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Analysis of structural-acoustic coupling problems by a two-level boundary integral equations method
Journal of Sound and Vibration (1995) 184(2), 213–228
ANALYSIS OF STRUCTURAL–ACOUSTIC COUPLING PROBLEMS BY A TWO-LEVEL BOUNDARY INTEGRAL EQUATIONS METHOD, PART 2: VIBRATIONS OF A CYLINDRICAL SHELL OF FINITE LENGTH IN AN ACOUSTIC MEDIUM S. V. S Department of Engineering Mechanics, Marine Technical University, Lotsmanskaya 3, St. Petersburg 190008, Russia (Received 6 April 1992, and in final form 15 March 1994) In Part 1 a method was proposed to evaluate the contact acoustic pressure and displacements of the surface of a composite thin-walled structure vibrating in an acoustic medium. Here the full formulation of a problem and its numerical aspect are examined in some detail. As an example, the method is applied to the analysis of vibrations of a reinforced cylindrical shell of finite length. 7 1995 Academic Press Limited
1. INTRODUCTION
The companion paper [1] has been devoted to a formulation of a new, two-level system, boundary integral equations method for computing the contact acoustic pressure on a surface of a thin-walled composite structure immersed in an acoustic medium. The displacements of the structure and the far field radiated sound pressure can then be readily deduced. Several simple test problems have been analyzed to validate the efficiency of the proposed method. This paper is concerned with the corresponding numerical technique with special reference to vibrations of a reinforced circular cylindrical shell with end caps. Attention is focused on the phenomenon of vibrations of the composite structure considered; the subsequent computation of the far field radiation pressure presents no difficulty. The analysis of the two-level system is performed based on the low frequency asymptotic representation of the kernels of the boundary integral equations. A classification of the parameters regime for structural–acoustic coupling is suggested. The notation used here, particularly for the geometry of the problem and the characteristics of the shell and the acoustic medium, is the same as in Part 1 [1]. 2. FORMULATION OF THE PROBLEM
The structure considered is a cylindrical shell with two end caps and three intermediate bulkheads. The longitudinal section of this composite structure is presented in Figure 1, with the enumeration of its elementary segments. The driving load arbitrarily distributed on the surface of the structure is assumed to be expanded into a series of trigonometric functions of the angular co-ordinate. Then, due to the axial symmetry of the structure, there is no coupling of the vibrational modes with different circumferential wavenumbers m and so each of them may be analyzed separately. 213 0022–460X/95/270213 + 16 $12.00/0
7 1995 Academic Press Limited
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214
Figure 1. The longitudinal section of a composite structure consisting of circular plates (1–5) and segments of cylindrical shell (6–9).
Vibrations of circular plates are considered in the framework of Kirchhoff theory [2]. The positive directions for displacements and forces are shown in Figure 2. The equations of the first level for a circular plate vibrating on the mth mode are presented in the companion paper [1]. There are two boundary integral equations for each plate. A Green function for vibrations in the mth mode is: W (p)(X, Y) =
1 [Y (sA) + (2/p)K0 (sA) − iJ0 (sA)], 8Ds 2 0
A = =X − Y=,
s 4 = rhv 2/D
(1)
(all the notations are defined in Appendix B of reference [1]). Vibrations of segments of cylindrical shells are considered in the framework of the Goldenvejzer–Novozhilov shell theory [3]. The positive directions for displacements and forces are shown in Figure 2. There are eight boundary equations of the first level for each segment of a cylindrical shell, which are also presented in reference [1]. A set of Green functions has the following form: 4
Wj1(c)(x, j) = s cjk ajk exp sk =x − j=, k=1 4
Wj2(c)(x, j) = s cjk bjk exp sk =x − j=, k=1 4
Wj3(c)(x, j) = s cjk exp sk =x − j=,
j = 1, 2, 3, 4.
(2)
k=1
Figure 2. Positive directions for generalized displacements (Figure 2(a)) and generalized forces (Figure 2(b)) of segments of the composite structure.
– , 2
215
There are 42 boundary integral equations of the first level for the whole composite structure shown in Figure 1. These equations contain the unknown contact acoustic pressure p and also 84 algebraic unknowns: displacements and forces at the interfacial points (points A, B, C, D and E of Figure 1). The conditions of continuity of the displacements and the equilibrium conditions should be formulated for each point of an intersection. They are presented here for the point B (see Figure 1). Since the Kirchhoff plate model is used, which postulates the absence of tangential deformations of the plate, the circumferential and radial displacements of cylindrical segments 6 and 7 are put to zero: w62 = w63 = w72 = w73 = 0.
