SOUND TRANSMISSION THROUGH A CYLINDRICAL SANDWICH SHELL WITH HONEYCOMB CORE

Transcription

1 SOUND TANSISSION THOUGH A CYLINDICAL SANDWICH SHELL WITH HONEYCOB COE Yvette Y. Tang* National esear Counil S 463, NASA Langley esear Center, Hampton, VA 2368 Jay H. obinson and iard J. Silox Strutural Aousti Bran S 463, NASA Langley esear Center, Hampton, VA 2368 Abstrat Sound transmission troug an infinite ylindrial sandwi sell is studied in te ontext of te transmission of airborne sound into airraft interiors. Te ylindrial sell is immersed in fluid media and exited by an oblique inident plane sound wave. Te internal and external fluids are different and tere is uniform airflow in te external fluid medium. An expliit expression of transmission loss is derived in terms of modal impedane of te fluids and te sell. Te results sow te effets of te inident angles of te plane wave; te fligt onditions of a number and altitude of te airraft; () te ratios between te ore tikness and te total tikness of te sell; and (d) te strutural loss fators on te transmission loss. Comparisons of te transmission loss are made among different sell onstrutions and different sell teories.. Introdution Te use of ig strengt-to-weigt ratio omposite materials in transport airraft may result in inreased interior noise levels. Previous studies ave sown tat more noise an be transmitted troug laminated fiber-reinfored strutures tan troug isotropi strutures in te ig frequeny range [7]. Noise transmission, measured by transmission loss (TL) troug te airraft, as been studied by Smit [3], Wite [4], Koval [4 to 7], and Blaise et al. [] for isotropi, ortotropi, and laminated fiber-reinfored omposite sells. Smit presented a teoretial study of transmission of sound energy troug a tin, isotropi elasti ylindrial sell from an oblique plane wave exitation. He defined a ross-setional absorption oeffiient tat is te ratio of te power absorbed to te inident power per unit lengt. Wite investigated sound transmission into an infinite and a finite * esear Assoiate, ember AIAA. Assistant Bran Head, ember AIAA. esear Engineer, ember AIAA. ylindrial sells and found two important arateristis, te ring and oinidene frequenies, at wi TL takes on minima. Koval extended Smit's work to present an analytial model for prediting of TL for isotropi, ortotropi, and laminated fiber-reinfored omposite sells. Te effet of external airflow was inluded and te fluids are different inside and outside te sell. Te signifiane of te ring and oinidene frequenies was revealed. Blaise et al. ten extended Kaval's work to onsider an ortotropi sell exited by an oblique plane sound wave wit two independent inident angles. Te diffuse field transmission oeffiient was examined. Two fluids inside and outside te sell are te same and te external airflow was not inluded in teir study. Te objetive of tis paper is to study sound transmission troug an infinite ylindrial sandwi sell under fligt onditions. An airraft in fligt wit an external airflow and a negative pressure differential between te inside and outside is modeled as an infinite ylindrial sell. Te sell is assumed to be immersed in a fluid media and exited by an inident oblique plane sound wave wit two independent angles. Te properties of te internal and external fluids surrounding te sell are different. Te sandwi sell onsists of a oneyomb ore and fae seets. Te fae seets are made of isotropi materials, ortotropi materials, or laminated fiber-reinfored omposite materials. An expliit expression of te frequeny spetrum of TL inluding te effet of te external airflow and two independent inident angles is derived in terms