AnalyticFormulafortheVibrationandSoundRadiationofa … · 2019. 7. 30. · structural-acoustic problems, plate vibration, and solution methods for nonlinear governing equations (e.g.,

Research ArticleAnalytic Formula for the Vibration and Sound Radiation of aNonlinear Duct

Yiu-Yin Lee

Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong

Correspondence should be addressed to Yiu-Yin Lee; [email protected]

Received 11 January 2019; Revised 27 March 2019; Accepted 14 April 2019; Published 28 April 2019

Academic Editor: Jean-Mathieu Mencik

Copyright © 2019 Yiu-Yin Lee.)is 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.

)is paper addresses the vibration and sound radiation of a nonlinear duct. Many related works assume that the boundaries arelinearly vibrating (i.e., their vibration amplitudes are small), or that the duct panels are rigid, and their vibrations can thus beneglected. A classic method combined with Vieta’s substitution technique is adopted to develop an analytic formula forcomputing the nonlinear structural and acoustic responses. )e development of the analytic formula is based on the classicalnonlinear thin plate theory and the three-dimensional wave equation. )e main advantage of the analytic formula is that nononlinear equation solver is required during the solution procedure. )e results obtained from the proposed classic method showreasonable agreement with those from the total harmonic balance method. )e effects of excitation magnitude, panel length,damping, and number of flexible panels on the sound and vibration responses are investigated.

1. Introduction

Over the past decades, many studies have consideredstructural-acoustic problems, plate vibration, and solutionmethods for nonlinear governing equations (e.g., [1–12]).Among various structural-acoustic problems, duct noise hasbeen a particular focus for many years. For example,Venkatesham et al. [13] studied the breakout noise from arectangular duct with compliant walls; they found that thelow-frequency breakout noise was important and should notbe neglected, and the coupling between the acoustic wavesand the structural waves must be considered in the pre-diction of the transverse transmission loss. Tang and Lin [14]proposed the use of stiff light composite panels for ductnoise reduction. )eir selling point was that the actuationstrategy would enable the creation of composite panels forduct noise control without the need for a traditional heavystructural mass. According to their results, the mass-springresonance absorption in the case of a stiff thick panel with athin flexible plate would be more efficient. Tiseo andConcilio [15] conducted a set of simulations for duct noisecontrol. )eir proposed feedback structural and acousticalstructural control strategies maximized the damping withinan acoustic duct and thus improved noise reduction. Jade

and Venkatesham [16] implemented intensity-based ex-perimental techniques for measuring breakout noise andvalidated “equivalent unfolded plate” analytical models.)ey developed an experimental test setup to measure theinput and radiated sound power and vibration level and thencalculated the transverse transmission loss and radiationefficiency. )ey adopted two measurement techniques: theintensity probe method (P-P method) and the microflowntechnique (P-U method). A numerical model was developedto predict transverse transmission loss and radiation effi-ciency, and the results were verified by the experimentalresults. In the aforementioned research works, nonlinearstructural vibration, which should be considered for thinpanel structures, was not the focus.)e number of publishedarticles on nonlinear structural acoustics is quite limitedalthough a few are worth mentioning (e.g., [17–20]).

In contrast, there are numerous studies on solutionmethods for nonlinear governing equations (e.g.,[21–28]). For example, Jacques et al. [29] adopted theharmonic balance and finite element methods for solvingthe problem of the nonlinear vibration of a viscoelasticsandwich beam. Chen et al. [30] analyzed the nonlinearsteady state vibrations of plane structures. Both the finiteelement method and incremental harmonic balance

HindawiShock and VibrationVolume 2019, Article ID 9685142, 12 pageshttps://doi.org/10.1155/2019/9685142

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method were used. )e nonlinear governing equationswere solved by the incremental harmonic balance method.Lee et al. [31] adopted the homotopy perturbation methodto solve parabolic partial differential equations withconstant coefficients for the nonlinear plate problem. Gaoet al. [32] adopted the multiple scales method to analyzethe nonlinear primary resonance of functionally gradedporous cylindrical shells. Hao et al. [33] developed thenonlinear governing equations of motion for the FGMplates using Hamilton's principle and adopted the as-ymptotic perturbation method to obtain four-di-mensional nonlinear averaged equations. When theaforementioned methods were applied to the nonlineargoverning equations, coupled nonlinear algebraic equa-tions were generated that needed to be solved by anonlinear equation solver. )is motivated the currentstudy to adopt the classical method and Vieta’s sub-stitution technique [34] and to develop analytic formu-lations that would not require an equation solver tocompute the nonlinear sound and vibration responses of arectangular duct. Note that, normally a nonlinear problemwould generate a set of nonlinear equations. A nonlinearequation solver is needed to solve them to obtain thenonlinear solutions (e.g. [35]). )e effects of excitationmagnitude, panel length, damping, and number of flexiblepanels on sound and vibration responses were also in-vestigated for various panel cases.

