SOUND ANSMISSION HOUGH WO CONCENIC CYLINDICAL SANDWICH SHELLS Yvette Y. ang, esearch Associate National esearch Council NASA esearch Center, MS 463, Hampton, VA 23681 ichard J. Silcox, Assistant Branch Head and Jay H. obinson, esearch Engineer Structural Acoustic Branch NASA esearch Center, MS 463, Hampton, VA 23681 ABSAC his paper solves the problem of sound transmission through a system of two infinite concentric cylindrical sandwich shells. he shells are surrounded by external and internal fluid media and there is fluid (air) in the annular space between them. An oblique plane sound wave is incident upon the surface of the outer shell. A uniform flow is moving with a constant velocity in the external fluid medium. Classical thin shell theory is applied to the inner shell and first-order shear deformation theory is applied to the outer shell. A closed form for transmission loss is derived based on modal analysis. Investigations have been made for the impedance of both shells and the transmission loss through the shells from the exterior into the interior. esults are compared for double sandwich shells and single sandwich shells. his study shows that (1) the impedance of the inner shell is much smaller than that of the outer shell so that the transmission loss is almost the same in both the annular space and the interior cavity of the shells; (2) the two concentric sandwich shells can produce an appreciable increase of transmission loss over single sandwich shells especially in the high frequency range; and (3) design guidelines may be derived with respect to the noise reduction requirement and the pressure in the annular space at a mid-frequency range. NOMENCLAUE a,b, : thickness and width of intrinsic cell material of honeycomb core c : sound speed in fluid D : displacement component along radial direction E, G, µ : elastic constants of the face sheets and honeycomb core of sandwich shell f : frequency and mode dependent function h, : thickness and radius of sandwich shells H, J : Hankel and Bessel functions i : -1 k : wave number L : differential operator M : Mach number n : mode number p : pressure in the external, annular, and internal cavities r, z, θ : cylindrical coordinates along radial, axial, and circumferential, directions S : cross area for annular or interior cavity t : time L : transmission loss u, v, w : displacement components along axial, circumferential, and radial directions V : velocity of the external flow W : power flow per unit length x, y, z : Cartesian coordinates Z : modal impedance γ : incident angle of the plane sound wave : gradient ε : Neumann factor ζ : local coordinate along thickness direction of shell ρ : mass density of fluid and shell ψ : rotation of the shell ω : angular or rotational frequency 1. INODUCION Sound transmission through double walls separated by an airgap has been investigated by many researchers [1 to 5, 7, 1 to 14, 16, 18, 19, 22 to 24] since the construction will produce both noise attenuation and thermal insulation. hese investigations have shown that double walls can increase transmission loss over a single wall. Among these studies, most of them assumed the double walls are flat with absorbent materials. Only a few focused on the study of double walls consisting of two concentric cylindrical shells. However, those investigations are mostly limited to the study of two isotropic shells and only thin shell theory is applied or to the use of the finite element method for composite double wall cylinders. Sandwich structures consisting of lightweight, flexible cores between relatively stiff skins can be used to increase the sound insulation [4, 6, 11, 2, 21]. he authors have studied sound transmission through single cylindrical sandwich shells with honeycomb core analytically. esults show that the sandwich shells can offer advantage over isotropic shells for noise reduction, especially at high frequencies [2]. In the aerospace and marine applications, cylindrical shells often occur as part of double shell constructions for the purpose of noise insulation, streamlining requirements, thermal shield, or interior finish requirements. he outer shell represents the exterior skin of an aircraft fuselage and the inner shell represents the trim panel in the aerospace application. In the marine application, the outer shell is thin to satisfy streamline requirements while the inner shell is thicker to withstand underwater pressures.
