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ANALYSIS OF FREE VIBRATION OF STRUCTURAL–ACOUSTIC COUPLED SYSTEMS, PART I: DEVELOPMENT AND VERIFICATION OF THE PROCEDURE
Journal of Sound and Vibration (1995) 188(4), 561–575
ANALYSIS OF FREE VIBRATION OF STRUCTURAL–ACOUSTIC COUPLED SYSTEMS, PART I: DEVELOPMENT AND VERIFICATION OF THE PROCEDURE 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) A new procedure to formulate and analyze the free vibration problem of structural–acoustic coupled systems is suggested. The system equations are formulated utilizing the concept of the equivalent mass source. The modal expansion method and a matrix transformation technique are used to solve the system equations to obtain the natural frequencies and modes of the system. The procedure is verified by comparing its result with the exact solution of a one-dimensional coupled system. Parameters dictating coupling effects are also identified and discussed. 7 1995 Academic Press Limited
1. INTRODUCTION
When an acoustic cavity is enclosed by a flexible structure, dynamic responses of the structure and the cavity are coupled together. When the coupling effect between the structure and acoustic cavity is strong, the dynamic equations which govern the structural motion and the cavity pulsation should be solved simultaneously. This type of problem has been called the ‘‘acousto-elasticity problem’’ or ‘‘structural–acoustic coupled problem’’. Practical examples are found in noise problems of automobile mufflers, hermetic compressors and passenger compartments of automobiles or aircraft. Dowell and Voss [1] expressed the acoustic velocity potential in a rectangular cavity in terms of an infinite Fourier series to solve the coupled problem. They derived an approximate method to predict the effect of the acoustic pulsations on the panel flutter of a supersonic aircraft. Later, Pretlove [2] used a similar technique to obtain an analytic solution using approximate acoustic modes to investigate the effect of the pressure pulsations of the cavity on the panel motion. It was shown that the acoustic medium in the cavity acts like a spring or virtual mass on the panel. The coupled system frequencies were obtained by solving a non-linear matrix characteristic equation using an iterative scheme. The method seems to be applicable only to a weakly coupled system because the cross-coupling terms were neglected. Au-Yang [3] considered a system composed of two coaxial cylindrical shells filled with fluid between the shells, and determined the virtual mass as a function of the system frequency. By assuming that the shell motions are coupled only with the fluid but not with each other, two sets of fluid–structure coupled equations were solved simultaneously. The resulting non-linear eigenvalue problem was solved by a matrix iteration technique. In 561 0022–460X/95/490561+15 $12.00/0
7 1995 Academic Press Limited
562
. . .
the papers by Bentley and Firth [4] and Fuller and Fahy [5], a cylindrical shell filled with fluid was considered. They used similar approaches, expanding the shell motion and the sound field using the natural modes in the Bessel function forms. The system characteristic equation was obtained as a transcendental equation, therefore a numerical root search scheme was required to find the eigenvalues. Narayanan and Shanbhag [6] studied the sound transmission and structural response of a damped sandwich panel backed by many small cavities. In their work, a matrix inversion scheme was used to obtain the forced response solutions. Pan and Bies [7] and Dowell et al. [8] expanded both the acoustic sound field and the structural displacement in terms of there uncoupled natural modes. In the work of the latter, a good discussion was made on the general theory of the acoustic-elasticity including the effect of the absorbent wall. Although both methods in references [7] and [8] allow one to solve the coupled system frequencies, the related formulation and solution procedures are very involved. Furthermore, the dimension of the matrices of the characteristic equation becomes twice the total degrees of freedom of the system. Wolf [9] and Nefske et al. [10] applied the mode synthesis technique with the finite element method which gives a linear eigenvalue equation in a matrix form. Their method is similar to the one proposed in this work, except that the formulation in the latter is done in analytical forms. In the first part of this paper, a new formulation procedure is proposed to obtain the dynamic equations of structural–acoustic coupled systems utilizing a concept of the equivalent mass source. Then, the equations are solved by the modal expansion method using uncoupled natural modes of the sub-systems. The validity of the procedure is proved by using a one-dimensional acoustic pipe as an example. Because the solution in this method is obtained in analytic forms, a complete proof of the procedure becomes possible. It will be demonstrated that the proposed method can be easily applied to more general problems with two or three dimensionality in the second paper from this work [11]. Because the procedure presented here does not use any simplifications, the accuracy of the result is constrained only by the capacity of the computer. The effort required to solve a structural–acoustic coupled system becomes approximately the same as that to solve a forced vibration problem of the system of similar complexity. 2. FORMULATION AND ANALYSIS OF THE COUPLED STRUCTURE–ACOUSTIC SYSTEM
Typically, formulation of the governing equations of the structural–acoustic coupled system has been done by an integral formulation utilizing Green’s theorem [8] or matching boundary conditions at the interface [3]. An alternative method is presented here,
Figure 1. One-dimensional acoustic–structural coupled system.
,
563
employing a concept of the equivalent mass source to treat the coupling effect between the structure and the acoustic medium. Let us consider a one-dimensional acoustic pipe whose one end is closed and the other end is attached to a one-degree-of-freedom mass–spring system as shown in Figure 1. External mass flow inputs into the system may also be considered at any arbitrary points as shown in the figure. Using such a simple system makes it possible to prove the validity of the procedure to be developed since the exact solution is available. As it will be seen, it also helps to discuss general characteristics of the structural–acoustic coupling effect with more insight. In this work, we limit the problem to the harmonic input case. Obviously, any other cases can be handled by the superposition of the harmonic responses. 2.1. The wave equation of the acoustic system shown in Figure 1 becomes 1 2p 1 1 2p 1 (x, t)=− (m˙ (x, t)), 2 (x, t)− 2 co 1t 2 1t 1x
(1)
where co is the speed of sound in the acoustic medium, p(x, t) is the acoustic pressure in the cavity and m˙ (x, t) is the harmonic mass flow source distributed in the acoustic medium. The equation of motion of the structure, which is a one-degree-of-freedom system in this example, is represented by Mj +Kj=−Sp(l, t)−F eivt ,
(2)
where M and K are the mass and the spring constants respectively, j is the displacement of the mass and i=z−1. Also, F is the amplitude of the external force applied to the structure and S is the cross-sectional area of the pipe. It is considered that the mass flow source is composed of the regular mass flow inputs and the effective flow input caused by the structural motion, as follows: N
m˙ (x, t)=ro j (t)d(x−l)+ s ro Qj d(x−xj ) eivt ,
(3)
j=1
where ro is the density of the acoustic medium, d(x) is the Dirac delta function and Qj is the external input flow at x=xj . Now, the solution of the combined system can be obtained by solving equations (1) and (2) simultaneously. As shown, the dynamic equations of the coupled system have been obtained simply in the form of two forced vibration equations. 2.2. Equations (1) and (2) are solved by the standard modal expansion method using the uncoupled natural modes of the structure and acoustic cavity. Here, the uncoupled mode is defined as the in vacuo mode of the structure and the rigid wall acoustic mode of the cavity. If the input is harmonic, equation (1) becomes
$
%
N d2P(x) 2 2 2 2 +k P(x)= ro co k j d(x−l)− s iro co kQj d(x−xj ) , dx j=1
(4)
. . .
