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Response of and sound radiation from a layered cylinder with regular axial stiffeners
Journal of Sound and Vibration (1985) 103(4), 519-531
RESPONSE OF AND S O U N D RADIATION FROM A LAYERED CYLINDER WITH REGULAR AXIAL STIFFENERS E. S. REDDY'~ Department of Mechanical and Aerospace Engineering, Arizona State University Tempe, Arizona 85287, U.S.A. AND
A. K. MALLIK Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208 016, India (Received 18 May 1984, and in revised form 12 December 1984) Response and sound radiation characteristics of a three-layered cylinder with equispaced (periodic) axial stiffeners are obtained using the "wave" approach. A two-dimensional situation with no variation in the axial direction is considered. Numerical results are presented for the case of three stiffeners, which are assumed to prevent both radial and tangential displacements of the base layer. The pressure harmonics exciting various modes are analyzed. Three ~litIerent types of interior, namely, totally resonant or absorbent and partially absorbent, are investigated and their effects on the response and sound power radiation are studied. 1. INTRODUCTION Axially stiffened cylinders are used in aerospace and underwater structures. These applications have led researchers to study the response and sound transmission characteristics of such structures. Relatively simplified problems have been studied by m a n y investigators as the presence of stiffeners makes the exact analysis difficult. Foxwell and Franklin [1] have studied an idealized problem of sound transmission in which a plane wave incident normally on an infinitely long cylinder was considered. For simplicity, only the ring (frame) type stiffeners were included in the model and the interior of the structure was assumed to be totally resonant. The problem of a totally absorbent interior was analysed by Smith [2] and Koval [3, 4]. By improving the smeared stiffener theory, Koval [5, 6] included the effect of axial stiffeners. In actual practice, however, the interior is neither totally resonant nor totally absorbent. In a later p a p e r [7], Koval considered the partially absorbing nature of the cavity while studying the effect of cavity resonances on sound transmission. Sound-structure interaction studies of periodic structures have been attempted by only a few investigators [8-11]. The response o f and sound radiated from a periodic plate have been solved exactly by introducing a space harmonic series in reference [9]. In this approach, the number o f simultaneous equations to be solved is equal to the number o f terms chosen in the infinite series. For adequate convergence, however, the number o f terms required were shown to be eleven. By using basically the same approach, as used "~Now at Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, Pennsylvania 19104, U.S.A. 519 0022-460X/85/240519+ 11 $03.00/0
O 1985 Academic Press Inc. (London) Limited
520
E. S. R E D D Y A N D
A. K. M A L I K
in the present work, it is possible to formulate the problem in terms of unknown constants of the displacement solution. This renders the number of simultaneous equations to be solved equal to the number of support conditions. For the beam/plate problem considered in reference [9] this number is only four. The objective of the present work is to study the harmonic response and sound radiation characteristics of an infinitely long axially stiffened cylinder with constrained damping treatment. This is carded out by neglecting any variation in the axial direction. Thus the problem is simplified to a two dimensional version: i.e., to that of a three-layered ring. As the stiffeners are often regularly spaced, the ring is considered to be on periodic radial supports. The "wave" approach for vibration analysis of periodic structures used in the above literature [8-11] is extended to the analysis of the present problem. Aerospace and submerged structures are often excited by the loading due to pressure fluctuations in the turbulent boundary layer or impinging noise, etc. A harmonically time-varying radial loading can be expressed as an infinite series of circumferential space harmonics of integer wave numbers. In the present work, the analysis is carried out for a specific harmonic of the pressure loading, with consideration of the associated transmitted pressure into the cylindrical cavity and its coupling with the motion of the structure. Results are presented for the response (with and without the presence of the accoustic medium) and the sound power radiated with different interiors, namely, totally resonant, totally absorbent and partially absorbent. Numerical results are obtained for the case of three supports which are assumed to prevent both radial and tangential displacements of the base layer. Finally, the effectiveness of the constrained damping treatment is compared to that of an unconstrained (i.e., a two-layered case) -treatment. 2. THEORETICAL ANALYSIS 2.1. T R A N S M I ' V F E D ( I N T E R N A L ) A C O U S T I C F I E L D Consider an axially stiffened infinite cylinder subjected to an external harmonic pressure field po(O, t). The structure vibrates due to this pressure and thereby transmits a pressure field into its cavity called transmitted or internal acoustic field, p,(O, t) which is given by [1, 7]
p,( O, t)= ~ (an cos nO+ bn sin nO)Jn(klr) exp (itot),
(1)
n=O
where an and bn are constants to be determined, Jn is the Bessel function of first kind, k~ is the acoustic wave number of the inside medium, o~ is the exciting frequency and t is time. Equation (1) can be rewritten as
p,(O, t)= ~. Ptn e -in~ exp (itot) with P,n =A,Jlnl(klr),
(2,2a)
n~--oo
where the An's are complex constants anti the symbol Inl indicates the modulus of n. The pressure field given by equation (2) represents a resonant interior of the structure. For a non-resonant interior, the pressure field can be obtained by replacing the Bessel function (J,) in equation (2a) with the Hankel function of the first kind, tJtl) ,,n , [7] as
P,n = A~Hl~?(ktr),
(2b)
where H(~)= J. + iYn, with Y. t h e B essel function of the second kind. The solution given by equation (2b) represents an inward propagating wave that is totally absorbed in the
L A Y E R E D C Y L I N D E R WITH A X I A L S T I F F E N E R S
521
cylinder cavity. To study the partially absorbing cavity, a modified Hankel function is used [7]: i.e.,
Pt, =A,H.l~,~(klr),
where
H(,n = J . + i a Y . .
(2c, d)
In equation (2d), a = 1 corresponds to a totally absorbing interior and a = 0 to a totally resonant interior. Thus, the selection of an appropriate value of a can bridge the gap between the two extremes. 2.2. R E S P O N S E ANALYSIS Figure 1 shows a three-layered ring on N periodic radial supports. Layers 1 and 3 are elastic and layer 2 is assumed to be elastic/viscoelastic capable of transmitting only shear.
t
! r:uo'v
cLoyer
o(~, l, \ I i / /.-~,F'o~o,
I
"',,_ i. /"
-tX\~,~.
-'1t'- Zh,
!
<,~////"-
2 h2--dF-
Figure 1. Three-layered ring on equi-spaced radial supports.
With the lth (integer) space harmonic of the exciting pressure given by Poe i(''-z~ and the internal pressure taken into account, the governing differential equations for the ring in reference 1"12] can be modified to become
=(p,l,=R,-Poe i(<~t-to))bR3, (L2U+L4VI+LsVs)eI~'t=O, (LsU+LsV1+L6Vs)ei~~ (L~ U +
L 2 V I + L 3 V3) e ia't
(3a) (3b, c)
where the differential operators LI, L2,..., L6 and the other symbols are defined in Appendix A. The quantities U, V~, V3 respectively are the harmonic amplitudes of radial (u) and tangential (vx, v3) displacements oflayers 1 and 3: i.e., u = U exp(itot) and so on. It may be noted that in deriving the forcing function in equation (3a), the half thicknesses of the layers are assumed to be negligible compared to the midplane radius of layer 3.