Few studies have examined nonlinear structural acoustics,and studies of nonlinear duct sound and vibration are verylimited.)emodelling technique presented here is suitable forhandling more than one flexible panel. Other works (e.g., [36,37]) only considered one flexible panel.)emain advantage ofthe solution method is that the procedure does not require anonlinear equation solver; rather, the final nonlinear solutionscan be expressed in terms of a set of symbolic parameters withvarious physical meanings.)emethod is very suitable for fastengineering calculation purposes. If a numerical method isused, the final solutions are numerical and require morecomputation time. )e limitation of this solution method isthat it can only be used for a simply supported plate. For otherboundary conditions, the nonlinear modal differential equa-tions are more complicated. )us, no analytical solutions canbe obtained from the proposed method.

2. Theory and Formulation

Figure 1 shows an open-end duct made of flexible panels. Apiston-like excitation is generated at the other end. )eacoustic pressure within the rectangular cavity can be ob-tained from the well-known homogeneous wave equation[2, 11, 37]:

∇2 + k2 pQ(x, y, z) � 0, (1)

where pQ(x, y, z) is the acoustic pressure at the position of(x, y, z) induced by the Qth panel mode and k is the wavenumber.

)e solution form of the acoustic pressure and theacoustic cavity mode can be expressed in the following form:

pQ(x, y, z) �

J

J�1PQJφJ(x, y, z), (2)

φJ(x, y, z) � sinUπx2a

cosVπy

bcos

Wπzc

, (3)

where PQJ is the acoustic pressure amplitude of the Jth

acoustic mode; J is the general acoustic mode number; J isthe number of acoustic mode used; U, V, andW are theacoustic mode numbers for the x, y, and z directions; and a,b, and c are the dimensions of the cavity. Note that b and care much smaller than a. It is implied that the acousticresonant frequencies of nonzero V and W modes are muchhigher than the first several acoustic resonant frequencies ofthe duct cavity.

Putting equations (2) and (3) into (1) and using thetechnique of integration by parts in [37] yield the followingequation:

k2 − k2J αJJPQJ � −

pan

zpQ

znφJdx dy, (4)

where kJ ��

(U/2a)2 + (V/b)2 + (W/c)2

is the wave num-ber of the Jth acoustic mode and αJJ � volφJφJdx dy dy. )esubscripts “vol” and “pan” represent the cavity volume andpanel surface, respectively. zpQ/zn is the pressure gradienton the panel surface.

At the panel surface,zpQ,h

zn� ρaω

2AQϕQ(x, y), (5)

where ρa is the air density; ω is the excitation frequency; Aoand AQ are the displacement amplitudes of the source andpanel, respectively; ϕQ(x, y) is the Q

th panel mode shape,which is a double sine function, sin((mπx)/a)sin((nπy)/b)(the panel is simply supported with immovable edges, andhence, the model shape can be written as a double sinefunction); and m and n are the structural mode numbers.

Putting equation (5) into (4) yields the followingequation:

PQJ �−ρaω2

k2 − k2J αJJαQJAQ − αoJAo , (6)

where αQJ � panϕQφJdx dy and αoJ � souφJdx dy and thesubscript “sou” represents the vibration source.

In equation (6), the first term on the right side is themodal acoustic stiffness term, i.e.,

modal acoustic stiffness term, Kaco,J �ρaω2αQJ

k2 − k2J αJJ. (7)

If there are 2 or more flexible panels considered, theoverall acoustic stiffness is given by

overall acoustic stiffness term,

Kaco � Nρaω2

J

J�1

αQJk2 − k2J αJJ

αQJαQQ

,(8)

2 Shock and Vibration

where N is the number of panels and the subscript “aco”means acoustics. It is noted that the term on the right side isthe acoustic sti�ness force term. Note that in equation (8),αQJ/αQQ is the modal contribution factor and αQQ �∫panϕQϕQdx dy.

