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ANALYSIS OF FREE VIBRATION OF STRUCTURAL-ACOUSTIC COUPLED SYSTEMS, PART II: TWO- AND THREE-DIMENSIONAL EXAMPLES
Journal of Sound and Vibration (1995) 188(4), 577–600
ANALYSIS OF FREE VIBRATION OF STRUCTURAL–ACOUSTIC COUPLED SYSTEMS, PART II: TWO- AND THREE-DIMENSIONAL EXAMPLES K. L. H J. K Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 45221-0072, U.S.A. (Received 17 June 1994, and in final form 9 December 1994) Natural frequencies and modes of two- and three-dimensional cavities bounded by elastic structures are obtained. It is shown that the free vibration problems of the structural–acoustic coupled system of two- and three-dimensions can be easily formulated and solved by utilizing the proposed method. General characteristics of the dynamic coupling between the structure and the acoustic cavity are discussed based on the system natural frequencies and mode shapes. 7 1995 Academic Press Limited
1. INTRODUCTION
A new method to solve free vibration problems of an acoustic cavity bounded by elastic boundaries has been proposed by the authors in the first paper related to this work [1]. In the present paper, demonstrations are made to show advantages of the method by applying it to realistic problems of two and three dimensions. The complete analysis of such problems has been considered very involved due to the complexity in the related formulation and solution procedures. Examples found in most of the previous works in this area either used a limited number of modes in numerical calculations or made some simplifications which are not always valid [2–7]. The method in this work allows one to solve general structural–acoustic coupled problems without using any simplifications other than the number of modes to be used in the calculation. When the coupling effect is strong, natural frequencies and mode shapes of the combined system become quite different from those of the uncoupled sub-system modes. Therefore, simple extension of the conclusions based on the solution of the uncoupled systems will result in wrong conclusions. In this work, the general characteristics of the coupling effect between the structure and the acoustic cavity, and some key parameters are discussed. 2. TWO-DIMENSIONAL RECTANGULAR CAVITY BOUNDED BY AN ELASTIC BEAM
2.1. Figure 1 shows a two-dimensional acoustic cavity of rectangular shape. One side of the cavity is bounded by a flexibile beam of unit width, while all the other walls of the cavity are considered rigid. A harmonic volume flow input Q may be considered in the cavity 577 0022–460X/95/490577+24 $12.00/0
7 1995 Academic Press Limited
. . and .
578
at any arbitrary point (x¯ , y¯ ). A pratical example of such problems may be found in a very long, rectangular elastic panel backed by a cavity of a rectangular section. The wave equation of the acoustic cavity is represented as, 9 2p(x, y, t)−
1 1 2p 1 (x, y, t)=− m˙ (x, y, t), c02 1t 2 1t
(1)
where p(x, y, t) is the acoustic pressure in the cavity and c0 is the speed of sound in the acoustic medium. In equation (1), the mass source term includes the effect of the structural motion, as follows: m˙ (x, y, t)=−r0 j (x, t)d(y−H)+r0 Qd(x−x¯ )d(y−y¯ ) eivt ,
(2)
where r0 is the density of the acoustic medium, j the deflection of the beam, H is the height of the cavity, i=z−1, and d(x) is the Dirac delta function. The equation of motion of the beam is given by EI
1 4j 1 2j (x, t)+rh 2 (x, t)=p(x, H, t), 1x 4 1t
(3)
where EI is the stiffness of the beam, r the density of the beam material and h is the thickness of the beam. The effect of the pressure pulsation on the beam motion is included in the forcing term p(x, H, t) in equation (3). Structural forcing terms may be added as necessary, although they are not considered here. As in reference [1], the natural mode expansion method is used for the analysis of the problem. Assuming harmonic responses, the pressure and the beam motion can be expressed as
$
a a
%
p(x, y, t)= s s Amn P mn (x, y) eivt , m
n
$
a
%
j(x, t)= s Br j r (x) eivt , r
Figure 1. A two-dimensional cavity bounded by a beam.
(4, 5)
,
579
T 1 Frequency parameter b=kL of the uncoupled systems and the weakly coupled system, two-dimensional cavity Uncoupled systems
Coupled system
Mode number
31 terms’ approximation
Mode number
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
acoustic mode, m=0, n=0 acoustic mode, m=0, n=1 acoustic mode, m=0, n=2 acoustic mode, m=1, n=0 acoustic mode, m=1, n=1 acoustic mode, m=1, n=2 acoustic mode, m=0, n=3 acoustic mode, m=1, n=3 acoustic mode, m=2, n=0 acoustic mode, m=2, n=1 acoustic mode, m=2, n=2 acoustic mode, m=2, n=3 1, structural mode, r=1 13, acoustic mode, m=3, n=0 14, acoustic mode, m=3, n=1 15, acoustic mode, m=3, n=2 16, acoustic mode, m=3, n=3 2, structural mode, r=2 3, structural mode, r=3 4, structural mode, r=4 5, structural mode, r=5 6, structural mode, r=6 7, structural mode, r=7 8, structural mode, r=8 9, structural mode, r=9 10, structural mode, r=10 11, structural mode, r=11 12, structural mode, r=12 13, structural mode, r=13 14, structural mode, r=14 15, structural mode, r=15
0·0000 1·5708 3·1416 3·1416 3·5124 4·4429 4·7124 5·6636 6·2832 6·4766 7·0248 7·8540 8·8890 9·4248 9·5548 9·9346 10·537 35·556 80·001 142·22 222·22 320·00 435·56 568·90 720·01 888·90 1075·6 1280·0 1502·2 1742·2 2000·0
1, system mode (A1) 2, system mode (A2) 3, system mode (A3) 4, system mode (A4) 5, system mode (A5) 6, system mode (A6) 7, system mode (A7) 8, system mode (A8) 9, system mode (A9) 10, system mode (A10) 11, system mode (A11) 12, system mode (A12) 13, system mode (S1) 14, system mode (A13) 15, system mode (A14) 16, system mode (A15) 17, system mode (A16) 18, system mode (S2) 19, system mode (S3) 20, system mode (S4) 21, system mode (S5) 22, system mode (S6) 23, system mode (S7) 24, system mode (S8) 25, system mode (S9) 26, system mode (S10) 27, system mode (S11) 28, system mode (S12) 29, system mode (S13) 30, system mode (S14) 31, system mode (S15)
0·0000 1·5706 3·1412 3·1416 3·5124 4·4429 4·7116 5·6635 6·2830 6·4762 7·0243 7·8531 8·8954 9·4248 9·5548 9·9346 10·537 35·557 80·001 142·22 222·22 320·00 435·56 568·90 720·01 888·90 1075·6 1280·0 1502·2 1742·2 2000·0
where P mn (x, y) and j r (x) are the uncoupled natural modes of the acoustic cavity and beam. In other words, the former is the acoustic mode of the cavity when its boundaries are all rigid, and the latter is the in vacuo structural mode of the beam. Substituting equations (4) and (5) into equation (1), we obtain a a
s s Amn [9 2P mn (x, y)+k 2P mn (x, y)] m
n
$
a
%
=− r0 c02k 2 s Br j r (x)d(y−H)+ir0 c0 kQd(x−x¯ )d(y−y¯ ) . r
(6)
. . and .
580
Utilizing the orthogonality of the natural modes, equation (6) can be transformed as follows:
gg H
2 Amn (k 2−k˜ mn )
0
L 2 Pmn (x, y) dx dy
0
$ g a r
%
L
=−r0 c02k 2 s Br
j r (x)Pmn (x, H) dx −ir0 c0 kQPmn (x¯ , y¯ ).
0
(7)
In equations (6) and (7) the wave number of the uncoupled acoustic system is 2 k˜ mn =
2 vmn , c02
(8a)
T 2 Frequency parameter b=kL of the uncoupled systems and the strongly coupled system, two-dimensional cavity Uncoupled systems
Coupled system 33 66 99 terms’ terms’ terms’ Root number approx. approx. approx.
