A simplified approach for predicting interaction between flexible structures and acoustic enclosures

A simplified approach for predicting interaction between flexible structures and acoustic enclosures Contents lists available at ScienceDirect Journal of Fluids and Structures journal homepage www els[.]

Contents lists available atScienceDirect

Journal of Fluids and Structures journal homepage:www.elsevier.com/locate/jfs

structures and acoustic enclosures

R.B Davis

College of Engineering, University of Georgia, Athens, GA 30602, USA

A R T I C L E I N F O

Keywords:

Acoustoelasticity

Acoustic–structure interaction

Coupled mode theory

Eigenvalue veering

Acoustic enclosures

A B S T R A C T The natural frequencies of acoustic–structure systems can be approximated by a closed-form expression that accounts for the interaction between two uncoupled component modes: a given structural mode and the acoustic mode with which it couples most strongly This expression requires spatial integration of the component mode shapes In practice, the effort to determine the component mode shapes and compute the necessary integrals negates the simplicity afforded

by the closed-form expression Here, with the use of coupled mode theory, a new nondimen-sional expression for the coupled natural frequencies is derived The derivation includes the

definition of a new dimensionless number that quantifies the natural propensity of two component modes to couple, irrespective of the enclosure size or the fluid and structural properties Values of this dimensionless number are presented for common geometries and boundary conditions With these values, approximations of the coupled natural frequencies can

be calculated by hand without explicit knowledge of the component mode shapes or their spatial integrals The accuracy of these hand calculations is shown for two common acoustic–structure systems: a plate coupled to a rectangular air-filled enclosure and a cylindrical shell containing water

1 Introduction

Acoustic–structure interaction refers to the dynamic interplay between acoustic pressure fields and flexible structures Given the ubiquity of acoustic media and their inevitable contact with natural or man-made structures, it is not surprising that acoustic– structure interaction is relevant to the dynamic analysis of many physical systems When air is thefluid of interest, it is often convenient and accurate to solve the acoustic and structural problems independently (i.e., neglecting any mutual influence between the acousticfluid and the structure) This strategy involves formulating the acoustic problem under the assumption that any adjacent structure is perfectly rigid The corresponding structural dynamic problem is then solved by treating the structure as though it were

in a vacuum

While classical acoustic and vibration analysis treats the acoustic and structural problems independently, there are a host of common scenarios in which this strategy is not appropriate Such cases arise when thefluid of interest is dense or when the structure

is highlyflexible In these situations, the fluid and the structure influence each other in a non-negligible manner, and it may be necessary to solve the acoustic and structure problems simultaneously Excellent introductions to problems of this nature can be found in the texts ofJunger and Feit (1993)andFahy (1985) Classes of problems that may require an acoustic–structure interaction approach are numerous; here, the focus is on the dynamics of structures in contact with an enclosed acousticfluid, referred to here

as acoustoelasticity (Dowell et al., 1977; Dowell and Tang, 2003) (This is not to be confused with the acoustoelastic effect, a term

http://dx.doi.org/10.1016/j.j fluidstructs.2017.02.003

Received 19 July 2016; Received in revised form 16 November 2016; Accepted 9 February 2017

E-mail address: ben.davis@uga.edu

0889-9746/ © 2017 The Author Published by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

MARK

used to describe changes in wave velocity in elastic structures due to static stressfields.)

Many existing acoustoelasticity studies present a theoretical framework followed by results corresponding to systems with specific geometry, material, and fluid characteristics While it is possible to draw upon these studies to make qualitative observations

on the nature of acoustoelastic coupling, their results are generally not suitable for direct or broad application These studies can be used to implement a numerical algorithm that solves a particular problem of interest, but such an effort may not be efficient for the practicing engineer attempting to perform a preliminary design or troubleshoot a failing component In these time-critical situations,

it may be similarly impractical to employ discretization techniques While the ability of commercialfinite element software to model acoustoelasticity problems has advanced considerably, setting up and verifying a working model is not trivial Further, the acoustic– structure coupling inherent to these problems typically requires the solution of unsymmetric eigenvalue problems, which can be computationally expensive for models with a moderate or high number of degrees of freedom