564
where P(x) is the amplitude of the acoustic pressure, j is the amplitude of the structural motion, and k=v/co is the wave number of the acoustic medium. We expand the pressure in the above equation in terms of uncoupled acoustic modes, a
P(x)= s An P n (x),
(5)
n=0
where P n (x)=cos (npx/l) is the nth uncoupled acoustic mode with the associated natural frequency of v˜ n=co k˜ n=co (np/l). Then, equation (4) becomes a
s An n=0
$
%$
%
N d2P n (x) +k 2P n (x) = ro co2k 2j d(x−l)−i s ro co kQj d(x−xj ) . 2 dx j=1
(6)
Applying the orthogonal property of the uncoupled modes to equation (6), as in the standard natural mode expansion procedure [12], we obtain coupled modal equations as follows: An (k 2−k˜ n2 )
g
l
$
N
%
P n2 (x) dx= ro co2k 2j P n (l)−i s ro co kQj P n (xj ) ,
0
j=1
(7)
where n=0, 1, 2, 3, . . . Substituting equation (5) into equation (2), the equation of motion of the structure becomes a
(K−Mv 2 )j =−S s An P n (l)−F.
(8)
n=0
Now, equations (7) and (8) are the equations of motion of the coupled system in terms of the uncoupled natural modes of its sub-systems. It should be noted that the structural motion should also be expanded in terms of its natural modes if the structure is a continuous one as in reference [11]. After the integrations are evaluated, equation (7) becomes N
(An /en )(b 2−bn2 )+(−1)n+1 (ro co2 /l)j b 2+ibro co s Qj cos (npxj /l)=0,
(9)
j=1
where n=0, 1, 2, 3, . . . Also, en=1 for n=0 and en=2 for n=1, 2, 3, . . . , b=kl, and bn=kn l or bn=np. Equation (8) can be put into the form a
2 (Mg /M) s (−1)nAn−(ro co2 /l)(b 2−bM )j =−rlF/M,
(10)
n=0
where bM=vM l/Co , vM=zK/M which is the in vacuo natural frequency of the structure, and Mg=ro lS which is the total mass of the acoustic medium in the system.
,
565
Equations (9) and (10) are further rearranged in the following forms using two 2 non-dimensional parameters defined as A* n =An /ro co and j*=j /l: N
2 2 n+1 A* j*en b 2+iben s (Qj /co ) cos (npxj /l)=0, n (b −bn )+(−1)
(11)
j=1
for n=0, 1, 2, 3, . . . , and a
2 2 2 (Mg /M) s (−1)nA* n −(b −bM )j*=−Fl/Mco .
(12)
n=0
By taking only the first n acoustic equations from equation (11), equations (11) and (12) can be put into matrix form [W]{X}=−i{F }, (13) where the matrices can be referenced from the following equation:
K b2 G 0 G G 0 G 0 G .. G . G .. G . G 0 G Mg kM
where rj=xj /l.
0 0 2 b −b22
··· ··· ··· ··· ··· ···
0 0 0
0 . . . . . . 0 M − g M
0 . . . . . . 0 Mg M
··· ··· . . . . . . . . . . . . ··· ···
0 . . . . . . b 2−bn2 M (−1)n g M
··· ···
F G G G G G G G G G j =−i J G G G G G G G G G f
L G G G 2b 2 G . G . . G . G . . G 2(−1)n+1b 2G 2 G −(b 2−bM ) l −b 2 2b 2 −2b 2
0 b −b12 0 2
F A*0 J G A*G G 1G G A*2 G G A*3 G j .. f J .F G .. G G .G G A*n G G j*G f j
J G j=1 G G N 2b s (Qj /co ) cos (rj p) G j=1 G G N 2b s (Qj /co ) cos (2rj p) G j=1 G G N f , 2b s (Qj /co ) cos (3rj p) F j=1 G . . G . G . . G . G N G 2b s (Qj /co ) cos (nrj p) G j=1 G lF G −i 2 co M j N
s (Qj /co )
(14)
566
. . .
Forced response solutions can be obtained by solving equation (14) directly. The solution provides the relative strength of the participation of each uncoupled mode. Therefore, the actual pressure response can be obtained by substituting the solution vector to equation (5). If the only input to the system is input flow of strength Q1 at the left end of the pipe, then the forced response of the coupled system is obtained as
F A*0 J G A*1 G G . G . 1 }, {X}=g . h=−ib[W]−1 (Q1 /co ){F G A*n G G–––G f j* j
(15)
where {F 1 }=[1, 2, 2, . . . , 0]T. When [W]−1 is calculated, one can take the advantage of the fact that the matrix [W] has a large partition of the diagonal matrix to reduce the computation time. The work by Narayanan and Shanbhag [6] may be referred to for this purpose. The transfer functions TQ can be obtained by rearranging equation (15) as follows: A 0 (b) J F G G G A1 .(b) G A* G .. G −1 (co /Q1 ){X}=(co /Q1 ) ––– =g A 1 }, n (b) h=−ib[W] {F j* G G G ··· G G f TQ (b)/ib G j
89
(16)
where the transfer function TQ is defined as TQ=Q2 /Q1=−j /Q1 .
(17)
The transfer defined above reflects the ratio of the noise radiation from the system to the input flow. 2.3. For the free vibration case, equation (13) becomes [W]{X}={0}.
(18)
Noticing that the matrix [W] is mostly diagonal except the elements corresponding to the coupled terms, equation (18) can be put into the form [[L]−b 2[U]]{X}={0}, where the lower and upper triangular matrices [L] and [U] are
(19)
, 0 K 0 G 0 p2 G 0 0 G 0 0 G . . . G .. [L]= . G . . . G .. . G 0 G 0 G Mg Mg k− M M
K G G G G [U]=G G G G G k
567
0
0
···
···
0
0 (2p)2
0 0
··· ···
··· ···
0 0
0 . . . . . .
(3p)2 . . . . . .
···
···
0 . . . . . .
0
0
···
···
(np)2
Mg M
···
···
(−1)n+1
Mg M
−
. . . . . .