Root number 1, 1, 2, 3, 4, 5, 6, 7, 8, 2, 9, 10, 11, 3, 4, 12, 5, 13, 14, 6, 7, 15, 8, 16, 9, 10, 17, 11, 12, 13,
acoustic mode, m=0, n=0 structural mode, r=1 structural mode, r=2 structural mode, r=3 structural mode, r=4 structural mode, r=5 structural mode, r=6 structural mode, r=7 structural mode, r=8 acoustic mode, m=1, n=0 structural mode, r=9 structural mode, r=10 structural mode, r=11 acoustic mode, m=0, n=1 acoustic mode, m=2, n=0 structural mode, r=12 acoustic mode, m=1, n=1 structural mode, r=13 structural mode, r=14 acoustic mode, m=2, n=1 acoustic mode, m=3, n=0 structural mode, r=15 acoustic mode, m=3, n=1 structural mode, r=16 acoustic mode, m=0, n=2 acoustic mode, m=4, n=0 structural mode, r=17 acoustic mode, m=1, n=2 acoustic mode, m=2, n=2 acoustic mode, m=4, n=1
0·0000 0·0444 0·1778 0·4000 0·7111 1·1111 1·6000 2·1778 2·8445 3·1416 3·6000 4·4445 5·3778 6·2832 6·2832 6·4001 7·0248 7·5112 8·7112 8·8858 9·4248 10·000 11·327 11·378 12·566 12·566 12·845 12·953 14·050 14·050
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
system system system system system system system system system system system system system system system system system system system system system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode
(S1) (S2) (S3) (S4) (S5) (S6) (S7) (A1) (S8) (S9) (A2) (S10) (S11) (S12) (A3) (A4) (S13) (A5) (S14) (A6) (A7) (S15) (A8) (S16) (S17) (S10) (A9) (A11) (A12) (A13)
0·0000 0·1222 0·3159 0·6192 1·0661 1·5554 2·0784 2·3345 2·7964 3·5800 3·8494 4·4375 5·3434 6·3602 6·6732 7·3475 7·6133 8·0416 8·7319 9·6665 9·6989 10·076 11·331 12·055 12·808 13·332 13·665 14·715 16·293 19·392
0·0000 0·1208 0·3060 0·5996 0·9838 1·4691 2·0556 2·3098 2·7770 3·5655 3·8337 4·4233 5·3306 6·3484 6·6632 7·3181 7·5838 8·0025 8·7156 9·6367 9·6909 10·053 11·318 12·016 12·708 12·844 13·284 13·590 14·403 14·582
0·0000 0·1205 0·3046 0·5972 0·9629 1·4646 1·9677 2·2931 2·7759 3·5469 3·8298 4·4221 5·3182 6·3477 6·6606 7·3076 7·5732 7·9938 8·7135 9·6283 9·6891 10·043 11·317 12·009 12·702 12·838 13·274 13·582 14·401 14·576
,
581
and the wave number of the coupled system is k 2=
v2 . c02
(8b)
Equation (7) can be put into the form a
2 2 A* mn )Imn+b 2H s B* mn (b −b r Jrmn+i r
01
Q bLPmn (x¯ , y¯ )=0, C0
(9)
where
gg H
Imn=
0
L 2 Pmn (x, y) dx dy,
Jrmn=
0
g
L
j r (x)Pmn (x, H) dx,
(10a, b)
0
2 A* mn=Amn /r0 C0 ,
B* r =Br /H,
b=kL,
b mn=kmn L.
(10c–f)
By substituting equations (4) and (5) into equation (3), we obtain
$
a
EI s Br r
%
a a a d4j r (x) 2 −rhv s B j (x) = s s Amn P mn (x, H). r r dx 4 r m n
(11)
Since j r (x) is the in vacuo natural mode of the beam, it satisfies EI
d4j r (x)−rhv˜ r2j r (x)=0, dx 4
(12)
Figure 2. Mode shape of the sixth system mode. The mode is dominated by the sixth structural mode. (a) Beam mode; (b) acoustic mode.
. . and .
582
where v˜ r is the rth in vacuo natural frequency of the beam. By utilizing the orthogonality of the natural modes in equation (12), we obtain
rh(v˜ r2−v 2 )Br
g
a a
L
j r2 (x) dx=s s Amn m
0
n
g
L
j r (x)Pmn (x, H) dx.
(13)
0
Again, the above equation is rearranged: a a
(r/r0 )(h/L)(H/L)(b 2−b r2 )Ir B* r +s s A* mn Jrmn=0, m
(14)
n
where the following notations are used:
Ir=
g
L
j r2 (x) dx,
(15a)
0
b r2=k˜ r2L 2=(v˜ r2L 2/c02)=121 (c/c0 )2(h/L)2(hr p)4.
(15b)
In equation (15b), c=zE/r is the speed of the sound in the beam, and hr is a non-dimensional parameter which depends on the boundary condition of the beam.
Figure 3. Mode shape of the 18th system mode. The mode is dominated by the fifth acoustic model (i.e., m=1, n=1). Beam mode; (b) acoustic mode.
,
583
Figure 4. Mode shape of the 23rd system mode. The mode is dominated by the eighth acoustic mode (i.e., m=3, n=1). Beam mode; (b) acoustic mode.
Equations (9) and (14) can be put into non-dimensional forms: a
2 2 A* mn )+b 2 s Grmn B* ¯ , y¯ )=0, mn (b −b r +ibFmn (x
(16)
r
a
a
m
n
2 R s s Hrmn A* r2 )B* mn+(b −b r =0,
(17)
where
Fmn (x¯ , y¯ )=
01
LPmn (x¯ , y¯ ) Q , Imn C0
R=
Grmn=HJrmn /Imn ,
0 10 10 1 r0 r
L H
L h
Hrmn=Jrmn /Ir ,
(18a–d)
where (r0 /r) is the density ratio, (L/H) is the aspect ratio and (L/h) is the slenderness ratio. In the above equations, Grmn has the meaning of the generalized force exerted on the (m, n)th acoustic mode by the rth beam mode. Similarly, Hrmn can be considered as the gereralized force exerted on the rth beam mode by the (m, n)th acoustic mode.
584
. . and .
Figure 5. The mode shape of the 29th system mode. The mode is dominated by the 11th acoustic mode (i.e., m=1, n=2). (a) Beam mode; (b) acoustic mode.
Figure 6. Mode shape of the 33rd system mode. The mode is dominated by the 19th beam mode. (a) Beam mode; (b) acoustic mode.
,
585
Figure 7. Mode shape of the 44th system mode. The mode is dominated by the 23rd acoustic mode (i.e., m=5, n=2). (a) Beam mode; (b) acoustic mode.
Figure 8. Mode shape of the 48th system mode. The mode is dominated by the 25th acoustic mode (i.e., m=4, n=3). (a) Beam mode; (b) acoustic mode.
. . and .
586
Figure 9. Three-dimensional acoustic cavity bounded by a plate.
Equations (16) and (17) can be written in the following matrix form by retaining (m+1) by (n+1) acoustic modes and r structural modes:
Kb2−b 002 G G 0 G 0 G G * G 0 G G RH100 G RH200 G G * G * G k RHr00
... ...
0
b 2G100
b 2G200 . . . . . .
2 b 2−b 01
0 ...
0
b 2G101
b 2G201 . . . . . .
...
·· . . . ·
*
*
*
*
*
*
*
*
*
0
*
*
*
0
· ... ...
2 b 2−b mn
b 2Glmn
RH101
... ...
RHlmn
b 2−b 12
0
0 ...
RH201
... ...
RH2mn
0
b 2−b 22
0 ...
··
b 2G2mn . . . . . .
*
*
*
*
0
...
*
*
*
*
*
*
··· . . . * · ··
RHmn
0
0
... ...
RHr01
......
LF A*00 J GG G b 2Gr01 GG A* 01 G GG * * G GG G * GG * G G b 2Grmn GGA* mn Gg h 0 GG B* 1 G GG G 0 B* 2 GG G * GG * G * GG * G GG G b 2−b r2 lf B* r j b 2Gr00
FF00 (x¯ , y¯ )J G G GF01 (x¯ , y¯ )G G * G G G G * G GFmn (x¯ , y¯ )G =−ib g h. G 0 G G 0 G G G G * G G * G G G f 0 j
(19)
,
587
Equation (19) can be solved directly to obtain the responses of the cavity and the structure to the mass flow input. For example, transfer functions of the system can be obtained by solving equation (19) directly, as it was done for the one-dimensional problem in reference [1]. If we limit our discussion on the free vibration case, the right side of equation (19) becomes a zero vector. The coefficient matrix in equation (19) can be split into two 2 matrices, one with the natural frequency parameters b mn and b r2 of the uncoupled 2 sub-systems, the other with the system frequency parameter b which we want to find. That is,
$
[bA ] −R[H]
%
K b 002 0 0 . . . 0 L G G 2 G 0 b 01 0 . . . 0 G [bA ]=G * * ··· . . . * G , G G * * ··· * G G * 2 l k0 0 . . . . . . b mn K G G [H]=G G G k
H100
H101 . . . . . . H1mnL
H200
H201 . . . . . .
*
*
... ...
*
*
... ...
Hr00
Hr01 . . . . . .
$
[0 [I] {X}=b 2 [bB ] [0]
G H2mnG * G, G * G Hrmn l
%
[G] {X} [I]
K b 12 0 0 G 2 G 0 b 2 0 [bB ]= G * * ··· G G* * * k 0 0 ... K G100 G200 G G G101 G201 [G]= G * * G * G * k G1mn Gr01
(20) ... 0L
G
... 0G ...
* G,
G ·· · *G . . . b r2l
(21b, c)
. . . . . . Gr00 L
G
. . . . . . Gr01 G
G G ... ... * G . . . . . . Grmn l ... ...
*
(21d, e) [I]=Identity matrix,
[0]=Zero matrix.
(21f, g)
As can be seen in equations (19) and (20), the Nth column of matrix [G] represents the generalized forces exerted on the Nth acoustic mode by the beam modes. Similarly, the Mth row of matrix [H] is the generalized forces exerted on the Mth beam mode by all the acoustic modes. As a special case, if there are no interactions betwen the acoustic system and the structure, then all the elements in matrices [G] and [H] should become zero. Subsequently, equation (20) becomes
$
[bA ] [0]
%
$
[0] [I] {X}=b 2 [bB ] [0]
%
[0] {X}. [I]
(22)
This equation simply tells us that the system frequencies are the same as the uncoupled sub-system frequencies. It is informative to examine the associated eigenvectors in this case. For example, the system eigenvector associated with the second natural frequency will become [0, 1, 0, 0, . . . , 0]T. This means that the second acoustic mode is the only contributor to the system eigenvector. Therefore, the system eigenvector obtained from equation (20) defines the relative strength of the participation of each uncoupled structural and acoustic mode in the corresponding system mode.