Given these difficulties, there is a need for approximate techniques that can be used to quickly assess the extent to which a structure couples to an adjacent acoustic cavity Here, a theoretical framework to make these assessments is presented The approach begins by modifying the acoustoelastic equations of motion to consider two component modes: a single structural mode of interest and the acoustic mode with which it couples most strongly The modified equations of motion lead to a closed-form expression that can be used to approximate coupled natural frequencies This expression, which has been reported elsewhere (Fahy,

1985), can be cumbersome to implement because it requires spatial integration of the component mode shapes By writing the model equations in their so-called coupled mode form (Louisell, 1960), and invoking an approximation known as the weak coupling assumption, a new approximate acoustoelastic natural frequency expression can be derived This expression can then be cast in a simple nondimensional form that provides insight into the fundamental nature of acoustic–structure coupling The nondimensional expression includes the definition of a new dimensionless number quantifying the natural propensity of a structural mode to couple with a given acoustic mode, irrespective of cavity size,fluid properties, or structural properties Values of this dimensionless number are calculated for common geometries and boundary conditions With these values, approximations of the acoustoelastic natural frequencies can be calculated by hand without the need to compute integrals This approach is attractive to practitioners wishing to quickly determine the importance of acoustic–structure interaction before implementing a more rigorous analysis The presented approach can also be used to perform parametric studies or design optimization analyses that are too computationally expensive via any other method Another advantage of the approach is the physical insight it affords the analyst, thus providing a means by which

to perform physics-based checks on complex models

The remainder of this paper is organized as follows:Section 2discusses relevant previous work in the areas of acoustoelasticity and coupled mode theory.Section 3derives two approximate closed-form expressions for the coupled natural frequencies of an

Nomenclature

AF area offluid–structure interface

a0 radius of shell from origin to middle surface

aj acoustic modal coordinate

c0 acoustic speed of sound

D1,2,D *1,2 modal coordinates in coupled mode form

E Young's modulus

d0,1,2 coefficients of shell characteristic equation

Fj acoustic normal mode

g system equation coefficients in coupled mode form

Gk structural normal mode

H height of rectangular enclosure

h thickness of plate

J n a nath-order Bessel function

k n p a a roots of characteristic equation for rigid annular

cavity

L length of rectangular enclosure

Ljk coupling coefficient

ℓ cylindrical shell length

Mj, Mk acoustic, structural modal normalization factor

ma, na, pa acoustic wavenumber indices

ms, ns structural wavenumber indices

qk structural modal coordinate

Ri inner radius of cylindrical shell or annulus

Ro outer radius of annulus

r,θ, z cylindrical coordinates

u, v axial, circumferential displacement of cylindrical

shell

V volume offluid cavity

W width of rectangular enclosure

w displacement of structure in normal direction

x, y, z Cartesian coordinates

Y n nath-order Neumann function

α scaling parameter

α n p

a a roots of characteristic equation for rigid cylindri-cal duct

βjk coupling strength parameter

Γ radial variation of rigid wall acoustic mode in an

annulus

Δ dimensionless component frequency separation

δ n a = 1 if na= 0, = 0 otherwise

ηjk energy transfer factor

ϵ = 1 if subscript= 0, = 2 otherwise

λ n p a a =π n s +m s( )

W H

⎝

⎞

⎠

κ1,2 dimensionless separation between coupled and

uncoupled natural frequencies

ϕ fluid velocity potential

Ψ dimensionless coupling parameter

ρ, ρs density of structure per unit volume, per unit area

ρ0 density offluid per unit volume

ν Poisson's ratio

Ω cylindrical shell frequency parameter

ω c

1,2 coupled natural frequencies

ωj,ωk uncoupled natural frequency of cavity, structure

acoustoelastic system The so-called weak frequency expression is then non-dimensionalized, leading to insights into the nature of acoustoelastic coupling The non-dimensional acoustoelastic coupling number found inSection 3is then investigated inSection 4for some common acoustoelastic systems Two specific examples inSection 5illustrate how the expressions derived here can be used to make accurate predictions acoustoelastic natural frequencies without resorting to more complex methods Issues related to the accuracy and limitations of the approximate approach are presented inSection 6and general conclusions are discussed inSection 7

