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[.]
Journal of Fluids and Structures 70 (2017) 276–294 Contents lists available at ScienceDirect Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs A simplified approach for predicting interaction between flexible structures and acoustic enclosures MARK R.B Davis College of Engineering, University of Georgia, Athens, GA 30602, USA A R T I C L E I N F O ABSTRACT Keywords: Acoustoelasticity Acoustic–structure interaction Coupled mode theory Eigenvalue veering Acoustic enclosures 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 nondimensional 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 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 the fluid of interest, it is often convenient and accurate to solve the acoustic and structural problems independently (i.e., neglecting any mutual influence between the acoustic fluid 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 the fluid of interest is dense or when the structure is highly flexible 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 of Junger and Feit (1993) and Fahy (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 acoustic fluid, 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 E-mail address: ben.davis@uga.edu http://dx.doi.org/10.1016/j.jfluidstructs.2017.02.003 Received 19 July 2016; Received in revised form 16 November 2016; Accepted February 2017 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/) Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis Nomenclature V W w x, y, z Yna α αnap area of fluid–structure interface radius of shell from origin to middle surface acoustic modal coordinate acoustic speed of sound modal coordinates in coupled mode form Young's modulus coefficients of shell characteristic equation acoustic normal mode system equation coefficients in coupled mode form structural normal mode height of rectangular enclosure thickness of plate nath-order Bessel function roots of characteristic equation for rigid annular a 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 t time u, v axial, circumferential displacement of cylindrical AF a0 aj c0 * D1,2 , D1,2 E d 0,1,2 Fj g Gk H h Jna k nap a βjk Γ Δ δna ηjk ϵ λ nap a κ1,2 ϕ Ψ ρ, ρs ρ0 ν Ω ωc1,2 ωj, ωk shell volume of fluid cavity width of rectangular enclosure displacement of structure in normal direction Cartesian coordinates nath-order Neumann function scaling parameter roots of characteristic equation for rigid cylindrical duct coupling strength parameter radial variation of rigid wall acoustic mode in an annulus dimensionless component frequency separation = if na = 0, = otherwise energy transfer factor = if subscript= 0, = otherwise ⎛ W 2⎞ =π 2⎜ns2 + ms2( H ) ⎟ ⎝ ⎠ dimensionless separation between coupled and uncoupled natural frequencies fluid velocity potential dimensionless coupling parameter density of structure per unit volume, per unit area density of fluid per unit volume Poisson's ratio cylindrical shell frequency parameter coupled natural frequencies uncoupled natural frequency of cavity, structure used to describe changes in wave velocity in elastic structures due to static stress fields.) 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 commercial finite 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 discusses relevant previous work in the areas of acoustoelasticity and coupled mode theory Section derives two approximate closed-form expressions for the coupled natural frequencies of an 277 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis 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 in Section is then investigated in Section for some common acoustoelastic systems Two specific examples in Section illustrate 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 in Section and general conclusions are discussed in Section 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 the fluid 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 of fluid 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 by Pierce (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 by Louisell (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) and Scarpa (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 a finite 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 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, AF The remaining fluid–structure interface is rigid and denoted AR 278 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis 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 a fluid-filled enclosure of arbitrary geometry represented by Fig A quiescent, inviscid, compressible fluid fills a volume V and is coupled to the motion of a flexible 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 ) = ∑ aj(t )Fj, (1) j w (t ) = ∑ qk (t )Gk , (2) k where aj(t) and qk(t) are the time-dependent fluid and structural modal coordinates, respectively Fj denotes the shape of the jth uncoupled mode of the acoustic enclosure while Gk represents 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 Mj[aăj + j2aj ] = c02AF V Mk[qăk + k2qk ] = ρs ∑ Ljkq˙k , (3a) k ∑ Ljka˙j (3b) j Eqs (3) represent a system of gyroscopically coupled ordinary differential equations where c0, ρ0, and ρs are the speed of sound in the fluid, the fluid density, and the mass of the structure per unit area The natural frequencies of the uncoupled acoustic cavity are ωj and the in vacuo natural frequencies of the structure are given by ωk While cavity geometry is irrelevant at this stage, Eqs (3) are written here for an interior acoustic cavity Mj and Mk are the modal normalization factors defined as Mj ≡ V Mk ≡ AF ∫V Fj2dV , ∫A (4) Gk2dA (5) F The experienced researcher may observe that Mk is 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 to ρs in 