of te modal impedanes of te fluids and te sell. In order to study TL, two sell teories are applied. Te first is te tin sell teory in wi te eissner-nagdi- Berry teory [] is used. For tik sells or in te ig frequeny range of vibrational sells, te transverse sear deformation effet was found to be signifiant [2, 8,, 2]. Hene, te first-order sell teory, in wi te sear and rotation effets are taken into aount, is onsidered. Te effets of different soure onditions, strutural properties and fligt onditions on TL are studied for a range of values, speifially, inident angle of te plane wave; a number and te fligt altitude of te airraft; te ratios between te ore tikness and

2 te total tikness of te sell, and te strutural loss fators are sown. Comparisons of TL are made among an aluminum monooque sell, an aluminum sandwi sell wit aluminum fae seets and an aluminum ore, and a omposite sandwi sell wit laminated grapite/epoxy fae seets and te same aluminum ore as aluminum sandwi sell and between a tin sell teory and te first-order sell teory. 2. atematial model Figure sows an infinite ylindrial sandwi sell wit radius and wall tikness. Te sell is immersed in fluid medium (external fluid) and ontains fluid medium 2 (internal fluid). Te mass density and sound speed for fluid media and 2 are {ρ, } and {ρ 2, 2 }, respetively. An oblique plane sound wave p I is approaing te sell wit inident angles {, ψ }. Airflow in fluid medium (external) is moving wit a onstant veloity V = {V x, V y, V } in te x, y, and diretions. 2.. Governing equations of te fluids Due to te existene of airflow in te external fluid medium, te external pressure, wi is te summation of te inident wave p I and te refleted wave p, satisfies te onveted wave equation 2 2 (p I +p ) + t +V. 2 (p I +p ) = () were and 2 are te gradient operator and te Laplaian operator, respetively. Te internal pressure satisfies te aousti wave equation p T + 2 p T = (2) t 2 were p T is te transmitted wave. Note tat Eq. (2) implies tat te sell interior is totally absorptive. Terefore only an inward-traveling wave exists in te sell interior. Altoug tis is not representative of an airraft interior, te model allows te transmission loss of te ylindrial wall to be studied in a straigt forward manner Governing equations of te sandwi sell Let te non-dimensional axial oordinate be ξ =/, te irumferential diretion be θ, and te normal diretion to te middle surfae of te sell be ζ, as sown in Fig. 2. Denote te sell displaement omponents at te midsurfae in te axial, irumferential, and radial diretions as {u, v, w }, respetively. Let {ψ ξ, ψ θ } be te rotations of te normal diretion relative to te undeformed midsurfae. Te displaement omponents at an arbitrary distane ζ from te midsurfae along te axial, te irumferential, and te radial diretions are u = u +ζψ ξ, v = v +ζψ θ, w = w (3) It sould be mentioned tat if te tin sell teory is applied, ten te rotations are ψ ξ = w,ξ / and ψ θ = w,θ /, were te omma denotes te partial derivative wit respet to te named variable. Using te strain-displaement relations, te onstitutive laws, and te equations of motion of te tree-dimensional teory of elastiity in te ylindrial oordinate system {r, θ, ξ}, were r is te radial oordinate, one an obtain te following equations of motion for an N-layered ylindrial sandwi sell [2] N, ξ + N 6, θ = u,tt + ψ ξ,tt (4) N 2, θ + N 6,ξ Q 2 = v,tt + ψ θ,tt (5) Q,ξ +Q 2,θ +N 2 = w,tt +[p T (p I +p )] r = (6), ξ + 6,θ Q = u,tt + 2 ψ ξ,tt (7) 2, θ + 6,ξ Q 2 = v,tt + 2 ψ θ,tt (8) were t is time. Equations (4) to (6) present te equation of te translation motion and Eqs. (7) and (8) present te equations of