1
0
0
0 ··· ···
0
0 0 0 . . . . . . 0
1 0 0 . . . . . . 0
0 1 0 . . . . . . 0
0 0 1 . . . . . . 0
··· ···
0 0 0 . . . . . . 1
0
0
0
0 ··· ···
0
··· ··· ··· ··· ··· ··· ··· ··· ··· ···
L 0 G G 0 G 0 G . . G . , G . . G . G 0 G G 2 −bM l 0
Mg M
L G 2 G −2 G 2 G . G. . . G . G . . G 2(−1)n+1G −1 l
(20)
−1
(21)
Then, equation (19) can be made into the form [D]{X}=b 2{X},
(22)
where [D]=[U]−1 [L]. Inverting the matrix [U] is computationally very easy since it is a triangular matrix. In general, the matrix [D] in equation (22) becomes a non-symmetric matrix. Nefske et al. [10] obtained a similar equation, but based on the discrete modes from the finite element method solution. Now, equation (22) is a standard form of the eigenvalue problem whose solution algorithm is very well established and readily available in many commercial mathematic software. The eigenvalues and eigenvectors obtained from equation (22) give the approximate natural frequencies and mode shapes of the coupled system. The term ‘‘approximate’’ is used because only a finite number of terms are used in equation (22). Obviously, taking more terms will improve the accuracy of the solution assuming that uncoupled modes and the computational resource are available as necessary.
3. COUPLING EFFECT
Looking the coefficient matrix of equation (14), it is expected that the coupling effect will depend on the magnitude of the off-diagonal of the [W] matrix relative to the uncoupled frequency parameters. Therefore, the coupling effect will be stronger when the
568
. . .
parameter (Mg /M), which is the ratio of the mass of the acoustic medium to the mass of the structural system, is relatively large. It will also be shown later that the coupling effect becomes stronger and broader banded when the coupling occurs in the lower frequency range. The relative strength of the coupling effect can also be checked retrospectively (e.g., after the solution is obtained) from the system eigenvector. The components of the eigenvector represent the relative strength of the participation of each uncoupled mode as is seen from equation (15). Therefore, there are two or more relatively large components in the eigenvector when the coupling effect is strong at that system mode. On the contrary, there will be only one element which is dominantly larger than the others in the eigenvector when the coupling is weak. To illustrate the coupling effect, let the model considered in Figure 1 have the following data: l=0·12 m, co=162·9 m/s, rg=6·04 kg/m3; the radius of the pipe is r=0·008 m. The exact characteristic equation of such a system is given as [10, 13],
tan (b)=
Mg b 2 . M b 2−bM
(23)
3.1. 1: A case is considered where the physical size of the structure is much larger than that of the acoustic system and the structural natural frequency is much higher than the lowest natural frequency of the acoustic system. The stiffness of the structure is taken as K=3×105 N/m and the mass of the structure as twice the total mass of the acoustic medium. Figure 2 shows the graphical representation of equation (23) and system roots. The system roots are obtained as the points at which two curves corresponding to the two sides of equation (23) cross each other. As is shown in the figure, the uncoupled structural frequency is located between the eighth and ninth uncoupled acoustic frequencies. Figure 2 shows that the system frequencies are very close to the uncoupled frequencies except in the narrow frequency range around the uncoupled structural frequency. Therefore, the effect of coupling may be neglected in other frequency ranges for most practical purposes.
3.2. 2: If we choose the stiffness of K=2000 N/m and the structural mass of 30 per cent of the acoustic system, the graphical representation of equation (23) becomes that as shown in Figure 3. Now the uncoupled structural frequency is located between the second and the third uncoupled acoustic frequencies. In this case, the system characteristics change substantially in the lower frequency range. Therefore, the combined system will behave quite differently from what would have been expected from the result of the uncoupled analysis. Figure 3 also indicates that even with a strong coupling effect, both coupled and uncoupled systems have close natural frequencies in the higher frequency range.
,
569
Figure 2. Characteristic roots of a weakly coupled system: w, roots of the uncoupled system; ×, roots of 2 coupled system; ——, tan b; –––, (Mg /M)b/(bM −b 2 ).
4. VERIFICATION OF THE METHOD
The proposed method is verified in this section. One possible argument may be whether it is valid to use the rigid wall acoustic modes in the modal expansion, because a fluid velocity is not allowed to occur on the boundary in those modes while it should exist in the coupled system. The answer to the question is presented first, then the
Figure 3. Characteristic roots of a strongly coupled system: w, roots of the uncoupled system; ×, roots of 2 coupled system; ——, tan b; –––, (Mg /M)b/(bM −b 2 ).
. . .
570
eigenvalue solution algorithm is verified by showing that the solutions converge to the exact values.
4.1. The exact expression of the volume flow transfer function of the system in Figure 1 is obtained in a closed form as [13]
TQ=
Q2 cosec (b) = 2 Q1 cot (b)+(M/Mg )((bM −b 2 )/b),
(24)
where bM , Mg are the same as in equation (10). The exact characteristic equation shown in equation (23) is obtained by making the denominator of the transfer function TQ equal to zero. Exact system natural frequencies are obtained by finding the roots of this non-linear characteristic equation. In section 2, the model expansion method using uncoupled modes resulted in equations (9) and (10). Combining equations (9) and (10) with Q=F=0, we obtain
0 10 M Mg
1 $
%
a 2 b 2−bM 1 1 j
= +2 s j . 2 2 b2 b2 (b −(np) ) n=1
(25)
The series on the right side of equation (25) has been shown to have the following equality in reference [14] (equation 1.421–3):
a 1 1 1 = 2+2 s 2 2. b tan (b) b b −(np) n=1
(26)
Hence, equation (25) becomes the same as equation (23). This shows that the characteristic equation obtained from the proposed method is exactly the same as that obtained from the analytic solution. Thus, it proves that the in vacuo natural modes of the structure and the rigid wall acoustic modes of the cavity can be used for the analysis of the acoustic–structural coupled system analysis.
4.2. Numerical verification is made by comparing the solutions from the proposed method with the exact solutions from equation (23). Similar comparisons are found in reference [9] when the finite element method is used. The system is chosen to have the same properties and dimensions as the system discussed in the preceding section for the weak coupling case. Further comparisons are made for strongly coupled cases in the subsequent section. In Table 1 the natural frequency parameters obtained by the proposed procedure are listed for three cases when 10, 20 and 30 terms are used in equation (20). For example,
,
571
T 1 Non-dimensional frequency parameter b=kL obtained by different approaches when the coupling effect is weak Coupled analysis ZXXXXXXXXXXXXCXXXXXXXXXXXXV Selected Exact 20 terms 10 terms mode Root number analysis solution solution solution
Root number 1, acoustic mode 2, acoustic mode 3, acoustic mode 4, acoustic mode 5, acoustic mode 6, acoustic mode 7, acoustic mode 8, acoustic mode Structural mode 9, acoustic mode
0·00000 3·14159 6·28319 9·42478 12·5664 15·7079 18·8496 21·9911 23·6337 25·1327
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode
0·00000 3·13873 6·27714 9·41476 12·55072 15·68288 18·80372 21·85677 23·56194 25·28774
0·00000 3·13873 6·27714 9·41477 12·55076 15·68299 18·80416 21·86084 23·67432 25·29449
0·00000 3·13873 6·27715 9·41478 12·55080 15·68315 18·80484 21·86797 23·79803 25·31597
Void Void Void Void Void Void 18·80266 21·85534 23·63788 25·29359
a 10-term solution means that nine acoustic modes and one structural mode are used in the solution procedure, i.e., n=9 in equations (20) and (21). The results are compared with the exact frequencies obtained from equation (23) in Table 1. Table 1 shows that the solution converges to the exact solution as more modes are used. As was discussed for a general combined system in references [12] and [15], the acoustic modes with the uncoupled frequencies higher than the structural mode will have their frequencies increased, and the others will have theirs decreased. The comparison shows that the method gives a good agreement with the exact solution even when only 10 modes are used. More numerical examples in the next section will show that the solutions for strongly coupled cases also converge to the exact solution as more terms are used. The last column of Table 1 shows the solution obtained by using only four acoustic modes around the structural frequency. The result seems to be reasonable. Therefore, only a few modes in the vicinity of the coupling mode of interest need be used for the analysis. This will be useful for the analysis of coupling modes in the high frequency range. 5. MORE NUMERICAL EXAMPLES
5.1. 1: Figure 4 shows the transfer functions TQ defined in equation (17) based on various approaches. The exact transfer function in equation (24) is compared with the transfer function obtained by solving the matrix equation (15) at each frequency. The transfer function obtained by the uncoupled approach is also included in the comparison for reference. In the uncoupled approach, it is considered that the structure is excited by the pressure of the cavity, assuming that all of its boundaries are rigid. The function is obtained as Mg 1 b . 2 M sin b bM −b 2
TQ=−
(27)
The figure shows that the method in this work gives good results, except that the 10-term solution fails around the 10th system frequency, which is expected because all 10th or higher modes are truncated in the solution. Also, it shows that the result from the
. . .