According to [38, 39], the Airy stress function derivativescan be expressed as follows:

z2F

zξ2�zv

zη+12

zw

zη( )

2

+ ]zu

zξ+12

zw

zξ( )

2 , (9a)

z2F

zη2�zu

zξ+12

zw

zξ( )

2

+ ]zv

zη+12

zw

zη( )

2 , (9b)

−z2F

zξzη�1− v2

zu

zη+zv

zξ+zw

zηzw

zξ[ ]. (9c)

e solution forms of w, u, and v are given in [40]:

w � ΛQ sin(πζ)sin(πrη), (10a)

u �ΛQ( )

2π

16cos(2πrη)− 1 + ]r2( )sin(2πζ), (10b)

v �ΛQ( )

2π

16r cos(2πrζ)− r +

]r

( )sin(2πrη). (10c)

Hence, the governing equation of Airy stress function isgiven by

112

d2w

dτ2+ ∇4w( ) �

z2F

zη2z2w

zξ2+z2F

zξ2z2w

zη2− 2

z2F

zηzξz2w

zηzξ( ),

(11)

whereΛQ � (AQ/h)T(τ), T(τ) is a time function≤ 1; h is thepanel thickness; F is the Airy stress function; w, u, and v arethe dimensionless displacements along the z, x, and y di-rections, respectively; v is Poisson’s ratio; ζ � mx/a and η �ny/a are the dimensionless x and y coordinates; r� a/b isthe panel aspect ratio; τ � (Cpt)/(

��12

√a) is nondimensional

time; Cp ��E/(ρp(1− ]2))√

is the phase velocity of com-pressional waves; E is Young’s modulus; ρp is the paneldensity; and t is time.

Hence, the terms with F in equation (11) can be replacedby equations (9a)–(9c). en, according to [38], the highermode terms (i.e., sin(3πζ)sin(πrη), sin(πζ)sin(3πrη), . . .)are neglected, and the nonlinear governing equation of freeplate vibration is expressed in the following form:

D1d2ΛQdτ2

+D2ΛQ +D3Λ3Q � 0, (12a)

whereD1,D2, and D3 are the constants in terms of r, ], E, a,b, . . . etc. e variables in the nonlinear governing equation(i.e., equations (12a)–(12c)) are dimensionless. Note that theabove equation is for free vibration only (i.e., no force term).In this study, the governing equation of nonlinear forcedvibration is dimensional and shown as follows.

e following dimensional variables are introduced:

ΛQ � hΛQ, (12b)

t ��12

√a

Cpτ. (12c)

en, put equations (12b) and (12c) into (12a) toeliminate the two nondimensional variables (i.e., ΛQ and τ).Hence, the governing equation of nonlinear free vibration isgiven by the following equation:

ρpd2ΛQdt2

+ ρpω2QΛQ + βQΛ

3Q � 0. (13a)

Consider the acoustic sti�ness force term in equation (8)and a force term which represents uniform harmonic ex-citation. Equation (13a) can be rewritten as follows:

ρpd2ΛQdt2

+ ρpω2Q +Nρaω

2∑J

J�1

αQJk2 − k2J( )αJJ

αQJαQQ

ΛQ

+ βQΛ3Q + κρpg

αoJαQQ

sin(ωt) � 0,

(13b)

Openend

Y

ZFront view

Nonlinearpanel c

b

Openend

X

Z

Pistonexcitationat this end

Side view

Nonlinear panel c

a

Figure 1: A duct/tube formed by nonlinear exible panels.

Shock and Vibration 3

where κ� dimensionless excitation parameter, g � 9.81m·s−2,ΛQ � AQT(t), βQ � (Eh/4(1− v2))(mπ/b)

4[(1 + ((n/m)r)4)((3/4)− (v2/4)) + v((n/m)r)2], and note that again αQJ/αQQ isthe modal contribution factor, which is the same as the one inequation (8).

In equation (13b), the nonlinearity of the structural-acoustic system is represented by the term βQΛ3Q. Physi-cally, the nonlinearity is caused by the large amplitudevibration. In a typical linear vibration (or small vibration)analysis, the axial and transverse vibrations of a panel areindependent, whereas they are coupled in a nonlinear vi-bration analysis. )ey are coupled in the nonlinear analysisbecause in small deflection cases, it is assumed that thedirection of the panel’s transverse vibration is still per-pendicular to the surface; in large deflection cases, thetransverse vibration direction is no longer perpendicular tothe surface because the surface is subject to large de-formation. Mathematically, the nonlinear term in equa-tions (13a) and (13b) results in multiple solutions, whereasthere is only one solution in linear cases. In fact, equations(13a) and (13b) are in the form of the well-known Duffingequation.