522
E. S. R E D D Y
AND
A. K. MAL1K
The general solutions of equation (3) can be written as 8
U(O) = "~ Cj C,~ j=l
~
n=--co
)tln(Ptnlr=R3--PO~nl) e - i " ~
~7,(0)=j =Zl Bier~ n=--oo Z yI.(P,.I,=R-~oao,)e and
"r
where
y,. =
y3,(P,,I,=R,--PoS~) e-~"~
E /~C?+ j=l
-~"~
n=-oD
(4a) (4b) (4c)
"
[1619n4- (I6I,0+ I719)n2+ (I7I,0+I2)]/r
Y2. =[-I4Ign2-(IsIs-I4Ito)]in/r
Y3. =[-IsI6nI-(I4Is-IsIT)]in/r
r
U= U/R3, ~'l = VI/R3, "r = V3/R3, P,. = P,.b / D3,
Po = Pob/ D3,
The si"s are the roots of the auxiliary equation
$8-~ SIS6-1-$2s4.~-$3$2-~ S4~.0, 8,z is the K_ronecker delta function and the B/s, C/s and /~/s are constants to be determined. The I's and S's are defined in Appendix A. As mentioned in the previous section, the terms containing /st, in equation (4) are unknown and are related to the radial displacement, u(O, t) by a boundary condition on the transmitted pressure. By using this condition, / 5 can be eliminated from equation (4); the details are.given in the following section. 2.3. B O U N D A R Y C O N D I T I O N O N T R A N S M I T r E D PRESSURE The boundary condition on the transmitted pressure is given by [1]
ap,/ar],=R3 = -p~ a2u/at 2,
(5)
where p~ is the density of the acoustic medium inside the structure. For simplicity of analysis, the pressures are assumed to act on the midplane of the base layer and hence the pressure gradient is also evaluated at that plane. By assuming harmonic time variation of displacements and using equation (2), equation (5) can be transformed to (W
P,,lr=e, "HI"I(k, R3) e_i,, o = p~2FO. ....
(6)
.I:I t,, z (k~R3)
The following non-dimensional quantities are used in equation (6): the density parameter of the acoustic medium, fi~=p~bR~/m~; the frequency parameter, I't = tov'm'3/D3; the sound speed parameter, F = l'l/klR 3. The quantity m3 is defined in Appendix A. The prime (') denotes the derivative with respect to the argument. Since all the ring bays (beiween adjacent supports) are identical, and are subjected to the same exciting pressure field except for a phase difference, the responses in all bays must be identical apart from the same phase difference. In fact, considering any two adjacent bays, one can write u (0 + A, t) = e -u~ u (0, t), with )t = 2 ~r/N. This latter property, which applies to all adjacent pairs of bays, makes it possible to write U(0) in a series of space harmonics [9, 11], 0(0)
=
~ j=--eo
.~ - i . . o, , tJje
"
wherenj=l+jN.
(7, 8)
LAYERED
CYLINDER
WITH
AXIAL
523
STIFFENERS
Next, in the light of equation (7), equation (4a) is rewritten as follows where the transmitted pressure caused by U is also written in terms of the space harmonics: O(0)
=
Oue-i%~ - ~ "Ylnj('PmjIr=Rs-Pot~nt) e-i%0,
~ j ~ -co
(9)
n = -co
The Fourier coefficients Oij are given by
1 A Uu=A" fo
(k~='Cke'kO)ei%~
(10)
Similarly, the boundary condition given by equation (6) can be written as (I), .HI,: (k, R3) e_%O = ~I2FO. n ~'~ - o o /5t~l~=R3 HI,fi(gig3)
(11)
Substituting equation (9) in equation (11) and solving for/~,,,, one gets /Stnjlr=a~ = Xj( 01~ - Vl,,/508n,t), where
(12)
,.
Xj =
From here onwards the argument, klR3 (= I~/F), of the special functions is omitted. Now, with the pressure amplitudes known as functions of the C's the displacement solutions (equations (4a)-(4c)) can be written as 0(0)=
o
Ck e ' : + kffil
I?,(0)= ~ k=|
\
)
~]ke-i5~
E
--/5o~',,fae-i'~
(13a)
j=-oo
(Bke':+ ~
~:.~2ke-%~176
(13b)
~]k e-i'~ -/5o3'~,ft e -u~
(13c)
j=--eo
a(O)= ~ (Bk e "*~
~
k=l \
j=--oo
/
where ~r = ~,,,XjZsk, SC2k= V2~jX~Zjk,g:~k= Va~,XjZjk and fl = 1 + )'liXo, with Zl.k = [e O:i0x - 1 ] / ( S k "1"in/)A. It is interesting to note that these displacement solutions are very similar to those for the in vacuo response. The sum of an infinite series appearing as an additional term is due to the transmitted pressure and its coupling with the radial displacement. From the radial displacement the non-dimensional sound power radiated per unit radian from the ring of unit width into its cavity (SP) can be obtained as [11]
SP--Re [in k 2 j=_~ p,.,o;],
(14)
where U* is the complex conjugate of Oj and Re stands for "real part of". Substituting f o r / s j from equation (12) gives --a
2
9 (I) (I)~ - 2 sp_pla2 F./=-~o ~ Re[iHl,,jl//__/i,,jl]l~l '
or
-a
SP =p'I2 2
2
F
~.
a(Jl":Yl 3-J "'lYl'q), ,
j=-~o (JI-jl) + a (YI-:)
(15)
(16)
524
E . S . I~EDDY A N D A. K. MALIK
2.4. SUPPORT A N D WAVE C O N D I T I O N S
There can be different types of supports [13]: viz., (i) supports preventing both radial and tangential displacements, (ii) supports preventing only radial displacement, and (iii) flexible radial supports. The structure under consideration, being periodic in the 0-direction, has a set of coupling co-ordinates (between the adjacent periodic elements): namely, q,=Ou/O0, q2 = /)3, q3 = U and q4 =/)~- Then coupling co-ordinates and the associated coupling forces (the Qj's with j = 1, 2, 3, 4) can be identified from the boundary COnditions derived for a single periodic element through Hamilton's principle [12]. The expressions for these coupling forces are given in terms of the displacements and their derivatives in Appendix B.