572
Figure 4. Transfer functions based on different approaches when the coupling effect is weak. –· –· –, uncoupled approach; –––, exact solution; · · · ·, 20-term solution; ——, 10-term solution.
uncoupled approach would be acceptable in the lower frequency range. Figure 6 shows the pressure mode shape of the ninth system mode which is predominantly a structural mode. Very little change is observed from the corresponding uncoupled acoustic mode.
5.2. 2: Table 2 lists the system natural frequencies for the case shown in Figure 3. Again, frequencies obtained by the proposd method using different numbers of terms are compared with the frequencies of the uncoupled subsystems. Table 2 and Figure 3 indicate that a strong coupling effect exists around the structural mode. The third system mode corresponds predominantly to the structural mode of the uncoupled system. The second, third and fourth system modes are quite different from any of the uncoupled sub-system modes. It is shown that the proposed method provides reasonable results when only 10 terms are used, even if the coupling effect is strong. Again, only four acoustic modes around the structural frequency are used for the selected mode solutions in the table. T 2 Non-dimensional frequency parameter b=kL obtained by different approaches when the coupling effect is strong Coupled analysis ZXXXXXXXXXXXXCXXXXXXXXXXXXV Selected Exact 20 terms 10 terms mode Root number analysis solution solution solution
Root number 1, acoustic mode 2, acoustic mode Structural mode 3, acoustic mode 4, acoustic mode 5, acoustic mode 6, acoustic mode 7, acoustic mode 8, acoustic mode 9, acoustic mode
0·00000 3·14159 4·98242 6·28319 9·42478 12·5664 15·7079 18·8496 21·9911 25·1327
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode
0·00000 2·67439 4·71239 7·04184 9·85149 12·86232 15·93577 19·03537 22·14836 25·26914
0·00000 2·67904 4·84710 7·07303 9·86971 12·87463 15·94512 19·04298 22·15483 25·27481
0·00000 2·68447 4·88644 7·11368 9·89481 12·89263 15·95999 19·05668 22·16898 25·29294
0·00000 2·74835 5·12007 7·35099 10·09257 Void Void Void Void Void
,
573
Figure 5. Transfer functions based on different approaches when the coupling effect is strong. –· –· –, Uncoupled approach; –––, exact solution; · · · ·, 20-term solution; ——, 10-term solution.
Figure 5 shows the transfer functions TQ obtained by various approaches when the coupling effect is strong in the frequency range around the uncoupled structural mode. The figure shows that the combined system has substantially different characteristics compared to the characteristics of the sub-systems. Therefore, the transfer function obtained by the uncoupled approach would give completely wrong information in a wide frequency range. Finally, Figures 7 and 8 show the pressure mode shapes of the third and fourth system modes which have the strongest coupling effect. They are quite different from any of the uncoupled modes, which indicates that a complete coupled analysis has to be done to make a reasonably accurate prediction of the interior pressure field.
6. CONCLUSIONS
A procedure to formulate and analyze the free vibration problem of structural–acoustic coupled systems has been suggested and verified. The procedure is very simple and intuitive to implement. The related effort in the formulation and numerical computation to analyze the free-vibration problem of a structural–acoustic coupled system becomes roughly the same as the effort of solving a standard forced vibration problem of similar complexity. Further demonstrations of the computational advantage of the proposed method are made in the second part of this work, using two- and three-dimensional cases as examples.
Figure 6. Pressure profile of the ninth system mode of a weakly coupled system by different approaches: ——, uncoupled approach; · · · ·, exact mode shape of the coupled system; –––, mode shape obtained by using 19 acoustic modes; –· –· –, mode shape obtained by using 10 acoustic modes.
574
. . .
Figure 7. Pressure profile of the second system mode of a strongly coupled system by different approaches: ——, uncoupled approach; · · · ·, exact mode shape of the coupled system; –––, mode shape obtained by using 19 acoustic modes; –· –· –, mode shape obtained by using 10 acoustic modes.
Figure 8. Pressure profile of the third system mode of a strongly coupled system by different approaches: ——, uncoupled approach; · · · ·, exact mode shape of the coupled system; –––, mode shape obtained by using 19 acoustic modes; –· –· –, mode shape obtained by using 10 acoustic modes.
If the natural modes of the uncoupled sub-systems are known, the rest of the procedure of the proposed method becomes a purely computational task. Therefore, any analytical, numerical or experimental methods can be used to obtain the natural modes of the sub-systems to use in the proposed method. For example, one may use the boundary element method for the acoustic cavity and the finite element method for the structure to obtain the respective uncoupled natural modes.
ACKNOWLEDGMENT
The support for the first author from the University Research Council and the Structural Dynamics Research Laboratories at the University of Cincinnati is gratefully acknowledged.