he objective of this paper is to study noise transmission through a system of two infinite concentric cylindrical sandwich shells excited by an incident oblique plane sound wave analytically. he shells are surrounded with fluids and there is air in the annular space between them. A uniform flow is moving with a constant velocity in the external fluid medium. Each of the sandwich shells is made of honeycomb core and face sheets that may be isotropic, orthotropic, or laminated fiber-reinforced composite materials. In this paper, we will focus on the aerospace application in which the outer shell is a thick shell while the inner is thin. he effect of the shear deformation and rotation can not be neglected for a thick shell [15, 17, 2]. herefore, the first-order shear deformation theory is applied for the outer shell. For the inner shell, the classical thin shell theory is used. he concept under study is that a plane acoustic wave is incident and is reflected by the elastic shell in the exterior space, standing waves that consist of transmitted and reflected waves exist in the annular space, and transmitted waves exist in the interior cavity. o develop the solution, modal analysis is used to solve the coupled equations simultaneously including the convective wave equation for the external fluid, the wave equation for the annular and internal fluids, and the vibration equation for both outer and inner shells. A closed form expression for the transmission loss (L) is derived. Calculations have been carried out for the impedances of both shells and the L through the shells. Finally, comparisons of the L between the double sandwich shells and single sandwich shells are made. 2. MAHEMAICAL ANALYSIS 2.1 Governing equations Figure 1 shows a schematic of two infinite concentric cylindrical sandwich shells with radii 1 and 2 and wall thicknesses h 1 and h 2 for the outer and inner shells, respectively. he shells are surrounded by the external fluid and the internal fluid including that in the airgap (annular fluid) between them. he mass density and sound speed for the external, annular, and internal fluid media are {ρ 1, c 1 }, {ρ 2, c 2 } and {ρ 3, c 3 }. An oblique plane sound wave p I is incident upon the system from the exterior of the shells with incident angle γ 1 (measured from the axial coordinate z). An airflow in the external fluid medium is moving with a constant velocity V along z direction. Without loss of generality, the angles of the incident wave with respect to the axes x and y are and π/2, respectively. In the exterior space, the pressure p 1 = p I + p 1, where p I is the incident wave and p 1 is the reflected wave, satisfies the convected wave equation c 1 2 2 (p I + p 1 ) + t + V. 2 (p I + p 1 ) = (1) where the gradient and 2 =. the Laplacian operator. he pressures here and in the following represent the perturbation pressures. In the annular space, the pressure p 2 = p 2 + p 2, where p 2 is the transmitted wave and p 2 is the reflected wave, satisfies the acoustic wave equation c 2 2 (p 2 + p 2 ) + 2 (p 2 + p 2 ) = (2) t 2 In the interior cavity, the pressure p 3 = p 3, where p 3 is transmitted wave, satisfies the acoustic wave equation c 3 2 2 p 3 + 2 p 3 t 2 = (3) It is assumed here that the interior cavity inside the shells is totally absorptive. his indicates there exists only inwardtraveling wave. For the shells, let {u i, v i, w i } be the displacement components at the middle surface of the shell in the axial, circumferential, radial directions, where the subscript i denotes the variables associated with the outer shell (i = 1) and the inner shell (i = 2). Let {ψ z, ψ θ } be the rotations of the normal to the undeformed midsurface of the outer shell, where θ is the angle in the circumferential direction. he governing shell equations for both the thin shell and the first-order shell theories have been shown in the authors' previous study [2]. Eliminating u 1, v 1, ψ z, and ψ θ for the outer shell and u 2 and v 2 for the inner shell, one can obtain differential equations in terms of displacement components in the radial direction w 1 and w 2 L 1 (w 1 ) = p 12 = p 2 + p 2 (p 1 I + p 1 ) (4) L 2 (w 2 ) = p 23 = p 3 (p 2 + p 2 ) (5) where L 1 and L 2 are the differential operators. At the interfaces between the shells and fluids, the following equations must be satisfied (p I + p 1 ) r r = 1 (p 2 + p 2 ) r r = 1 (p 2 + p 2 ) r r = 2 p 3 r r = 2 = ρ 1 t + V. 2 w 1 (6) = ρ 2 2 w 1 t 2 (7) = ρ 2 2 w 2 t 2 (8) = ρ 3 2 w 2 t 2 (9) where r is the radial coordinate of the shells. 2.2 ransmission loss For a harmonic incident wave, assume that p I can be expanded as p I (r, z, θ, t) = p. ε n ( i) n J n (k 1r r)* n = (1) cos[nθ] exp[i(ωt k 1z z)] with p is the amplitude of the incident wave; n the number of the circumferential mode; ε = 1 and ε n = 2 otherwise; J n Bessel function of the first kind of order n; ω annular or rotational frequency; and k 1r = k 1 sin(γ 1 ), k 1z = k 1 cos(γ 1 ) (11) where k 1 = (ω/c 1 )/[1+M cos(γ 1 )] and M = V/c 1.