1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 3,
acoustic mode, m=0, n=0, l=0 structural mode, q=1, r=1 structural mode, q=1, r=2 structural mode, q=1, r=3 structural mode, q=2, r=1 structural mode, q=1, r=4 structural mode, q=2, r=2 structural mode, q=2, r=3 structural mode, q=1, r=5 structural mode, q=2, r=4 acoustic mode, m=0, n=0, l=1 structural mode, q=3, r=1 structural mode, q=1, r=6 structural mode, q=3, r=2 structural mode, q=2, r=5 structural mode, q=3, r=3 structural mode, q=2, r=6 structural mode, q=3, r=4 structural mode, q=1, r=7 structural mode, q=3, r=5 structural mode, q=2, r=7 structural mode, q=4, r=1 acoustic mode, m=0, n=0, l=2
Mode number
Uncoupled systems
0·0000 0·2372 0·3796 0·6168 0·8066 0·9489 0·9489 1·1861 1·3759 1·5183 1·5708 1·7555 1·8978 1·8978 1·9453 2·1351 2·4672 2·4672 2·5146 2·8942 3·0840 3·0840 3·1416
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
system system system system system system system system system system system system system system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode
(S1) (S2) (A1) (S4) (S3) (S6) (S5) (S7) (S8) (S9) (S10) (S12) (S11) (S13) (S14) (A2) (S16) (S15) (S17) (S20) (S18) (S22) (S19)
Mode number 0·0000 0·3322 0·5426 0·6516 0·7031 0·8596 0·9148 1·1208 1·3316 1·4733 1·5278 1·7695 1·8791 1·9026 2·0285 2·1526 2·4086 2·4471 2·5024 2·8279 2·8461 3·0618 3·0857
0·0000 0·3283 0·5382 0·6439 0·6979 0·8503 0·9066 1·1125 1·3274 1·4607 1·5131 1·7538 1·8622 1·8966 2·0159 2·1384 2·3903 2·4258 2·5000 2·7908 2·8383 3·0433 3·0582
Coupled system 125 acoustic & 216 acoustic & 64 structural 100 structural modes’ approx. modes’ approx.
0·0000 0·3255 0·5344 0·6385 0·6941 0·8434 0·9023 1·1055 1·3191 1·4542 1·5008 1·7406 1·8582 1·8847 2·0042 2·1276 2·3801 2·4209 2·4828 2·7731 2·8209 3·0253 3·0371
343 acoustic & 121 structural modes’ approx.
T 3 Frequency parameter b=kL of the uncoupled systems and the strongly coupled system, three-dimensional cavity
588 . . and .
4, 21, 22, 23, 24, 5, 25, 26, 27, 28, 29, 6, 30, 7, 8, 31, 32, 33, 34, 35, 9, 36,
acoustic mode, m=1, n=0, l=0 structural mode, q=1, r=8 structural mode, q=4, r=2 structural mode, q=3, r=6 structural mode, q=4, r=3 acoustic mode, m=1, n=0, l=1 structural mode, q=2, r=8 structural mode, q=4, r=4 structural mode, q=1, r=9 structural mode, q=3, r=7 structural mode, q=4, r=5 acoustic mode, m=1, n=0, l=2 structural mode, q=2, r=9 acoustic mode, m=0, n=0, l=3 acoustic mode, m=0, n=1, l=0 structural mode, q=3, r=8 structural mode, q=4, r=6 structural mode, q=5, r=1 structural mode, q=1, r=10 structural mode, q=5, r=2 acoustic mode, m=0, n=1, l=1 structural mode, q=5, r=3
3·1416 3·2263 3·2263 3·4161 3·4636 3·5124 3·3957 3·7957 4·0329 4·0329 4·2227 4·4429 4·6023 4·7124 4·7124 4·7446 4·7446 7·7920 4·9344 4·9344 4·9673 5·1716
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
system system system system system system system system system system system system system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode
(S21) (A4) (S24) (S23) (A3) (S26) (S25) (A5) (S28) (S27) (S29) (S33) (S30) (S35) (S32) (S31) (A6) (A8) (S34) (A7) (S36) (A9)
3·2065 3·2283 3·3674 3·3950 3·5562 3·7349 3·7787 3·8093 4·0148 4·1769 4·5535 4·69581 4·7242 4·7461 4·7532 4·7884 5·0363 5·1578 5·1808 5·3445 5·4325 5·7045
3·2015 3·2250 3·3321 3·3679 3·5284 3·6961 3·7727 3·7951 4·0111 4·0238 4·1551 4·5315 4·5620 4·6782 4·6863 4·7362 4·7582 4·7813 4·9019 5·0183 5·1297 5·1862
3·1999 3·2234 3·3165 3·3611 3·5159 3·6833 3·7708 3·7891 3·9851 4·0181 4·1326 4·4836 4·5567 4·6459 4·6784 7·7268 4·7475 4·7720 4·8996 4·9748 5·0953 5·1795
, 589
. . and .
590
By a procedure similar to reference [1], equation (20) can be transformed into the standard linear eigenvalue problem [D]{X}=b 2{X},
(23)
where the dynamic matrix [D] is obtained as [D]=
$
[G] [I]
=
$
−[G] [I]
[I] [0] [I] [0]
%$ −1
[bA ] −R[H]
%$
[bA ] −R[H]
[0] [bB ]
% %$
[0] [b ]+R[G][H] = A [bB ] −R[H]
%
−[G][bB ] . [bB ]
(24)
It should be noted that the matrix [D] is obtained simply by multiplying submatrices in equation (20) without any inversion process due to the unique structure of the system matrix. The matrix [D] becomes a non-symmetric matrix in general. The eigenvector {X} of equation (20) is composed of non-dimensional generalized co-ordinates which define the relative strength of the participation of each uncoupled mode in the system mode, as mentioned before. The system eigenvectors are orthogonal not to each other, but to the eigenvectors associated with the adjoint eigenvalue problem: [D]T{Y}=b 2{Y}.
(25)
That is, the following relationship is valid: {X}Ti {Y}j=0,
if bi2$bj2 ,
(26)
Figure 10. Mode shape of the eighth system mode. The mode shape is dominated by the seventh plate mode (i.e., m=2, m=3). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
591
Figure 11. Mode shape of the 18th system mode. The mode shape is dominated by the 15th plate mode (i.e., m=2, m=6). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
where bi is the ith eigenvalue of the equations in equations (23) and (25). The eigenvalue problems of equations (23) and (25) share the same eigenvalue but different eigenvectors. This modified orthogonality property will be helpful to check the accuracy of numerical analyses or experimental measurement. In the dynamic matrix [D] shown in equation (24), bA and bB are frequency parameters of the uncoupled acoustic and structural systems, respectively. Considering that G and H define only the relationships between sub-system modes, this means that the coupling effect is expected to be stronger as R increases. From the definition of R in equation (18d), this suggests that a strong coupling effect would exist when the total mass of the acoustic system is large and the cavity is relatively shallow. This is consistent with the one-dimensional case which we discussed in reference [1]. It is also expected that the deviation of the system responses from those of the corresponding uncoupled subsystems will be larger if the coupling occurs in the lower frequency range. This will be confirmed by the following two examples. 2.2. : - If both ends of the beam in Figure 1 are simply supported, the uncoupled structural modes are given as j r (x)=sin (rpx/L),
r=1, 2, 3, . . .
(27)
. . and .
592
Also, the uncoupled structural frequency parameters are b r2=kr2L 2=121
0 10 1 c c0
2
2
h (rp)4, L
r=1, 2, 3, . . .
(28)
The uncoupled natural modes and frequency parameters of the acoustic cavity become, Pmn (x, y)=cos (mpx/L) cos (npy/H), 2 2 =kmn L 2=[m 2+n 2(L/H )2]p 2, b mn
m, n=0, 1, 2, . . . , m, n=0, 1, 2, . . .
(29) (30)
Generalized forces in equation (18) have the following expressions: Fmn (x¯ , y¯ )=(Qem en /HC0 ) cos (mpx¯ /L) cos (npy¯ /H), Grmn=
Hrmn=
8
8
m, n=0, 1, 2, . . . ,
0,
if
r=m,
re e (−1)n 2 m n2 (1−(−1)r−m ), (r −m )p
if
r$m,
0,
01
2 r (−1)n (1−(−1)r−m ), p (r 2−m 2 )
if
r=m,
if
r$m,
(31) (32)
(33)
Figure 12. Mode shape of the 31st system mode. The mode shape is dominated by the fifth acoustic mode (i.e., p=1, q=1, r=0). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
593
Figure 13. Mode shape of the 32nd system mode. The mode shape is dominated by the 28th plate mode (i.e., m=3, n=7). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
where eq in equation (32) and (33) is defined as eq =
6
2, 1,
if q$0, if q=0.
(34)
From equations (32) and (33), it is seen that Grmn and Hrmn are independent of any of the system parameters. Instead, they give the information on how the acoustic modes are coupled with the structural system. For example, the above equations show that there is no coupling between the structural modes of the odd wave number (symmetric modes) and the acoustic modes which have the odd wave number in the horizontal direction (anti-symmetric modes), as it should be. As discussed, the main factors that decide the coupling strength are the non-dimensional parameter R=(r0 /r)(L/H)(L/h) and the frequency range where the lowest system natural frequency exists. With this information in mind, let us consider two cases.