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 Mk in the manner of Eq (5) aids the forthcoming derivation of a nondimensionalized coupled frequency expression The Ljk terms in Eqs (3) represent coupling coefficients given by Ljk = AF ∫A FjGkdA (6) F Ljk can 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˙j dt daj dt dq˙k dt dqk dt = c02AF Ljk VMj q˙k − ωj2aj , (7a) = a˙j , =− (7b) ρ0 Ljk ρs Mk a˙j − ωk2qk , (7c) = q˙k (7d) Following the procedure outlined by Louisell (1960), Eq (7b) is multiplied by ±iωj and added to Eq (7a) Similarly, Eq (7d) is multiplied by ±iωk and added to Eq (7c) The result is the following system of equations: 279 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis c02AF Ljk dD1 = q˙ − ωj2aj + iωja˙j , dt VMj k (8a) c02AF Ljk dD1* = q˙ − ωj2aj − iωja˙j , dt VMj k (8b) ρ Ljk dD2 = − a˙j − ωk2qk + iωk q˙k , dt ρs Mk (8c) ρ Ljk dD2* = − a˙j − ωk2qk − iωk q˙k , dt ρs Mk (8d) where D1 ≡ (a˙j + iωjaj ), D1* ≡ (a˙j − iωjaj ), D2 ≡ (q˙k + iωk qk ), and D2* ≡ (q˙k − iωk qk ) Eqs (8) can now be written in their coupled mode form: dD1 = g11D1 + g12D2 + g13D1* + g14D2*, dt (9a) dD2 = g21D1 + g22D2 + g23D1* + g24D2*, dt (9b) dD1* = g31D1 + g32D2 + g33D1* + g34D2*, dt (9c) dD2* = g41D1 + g42D2 + g43D1* + g44D2*, dt (9d) where g11 = − g33 = iωj , g12 = g14 = g32 = g34 = g21 = g23 = g41 = g43 = − ρ0 Ljk 2ρs Mk c02AF Ljk 2VMj , g22 = − g44 = iωk , , g13 = g24 = g31 = g42 = (10) By assuming harmonic solutions to Eqs (9) and solving the associated eigenvalue problem, the two system natural frequencies, ωc1,2 , are found: ωc1,2 = ωj2 + ωk2 + βjk ∓ ⎞2 ⎛ ⎜ωj + ωk2 + βjk ⎟ − (2ωjωk )2 , ⎠ ⎝ (11) where βjk = ρ0 c02AF Ljk2 ρs VMjMk (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 dD1 = g11D1 + g12D2 , dt (13a) dD2 = g21D1 + g22D2 , dt (13b) and a second approximate expression for the coupled natural frequencies can be found: ωc1,2 = 1⎛ ⎜ωj + ωk ∓ 2⎝ ⎞ (ωj − ωk )2 + βjk ⎟ ⎠ (14) While this expression is less accurate than Eq (11), it permits a simple nondimensionalization as will be shown in Section 3.3 280 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis 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 −1 ⎡ ⎛ ωj − ωk ⎞2 ⎤ ⎥ ⎢ ηjk = + ⎜ , ⎟ ⎢⎣ ⎠ βjk ⎥⎦ ⎝ (15) where ηjk is a dimensionless quantity that varies between zero and unity A value of ηjk that 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 enclosures filled with air The transfer factor depends on two quantities: (ωj − ωk ) and βjk The quantity (ωj − ωk ) is simply the separation of the component mode frequencies For a given value of (ωj − ωk ), βjk determines the extent to which the two component modes interact By inspection of Eq (12), βjk is comprised of several dimensional parameters that describe the physical properties and geometry of the structure and the cavity (namely, c0, AF, V, and ρs) 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 Ψ≡ Ljk2 MjMk (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 of Ψ for some common geometries and boundary conditions are presented in Section 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 is 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 depicts 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 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 A representative depiction of coupled and uncoupled frequency behavior 281 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis 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 ωj ≤ ωk , the difference between the lower of the two system frequencies, ωc1, and its nearest component frequency, ωj, is found by subtracting ωj from both sides of Eq (14) ωc1 − ωj = 1⎛ ⎜ωk − ωj − 2⎝ ⎞ (ωj − ωk )2 + βjk ⎟ ⎠ (17) Multiplying Eq (17) by the following parameter: α≡ ρs V ρ0 c02AF , (18) recasts the equation in a nondimensional form κ1 = (−Δ − Δ2 + Ψ ) for ωj ≤ ωk , (19) where Δ is the nondimensional frequency separation between the two component modes Δ ≡ α(ωj − ωk ), (20) and where κ1 represents the nondimensional frequency reduction due to the acoustic–structure coupling κ1 ≡ α(ωc1 − ωj ) for ωj ≤ ωk (21) Now assuming that ωj > ωk , κ1 is defined such that it represents the nondimensional difference between ωc1 and ωk κ1 ≡ α(ωc1 − ωk ) for ωj > ωk (22) The nondimensional frequency relationship in this case is κ1 = (Δ − Δ2 + Ψ ) for ωj > ωk (23) Arguments similar to those outlined above are used to find analogous expressions related to ωc2 , the greater of the two system frequencies Upon doing so, κ2 is defined in the following piecewise manner: Fig (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 282 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis ⎧ α(ωc − ωk ) for ωj ≤ ωk κ2 ≡ ⎨ ⎩ α(ωc2 − ωj ) for ωj > ωk , ⎪ ⎪ (24) and the corresponding nondimensional frequency relation is ⎧1 ⎪ ⎪ (Δ + Δ + Ψ ) for ωj ≤ ωk κ2 = ⎨ ⎪ (−Δ + Δ2 + Ψ ) for ω > ω ⎪ j k ⎩2 (25) Note that Eq (11) can also be cast in a nondimensional form by multiplying both sides by α However, the expression does not reduce to a simple dependence on Δ and Ψ due to the presence of cross-terms Eqs (19), (23), and (25) are plotted in three dimensions against Δ and Ψ in Fig 3(a), while Fig 3(b) is a two-dimensional projection of these equations that shows several curves with constant Ψ 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 the Ψ 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 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 the Ψ parameter as it is given by Eq (16) 4.1 Rectangular enclosure The system of a flexible 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 (see Fig 4), these mode shapes are given by ⎛ m πx ⎞ ⎛ n πy ⎞ Gk = sin⎜ s ⎟sin⎜ s ⎟ , ⎝ W ⎠ ⎝ H ⎠ (26) where the modal index k includes indices, ms and 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 ⎛ m πx ⎞ ⎛ n πy ⎞ ⎛ p πz ⎞ Fj = cos⎜ a ⎟cos⎜ a ⎟cos⎜ a ⎟ , ⎝ W ⎠ ⎝ H ⎠ ⎝ L ⎠ (27) 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 Mj = , εmaεnaε p (28a) , (28b) a Mk = where ε 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 Schematic of a flexible plate backed by a rectangular enclosure 283 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis orthogonality of certain component mode shapes, the coupling coefficient will be identically zero if ms + ma or ns + na is even In all other cases, the coupling coefficients are given by Ljk = msns((−1)ma + ms − 1)((−1)na + ns − 1) π 2(ma2 − ms2 )(na2 − ns2 ) (29) Substituting Eq (28) and Eq (29) into Eq (16) leads to the following expression for Ψ: Ψ= 4εmaεnaε p [msns((−1)ma + ms − 1)((−1)na + ns − 1)] a π 4[(ma2 − ms2 )(na2 − ns2 )] (30) Table lists the component mode combinations that result in the ten highest values of Ψ In all cases, Ψ assumes the given value provided that pa ≥ For plates that are not simply supported on all edges, such an exact statement of Ψ 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 of finite 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 length ℓ the structural mode shapes can be expressed as ⎧Gmu n ⎫ ⎧ (∂/∂z )cos(nsθ )⎫ ⎪ ⎛ msπ ⎞⎪ ⎪ ⎪ ⎪ vs s ⎪ G Gk = ⎨ msns ⎬ = sin⎜ z⎟⎨ sin(nsθ ) ⎬, ⎠ ⎝ ℓ ⎪ ⎪ ⎪ ⎪ ⎪Gmw n ⎪ ⎩ cos(nsθ ) ⎭ ⎩ s s⎭ (31) where the u, v, and w superscripts denote axial, circumferential, and radial components of the modes, and ms and ns are the axial and circumferential wavenumbers, respectively Fig displays the in vacuo mode shapes of a simply supported shell for various values of ms and 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) Substituting Gmsnseiωmsnst into Flügge's equations leads to the following bi-cubic characteristic equation: Ω + d 2Ω + d1Ω + d = 0, (32) where the frequency parameter is Ω = ωmsnsa (ρ(1 − ν )/ E ) Here, ρ is the material density, ν is Poisson's ratio, and E denotes Young's modulus The coefficients d0, d1, and d2 are lengthy expressions (see Davis et al., 2008 for 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 length ℓ and inner radius Ri are ⎛ αnapa ⎞2 ⎛ maπ ⎞2 ωj = c0 ⎜ ⎟ +⎜ ⎟ , ⎝ ℓ ⎠ ⎝ Ri ⎠ (33) ⎛ ⎛m π ⎞ r ⎞ Fj = Jna⎜αnap ⎟cos(naθ )cos⎜ a z⎟ , aR ⎝ ℓ ⎠ ⎝ i⎠ (34) Table 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 4Ψ for the sake of exactness.) ms ns ma na pa π 4Ψ 1 2 2 0 1 1 ≥1 ≥1 ≥1 ≥1 ≥1 ≥1 ≥1 ≥1 ≥1 ≥1 128 1024/9 1024/9 8192/81 2304/25 2304/25 4096/49 4096/49 2048/25 2048/25 284 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis Fig In vacuo radial mode shapes and nodal pattern for a simply supported cylindrical shell where ma, na and pa are the axial, circumferential and radial wavenumbers, and αnap represents the roots of the characteristic a equation J ′na (αnap ) = (35) a Jna is 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 Ψ values for the ms = structural modes coupled to some ma = and ma = families of acoustic modes for a simply supported cylindrical shell filled with fluid 285 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis Mj = Mk = ⎛ ⎛ ⎞2 ⎞ na ⎜ ⎜ ⎟⎟ − ⎜ α + δ ⎟ ⎟⎟Jna(αnapa ), εnaεma ⎜⎜ napa na ⎠ ⎝ ⎝ ⎠ (36) a0 ≈ , 2εnsRi 2εns (37) where δna is the Kronecker delta function (i.e., δna = if na = 0, and δna = otherwise) It is noted that for thin shells, a ≈ Ri Substituting Eqs (31) and (34) into Eq (6) gives the coupling coefficients for this system Ljk = Jna(αnap )ms(( − 1)ma + ms − 1) a εnsπ (ma2 − ms2 ) (38) Next, substituting Eqs (36)–(38) into Eq (16) results in an expression for Ψ that depends solely on component modes wavenumbers and the roots of the characteristic equation for a rigid cylindrical duct Ψ= 2εmams2((−1)ma + ms − 1) ⎛ ⎛ ⎞2 ⎞ n ⎟ 2 2⎜ a ⎟ π (ma − ms ) ⎜1 − ⎜⎜ ⎟⎟ ⎜ ⎝ αnapa + δna ⎠ ⎟⎠ ⎝ (39) With the aid of Eq (39) and a knowledge of the component mode frequencies, it is possible to use Eq (11) or Eq (14) to arrive at an approximation of the system frequencies of the fluid-filled shell Note that the derivation of Eq (39) includes the approximation that a0—the radius of the shell measured from the origin to the middle surface—is equal to the acoustic radius of the duct, Ri (i.e., h ≪ a ) Fig uses Eq (39) to plot values of Ψ for the first family of structural modes (i.e., ms = 1) coupled to ma = and ma = families of acoustic modes with various radial wavenumbers The plot indicates that, ceteris paribus, rigid wall acoustic modes of high circumferential wavenumber and low radial wavenumber have a greater propensity to couple to the first family of structural modes Simply supported boundary conditions represent a special case in plate and shell dynamics When all edges of a plate or both ends of a shell are simply supported, the structure's in vacuo modes can be expressed exactly with trigonometric functions This is not the case for plates and shells possessing other boundary conditions In such cases, the modal normalization factors, the coupling coefficients, and consequently, the values of Ψ cannot be expressed with exact expressions For a plate with one or more edges that are not simply supported, the in vacuo mode shapes are only constant for fixed aspect ratios Similarly, when one or both ends of a shell are not simply supported, the mode shapes are only constant for fixed values of a / h , ℓ/ a , and Poisson's ratio, υ To determine the in vacuo natural frequencies and mode shapes of shells with various boundary conditions, the analytical procedure of Callahan and Baruh (1999) is implemented Fig assesses the effect that boundary conditions have on a shell's ability to couple with certain acoustic modes The figure contains Ψ values for the ms = family of structural modes coupled to the ma = 0, pa = family of acoustic modes for four common structural boundary conditions: simply supported on both ends (SS–SS), clamped on one end and simply supported