REFERENCES 1. E. H. D and H. M. V 1963 AIAA Journal 1, 476. The effect of a cavity on panel vibration. 2. 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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3. M. K. A-Y 1976 Journal of Applied Mechanics 43, 480–484. Free vibrations of fluid-coupled coaxial cylindrical shells of different lengths. 4. P. G. B and D. F 1971 Journal of Sound and Vibration 19, 179–191. Acoustically excited vibrations in a liquid-filled cylindrical tank. 5. C. R. F and F. J. F 1982 Journal of Sound and Vibration 81, 501–518. Characteristics of wave propagation and energy distributions in cylindrical elastic shells filled with fluid. 6. 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. 7. 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. 8. 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. 9. J. A. W 1977 American Institute of Aeronautics and Astronautics Journal 15, 743–745. Modal synthesis for combined structural–acoustic systems. 10. D. J. N, J. A. W and L. J. H 1982 Journal of Sound and Vibration 80(2), 247–266. Structural–acoustic finite element analysis of the automotive passenger compartment: review of current practice 11. K. L. H and J. K 1995 Journal of Sound and Vibration (in press). Analysis of free-vibration of structural–acoustic coupled systems, part II: two and three-dimensional examples. 12. W. S 1993 Vibrations of Shells and Plates, second edition. New York: Marcel Dekker. 13. W. S 1978 Gas Pulsations in Compressor and Engine Manifold. West Lafayette, In: Ray W. Herric Laboratories, Purdue University. 14. I. S. G and I. M. R 1965 Table of Integrals Series and Products. New York: Academic Press. 15. E. H. D 1979 Journal of Applied Mechanics 46, 206–209. On some general properties of combined dynamic systems.
ANALYSIS OF FREE VIBRATION OF STRUCTURAL–ACOUSTIC COUPLED SYSTEMS, PART I: DEVELOPMENT AND VERIFICATION OF THE PROCEDURE 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) A new procedure to formulate and analyze the free vibration problem of structural–acoustic coupled systems is suggested. The system equations are formulated utilizing the concept of the equivalent mass source. The modal expansion method and a matrix transformation technique are used to solve the system equations to obtain the natural frequencies and modes of the system. The procedure is verified by comparing its result with the exact solution of a one-dimensional coupled system. Parameters dictating coupling effects are also identified and discussed. 7 1995 Academic Press Limited
1. INTRODUCTION
When an acoustic cavity is enclosed by a flexible structure, dynamic responses of the structure and the cavity are coupled together. When the coupling effect between the structure and acoustic cavity is strong, the dynamic equations which govern the structural motion and the cavity pulsation should be solved simultaneously. This type of problem has been called the ‘‘acousto-elasticity problem’’ or ‘‘structural–acoustic coupled problem’’. Practical examples are found in noise problems of automobile mufflers, hermetic compressors and passenger compartments of automobiles or aircraft. Dowell and Voss [1] expressed the acoustic velocity potential in a rectangular cavity in terms of an infinite Fourier series to solve the coupled problem. They derived an approximate method to predict the effect of the acoustic pulsations on the panel flutter of a supersonic aircraft. Later, Pretlove [2] used a similar technique to obtain an analytic solution using approximate acoustic modes to investigate the effect of the pressure pulsations of the cavity on the panel motion. It was shown that the acoustic medium in the cavity acts like a spring or virtual mass on the panel. The coupled system frequencies were obtained by solving a non-linear matrix characteristic equation using an iterative scheme. The method seems to be applicable only to a weakly coupled system because the cross-coupling terms were neglected. Au-Yang [3] considered a system composed of two coaxial cylindrical shells filled with fluid between the shells, and determined the virtual mass as a function of the system frequency. By assuming that the shell motions are coupled only with the fluid but not with each other, two sets of fluid–structure coupled equations were solved simultaneously. The resulting non-linear eigenvalue problem was solved by a matrix iteration technique. In 561 0022–460X/95/490561+15 $12.00/0
7 1995 Academic Press Limited
562
. . .
the papers by Bentley and Firth [4] and Fuller and Fahy [5], a cylindrical shell filled with fluid was considered. They used similar approaches, expanding the shell motion and the sound field using the natural modes in the Bessel function forms. The system characteristic equation was obtained as a transcendental equation, therefore a numerical root search scheme was required to find the eigenvalues. Narayanan and Shanbhag [6] studied the sound transmission and structural response of a damped sandwich panel backed by many small cavities. In their work, a matrix inversion scheme was used to obtain the forced response solutions. Pan and Bies [7] and Dowell et al. [8] expanded both the acoustic sound field and the structural displacement in terms of there uncoupled natural modes. In the work of the latter, a good discussion was made on the general theory of the acoustic-elasticity including the effect of the absorbent wall. Although both methods in references [7] and [8] allow one to solve the coupled system frequencies, the related formulation and solution procedures are very involved. Furthermore, the dimension of the matrices of the characteristic equation becomes twice the total degrees of freedom of the system. Wolf [9] and Nefske et al. [10] applied the mode synthesis technique with the finite element method which gives a linear eigenvalue equation in a matrix form. Their method is similar to the one proposed in this work, except that the formulation in the latter is done in analytical forms. In the first part of this paper, a new formulation procedure is proposed to obtain the dynamic equations of structural–acoustic coupled systems utilizing a concept of the equivalent mass source. Then, the equations are solved by the modal expansion method using uncoupled natural modes of the sub-systems. The validity of the procedure is proved by using a one-dimensional acoustic pipe as an example. Because the solution in this method is obtained in analytic forms, a complete proof of the procedure becomes possible. It will be demonstrated that the proposed method can be easily applied to more general problems with two or three dimensionality in the second paper from this work [11]. Because the procedure presented here does not use any simplifications, the accuracy of the result is constrained only by the capacity of the computer. The effort required to solve a structural–acoustic coupled system becomes approximately the same as that to solve a forced vibration problem of the system of similar complexity. 2. FORMULATION AND ANALYSIS OF THE COUPLED STRUCTURE–ACOUSTIC SYSTEM
Typically, formulation of the governing equations of the structural–acoustic coupled system has been done by an integral formulation utilizing Green’s theorem [8] or matching boundary conditions at the interface [3]. An alternative method is presented here,
Figure 1. One-dimensional acoustic–structural coupled system.
,
563
employing a concept of the equivalent mass source to treat the coupling effect between the structure and the acoustic medium. Let us consider a one-dimensional acoustic pipe whose one end is closed and the other end is attached to a one-degree-of-freedom mass–spring system as shown in Figure 1. External mass flow inputs into the system may also be considered at any arbitrary points as shown in the figure. Using such a simple system makes it possible to prove the validity of the procedure to be developed since the exact solution is available. As it will be seen, it also helps to discuss general characteristics of the structural–acoustic coupling effect with more insight. In this work, we limit the problem to the harmonic input case. Obviously, any other cases can be handled by the superposition of the harmonic responses. 2.1. The wave equation of the acoustic system shown in Figure 1 becomes 1 2p 1 1 2p 1 (x, t)=− (m˙ (x, t)), 2 (x, t)− 2 co 1t 2 1t 1x
(1)
where co is the speed of sound in the acoustic medium, p(x, t) is the acoustic pressure in the cavity and m˙ (x, t) is the harmonic mass flow source distributed in the acoustic medium. The equation of motion of the structure, which is a one-degree-of-freedom system in this example, is represented by Mj +Kj=−Sp(l, t)−F eivt ,
(2)
where M and K are the mass and the spring constants respectively, j is the displacement of the mass and i=z−1. Also, F is the amplitude of the external force applied to the structure and S is the cross-sectional area of the pipe. It is considered that the mass flow source is composed of the regular mass flow inputs and the effective flow input caused by the structural motion, as follows: N
m˙ (x, t)=ro j (t)d(x−l)+ s ro Qj d(x−xj ) eivt ,
(3)
j=1
where ro is the density of the acoustic medium, d(x) is the Dirac delta function and Qj is the external input flow at x=xj . Now, the solution of the combined system can be obtained by solving equations (1) and (2) simultaneously. As shown, the dynamic equations of the coupled system have been obtained simply in the form of two forced vibration equations. 2.2. Equations (1) and (2) are solved by the standard modal expansion method using the uncoupled natural modes of the structure and acoustic cavity. Here, the uncoupled mode is defined as the in vacuo mode of the structure and the rigid wall acoustic mode of the cavity. If the input is harmonic, equation (1) becomes
$
%
N d2P(x) 2 2 2 2 +k P(x)= ro co k j d(x−l)− s iro co kQj d(x−xj ) , dx j=1
(4)
. . .
