump dn freq = 524Hz

Jump dn freq = 454 Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120581 = 10

120581 = 1

120581 = 5

Peak freq = 178Hz

Jump up freq = 410Hz

Jump up freq = 424Hz

(b)

Figure 5 (a) The effect of excitation level (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01 m 119863 = 250mm

120585 = 001 120590 = 05 and 119889 = 04mm) (b) The effect of excitation level (high to low frequency excitation 120596119900= 38725 times 2120587119867119911 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 and 119889 = 04mm)

8 Abstract and Applied Analysis

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump dn freq = 438Hz

Jump dn freq= 524Hz

Jump dn freq= 552Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump up freq = 424Hz

Jump up freq = 424Hz

Jump up freq = 424Hz

(b)

Figure 6 (a) The effect of damping ratio (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm

120590 = 05 119889 = 04mm and 120581 = 10) (b) The effect of damping ratio (high to low frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120590 = 05 119889 = 04mm and 120581 = 10)

00

02

04

06

08

10

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 286Hz

(a)

00

05

10

15

20

25

30

0 100 200 300 400 500 600 700Frequency (Hz)

Vibr

atio

n am

plitu

dep

anel

thic

knes

s

Jump dn freq = 286Hz

(b)

Figure 7 (a) The absorption peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm 119886 = 119887 = 015m119863 = 250mm and 119896 = 5) (b) The vibration peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm119886 = 119887 = 015m119863 = 250mm and 120581 = 5)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Air pumping

Both effects

Microperforation

Peak freq = 102Hz

Peak freq = 100Hz

Peak freq = 100Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Both effectsMicroperforation

Air pumping

Peak freq = 324Hz

Peak freq = 308Hz

Peak freq= 116Hz

(b)

Figure 8 (a) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 150mm 120590 = 05 119889 = 03mm119871 = 10mm and 119886

119905= 5mm) (b) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 100mm 120590 = 028

119889 = 03mm 119871 = 10mm and 119886119905= 5mm)

Abstract and Applied Analysis 9

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 9 (a) The effects of linear wall vibration and microperforation (120596119882= 80 times 2120587Hz 120588

119908= 240m2 119905 = 08mm 119886 = 119887 = 01m

119863 = 250mm 120585119882= 002 120590 = 05 and 119889 = 04mm) (b) The effects of linear wall vibration and microperforation (120596

119882= 80 times 2120587Hz

120588119908= 240m2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585

119882= 001 120590 = 05 and 119889 = 04mm)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 524Hz

Peak freq = 178Hz

Dip freq = 80Hz

Peak freq = 452Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump up freq = 424Hz

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 10 (a) The effects of microperforation and linear and nonlinear vibrations (low to high frequency excitation 120596119900= 38725 times 2120587Hz

120596119882= 80 times 2120587Hz 120588

119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10) (b) The

effects of microperforation and linear and nonlinear vibrations (high to low frequency excitation 120596119900= 38725 times 2120587Hz 120596

119882= 80 times 2120587Hz

120588119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10)

each other the enhancement of the absorption performanceis small and (6) the wall vibration does not affect theabsorption performance significantly and just induces a smallabrupt change on the absorption bandwidth around the wallresonant frequency

Appendices

A Derivation of the NonlinearDifferential Equation

The explanation of how the formulation of a nonlinearpanel is expressed into the form of Duffing equation whichis abstracted from [30] is shown here The flexible panelvibration is governed by the Von-Karman plate theory Thewell-known partial differential equation of the nonlinear

panel can be presented in terms of its displacement and theAiry stress function as follows

1

12(1198892119908

1198891205852+ nabla4119908)

= (1205972119865

1205971205782

1205972119908

1205971205852+1205972119865

1205971205852

1205972119908

1205971205782minus 2

1205972119865

120597120578120597120585

1205972119908

120597120578120597120585)

(A1)

Note that (A1) is abstracted [30] and the notations are thesame as those in [30] but the definitions of some notationsare different from those in the main text (see the followingdefinitions) 119908 = panel displacement 120585 and 120578 are thedimensionless space variables of the 119909 and 119910 directions 120585is the dimensionless time variable ] is Poissonrsquos ratio 119865 isthe Airy stress function According to [30] the derivativesof 119865 can be expressed in terms of the in-plane displacements

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

structural vibration in Figure 3 is less nonlinear (ie 120596119873is

closer to 120596119900) and the deviation between the peak values from

the two methods is smaller Note that in [32] the numericalintegration method did not deliver the unstable solutionsTherefore only stable results obtained by the proposedmethod in Figure 3 are compared with the numerical resultsin [32] Besides the unstable results are not obtainable inpractical environments and not very useful for practicalengineering purpose

Figures 4(a)ndash4(c) present the sound absorption plottedagainst the excitation frequency for various panel resonantfrequencies In each of these three figures there are twononlinear solution types The ldquojump uprdquo or ldquojump downrdquosolution is obtained by sweeping the excitation frequencyfrom ldquohigh to lowrdquo or ldquolow to highrdquo In Figures 4(a)ndash4(c) thelinear (11) mode panel resonant frequencies of the three casesare 387Hz 172Hz and 97Hz respectively The absorptionpeak frequency due to the microperforation effect is 178HzIt can be seen that (1) if the panel resonant frequencyis higher than the absorption peak frequency due to themicroperforation effect the panel vibration can enhanceand widen the absorption bandwidth and (2) if the panelresonant frequency is close to or lower than the absorptionpeak frequency the panel vibration degrades the absorptionperformance and the peak absorption is deteriorated