Following the procedures presented in the authors' previous paper [2], one can expand the pressures p 1, p 2, p 2, and p 3 which satisfy Eqs. (1) to (3) and displacements w 1 and w 2 in terms of cos[nθ] exp[i(ωt k 1z z)]. hen substitute these results into Eqs. (4) to (9) to solve for unknowns coefficients of p 1, p 2, p 2, p 3, w 1, and w 2. he derivation and the closed form solutions are given in the appendix. In order to define the transmission loss, consider the transmitted power flow per unit length along the axial direction of the shells in the interior cavity inside both shells, W, which is given by the following W = 1 2 e { S2(p 3 w 2 * ) r = 2 ds } (12) where S 2 = 2π 2 and e{.} and the superscript * represent real part and the complex conjugate of the argument, respectively. Substitution of Eqs. (A1) and (A12) for p 3 and w 2 into above equation yields an expression for the components of W W n = 2 p 2 ρ 3 ε n ω f 2 (13) Here,. is the absolute value of the argument and f is a frequency and modal dependent function, f = ρ 3k 1r H n 1' (k 2r 2 ) ρ 1 k 3r H 1' n (k 2r 1 ) J n ' (k 1r 1 ) H 1' n (k 3r 2 ) (Z 11 I +Z 11 )(Z 22 +Z 22 )ω 2 (14) 2 (c 1 k 2 1 ) = (Z s 1 +Z 21 +Z 11 )(Z s 2 +Z 32 +Z 22 ) (Z s 1 +Z 11 Z 21 )(Z s 2 +Z 32 Z 22 ) (15) H 1 n (k 2r 2 ) H 2 n (k 2r 1 )/[H 1 n (k 2r 1 )H 2 n (k 2r 2 )] where primes represent the derivative, J n are Bessel functions of the first kind of order n, and H n 1 and H n 2 are Hankel functions of the first and second kinds of order n, respectively, k 2 = ω/c 2, k 2z = k 1z, k 2r = k 3 = ω/c 3, k 3z = k 1z, k 3r = Z I 11 (n, ω) = iω ρ 1 k 1 c 2 1 J n (k 1r 1 ) k 1r ω J ' n (k 1r 1 ) Z 11 (n, ω) = iω ρ 1 k 1 c 2 1 H 2 n (k 1r 1 ) k 1r ω H 2' n (k 1r 1 ) 1 (k 2r 1 ) k 2 2 2 k 2z k 2 2 3 k 3z 2 (k 2r 1 ) (16) (17) (18) (19) Z 21 (n, ω) = iω ρ 2 H n k 2r H 1' n (k 2r 1 ), Z 21 (n, ω) = iω ρ 2 H n k 2r H 2' n (k 2r 1 ) (2) 1 (k 2r 2 ) 2 (k 2r 2 ) Z 22 (n, ω) = iω ρ 2 H n k 2r H 1' n (k 2r 2 ), Z 22 (n, ω) = iω ρ 2 H n k 2r H 2' n (k 2r 2 ) (21) 1 (k 3r 2 ) Z 32 (n, ω) = iω ρ 3 H n k 3r H 1' n (k 3r 2 ) (22) In above equations, Z s s 1 and Z 2 are the modal impedances of the outer and inner shells defined by Z 1 s = p 12 /w 1, Z 2 s = p 23 /w 2 (23) he transmission loss is, therefore, defined by L = 1 log 1 W n (24) n = W I where W I is the incident power flow per unit length along the axial direction of the shells W I = 1 cos(γ 1 ) ρ 1 c 1 p 2 (25) hen, a closed form for the transmission loss can be obtained by substituting Eqs. (13) and (25) into Eq. (24) 2ρ 1 c 1 L = 1 log 1 f 2 (26) n = ε n ρ 3 ω 1 cos(γ 1 ) When 1 = 2, the expression can be decomposed into the L for single shell [2] with shell modal impedance Z 1 s + Z 2 s. 2.3 Equivalent constants of the honeycomb core o calculate the equivalent mass density and elastic material constants of the honeycomb core, assume that the core is constructed from a regular hexagonal structure. he equivalent mass density is given by [8] ρ = ρ s 2 a (27) 3 b where ρ s is the intrinsic cell wall material mass density of the honeycomb core and a and b are its thickness and width, respectively, as shown in Fig. 1. he in-plane elastic constants of the honeycomb, E z, E θ, µ zζ, and µ θζ (ζ is the local coordinate along the radial direction of the considered shell) are also given by [8] E z = E θ = E s 4 3 a 3 b 3, µ zζ = µ θζ = 1 (28) where E s is the intrinsic Young's modulus. Since the outer shell is a thick shell, the transverse mechanical properties need to be known. he paper [9] presented a formula for the transverse shear modula of a honeycomb core of a regular hexagonal structure G zζ = G θζ = 3 G s a b (29) where G s is the intrinsic shear modulus. he third