rotational motion. Te meaning of ea term in tese equations is explained in te following. Te inertia terms,, and 2 in Eqs. (4) to (8) are defined by N {,, 2 } = k = ρ k {, ζ, ζ 2 }dζ (9) k were ρ k and k are te mass density and te tikness of te k-t layer. In Eqs. (4) to (8), te stress resultants N, N 2, and N 6 and ouples, 2, and 6 are given by N i = A ij ε j +B ij χ j, i = B ij ε j +D ij χ j () were te subsripts (i, j) (, 2, 6) follow Voigt's notations and te streting, bending-streting oupling, and bending stiffness onstants A ij, B ij, and D ij are N {A ij, B ij, D ij } = k = k ij {, ζ, ζ 2 }dζ () k and ij k are te transformed redued stiffness onstants of te k-t layer [3]. Te strains at te midsurfae ε i in Eq. () are { ε, ε 2, ε 6 } = {u,ξ, (v,θ w ), (v,ξ +u,θ )}/ (2) and te urvatures at te midsurfae χ i for te tin sell teory are {χ, χ 2, χ 6 } = {w,ξξ, (w,θθ v,θ ), (2w,ξθ v,ξ )}/ 2 (3) and for te first-order sell teory are {χ, χ 2, χ 6 } = {ψ ξ,ξ, ψ θ,θ, (ψ ξ,θ +ψ θ,ξ )}/ (4) Te transverse sear fores Q and Q 2 in Eqs. (5) to (8) are obtained differently for te tin sell teory and te first-order sell teory. For te tin sell teory, beause te effets of sear and rotation are negleted, te inertial terms = 2 = and te sear fores Q and Q 2 an be obtained diretly from Eqs. (7) 2

3 and (8). However, for te first-order sell teory, te sear fores Q and Q 2 are Q = k 4 2 A 44 (w,ξ /+ψ ξ )+k 4 k 5 A 45 (w,θ /+ψ θ ) (5) Q 2 = k 2 5 A 55 (w,θ /+ψ θ )+k 4 k 5 A 45 (w,ξ /+ψ ξ ) (6) Here, in Eqs. (5) and (6), k 4 and k 5 are te sear orretion fators and te sear elasti onstants A 44, A 45, and A 55 are N {A 44, A 45, A 55 } = k { k 44, k 45, k 55 }dζ (7) = k were k 44, k k 45, and 55 are te transformed sear stiffness onstants of te k-t layer [3] Boundary onditions Te boundary onditions inside and outside te sell are given by following (p I + p ) = ρ r r = t + V. 2 w (8) p T = ρ 2 w 2 (9) r r = t 2 wi desribe te effet of te fluid pressure on te sell motion Solutions To obtain te solution of Eq. (), te given inident wave p I an be expanded as p I = p ε n ( i) n J n os[n(θ ψ )]exp[i(ωt k )] (2) were p is te amplitude of te inident wave, n te order of te irumferential mode, ε n = wen and ε n = 2 wen n, and J n = J n (k r r) te Bessel funtion of te first kind of order n. Te wave numbers k r and k are to be determined. Te refleted wave p an be obtained by following p = A n H n 2 os[n(θ ψ )]exp[i(ωt k )] (2) were H 2 n = H 2 n (k r r) is te Hankel funtion of te seond kind of order n and A n are yet-to-be-determined omplex amplitude fators. Substituting Eqs. (2) and (2) into Eq. (), one an obtain te wave numbers k = k os, k r = k sin (22) were k = ω ( x sin osψ + y sin sinψ + os ) (23) x = V x /, y = V y /, = V / (24) Te transmitted wave p T, wi satisfies Eq. (2), is p T = B n H n os[n(θ ψ )]exp[i(ωt k 2 )] (25) were H n = H n (k 2 r r) is te Hankel funtion of te first kind of order n, k 2 = ω/ 2, k 2 = k, k 2 r = k k 2 (26) and B n are yet-to-be-determined omplex amplitude fators. Te displaement omponents u, v, and w are also assumed as u = v = w = iu n os[n(θ ψ )]exp[(ωt k )] v n sin[n(θ ψ )]exp[(ωt k )] w n os[n(θ ψ )]exp[(ωt k )] (27) (28) (29) For te first-order sell teory, te rotation omponents ψ ξ and ψ θ are assumed as ψ ξ = ψ θ = iu n os[n(θ ψ )]exp[(ωt k )] v n sin[n(θ ψ )]exp[(ωt k )] (3) (3) Here, in Eqs. (27) to (3), {u n, v n, w n } and { u n, v n } are unknown omplex amplitudes of te displaement and rotation omponents, respetively. It an be seen tat for ea mode number n, tere are five unknowns {u n, v n, w n, A n, B n } and five equations (4) to (6), (8), and (9) for te tin sell teory