Figures 5(a) and 5(b) present the sound absorption plot-ted against the excitation frequency for various excitation lev-els The ldquojump downrdquo solutions are shown in Figure 5(a) (iesweeping the excitation frequency from ldquohigh to lowrdquo) whilethe ldquojump uprdquo solutions are in Figure 5(b) (ie sweepingthe excitation frequency from ldquolow to highrdquo) The absorptionpeak around 178Hz is due to the microperforation effect andis not significantly affected by the excitation level It is notedthat in the cases of 120581 = 1 there is no jump phenomenon as theexcitation force is quite small (ie linear case) Besides it canbe seen that (1) in Figure 5(a) the higher the excitation forcethe wider the absorption peak due to the panel vibrationand (2) in Figure 5(b) if the panel vibration is linear (or lowexcitation level) the absorption peak is very narrow and notvery useful for widening the absorption bandwidth Figures6(a) and 6(b) present the sound absorption plotted against theexcitation frequency for various damping ratios The ldquojumpdownrdquo solutions are shown in Figure 6(a) while the ldquojumpuprdquo solutions are in Figure 6(b) The absorption peak dueto the microperforation effect is not significantly affected bythe panel damping ratio Besides it can be seen that (1) inFigure 6(a) the higher the damping ratio the narrower theabsorption peak due to the ldquojump downrdquo panel vibration andthe lower the ldquojump downrdquo frequency and (2) in Figure 6(b)the absorption peak due to the ldquojump uprdquo panel vibration isnot significantly affected by the damping ratio

Figures 7(a) and 7(b) present the sound absorption andvibration amplitude plotted against the excitation frequencyNote that this case does not consider any perforation effectand other effects There are two peaks in each of the twofigures which are due to the (11) and (13) mode resonancesrespectively It can be seen that the peaks due to the (11)mode resonance are more nonlinear As the two peaks are farfrom each other no strong mode coupling effect is induced

In Figure 7(b) the vibration amplitude is almost linearlyincreasing from 100Hz to 286Hz while the sound absorptionis almost linearly increasing from 100Hz to 150Hz only andslowly and nonlinearly increasing from 150Hz to 286HzBesides the absorption peak due to the (13) mode resonanceis much more significant than the vibration peak It canbe implied that higher mode responses are important forabsorption but not for vibration

Figures 8(a) and 8(b) present the sound absorptionplotted against the excitation frequency and show the effectsof air pumping (Helmholtz resonance) andmicroperforationIn Figure 8(a) the peak frequencies of air pumping andmicroperforation are around 100Hz It can be seen that theabsorption bandwidth of the case with both effects is muchwider and its peak absorption is much higher than the caseswith air pumping or microperforation only The two effectscan interact with each other and significantly enhance theabsorption performance In Figure 8(b) the peak frequenciesof air pumping and microperforation are quite far from eachother The enhancement of the absorption performance dueto the two effects is not as good as that in Figure 8(a)

Figures 9(a) and 9(b) present the sound absorptionplotted against the excitation frequency for various dampingratios and show the effect of linear wall vibration Thepeak frequencies of wall panel and microperforation arearound 80Hz and 178Hz respectively It can be seen thatthe wall vibration does not affect the absorption performancesignificantly and just induces an abrupt change around 80HzThemagnitude of the abrupt change depends on the dampingratio (ie the higher the damping ratio the smaller the abruptchange) Figures 10(a) and 10(b) present the sound absorptionplotted against the excitation frequency and show the effectsof microperforation and linear and nonlinear vibrations Itcan be seen that the effects ofmicroperforation and nonlinearpanel vibration are positive and significant while the effect oflinear wall vibration is negative but not significant

4 Conclusions

In this study the analytic absorption formulation has beendeveloped for the panel absorber under the effects of microp-erforation air pumping and linear and nonlinear vibrationFrom the predictions it can be concluded that (1) if thepanel resonant frequency is higher than the absorptionpeak frequency due to the microperforation effect the panelvibration can enhance and widen the absorption bandwidth(2) if the panel resonant frequency is close to or lowerthan the absorption peak frequency the panel vibrationdegrades the absorption performance or the peak absorptionis deteriorated (3) the higher the excitation force the widerthe absorption peak due to the panel vibration (4) if the panelvibration is linear (or low excitation level) the absorptionpeak is very narrow and not very useful for widening theabsorption bandwidth (5) the absorption bandwidth of thecase with the air pumping and microperforation effects ismuch wider and its peak absorption is much higher thanthe cases with any one of these two effects If the peakfrequencies of air pumping andmicroperforation are far from

Abstract and Applied Analysis 7

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump up

Jump down

Peak freq = 178HzPeak freq = 452Hz

Jump up freq = 422HzJump dn freq = 522Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump downJump down