2.2.1 . Case 1: weak coupling A cavity of the size given as H=0·1 m and L=0·05 m is bounded by an aluminum beam with E=70×109 N/m2, r=2710 kg/m3 and h=0·005 m. Also, let the density and the speed of sound in the acoustic medium be r0=6·04 kg/m3 and c0=162·9 m/s. The system natural frequency parameters calculated for such a case are shown in Table 1. Non-dimensional frequency parameters b mn and b r of the uncoupled sub-systems, shown in each row are compared with the natural frequency parameters of the coupled system in ascending order. The first substantial coupling is observed around the first beam mode which is between the 12th and 13th acoustic modes. The solution was obtained using 31
594
. . and .
terms (16 acoustic modes and 15 structural modes) which gives well-converged results. The numbers of parentheses following the system mode number indicate the uncoupled sub-system mode which is the strongest contributor to the system mode. For example, (A15) means that the mode is a perturbation from the 15th acoustic mode. One would also notice that the result would have been virtually the same even if the last 10 or so structural modes had been omitted. This is because all are substantially higher than the frequency range considered here.
2.2.2 . Case 2: strong coupling The beam and acoustic activity have the same properties as the previous case, except that the size of the cavity is L=0·2 m, H=0·1 m and h=0·0001 m. This case corresponds to a very thin beam backed by a shallow acoustic cavity. From Table 2, it is seen that the first strong coupling occurs around the first beam frequency. The last three columns of the table compare the effect of number of modes used on the accuracy of the calculation. It shows that the solutions are consistently convergent as the number of terms increase. The difference between the system natural frequencies and the uncoupled sub-system frequencies becomes large especially in the lower frequency range. As before, the numbers in brackets in the third column indicate the corresponding uncoupled mode numbers. For example, (12S) means that the system mode is a perturbation from the 12th structural mode. These modes can be identified by checking the associated system eigenvector because its components represent the relative magnitudes of the participation of the sub-system modes.
Figure 14. Mode shape of the 38th system mode. The mode shape is dominated by the 32nd plate mode (i.e., m=4, n=6). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
595
Figure 15. Mode shape of the 45th system mode. The mode shape is dominated by the ninth acoustic mode (i.e., p=0, q=1, r=1). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
Figures 2–8 show some of the system mode shapes of the problem being discussed in this section. It is seen that each coupled mode shape is dominated by one or two uncoupled modes. For example, the sixth system mode shown in Figure 2 is a slightly perturbed form of the sixth structural mode and the second acoustic mode.
3. THREE-DIMENSIONAL RECTANGULAR CAVITY BOUNDED BY A PLATE
3.1. Now let us consider a flexible plate backed by a rectangular acoustic cavity of rigid walls, as shown in Figure 9. The plate has the size L×W and thickness h, while the cavity has a depth of H. A harmonic volume flow input to the system at point (x¯ , y¯ , z¯ ) inside the cavity may also be considered. The interaction between these two sub-systems is described by the following two coupled equations, as before:
9 2p(x, y, z)−
1 1 2p 1 (x, y, z, t)=− m˙ (x, y, z, t), c02 1t 2 1t
D9 4j(x, z, t)+rh
1 2j (x, z, t)=p(x, H, z, t), 1t 2
(35)
(36)
. . and .
596
where the flexural rigidity of the plate is D=
Eh 3 . 12(1−n 2 )
(37)
The mass flow source term is, m˙ (x, y, z, t)=−r0 j (x, z, t)d( y−H)+r0 Qd(x−x¯ )d( y−y¯ )d(z−z¯ ) eivt .
(38)
Again, we express the acoustic pressure p and plate deflection j in term of their uncoupled modes.
$
a a
%
a
p(x, y, z, t)= s s s Apqr P pqr (x, y, z) eivt , p
q
r
$
a a
%
j(x, z, t)= s s Bmn j mn (x, z) eivt . m
n
(39, 40) A procedure similar to the one used in the previous section leads us to the following two coupled equations: a a
2 2 A* pqr )+b 2 s s B* ¯ , y¯ , z¯ )=0, pqr (b −b mn Gmnpqr+ibFpqr (x m
a
a
a
p
q
r
(41)
n
2 2 R s s s A* mn )=0. pqr Hmnpqr+B* mn (b −b
(42)
Figure 16. Mode shape of the 81st system mode. The mode shape is dominated by the 24th acoustic mode (i.e., p=2, q=3, r=0). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
597
Figure 17. Mode shape of the 92nd system mode. The mode shape is dominated by the 30th acoustic mode (i.e., p=2, q=4, r=0). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
2 2 A* pqr and b mn are the pqr and B* mn are similarly defined as those of equations (9b) and (9c). b frequency parameters of the sub-systems, also similarly defined as in equations (9) and (15). Further, we define
gg ggg W
L
j mn (x, z)P pqr (x, H, z) dx dz
H HJ Gmnpqr= mnpqr= Ipqr
0
0
W
H
0
gg W
J Hmnpqr= mnpqr= Imn
0
0
,
(43)
L 2 P pqr (x, y, z) dx dy dz
0
L
j mn (x, z)P pqr (x, H, z) dx dz
0
gg W
0
Fpqr (x¯ , y¯ , z¯ )=
,
(44)
L 2 j mn (x, z) dx dz
0
01
LP pqr (x¯ , y¯ , z¯ ) Q . Ipqr C0
(45)
Equations (41) and (42) can be put into the same matrix form given by equation (20), with the elements of matrix [G], [H] and {F} defined by equations (43)–(45). After the equation is transformed into a standard eigenvalue form as shown in equation (20), the natural frequencies and mode shapes of the coupled system can be easily obtained.
. . and .
598
3.2. : - A simply supported ractangular plate backed by a rectangular acoustic cavity as shown in Figure 9 is used as an example. For the plate, we have j mn (x, z)=sin (mpx/L) sin (npz/W),
m, n=1, 2, 3, . . . ,
2 2 b mn =k˜ mn L 2={(p 4/12)[m 2+(L/W)2n 2 ]2}(c/c0 )2(h/L)2,
(46)
m, n=1, 2, 3, . . . ,
(47)
p, q, r=0, 1, 2, . . . ,
(48)
where c 2=E/r(1−v 2 ). For the rectangular cavity with rigid boundaries, we have P pqr (x, y, z)=cos ( ppx/L) cos (qpy/H) cos (rpz/W), 2 2 b pqr =k˜ pqr L 2=[ p 2+(L/H)2q 2+(L/W)2r 2 ]p 2,
p, q, r=0, 1, 2, . . .
(49)
Also, we define Gmnpqr
=
8
0,
if m=p or n=r,
(−1)q mnep eq er (1−(−1))m−p (1−(−1)n−r ), p 2 (m 2−p 2 )(n 2−r 2 )
if else. (50)
Figure 18. Mode shape of the 100th system mode. The mode shape is dominated by the 94th plate mode (i.e., m=7, n=2). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
599
Hmnpqr
8
0,
=
if m=p or n=r,
4(−1)q mn (1−(−1))m−p (1−(−1)n−r ), p 2 (m 2−p 2 )(n 2−r 2 )
if else. (51)
The plate and the acoustic medium are considered to have the same properties as in the previous two dimensional example. The cavity has the dimension of L=1 m, H=(2/3) m, W=0·5 m and the plate has the thickness of h=0·002 m. This configuration is expected to give the system a relatively strong coupling effect. By a similar procedure as before, we obtain the non-dimensional frequency b of the coupled system as shown in Table 3. Obviously, more terms have to be used because of the added dimensionality. However, the table shows that the results from 189-term solutions (the second column of the coupled solution) are accurate enough for practical purposes. In fact, the algorithm developed in this work can easily handle fairly large problems. For example the calculation using 464 terms (the last column) was done on a mid-range personal computer. Figures 10–18 show some of the coupled mode shapes of the system. 4. CONCLUDING REMARKS
Demonstrations of the new analysis procedure to solve free vibration problems of dynamic systems with elasto-acoustic coupling have been made. Two- and threedimensional rectangular cavities bounded by an elastic plate were used as examples. It has been shown that the eigenvalue problems of such systems can be easily formulated and solved utilizing the proposed method. Expanding the system responses in terms of uncoupled natural modes of the sub-system, the system characteristic equation has been obtained as a matrix eigenvalue equation. It has been shown that the resulting non-standard eigenvalue problem can be transformed into a standard linear eigenvalue problem by a simple matrix transformation which reduces the necessary computational effort drastically. The best application of this work may be found in the interior or exterior noise problems of various industrial equipment. For the exterior noise problems, the procedure will have to be extended to include the noise radiation model. However, in such a case, the coupling effect will have to be considered only between the structure and the interior cavity because the exterior pressure level is very low for most practical cases [8]. ACKNOWLEDGMENT
The support for the first author from the University Research Council and the Structural Dynamics Research Laboratories of the University of Cincinnati is gratefully acknowledged. REFERENCES 1. K. L. H and J. K 1995 Journal of Sound and Vibration 188, 561–575. Analysis of free vibration of structural–acoustic coupled systems, part I: development and verification of the procedure. 2. E. H. D and H. M. V 1963 AIAA Journal 1, 476. The effect of a cavity on panel vibration. 3. A. J. P 1965 Journal of Sound and Vibration 2, 197–209. Free vibrations of a rectangular panel backed by a closed rectangular cavity.
600
. . and .