For the pressure loading with an integer wave number I, the phase difference between pressures at two points a distance 3. radians apart is 13.. This phase difference may be regarded as the propagation constant of the loading [9]. At steady state, as stated earlier, the flexural wave motion excited by this loading has this propagation constant imposed upon it. The response quantities at corresponding points in adjacent bays then have the same amplitude but they differ in phase by 13.. In other words, Wave conditions to be satisfied by the non-zero coupling co-ordinates and the associated COupling forces at the supports are _ e-i/x q~lo~xqj o~o
and
Qjlo=x=
e-ilx
Qilo~o,
(17)
where the suffixj refers to a particular non-zero coupling co-ordinate or the corresponding coupling force. Depending upon the types of supports, zero-displacement conditions and the wave conditions can be applied. The use of these conditions in equation (13) results in a set of linear non-homogeneous equations which can be solved for eight independent unknown constants, the C's appearing in the displacement solutions. 3. RESULTS AND DISCUSSION Numerical results are presented for a ring on three supports. The SUpports are assumed to prevent both radial and tangential displacements of the base layer. This implies both q2 and q3 = 0 at 0 = 0 and 0 = 3.. The remaining four support conditions are obtained from equations (17). The following data has been used in the computation: 6,=0.015, 82= 0 . 0 3 0 , ~ 3 = 0 . 0 3 0 , E~/E3=I'O, p~/p3=l'O, G/E3=O'O001, p2/P3=0.2, N = 3 , /50= 1.0, /~ = 0.05, F = 5.0, a = 0.0, 0.25 and 1.0 and l = 0, 1,2 and 3. In the infinite series (equations (13) and (16)), 11 terms were considered in the computations. 3.1. I N VACUO RESPONSE With the effect of the transmitted pressure on the structure neglected, the response was computed for different harmonics, 1, of the external pressure. For a given value of 1, the structure does not resonate at every natural frequency. Table 1 shows various pressure harmonics which excite different modes for a three-layered ring. The natures of different modes mentio'ned in the table are based on the mode shapes obtained in reference [13]. It may be seen that only the modes which correspond to a free phase constant (/x~) equal to zero are excited by the wave numbers l = 0 and 3, whereas the other modes (with/.ti # 0) are excited with l = 1 and 2. This is because only an identical pressure distribution (produced by I = 0, 3, 6,...) in all the bays can excite the modes having identical deformation (implied by #~ = 0) in all the bays. Moreover, the wave numbers generating identical pressure distributions have both symmetric and antisymmetric components in their complex forms, and so both symmetric and antisym-
LAYERED
CYLINDER
525
WITH AXIAL STIFFENERS
TABLE 1
Pressure harmonics exciting various modes
Mode no.
Free phase constant
Natural frequency
1
0.0
5.522
2 3
2~'/3 0.0
8.262 11.768
4 5
27r/3 0.0
14.920 18.302
Nature of modet (radial disp.)
Pressure harmonics (l)
Antisymmetric and identical Degenerate Symmetric and identical Degenerate Symmetric and identical
--
--
--
3
---
1 --
2 --
-3
-0
1 --
2 --
-3
t Symmetric/antisymmetric means symmetric/antisymmetricin a bay; identical means identical deformation in all bays. m e t r i c m o d e s a r e e x c i t e d in a bay. T h e u n i f o r m p r e s s u r e (I = 0), h o w e v e r , b e i n g s y m m e t r i c in a b a y , c a n excite o n l y s y m m e t r i c m o d e s in a bay. S i m i l a r c h a r a c t e r i s t i c s were o b s e r v e d for a t w o - l a y e r e d ring a n d with o t h e r t y p e s o f s u p p o r t c o n d i t i o n s [13]. In g e n e r a l it can be c o n c l u d e d that (i) the p r e s s u r e h a r m o n i c s 1 = i N , j = 1, 2 . . . . . c a n excite m o d e s having i d e n t i c a l s h a p e s in e a c h b a y w h e r e a s the o t h e r h a r m o n i c s can excite the r e m a i n i n g d e g e n e r a t e m o d e s [12, 13], a n d (ii) a u n i f o r m presstire d i s t r i b u t i o n (i.e., 1= 0) c a n excite o n l y ! h e s y m m e t r i c m o d e s with i d e n t i c a l d e f o r m a t i o n s in each bay. T h e r e s p o n s e o f the structure with a n d w i t h o u t d a m p i n g in the core is s h o w n in F i g u r e 2. F o r a relatively t h i c k core, the structure u n d e r c o n s i d e r a t i o n has a p r e d o m i n a n t t a n g e n t i a l m o d e [12]. As this m o d e is also a s s o c i a t e d with s o m e r a d i a l d i s p l a c e m e n t s , it 10 0
10-1 ~r -%
1
I
I
I
I
I
I
I
I
I
1
I
I
I
9
11
15
15
17
19
21
A
I:D
10-2
10-3 5
7
Figure 2. In vacuo response of three-layered ring, i = 3.
, fl = 0, - -~-, fl = 1.0.
526
E. S. R E D D Y
AND
A, K. M A L 1 K
gets excited by the radial pressure loading. The centre peak in the figure corresponds to this mode. It can be clearly seen that when viscoelastic damping (characterized by the loss factor/3) is introduced in the core this peak vanishes altogether, indicating the higher effectiveness of damping in the tangential mode. At certain frequencies, depending on the value o f l, the response is negligible. These frequencies are called antiresonances [14]. It may also be noted that viscoelastic damping is effective in controlling the resonance response whereas it is detrimental near the antiresonances as expected.
I~"
"~" 10-3
I0-4[ 3
I 5
I
I
I
I II
7
9
11
13
II 15
I
I
I
17
19
21
F i g u r e 3. C o m p a r i s o n o f r e s p o n s e f o r t w o - a n d t h r e e - l a y e r e d t i n g s , l = 1,
. , / 3 = 0 . 0 ; - - - , / 33 = 1 . 0 .
Figure 3 compares the effectiveness o f damping treatment with unconstrained and constrained layers. For the sake of comparison, thicknesses of the core and base layers in the three-layered ring are chosen to be the same as those of the outer and base layers in the two-layered ring. The thickness of the outer layer o f the three-layered ring is arbitrarily taken as half of that for the base layer. Even with other types of support conditions [ 13], it was found that the constrained layer damping treatment is more effective in controlling the resonant response. 3.2. RESPONSE AND SOUND RADIATION IN AN ACOUSTIC FIELD Figures 4(a) and (b) show the response of the ring in the presence of an acoustic field inside the structure. The ettect of the core damping on the response was found to be the same as that for the in v a c u o response and hence is not shown. When the interior is considered to be totally resonant (a = 0.0) it can be seen that resonance peaks in addition to the ones shown for the in vacuo response are present. These are due to the coupling between the vibration of the structure and the wav e motion of the acoustic medium within the structure. The frequencies corresponding to these additional peaks can be identified to be the cavity resonances which are given by J ' ( k t R 3 ) = 0 [1,7]. The other peak frequencies corresponding to the natural modes of the structure are shifted to values
527
LAYERED C Y L I N D E R WITH AXIAL STIFFENERS 10 -1
I
I
d,
I
I
I
I
I
I
10-2
10-3 _
1 0 -4
lOI~
I
I
I
I
(b)
10-1~
10-21 10-33/
5I
7
/ 13 15 17 19 21 9 11 .t2
Figure 4. Response of three-layered ring in the presence of the acoustic medium. (a) l = 1;/% a = 0.25; O, , a=0-0;/3 =0.
a=l.OO;(b) l= 3; O, a=O.25; tq, a = l.O0.
lower than the natural frequencies. This shift is marginal if a cavity resonance frequency is close to that natural frequency and is negligible otherwise. However, with an absorbing interior the cavity resonances disappear and only the natural modes are excited exactly at the natural frequencies of the structure. A totally absorbing interior (a = 1-0) is seen to result in higher peaks as compared to a partially absorbing interior (a = 0.25). Away from the peaks, the response for both values o f a is nearly the same. The peak corresponding to the tangential mode discussed in the previous section is present irrespective of the characteristics of the interior. From equation (16), it is obvious that for a resonant interior (a = 0.0) the sound power radiated ( S P ) is zero. With a = 0.25 and a = 1-0 the variation of SP with frequency for l = 3 is shown in Figure 5. It may be observed that S P is maximum at the resonance frequencies where the in vacuo response is also a maximum. Moreover, this maximum has an optimum for an intermediate value of a ( 0 < a < 1.0). With the introduction o f damping (fl = 1.0) in the structure, S P is reduced considerably only at the peaks, as expected.