Jump dn freq = 218HzJump dn freq = 366Hz

(b)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump upJump down

Jump up freq = 294Hz

Jump dn freq = 144Hz

(c)

Figure 4 (a) The effect of the (11) mode vibration (120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 001 120590 = 05

119889 = 04mm and 120581 = 10) (b) The effect of the (11) mode vibration (120596119900= 17211 times 2120587Hz 119905 = 08mm 119886 = 119887 = 015m 119863 = 250mm 120585 = 001

120590 = 05 119889 = 04mm and 120581 = 10) (c) The effect of the (11) mode vibration (120596119900= 9681 times 2120587Hz 119905 = 08mm 119886 = 119887 = 02m 119863 = 250mm

120585 = 001 120590 = 05 119889 = 04mm and 120581 = 10)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178 Hz

120581 = 10

120581 = 5

120581 = 1

Jump dn freq = 524Hz

Jump dn freq = 454 Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120581 = 10

120581 = 1

120581 = 5

Peak freq = 178Hz

Jump up freq = 410Hz

Jump up freq = 424Hz

(b)

Figure 5 (a) The effect of excitation level (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01 m 119863 = 250mm

120585 = 001 120590 = 05 and 119889 = 04mm) (b) The effect of excitation level (high to low frequency excitation 120596119900= 38725 times 2120587119867119911 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 and 119889 = 04mm)

8 Abstract and Applied Analysis

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump dn freq = 438Hz

Jump dn freq= 524Hz

Jump dn freq= 552Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump up freq = 424Hz

Jump up freq = 424Hz

Jump up freq = 424Hz

(b)

Figure 6 (a) The effect of damping ratio (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm

120590 = 05 119889 = 04mm and 120581 = 10) (b) The effect of damping ratio (high to low frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120590 = 05 119889 = 04mm and 120581 = 10)

00

02

04

06

08

10

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 286Hz

(a)

00

05

10

15

20

25

30

0 100 200 300 400 500 600 700Frequency (Hz)

Vibr

atio

n am

plitu

dep

anel

thic

knes

s

Jump dn freq = 286Hz

(b)

Figure 7 (a) The absorption peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm 119886 = 119887 = 015m119863 = 250mm and 119896 = 5) (b) The vibration peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm119886 = 119887 = 015m119863 = 250mm and 120581 = 5)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Air pumping

Both effects

Microperforation

Peak freq = 102Hz

Peak freq = 100Hz

Peak freq = 100Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Both effectsMicroperforation

Air pumping

Peak freq = 324Hz

Peak freq = 308Hz

Peak freq= 116Hz

(b)

Figure 8 (a) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 150mm 120590 = 05 119889 = 03mm119871 = 10mm and 119886

119905= 5mm) (b) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 100mm 120590 = 028

119889 = 03mm 119871 = 10mm and 119886119905= 5mm)

Abstract and Applied Analysis 9

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 9 (a) The effects of linear wall vibration and microperforation (120596119882= 80 times 2120587Hz 120588

119908= 240m2 119905 = 08mm 119886 = 119887 = 01m

119863 = 250mm 120585119882= 002 120590 = 05 and 119889 = 04mm) (b) The effects of linear wall vibration and microperforation (120596

119882= 80 times 2120587Hz

120588119908= 240m2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585

119882= 001 120590 = 05 and 119889 = 04mm)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 524Hz

Peak freq = 178Hz

Dip freq = 80Hz

Peak freq = 452Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump up freq = 424Hz

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 10 (a) The effects of microperforation and linear and nonlinear vibrations (low to high frequency excitation 120596119900= 38725 times 2120587Hz

120596119882= 80 times 2120587Hz 120588

119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10) (b) The

effects of microperforation and linear and nonlinear vibrations (high to low frequency excitation 120596119900= 38725 times 2120587Hz 120596

119882= 80 times 2120587Hz

120588119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10)

each other the enhancement of the absorption performanceis small and (6) the wall vibration does not affect theabsorption performance significantly and just induces a smallabrupt change on the absorption bandwidth around the wallresonant frequency

Appendices

A Derivation of the NonlinearDifferential Equation

The explanation of how the formulation of a nonlinearpanel is expressed into the form of Duffing equation whichis abstracted from [30] is shown here The flexible panelvibration is governed by the Von-Karman plate theory Thewell-known partial differential equation of the nonlinear

panel can be presented in terms of its displacement and theAiry stress function as follows

1

12(1198892119908

1198891205852+ nabla4119908)

= (1205972119865

1205971205782

1205972119908

1205971205852+1205972119865

1205971205852

1205972119908

1205971205782minus 2

1205972119865

120597120578120597120585

1205972119908

120597120578120597120585)

(A1)

Note that (A1) is abstracted [30] and the notations are thesame as those in [30] but the definitions of some notationsare different from those in the main text (see the followingdefinitions) 119908 = panel displacement 120585 and 120578 are thedimensionless space variables of the 119909 and 119910 directions 120585is the dimensionless time variable ] is Poissonrsquos ratio 119865 isthe Airy stress function According to [30] the derivativesof 119865 can be expressed in terms of the in-plane displacements