shear modulus in this study is defined by G ξθ = Es 3 a 3 b 3 (3) 3. NUMEICAL ANALYSIS 3.1 Given constants Numerical studies will illustrate the analysis presented here by considering a typical aircraft fuselage made from two concentric cylindrical sandwich shells. he outer shell consists of titanium face sheets and titanium honeycomb core and the inner shell consists of four layer laminated cross-ply graphite/epoxy face sheets and aluminum honeycomb core. he fiber orientation for the inner shell is {9,, 9,, honeycomb core,, 9,, 9 } with axial direction measured from the exterior surface of the inner shell. he radius and wall thickness are 1 = 1.88m and h 1 = 5.79cm for the outer shell and 2 = 1.84m and h 2 =.635cm for the inner shell. he face sheets are made from the same material
and with the same thickness for each shell. he thickness ratio of the core and the total for each shell is.84. he structural loss factor is η =.1. he material properties of the face sheet are given in able. 1, where α is the fiber direction and β is the direction perpendicular to the fiber. he equivalent material properties of honeycomb core can be obtained by substitution of a/b =.1 and the material properties of the intrinsic core materials, titanium (given in able. 1) and aluminum (ρ s = 275kg/m 3, E = 72GPa, µ =.3) into Eqs. (27) to (3), as given in able. 2. able 1. Material properties of the face sheet of the sandwich shell. titanium mass density: ρ s 1 (kg/m 3 ) 451 elastic modulus: E (GPa) 12.2 Poison's ratio: µ.361 graphite/epoxy layer mass density: ρ s 2 (kg/m 3 ) 158 elastic modulus: E α (GPa) 181 elastic modulus: E β = E z (GPa) 1.3 shear modulus: G αβ = G zα (GPa) 7.17 shear modulus: G βz (GPa) 2.87 Poison's ratio: µ βα.33 Poison's ratio: µ zα = µ zβ.28 able 2. Equivalent material properties of the honeycomb core. 3.2 esults Since the structural impedance of each shell Z s 1 and Z s 2 plays an important role in calculating the L, figure 2 shows the modulus of the shell impedance versus frequency for M =.5, 1., and 2.. For the outer shell, the resonances do not influence the impedance for M =.5. However, the resonances will result in peaks in the impedance for M = 2. at low frequencies, for instance, less than 1.5kHz. With an increase of the Mach numbers, the effect of the resonances on the impedance becomes significance. For the inner shell, the impedance is strongly affected by the resonances for both subsonic and supersonic Mach numbers at low frequencies. he minima in the impedance at the coincidence are shown in the inner shell and they are shifted upwards with increasing Mach number. he minima can not be observed in the outer shell for M =.5 and 1.. Comparison of (a) and (b) reveals that the impedance is much larger in the outer shell and than in the inner shell. his illustrates that the outer shell transmits much less incident energy than the inner shell so that noise will be reduced largely after it transmits the outer shell. Figure 3 shows the L in the annular space and interior cavity of the shells. he formula for the L in the cavity is given by Eq. (26) while in the space it is defined by L = 1 log 1 W W I (31) he transmitted power flow W is defined by W = 1 2 e { S1[(p 2 + p 2 )w 1 * ] r = 1 ds} (32) titanium honeycomb core mass density: ρ s 1 (kg/m 3 ) 52.77 elastic modulus: E z = E θ (GPa).277 elastic modulus: E ζ (GPa) 12.2 shear modulus: G zζ =G θζ (GPa) 2.545 shear modulus: G zθ (GPa).69 Poison's ratio: µ zθ 1 Poison's ratio: µ zζ = µ θζ.361 aluminum honeycomb core mass density: ρ s 2 (kg/m 3 ) 317.54 elastic modulus: E z = E θ (GPa).166 elastic modulus: E ζ (GPa) 72 shear modulus: G zζ =G θζ (GPa) 1.599 shear modulus: G zθ (GPa).42 Poison's