on the other (C–SS), clamped on both ends (C–C), and clamped on one end and free on the other (C–F) In all cases, the fluid is assumed to have rigid boundary conditions at both shell ends The Ψ values corresponding to the ms = family of Fig Ψ values for the ms = and ms = families of structural modes coupled to the ma = pa = family of acoustic modes for a fluid-filled cylindrical shell with common boundary conditions SS, C, and F denote simply supported, clamped, and free boundary conditions, respectively 286 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis structural modes coupled to the ma = 0, pa = acoustic family are also plotted, but with the exception of the C–F case, these values are either identically, or effectively, zero While the Ψ values corresponding to the SS–SS shell are independent of shell geometry and material properties, Ψ values for shells with other boundary conditions are calculated based on a shell with a / h = 20 , ℓ/ a = , and υ = 0.3 Parametric studies indicate that values of Ψ not vary significantly across realistic ranges of a0/h, ℓ/a0, and υ Furthermore, Fig indicates that shell boundary conditions not appear to have a substantial impact on the values of Ψ It may be reasonable to use the Ψ values corresponding to a simply supported shell—i.e., Eq (39)—when approximating the system frequencies of a fluid-filled cylindrical shell possessing other boundary conditions Doing so would allow one to estimate system frequencies by hand and without explicit knowledge of the in vacuo structural mode shapes The accuracy of using this approach when approximating the system frequencies of a fluid-filled clamped–free shell is shown in Section 4.3 Annular enclosure The coupling capacity of fluid-filled annular enclosures is now considered A rigid outer cylinder of radius Ro bounds an annulus of acoustic fluid At the inner radius, Ri, the fluid is in contact with a simply supported cylindrical shell The ends of the enclosure are both modeled as rigid Calculating the structural and acoustic modal masses and their coupling coefficient leads to the following expression for Ψ (Davis, 2008): ⎞2 ⎛ ⎛ ⎞ ⎜ Γ ⎜k napa⎟ms((−1)ma + ms − 1) ⎟ ⎛ ⎠ ⎟ ⎜ ⎝ ⎜2 × Ψ=⎜ ⎟ ⎜V π (ma2 − ms2 ) ⎝ ⎟ ⎜ ⎠ ⎝ ℓ 2π ∫0 ∫0 ∫R ⎞ ⎛ ⎛ m π ⎞⎤ r ⎞ ⎢Γ ⎜k n p ⎟cos(naθ )cos⎜ a z⎟⎥ rdrdθdz⎟ , ⎟ ⎢⎣ ⎝ a a Ri ⎠ ⎝ ℓ ⎠⎥⎦ ⎠ Ro ⎡ i (40) where the volume integral is calculated numerically and k napa are the roots of the characteristic equation ⎡ ⎛ ⎛ ⎞⎤ ⎞ ⎛ R ⎞ ⎛ R ⎞ k n2ap ⎢Y ′na ⎜k nap o ⎟J ′na ⎜k nap ⎟ − J ′na ⎜k nap o ⎟Y ′na ⎜k nap ⎟⎥ = a⎢ aR a a a ⎠⎥⎦ ⎠ ⎝ ⎝ Ri ⎠ ⎝ ⎝ ⎣ i⎠ (41) Here, Yna is the na-order Neumann function The function, Γ, represents the radial dependance of the rigid wall acoustic mode shape and is given by ⎛ R ⎞ J ′na ⎜k nap o ⎟ a ⎛ ⎛ Ri ⎠ ⎛ ⎝ r ⎞ r ⎞ r ⎞ Γ ⎜k nap ⎟ ≡ Jna⎜k nap ⎟ − Yn ⎜k n p ⎟ ⎛ ⎝ a Ri ⎠ ⎝ a Ri ⎠ Ro ⎞ a⎝ a a Ri ⎠ Y ′na ⎜k nap ⎟ ⎝ a Ri ⎠ (42) Fig graphs values of Ψ versus circumferential wavenumber and the ratio of Ri to Ro These Ψ values quantify the capacity of the (0,na,1) family of acoustic modes to couple to the (1,ns) family of structural modes Ψ values are calculated for Ri/Ro ratios as high as 0.99 and are found to quickly increase as Ri/Ro approaches unity This is consistent with previous observations that very thin fluid cavities have a strong propensity to couple to adjacent structures (Dowell and Voss, 1963) It is also observed from Fig that the Ψ values related to modes with circumferential wavenumbers of zero are equal to 0.81 and are independent of the Ri/Ro ratio To assess the capacity of the (0,na,1) family of component modes to couple to structural modes with higher axial wavenumbers, the Ψ values in Fig need to be multiplied by constant factors For the (3,ns) family of modes the appropriate factor is 0.11 The factor corresponding to the (5,ns) family is 0.04 Thus, the propensity of the (0,na,1) modes to couple quickly diminishes with an increase Fig Ψ values for the (1,ns) family of structural modes coupled to the (0,na,1) family of acoustic modes for a fluid-filled annular enclosure 287 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis in the axial wavenumber of the structure (Note that the (0,na,1) modes not couple to the (2,ns) and (4,ns) modes because for a simply supported shell coupled to a fluid enclosure with rigid ends, ma and ms must differ by an odd number in order for the coupling coefficient to be non-zero.) Examples It will now be shown that known values of Ψ can be used to hand-calculate acoustoelastic natural frequencies The approach— referred to as the coupled mode method (CMM)—involves using Eq (11) to obtain system frequency predictions However, instead of calculating the integrals needed to find the coupling coefficients and component modal masses, the method uses appropriate values of Ψ, such as those found in Section With use of these Ψ values, Eq (11) reduces to a simple hand calculation To assess accuracy of the CMM, the results are compared to results from a finite element acoustoelastic analysis that uses the commercial software program, ANSYS (Note that in ANSYS, acoustoelastic analyses are termed fluid–structure interaction (FSI) analyses.) Two examples are presented and discussed: (1) a simply supported plate coupled to an air-filled rectangular enclosure and (2) a clamped– free cylindrical shell filled with water Each example is designed to illustrate specific capabilities and limitations of the CMM 5.1 Simply supported plate as one side of a rectangular enclosure Consider the rectangular enclosure shown in Fig On the end of the enclosure coinciding with the XY plane is an aluminum (E = 69 GPa, ρ = 2768 kg/m3, ν = 0.3) plate that is simply supported on all four edges The box is filled with air (ρ0 = 1.21 kg/m3, c0 = 343 m/s) and shares a width