and seven unknowns {u n, v n, w n, u n, v n, A n, B n } and seven equations (4) to (8), (8), and (9) for te first-order sell teory. To obtain simplified and meaningful results, one an get a set of equations in terms of A n B n and w n by eliminating u n and v n for te tin sell teory and u n, v n, u n, and v n for te first-order sell teory. Solving tese oupled equations yields A n = p ε n ( i) n J n (k r ) H n 2 (k r ) Z S n +Z T I n +Z n Z S n +Z T n +Z n (32) B n = p ε n ( i) n ρ 2k r J n (k r ) Z n +Z I n (k /ω) 2 ρ k 2 r H n (k 2 r ) Z S n +Z T n +Z n (33) w n = p ε n ( i) nk rj n (k r ) Z n +Z I n (k /ω) 2 ρ ω 2 Z S n +Z T n +Z n (34) were I Z n = iω ρ J n (k r ) k r J n (k r ) (35) 2 (k r ) 2 Z n Z n T = iω = iω ρ H n k r ρ 2 H n k 2 r H 2 n (k r ) (k 2 r ) k ω (36) (37) H n (k 2 r ) Z S n = (38) iω D n Here, in Eqs. (32) to (38), te primes denote derivatives wit respet to te argument; Z I n, Z n, and Z T n are modal arateristi aousti impedanes of te fluids; Z S n is te modal impedane of te sell; and D n is te modal amplitude of te displaement omponent of te sell in te radial diretion wit a unit pressure. 3

4 2.5. Transmission loss Te transmission loss of te sell is defined by te ratio of te transmitted power W T and te inident power W I per unit lengt of axial setion of te sell as follows [9] TL = log W T W I (39) were W I = p 2 os ρ (4) W T = 2 e S(p T w * ) r = ds (4) in wi e[.] is te real part, te supersript * represents te omplex onjugate of te argument, and S is te area of te ross setion of te sell. Substituting Eqs. (25) and (29) by using Eqs. (33) and (34) into Eq. (39) leads to te expression of TL in terms of te modal impedanes TL = log 2 ε n ω 2 * k r k e[z n ] e[z T (42) n ] Z n + Z T n + Z S n 2 were. is te absolute value of te argument. 3. Parametri studies Parametri numerial studies of transmission loss (TL) are onduted for frequeny from. to 2 times te ring frequeny f. Tese studies provide insigt into te effet of te aousti properties of te fluids and te strutural or material parameters of te sells on TL. A typial narrow-bodied jet fuselage made of te sandwi sells wit radius =.83m is onsidered. Te inner and outer fae seets are assumed to be te same. Te properties of te fluids and te sells are as follows: Te airraft is in ruising fligt at,ft (ρ =.94kg/m 3, = m/s), 25,ft (ρ =.5489kg/m 3, = m/s), and 35,ft (ρ =.3795kg/m 3, = m/s) wit a number x = y =, and =,.5,.,.5. Te fuselage interior is pressuried to,ft, su tat no pressure differential exists at,ft. Te plane sound wave approaes te sell wit inident angles = 5, 35, and 2 and ψ =, respetively. Two kinds of te sandwi sells wit te same weigt, radius, and aluminum ore are onsidered. One is an aluminum sandwi sell wose fae seets are made of aluminum, and te oter is a omposite sandwi sell wose fae seets are made of laminated ross-ply grapite/epoxy layers. Te ore as low density and ig out-of-plane modulus. Terefore, assume te material onstants of te aluminum ore inluding te mass density, Young's modulus, Poison's ratio, and sear modulus to be ρ = 55kg/m 3, E, µ, and G =.38GPa, respetively. Te ore tikness of bot aluminum and omposite sandwi sells is denoted as. Te material onstants of te aluminum fae seets are ρ = 275kg/m 3, E = 72GPa, and µ =.3. Te material onstants of te laminated grapite/epoxy fae seets are ρ = 6kg/m 3, E α = 22GPa, E β = 6.9GPa, µ αβ =.25, and G αβ = 4.8GPa, were α is te fiber diretion and β is te diretion perpendiular to te fiber. Bot inner and outer laminated grapite/epoxy fae seets onsist of four layers wit te fiber orientations {9,, 9, } and {, 9,, 9 } wit ξ diretion on te sell outer