528
E. S. R E D D Y
i
A N D A. K. M A L I K
i
I
II
D __1
6
0 --I
o
~
II
o
I
._1 ._1
I To
i b
I b
I b
b
~" II
dF
cq
c~ II
lili
-
,
~o
~
}
~ .~ ~ < 3
! 0 9. ~
.,./
.
oo
))
O
II
~
~
o
~
o.~ o~ ~
II
I
I
I
I
o
b
b
b
i
o
II
LAYERED
CYLINDER
WITH
AXIAL STIFFENERS
529
The performances of unconstrained and constrained layer damping treatments in the reduction of S P are compared in Figure 6. The constrained layer treatment is obviously seen to be more effective in reducing the sound power radiated at resonances. 4. CONCLUSIONS Not all the modes of a ring on periodic radial supports can be excited by a single space harmonic of the pressure loading. Only identical pressure distributions in each bay (i.e., the pressure harmonic is equal to an integral multiple of the number of supports) can excite the unique modes (modes with a phase difference of zero). A uniform pressure distribution (i.e., the zeroth harmonic) excites only those modes which are identical in each bay and symmetric between the supports within one bay. The remaining degenerate modes are excited by the other pressure harmonics as their shapes are neither unique nor identical in all the bays. For a three-layered ring, the predominant tangential mode responds to the pressure loading depending on the mode shape of the radial displacement associated with it. However, the corresponding resonance is readily controlled by the presence ofviscoelastic damping in the core layer. The space harmonic series is useful for studying the sound-structure interaction if the structure is a so called "periodic structure". In the presence of a resonant acoustic medium within the structure, the resonance peaks are shifted to values lower than the structure's natural frequencies. This shift is very marginal and its amount depends on the proximity of a cavity resonance. However, this is not the case xvith an absorbing interior. The sound power radiation is m a x i m u m at the resonance frequencies where the in vacuo response is also maximum. Introduction of damping in the structure effectively reduces the response as well as the sound power radiation near the resonances. REFERENCES 1. J. H. FOXWELL and R. E. FRANKLIN 1959 The Aeronautical Quarterly 10, 47-64. The vibration of a thin-walled stiffened cylinder in an acoustic field. 2. P. W. SMITH JR. 1957 Journal of the Acoustical Society of America 29, 721-729. Sound transmission through thin cylindrical shells. 3. L. R. KOVAL 1976 Journal of Sound and Vibration 48, 265-275. On sound transmission into cylindrical shell under "flight conditions". 4. L. R. KOVAL 1978 Journal of Sound and Vibration 57, 155-156. On sound transmission into heavily-damped cylinder. 5. L. R. KOVAL 1977 American Institute of Aeronautics and Astronautics Journal 15, 899-900. Effect of stiffening on sound transmission into a cylindrical shell in flight. 6. L. R. KOVAL 1978 Journal of Aircraft 15, 816-821. Effect of longitudinal stringers on sound transmission into a thin cylindrical shell. 7. L. R. KOVAL 1978 Journal of Sound and Vibration 59, 23-33. Effect of cavity resonances on sound transmission into a thin cylindrical shell. 8. D. J. MEAD 1971 Journal of Engineering for'Industry, Transactions of the American Society of Mechanical Engineers 93, 783-792. Vibration response and wave propagation in periodic structures. 9. D. J. MEAD and K. K. PUJARA 1971 Journal of Sound and Vibration 14, 525-541. Space harmonic analYsis of periodically supported beams: Response to convected random loading. 10. D.J. MEAD and A. K. MALLIK 1976 Journal of Sound and Vibration 47,457-471. An approximate method of predicting the response of periodically supported beams subjected to random eonvected loading. 11. D.J. MEAD and A. K. MALLIK 1978 Journal of Sound Vibration 61,315-326. An approximate theory for the sound radiated from a periodic line-supported plate.
530
E.S. REDDY AND A. K. MALIK
12. E. S. REDDY and A. K. MALLIK 1984 American Institute of Aeronautics and Astronautics Journal 22, 543-551. Vibration of a three-layered ring on periodic radial supports. 13. E.S. REDDY 1982 Ph.D. Thesis, Indian Institute of Technology, Kanpur, India. In-plane vibrations of layered rings on periodic radial supports. 14. J. C. SNOX,VDON 1968 Vibration and Shock in Damped Mechanical Systems. New York: John Wiley. APPENDIX A The differential operators L,, L 2 , . . . , L6 are
L~= l t - d ' / d O ' + 1 2 d 2 / d O 2 + L3=Isd/dO,
h,
L4=16d/dO+lT,
L2= I4d/dO,
Ls=Is, L6=Igd2/d02+Ilo 9
The constant S's and I ' s are given by & = (/d6/~o+ I d d g + 1d619)/Iddg,
$2 = ( I~ITI~o + I316Is + I216I~0 + I21719- I l I 2 - I2419- I216)/ I~I619, $3 = (i217110 + i316ito+ i31719 _ i212_ I , l 2~o- I 2sI~ + 2IAsls) I d6t~,
& = (I3Id,o" I3I~)/Idd~, t~ = D, I 2 = 2 D - g * d 2 + ( ( 0 2 / 2 ) m 2 d 2,
13 = D § (0 2
I4= K ~ - g * d l d 2 + ( w 2 / 2 ) m d l d 4 , I7 = (02(ml +89
19 = K~, Here D = Di + D3, d2 = t~l + 2t~2+ t~3,
mj = 2blb/~Rj,
-
I5 = K 3 + g * d 2 d 3 + ' -~ m2d3d4,
d2) - g*d 2,
I6 ~ g l ,
Is = (g* +89
1,o = (0:(m~ §189
g*d~.
Kj = EiA ff Rj,
i)i = ~EjbSj, 2 3
d,,= 8 ~ - 83,
d 3 = l+t~3,
g* = g(1 + ifl),
m = mt + m2 + m3,
dl = 1 - ~ , 8j=hj/Rj, g = Gb/262.
Ej is the Young's modulus, Aj is the area of cross section, Rj is the radius o f the midplane, hj is the half thickness, pj is the density, and the subscript j indicates the j t h layer. G and fl are respectively the shear modulus and the shear loss factor of the second layer and b is the width of the ring.