of W = 0.91 m and a height of H = 0.31 m high with the plate The acoustic natural frequencies of a rigid rectangular enclosure are (Blackstock, 2000) ⎛ m ⎞2 ⎛ n ⎞2 ⎛ p ⎞2 ωj = ωmanap = πc0 ⎜ a ⎟ + ⎜ a ⎟ + ⎜ a ⎟ a ⎝L⎠ ⎝H⎠ ⎝W ⎠ (43) The in vacuo structural natural frequency is given by (Blevins, 2015) ωk = ωnsms = λ n2sms W Eh2 , 12ρ(1 − ν ) where h is the plate thickness and λ n2sms βjk = ρ0 c02 ρhL (44) ⎛ W 2⎞ = π 2⎜ns2 + ms2( H ) ⎟ For this geometry, the coupling parameter β is ⎝ ⎠ Ψjk , (45) where the value of Ψjk can be calculated using Eq (29) or Table For a given mode pair of interest, Eqs (43)–(45) are calculated and substituted into Eq (11) to obtain predictions of the two associated acoustoelastic natural frequencies Acoustoelastic natural frequencies are calculated for two different control parameters with results shown in Figs and 10 In Fig 9, the length of the enclosure is fixed at L = 1.52 m while the thickness of the plate is varied from 0.1 mm to 6.0 mm The Fig Natural frequency map for the (1,1) mode of a simply supported plate coupled to a rectangular enclosure with varying plate thickness, h 288 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis coupling between the (0,0,1) acoustic mode and the (1,1) plate mode is considered From Table 1, the Ψ value associated with this mode pair is 128/ π ≈ 1.314 The in vacuo natural frequency of the plate increases linearly with the thickness, h, while the uncoupled acoustic natural frequencies are independent of plate geometry Inspection of Eq (45) reveals that the coupling parameter, β, will be inversely proportional to thickness, thus verifying the intuitive expectation that there will be stronger interaction between the structure and the fluid when the plate is thin The predicted acoustoelastic natural frequencies are shown in Fig for both the CMM and the ANSYS FSI model Note that the ANSYS solution predicts several other acoustoelastic natural modes in the plotted frequency range; however, since only the (0,0,1) acoustic mode and the (1,1) plate mode are of interest, only the modes in which the structural potion of the system exhibits a shape consistent with the (1,1) mode are shown At a plate thickness of approximately mm, the uncoupled frequencies coincide and the predicted acoustoelastic frequencies clearly exhibit curve veering Despite the fact that the mass of the plate in this case is approximately six times that of the enclosed air, the effects of the fluid presence are not negligible for plate thickness in which the component frequencies nearly coincide When h = mm, the difference between the lower acoustoelastic frequency and its nearest component frequency is 6.8% This difference could be important to consider in a forced response analysis In a classical analysis, one might solve the uncoupled acoustic response of the system and apply it as a forcing on the plate This approach would lead to a large resonant response at the frequency of uncoupled mode coincidence In reality, the system will exhibit two resonances at frequencies on either side of the coincidence Additionally, since the coupled modes store energy in both the fluid and structural portions of the system mode, the resonant response of the coupled structure (or the fluid) will be less than what would be predicted by an uncoupled analysis Excellent agreement can be observed between the ANSYS FSI results and the results that were hand calculated using the CMM Using the (1,1) and (0,0,1) component modes with plate thicknesses of mm and above, the average absolute difference between the ANSYS and the CMM solution is just 0.49% across both branches of acoustoelastic frequencies For thicknesses below mm, there is a noticeable discrepancy between the ANSYS results and the lower of the two CMM frequencies when the (1,1) and (0,0,1) component modes are used This can be explained by recalling that for closed acoustic cavities with entirely rigid boundaries, there exists a (0,0,0) mode that has a zero frequency (analogous to a rigid body mode in structural dynamics) This mode is sometimes called the constant pressure mode Now consider the shape of the (1,1) plate mode This mode comprises a half-sine wave in both dimensions, and is unique because all points on the plate displace in the same direction This is not true for higher modes in which more than one half-sine waves set up in the plate Thus, as the plate vibrates into the cavity in the (1,1) mode, its vibration is opposed at all locations by the presence of the fluid This opposition can be modeled as an interaction of the (1,1) plate mode with the (0,0,0) constant pressure fluid mode The consequence of this interaction is an effective stiffening of the (1,1) plate mode This is sometimes called the Helmholtz stiffening effect For increasingly thin plates, the uncoupled frequency of the (1,1) mode becomes more separated from the (1,0,0) acoustic mode and closer to the zero-frequency (0,0,0) mode This results in a non-negligible interaction with the (0,0,0) and an increase in the coupled natural frequency of the structurally dominated (1,1) mode To capture this interaction, the CMM was applied using the (1,1) and (0,0,0) modes, and the non-zero frequency results are also shown in Fig The Ψ value for this component mode pair was found to be 64/ π ≈ 0.657 using Eq (30) The CMM that uses the (1,1) and (0,0,0) component modes is able to predict the Helmholtz stiffening effect for plate thicknesses below mm, though the agreement deteriorates with decreasing plate thickness At low thicknesses, the upper branch of acoustoelastic natural frequencies is observed to deviate from the component acoustic frequency The deviation of these acoustically dominated coupled frequencies is well-predicted by the CMM (at least for the plate thicknesses