surfae, respetively. Te ratio of te ore tikness and te total tikness of te sell is / =.8,.9, and.95. Te strutural loss fators are assumed as η =.,.5, and.. For referene purposes, te same weigt and radius aluminum monooque sell wit te wall tikness =.59m is onsidered (oter material onstants are given above). Ten te wall tiknesses of te aluminum and omposite sandwi sells are =.736m and =.2m, respetively. Te frequeny parameter of all results in te following figures is normalied by te ring frequeny of te aluminum monooque sell, f = (2π) E/ρ = 445H. Te effet of te pressure on TL is negleted sine ontribution of te pressure to te sell mode impedane is less tan %. 4. esults Te effets of te inident angles of te plane sound wave, te a numbers, te fligt altitudes, te ratios of te ore tikness to te total tikness of te sell, and te loss fators of te sell material on TL are sown in Figs. 3 troug 7 based on te tin sell teory. Figures and represent TL versus te normalied frequeny for te aluminum sandwi sell and te omposite sandwi sell, respetively. 4.. Te effet of te fluid properties on TL Figure 3 sows te effet of te inident angles wit = 5, 35, and 2 on TL. Tere are two major minima in TL for te aluminum sandwi sell at te ring frequeny f and te ritial frequeny f, (>f ) wi orresponds to te oinidene frequeny of an infinite plate wit te same sell tikness, as sown in Fig. 3. Te frequeny f is sifted upwards wen te plane sound wave is lose to te normal inidene ( = 9 ) (see Fig. 3). However, only one major minimum in TL at f is observed for te omposite sandwi sell, as sown in Fig. 3. In tis ase, te bending wave of te sell tat orresponds to f beomes unimportant for sound transmission in te omposite sandwi sell wen f < f. Figure 4 sows te effet of a numbers wit =,.5,., and.5 on TL. Two major minima of TL at f and f our for te aluminum sandwi 4

5 sell. Only one major minimum in TL at f is seen in te omposite sandwi sell, as sown in Fig. 4. Te frequeny f is sifted upwards wit inrease of a number for bot sandwi sells. Figure 5 sows te effet on TL of altitudes of,ft, 25,ft, and 35,ft altitudes wit te same interior pressure at,ft. Te effet of te pressure differential inside and outside te sell on te model impedane of te sell is small so tat it an be negleted. A iger fligt altitude will lead to a larger aousti impedane ( i ρ i ) mismat between te fluids inside and outside te sell. Te urves in Figs. 5a and 5b illustrate tat wit an inrease of te aousti impedane mismat, te transmission loss inreases exept near f (see Fig. 5). Figure 5 sows tat f is sifted downwards sligtly wen te fligt altitude inreases. Figure 5 sows only one major minimum in TL at f for te omposite sandwi sell Te effet of te strutural properties on TL Figure 6 sows te effet of te different tikness ratios wit / =.8,.9, and.95 on TL. For te ratio / inreasing, tis indiates tat te ore tikness inreases wile te tiknesses of te fae seets derease sine te total tikness is kept onstant for ea sell. Te inrease of / will results in tat te bending stiffness is improved and te in-lane stiffness is te same sine bending modes are more important at iger frequenies to transmit wave. Te minimum in TL at f is observed only wen / =.8 for te aluminum sandwi sell, as sown in Fig. 6. Te urves in Fig. 6 reveal tat wen f < f te transmission loss follows a stiffness-ontrolled law in wi TL dereases 5-6 db/otave and wen f > f for te aluminum sandwi sell and f > f for te omposite sandwi sell, te transmission loss inreases about 2 db/otave. Figure 7 sows te effet of te strutural loss fators wit η =.,.5, and. on TL. Te strutural damping beomes unimportant in te study