564
where P(x) is the amplitude of the acoustic pressure, j is the amplitude of the structural motion, and k=v/co is the wave number of the acoustic medium. We expand the pressure in the above equation in terms of uncoupled acoustic modes, a
P(x)= s An P n (x),
(5)
n=0
where P n (x)=cos (npx/l) is the nth uncoupled acoustic mode with the associated natural frequency of v˜ n=co k˜ n=co (np/l). Then, equation (4) becomes a
s An n=0
$
%$
%
N d2P n (x) +k 2P n (x) = ro co2k 2j d(x−l)−i s ro co kQj d(x−xj ) . 2 dx j=1
(6)
Applying the orthogonal property of the uncoupled modes to equation (6), as in the standard natural mode expansion procedure [12], we obtain coupled modal equations as follows: An (k 2−k˜ n2 )
g
l
$
N
%
P n2 (x) dx= ro co2k 2j P n (l)−i s ro co kQj P n (xj ) ,
0
j=1
(7)
where n=0, 1, 2, 3, . . . Substituting equation (5) into equation (2), the equation of motion of the structure becomes a
(K−Mv 2 )j =−S s An P n (l)−F.
(8)
n=0
Now, equations (7) and (8) are the equations of motion of the coupled system in terms of the uncoupled natural modes of its sub-systems. It should be noted that the structural motion should also be expanded in terms of its natural modes if the structure is a continuous one as in reference [11]. After the integrations are evaluated, equation (7) becomes N
(An /en )(b 2−bn2 )+(−1)n+1 (ro co2 /l)j b 2+ibro co s Qj cos (npxj /l)=0,
(9)
j=1
where n=0, 1, 2, 3, . . . Also, en=1 for n=0 and en=2 for n=1, 2, 3, . . . , b=kl, and bn=kn l or bn=np. Equation (8) can be put into the form a
2 (Mg /M) s (−1)nAn−(ro co2 /l)(b 2−bM )j =−rlF/M,
(10)
n=0
where bM=vM l/Co , vM=zK/M which is the in vacuo natural frequency of the structure, and Mg=ro lS which is the total mass of the acoustic medium in the system.
,
565
Equations (9) and (10) are further rearranged in the following forms using two 2 non-dimensional parameters defined as A* n =An /ro co and j*=j /l: N
2 2 n+1 A* j*en b 2+iben s (Qj /co ) cos (npxj /l)=0, n (b −bn )+(−1)
(11)
j=1
for n=0, 1, 2, 3, . . . , and a
2 2 2 (Mg /M) s (−1)nA* n −(b −bM )j*=−Fl/Mco .
(12)
n=0
By taking only the first n acoustic equations from equation (11), equations (11) and (12) can be put into matrix form [W]{X}=−i{F }, (13) where the matrices can be referenced from the following equation:
K b2 G 0 G G 0 G 0 G .. G . G .. G . G 0 G Mg kM
where rj=xj /l.
0 0 2 b −b22
··· ··· ··· ··· ··· ···
0 0 0
0 . . . . . . 0 M − g M
0 . . . . . . 0 Mg M
··· ··· . . . . . . . . . . . . ··· ···
0 . . . . . . b 2−bn2 M (−1)n g M
··· ···
F G G G G G G G G G j =−i J G G G G G G G G G f
L G G G 2b 2 G . G . . G . G . . G 2(−1)n+1b 2G 2 G −(b 2−bM ) l −b 2 2b 2 −2b 2
0 b −b12 0 2
F A*0 J G A*G G 1G G A*2 G G A*3 G j .. f J .F G .. G G .G G A*n G G j*G f j
J G j=1 G G N 2b s (Qj /co ) cos (rj p) G j=1 G G N 2b s (Qj /co ) cos (2rj p) G j=1 G G N f , 2b s (Qj /co ) cos (3rj p) F j=1 G . . G . G . . G . G N G 2b s (Qj /co ) cos (nrj p) G j=1 G lF G −i 2 co M j N
s (Qj /co )
(14)
566
. . .
Forced response solutions can be obtained by solving equation (14) directly. The solution provides the relative strength of the participation of each uncoupled mode. Therefore, the actual pressure response can be obtained by substituting the solution vector to equation (5). If the only input to the system is input flow of strength Q1 at the left end of the pipe, then the forced response of the coupled system is obtained as
F A*0 J G A*1 G G . G . 1 }, {X}=g . h=−ib[W]−1 (Q1 /co ){F G A*n G G–––G f j* j
(15)
where {F 1 }=[1, 2, 2, . . . , 0]T. When [W]−1 is calculated, one can take the advantage of the fact that the matrix [W] has a large partition of the diagonal matrix to reduce the computation time. The work by Narayanan and Shanbhag [6] may be referred to for this purpose. The transfer functions TQ can be obtained by rearranging equation (15) as follows: A 0 (b) J F G G G A1 .(b) G A* G .. G −1 (co /Q1 ){X}=(co /Q1 ) ––– =g A 1 }, n (b) h=−ib[W] {F j* G G G ··· G G f TQ (b)/ib G j
89
(16)
where the transfer function TQ is defined as TQ=Q2 /Q1=−j /Q1 .
(17)
The transfer defined above reflects the ratio of the noise radiation from the system to the input flow. 2.3. For the free vibration case, equation (13) becomes [W]{X}={0}.
(18)
Noticing that the matrix [W] is mostly diagonal except the elements corresponding to the coupled terms, equation (18) can be put into the form [[L]−b 2[U]]{X}={0}, where the lower and upper triangular matrices [L] and [U] are
(19)
, 0 K 0 G 0 p2 G 0 0 G 0 0 G . . . G .. [L]= . G . . . G .. . G 0 G 0 G Mg Mg k− M M
K G G G G [U]=G G G G G k
567
0
0
···
···
0
0 (2p)2
0 0
··· ···
··· ···
0 0
0 . . . . . .
(3p)2 . . . . . .
···
···
0 . . . . . .
0
0
···
···
(np)2
Mg M
···
···
(−1)n+1
Mg M
−
. . . . . .