APPENDIX B The harmonic amplitudes .of the coupling forces Qj are d2
Oleo [ ~ _
0=0
3,o o [ K3(-+ dO,j 11o o
d 89
LAYERED
-
CYLINDER
r
((
L
\dO
WITH
AXIAL STIFFENERS
d _____~ U3 d
da+dUd2~
dO3]-gd2\V'd'-
H e r e Qs = 0j ei~~ (J = 1, 2, 3, 4), u = U e i~ot, vl = V I e i'~ a n d v3 = V3 e i'~
531
RESPONSE OF AND S O U N D RADIATION FROM A LAYERED CYLINDER WITH REGULAR AXIAL STIFFENERS E. S. REDDY'~ Department of Mechanical and Aerospace Engineering, Arizona State University Tempe, Arizona 85287, U.S.A. AND
A. K. MALLIK Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208 016, India (Received 18 May 1984, and in revised form 12 December 1984) Response and sound radiation characteristics of a three-layered cylinder with equispaced (periodic) axial stiffeners are obtained using the "wave" approach. A two-dimensional situation with no variation in the axial direction is considered. Numerical results are presented for the case of three stiffeners, which are assumed to prevent both radial and tangential displacements of the base layer. The pressure harmonics exciting various modes are analyzed. Three ~litIerent types of interior, namely, totally resonant or absorbent and partially absorbent, are investigated and their effects on the response and sound power radiation are studied. 1. INTRODUCTION Axially stiffened cylinders are used in aerospace and underwater structures. These applications have led researchers to study the response and sound transmission characteristics of such structures. Relatively simplified problems have been studied by m a n y investigators as the presence of stiffeners makes the exact analysis difficult. Foxwell and Franklin [1] have studied an idealized problem of sound transmission in which a plane wave incident normally on an infinitely long cylinder was considered. For simplicity, only the ring (frame) type stiffeners were included in the model and the interior of the structure was assumed to be totally resonant. The problem of a totally absorbent interior was analysed by Smith [2] and Koval [3, 4]. By improving the smeared stiffener theory, Koval [5, 6] included the effect of axial stiffeners. In actual practice, however, the interior is neither totally resonant nor totally absorbent. In a later p a p e r [7], Koval considered the partially absorbing nature of the cavity while studying the effect of cavity resonances on sound transmission. Sound-structure interaction studies of periodic structures have been attempted by only a few investigators [8-11]. The response o f and sound radiated from a periodic plate have been solved exactly by introducing a space harmonic series in reference [9]. In this approach, the number o f simultaneous equations to be solved is equal to the number o f terms chosen in the infinite series. For adequate convergence, however, the number o f terms required were shown to be eleven. By using basically the same approach, as used "~Now at Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, Pennsylvania 19104, U.S.A. 519 0022-460X/85/240519+ 11 $03.00/0
O 1985 Academic Press Inc. (London) Limited
520
E. S. R E D D Y A N D
A. K. M A L I K
in the present work, it is possible to formulate the problem in terms of unknown constants of the displacement solution. This renders the number of simultaneous equations to be solved equal to the number of support conditions. For the beam/plate problem considered in reference [9] this number is only four. The objective of the present work is to study the harmonic response and sound radiation characteristics of an infinitely long axially stiffened cylinder with constrained damping treatment. This is carded out by neglecting any variation in the axial direction. Thus the problem is simplified to a two dimensional version: i.e., to that of a three-layered ring. As the stiffeners are often regularly spaced, the ring is considered to be on periodic radial supports. The "wave" approach for vibration analysis of periodic structures used in the above literature [8-11] is extended to the analysis of the present problem. Aerospace and submerged structures are often excited by the loading due to pressure fluctuations in the turbulent boundary layer or impinging noise, etc. A harmonically time-varying radial loading can be expressed as an infinite series of circumferential space harmonics of integer wave numbers. In the present work, the analysis is carried out for a specific harmonic of the pressure loading, with consideration of the associated transmitted pressure into the cylindrical cavity and its coupling with the motion of the structure. Results are presented for the response (with and without the presence of the accoustic medium) and the sound power radiated with different interiors, namely, totally resonant, totally absorbent and partially absorbent. Numerical results are obtained for the case of three supports which are assumed to prevent both radial and tangential displacements of the base layer. Finally, the effectiveness of the constrained damping treatment is compared to that of an unconstrained (i.e., a two-layered case) -treatment. 2. THEORETICAL ANALYSIS 2.1. T R A N S M I ' V F E D ( I N T E R N A L ) A C O U S T I C F I E L D Consider an axially stiffened infinite cylinder subjected to an external harmonic pressure field po(O, t). The structure vibrates due to this pressure and thereby transmits a pressure field into its cavity called transmitted or internal acoustic field, p,(O, t) which is given by [1, 7]
p,( O, t)= ~ (an cos nO+ bn sin nO)Jn(klr) exp (itot),
(1)
n=O
where an and bn are constants to be determined, Jn is the Bessel function of first kind, k~ is the acoustic wave number of the inside medium, o~ is the exciting frequency and t is time. Equation (1) can be rewritten as
p,(O, t)= ~. Ptn e -in~ exp (itot) with P,n =A,Jlnl(klr),
(2,2a)
n~--oo
where the An's are complex constants anti the symbol Inl indicates the modulus of n. The pressure field given by equation (2) represents a resonant interior of the structure. For a non-resonant interior, the pressure field can be obtained by replacing the Bessel function (J,) in equation (2a) with the Hankel function of the first kind, tJtl) ,,n , [7] as
P,n = A~Hl~?(ktr),
(2b)
where H(~)= J. + iYn, with Y. t h e B essel function of the second kind. The solution given by equation (2b) represents an inward propagating wave that is totally absorbed in the
L A Y E R E D C Y L I N D E R WITH A X I A L S T I F F E N E R S
521
cylinder cavity. To study the partially absorbing cavity, a modified Hankel function is used [7]: i.e.,
Pt, =A,H.l~,~(klr),
where
H(,n = J . + i a Y . .
(2c, d)
In equation (2d), a = 1 corresponds to a totally absorbing interior and a = 0 to a totally resonant interior. Thus, the selection of an appropriate value of a can bridge the gap between the two extremes. 2.2. R E S P O N S E ANALYSIS Figure 1 shows a three-layered ring on N periodic radial supports. Layers 1 and 3 are elastic and layer 2 is assumed to be elastic/viscoelastic capable of transmitting only shear.
t
! r:uo'v
cLoyer
o(~, l, \ I i / /.-~,F'o~o,
I
"',,_ i. /"
-tX\~,~.
-'1t'- Zh,
!
<,~////"-
2 h2--dF-
Figure 1. Three-layered ring on equi-spaced radial supports.
With the lth (integer) space harmonic of the exciting pressure given by Poe i(''-z~ and the internal pressure taken into account, the governing differential equations for the ring in reference 1"12] can be modified to become
=(p,l,=R,-Poe i(<~t-to))bR3, (L2U+L4VI+LsVs)eI~'t=O, (LsU+LsV1+L6Vs)ei~~ (L~ U +
L 2 V I + L 3 V3) e ia't
(3a) (3b, c)
where the differential operators LI, L2,..., L6 and the other symbols are defined in Appendix A. The quantities U, V~, V3 respectively are the harmonic amplitudes of radial (u) and tangential (vx, v3) displacements oflayers 1 and 3: i.e., u = U exp(itot) and so on. It may be noted that in deriving the forcing function in equation (3a), the half thicknesses of the layers are assumed to be negligible compared to the midplane radius of layer 3.