2 Background

Acoustoelasticity has been of interest to researchers for well over 50 years.Warburton (1961)noticed that a cylindrical shell containing air possesses resonant frequencies which are close to either the in vacuo structural natural frequencies or the rigid wall acoustic natural frequencies of the enclosed air.Dowell et al (1977)expanded upon this idea to develop a theoretical formulation that combines the uncoupled acoustic enclosure modes and the in vacuo structural modes into a system of coupled ordinary

differential equations This formulation, which serves as the theoretical basis for much of the present work, has been used to investigate acoustic–structure interaction in a variety of systems of practical interest including rectangular enclosures (Bokil and Shirahatti, 1994), airplane fuselages (Dowell, 1980), cylindrical and annular enclosures (Davis et al., 2008), and computer disk drives (Kang and Raman, 2004) Dowell's formulation is more easily implemented than alternative methods because it uses uncoupled component modes as its basis functions These modes are familiar to many investigators and can be readily obtained analytically or with discretization methods Researchers have noted (Ginsberg, 2010) that because of its use of uncoupled acoustic modes as basis functions, Dowell's formulation cannot give correct results for thefluid velocity on the surface of the flexible structure While this is true, Dowell's formulation does provide accurate calculations of the structural wall velocity, which can then

be used to recover the normal component offluid velocity at the wall, if desired (Dowell, 2010) In any case, Dowell's formulation has been shown repeatedly to provide excellent predictions of acoustoelastic natural frequencies, which are the principle quantities of interest in this study

Here, the theoretical development begins with Dowell's acoustoelasticity equations and writes them in what is known as coupled mode form Coupled mode theory was introduced byPierce (1954)and has most often been used in the context of coupled electronic transmission lines (Louisell, 1960), but has also been used to investigate the dynamics of coupled pendula (Teoh and Davis, 1996) Pan and Bies (1990)are the only other researchers to apply coupled mode theory to acoustoelastic systems As part of a larger study concerning air-filled rectangular cavities treated with sound-absorbing material, Pan and Bies employ an energy transfer factor derived byLouisell (1960)(and presented here as Eq.(15)) to quantify the interaction between pairs of uncoupled structural and acoustic modes The Pan and Bies study does not, however, investigate the ways in which coupled mode theory can be used to approximate the natural frequencies of acoustic–structure systems

Other related work involves the sensitivity analysis of coupled acoustoelastic frequencies.Scarpa and Curti (1999)andScarpa (2000)begin with Dowell's formulation to obtain expressions for the derivatives of system frequencies These expressions are then used to investigate how the system frequencies are altered with changes to various design parameters Specifically, the studies consider frequency sensitivity in the context of air-filled rectangular cavities coupled to simply supported plates The design variables

of interest are the thickness of the plate and the length of the cavity Results are compared to those obtained from afinite element model By casting system frequency behavior in a nondimensional form, the approach employed here can be used to inform and explain many of the results presented by Scarpa and Curti

Fig 1 Schematic of a fluid-filled enclosure of arbitrary geometry The acoustic fluid fills a volume V and is coupled to the motion of a flexible structural surface of area, A F The remaining fluid–structure interface is rigid and denoted A R

3 Theoretical model

3.1 Approximate expressions of system natural frequency

In this section, two approximate closed-form expressions for the coupled natural frequencies of an acoustoelastic system are derived The derivation involves writing the system equations of motion in Hamiltonian form and then transforming them into coupled mode form The full equations in their coupled mode form allow for the derivation of a previously known expression for system natural frequency The application of the weak coupling assumption then leads to a new expression that is less accurate, but permits a simple nondimensionalization and affords useful physical insights