1
0
0
0 ··· ···
0
0 0 0 . . . . . . 0
1 0 0 . . . . . . 0
0 1 0 . . . . . . 0
0 0 1 . . . . . . 0
··· ···
0 0 0 . . . . . . 1
0
0
0
0 ··· ···
0
··· ··· ··· ··· ··· ··· ··· ··· ··· ···
L 0 G G 0 G 0 G . . G . , G . . G . G 0 G G 2 −bM l 0
Mg M
L G 2 G −2 G 2 G . G. . . G . G . . G 2(−1)n+1G −1 l
(20)
−1
(21)
Then, equation (19) can be made into the form [D]{X}=b 2{X},
(22)
where [D]=[U]−1 [L]. Inverting the matrix [U] is computationally very easy since it is a triangular matrix. In general, the matrix [D] in equation (22) becomes a non-symmetric matrix. Nefske et al. [10] obtained a similar equation, but based on the discrete modes from the finite element method solution. Now, equation (22) is a standard form of the eigenvalue problem whose solution algorithm is very well established and readily available in many commercial mathematic software. The eigenvalues and eigenvectors obtained from equation (22) give the approximate natural frequencies and mode shapes of the coupled system. The term ‘‘approximate’’ is used because only a finite number of terms are used in equation (22). Obviously, taking more terms will improve the accuracy of the solution assuming that uncoupled modes and the computational resource are available as necessary.
3. COUPLING EFFECT
Looking the coefficient matrix of equation (14), it is expected that the coupling effect will depend on the magnitude of the off-diagonal of the [W] matrix relative to the uncoupled frequency parameters. Therefore, the coupling effect will be stronger when the
568
. . .
parameter (Mg /M), which is the ratio of the mass of the acoustic medium to the mass of the structural system, is relatively large. It will also be shown later that the coupling effect becomes stronger and broader banded when the coupling occurs in the lower frequency range. The relative strength of the coupling effect can also be checked retrospectively (e.g., after the solution is obtained) from the system eigenvector. The components of the eigenvector represent the relative strength of the participation of each uncoupled mode as is seen from equation (15). Therefore, there are two or more relatively large components in the eigenvector when the coupling effect is strong at that system mode. On the contrary, there will be only one element which is dominantly larger than the others in the eigenvector when the coupling is weak. To illustrate the coupling effect, let the model considered in Figure 1 have the following data: l=0·12 m, co=162·9 m/s, rg=6·04 kg/m3; the radius of the pipe is r=0·008 m. The exact characteristic equation of such a system is given as [10, 13],
tan (b)=
Mg b 2 . M b 2−bM
(23)
3.1. 1: A case is considered where the physical size of the structure is much larger than that of the acoustic system and the structural natural frequency is much higher than the lowest natural frequency of the acoustic system. The stiffness of the structure is taken as K=3×105 N/m and the mass of the structure as twice the total mass of the acoustic medium. Figure 2 shows the graphical representation of equation (23) and system roots. The system roots are obtained as the points at which two curves corresponding to the two sides of equation (23) cross each other. As is shown in the figure, the uncoupled structural frequency is located between the eighth and ninth uncoupled acoustic frequencies. Figure 2 shows that the system frequencies are very close to the uncoupled frequencies except in the narrow frequency range around the uncoupled structural frequency. Therefore, the effect of coupling may be neglected in other frequency ranges for most practical purposes.
3.2. 2: If we choose the stiffness of K=2000 N/m and the structural mass of 30 per cent of the acoustic system, the graphical representation of equation (23) becomes that as shown in Figure 3. Now the uncoupled structural frequency is located between the second and the third uncoupled acoustic frequencies. In this case, the system characteristics change substantially in the lower frequency range. Therefore, the combined system will behave quite differently from what would have been expected from the result of the uncoupled analysis. Figure 3 also indicates that even with a strong coupling effect, both coupled and uncoupled systems have close natural frequencies in the higher frequency range.
,
569
Figure 2. Characteristic roots of a weakly coupled system: w, roots of the uncoupled system; ×, roots of 2 coupled system; ——, tan b; –––, (Mg /M)b/(bM −b 2 ).
4. VERIFICATION OF THE METHOD
The proposed method is verified in this section. One possible argument may be whether it is valid to use the rigid wall acoustic modes in the modal expansion, because a fluid velocity is not allowed to occur on the boundary in those modes while it should exist in the coupled system. The answer to the question is presented first, then the
Figure 3. Characteristic roots of a strongly coupled system: w, roots of the uncoupled system; ×, roots of 2 coupled system; ——, tan b; –––, (Mg /M)b/(bM −b 2 ).
. . .
570
eigenvalue solution algorithm is verified by showing that the solutions converge to the exact values.
4.1. The exact expression of the volume flow transfer function of the system in Figure 1 is obtained in a closed form as [13]
TQ=
Q2 cosec (b) = 2 Q1 cot (b)+(M/Mg )((bM −b 2 )/b),
(24)
where bM , Mg are the same as in equation (10). The exact characteristic equation shown in equation (23) is obtained by making the denominator of the transfer function TQ equal to zero. Exact system natural frequencies are obtained by finding the roots of this non-linear characteristic equation. In section 2, the model expansion method using uncoupled modes resulted in equations (9) and (10). Combining equations (9) and (10) with Q=F=0, we obtain
0 10 M Mg
1 $
%
a 2 b 2−bM 1 1 j
= +2 s j . 2 2 b2 b2 (b −(np) ) n=1
(25)
The series on the right side of equation (25) has been shown to have the following equality in reference [14] (equation 1.421–3):
a 1 1 1 = 2+2 s 2 2. b tan (b) b b −(np) n=1
(26)
Hence, equation (25) becomes the same as equation (23). This shows that the characteristic equation obtained from the proposed method is exactly the same as that obtained from the analytic solution. Thus, it proves that the in vacuo natural modes of the structure and the rigid wall acoustic modes of the cavity can be used for the analysis of the acoustic–structural coupled system analysis.