522
E. S. R E D D Y
AND
A. K. MAL1K
The general solutions of equation (3) can be written as 8
U(O) = "~ Cj C,~ j=l
~
n=--co
)tln(Ptnlr=R3--PO~nl) e - i " ~
~7,(0)=j =Zl Bier~ n=--oo Z yI.(P,.I,=R-~oao,)e and
"r
where
y,. =
y3,(P,,I,=R,--PoS~) e-~"~
E /~C?+ j=l
-~"~
n=-oD
(4a) (4b) (4c)
"
[1619n4- (I6I,0+ I719)n2+ (I7I,0+I2)]/r
Y2. =[-I4Ign2-(IsIs-I4Ito)]in/r
Y3. =[-IsI6nI-(I4Is-IsIT)]in/r
r
U= U/R3, ~'l = VI/R3, "r = V3/R3, P,. = P,.b / D3,
Po = Pob/ D3,
The si"s are the roots of the auxiliary equation
$8-~ SIS6-1-$2s4.~-$3$2-~ S4~.0, 8,z is the K_ronecker delta function and the B/s, C/s and /~/s are constants to be determined. The I's and S's are defined in Appendix A. As mentioned in the previous section, the terms containing /st, in equation (4) are unknown and are related to the radial displacement, u(O, t) by a boundary condition on the transmitted pressure. By using this condition, / 5 can be eliminated from equation (4); the details are.given in the following section. 2.3. B O U N D A R Y C O N D I T I O N O N T R A N S M I T r E D PRESSURE The boundary condition on the transmitted pressure is given by [1]
ap,/ar],=R3 = -p~ a2u/at 2,
(5)
where p~ is the density of the acoustic medium inside the structure. For simplicity of analysis, the pressures are assumed to act on the midplane of the base layer and hence the pressure gradient is also evaluated at that plane. By assuming harmonic time variation of displacements and using equation (2), equation (5) can be transformed to (W
P,,lr=e, "HI"I(k, R3) e_i,, o = p~2FO. ....
(6)
.I:I t,, z (k~R3)
The following non-dimensional quantities are used in equation (6): the density parameter of the acoustic medium, fi~=p~bR~/m~; the frequency parameter, I't = tov'm'3/D3; the sound speed parameter, F = l'l/klR 3. The quantity m3 is defined in Appendix A. The prime (') denotes the derivative with respect to the argument. Since all the ring bays (beiween adjacent supports) are identical, and are subjected to the same exciting pressure field except for a phase difference, the responses in all bays must be identical apart from the same phase difference. In fact, considering any two adjacent bays, one can write u (0 + A, t) = e -u~ u (0, t), with )t = 2 ~r/N. This latter property, which applies to all adjacent pairs of bays, makes it possible to write U(0) in a series of space harmonics [9, 11], 0(0)
=
~ j=--eo
.~ - i . . o, , tJje
"
wherenj=l+jN.
(7, 8)
LAYERED
CYLINDER
WITH
AXIAL
523
STIFFENERS
Next, in the light of equation (7), equation (4a) is rewritten as follows where the transmitted pressure caused by U is also written in terms of the space harmonics: O(0)
=
Oue-i%~ - ~ "Ylnj('PmjIr=Rs-Pot~nt) e-i%0,
~ j ~ -co
(9)
n = -co
The Fourier coefficients Oij are given by
1 A Uu=A" fo
(k~='Cke'kO)ei%~
(10)
Similarly, the boundary condition given by equation (6) can be written as (I), .HI,: (k, R3) e_%O = ~I2FO. n ~'~ - o o /5t~l~=R3 HI,fi(gig3)
(11)
Substituting equation (9) in equation (11) and solving for/~,,,, one gets /Stnjlr=a~ = Xj( 01~ - Vl,,/508n,t), where
(12)
,.
Xj =
From here onwards the argument, klR3 (= I~/F), of the special functions is omitted. Now, with the pressure amplitudes known as functions of the C's the displacement solutions (equations (4a)-(4c)) can be written as 0(0)=
o
Ck e ' : + kffil
I?,(0)= ~ k=|
\
)
~]ke-i5~
E
--/5o~',,fae-i'~
(13a)
j=-oo
(Bke':+ ~
~:.~2ke-%~176
(13b)
~]k e-i'~ -/5o3'~,ft e -u~
(13c)
j=--eo
a(O)= ~ (Bk e "*~
~
k=l \
j=--oo
/
where ~r = ~,,,XjZsk, SC2k= V2~jX~Zjk,g:~k= Va~,XjZjk and fl = 1 + )'liXo, with Zl.k = [e O:i0x - 1 ] / ( S k "1"in/)A. It is interesting to note that these displacement solutions are very similar to those for the in vacuo response. The sum of an infinite series appearing as an additional term is due to the transmitted pressure and its coupling with the radial displacement. From the radial displacement the non-dimensional sound power radiated per unit radian from the ring of unit width into its cavity (SP) can be obtained as [11]
SP--Re [in k 2 j=_~ p,.,o;],
(14)
where U* is the complex conjugate of Oj and Re stands for "real part of". Substituting f o r / s j from equation (12) gives --a
2
9 (I) (I)~ - 2 sp_pla2 F./=-~o ~ Re[iHl,,jl//__/i,,jl]l~l '
or
-a
SP =p'I2 2
2
F
~.
a(Jl":Yl 3-J "'lYl'q), ,
j=-~o (JI-jl) + a (YI-:)
(15)
(16)
524
E . S . I~EDDY A N D A. K. MALIK
2.4. SUPPORT A N D WAVE C O N D I T I O N S
There can be different types of supports [13]: viz., (i) supports preventing both radial and tangential displacements, (ii) supports preventing only radial displacement, and (iii) flexible radial supports. The structure under consideration, being periodic in the 0-direction, has a set of coupling co-ordinates (between the adjacent periodic elements): namely, q,=Ou/O0, q2 = /)3, q3 = U and q4 =/)~- Then coupling co-ordinates and the associated coupling forces (the Qj's with j = 1, 2, 3, 4) can be identified from the boundary COnditions derived for a single periodic element through Hamilton's principle [12]. The expressions for these coupling forces are given in terms of the displacements and their derivatives in Appendix B.