Consider the approximation of T(t) ≈ sin(ωt), neglectthe higher harmonic terms, and then equation (13b) is re-written in the following form:

34βQA

3Q + ΠQAQ + ΓQ � 0, (14)

where ΓQ � κρpg(αQJ/αQQ) and ΠQ � ρp(ω2Q −ω2) + Nρaω2

JJ�1(αQJ/((k2 − k

2J)αJJ))(αQJ/αQQ). ωQ is the resonant fre-

quency of the Qth panel mode.Note that there is a limitation in the proposed solution

form. In highly nonlinear cases, superharmonic and sub-harmonic responses occur. As the solution form does notinclude the components of sin(3ωt) or sin(1/3ωt), the re-sults cannot show these phenomena.

For damped vibration, AQ in equation (14) is rewritteninto the following complex form:

AQ

�ΓQ

ΠQ +(3/4)βQ AQ

2

, (15)

⟹ AQ

2

�ΓQ

2

Re〈ΠQ〉 +(3/4)βQ AQ

2

2

+ Im〈ΠQ〉 2.

(16)

)en, substituting equation (16) in (14), we get thefollowing equation:

G3c3

+ G2c2

+ G1c + G0 � 0, (17)

where c � |AQ|2; G3 � ((3/4)βQ)

2; G2 � (3/2)βQRe〈ΠQ〉;G1 � |ΠQ|

2; G0 � |ΓQ|2; Re〈ΠQ〉 is the real part of ΠQ;

Im〈ΠQ〉 is the imaginary part of ΠQ; ΠQ � ρp(ω2Q −ω2) +i(2ζωωQ) + Nρaω2

JJ�1(αQJ/((k

2 + i(2ζkkJ)− k2J)αJJ))(αQJ/

αQQ); i is equal to��−1

√; and ζ is the damping ratio. Physically,

the damping force is assumed to be proportional to thedamping ratio× the velocity (i.e., dΛQ/dt). )e damping ratio

represents the level of damping relative to the critical damping(i.e., ζ � 1).

According to Vieta’s substitution [34], equation (17) canbe simplified using the substitution of c � λ− (G2/3G3):

λ3 + ε1λ + ε0 � 0, (18)

where ε1 � (3G3G1 −G22)/3G23 and ε0 � (2G

32 − 9G3G2G1 +

27G23G0)/27G33.

Equation (18) can be further simplified using the sub-stitutions of λ � B− (ε1/3B) and χ � B3:

χ2 + ε0χ −ε3127

� 0. (19)

)e exact solutions of equation (19) are easily obtainablewithout any nonlinear equation solver. Finally, the solutionsof λ are given below:

λ1 � −13

��12

δ1 + δ2( 3

−13

��12

δ1 − δ2( 3

, (20a)

λ2 �1 + i

�3

√

6

��12

δ1 + δ2( 3

+1− i

�3

√

6

��12

δ1 − δ2( 3

, (20b)

λ3 �1− i

�3

√

6

��12

δ1 + δ2( 3

+1 + i

�3

√

6

��12

δ1 − δ2( 3

, (20c)

where δ1 ��

(27ε1)2 + 4(3ε0)

3

and δ2 � 27ε1.Hence, |AQ| can be found from c � λ− (G2/3G3) and

c � |AQ|2. )en, for the calculation of sound radiation, the

following equations can be used to find the radiation effi-ciency [41]:

σQ �32k2abm2n2π5

⎧⎨

⎩1−k2ab

12⎡⎣ 1−

8(mπ)2

a

b

+ 1−8

(nπ)2

b

a⎤⎦⎫⎬

⎭, for symmetricmodes,

(21a)

σQ �8k4a3b3m2n2π5

⎧⎨

⎩1−k2ab

20⎡⎣ 1−

24(mπ)2

a

b

+ 1−8

(nπ)2

b

a⎤⎦⎫⎬

⎭, for antisymmetricmodes,

(21b)

where symmetric modes are the modes ofm and n set to oddnumbers and antisymmetric modes are the modes ofm and/or n set to even numbers.