considered) This observed increase in acoustoelastic frequencies is explained by considering the fact that as thickness of Fig 10 Natural frequency map for the (1,1) mode of a simply supported plate coupled to a rectangular enclosure as a function of enclosure length, L 289 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis the plate decreases, the plate becomes less acoustically rigid In the zero thickness limit, the plate provides no impedance to the fluid and the acoustic system is best modeled as a rigid box with an open end Making the approximation that the open end represents a pressure-release boundary condition, the uncoupled acoustic frequencies are given by ⎛ m ⎞2 ⎛ n ⎞2 ⎛ (2p + 1) ⎞2 ωj = ωmanap = πc0 ⎜ a ⎟ + ⎜ a ⎟ + ⎜ a ⎟ a ⎝H⎠ ⎝W ⎠ ⎠ ⎝ 2L (46) For the example system, the frequency associated with the (0,0,1) closed-open mode is 169 Hz In the limit as plate thickness goes to zero, the upper branch of acoustoelastic frequencies is expected to approach this value, and indeed, the ANSYS model can demonstrate this However, as thickness decreases, the CMM will calculate an upper branch of acoustoelastic natural frequencies that increases without bound Therefore, in instances in which plate thickness is very small, it would be advisable to model the system as closed–open for the purpose of computing the component acoustic frequencies and the nondimensional coupling parameter, Ψ Fig 10 shows the acoustoelastic frequency behavior of the system when plate thickness is fixed at h = mm and the length of the enclosure is the control parameter In this case, the in vacuo natural frequencies of the plate remain constant while the uncoupled longitudinal acoustic natural frequencies are inversely proportional to the enclosure length The length of the enclosure is varied from 0.2 m to m Only the ANSYS frequencies for which the structural portion of the system mode exhibits a shape consistent with the (1,1) mode are shown At a enclosure length of approximately 1.5 m, the uncoupled (0,0,1) acoustic frequency intersects the line representing the (1,1) plate frequency At twice this length, the structural frequency intersects (0,0,2) acoustic frequency Two sets of acoustoelastic natural frequencies are calculated using the CMM The coupling between the (1,1) plate mode and the (0,0,1) acoustic mode is considered in one case, while the coupling with the (0,0,2) mode is considered in the other According to Table 1, both pairs of component modes have a Ψ value of 128/π4 Note that the Helmholtz stiffening effect can also be observed here for small enclosure lengths It is also observed that the extent of curve veering is somewhat greater near the first component mode frequency intersection than it is near the second This can be explained by noting in Eq (45) that the coupling strength parameter, β, is inversely proportional to enclosure length 5.2 Cylindrical shell filled with water This example is designed to demonstrate two features of the CMM First, it shows that precise knowledge of the nondimensional coupling parameter, Ψ, may not be necessary to obtain acceptable estimates of acoustoelastic frequencies This is important because for many geometries or boundary conditions, Ψ values will not be known a priori, and a rigorous calculation of these values requires spatial integration over the component mode shapes which may require a non-trivial effort However, if a reasonable estimate of Ψ can result in acceptably accurate acoustoelastic frequency predictions, then the process of obtaining these predictions reduces to a simple hand calculation Second, by considering water as the fluid medium, this example illustrates that it is possible to use the coupled mode approach to obtain reasonably accurate natural frequencies predictions for shells in contact with dense fluids While fluid added mass formulae for cylindrical shells can be found in standard references (Blevins, 2015), the formulae are cumbersome These formulae assume that the fluid is incompressible and that the shell is simply supported on both ends The CMM offers a simpler way of approximating fluid-loaded natural frequencies and can be easily adapted for use with structures of different geometries and boundary conditions Consider a clamped–free steel (E = 200 GPa, ρ = 7700 kg/m3, υ = 0.3) cylindrical shell filled with water (ρ0 = 1000 kg/m3, c0 = 1483 m/s) The shell has a radius of a0 = 10.5 cm, a length of 12.6 cm and a thickness of 1.05 cm Fig 11 Ψ values for each of the three sets consider in the water-filled shell example 290 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis The ratio of the mass of the enclosed fluid to the mass of the structure is 0.65 The in vacuo frequencies and their corresponding circumferential wavenumbers are calculated for the ms = family of modes while the uncoupled frequencies of the (0,na,1) acoustic mode were directly computed with Eq (33) The first branch of structurally dominated acoustoelastic frequencies are calculated using three sets of Ψ values Each successive set represents an additional level of analytical refinement The three sets are as follows: (1) Values denoted Ψ1 correspond to a simply supported shell and are calculated with Eq (39) (2) Values denoted Ψ2 correspond to a clamped–free shell with a / h = 20, ℓ/ a = 2, and υ = 0.3 These are the same Ψ values presented in Fig for a clamped–free shell (3) Values denoted Ψ3 correspond to the specific shell considered here, namely, a clamped–free shell with a0/h = 10, ℓ/a0 = 1.2, and υ = 0.3 The three families of Ψ values are plotted versus circumferential wavenumber in Fig 11 Note that the values of Ψ2 and Ψ3 are very similar Since the values of Ψ1 and Ψ2 can also be found in Fig 7, only the Ψ3 values need to be computed This example assesses whether