ratio: µ zθ 1 Poison's ratio: µ zζ = µ θζ.3 he properties of the fluids are given in the followings: he aircraft is in cruising flight at 25,ft altitude (ρ 1 =.5489kg/m 3, c 1 = 39.966m/s) with interior pressurized to 1,ft altitude (ρ 3 =.941kg/m 3, c 3 = 328.558m/s). he mass density and sound speed of the fluid in the annular space between two shells are the same as that in the interior cavity. hree flight conditions encompassing subsonic, transonic, and supersonic conditions with Mach numbers M =.5, 1., and 2. are considered. he incident angle of the oblique plane sound wave is γ 1 = 3. in which S 1 = 2π 1. he closed form solutions for p 2, p 2, and w 1 are given in the appendix. he power flow can be obtained by substituting the expressions of p 2, p 2, and w 1 into above equations. Major minima in the L corresponding to coincidence frequencies are shown in these figures. hese minima are shifted upwards with increasing the Mach numbers. he effect of the shell resonances on the L can be observed only for M = 1. and 2.. he transmission loss is almost the same in the interior cavity and annular space at low frequencies. his is what we have expected because the impedance of the inner shell is much smaller than that of the outer shell, as shown in Fig. 2. Figure 4 shows a comparison of the L between the double sandwich shells and single sandwich shells. he single sandwich shell consists of either the outer shell or the inner shell. At low frequencies, not much difference exists for the L between the double sandwich shells and single outer sandwich shell. However, with increasing frequencies, the advantage that the double sandwich shells can offer more noise reduction than single shells is revealed except near coincidence. Double sandwich shells can produce an appreciable increase of the L over single sandwich shells at high frequencies. he effect of the pressure of the fluid in the annular space between the shells on the L is shown in Fig. 5. he space is pressurized to 1,ft, 15,ft (ρ 2 =.778kg/m3, c 2 = 322.463m/s), 2,ft (ρ 2 =.6523kg/m3, c 2 = 316.62m/s),
or unpressurized at 25,ft. he interior cavity is pressurized to 1,ft in all cases. esults demonstrate that the variation of the pressures will lead to the difference in the L within 2dB when the frequency is less about 16Hz. he difference will increase gradually to 12dB in the midfrequency range, i. e., between 16Hz and 5Hz, so that a criterion can be made according to noise reduction requirement by selecting the pressure in the annular space at this range. When the frequency is greater than 5Hz, the difference of the L is within 6dB except near coincidence. he minima in the L are shifted upwards slightly with the decrease of the mass densities. 4. CONCLUDING EMAKS his paper develops a mathematical model for prediction of sound transmission through two infinite concentric cylindrical sandwich shells excited by an incoming oblique plane sound wave. he shells are surrounded by fluid media and there is air in the annular space between them. Each sandwich shell is made of the honeycomb core and face sheets which can be isotropic, orthotropic, or laminated fiberreinforce composite materials. he first-order shear deformation theory is applied for the outer shell and the classical thin shell theory is applied for the inner shell. A closed form for the transmission loss is derived including the effect of the external flow based the modal analysis. he following conclusions can be drawn: (i). he transmission loss is almost the same in both the annular space and the interior cavity of the shells except near the coincidence since the impedance of the inner shell is much smaller than that of the outer shell. (ii). he two concentric sandwich shells can offer appreciable advantage over the single outer sandwich shell and the single inner sandwich shell for noise reductions especially at high frequencies. (iii). he transmission loss is not sensitive to the change of the pressure in the annular space in the low frequency range. However, in the mid-frequency range, an enhancement of the L is achieved by selecting the pressure in the annular space. 