3. Numerical Results and Discussion

)e material and physical properties adopted in the fol-lowing numerical cases are as follows: Young’s modulus�7.1× 1010N/m2; Poisson’s ratio� 0.3; panel thickness�2mm; panel density� 2700 kg/m3; air density� 1.2 kg/m3;sound speed� 340m/s; and cross-sectional dimensions ofthe duct� 0.5m× 0.5m. Tables 1–3 show the acoustic modeconvergence studies for various excitation levels. )e


damping ratio, ζ, is 0.02. )e seven mode solutions arenormalized as 100. In the (1,1) and (2,1) mode cases, the fouracoustic mode solutions achieve an error rate of less than 1%for different excitation levels (κ� 0.05 to 0.5), while in the(3,1) mode case, the six mode solutions achieve an error rateof around 3.5%. Tables 4–6 show the structural modecontributions for various excitation levels: the larger theexcitation level is, the higher the contributions of the twohigher modes are. In other words, when the excitation ishigh, more modes are needed. )e comparisons betweenTables 4–6 show the modal contributions at the corre-sponding peak frequencies. )e contributions of the dom-inant modes at the corresponding peak frequencies arealways higher than 94%. )us, the use of two structuralmodes is appropriate for obtaining solutions with goodaccuracy. Figures 2(a) and 2(b) present the comparisonsbetween the first symmetric and first antisymmetric modalamplitude solutions obtained from the proposed methodand the classical harmonic balance method (e.g., [20]). )enondimensional excitation parameter value, κ, is 0.5. )edamping ratio, ζ, is 0.02. )e panel length a is 3m, and thenumber of flexible panels is four. Two harmonic compo-nents, sin(ωt) and sin(3ωt), are adopted in the classicalharmonic balance method. )e results obtained from thetwo methods are generally in good agreement for the firstsymmetric and first antisymmetric modal amplitude solu-tions. )e only slight differences are found at the firstnonlinear peak values, around the frequencies ω/ω1 � 0.9 inFigure 2(a) and 1.15 in Figure 2(b). )e differences arecaused by the different definitions of the damping terms inthe two methods. )ere is a limitation in the proposedsolution form. In highly nonlinear cases, superharmonic andsubharmonic responses occur. As the solution form does notinclude the sin(3ωt) or sin(1/3ωt) components, the resultscannot show these phenomena.

Figures 3(a)–3(d) show the symmetric and antisym-metric vibration amplitudes and sound powers plottedagainst the excitation frequency for various excitationmagnitudes. )e damping ratio ζ is 0.02; the first twostructural modes and seven acoustic modes are adopted; fourflexible panels are considered (N� 4); and the panel lengtha� 3m. )e peak frequencies and peak values of thestructural resonances (i.e., the first two peaks in the figures)increase with the excitation magnitude. )e peak values ofthe other resonances (i.e., acoustic resonances) also increasewith the excitation magnitude. )e peak frequencies remainalmost unchanged, and the peaks are generally more linear,except those under the high excitation level κ� 0.5, whichare more nonlinear, and their peak frequencies are aroundω/ω1 � 2.3. )e comparisons between the vibration ampli-tude and sound power curves show that the high frequencyacoustic peaks are more important in the sound powerfigures while the low frequency structural peaks are moreimportant in the vibration amplitude figures because theradiation efficiencies are monotonically increasing with theexcitation frequency (Figure 3(e)). )e higher the excitationfrequency, the higher the radiation efficiency. )e com-parisons between the curves of symmetric and antisym-metric cases show that the second or higher acoustic peaks

Table 1: (1,1) mode peak vibration amplitude convergence forvarious excitation magnitudes (3 symmetric structural modes,ζ � 0.02).

No. of acoustic modes κ� 0.05 0.2 0.51 78.7 82.4 84.74 99.9 99.9 99.76 100.0 100.0 100.07 100.0 100.0 100.0

Table 2: (2,1) mode peak vibration amplitude convergence forvarious excitation magnitudes (3 antisymmetric structural modes,ζ � 0.02).


Table 3: (3,1) mode peak vibration amplitude convergence forvarious excitation magnitudes (3 symmetric structural modes,ζ � 0.02).


Table 4: Mode contribution at the (1,1) mode peak frequency(κ� 0.2, ζ � 0.02).

No. of acoustic modes (1,1) mode (3,1) mode (5,1) mode1 95.6 3.1 1.24 96.2 2.7 1.16 95.8 3.1 1.17 95.8 3.1 1.1






k = 0.5

0.2

0.05

1 2 3 4 5 6 70Nondimensional excitation frequency, ω/ω1

1E – 03

1E – 02

1E – 01

1E + 00

Vibr

atio

n am

plitu

de/th

ickn

ess

(a)

k = 0.5

0.2

0.05


1E – 08

1E – 07

1E – 06

1E – 05

1E – 04

1E – 03

1E – 02

1E – 01

1E + 00

Soun

d po

wer

/(th

ickn

ess ×

ω1)

2

(b)

k = 0.5

0.2

0.05


1E – 03

1E – 02

1E – 01

1E + 00

Vibr

atio

n am

plitu

de/th

ickn

ess

(c)

k = 0.5

0.2

0.05


1E – 08

1E – 07

1E – 06

1E – 05

1E – 04

1E – 03

1E – 02

1E – 01

1E + 00

Soun

d po

wer

/(th

ickn

ess ×

ω1)

2

(d)

Figure 3: Continued.