Consider afluid-filled enclosure of arbitrary geometry represented by Fig 1 A quiescent, inviscid, compressiblefluid fills a volume V and is coupled to the motion of aflexible structural surface of area AF The system equations of motion can be found by first expanding the fluid velocity potential, ϕ, and the structural displacement, w, in terms of their normal modes

∑

ϕ t( ) = a t F( ) ,

j

(1)

∑

w t( ) = q t G( ) ,

k

(2) where aj(t) and qk(t) are the time-dependentfluid and structural modal coordinates, respectively Fjdenotes the shape of the jth uncoupled mode of the acoustic enclosure while Gkrepresents the kth in vacuo mode of the structure It can be shown (Dowell et al., 1977; Dowell and Tang, 2003; Fahy, 1985) that the equations modeling the free vibration of the acoustoelastic system are

∑

M a ω a c A

j j j j

F k

jk k

(3a)

∑

ρ L a

s j

jk j

(3b) Eqs.(3)represent a system of gyroscopically coupled ordinary differential equations where c0,ρ0, andρsare the speed of sound

in thefluid, the fluid density, and the mass of the structure per unit area The natural frequencies of the uncoupled acoustic cavity are

ωjand the in vacuo natural frequencies of the structure are given byωk While cavity geometry is irrelevant at this stage, Eqs.(3)are written here for an interior acoustic cavity Mjand Mkare the modal normalization factors defined as

∫

M

V F dV

j

V j

2

(4)

∫

M

A G dA

k

F A

k2

The experienced researcher may observe that Mkis traditionally (Dowell et al., 1977; Dowell and Tang, 2003; Fahy, 1985; Bokil and Shirahatti, 1994) expressed as a generalized modal mass and thereby includes a term analogous toρsin its integrand Here, the structural mass density is assumed to be spatially uniform and is included on the right-hand side of Eq.(3b) Making this assumption and defining Mkin the manner of Eq.(5)aids the forthcoming derivation of a nondimensionalized coupled frequency expression The Ljkterms in Eqs.(3)represent coupling coefficients given by

∫

L

A F G dA

jk

F A

j k

Ljkcan thus be interpreted as a measure of the spatial similarity between the component mode shapes Rewriting Eqs.(3)to consider the coupling of just two component modes and then expressing the result in Hamiltonian form gives

da

dt

c A L k

VM q ω a

˙

j

(7a)

da

dt = ˙ ,a

j

dq

dt

ρ L

ρ M a ω q

˙

s k

(7c)

dq

dt = ˙ q

k

Following the procedure outlined byLouisell (1960), Eq.(7b)is multiplied by iω± jand added to Eq.(7a) Similarly, Eq.(7d)is multiplied by±iω kand added to Eq.(7c) The result is the following system of equations:

dt

c A L

VM q ω a iω a

j

(8a)

dD

dt

c A L

VM q ω a iω a

*

j

(8b)

dD

dt

ρ L

ρ M a ω q iω q

s k

(8c)

dD

dt

ρ L

ρ M a ω q iω q

*

s k

(8d)

where D1≡ ( ˙ +a j iω a j j), D1* ≡ ( ˙ −a j iω a j j), D2≡ ( ˙ +q k iω q k k), and D2* ≡ ( ˙ −q k iω q k k) Eqs.(8)can now be written in their coupled mode form:

dD

dt1 =g D +g D +g D* +g D*,

dD

dt2 =g D +g D +g D* +g D*,

dD

*

1

dD

*

2

where

j

F jk j

k

jk

s k

0

0

(10)

By assuming harmonic solutions to Eqs.(9)and solving the associated eigenvalue problem, the two system natural frequencies,

ω c1,2, are found:

2

2 1,2

⎛

⎝

⎞

where

β ρ c A L

ρ VM M

jk

F jk

s j k

0 0

(12) Fahy (1985)provides an expression equivalent to Eq.(11), though he derives it differently The utility of writing the equations of motion in the seemingly cumbersome form given by Eqs.(9)will now become apparent By inspection of Eqs.(8)it can be seen that