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump up

Jump down

Peak freq = 178HzPeak freq = 452Hz

Jump up freq = 422HzJump dn freq = 522Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump downJump down

Jump dn freq = 218HzJump dn freq = 366Hz

(b)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Low to high freqHigh to low freq

Jump upJump down

Jump up freq = 294Hz

Jump dn freq = 144Hz

(c)

Figure 4 (a) The effect of the (11) mode vibration (120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 001 120590 = 05

119889 = 04mm and 120581 = 10) (b) The effect of the (11) mode vibration (120596119900= 17211 times 2120587Hz 119905 = 08mm 119886 = 119887 = 015m 119863 = 250mm 120585 = 001

120590 = 05 119889 = 04mm and 120581 = 10) (c) The effect of the (11) mode vibration (120596119900= 9681 times 2120587Hz 119905 = 08mm 119886 = 119887 = 02m 119863 = 250mm

120585 = 001 120590 = 05 119889 = 04mm and 120581 = 10)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178 Hz

120581 = 10

120581 = 5

120581 = 1

Jump dn freq = 524Hz

Jump dn freq = 454 Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120581 = 10

120581 = 1

120581 = 5

Peak freq = 178Hz

Jump up freq = 410Hz

Jump up freq = 424Hz

(b)

Figure 5 (a) The effect of excitation level (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01 m 119863 = 250mm

120585 = 001 120590 = 05 and 119889 = 04mm) (b) The effect of excitation level (high to low frequency excitation 120596119900= 38725 times 2120587119867119911 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120585 = 001 120590 = 05 and 119889 = 04mm)

8 Abstract and Applied Analysis

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump dn freq = 438Hz

Jump dn freq= 524Hz

Jump dn freq= 552Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump up freq = 424Hz

Jump up freq = 424Hz

Jump up freq = 424Hz

(b)

Figure 6 (a) The effect of damping ratio (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm

120590 = 05 119889 = 04mm and 120581 = 10) (b) The effect of damping ratio (high to low frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120590 = 05 119889 = 04mm and 120581 = 10)

00

02

04

06

08

10

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 286Hz

(a)

00

05

10

15

20

25

30

0 100 200 300 400 500 600 700Frequency (Hz)

Vibr

atio

n am

plitu

dep

anel

thic

knes

s

Jump dn freq = 286Hz

(b)

Figure 7 (a) The absorption peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm 119886 = 119887 = 015m119863 = 250mm and 119896 = 5) (b) The vibration peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm119886 = 119887 = 015m119863 = 250mm and 120581 = 5)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Air pumping

Both effects

Microperforation

Peak freq = 102Hz

Peak freq = 100Hz

Peak freq = 100Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Both effectsMicroperforation

Air pumping

Peak freq = 324Hz

Peak freq = 308Hz

Peak freq= 116Hz

(b)

Figure 8 (a) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 150mm 120590 = 05 119889 = 03mm119871 = 10mm and 119886

119905= 5mm) (b) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 100mm 120590 = 028

119889 = 03mm 119871 = 10mm and 119886119905= 5mm)

Abstract and Applied Analysis 9

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 9 (a) The effects of linear wall vibration and microperforation (120596119882= 80 times 2120587Hz 120588

119908= 240m2 119905 = 08mm 119886 = 119887 = 01m

119863 = 250mm 120585119882= 002 120590 = 05 and 119889 = 04mm) (b) The effects of linear wall vibration and microperforation (120596

119882= 80 times 2120587Hz

120588119908= 240m2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585

119882= 001 120590 = 05 and 119889 = 04mm)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 524Hz

Peak freq = 178Hz

Dip freq = 80Hz

Peak freq = 452Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump up freq = 424Hz

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 10 (a) The effects of microperforation and linear and nonlinear vibrations (low to high frequency excitation 120596119900= 38725 times 2120587Hz

120596119882= 80 times 2120587Hz 120588

119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10) (b) The

effects of microperforation and linear and nonlinear vibrations (high to low frequency excitation 120596119900= 38725 times 2120587Hz 120596

119882= 80 times 2120587Hz

120588119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10)

each other the enhancement of the absorption performanceis small and (6) the wall vibration does not affect theabsorption performance significantly and just induces a smallabrupt change on the absorption bandwidth around the wallresonant frequency

Appendices

A Derivation of the NonlinearDifferential Equation

The explanation of how the formulation of a nonlinearpanel is expressed into the form of Duffing equation whichis abstracted from [30] is shown here The flexible panelvibration is governed by the Von-Karman plate theory Thewell-known partial differential equation of the nonlinear

panel can be presented in terms of its displacement and theAiry stress function as follows

1

12(1198892119908

1198891205852+ nabla4119908)

= (1205972119865

1205971205782

1205972119908

1205971205852+1205972119865

1205971205852

1205972119908

1205971205782minus 2

1205972119865

120597120578120597120585

1205972119908

120597120578120597120585)

(A1)

Note that (A1) is abstracted [30] and the notations are thesame as those in [30] but the definitions of some notationsare different from those in the main text (see the followingdefinitions) 119908 = panel displacement 120585 and 120578 are thedimensionless space variables of the 119909 and 119910 directions 120585is the dimensionless time variable ] is Poissonrsquos ratio 119865 isthe Airy stress function According to [30] the derivativesof 119865 can be expressed in terms of the in-plane displacements