5.. ACKNOWLEDGMENS Authors would like to thank Dr. obert G. ackl, Boeing Commercial Airplane Group, for recommending this mathematical model. 6. EFEENCE [1] Beranek, L. L., Sound transmission through multiple structures containing flexible blankets, J. Acoust. Soc. Am., Vol. 21, No. 4, pp. 419-428, 1949. [2] Beranek, L. L. (ed), Noise and vibration control, McGraw-Hill, 1988. [3] Chen, S. S. and osenberg, G. S., Dynamics of a coupled shell-fluid system, Nucl. Eng. Des., Vol. 32, pp. 34-31, 1974. [4] Coats,. J., Silcox,. J., and Lester, H. C., Numerical study of active structural acoustic control in a doublewalled cylinder, Second Conference on ecent Advances in Active Control of Sound and Vibration, Blacksburg, Virginia, April 28-3, s11-s27, 1993. [5] Desmet, W. and Sas, P., Sound transmission of finite double-panel partitions with sound absorbing material and panel stiffeners, he CEAS/AIAA 1st Joint Aeroacoustic Conference, Munich, Germany, Jun. 12-15, pp. 149-153, 1995. [6] Dym, C. L. and Lang, M. A., ransmission of sound through sandwich panels, J. Acoust. Soc. Am., Vol. 56, No. 5, pp. 1523-1532, 1974. [7] Ford,. D., Lord, P., and Willams, P. C., he influence of absorbent linings on the transmission loss of doubleleaf partitions, J. Sound Vib., Vol. 5, No. 1, pp. 22-28, 1967. [8] Gibson, L. J., Ashby, M. F., Shajer, G. S., and obertson, C. I., he mechanics of two-dimensional cellular materials, Pro.. Soc. Lond., A 382, pp. 25-42, 1982. [9] Grediac, M., A finite element study of the transverse shear in honeycomb cores, Int. J. Solids Structures, Vol. 13, No. 13, pp. 1777-1788, 1993. [1] Grooteman, F. P., Boer, D., and Schippers, H., Vibroacoustic analysis of double wall structures, he CEAS/AIAA 1st Joint Aeroacoustic Conference, Munich, Germany, Jun. 12-15, pp. 149-153, 1995. [11] Grosveld, F. W., Coats,. J., Lester, H. C., and Silcox,. J., A numerical study of active structural acoustic control in a stiffened, double wall cylinder, Noise-Con 94, Ft. Lauderdale, Florida, May 1-4, 1994. [12] Krajcinovic, D., Vibrations of two coaxial cylindrical shells containing fluid, Nucl. Eng. Des., Vol 32, pp. 242-248, 1974. [13] Levin, L. and Milan, D.,Coupled breathing vibrations of two thin cylindrical coaxial shells in fluids, Proc. Vib. Probl. Ind., Paper No. 616, Keswick, England, 1973. [14] London, A., ransmission of reverberate sound through double walls, J. Acoust. Soc. Am., Vol. 22, No. 2, pp. 27-279, 195. [15] Mindlin,. D., Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. Mech. Vol. 18 (1), 1951, pp. 31-38. [16] Mulholland, K. A. and Parbrook, H. D., and Cummings, A., he transmission loss of double panels, J. Sound Vib., Vol. 6, No. 3, pp. 324-334, 1967. [17] eissner, E., he effect of transverse shear deformation on the bending of elastic plate, J. Appl. Mech. Vol. 12, 1945, A69-A77. [18] Skelton, E. A., Acoustic scattering by a disk or annulus linking two concentric cylindrical shells, Part I: theory and results for heavy exterior fluid loading, J. Sound Vib., Vol. 154, No. 2, pp. 25-22, 1992. [19] Skelton, E. A., Acoustic scattering by a disk or annulus linking two concentric cylindrical shells, Part II: results for heavy exterior fluid loading on both shells, J. Sound Vib., Vol. 154, No. 2, pp. 221-248, 1992.