the additional effort required to calculate the Ψ3 values results in more accurate acoustoelastic frequency predictions The accuracy of the acoustoelastic frequency predictions is assessed by comparing them to the output of an ANSYS FSI model Fig 12 shows the calculated component frequencies together with the lowest family of structurally dominated system frequencies for the water-filled shell The solid line connects the system frequencies calculated by the ANSYS model Fig 13 displays the percent errors associated with each of the three approximate calculations at each of the considered circumferential wavenumbers Across all values of Ψ and circumferential wavenumber, the percent errors associated with the coupled mode approach generally decrease with increasing circumferential wavenumber The relatively less accurate predictions at lower circumferential wavenumbers can be explained by considering the relative frequency proximity of higher-order acoustic modes at these lower wavenumbers and recalling that coupling effects associated with these higher-order modes are not considered in the CMM The average absolute error across all circumferential wavenumbers was 3.1%, 3.7%, and 3.6% for Ψ1, Ψ2, and Ψ3, respectively This indicates that the CMM approach provides reasonable fluid-loaded frequency estimates, even in the presence of a dense fluid such as water It also indicates that the additional effort required to obtain analytically refined values of Ψ does not improve accuracy In present example (and presumably in similar examples), it would be acceptable to estimate system frequencies using the easily obtainable Ψ1 values Accuracy and limitations of the CMM The two examples demonstrate that the CMM is capable of providing accurate estimates of acoustoelastic natural frequencies However, the examples little to address the general accuracy of the CMM and the limits of its applicability Recognizing that the utility of the CMM arises from the ability to apply it quickly and easily without resorting to more robust methods, it is helpful to know a priori the situations in which the approach will yield reasonable results While the accuracy of the approach will be system dependent, some general observations can be made to help guide its application Since the CMM considers just a single pair of component modes, its accuracy will ultimately depend on how strongly a given component mode interacts with all of the remaining—i.e., neglected—component modes Identifying the neglected component modes and calculating their capacity to interact with the component mode of interest somewhat defeats the purpose of an approximate approach However, one potential way to assess the extent of the contribution of these neglected component modes is to individually pair them with the component mode of interest and compute the corresponding energy transfer factor, η (see Eq (15)) If the resulting η values are well below the η value for the retained pair of modes, then the CMM will likely produce accurate system Fig 12 Families of component and system natural frequencies for a clamped–free cylindrical shell filled with water The solid line connects the system frequencies calculated by the ANSYS FSI model The markers indicate the frequencies calculated using the coupled mode method with various values of Ψ 291 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis Fig 13 Percent error of the CMM associated with using each set of Ψ values frequencies More specific guidance on the limitations of the CMM is possible by considering how its accuracy relates to the quantities comprising the energy transfer factor, η Recall that η depends on the component mode frequency separation and the β parameter The relationship between the accuracy of the CMM and these two quantities is assessed by performing a series of randomly sampled simulations The simulations consider simply supported fluid-filled shells with rigid fluid boundary conditions at each end During each simulation, random components are added to the various physical parameters of the structure (i.e., shell radius, length, thickness, Young's modulus, Poisson's ratio, and density) and the fluid (i.e., density and speed of sound) For any given simulation, each parameter value is random within predefined limits The limits are set to correspond to values typical of engineering structures and fluids For example, the material properties associated with the shell structure are constrained such that they assume random values that fall between the material properties associated with aluminum and steel The mean value of β over all simulations is found to be 1.3 × 108 s−2 Three thousand simulations are performed Within each simulation, system frequencies are calculated for circumferential wavenumbers ranging from one to ten A total of 30,000 system frequencies are thus considered At each circumferential wavenumber, the lowest system frequency is calculated using Eqs (11) and (14) and then compared to the system frequency obtained from a well-converged modal expansion The modal expansion is formulated according to Dowell's acoustoelasticity approach (Dowell et al., 1977) and retains a total of 240 acoustic component modes for each component structural mode A convergence study revealed that the retention of this number acoustic modes results in system frequencies that are converged to well less than 1% (Davis, 2008) Fig 14 The absolute percent errors associated with the CMM as determined from a series of randomized simulations using (a) Eq (11) and (b) Eq (14) Errors are calculated by comparing approximate frequencies to the system frequency values obtained from a well-converged modal expansion 292 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis Fig 14 displays the percent errors associated