of TL exept at frequenies f and f were magnitude of TL inreases wit inrease of te loss fator near f and f as sown in Fig Comparison of TL among different sells Figures 8 and sow TL of te aluminum monooque sell, aluminum sandwi sell, and omposite sandwi sell wit =.8 and =.95, respetively. Below f, transmission loss is almost te same in te aluminum monooque sell and in te aluminum sandwi sell. It an be seen tat wen f>f, te transmission loss of te omposite sandwi sell is te largest among te sells. However, wen f<f, te transmission loss for te omposite sandwi sell is te least of te tree sells. It an also be seen tat at ig frequenies, an inrease in wall tikness ratio ( /) as a large effet on TL for te sandwi sell tan te monooque sell. Figures 9 and sow TL te aluminum monooque sell, aluminum sandwi sell, and omposite sandwi sell at altitudes 25,ft and 35,ft wit te same interior pressure (at,ft altitude), respetively. Below f, transmission loss is almost te same in te aluminum monooque sell and in te aluminum sandwi sell. esults illustrate tat te omposite sandwi sell offers an advantage in noise redution over te aluminum monooque and sandwi sells in te ig frequeny range Comparison of TL for different sell teories Figures and sow a omparison on TL for te tin sell teory and te first-order sell teory. Te seleted sear orretion fators k 4 = k 5 = 5/6. At te lower frequeny, te effets of sear and rotation on TL are negligible. However, in te ig frequeny range, te sear waves transmit sound troug te sell resulting in a derease of TL. Te effet of te sear wave on sound transmission beomes more important for te omposite sandwi sell tan te aluminum sandwi sell beause te omposite sandwi sell is tiker tan te oter. 5. Conlusion In tis paper, we studied sound transmission troug an infinite ylindrial sandwi sell. An expliit expression for te sell transmission loss (TL) inluding te effet of te external airflow and two independent inident angles is derived. Te frequeny spetrum of TL versus te normalied frequeny is studied by varying parameters of te fluids and sells. Comparison of TL is made among te aluminum monooque sell, aluminum sandwi sell, and omposite sandwi sell based on te tin sell teory and between te tin and first order sell teories, respetively. Te following onlusions an be drawn from tis numerial study:. Two major minima in TL at f and f are present in te aluminum sandwi sell. However, only one major minimum in TL at f is sown in te omposite sandwi sell. In tis ase, te bending wave of te omposite sandwi sell beomes unimportant for sound transmission beause no major minimum in TL is sown at f. 2. Inrease of te ratio / will lead to an inrease of TL in te ig frequeny range. Te strutural damping is unimportant in te study of TL altoug te magnitude of TL varies sligtly for different dampings near at f and f. 3. Te omposite sandwi sell offers an advantage in noise redution over bot te aluminum monooque sell and te aluminum sandwi sell for te larger impedane mismat between te fluids inside and outside te sell and te larger 5

6 ratio of te ore tikness and total tikness in ig frequeny range based on tin sell teory at te ig frequeny. 