[2] ang, Y. Y., obinson, J. H., and Silcox,. J., Sound transmission through a cylindrical sandwich shell, accepted by 34th AIAA Aerospace Science Meeting and Exhibit, eno, NV, 1996. [21] Vaicaitis,. and Bofilios, D. A., Noise transmission of double wall composite shells, AIAA 1th Aeroacoustics Conference, Paper No. 85-786-CP, Seattle, WA, July 9-11, 1986. [22] Ver, I. L. and Holmer, C. I., Interaction of sound waves with solid structures, Noise and vibration control, ed. by Leo L. Beranek, pp. 27-361, 1988. [23] White, P. H. and Powell, A., ransmission of random sound and vibration through a rectangular double wall, J. Acoust. Soc. Am., Vol. 44, No. 4, pp. 821-832, 1966. w 2n = k 1r ρ 1 ω 2 ω 2 k 1 2 c 1 2 J n (k 1r 1 ) H n 1 (k 2r 2 ) H n 1 (k 2r 1 ) 6 p ε n (-i) n where are given in the text and 1 = (Z s 1 +Z 21 -Z I 11 )(Z s 2 +Z 32 +Z 22 ) (Z s 1 -Z I 11 Z 21 )(Z s 2 +Z 32 Z 22 ) H 1 n (k 2r 2 ) H 2 n (k 2r 1 )/[H 1 n (k 2r 1 )H 2 n (k 2r 2 )] 2 = (1+Z I 11 /Z 11 )(Z s 2 +Z 32 +Z 22 ) 3 = (1+Z I 11 /Z 11 )(Z s 2 +Z 32 Z 22 ) 4 = (1+Z I 11 /Z 11 )(Z 22 +Z 22 )Z 11 5 = (1+Z I 11 /Z 11 ){(Z s 2 +Z 32 +Z 22 ) (Z s 2 +Z 32 Z 22 ) H 1 n (k 2r 2 ) H 2 n (k 2r 1 )/[H 1 n (k 2r 1 )H 2 n (k 2r 2 )]} 6 = (1+Z I 11 /Z 11 )(Z 22 +Z 22 )Z 11 (A12) (A13) (A14) (A15) (A16) (A17) (A18) [24] Yeh,.. and Chen, S. S., Dynamics of a cylindrical shell system coupled by viscous fluid, J. Acoust. Soc. Am., Vol. 62, No. 2, pp. 262-27, 1977. APPENDIX: DEIVAION OF HE SOLUIONS o develop the solutions, assume the pressures p 1, p 2, p 2, and p 3 which satisfy Eqs. (1) to (3) as p 1 (r, z, θ, t) = p 1n H 2 n (k 1r r) cos[nθ] exp[i(ωt k 1z z)] (A1) n = p 2 (r, z, θ, t) = n = p 2 (r, z, θ, t) = n = p 3 (r, z, θ, t) = n = p 2n H 1 n (k 2r r) cos[nθ] exp[i(ωt k 1z z)] (A2) p 2n H 2 n (k 2r r) cos[nθ] exp[i(ωt k 1z z)] (A3) p 3n H 1 n (k 3r r) cos[nθ] exp[i(ωt k 1z z)] (A4) where p 1n, p 2n, p 2n, and p 3n are yet-to-be-determined complex amplitude factors. he displacement components w 1, and w 2 are assumed as w 1 (z, θ, t) = w 1n cos[nθ] exp[i(ωt k 1z z)] (A5) w 2 (z, θ, t) n = = n = w 2n cos[nθ] exp[i(ωt k 1z z)] (A6) incident