4.2. Numerical verification is made by comparing the solutions from the proposed method with the exact solutions from equation (23). Similar comparisons are found in reference [9] when the finite element method is used. The system is chosen to have the same properties and dimensions as the system discussed in the preceding section for the weak coupling case. Further comparisons are made for strongly coupled cases in the subsequent section. In Table 1 the natural frequency parameters obtained by the proposed procedure are listed for three cases when 10, 20 and 30 terms are used in equation (20). For example,
,
571
T 1 Non-dimensional frequency parameter b=kL obtained by different approaches when the coupling effect is weak Coupled analysis ZXXXXXXXXXXXXCXXXXXXXXXXXXV Selected Exact 20 terms 10 terms mode Root number analysis solution solution solution
Root number 1, acoustic mode 2, acoustic mode 3, acoustic mode 4, acoustic mode 5, acoustic mode 6, acoustic mode 7, acoustic mode 8, acoustic mode Structural mode 9, acoustic mode
0·00000 3·14159 6·28319 9·42478 12·5664 15·7079 18·8496 21·9911 23·6337 25·1327
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode
0·00000 3·13873 6·27714 9·41476 12·55072 15·68288 18·80372 21·85677 23·56194 25·28774
0·00000 3·13873 6·27714 9·41477 12·55076 15·68299 18·80416 21·86084 23·67432 25·29449
0·00000 3·13873 6·27715 9·41478 12·55080 15·68315 18·80484 21·86797 23·79803 25·31597
Void Void Void Void Void Void 18·80266 21·85534 23·63788 25·29359
a 10-term solution means that nine acoustic modes and one structural mode are used in the solution procedure, i.e., n=9 in equations (20) and (21). The results are compared with the exact frequencies obtained from equation (23) in Table 1. Table 1 shows that the solution converges to the exact solution as more modes are used. As was discussed for a general combined system in references [12] and [15], the acoustic modes with the uncoupled frequencies higher than the structural mode will have their frequencies increased, and the others will have theirs decreased. The comparison shows that the method gives a good agreement with the exact solution even when only 10 modes are used. More numerical examples in the next section will show that the solutions for strongly coupled cases also converge to the exact solution as more terms are used. The last column of Table 1 shows the solution obtained by using only four acoustic modes around the structural frequency. The result seems to be reasonable. Therefore, only a few modes in the vicinity of the coupling mode of interest need be used for the analysis. This will be useful for the analysis of coupling modes in the high frequency range. 5. MORE NUMERICAL EXAMPLES
5.1. 1: Figure 4 shows the transfer functions TQ defined in equation (17) based on various approaches. The exact transfer function in equation (24) is compared with the transfer function obtained by solving the matrix equation (15) at each frequency. The transfer function obtained by the uncoupled approach is also included in the comparison for reference. In the uncoupled approach, it is considered that the structure is excited by the pressure of the cavity, assuming that all of its boundaries are rigid. The function is obtained as Mg 1 b . 2 M sin b bM −b 2
TQ=−
(27)
The figure shows that the method in this work gives good results, except that the 10-term solution fails around the 10th system frequency, which is expected because all 10th or higher modes are truncated in the solution. Also, it shows that the result from the
. . .
572
Figure 4. Transfer functions based on different approaches when the coupling effect is weak. –· –· –, uncoupled approach; –––, exact solution; · · · ·, 20-term solution; ——, 10-term solution.
uncoupled approach would be acceptable in the lower frequency range. Figure 6 shows the pressure mode shape of the ninth system mode which is predominantly a structural mode. Very little change is observed from the corresponding uncoupled acoustic mode.
5.2. 2: Table 2 lists the system natural frequencies for the case shown in Figure 3. Again, frequencies obtained by the proposd method using different numbers of terms are compared with the frequencies of the uncoupled subsystems. Table 2 and Figure 3 indicate that a strong coupling effect exists around the structural mode. The third system mode corresponds predominantly to the structural mode of the uncoupled system. The second, third and fourth system modes are quite different from any of the uncoupled sub-system modes. It is shown that the proposed method provides reasonable results when only 10 terms are used, even if the coupling effect is strong. Again, only four acoustic modes around the structural frequency are used for the selected mode solutions in the table. T 2 Non-dimensional frequency parameter b=kL obtained by different approaches when the coupling effect is strong Coupled analysis ZXXXXXXXXXXXXCXXXXXXXXXXXXV Selected Exact 20 terms 10 terms mode Root number analysis solution solution solution
Root number 1, acoustic mode 2, acoustic mode Structural mode 3, acoustic mode 4, acoustic mode 5, acoustic mode 6, acoustic mode 7, acoustic mode 8, acoustic mode 9, acoustic mode
0·00000 3·14159 4·98242 6·28319 9·42478 12·5664 15·7079 18·8496 21·9911 25·1327
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
system system system system system system system system system system
mode mode mode mode mode mode mode mode mode mode
0·00000 2·67439 4·71239 7·04184 9·85149 12·86232 15·93577 19·03537 22·14836 25·26914
0·00000 2·67904 4·84710 7·07303 9·86971 12·87463 15·94512 19·04298 22·15483 25·27481
0·00000 2·68447 4·88644 7·11368 9·89481 12·89263 15·95999 19·05668 22·16898 25·29294
0·00000 2·74835 5·12007 7·35099 10·09257 Void Void Void Void Void
,
573
Figure 5. Transfer functions based on different approaches when the coupling effect is strong. –· –· –, Uncoupled approach; –––, exact solution; · · · ·, 20-term solution; ——, 10-term solution.
Figure 5 shows the transfer functions TQ obtained by various approaches when the coupling effect is strong in the frequency range around the uncoupled structural mode. The figure shows that the combined system has substantially different characteristics compared to the characteristics of the sub-systems. Therefore, the transfer function obtained by the uncoupled approach would give completely wrong information in a wide frequency range. Finally, Figures 7 and 8 show the pressure mode shapes of the third and fourth system modes which have the strongest coupling effect. They are quite different from any of the uncoupled modes, which indicates that a complete coupled analysis has to be done to make a reasonably accurate prediction of the interior pressure field.
6. CONCLUSIONS
A procedure to formulate and analyze the free vibration problem of structural–acoustic coupled systems has been suggested and verified. The procedure is very simple and intuitive to implement. The related effort in the formulation and numerical computation to analyze the free-vibration problem of a structural–acoustic coupled system becomes roughly the same as the effort of solving a standard forced vibration problem of similar complexity. Further demonstrations of the computational advantage of the proposed method are made in the second part of this work, using two- and three-dimensional cases as examples.
Figure 6. Pressure profile of the ninth system mode of a weakly coupled system by different approaches: ——, uncoupled approach; · · · ·, exact mode shape of the coupled system; –––, mode shape obtained by using 19 acoustic modes; –· –· –, mode shape obtained by using 10 acoustic modes.
574
. . .
Figure 7. Pressure profile of the second system mode of a strongly coupled system by different approaches: ——, uncoupled approach; · · · ·, exact mode shape of the coupled system; –––, mode shape obtained by using 19 acoustic modes; –· –· –, mode shape obtained by using 10 acoustic modes.
Figure 8. Pressure profile of the third system mode of a strongly coupled system by different approaches: ——, uncoupled approach; · · · ·, exact mode shape of the coupled system; –––, mode shape obtained by using 19 acoustic modes; –· –· –, mode shape obtained by using 10 acoustic modes.
If the natural modes of the uncoupled sub-systems are known, the rest of the procedure of the proposed method becomes a purely computational task. Therefore, any analytical, numerical or experimental methods can be used to obtain the natural modes of the sub-systems to use in the proposed method. For example, one may use the boundary element method for the acoustic cavity and the finite element method for the structure to obtain the respective uncoupled natural modes.
ACKNOWLEDGMENT
The support for the first author from the University Research Council and the Structural Dynamics Research Laboratories at the University of Cincinnati is gratefully acknowledged.
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