For the pressure loading with an integer wave number I, the phase difference between pressures at two points a distance 3. radians apart is 13.. This phase difference may be regarded as the propagation constant of the loading [9]. At steady state, as stated earlier, the flexural wave motion excited by this loading has this propagation constant imposed upon it. The response quantities at corresponding points in adjacent bays then have the same amplitude but they differ in phase by 13.. In other words, Wave conditions to be satisfied by the non-zero coupling co-ordinates and the associated COupling forces at the supports are _ e-i/x q~lo~xqj o~o
and
Qjlo=x=
e-ilx
Qilo~o,
(17)
where the suffixj refers to a particular non-zero coupling co-ordinate or the corresponding coupling force. Depending upon the types of supports, zero-displacement conditions and the wave conditions can be applied. The use of these conditions in equation (13) results in a set of linear non-homogeneous equations which can be solved for eight independent unknown constants, the C's appearing in the displacement solutions. 3. RESULTS AND DISCUSSION Numerical results are presented for a ring on three supports. The SUpports are assumed to prevent both radial and tangential displacements of the base layer. This implies both q2 and q3 = 0 at 0 = 0 and 0 = 3.. The remaining four support conditions are obtained from equations (17). The following data has been used in the computation: 6,=0.015, 82= 0 . 0 3 0 , ~ 3 = 0 . 0 3 0 , E~/E3=I'O, p~/p3=l'O, G/E3=O'O001, p2/P3=0.2, N = 3 , /50= 1.0, /~ = 0.05, F = 5.0, a = 0.0, 0.25 and 1.0 and l = 0, 1,2 and 3. In the infinite series (equations (13) and (16)), 11 terms were considered in the computations. 3.1. I N VACUO RESPONSE With the effect of the transmitted pressure on the structure neglected, the response was computed for different harmonics, 1, of the external pressure. For a given value of 1, the structure does not resonate at every natural frequency. Table 1 shows various pressure harmonics which excite different modes for a three-layered ring. The natures of different modes mentio'ned in the table are based on the mode shapes obtained in reference [13]. It may be seen that only the modes which correspond to a free phase constant (/x~) equal to zero are excited by the wave numbers l = 0 and 3, whereas the other modes (with/.ti # 0) are excited with l = 1 and 2. This is because only an identical pressure distribution (produced by I = 0, 3, 6,...) in all the bays can excite the modes having identical deformation (implied by #~ = 0) in all the bays. Moreover, the wave numbers generating identical pressure distributions have both symmetric and antisymmetric components in their complex forms, and so both symmetric and antisym-
LAYERED
CYLINDER
525
WITH AXIAL STIFFENERS
TABLE 1
Pressure harmonics exciting various modes
Mode no.
Free phase constant
Natural frequency
1
0.0
5.522
2 3
2~'/3 0.0
8.262 11.768
4 5
27r/3 0.0
14.920 18.302
Nature of modet (radial disp.)
Pressure harmonics (l)
Antisymmetric and identical Degenerate Symmetric and identical Degenerate Symmetric and identical
--
--
--
3
---
1 --
2 --
-3
-0
1 --
2 --
-3
t Symmetric/antisymmetric means symmetric/antisymmetricin a bay; identical means identical deformation in all bays. m e t r i c m o d e s a r e e x c i t e d in a bay. T h e u n i f o r m p r e s s u r e (I = 0), h o w e v e r , b e i n g s y m m e t r i c in a b a y , c a n excite o n l y s y m m e t r i c m o d e s in a bay. S i m i l a r c h a r a c t e r i s t i c s were o b s e r v e d for a t w o - l a y e r e d ring a n d with o t h e r t y p e s o f s u p p o r t c o n d i t i o n s [13]. In g e n e r a l it can be c o n c l u d e d that (i) the p r e s s u r e h a r m o n i c s 1 = i N , j = 1, 2 . . . . . c a n excite m o d e s having i d e n t i c a l s h a p e s in e a c h b a y w h e r e a s the o t h e r h a r m o n i c s can excite the r e m a i n i n g d e g e n e r a t e m o d e s [12, 13], a n d (ii) a u n i f o r m presstire d i s t r i b u t i o n (i.e., 1= 0) c a n excite o n l y ! h e s y m m e t r i c m o d e s with i d e n t i c a l d e f o r m a t i o n s in each bay. T h e r e s p o n s e o f the structure with a n d w i t h o u t d a m p i n g in the core is s h o w n in F i g u r e 2. F o r a relatively t h i c k core, the structure u n d e r c o n s i d e r a t i o n has a p r e d o m i n a n t t a n g e n t i a l m o d e [12]. As this m o d e is also a s s o c i a t e d with s o m e r a d i a l d i s p l a c e m e n t s , it 10 0
10-1 ~r -%
1
I
I
I
I
I
I
I
I
I
1
I
I
I
9
11
15
15
17
19
21
A
I:D
10-2
10-3 5
7
Figure 2. In vacuo response of three-layered ring, i = 3.
, fl = 0, - -~-, fl = 1.0.
526
E. S. R E D D Y
AND
A, K. M A L 1 K
gets excited by the radial pressure loading. The centre peak in the figure corresponds to this mode. It can be clearly seen that when viscoelastic damping (characterized by the loss factor/3) is introduced in the core this peak vanishes altogether, indicating the higher effectiveness of damping in the tangential mode. At certain frequencies, depending on the value o f l, the response is negligible. These frequencies are called antiresonances [14]. It may also be noted that viscoelastic damping is effective in controlling the resonance response whereas it is detrimental near the antiresonances as expected.
I~"
"~" 10-3
I0-4[ 3
I 5
I
I
I
I II
7
9
11
13
II 15
I
I
I
17
19
21
F i g u r e 3. C o m p a r i s o n o f r e s p o n s e f o r t w o - a n d t h r e e - l a y e r e d t i n g s , l = 1,
. , / 3 = 0 . 0 ; - - - , / 33 = 1 . 0 .
Figure 3 compares the effectiveness o f damping treatment with unconstrained and constrained layers. For the sake of comparison, thicknesses of the core and base layers in the three-layered ring are chosen to be the same as those of the outer and base layers in the two-layered ring. The thickness of the outer layer o f the three-layered ring is arbitrarily taken as half of that for the base layer. Even with other types of support conditions [ 13], it was found that the constrained layer damping treatment is more effective in controlling the resonant response. 3.2. RESPONSE AND SOUND RADIATION IN AN ACOUSTIC FIELD Figures 4(a) and (b) show the response of the ring in the presence of an acoustic field inside the structure. The ettect of the core damping on the response was found to be the same as that for the in v a c u o response and hence is not shown. When the interior is considered to be totally resonant (a = 0.0) it can be seen that resonance peaks in addition to the ones shown for the in vacuo response are present. These are due to the coupling between the vibration of the structure and the wav e motion of the acoustic medium within the structure. The frequencies corresponding to these additional peaks can be identified to be the cavity resonances which are given by J ' ( k t R 3 ) = 0 [1,7]. The other peak frequencies corresponding to the natural modes of the structure are shifted to values
527
LAYERED C Y L I N D E R WITH AXIAL STIFFENERS 10 -1
I
I
d,
I
I
I
I
I
I
10-2
10-3 _
1 0 -4
lOI~
I
I
I
I
(b)
10-1~
10-21 10-33/
5I
7
/ 13 15 17 19 21 9 11 .t2
Figure 4. Response of three-layered ring in the presence of the acoustic medium. (a) l = 1;/% a = 0.25; O, , a=0-0;/3 =0.
a=l.OO;(b) l= 3; O, a=O.25; tq, a = l.O0.
lower than the natural frequencies. This shift is marginal if a cavity resonance frequency is close to that natural frequency and is negligible otherwise. However, with an absorbing interior the cavity resonances disappear and only the natural modes are excited exactly at the natural frequencies of the structure. A totally absorbing interior (a = 1-0) is seen to result in higher peaks as compared to a partially absorbing interior (a = 0.25). Away from the peaks, the response for both values o f a is nearly the same. The peak corresponding to the tangential mode discussed in the previous section is present irrespective of the characteristics of the interior. From equation (16), it is obvious that for a resonant interior (a = 0.0) the sound power radiated ( S P ) is zero. With a = 0.25 and a = 1-0 the variation of SP with frequency for l = 3 is shown in Figure 5. It may be observed that S P is maximum at the resonance frequencies where the in vacuo response is also a maximum. Moreover, this maximum has an optimum for an intermediate value of a ( 0 < a < 1.0). With the introduction o f damping (fl = 1.0) in the structure, S P is reduced considerably only at the peaks, as expected.