plane wave y 2 1 face sheet core face sheet face sheet core face sheet x γ 1 cross section of the shells 2 external flow inner shell airgap outer shell 1 3 a b honeycomb core z where w 1n and w 2n are unknown complex amplitude factors. Substitution of Eqs. (A1) to (A6) into Eqs. (4) to (9) yields the solutions for p 1n, p 2n, p 2n, p 3n, w 1n and w 2n Figure 1. Schamatic of two concentric shells and their geometry construction. p 1n = J n(k 1r 1 ) H 2 n (k 1r 1 ) 1 p ε n (-i) n p 2n = iρ 2ω J n (k 1r 1 ) k 2r H 1 n (k 2r 1 ) H n 2 (k 1r 1 ) H 2 n (k 1r 1 ) 2 p ε n (-i) n iρ 2 ω p 2n = J n (k 1r 1 ) k 2r H 1 n (k 2r 1 ) H n 2 (k 1r 1 ) H 2 n (k 1r 1 ) H n 1 (k 2r 2 ) H 2 n (k 2r 2 ) 3 p ε n (-i) n ρ 3 k 1r p 3n = ω 2 ρ 1 k 3r k 2 1 c J n (k 1r 1 ) 2 1 H 1 n (k 3r 2 ) H n 1 (k 2r 2 ) H 1 n (k 2r 1 ) 4 p ε n (-i) n w 1n = i ω J n (k 1r 1 ) H 1 n (k 1r 1 ) 5 p ε n (-i) n H 2 n (k 1r 1 ) (A7) (A8) (A9) (A1) (A11)
Impedance of the outer shell 1 7 1 6 1 5 1 4 1 3 1 2 (a) M =.5 M = 1. M = 2. ransmission loss (L) 2 15 1 5 (a) interior cavity annular space Impedance of the inner shell 1 5 1 4 1 3 1 2 1 1 (b) M =.5 M = 1. M = 2. 1 2 1 3 1 4 Frequency (Hz) Figure 2. Modulus of impedance of the sandwich shells. ransmission loss (L) ransmission loss (L) 2 15 1 5 2 15 1 5 (b) (c) interior cavity annular space interior cavity annular space 1 2 1 3 1 4 Frequency (Hz) Figure 3. he L at annular space and interior cavity: (a) M =.5; (b) M = 1.; (c) M = 2..
ransmission loss (L) 2 15 1 5 (a) double shells single outer shell single inner shell ransmission loss (L) 2 15 1 5 (a) 1,ft 15,ft 2,ft 25,ft ransmission loss (L) 2 15 1 5 (b) double shells single outer shell single inner shell ransmission loss (L) 2 15 1 5 (b) 1,ft 15,ft 2,ft 25,ft ransmission loss (L) 2 15 1 5 (c) double shells single outer shell single inner shell ransmission loss (L) 2 15 1 5 (c) 1,ft 15,ft 2,ft 25,ft 1 2 1 3 1 4 Frequency (Hz) 1 2 1 3 1 4 Frequency (Hz) Figure 4 A comparison of the L between double shells and single shells: (a) M =.5; (b) M = 1.; (c) M = 2.. Figure 5. he effect of the annular pressure on the L: (a) M =.5; (b) M = 1.; (c) M = 2..