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump dn freq = 438Hz

Jump dn freq= 524Hz

Jump dn freq= 552Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

120582 = 0008

120582 = 001

120582 = 0025

Peak freq = 178HzJump up freq = 424Hz

Jump up freq = 424Hz

Jump up freq = 424Hz

(b)

Figure 6 (a) The effect of damping ratio (low to high frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm

120590 = 05 119889 = 04mm and 120581 = 10) (b) The effect of damping ratio (high to low frequency excitation 120596119900= 38725 times 2120587Hz 119905 = 08mm

119886 = 119887 = 01m119863 = 250mm 120590 = 05 119889 = 04mm and 120581 = 10)

00

02

04

06

08

10

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 286Hz

(a)

00

05

10

15

20

25

30

0 100 200 300 400 500 600 700Frequency (Hz)

Vibr

atio

n am

plitu

dep

anel

thic

knes

s

Jump dn freq = 286Hz

(b)

Figure 7 (a) The absorption peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm 119886 = 119887 = 015m119863 = 250mm and 119896 = 5) (b) The vibration peaks due to the (11) and (13) mode vibrations (low to high frequency excitation 119905 = 05mm119886 = 119887 = 015m119863 = 250mm and 120581 = 5)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Air pumping

Both effects

Microperforation

Peak freq = 102Hz

Peak freq = 100Hz

Peak freq = 100Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Both effectsMicroperforation

Air pumping

Peak freq = 324Hz

Peak freq = 308Hz

Peak freq= 116Hz

(b)

Figure 8 (a) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 150mm 120590 = 05 119889 = 03mm119871 = 10mm and 119886

119905= 5mm) (b) The effects of the air pumping and microperforation (119905 = 03mm 119886 = 119887 = 01m 119863 = 100mm 120590 = 028

119889 = 03mm 119871 = 10mm and 119886119905= 5mm)

Abstract and Applied Analysis 9

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 9 (a) The effects of linear wall vibration and microperforation (120596119882= 80 times 2120587Hz 120588

119908= 240m2 119905 = 08mm 119886 = 119887 = 01m

119863 = 250mm 120585119882= 002 120590 = 05 and 119889 = 04mm) (b) The effects of linear wall vibration and microperforation (120596

119882= 80 times 2120587Hz

120588119908= 240m2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585

119882= 001 120590 = 05 and 119889 = 04mm)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 524Hz

Peak freq = 178Hz

Dip freq = 80Hz

Peak freq = 452Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump up freq = 424Hz

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 10 (a) The effects of microperforation and linear and nonlinear vibrations (low to high frequency excitation 120596119900= 38725 times 2120587Hz

120596119882= 80 times 2120587Hz 120588

119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10) (b) The

effects of microperforation and linear and nonlinear vibrations (high to low frequency excitation 120596119900= 38725 times 2120587Hz 120596

119882= 80 times 2120587Hz

120588119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10)

each other the enhancement of the absorption performanceis small and (6) the wall vibration does not affect theabsorption performance significantly and just induces a smallabrupt change on the absorption bandwidth around the wallresonant frequency

Appendices

A Derivation of the NonlinearDifferential Equation

The explanation of how the formulation of a nonlinearpanel is expressed into the form of Duffing equation whichis abstracted from [30] is shown here The flexible panelvibration is governed by the Von-Karman plate theory Thewell-known partial differential equation of the nonlinear

panel can be presented in terms of its displacement and theAiry stress function as follows

1

12(1198892119908

1198891205852+ nabla4119908)

= (1205972119865

1205971205782

1205972119908

1205971205852+1205972119865

1205971205852

1205972119908

1205971205782minus 2

1205972119865

120597120578120597120585

1205972119908

120597120578120597120585)

(A1)

Note that (A1) is abstracted [30] and the notations are thesame as those in [30] but the definitions of some notationsare different from those in the main text (see the followingdefinitions) 119908 = panel displacement 120585 and 120578 are thedimensionless space variables of the 119909 and 119910 directions 120585is the dimensionless time variable ] is Poissonrsquos ratio 119865 isthe Airy stress function According to [30] the derivativesof 119865 can be expressed in terms of the in-plane displacements

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 9 (a) The effects of linear wall vibration and microperforation (120596119882= 80 times 2120587Hz 120588

119908= 240m2 119905 = 08mm 119886 = 119887 = 01m

119863 = 250mm 120585119882= 002 120590 = 05 and 119889 = 04mm) (b) The effects of linear wall vibration and microperforation (120596

119882= 80 times 2120587Hz

120588119908= 240m2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585

119882= 001 120590 = 05 and 119889 = 04mm)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump dn freq = 524Hz

Peak freq = 178Hz

Dip freq = 80Hz

Peak freq = 452Hz

(a)

0

02

04

06

08

1

0 100 200 300 400 500 600 700Frequency (Hz)

Abso

rptio

n co

effici

ent120572o

Jump up freq = 424Hz

Peak freq = 178Hz

Dip freq = 80Hz

(b)