4. M. K. A-Y 1976 Journal of Applied Mechanics 43, 480–484. Free vibration of fluid-coupled coaxial cylindrical shells of different lengths. 5. S. N and R. L. S 1981 Journal of Sound and Vibration 78, 453–473. Acoustoelasticity of a damped sandwich panel backed by a cavity. 6. J. P and D. A. B 1990 Journal of Acoustical Society of America 87, 691–707. The effect of fluid–structural coupling on sound waves in an enclosure—theoretical part. 7. E. H. D, G. F. G, III and D. A. S 1977 Journal of Sound and Vibration 52, 519–542. Acoustoelasticity: general theory, acoustic natural modes and forced response to sinusoidal excitation, including comparisons with experiment. 8. W. S 1993 Vibrations of Shells and Plates, second edition. New York: Marcel Dekker.
ANALYSIS OF FREE VIBRATION OF STRUCTURAL–ACOUSTIC COUPLED SYSTEMS, PART II: TWO- AND THREE-DIMENSIONAL EXAMPLES K. L. H J. K Mechanical, Industrial and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 45221-0072, U.S.A. (Received 17 June 1994, and in final form 9 December 1994) Natural frequencies and modes of two- and three-dimensional cavities bounded by elastic structures are obtained. It is shown that the free vibration problems of the structural–acoustic coupled system of two- and three-dimensions can be easily formulated and solved by utilizing the proposed method. General characteristics of the dynamic coupling between the structure and the acoustic cavity are discussed based on the system natural frequencies and mode shapes. 7 1995 Academic Press Limited
1. INTRODUCTION
A new method to solve free vibration problems of an acoustic cavity bounded by elastic boundaries has been proposed by the authors in the first paper related to this work [1]. In the present paper, demonstrations are made to show advantages of the method by applying it to realistic problems of two and three dimensions. The complete analysis of such problems has been considered very involved due to the complexity in the related formulation and solution procedures. Examples found in most of the previous works in this area either used a limited number of modes in numerical calculations or made some simplifications which are not always valid [2–7]. The method in this work allows one to solve general structural–acoustic coupled problems without using any simplifications other than the number of modes to be used in the calculation. When the coupling effect is strong, natural frequencies and mode shapes of the combined system become quite different from those of the uncoupled sub-system modes. Therefore, simple extension of the conclusions based on the solution of the uncoupled systems will result in wrong conclusions. In this work, the general characteristics of the coupling effect between the structure and the acoustic cavity, and some key parameters are discussed. 2. TWO-DIMENSIONAL RECTANGULAR CAVITY BOUNDED BY AN ELASTIC BEAM
2.1. Figure 1 shows a two-dimensional acoustic cavity of rectangular shape. One side of the cavity is bounded by a flexibile beam of unit width, while all the other walls of the cavity are considered rigid. A harmonic volume flow input Q may be considered in the cavity 577 0022–460X/95/490577+24 $12.00/0
7 1995 Academic Press Limited
. . and .
578
at any arbitrary point (x¯ , y¯ ). A pratical example of such problems may be found in a very long, rectangular elastic panel backed by a cavity of a rectangular section. The wave equation of the acoustic cavity is represented as, 9 2p(x, y, t)−
1 1 2p 1 (x, y, t)=− m˙ (x, y, t), c02 1t 2 1t
(1)
where p(x, y, t) is the acoustic pressure in the cavity and c0 is the speed of sound in the acoustic medium. In equation (1), the mass source term includes the effect of the structural motion, as follows: m˙ (x, y, t)=−r0 j (x, t)d(y−H)+r0 Qd(x−x¯ )d(y−y¯ ) eivt ,
(2)
where r0 is the density of the acoustic medium, j the deflection of the beam, H is the height of the cavity, i=z−1, and d(x) is the Dirac delta function. The equation of motion of the beam is given by EI
1 4j 1 2j (x, t)+rh 2 (x, t)=p(x, H, t), 1x 4 1t
(3)
where EI is the stiffness of the beam, r the density of the beam material and h is the thickness of the beam. The effect of the pressure pulsation on the beam motion is included in the forcing term p(x, H, t) in equation (3). Structural forcing terms may be added as necessary, although they are not considered here. As in reference [1], the natural mode expansion method is used for the analysis of the problem. Assuming harmonic responses, the pressure and the beam motion can be expressed as
$
a a
%
p(x, y, t)= s s Amn P mn (x, y) eivt , m
n
$
a
%
j(x, t)= s Br j r (x) eivt , r
Figure 1. A two-dimensional cavity bounded by a beam.
(4, 5)
,
579
T 1 Frequency parameter b=kL of the uncoupled systems and the weakly coupled system, two-dimensional cavity Uncoupled systems
Coupled system
Mode number
31 terms’ approximation
Mode number
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
acoustic mode, m=0, n=0 acoustic mode, m=0, n=1 acoustic mode, m=0, n=2 acoustic mode, m=1, n=0 acoustic mode, m=1, n=1 acoustic mode, m=1, n=2 acoustic mode, m=0, n=3 acoustic mode, m=1, n=3 acoustic mode, m=2, n=0 acoustic mode, m=2, n=1 acoustic mode, m=2, n=2 acoustic mode, m=2, n=3 1, structural mode, r=1 13, acoustic mode, m=3, n=0 14, acoustic mode, m=3, n=1 15, acoustic mode, m=3, n=2 16, acoustic mode, m=3, n=3 2, structural mode, r=2 3, structural mode, r=3 4, structural mode, r=4 5, structural mode, r=5 6, structural mode, r=6 7, structural mode, r=7 8, structural mode, r=8 9, structural mode, r=9 10, structural mode, r=10 11, structural mode, r=11 12, structural mode, r=12 13, structural mode, r=13 14, structural mode, r=14 15, structural mode, r=15
0·0000 1·5708 3·1416 3·1416 3·5124 4·4429 4·7124 5·6636 6·2832 6·4766 7·0248 7·8540 8·8890 9·4248 9·5548 9·9346 10·537 35·556 80·001 142·22 222·22 320·00 435·56 568·90 720·01 888·90 1075·6 1280·0 1502·2 1742·2 2000·0
1, system mode (A1) 2, system mode (A2) 3, system mode (A3) 4, system mode (A4) 5, system mode (A5) 6, system mode (A6) 7, system mode (A7) 8, system mode (A8) 9, system mode (A9) 10, system mode (A10) 11, system mode (A11) 12, system mode (A12) 13, system mode (S1) 14, system mode (A13) 15, system mode (A14) 16, system mode (A15) 17, system mode (A16) 18, system mode (S2) 19, system mode (S3) 20, system mode (S4) 21, system mode (S5) 22, system mode (S6) 23, system mode (S7) 24, system mode (S8) 25, system mode (S9) 26, system mode (S10) 27, system mode (S11) 28, system mode (S12) 29, system mode (S13) 30, system mode (S14) 31, system mode (S15)
0·0000 1·5706 3·1412 3·1416 3·5124 4·4429 4·7116 5·6635 6·2830 6·4762 7·0243 7·8531 8·8954 9·4248 9·5548 9·9346 10·537 35·557 80·001 142·22 222·22 320·00 435·56 568·90 720·01 888·90 1075·6 1280·0 1502·2 1742·2 2000·0
where P mn (x, y) and j r (x) are the uncoupled natural modes of the acoustic cavity and beam. In other words, the former is the acoustic mode of the cavity when its boundaries are all rigid, and the latter is the in vacuo structural mode of the beam. Substituting equations (4) and (5) into equation (1), we obtain a a
s s Amn [9 2P mn (x, y)+k 2P mn (x, y)] m
n
$
a
%
=− r0 c02k 2 s Br j r (x)d(y−H)+ir0 c0 kQd(x−x¯ )d(y−y¯ ) . r
(6)
. . and .
580
Utilizing the orthogonality of the natural modes, equation (6) can be transformed as follows:
gg H
2 Amn (k 2−k˜ mn )
0
L 2 Pmn (x, y) dx dy
0
$ g a r
%
L
=−r0 c02k 2 s Br
j r (x)Pmn (x, H) dx −ir0 c0 kQPmn (x¯ , y¯ ).
0
(7)
In equations (6) and (7) the wave number of the uncoupled acoustic system is 2 k˜ mn =
2 vmn , c02
(8a)
T 2 Frequency parameter b=kL of the uncoupled systems and the strongly coupled system, two-dimensional cavity Uncoupled systems
Coupled system 33 66 99 terms’ terms’ terms’ Root number approx. approx. approx.