D and D*represent identical, but out-of-phase modes One can then make the assumption that only those modes oscillating in-phase with each other will couple appreciably Thus, the D*equations can be neglected under this so-called weak coupling assumption When this is done, Eqs.(9)simplify to

dD

dt =g D +g D,

1

dD

dt =g D +g D,

2

and a second approximate expression for the coupled natural frequencies can be found:

1,2

⎛

⎝

⎞

While this expression is less accurate than Eq.(11), it permits a simple nondimensionalization as will be shown inSection 3.3

3.2 Dimensionless coupling parameter

Beginning with expressions in the form of Eqs.(13),Louisell (1960)derives a general transfer factor to quantify the energy exchange between two weakly coupled modes For the acoustoelastic systems of interest here, this factor is given by

β

2

1 ,

jk

jk

⎡

⎣

⎢

⎢

⎛

⎝

⎞

⎠

⎤

⎦

⎥

whereηjkis a dimensionless quantity that varies between zero and unity A value ofηjkthat is close to unity indicates strong coupling between the component modes A value near zero suggests that the component modes are not appreciably affected by one another.Pan and Bies (1990)used this factor to investigate modal coupling in rectangular enclosuresfilled with air The transfer

factor depends on two quantities: ω( j−ω k) and βjk The quantity ω( j−ω k) is simply the separation of the component mode

frequencies For a given value of ω( j−ω k),βjkdetermines the extent to which the two component modes interact By inspection of

Eq.(12),βjkis comprised of several dimensional parameters that describe the physical properties and geometry of the structure and the cavity (namely, c0, AF, V, andρs) For any given system, these physical parameters are constant regardless of the two component modes being considered The remaining parameters (Ljk, Mj, and Mk) are all nondimensional quantities related to the shapes of the component modes A new parameter,Ψ, can therefore be defined as

M M

≡ jk

j k

2

(16)

Ψ can be interpreted as a measure of the natural propensity of a structural mode to couple with a given acoustic mode, irrespective of cavity size,fluid properties, or structural density Due to its dependence on the component mode shapes however, the value of Ψ is sensitive to the geometry and boundary conditions of the structure and of the acoustic cavity Studies investigating the values ofΨ for some common geometries and boundary conditions are presented inSection 4

3.3 Dimensionless system frequency expression

Before arriving at the dimensionless version of Eq.(14), it is helpful to qualitatively understand how system frequencies evolve as the frequency separation between the component modes is varied.Fig 2is representative of how acoustic and structural component frequencies might vary with respect to some control parameter (e.g., time, wavenumber, and enclosure size) The system frequencies resulting from the coupling of the two branches of component frequencies are also shown In an actual physical system, many branches of component frequencies may occupy a given frequency range For clarity, only two branches of component frequencies are shown

Fig 2depicts a scenario in which the two component curves intersect at some value of the control parameter As evidenced by Eq (15), two component modes that are capable of coupling will do so most strongly when the frequency separation between them is small The system frequency curves demonstrate eigenvalue veering near the intersection of the component frequency curves Eigenvalue veering is a documented feature of gyroscopically coupled systems (Vidoli and Vestroni, 2005) that is characterized by two eigenvalue loci approaching each other and then diverging Near these intersections, the system natural modes are well coupled and receive significant contribution in terms of energy from both the acoustic cavity and the structure The separation between the

Fig 2 A representative depiction of coupled and uncoupled frequency behavior.

two system frequency curves at their closest point of approach can be interpreted as an indication of coupling strength Large system frequency separations at this point suggest strong tendency for the component modes to interact Away from the intersection, the component mode frequency separation increases and the system frequencies assume values that are comparatively close to those of the nearest component mode At these control parameter values, the system is only lightly coupled and exhibits modes that are dominated by a contribution from the component mode that is closest in frequency