Proposed methodHarmonic balance method

1E – 04

1E – 03

1E – 02

1E – 01

1E + 00

Vibr

atio

n am

plitu

de/th

ickn

ess

(a)

Proposed methodHarmonic balance method


1E – 04

1E – 03

1E – 02

1E – 01

1E + 00

Vibr

atio

n am

plitu

de/th

ickn

ess

(b)

Figure 2: (a) Comparison of the symmetric and (b) antisymmetric vibration amplitude results from the proposed method and classicalharmonic balance method (κ� 0.5, a� 3m, b� c� 0.5m, ξ � 0.02, and N� 4).


are much less signicant in the antisymmetric case, while therst acoustic peaks are important in both the symmetric andantisymmetric cases.

Figures 4(a)–4(d) show the symmetric and antisym-metric vibration amplitudes and sound powers plottedagainst the excitation frequencies for various dampingratios. e excitation level κ� 0.2; the rst two structuralmodes and seven acoustic modes are adopted; four exiblepanels are considered (N � 4); and the panel lengtha � 3m. Obviously, the damping only a�ects the peak andtrough values. In the o�-resonance range, the dampinghas no signicant e�ect. If the damping ratio is small(ζ � 0.005), the two structural peaks and rst acousticpeaks look more nonlinear, the peak values are the highestamong the three damping cases, and the trough values (orantiresonant values) are the lowest. If the damping ratio ishigh (ζ � 0.05), (1) the two structural peaks and rstacoustic peaks, which look more nonlinear in the case ofζ � 0.005, become more linear; and (2) the peak values arethe lowest and the trough values the highest.Figures 5(a)–5(d) show the symmetric and antisymmetricvibration amplitudes and sound powers plotted againstthe excitation frequencies for various panel lengths. eexcitation level κ� 0.2; the damping ratio ζ � 0.02; the rsttwo structural modes and seven acoustic modes areadopted; and four exible panels are considered (N � 4).Obviously, the panel length (or cavity length) a�ects thestructural and acoustic resonant frequencies in all vi-bration amplitude and sound power gures. Generally, theshorter the panel length is, the higher the resonant fre-quencies and peak values are. Note that, in the case ofa � 2, some acoustic peaks are outside the frequency range.In Figures 5(c) and 5(d), the resonant frequencies of theacoustic peaks around ω/ω1 � 6.4 to 6.6 are not signi-cantly a�ected by the panel length.e rst antisymmetricacoustic peaks are more nonlinear than the rst sym-metric acoustic peaks. is phenomenon can also beobserved in Figures 6(c) and 6(d).

Figures 6(a)–6(d) show the symmetric and antisym-metric vibration amplitudes and sound powers plottedagainst the excitation frequencies for various numbers ofexible panels. e excitation level κ� 0.2; the dampingratio ζ � 0.02; the rst two structural modes and sevenacoustic modes are adopted; and the panel length a � 3m.Figures 6(a) and 6(c) show that the number of panels a�ectsthe peak values, peak frequencies, and o�-resonance re-sponses in all cases. If only one exible panel is installed, (1)the rst and second symmetric and antisymmetric struc-tural resonant frequencies among the three cases (i.e.,N � 1, 2, 4) are highest in both the vibration amplitude andthe sound power gures; (2) the rst symmetric and an-tisymmetric structural resonant peak values in the soundpower gures are also the highest, while the rst symmetricand antisymmetric structural resonant peak values in thevibration amplitude gures are nearly the same among thethree cases; (3) the second symmetric and antisymmetricstructural resonant peak values in the vibration amplitudeand sound power gures are the lowest among the threecases; and (4) the rst antisymmetric acoustic peaks aremore nonlinear (the jump phenomenon can be seen), whilethe second antisymmetric structural peaks look morelinear.