Root number 1, 1, 2, 3, 4, 5, 6, 7, 8, 2, 9, 10, 11, 3, 4, 12, 5, 13, 14, 6, 7, 15, 8, 16, 9, 10, 17, 11, 12, 13,
acoustic mode, m=0, n=0 structural mode, r=1 structural mode, r=2 structural mode, r=3 structural mode, r=4 structural mode, r=5 structural mode, r=6 structural mode, r=7 structural mode, r=8 acoustic mode, m=1, n=0 structural mode, r=9 structural mode, r=10 structural mode, r=11 acoustic mode, m=0, n=1 acoustic mode, m=2, n=0 structural mode, r=12 acoustic mode, m=1, n=1 structural mode, r=13 structural mode, r=14 acoustic mode, m=2, n=1 acoustic mode, m=3, n=0 structural mode, r=15 acoustic mode, m=3, n=1 structural mode, r=16 acoustic mode, m=0, n=2 acoustic mode, m=4, n=0 structural mode, r=17 acoustic mode, m=1, n=2 acoustic mode, m=2, n=2 acoustic mode, m=4, n=1
0·0000 0·0444 0·1778 0·4000 0·7111 1·1111 1·6000 2·1778 2·8445 3·1416 3·6000 4·4445 5·3778 6·2832 6·2832 6·4001 7·0248 7·5112 8·7112 8·8858 9·4248 10·000 11·327 11·378 12·566 12·566 12·845 12·953 14·050 14·050
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
system system system system system system system system system system system system system system system system system system system system system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode
(S1) (S2) (S3) (S4) (S5) (S6) (S7) (A1) (S8) (S9) (A2) (S10) (S11) (S12) (A3) (A4) (S13) (A5) (S14) (A6) (A7) (S15) (A8) (S16) (S17) (S10) (A9) (A11) (A12) (A13)
0·0000 0·1222 0·3159 0·6192 1·0661 1·5554 2·0784 2·3345 2·7964 3·5800 3·8494 4·4375 5·3434 6·3602 6·6732 7·3475 7·6133 8·0416 8·7319 9·6665 9·6989 10·076 11·331 12·055 12·808 13·332 13·665 14·715 16·293 19·392
0·0000 0·1208 0·3060 0·5996 0·9838 1·4691 2·0556 2·3098 2·7770 3·5655 3·8337 4·4233 5·3306 6·3484 6·6632 7·3181 7·5838 8·0025 8·7156 9·6367 9·6909 10·053 11·318 12·016 12·708 12·844 13·284 13·590 14·403 14·582
0·0000 0·1205 0·3046 0·5972 0·9629 1·4646 1·9677 2·2931 2·7759 3·5469 3·8298 4·4221 5·3182 6·3477 6·6606 7·3076 7·5732 7·9938 8·7135 9·6283 9·6891 10·043 11·317 12·009 12·702 12·838 13·274 13·582 14·401 14·576
,
581
and the wave number of the coupled system is k 2=
v2 . c02
(8b)
Equation (7) can be put into the form a
2 2 A* mn )Imn+b 2H s B* mn (b −b r Jrmn+i r
01
Q bLPmn (x¯ , y¯ )=0, C0
(9)
where
gg H
Imn=
0
L 2 Pmn (x, y) dx dy,
Jrmn=
0
g
L
j r (x)Pmn (x, H) dx,
(10a, b)
0
2 A* mn=Amn /r0 C0 ,
B* r =Br /H,
b=kL,
b mn=kmn L.
(10c–f)
By substituting equations (4) and (5) into equation (3), we obtain
$
a
EI s Br r
%
a a a d4j r (x) 2 −rhv s B j (x) = s s Amn P mn (x, H). r r dx 4 r m n
(11)
Since j r (x) is the in vacuo natural mode of the beam, it satisfies EI
d4j r (x)−rhv˜ r2j r (x)=0, dx 4
(12)
Figure 2. Mode shape of the sixth system mode. The mode is dominated by the sixth structural mode. (a) Beam mode; (b) acoustic mode.
. . and .
582
where v˜ r is the rth in vacuo natural frequency of the beam. By utilizing the orthogonality of the natural modes in equation (12), we obtain
rh(v˜ r2−v 2 )Br
g
a a
L
j r2 (x) dx=s s Amn m
0
n
g
L
j r (x)Pmn (x, H) dx.
(13)
0
Again, the above equation is rearranged: a a
(r/r0 )(h/L)(H/L)(b 2−b r2 )Ir B* r +s s A* mn Jrmn=0, m
(14)
n
where the following notations are used:
Ir=
g
L
j r2 (x) dx,
(15a)
0
b r2=k˜ r2L 2=(v˜ r2L 2/c02)=121 (c/c0 )2(h/L)2(hr p)4.
(15b)
In equation (15b), c=zE/r is the speed of the sound in the beam, and hr is a non-dimensional parameter which depends on the boundary condition of the beam.
Figure 3. Mode shape of the 18th system mode. The mode is dominated by the fifth acoustic model (i.e., m=1, n=1). Beam mode; (b) acoustic mode.
,
583
Figure 4. Mode shape of the 23rd system mode. The mode is dominated by the eighth acoustic mode (i.e., m=3, n=1). Beam mode; (b) acoustic mode.
Equations (9) and (14) can be put into non-dimensional forms: a
2 2 A* mn )+b 2 s Grmn B* ¯ , y¯ )=0, mn (b −b r +ibFmn (x
(16)
r
a
a
m
n
2 R s s Hrmn A* r2 )B* mn+(b −b r =0,
(17)
where
Fmn (x¯ , y¯ )=
01
LPmn (x¯ , y¯ ) Q , Imn C0
R=
Grmn=HJrmn /Imn ,
0 10 10 1 r0 r
L H
L h
Hrmn=Jrmn /Ir ,
(18a–d)
where (r0 /r) is the density ratio, (L/H) is the aspect ratio and (L/h) is the slenderness ratio. In the above equations, Grmn has the meaning of the generalized force exerted on the (m, n)th acoustic mode by the rth beam mode. Similarly, Hrmn can be considered as the gereralized force exerted on the rth beam mode by the (m, n)th acoustic mode.
584
. . and .
Figure 5. The mode shape of the 29th system mode. The mode is dominated by the 11th acoustic mode (i.e., m=1, n=2). (a) Beam mode; (b) acoustic mode.
Figure 6. Mode shape of the 33rd system mode. The mode is dominated by the 19th beam mode. (a) Beam mode; (b) acoustic mode.
,
585
Figure 7. Mode shape of the 44th system mode. The mode is dominated by the 23rd acoustic mode (i.e., m=5, n=2). (a) Beam mode; (b) acoustic mode.
Figure 8. Mode shape of the 48th system mode. The mode is dominated by the 25th acoustic mode (i.e., m=4, n=3). (a) Beam mode; (b) acoustic mode.
. . and .
586
Figure 9. Three-dimensional acoustic cavity bounded by a plate.
Equations (16) and (17) can be written in the following matrix form by retaining (m+1) by (n+1) acoustic modes and r structural modes:
Kb2−b 002 G G 0 G 0 G G * G 0 G G RH100 G RH200 G G * G * G k RHr00
... ...
0
b 2G100
b 2G200 . . . . . .
2 b 2−b 01
0 ...
0
b 2G101
b 2G201 . . . . . .
...
·· . . . ·
*
*
*
*
*
*
*
*
*
0
*
*
*
0
· ... ...
2 b 2−b mn
b 2Glmn
RH101
... ...
RHlmn
b 2−b 12
0
0 ...
RH201
... ...
RH2mn
0
b 2−b 22
0 ...
··
b 2G2mn . . . . . .
*
*
*
*
0
...
*
*
*
*
*
*
··· . . . * · ··
RHmn
0
0
... ...
RHr01
......
LF A*00 J GG G b 2Gr01 GG A* 01 G GG * * G GG G * GG * G G b 2Grmn GGA* mn Gg h 0 GG B* 1 G GG G 0 B* 2 GG G * GG * G * GG * G GG G b 2−b r2 lf B* r j b 2Gr00
FF00 (x¯ , y¯ )J G G GF01 (x¯ , y¯ )G G * G G G G * G GFmn (x¯ , y¯ )G =−ib g h. G 0 G G 0 G G G G * G G * G G G f 0 j
(19)
,
587
Equation (19) can be solved directly to obtain the responses of the cavity and the structure to the mass flow input. For example, transfer functions of the system can be obtained by solving equation (19) directly, as it was done for the one-dimensional problem in reference [1]. If we limit our discussion on the free vibration case, the right side of equation (19) becomes a zero vector. The coefficient matrix in equation (19) can be split into two 2 matrices, one with the natural frequency parameters b mn and b r2 of the uncoupled 2 sub-systems, the other with the system frequency parameter b which we want to find. That is,
$
[bA ] −R[H]
%
K b 002 0 0 . . . 0 L G G 2 G 0 b 01 0 . . . 0 G [bA ]=G * * ··· . . . * G , G G * * ··· * G G * 2 l k0 0 . . . . . . b mn K G G [H]=G G G k
H100
H101 . . . . . . H1mnL
H200
H201 . . . . . .
*
*
... ...
*
*
... ...
Hr00
Hr01 . . . . . .
$
[0 [I] {X}=b 2 [bB ] [0]
G H2mnG * G, G * G Hrmn l
%
[G] {X} [I]
K b 12 0 0 G 2 G 0 b 2 0 [bB ]= G * * ··· G G* * * k 0 0 ... K G100 G200 G G G101 G201 [G]= G * * G * G * k G1mn Gr01
(20) ... 0L
G
... 0G ...
* G,
G ·· · *G . . . b r2l
(21b, c)
. . . . . . Gr00 L
G
. . . . . . Gr01 G
G G ... ... * G . . . . . . Grmn l ... ...
*
(21d, e) [I]=Identity matrix,
[0]=Zero matrix.
(21f, g)
As can be seen in equations (19) and (20), the Nth column of matrix [G] represents the generalized forces exerted on the Nth acoustic mode by the beam modes. Similarly, the Mth row of matrix [H] is the generalized forces exerted on the Mth beam mode by all the acoustic modes. As a special case, if there are no interactions betwen the acoustic system and the structure, then all the elements in matrices [G] and [H] should become zero. Subsequently, equation (20) becomes
$
[bA ] [0]
%
$
[0] [I] {X}=b 2 [bB ] [0]
%
[0] {X}. [I]
(22)
This equation simply tells us that the system frequencies are the same as the uncoupled sub-system frequencies. It is informative to examine the associated eigenvectors in this case. For example, the system eigenvector associated with the second natural frequency will become [0, 1, 0, 0, . . . , 0]T. This means that the second acoustic mode is the only contributor to the system eigenvector. Therefore, the system eigenvector obtained from equation (20) defines the relative strength of the participation of each uncoupled structural and acoustic mode in the corresponding system mode.