From the foregoing discussion and inspection of Eq.(14), it is clear that the difference between a system frequency and its nearest component frequency depends largely on two quantities: the frequency separation between the two component modes and the natural propensity of these modes to couple The task now is to derive a functional relationship between system frequency and these two quantities Expressing this relationship in nondimensional terms removes any dependence on parameters specific to physical configuration and thereby reveals the fundamental nature of system frequency behavior in acoustoelastic systems

Assuming ω j≤ω k, the difference between the lower of the two system frequencies,ω c, and its nearest component frequency,ωj, is found by subtractingωjfrom both sides of Eq.(14)

2 1

⎛

⎝

⎞

Multiplying Eq.(17)by the following parameter:

ρ c A

F

0 0

2

(18) recasts the equation in a nondimensional form

2(− − + ) for j≤ k,

whereΔ is the nondimensional frequency separation between the two component modes

and whereκ1represents the nondimensional frequency reduction due to the acoustic–structure coupling

Now assuming that ω j>ω k,κ1is defined such that it represents the nondimensional difference betweenω c andωk

κ1≡α ω( c −ω k) forω j>ω k

The nondimensional frequency relationship in this case is

2( − + ) for j> k.

Arguments similar to those outlined above are used tofind analogous expressions related toω c, the greater of the two system frequencies Upon doing so,κ2is defined in the following piecewise manner:

Fig 3 (a) A 3D plot of nondimensional frequency augmentation and reduction versus nondimensional component frequency separation and the nondimensional parameter Ψ (b) A 2D projection of part (a) in which several curves with constant Ψ value are shown.

κ α ω ω ω ω

( − ) for > ,

2

2

2

⎪

⎪

⎧

⎨

and the corresponding nondimensional frequency relation is

κ

=

1

1

2(− + + ) for > .

2

2

2

⎧

⎨

⎪⎪

⎩

Note that Eq.(11)can also be cast in a nondimensional form by multiplying both sides byα However, the expression does not reduce to a simple dependence onΔ and Ψ due to the presence of cross-terms Eqs.(19), (23), and (25)are plotted in three dimensions againstΔ and Ψ inFig 3(a), whileFig 3(b) is a two-dimensional projection of these equations that shows several curves with constantΨ value These figures depict the fundamental nature of system frequency behavior in acoustoelastic systems Specifically, they show that the nondimensional separation between a system frequency and its nearest component frequency will be maximum when component frequency separation is zero (Δ = 0), with a maximum value given by Ψ Therefore, with a knowledge

of theΨ value for a particular component mode pair of interest, one can quickly bound the frequency shift that would result from acoustic–structure coupling The next section investigates values of Ψ for common configurations

4 Coupling behavior of systems with common geometries

In this section, the natural tendency for certain structural and acoustic component modes to couple is assessed for rectangular, cylindrical, and annular enclosures This is accomplished through the calculation of theΨ parameter as it is given by Eq.(16) 4.1 Rectangular enclosure

The system of aflexible plate backed by a fluid-filled rectangular enclosure is one that has been investigated extensively (see e.g., Lyon, 1963; Dowell and Voss, 1963; Pretlove, 1965; Pan and Bies, 1990; Bokil and Shirahatti, 1994) For the case in which the plate

is simply supported on all four edges, the structural mode shapes can be expressed exactly For a plate with width W and height H (seeFig 4), these mode shapes are given by

W

n πy H

k

⎛

⎝

⎞

⎠

⎛

⎝

⎞

where the modal index k includes indices, msand ns, with the s subscript denoting that they are structural wavenumber indices The flexible plate is considered to comprise one face of a rectangular enclosure with length L All other faces are assumed rigid The rigid wall acoustic modes for the enclosure are

W

n πy H

p πz L

j

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

where the index j consists of the acoustic wavenumber indices ma, na, and pa Eqs.(27) and (26)are applied to Eqs.(4) and (5)to find simple expressions for the modal normalization factors

M

ε ε ε

j

M = 1

4,

whereε is a parameter that equals unity only if the wavenumber given by its subscript equals zero The parameter ε equals two in all other cases An expression for the coupling coefficients can be found by computing the integral given by Eq.(6) Due to the