Figure 10 (a) The effects of microperforation and linear and nonlinear vibrations (low to high frequency excitation 120596119900= 38725 times 2120587Hz

120596119882= 80 times 2120587Hz 120588

119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m 119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10) (b) The

effects of microperforation and linear and nonlinear vibrations (high to low frequency excitation 120596119900= 38725 times 2120587Hz 120596

119882= 80 times 2120587Hz

120588119908= 240 kgm2 119905 = 08mm 119886 = 119887 = 01m119863 = 250mm 120585 = 120585

119882= 001 120590 = 05 119889 = 04mm and 120581 = 10)

each other the enhancement of the absorption performanceis small and (6) the wall vibration does not affect theabsorption performance significantly and just induces a smallabrupt change on the absorption bandwidth around the wallresonant frequency

Appendices

A Derivation of the NonlinearDifferential Equation

The explanation of how the formulation of a nonlinearpanel is expressed into the form of Duffing equation whichis abstracted from [30] is shown here The flexible panelvibration is governed by the Von-Karman plate theory Thewell-known partial differential equation of the nonlinear

panel can be presented in terms of its displacement and theAiry stress function as follows

1

12(1198892119908

1198891205852+ nabla4119908)

= (1205972119865

1205971205782

1205972119908

1205971205852+1205972119865

1205971205852

1205972119908

1205971205782minus 2

1205972119865

120597120578120597120585

1205972119908

120597120578120597120585)

(A1)

Note that (A1) is abstracted [30] and the notations are thesame as those in [30] but the definitions of some notationsare different from those in the main text (see the followingdefinitions) 119908 = panel displacement 120585 and 120578 are thedimensionless space variables of the 119909 and 119910 directions 120585is the dimensionless time variable ] is Poissonrsquos ratio 119865 isthe Airy stress function According to [30] the derivativesof 119865 can be expressed in terms of the in-plane displacements

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

along the 119909 and 119910 directions (ie 119906 and V) and transversedisplacement (ie 119908)Consider

1205972119865

1205971205852=120597V120597120578

+1

2(120597119908

120597120578)

2

+ ][120597119906

120597120585+1

2(120597119908

120597120585)

2

] (A2a)

1205972119865

1205971205782=120597119906

120597120585+1

2(120597119908

120597120585)

2

+ ][120597V120597120578

+1

2(120597119908

120597120578)

2

] (A2b)

minus1205972119865

120597120585120597120578=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585] (A2c)

As it is assumed that the panel is simply supported the follow-ing displacement functions which can satisfy the boundaryconditions are adopted

119908 = 120576 sin (119898120587120585) sin (119899120587119903120578) (A3a)

119906 =1205762120587

16(cos (2120587119899119903120578) minus 1 + ]

11989921199032

1198982)119898 sin (2120587119898120585)

(A3b)

V =1205762120587

16(119899119903 cos (2120587119898120585) minus 119899119903 + ]

1198982

119899119903) sin (2120587119899119903120578)

(A3c)

Again (A3a)ndash(A3c) originate from [30] and the definitionsof some notations are different from those in the maintext (see the following definitions) 119898 and 119899 are the modalnumber 120576 is the modal amplitude Consider 119903 = 119886119887

Then consider putting (A2a)ndash(A2c) and (A3a)ndash(A3c)into the right terms of (A1)

1205972119865

1205971205852

1205972119908

1205971205782

= minus1205972119865

1205971205852120587211989921199032119872 sin (119898120587120585) sin (120587119899119903120578)

= minus12057631205874

811989941199034(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus1

21205763120587411989941199034sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (119898120587120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (119898120587120585) sin (120587119899119903120578) (A4a)

1205972119865

1205971205782

1205972119908

1205971205852

= minus1205972119865

120597120578211989821205872120576 sin (120587119898120585) sin (120587119899119903120578)

= minus12057631205874

81198984(cos (2120587119903120578) minus 1 + ]

11989921199032

1198982)

times cos (2120587119898120585) sin (120587119898120585) sin (120587119899119903120578) minus 12120576312058741198984

times cos2 (120587119898120585) sin2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]12057631205874

8119899211990321198982(cos (2120587119898120585) minus 1 + ]

1198982

11989921199032)

times cos (2120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578)

minus ]1

212057631205874119898211989921199032sin2 (120587119898120585)

times cos2 (120587119899119903120578) sin (120587119898120585) sin (120587119899119903120578) (A4b)

minus1205972119865

120597120585120597120578

=1 minus ]2

[120597119906

120597120578+120597V120597120585+120597119908

120597120578

120597119908

120597120585]

=1 minus ]2

[minus1205762119898119899119903120587

2

4sin (2120587119899119903120578) sin (2120587119898120585)

+ 1205762119898119899119903120587

2 sin (120587119899119903120578)

times cos (120587119899119903120578) sin (120587119898120585) cos (120587119898120585)] = 0

(A4c)

Hence (A1) can be rewritten in the following form

[1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763] sin (120587119898120585) sin (120587119899119903120578)