1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 3,
acoustic mode, m=0, n=0, l=0 structural mode, q=1, r=1 structural mode, q=1, r=2 structural mode, q=1, r=3 structural mode, q=2, r=1 structural mode, q=1, r=4 structural mode, q=2, r=2 structural mode, q=2, r=3 structural mode, q=1, r=5 structural mode, q=2, r=4 acoustic mode, m=0, n=0, l=1 structural mode, q=3, r=1 structural mode, q=1, r=6 structural mode, q=3, r=2 structural mode, q=2, r=5 structural mode, q=3, r=3 structural mode, q=2, r=6 structural mode, q=3, r=4 structural mode, q=1, r=7 structural mode, q=3, r=5 structural mode, q=2, r=7 structural mode, q=4, r=1 acoustic mode, m=0, n=0, l=2
Mode number
Uncoupled systems
0·0000 0·2372 0·3796 0·6168 0·8066 0·9489 0·9489 1·1861 1·3759 1·5183 1·5708 1·7555 1·8978 1·8978 1·9453 2·1351 2·4672 2·4672 2·5146 2·8942 3·0840 3·0840 3·1416
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
system system system system system system system system system system system system system system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode
(S1) (S2) (A1) (S4) (S3) (S6) (S5) (S7) (S8) (S9) (S10) (S12) (S11) (S13) (S14) (A2) (S16) (S15) (S17) (S20) (S18) (S22) (S19)
Mode number 0·0000 0·3322 0·5426 0·6516 0·7031 0·8596 0·9148 1·1208 1·3316 1·4733 1·5278 1·7695 1·8791 1·9026 2·0285 2·1526 2·4086 2·4471 2·5024 2·8279 2·8461 3·0618 3·0857
0·0000 0·3283 0·5382 0·6439 0·6979 0·8503 0·9066 1·1125 1·3274 1·4607 1·5131 1·7538 1·8622 1·8966 2·0159 2·1384 2·3903 2·4258 2·5000 2·7908 2·8383 3·0433 3·0582
Coupled system 125 acoustic & 216 acoustic & 64 structural 100 structural modes’ approx. modes’ approx.
0·0000 0·3255 0·5344 0·6385 0·6941 0·8434 0·9023 1·1055 1·3191 1·4542 1·5008 1·7406 1·8582 1·8847 2·0042 2·1276 2·3801 2·4209 2·4828 2·7731 2·8209 3·0253 3·0371
343 acoustic & 121 structural modes’ approx.
T 3 Frequency parameter b=kL of the uncoupled systems and the strongly coupled system, three-dimensional cavity
588 . . and .
4, 21, 22, 23, 24, 5, 25, 26, 27, 28, 29, 6, 30, 7, 8, 31, 32, 33, 34, 35, 9, 36,
acoustic mode, m=1, n=0, l=0 structural mode, q=1, r=8 structural mode, q=4, r=2 structural mode, q=3, r=6 structural mode, q=4, r=3 acoustic mode, m=1, n=0, l=1 structural mode, q=2, r=8 structural mode, q=4, r=4 structural mode, q=1, r=9 structural mode, q=3, r=7 structural mode, q=4, r=5 acoustic mode, m=1, n=0, l=2 structural mode, q=2, r=9 acoustic mode, m=0, n=0, l=3 acoustic mode, m=0, n=1, l=0 structural mode, q=3, r=8 structural mode, q=4, r=6 structural mode, q=5, r=1 structural mode, q=1, r=10 structural mode, q=5, r=2 acoustic mode, m=0, n=1, l=1 structural mode, q=5, r=3
3·1416 3·2263 3·2263 3·4161 3·4636 3·5124 3·3957 3·7957 4·0329 4·0329 4·2227 4·4429 4·6023 4·7124 4·7124 4·7446 4·7446 7·7920 4·9344 4·9344 4·9673 5·1716
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
system system system system system system system system system system system system system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode mode
(S21) (A4) (S24) (S23) (A3) (S26) (S25) (A5) (S28) (S27) (S29) (S33) (S30) (S35) (S32) (S31) (A6) (A8) (S34) (A7) (S36) (A9)
3·2065 3·2283 3·3674 3·3950 3·5562 3·7349 3·7787 3·8093 4·0148 4·1769 4·5535 4·69581 4·7242 4·7461 4·7532 4·7884 5·0363 5·1578 5·1808 5·3445 5·4325 5·7045
3·2015 3·2250 3·3321 3·3679 3·5284 3·6961 3·7727 3·7951 4·0111 4·0238 4·1551 4·5315 4·5620 4·6782 4·6863 4·7362 4·7582 4·7813 4·9019 5·0183 5·1297 5·1862
3·1999 3·2234 3·3165 3·3611 3·5159 3·6833 3·7708 3·7891 3·9851 4·0181 4·1326 4·4836 4·5567 4·6459 4·6784 7·7268 4·7475 4·7720 4·8996 4·9748 5·0953 5·1795
, 589
. . and .
590
By a procedure similar to reference [1], equation (20) can be transformed into the standard linear eigenvalue problem [D]{X}=b 2{X},
(23)
where the dynamic matrix [D] is obtained as [D]=
$
[G] [I]
=
$
−[G] [I]
[I] [0] [I] [0]
%$ −1
[bA ] −R[H]
%$
[bA ] −R[H]
[0] [bB ]
% %$
[0] [b ]+R[G][H] = A [bB ] −R[H]
%
−[G][bB ] . [bB ]
(24)
It should be noted that the matrix [D] is obtained simply by multiplying submatrices in equation (20) without any inversion process due to the unique structure of the system matrix. The matrix [D] becomes a non-symmetric matrix in general. The eigenvector {X} of equation (20) is composed of non-dimensional generalized co-ordinates which define the relative strength of the participation of each uncoupled mode in the system mode, as mentioned before. The system eigenvectors are orthogonal not to each other, but to the eigenvectors associated with the adjoint eigenvalue problem: [D]T{Y}=b 2{Y}.
(25)
That is, the following relationship is valid: {X}Ti {Y}j=0,
if bi2$bj2 ,
(26)
Figure 10. Mode shape of the eighth system mode. The mode shape is dominated by the seventh plate mode (i.e., m=2, m=3). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
591
Figure 11. Mode shape of the 18th system mode. The mode shape is dominated by the 15th plate mode (i.e., m=2, m=6). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
where bi is the ith eigenvalue of the equations in equations (23) and (25). The eigenvalue problems of equations (23) and (25) share the same eigenvalue but different eigenvectors. This modified orthogonality property will be helpful to check the accuracy of numerical analyses or experimental measurement. In the dynamic matrix [D] shown in equation (24), bA and bB are frequency parameters of the uncoupled acoustic and structural systems, respectively. Considering that G and H define only the relationships between sub-system modes, this means that the coupling effect is expected to be stronger as R increases. From the definition of R in equation (18d), this suggests that a strong coupling effect would exist when the total mass of the acoustic system is large and the cavity is relatively shallow. This is consistent with the one-dimensional case which we discussed in reference [1]. It is also expected that the deviation of the system responses from those of the corresponding uncoupled subsystems will be larger if the coupling occurs in the lower frequency range. This will be confirmed by the following two examples. 2.2. : - If both ends of the beam in Figure 1 are simply supported, the uncoupled structural modes are given as j r (x)=sin (rpx/L),
r=1, 2, 3, . . .
(27)
. . and .
592
Also, the uncoupled structural frequency parameters are b r2=kr2L 2=121
0 10 1 c c0
2
2
h (rp)4, L
r=1, 2, 3, . . .
(28)
The uncoupled natural modes and frequency parameters of the acoustic cavity become, Pmn (x, y)=cos (mpx/L) cos (npy/H), 2 2 =kmn L 2=[m 2+n 2(L/H )2]p 2, b mn
m, n=0, 1, 2, . . . , m, n=0, 1, 2, . . .
(29) (30)
Generalized forces in equation (18) have the following expressions: Fmn (x¯ , y¯ )=(Qem en /HC0 ) cos (mpx¯ /L) cos (npy¯ /H), Grmn=
Hrmn=
8
8
m, n=0, 1, 2, . . . ,
0,
if
r=m,
re e (−1)n 2 m n2 (1−(−1)r−m ), (r −m )p
if
r$m,
0,
01
2 r (−1)n (1−(−1)r−m ), p (r 2−m 2 )
if
r=m,
if
r$m,
(31) (32)
(33)
Figure 12. Mode shape of the 31st system mode. The mode shape is dominated by the fifth acoustic mode (i.e., p=1, q=1, r=0). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
593
Figure 13. Mode shape of the 32nd system mode. The mode shape is dominated by the 28th plate mode (i.e., m=3, n=7). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
where eq in equation (32) and (33) is defined as eq =
6
2, 1,
if q$0, if q=0.
(34)
From equations (32) and (33), it is seen that Grmn and Hrmn are independent of any of the system parameters. Instead, they give the information on how the acoustic modes are coupled with the structural system. For example, the above equations show that there is no coupling between the structural modes of the odd wave number (symmetric modes) and the acoustic modes which have the odd wave number in the horizontal direction (anti-symmetric modes), as it should be. As discussed, the main factors that decide the coupling strength are the non-dimensional parameter R=(r0 /r)(L/H)(L/h) and the frequency range where the lowest system natural frequency exists. With this information in mind, let us consider two cases.