Fig 4 Schematic of a flexible plate backed by a rectangular enclosure.

orthogonality of certain component mode shapes, the coupling coefficient will be identically zero if ms+ maor ns+ nais even In all other cases, the coupling coefficients are given by

L m n

jk

s s

(29) Substituting Eq.(28)and Eq.(29)into Eq.(16)leads to the following expression forΨ:

Ψ ε ε ε m n

m n p s s

a a a

(30) Table 1lists the component mode combinations that result in the ten highest values ofΨ In all cases, Ψ assumes the given value

provided that p ≥ 1 a For plates that are not simply supported on all edges, such an exact statement ofΨ is not possible The next section studies the effect that various boundary conditions have on the values of Ψ in the context of a fluid-filled cylindrical shell 4.2 Cylindrical enclosure

Consider a cylindrical shell offinite length that is simply supported on both its ends and filled with acoustic fluid Simply supported boundary conditions—sometimes known as freely supported or shear diaphragm boundary conditions—indicate that the internal bending moment, the membrane normal force as well as the radial and circumferential (but not the axial) displacements are all equal to zero at both shell ends For a simply supported shell of lengthℓ the structural mode shapes can be expressed as

G

G

G

m π z

z n θ

n θ

n θ

ℓ

(∂/∂ )cos( ) sin( ) cos( )

,

k

m n

u

m n

v

m n

w

s

s s s

s s

s s

s s

⎧

⎨

⎪⎪

⎩

⎪⎪

⎫

⎬

⎪⎪

⎭

⎪⎪

⎛

⎝

⎞

⎠

⎧

⎨

⎩⎪

⎫

⎬

where the u, v, and w superscripts denote axial, circumferential, and radial components of the modes, and msand nsare the axial and circumferential wavenumbers, respectively.Fig 5displays the in vacuo mode shapes of a simply supported shell for various values of msand ns Because they are accurate across a wide range of practical configurations, Flügge's thin shell theory is chosen as the structural model (Leissa, 1973) SubstitutingGm n s s e iω msns t into Flügge's equations leads to the following bi-cubic characteristic equation:

where the frequency parameter is Ω =ω m n s s a0 ( (1 −ρ ν2)/ )E Here,ρ is the material density, ν is Poisson's ratio, and E denotes Young's modulus The coefficients d0, d1, and d2are lengthy expressions (seeDavis et al., 2008for their complete form) which depend on shell geometry, Poisson's ratio, as well as axial and circumferential wavenumbers For given axial and circumferential wavenumbers, Eq.(32)returns three natural frequencies The modes corresponding to these frequencies are typically dominated by either axial, circumferential, or radial contributions The radially dominant natural frequency (often the lowest of the three) is of interest here because it is the only frequency that is appreciably affected by the presence of the fluid

The rigid wall acoustic natural frequency modes of the cylindrical duct of lengthℓ and inner radius Riare

R

m π

j

n p

i

a

0

a a

⎛

⎝

⎠

⎝

⎞

m π z

i a a

a a a

⎛

⎝

⎠

⎝

⎞

Table 1

The highest Ψ values (and their associated wavenumbers) for a plate that is simply supported on all edges and backed by a rectangular fluid-filled cavity (Note that all

values are tabulated as π Ψ4 for the sake of exactness.)

where ma, naand paare the axial, circumferential and radial wavenumbers, and α n p

a arepresents the roots of the characteristic equation

J n ais the nath-order Bessel function and the prime denotes a spatial derivative Substituting Eq.(34)into Eq.(4)and the radial component of Eq.(31)into Eq.(5)results in the following modal normalization factors:

Fig 5 In vacuo radial mode shapes and nodal pattern for a simply supported cylindrical shell.

Fig 6 Ψ values for the m s = 1 structural modes coupled to some m a = 0 and m a = 2 families of acoustic modes for a simply supported cylindrical shell filled with fluid.