528
E. S. R E D D Y
i
A N D A. K. M A L I K
i
I
II
D __1
6
0 --I
o
~
II
o
I
._1 ._1
I To
i b
I b
I b
b
~" II
dF
cq
c~ II
lili
-
,
~o
~
}
~ .~ ~ < 3
! 0 9. ~
.,./
.
oo
))
O
II
~
~
o
~
o.~ o~ ~
II
I
I
I
I
o
b
b
b
i
o
II
LAYERED
CYLINDER
WITH
AXIAL STIFFENERS
529
The performances of unconstrained and constrained layer damping treatments in the reduction of S P are compared in Figure 6. The constrained layer treatment is obviously seen to be more effective in reducing the sound power radiated at resonances. 4. CONCLUSIONS Not all the modes of a ring on periodic radial supports can be excited by a single space harmonic of the pressure loading. Only identical pressure distributions in each bay (i.e., the pressure harmonic is equal to an integral multiple of the number of supports) can excite the unique modes (modes with a phase difference of zero). A uniform pressure distribution (i.e., the zeroth harmonic) excites only those modes which are identical in each bay and symmetric between the supports within one bay. The remaining degenerate modes are excited by the other pressure harmonics as their shapes are neither unique nor identical in all the bays. For a three-layered ring, the predominant tangential mode responds to the pressure loading depending on the mode shape of the radial displacement associated with it. However, the corresponding resonance is readily controlled by the presence ofviscoelastic damping in the core layer. The space harmonic series is useful for studying the sound-structure interaction if the structure is a so called "periodic structure". In the presence of a resonant acoustic medium within the structure, the resonance peaks are shifted to values lower than the structure's natural frequencies. This shift is very marginal and its amount depends on the proximity of a cavity resonance. However, this is not the case xvith an absorbing interior. The sound power radiation is m a x i m u m at the resonance frequencies where the in vacuo response is also maximum. Introduction of damping in the structure effectively reduces the response as well as the sound power radiation near the resonances. REFERENCES 1. J. H. FOXWELL and R. E. FRANKLIN 1959 The Aeronautical Quarterly 10, 47-64. The vibration of a thin-walled stiffened cylinder in an acoustic field. 2. P. W. SMITH JR. 1957 Journal of the Acoustical Society of America 29, 721-729. Sound transmission through thin cylindrical shells. 3. L. R. KOVAL 1976 Journal of Sound and Vibration 48, 265-275. On sound transmission into cylindrical shell under "flight conditions". 4. L. R. KOVAL 1978 Journal of Sound and Vibration 57, 155-156. On sound transmission into heavily-damped cylinder. 5. L. R. KOVAL 1977 American Institute of Aeronautics and Astronautics Journal 15, 899-900. Effect of stiffening on sound transmission into a cylindrical shell in flight. 6. L. R. KOVAL 1978 Journal of Aircraft 15, 816-821. Effect of longitudinal stringers on sound transmission into a thin cylindrical shell. 7. L. R. KOVAL 1978 Journal of Sound and Vibration 59, 23-33. Effect of cavity resonances on sound transmission into a thin cylindrical shell. 8. D. J. MEAD 1971 Journal of Engineering for'Industry, Transactions of the American Society of Mechanical Engineers 93, 783-792. Vibration response and wave propagation in periodic structures. 9. D. J. MEAD and K. K. PUJARA 1971 Journal of Sound and Vibration 14, 525-541. Space harmonic analYsis of periodically supported beams: Response to convected random loading. 10. D.J. MEAD and A. K. MALLIK 1976 Journal of Sound and Vibration 47,457-471. An approximate method of predicting the response of periodically supported beams subjected to random eonvected loading. 11. D.J. MEAD and A. K. MALLIK 1978 Journal of Sound Vibration 61,315-326. An approximate theory for the sound radiated from a periodic line-supported plate.
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E.S. REDDY AND A. K. MALIK
12. E. S. REDDY and A. K. MALLIK 1984 American Institute of Aeronautics and Astronautics Journal 22, 543-551. Vibration of a three-layered ring on periodic radial supports. 13. E.S. REDDY 1982 Ph.D. Thesis, Indian Institute of Technology, Kanpur, India. In-plane vibrations of layered rings on periodic radial supports. 14. J. C. SNOX,VDON 1968 Vibration and Shock in Damped Mechanical Systems. New York: John Wiley. APPENDIX A The differential operators L,, L 2 , . . . , L6 are
L~= l t - d ' / d O ' + 1 2 d 2 / d O 2 + L3=Isd/dO,
h,
L4=16d/dO+lT,
L2= I4d/dO,
Ls=Is, L6=Igd2/d02+Ilo 9
The constant S's and I ' s are given by & = (/d6/~o+ I d d g + 1d619)/Iddg,
$2 = ( I~ITI~o + I316Is + I216I~0 + I21719- I l I 2 - I2419- I216)/ I~I619, $3 = (i217110 + i316ito+ i31719 _ i212_ I , l 2~o- I 2sI~ + 2IAsls) I d6t~,
& = (I3Id,o" I3I~)/Idd~, t~ = D, I 2 = 2 D - g * d 2 + ( ( 0 2 / 2 ) m 2 d 2,
13 = D § (0 2
I4= K ~ - g * d l d 2 + ( w 2 / 2 ) m d l d 4 , I7 = (02(ml +89
19 = K~, Here D = Di + D3, d2 = t~l + 2t~2+ t~3,
mj = 2blb/~Rj,
-
I5 = K 3 + g * d 2 d 3 + ' -~ m2d3d4,
d2) - g*d 2,
I6 ~ g l ,
Is = (g* +89
1,o = (0:(m~ §189
g*d~.
Kj = EiA ff Rj,
i)i = ~EjbSj, 2 3
d,,= 8 ~ - 83,
d 3 = l+t~3,
g* = g(1 + ifl),
m = mt + m2 + m3,
dl = 1 - ~ , 8j=hj/Rj, g = Gb/262.
Ej is the Young's modulus, Aj is the area of cross section, Rj is the radius o f the midplane, hj is the half thickness, pj is the density, and the subscript j indicates the j t h layer. G and fl are respectively the shear modulus and the shear loss factor of the second layer and b is the width of the ring.
APPENDIX B The harmonic amplitudes .of the coupling forces Qj are d2
Oleo [ ~ _
0=0
3,o o [ K3(-+ dO,j 11o o
d 89
LAYERED
-
CYLINDER
r
((
L
\dO
WITH
AXIAL STIFFENERS
d _____~ U3 d
da+dUd2~
dO3]-gd2\V'd'-
H e r e Qs = 0j ei~~ (J = 1, 2, 3, 4), u = U e i~ot, vl = V I e i'~ a n d v3 = V3 e i'~
531