2.2.1 . Case 1: weak coupling A cavity of the size given as H=0·1 m and L=0·05 m is bounded by an aluminum beam with E=70×109 N/m2, r=2710 kg/m3 and h=0·005 m. Also, let the density and the speed of sound in the acoustic medium be r0=6·04 kg/m3 and c0=162·9 m/s. The system natural frequency parameters calculated for such a case are shown in Table 1. Non-dimensional frequency parameters b mn and b r of the uncoupled sub-systems, shown in each row are compared with the natural frequency parameters of the coupled system in ascending order. The first substantial coupling is observed around the first beam mode which is between the 12th and 13th acoustic modes. The solution was obtained using 31
594
. . and .
terms (16 acoustic modes and 15 structural modes) which gives well-converged results. The numbers of parentheses following the system mode number indicate the uncoupled sub-system mode which is the strongest contributor to the system mode. For example, (A15) means that the mode is a perturbation from the 15th acoustic mode. One would also notice that the result would have been virtually the same even if the last 10 or so structural modes had been omitted. This is because all are substantially higher than the frequency range considered here.
2.2.2 . Case 2: strong coupling The beam and acoustic activity have the same properties as the previous case, except that the size of the cavity is L=0·2 m, H=0·1 m and h=0·0001 m. This case corresponds to a very thin beam backed by a shallow acoustic cavity. From Table 2, it is seen that the first strong coupling occurs around the first beam frequency. The last three columns of the table compare the effect of number of modes used on the accuracy of the calculation. It shows that the solutions are consistently convergent as the number of terms increase. The difference between the system natural frequencies and the uncoupled sub-system frequencies becomes large especially in the lower frequency range. As before, the numbers in brackets in the third column indicate the corresponding uncoupled mode numbers. For example, (12S) means that the system mode is a perturbation from the 12th structural mode. These modes can be identified by checking the associated system eigenvector because its components represent the relative magnitudes of the participation of the sub-system modes.
Figure 14. Mode shape of the 38th system mode. The mode shape is dominated by the 32nd plate mode (i.e., m=4, n=6). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
595
Figure 15. Mode shape of the 45th system mode. The mode shape is dominated by the ninth acoustic mode (i.e., p=0, q=1, r=1). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
Figures 2–8 show some of the system mode shapes of the problem being discussed in this section. It is seen that each coupled mode shape is dominated by one or two uncoupled modes. For example, the sixth system mode shown in Figure 2 is a slightly perturbed form of the sixth structural mode and the second acoustic mode.
3. THREE-DIMENSIONAL RECTANGULAR CAVITY BOUNDED BY A PLATE
3.1. Now let us consider a flexible plate backed by a rectangular acoustic cavity of rigid walls, as shown in Figure 9. The plate has the size L×W and thickness h, while the cavity has a depth of H. A harmonic volume flow input to the system at point (x¯ , y¯ , z¯ ) inside the cavity may also be considered. The interaction between these two sub-systems is described by the following two coupled equations, as before:
9 2p(x, y, z)−
1 1 2p 1 (x, y, z, t)=− m˙ (x, y, z, t), c02 1t 2 1t
D9 4j(x, z, t)+rh
1 2j (x, z, t)=p(x, H, z, t), 1t 2
(35)
(36)
. . and .
596
where the flexural rigidity of the plate is D=
Eh 3 . 12(1−n 2 )
(37)
The mass flow source term is, m˙ (x, y, z, t)=−r0 j (x, z, t)d( y−H)+r0 Qd(x−x¯ )d( y−y¯ )d(z−z¯ ) eivt .
(38)
Again, we express the acoustic pressure p and plate deflection j in term of their uncoupled modes.
$
a a
%
a
p(x, y, z, t)= s s s Apqr P pqr (x, y, z) eivt , p
q
r
$
a a
%
j(x, z, t)= s s Bmn j mn (x, z) eivt . m
n
(39, 40) A procedure similar to the one used in the previous section leads us to the following two coupled equations: a a
2 2 A* pqr )+b 2 s s B* ¯ , y¯ , z¯ )=0, pqr (b −b mn Gmnpqr+ibFpqr (x m
a
a
a
p
q
r
(41)
n
2 2 R s s s A* mn )=0. pqr Hmnpqr+B* mn (b −b
(42)
Figure 16. Mode shape of the 81st system mode. The mode shape is dominated by the 24th acoustic mode (i.e., p=2, q=3, r=0). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
597
Figure 17. Mode shape of the 92nd system mode. The mode shape is dominated by the 30th acoustic mode (i.e., p=2, q=4, r=0). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
2 2 A* pqr and b mn are the pqr and B* mn are similarly defined as those of equations (9b) and (9c). b frequency parameters of the sub-systems, also similarly defined as in equations (9) and (15). Further, we define
gg ggg W
L
j mn (x, z)P pqr (x, H, z) dx dz
H HJ Gmnpqr= mnpqr= Ipqr
0
0
W
H
0
gg W
J Hmnpqr= mnpqr= Imn
0
0
,
(43)
L 2 P pqr (x, y, z) dx dy dz
0
L
j mn (x, z)P pqr (x, H, z) dx dz
0
gg W
0
Fpqr (x¯ , y¯ , z¯ )=
,
(44)
L 2 j mn (x, z) dx dz
0
01
LP pqr (x¯ , y¯ , z¯ ) Q . Ipqr C0
(45)
Equations (41) and (42) can be put into the same matrix form given by equation (20), with the elements of matrix [G], [H] and {F} defined by equations (43)–(45). After the equation is transformed into a standard eigenvalue form as shown in equation (20), the natural frequencies and mode shapes of the coupled system can be easily obtained.
. . and .
598
3.2. : - A simply supported ractangular plate backed by a rectangular acoustic cavity as shown in Figure 9 is used as an example. For the plate, we have j mn (x, z)=sin (mpx/L) sin (npz/W),
m, n=1, 2, 3, . . . ,
2 2 b mn =k˜ mn L 2={(p 4/12)[m 2+(L/W)2n 2 ]2}(c/c0 )2(h/L)2,
(46)
m, n=1, 2, 3, . . . ,
(47)
p, q, r=0, 1, 2, . . . ,
(48)
where c 2=E/r(1−v 2 ). For the rectangular cavity with rigid boundaries, we have P pqr (x, y, z)=cos ( ppx/L) cos (qpy/H) cos (rpz/W), 2 2 b pqr =k˜ pqr L 2=[ p 2+(L/H)2q 2+(L/W)2r 2 ]p 2,
p, q, r=0, 1, 2, . . .
(49)
Also, we define Gmnpqr
=
8
0,
if m=p or n=r,
(−1)q mnep eq er (1−(−1))m−p (1−(−1)n−r ), p 2 (m 2−p 2 )(n 2−r 2 )
if else. (50)
Figure 18. Mode shape of the 100th system mode. The mode shape is dominated by the 94th plate mode (i.e., m=7, n=2). (a) Plate mode; (b) three-dimensional view of the plate mode; (c) acoustic mode at y=H; (d) acoustic mode at z=W/2.
,
599
Hmnpqr
8
0,
=
if m=p or n=r,
4(−1)q mn (1−(−1))m−p (1−(−1)n−r ), p 2 (m 2−p 2 )(n 2−r 2 )
if else. (51)
The plate and the acoustic medium are considered to have the same properties as in the previous two dimensional example. The cavity has the dimension of L=1 m, H=(2/3) m, W=0·5 m and the plate has the thickness of h=0·002 m. This configuration is expected to give the system a relatively strong coupling effect. By a similar procedure as before, we obtain the non-dimensional frequency b of the coupled system as shown in Table 3. Obviously, more terms have to be used because of the added dimensionality. However, the table shows that the results from 189-term solutions (the second column of the coupled solution) are accurate enough for practical purposes. In fact, the algorithm developed in this work can easily handle fairly large problems. For example the calculation using 464 terms (the last column) was done on a mid-range personal computer. Figures 10–18 show some of the coupled mode shapes of the system. 4. CONCLUDING REMARKS
Demonstrations of the new analysis procedure to solve free vibration problems of dynamic systems with elasto-acoustic coupling have been made. Two- and threedimensional rectangular cavities bounded by an elastic plate were used as examples. It has been shown that the eigenvalue problems of such systems can be easily formulated and solved utilizing the proposed method. Expanding the system responses in terms of uncoupled natural modes of the sub-system, the system characteristic equation has been obtained as a matrix eigenvalue equation. It has been shown that the resulting non-standard eigenvalue problem can be transformed into a standard linear eigenvalue problem by a simple matrix transformation which reduces the necessary computational effort drastically. The best application of this work may be found in the interior or exterior noise problems of various industrial equipment. For the exterior noise problems, the procedure will have to be extended to include the noise radiation model. However, in such a case, the coupling effect will have to be considered only between the structure and the interior cavity because the exterior pressure level is very low for most practical cases [8]. ACKNOWLEDGMENT
The support for the first author from the University Research Council and the Structural Dynamics Research Laboratories of the University of Cincinnati is gratefully acknowledged. REFERENCES 1. K. L. H and J. K 1995 Journal of Sound and Vibration 188, 561–575. Analysis of free vibration of structural–acoustic coupled systems, part I: development and verification of the procedure. 2. E. H. D and H. M. V 1963 AIAA Journal 1, 476. The effect of a cavity on panel vibration. 3. A. J. P 1965 Journal of Sound and Vibration 2, 197–209. Free vibrations of a rectangular panel backed by a closed rectangular cavity.
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