with (a) Eq (11) and (b) Eq (14) The figures are truncated to include only those system frequencies with Δ values between −5 and 10 and β values less than × 108 s−2 The 23,812 points that meet these criteria are displayed in Figs 14(a) and (b) The percent errors are displayed as absolute values In some cases, Eq (14) results in very large percent errors, but in order to maintain identical color scales between Figs 14 (a) and (b), the cases for which Eq (14) results in percent errors greater than 20% are each displayed using the same dark red color In all observed cases, the full expression, Eq (11), overestimates system frequency while the weak expression, Eq (14), results in an underestimate In general, both approximate frequency expressions are more accurate for low values of β and for component frequency separations that are large in magnitude High values of β are indicative of (among other things) a system in which the mass of the enclosed fluid is high The fact that the accuracy of the approximate expressions decreases with an increase in the value of β is thus consistent with the results of the example problems In the example of the air-filled rectangular enclosure, typical β values are on the order of 105 s−2 with typical errors of less than 0.5% In the water-filled shell example, β values are on the order of 108 s−2 with average errors of about 3.5% The CMM is therefore not recommended for configurations that have high β values Based on Fig 14, an analyst might reasonably expect Eq (11) to be accurate to within 10% when β < × 108 s−2 Eq (14) is similarly accurate up to β values of × 107 s−2 The more limited applicability of Eq (14) is not especially consequential in practice because the weak expression primarily serves as a means by which to gain physical insight into the general nature of acoustic–structure systems It is not recommended for use in cases where the natural frequencies of a specific system are sought Conclusion The theoretical results presented here enable some revealing observations concerning the interaction of structures with acoustic enclosures: (1) The nondimensionalization of the weak system frequency expression provides a simple relationship that casts system frequency augmentation or reduction as a function of the dimensionless component frequency separation, Δ, and the natural propensity of the component modes to couple, Ψ This relationship is independent of system properties For fixed values of component frequency separation and Ψ, the dimensional system frequency will scale according to 1/ α However, changing any of the parameters that comprise α will correspond to changes to the component frequencies, which will almost certainly prompt a change in their separation (2) The nondimensional Ψ parameter allows for a direct calculation of the fundamental acoustic–structure coupling capacity between any pair of component modes Not only does Ψ allow one to compare the coupling capacity of modal pairs in a single system, but it also allows for these comparisons to be made across systems with dissimilar geometries and/or dissimilar boundary conditions While keeping in mind that component frequency separation plays an important role in determining system frequencies, analysts can potentially use the Ψ parameter as an efficient means of optimizing system designs such that they minimize or maximize acoustic–structure coupling (3) In general, computing Ψ values requires explicit knowledge of the component mode shapes Finding these mode shapes and then computing the integrals necessary to calculate Ψ is an eminently tractable, yet potentially time-consuming, task The rectangular enclosure example presented in Section demonstrates that it is possible to obtain accurate system frequency calculations by hand using tabulated or easily calculated values of Ψ The cylindrical shell example shows that accurate system frequency results can be obtained for structures containing dense fluids and demonstrates that it may be possible to achieve acceptable results by using readily available values of Ψ in lieu of more analytically refined values In a typical use case, one might perform a finite element analysis to calculate the uncoupled acoustic and structural natural frequencies and then use estimated Ψ values to arrive at good approximation of the coupled system frequencies This approximation may serve as a verification check—or even a substitute for—more complex analysis models Acknowledgments This work was completed with support from NASA Contract NNM05ZA07H and start-up funding furnished to the author by the University of Georgia Research Foundation The author would like to thank Sarah Signal, an undergraduate student at the University of Georgia, who assisted with the creation of Figs and and also proofread the manuscript References Blackstock, D.T., 2000 Fundamentals of Physical Acoustics Wiley, 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39 (4), 548–557 Vidoli, S., Vestroni, F., 2005 Veering phenomena in systems with gyroscopic coupling J Appl Mech 72 (5), 641–647 Warburton, G.B., 1961 Vibration of a cylindrical shell in an acoustic medium J Mech Eng Sci (1), 69–79 294 ... and Shirahatti, 1994) expressed as a generalized modal mass and thereby includes a term analogous to ρs in its integrand Here, the structural mass density is assumed to be spatially uniform and. .. 284 Journal of Fluids and Structures 70 (2017) 276–294 R.B Davis Fig In vacuo radial mode shapes and nodal pattern for a simply supported cylindrical shell where ma, na and pa are the axial, circumferential... circumferential and radial wavenumbers, and αnap represents the roots of the characteristic a equation J ′na (αnap ) = (35) a Jna is the nath-order Bessel function and the prime denotes a spatial derivative