Figures 7(a)–7(d) show the structural-acoustic peakratios plotted against the dimensionless excitation level forvarious structural modes. e damping ratio ζ � 0.02; therst seven acoustic modes are adopted; four exible panelsare considered (N� 4); and the panel length a� 2m. estructural-acoustic peak ratio is dened as the ratio of thestructural peak value to the highest acoustic peak value. InFigure 7(a), the two structural-acoustic peak ratios aremonotonically decreasing with the increasing excitationlevel. is implies that the acoustic peak is more importantor signicant when the excitation level is higher. InFigures 7(b) and 7(c), the structural-acoustic peak ratios ofthe vibration amplitude are almost constant when the ex-citation level ranges from κ� 0.05 to 0.12. ere is an abrupt

(1,1) mod

e

(3,1)(2,1

)

(4,1)


1E – 07

1E – 06

1E – 05

1E – 04

1E – 03

1E – 02

1E – 01

1E + 00

Radi

atio

n ef

ficie

ncy

(e)

Figure 3: (a) Symmetric and (c) antisymmetric vibration amplitude versus excitation frequency for various source excitation magnitudes(a� 3m, b� c� 0.5m, ξ � 0.02, and N� 4); (b) symmetric and (d) antisymmetric sound radiation versus excitation frequency for varioussource excitation magnitudes (a� 3m, b� c� 0.5m, ξ � 0.02, and N� 4); (e) radiation e¡ciency versus excitation frequency for variousstructural modes.


ζ = 0.005

0.02

0.05


1E – 03

1E – 02

1E – 01

1E + 00

Vibr

atio

n am

plitu

de/th

ickn

ess

(a)

ζ = 0.005

0.020.05


1E – 08

1E – 07

1E – 06

1E – 05

1E – 04

1E – 03

1E – 02

1E – 01

1E + 00

Soun

d po

wer

/(th

ickn

ess ×

ω1)

2

(b)

ζ = 0.005

0.020.05


1E – 03

1E – 02

1E – 01

1E + 00

Vibr

atio

n am

plitu

de/th

ickn

ess

(c)

ζ = 0.005

0.020.05


1E – 08

1E – 07

1E – 06

1E – 05

1E – 04

1E – 03

1E – 02

1E – 01

1E + 00

Soun

d po

wer

/(th

ickn

ess ×

ω1)

2

(d)

Figure 4: (a) Symmetric and (c) antisymmetric vibration amplitude versus excitation frequency for various damping ratios (κ� 0.2, a� 3m,b� c� 0.5m, and N� 4); (b) symmetric and (d) antisymmetric sound radiation versus excitation frequency for various damping ratios(κ� 0.2, a� 3m, b� c� 0.5m, and N� 4).

1E + 00 4m 3mα = 2m

1E – 01

1E – 02

1E – 031 2 3

Nondimensional excitation frequency, ω/ω14 5 6 70

Vibr

atio

n am

plitu

de/th

ickn

ess

(a)

1 2 3Nondimensional excitation frequency, ω/ω1

4 5 6 70

1E + 00

1E – 01

1E – 02

1E – 03

1E – 04

1E – 05

1E – 06

1E – 07

1E – 08

Soun

d po

wer

/(th

ickn

ess ×

ω1)

2

4m3m

α = 2m

(b)

Figure 5: Continued.


jump in each curve at around κ� 0.12 to 0.15. e abruptjumps are due to the jump-up phenomenon, which is wellknown in forced nonlinear panel vibration [34]. en, the

structural-acoustic peak ratios of vibration amplitude aremonotonically decreasing with the increasing excitationlevel. e structural-acoustic peak ratio of sound power is

4m3mα = 2m


4 5 6 70

1E + 00

1E – 01

1E – 02

1E – 03

Vibr

atio

n am

plitu

de/th

ickn

ess

(c)

4m

3m

α = 2m


4 5 6 70

1E + 00

1E – 01

1E – 02

1E – 03

1E – 04

1E – 05

1E – 06

1E – 07

1E – 08

Soun

d po

wer

/(th

ickn

ess ×

ω1)

2

(d)

Figure 5: (a) Symmetric and (c) antisymmetric vibration amplitude versus excitation frequency for various panel lengths (κ� 0.2,b� c� 0.5m, ξ � 0.02, andN� 4); (b) symmetric and (d) antisymmetric sound radiation versus excitation frequency for various panel lengths(κ� 0.2, b� c� 0.5m, ξ � 0.02, and N� 4).