+ higher mode terms = 0

(A5)

where 1198621 1198622 and 119862

3are the constants dependent of 119898

119899 119903 ] and the material and dimension parameters and soforth According to [30] the higher mode terms in (A5)are neglected For forced vibration cases a forcing term 119875 isconsidered That is

1198621

1198892120576

1198891205852+ 1198622120576 + 11986231205763+ 119875 = 0 (A6)

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

where (A6) is a Duffing equation which form is the same asthat of (1)

B Derivation of the Peak Frequency

Consider a damping term and rewrite (1)

1205881198892120576

1198891205912+ 2120585120588120596

119900

119889120576

119889120591+ 1205881205962

119900120576 + 120573120576

3+ 119865119900119898119899

sin (120596120591) = 0 (B1)

Take the approximation of 120576 = 119860cos cos(120596119905)+119860 sin sin(120596119905) andconsider the harmonic balance of sin(120596119905) and cos(120596119905)

minus 1205881205962119860 sin minus 2120585120588120596119900120596119860cos + 120588120596

2

119900119860 sin

+3

41205731198603

sin +3

4120573119860 sin119860

2

cos + 119865119900119898119899 = 0(B2a)

minus 1205881205962119860cos + 2120585120588120596119900120596119860 sin + 120588120596

2

119900119860cos

+3

41205731198881198603

cos +3

41205731198881198602

sin119860cos = 0(B2b)

Let 1198602119900= 1198602

cos + 1198602

sin Multiplying (B2a) and (B2b) by 119860cosand119860 sin respectively and then subtracting the new (B2a) bythe new (B2b) yield

minus21205851205881205961199001205961198602

119900

119865119900119898119899

= 119860cos (B3a)

119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

= 119860 sin (B3b)

Then inputting (B3a)-(B3b) into (B2b) yields

120588 (1205962minus 1205962

119900)21205851205881205961199001205961198602

119900

119865119900119898119899

+ 2120585120588120596119900120596119860119900radic1 minus (

2120585120588120596119900120596119860119900

119865119900119898119899

)

2

minus3

4120573(119860119900)4 2120585120588120596119900120596

119865119900119898119899

= 0

(B4)

Consider the equation representing the backbone curve bytaking the forcing and damping terms out of (B1)

120588 (minus1205962+ 1205962

119900)119860119900+3

41205731198603

119900= 0 (B5)

rArr

1198602

119900=120588 (1205962minus 1205962

119900)

(34) 120573 (B6)

Inputting (B6) into (B4) yields

1 minus (2120585120588120596119900120596

119865119900119898119899

)

2120588 (1205962minus 1205962

119900)

(34) 120573= 0 (B7)

where (B7) contains only one unknown 120596 It is obtainableand represents the frequency at which the vibration ampli-tude peaks and is equal to 120596

119873in (9)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work in this paper was fully supported by the CityUgrant ldquoNonlinear Dropped Ceiling for Noise and VibrationReduction Project no 7004039rdquo The author would like toexpress the sincere gratitude and appreciation to Dr WYPoon for his advice about this research

References

[1] Y Y Lee and C F Ng ldquoSound insertion loss of stiffenedenclosure plates using the finite element method and theclassical approachrdquo Journal of Sound and Vibration vol 217 no2 pp 239ndash260 1998

[2] Y Y Lee ldquoThe response frequency conversion characteristic ofa nonlinear curved panel with a centre mass and the soundradiationsrdquo Mathematical Problems in Engineering vol 2012Article ID 298413 11 pages 2012

[3] S Y Hsia and Y T Chou ldquoTraffic noise propagating from vibra-tion of railway wagonrdquo Mathematical Problems in Engineeringvol 2013 Article ID 651797 7 pages 2013

[4] J Feng X P Zheng H T Wang Y J Zou Y H Liu and Z HYao ldquoLow-frequency acoustic-structure analysis using coupledFEM-BEM methodrdquo Mathematical Problems in Engineeringvol 2013 Article ID 583079 8 pages 2013

[5] S M Chen D F Wang and J M Zan ldquoInterior noiseprediction of the automobile based on hybrid FE-SEAmethodrdquoMathematical Problems in Engineering vol 2011 Article ID327170 20 pages 2011

[6] R V Craster and S G L Smith ldquoA class of expansion functionsfor finite elastic plates in structural acousticsrdquo Journal of theAcoustical Society of America vol 106 no 6 pp 3128ndash3134 1999

[7] K A Mulholland and H D Parbrook ldquoTransmission of soundthrough apertures of negligible thicknessrdquo Journal of Sound andVibration vol 5 no 3 pp 499ndash508 1967

[8] R D Ford and M A McCormick ldquoPanel sound absorbersrdquoJournal of Sound and Vibration vol 10 no 3 pp 411ndash423 1969

[9] D Maa ldquoMicroperforated-panel wideband absorbersrdquo NoiseControl Engineering Journal vol 29 no 3 pp 77ndash84 1987

[10] D Maa ldquoPotential of microperforated panel absorberrdquo Journalof the Acoustical Society of America vol 104 no 5 pp 2861ndash2866 1998

[11] Y Wu and G Han ldquoInfinitely many sign-changing solutionsfor some nonlinear fourth-order beam equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 635265 11 pages 2013