4. Te effets of sear and rotation of te sandwi sells on TL an be negleted only in te lower frequeny range. In te ig frequeny range, te sear wave's transmission of sound will result in dereased TL. eferenes. Blaise, A., Lesueur, C., Gotteland,., and Barbe,., "On sound transmission into an ortotropi infinite sell: Comparison wit Koval's results and understanding of penomena", Journal of Sound and Vibration, Vol. 5, No. 2, 99, pp Greenberg, J. B. and Stavsky, Y., "Vibration of axially ompressed laminated ortotropi ylindrial sells, inluding transverse sear deformation", Ata eania, Vol. 37, 98, pp Jones,.., "eanis of omposite materials", Hemispere publising orporation", New York, 975, pp Koval, L.., "On sound transmission into a tin ylindrial sell under 'fligt onditions'", Journal of Sound and Vibration, Vol. 48, No. 2, 975, pp Koval, L.., "Effet of stiffening on sound transmission into a ylindrial sell in fligt", AIAA Journal, Vol. 5, No. 7, 977, pp Koval, L.., "On Sound transmission into an ortotropi sell", Journal of Sound and Vibration, Vol. 63, No., 979, pp Koval, L.., "Sound transmission into a laminated omposite ylindrial sell", Journal of Sound and Vibration, Vol. 7, No. 4, 98, pp Leissa, A. W., "Vibration of sells", NASA SP- 288, 973, pp Lesueur, C., "ayonnement aoustique des strutures-vibroaoutique-interations fluidestruture", Paris: Colletion EDF-DE, Editions Eyrolles, 988, pp arkus, S., "Te meanis of vibrations of ylindrial sells", Elsevier, 988, pp indlin,. D., "Influene of rotatory inertia and sear on flexural motions of isotropi elasti plates", J. e., Vol. 8, No., 95, pp eissner, E., "Te effet of transverse sear deformation on te bending of elasti plate", J. Appl. e., Vol. 2, 945, A69-A Smit, P. W., Jr., "Sound transmission troug tin ylindrial sells", Journal of te Aoustial Soiety of Ameria, Vol. 29, No. 6, 957, pp Wite, P. "Sound transmission troug an finite, losed, ylindrial sell", Journal of te Aoustial Soiety of Ameria, Vol. 4, No. 5., 966, pp y y y x ψ θ x x pi Vy 2 Vx Figure. System onfiguration. θ ζ ζ aluminum sandwi ξ fae seet oneyomb ore fae seet ζ V omposite sandwi Figure 2. Geometry of te sandwi sells. 6

7 = 5 = 35 = =.5 =. = = 5 = 35 = =.5 =. =.5.. Figure 3. Te effet of te inident angles on te TL at no airflow wit te fligt altitude (25,ft) and te interior pressure (,ft): η =.; / =.8.. aluminum sandwi sell; omposite sandwi sell. Figure 4. Te effet of te a numbers on te TL wit te fligt altitude (25,ft) and te interior pressure (,ft): η =.; = 5 ; / =.8.. aluminum sandwi sell; omposite sandwi sell. 7

8 , ft 25, ft 35, ft / =.8 / =.9 / = , ft 25, ft 35, ft / =.8 / =.9 / =.95.. Figure 5. Te effet of te fligt altitudes on te TL at no airflow wit te interior pressure (,ft): η =.; = 5 ; / =.8.. aluminum sandwi sell; omposite sandwi sell. Figure 6. Te effet of te tikness on te TL ratios at no airflow wit te fligt altitude (25,ft) and te interior pressure (,ft): η =.; = 5.. aluminum sandwi sell; omposite sandwi sell. 8

9 η =. η =.5 η = sell sell 2 sell η =. η =.5 η = sell sell 2 sell 3.. Figure 7. Te effet of te strutural loss fators on te TL at no airflow wit te fligt altitude (25,ft) and te interior pressure (,ft): = 5 ; / =.8.. aluminum sandwi sell; omposite sandwi sell. Figure 8. Comparison of te effet of te tikness ratio on te TL at no airflow wit te fligt altitude (25,ft) and te interior pressure (,ft): η =.; = 5. sell : aluminum monooque sell; sell 2: aluminum sandwi sell; sell 3: omposite sandwi sell.. / =.8; / =.95. 9

10 sell sell 2 sell witout sear effet wit sear effet sell sell 2 sell witout sear effet wit sear effet.. Figure 9. Comparison of te effet of te fligt altitudes on te TL at no airflow wit te interior pressure (,ft): η =.; = 5 ; / =.8. sell : aluminum monooque sell; sell 2: aluminum sandwi sell; sell 3: omposite sandwi sell.. fligt altitude 25, ft; fligt altitude 35, ft. Figure. Comparison of te TL between te tin and te first-order deformation sell teories at no airflow wit te fligt altitude (25,ft) and te interior pressure (,ft): η =.; = 5 ; / =.8.. aluminum sandwi sell; omposite sandwi sell.