1E + 00

1E – 01

42 N = 1

1E – 02

1E – 031 2 3

Nondimensional excitation frequency, ω/ω14 5 6 70

Vibr

atio

n am

plitu

de/th

ickn

ess

(a)

42 N = 1

1E + 00


4 5 6 70

1E – 01

1E – 02

1E – 03

1E – 04

1E – 05

1E – 06

1E – 07

1E – 08

Soun

d po

wer

/(th

ickn

ess ×

ω1)

2

(b)

4 2N = 1


4 5 6 70

1E + 00

1E – 01

1E – 02

1E – 03

Vibr

atio

n am

plitu

de/th

ickn

ess

(c)

42 N = 1


4 5 6 70

1E + 00

1E – 01

1E – 02

1E – 03

1E – 04

1E – 05

1E – 06

1E – 07

1E – 08

Soun

d po

wer

/(th

ickn

ess ×

ω1)

2

(d)

Figure 6: (a) Symmetric and (c) antisymmetric vibration amplitude versus excitation frequency for various numbers of exible panels(κ� 0.2, a� 3m, b� c� 0.5m, and ξ � 0.02); (b) symmetric and (d) antisymmetric sound radiation versus excitation frequency for variousnumbers of exible panels (κ� 0.2, a� 3m, b� c� 0.5m, and ξ � 0.02).


similar to the other ratios, except it is monotonically in-creasing when the excitation level is from κ� 0.05 to 0.12. InFigure 7(d), the two structural-acoustic peak ratios aremonotonically increasing for κ� 0.05 to 0.055. ere is anabrupt jump around κ� 0.06. en, the two ratios aremonotonically decreasing until κ� 0.17, at which pointanother abrupt jump can be seen. e abrupt jumps are alsodue to the jump-up phenomenon in the acoustic resonance.For κ> 0.17, the structural-acoustic peak ratio of vibrationamplitude is almost constant, while the structural-acousticpeak ratio of sound power is mildly decreasing.

4. Conclusions

is study investigated the vibration and sound radiation ofnonlinear duct panels using the analytic formula based onthe classic method combined with Vieta’s substitutiontechnique. e main advantage of the analytic formula isthat, during the solution procedure, no nonlinear equationsolver is required. In other words, the analytic formula is aset of symbolic parameters with various physical meanings.e results obtained from the proposed classic method showreasonable agreement with those from the total harmonicbalance method. e main ndings can be summarized as

follows: (1) the damping ratio and excitation level are the twokey factors to determine the nonlinearities of the resonantpeaks of nonlinear panels. (2)e acoustic peak values in thesound power gures are generally higher than those in thevibration gures. is implies that the high-frequencycomponents in the sound responses are more signicantthan those in the vibration responses because the soundradiation e¡ciency is higher when the response frequency ishigher. (3) e antisymmetric modal responses, which havebeen considered in only a few structural acoustic studies, arefound to be as important as (or as large as) the symmetricmodal responses. e reason is that the rst antisymmetricstructural mode is coupled with the rst antisymmetricacoustic mode, which is the most dominant acoustic modeeven though it is less important than the rst symmetricmode. (4) e well-known jump phenomenon is found inthe peaks of the structural modes coupled with the acousticmodes, whereas it has previously been found only in theresonant peaks of nonlinear structural vibrations.

Data Availability

e data used to support the ndings of this study are in-cluded within the article.

Vibration

Sound

500

50

Stru

ctur

al-a

cous

tic p

eak

ratio

50 0.1 0.2 0.3

Nondimensional excitation level, k0.4 0.5

(a)

Vibration

Sound

500

50

Stru

ctur

al-a

cous

tic p

eak

ratio

50 0.1 0.2 0.3


(b)

Vibration

Sound

500

50

Stru

ctur

al-a

cous

tic p

eak

ratio

50 0.1 0.2 0.3


(c)

Vibration

Sound

Stru

ctur

al-a

cous

tic p

eak

ratio 10

1

0.1

0.010 0.1 0.2 0.3


(d)

Figure 7: (a) (1,1) mode, (b) (2,1) mode, (c) (3,1) mode, and (d) (4,1) mode structural-acoustic peak ratios versus excitation level (a� 2m,b� c� 0.5m, ξ � 0.02, and N� 4).


Conflicts of Interest

)e author declares that there are no conflicts of interest.

Acknowledgments

)e author thanks the financial and technical supports fromthe City University of Hong Kong. )e work described inthis paper was fully supported by the CityU SRG (project no.7004701).

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AnalyticFormulafortheVibrationandSoundRadiationofa … · 2019. 7. 30. · structural-acoustic problems, plate vibration, and solution methods for nonlinear governing equations (e.g.,