[12] X Liu B Huo and S Zhang ldquoNonlinear dynamic analysison the rain-wind-induced vibration of cable considering theequilibrium position of rivuletrdquo Abstract and Applied Analysisvol 2013 Article ID 927632 10 pages 2013

[13] X Lin and F Li ldquoAsymptotic energy estimates for nonlinearPetrovsky platemodel subject to viscoelastic dampingrdquoAbstractandApplied Analysis vol 2012 Article ID 419717 25 pages 2012

[14] J A Esquivel-Avila ldquoDynamic analysis of a nonlinear Tim-oshenko equationrdquo Abstract and Applied Analysis vol 2011Article ID 724815 36 pages 2011

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Abstract and Applied Analysis

[15] M L Santos J Ferreira and C A Raposo ldquoExistence anduniform decay for a nonlinear beam equation with nonlinearityof Kirchhoff type in domains with moving boundaryrdquo Abstractand Applied Analysis no 8 pp 901ndash919 2005

[16] X He ldquoModal decoupling using the method of weightedresiduals for the nonlinear elastic dynamics of a clampedlaminated compositerdquo Mathematical Problems in Engineeringvol 2009 Article ID 972930 19 pages 2009

[17] D Wang J Zhang and Y Wang ldquoStrong attractor of beamequation with structural damping and nonlinear dampingrdquoMathematical Problems in Engineering vol 2013 Article ID769514 8 pages 2013

[18] Y Y Lee ldquoStructural-acoustic coupling effect on the nonlinearnatural frequency of a rectangular box with one flexible platerdquoApplied Acoustics vol 63 no 11 pp 1157ndash1175 2002

[19] Y Y Lee ldquoAnalysis of the nonlinear structural-acoustic resonantfrequencies of a rectangular tube with a flexible end using har-monic balance and homotopy perturbation methodsrdquo AbstractandAppliedAnalysis vol 2012 Article ID 391584 13 pages 2012

[20] C K Hui Y Y Lee and J N Reddy ldquoApproximate ellipticalintegral solution for the large amplitude free vibration of arectangular single mode plate backed by a multi-acoustic modecavityrdquoThin-Walled Structures vol 49 no 9 pp 1191ndash1194 2011

[21] Y Y Lee Q S Li A Y T Leung and R K L Su ldquoThe jumpphenomenon effect on the sound absorption of a nonlinearpanel absorber and sound transmission loss of a nonlinear panelbacked by a cavityrdquoNonlinear Dynamics vol 69 no 1-2 pp 99ndash116 2012

[22] Y Y Lee R K L Su C F Ng and C K Hui ldquoThe effect ofmodal energy transfer on the sound radiation and vibration ofa curved panel theory and experimentrdquo Journal of Sound andVibration vol 324 no 3ndash5 pp 1003ndash1015 2009

[23] A Mamandi M H Kargarnovin and S Farsi ldquoDynamicanalysis of a simply supported beam resting on a nonlinearelastic foundation under compressive axial load using nonlinearnormal modes techniques under three-to-one internal reso-nance conditionrdquo Nonlinear Dynamics vol 70 no 2 pp 1147ndash1172 2012

[24] A H Nayfeh The Method of Normal Forms Wiley-VCHWeinheim Germany 2nd edition 2011

[25] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability of axially moving visco-elastic Rayleigh beamsrdquoInternational Journal of Solids and Structures vol 45 no 25-26pp 6451ndash6467 2008

[26] S W Shaw and C Pierre ldquoNormal modes of vibration fornonlinear continuous systemsrdquo Journal of Sound and Vibrationvol 169 no 3 pp 319ndash347 1994

[27] M E King and A F Vakakis ldquoEnergy-based formulation forcomputing nonlinear normal modes in undamped continuoussystemsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 116 no 3 pp 332ndash340 1994

[28] J Kang and H V Fuchs ldquoPredicting the absorption of openweave textiles and micro-perforated membranes backed by anair spacerdquo Journal of Sound and Vibration vol 220 no 5 pp905ndash920 1999

[29] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics JohnWileyampSons 4th edition 1999

[30] H N Chu and G Herrmann ldquoInfluence of large amplitudes onfree flexural vibrations of rectangular elastic platesrdquo Journal ofApplied Mechanics vol 23 pp 532ndash540 1956

[31] W Weisstein Eric ldquoVietarsquos Substitutionrdquo From MathWorldmdashAWolfram Web Resource httpmathworldwolframcomVie-tasSubstitutionhtml

[32] Y Y Lee X Guo and EWM Lee ldquoEffect of the large amplitudevibration of a finite flexible micro-perforated panel absorber onsound absorptionrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 1 pp 41ndash44 2007

[33] Y Y Lee E W M Lee and C F Ng ldquoSound absorption of afinite flexible micro-perforated panel backed by an air cavityrdquoJournal of Sound and Vibration vol 287 no 1-2 pp 227ndash2432005

[34] Y Y Lee and E W M Lee ldquoWidening the sound absorptionbandwidths of flexiblemicro-perforated curved absorbers usingstructural and acoustic resonancesrdquo International Journal ofMechanical Sciences vol 49 no 8 pp 925ndash934 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of