(3a)
The first index is the number of the segment considered; the second one is the number of the component of the generalized displacements vector (see Figure 2(a)). For plates there is a sole component of this vector: wk3 , k = 1, 2, 3, 4, 5; w'k3 is its first derivative. The corresponding generalized forces are denoted as Qk1 and Qk2 respectively (see Figure 2(b)). For segments of the shell there are four components of the vector of generalized displacements: wkn , k = 6, 7, 8, 9 and n = 1, 2, 3, 4; wk4 = w'k3 . The corresponding generalized forces are denoted as Qkn . The remaining conditions ensure continuity of the axial displacements, w61 = w71 = w23 ,
(3b)
w64 = w74 = w'23 .
(3c)
and continuity of the angles, Equilibrium conditions for point B (they are similar for points C and D) have the following form (see Figure 2(b)): equilibrium of forces acting upon the edges of cylindrical segments 6 and 7 and plate 2, Q61 = Q71 + Q21 ;
(3d)
Q64 = Q74 + Q22 .
(3e)
equilibrium of bending moments, There are ten conditions (3) for each intermediate point B, C or D. The conditions for the end caps, say, for the point A, are w62 = w63 = 0,
w61 = w13 ,
Q61 = Q11 ,
w64 = w'13 ,
Q64 = Q12 .
(4a–c) (4d, e)
Thus one has 42 conditions of compatibility for the generalized displacements and generalized forces at the interfacial points for all the segments of the structure. The equation of the second level has the following form (lengths of cylindrical segments are denoted as l; the radius of the cylindrical shell is taken as unity: R = 1): 1 2
p(x, r) +
g g
1
F(x, 0, r, h)p(0, h)h dh +
0
+
1
g
4l
F(x, j, r, 1)p(j, 1) dj
0
F(x, 4l, r, h)p(4l, h)h dh − rv 2
0
g
1
G(x, 0, r, h){r1 [Q11 (r1 )W (p)(h, r1 )
0
+ Q20(h, r1 )w'13 (r1 ) − Q12 (r1 )1W (p)(h, r1 )/1r1 − Q10(h, r1 )w13 (r1 )]=r 1 = 1 +
g
1
0
9
p(0, r1 )W (p)(h, r1 )r1 dr1 }h dh − rv 2 s k=6
g
(lk )
G(x, j, r, 1)
. .
216 ×
6g
(c) (c) W33 (j, x1 )p(x1 , 1) dx1 + [Qk1 (j)W31 (x1 , j)
(lk )
(c) (c) (c) (j, x1 ) + Qk3 (j)W33 (j, x1 ) − Qk4 (j)W34 (j, x1 ) + Qk2 (j)W32 0 0 0 (j, x1 ) − wk2 (j)Q32 (j, x1 ) − wk3 (j)Q33 (j, x1 ) − wk1 (j)Q31 0 (k − 5)l 2 (j, x1 )]=xx11 = + wk4 (j)Q34 = (k − 6)l } dj − rv
g
1
G(x, 4l, r, h){r1 [Q51 (r1 )
0
× W (p)(h, r1 ) + Q20(h, r1 )w'53 (r1 ) − Q52 (r1 )1W (p)(h, r1 )/1r1 − Q10(h, r1 )w53 (r1 )]=r 1 = 1 +
g
1
p(4l, r1 )W (p)(h, r1 )r1 dr1 }h dh
0
= rv 2
g
1
G(x, 0, r, h)
0
q(0, r1 )W (p)(h, r1 )h dh r1 dr1 + rv 2
0
g 6 g 1
×
g
1
9
k=6
0
7
3
j=1
G(x, 4l, r, h)
0
q(4l, r1 )W (p)(h, r1 )h dh r1 dr1 + rv 2 s
× s
g
1
g
G(x, j, r, 1)
(lk )
W3j(c) (j, x1 )qj (x1 , 1) dx1 dj.
(lk )
(5)
Here G (the index m is omitted) is the Green function corresponding to a ring source (see reference [1]): Gm (=X − Y=) =
R 4p
g
2p
0
exp(ivA/c) cos mu du; A
A = =X − Y= = z(x − j)2 + R 2 + r 2 − 2Rr cos u .
(6)
F is a derivative of G: F = 1G/1n+ (n+ is distance along the outward unit normal to the fluid domain at the surface of the structure). All the rest of the notation is the same as in reference [1]. A piecewise constant approximation of the acoustic pressure along the surface of the structure is introduced. Then the standard collocation procedure is used to solve the integral equation (5) (the points of collocations are at the centers of the elements) and the two-level system of boundary equations is transformed to a system of linear algebraic equations (SLAE): [B]{Y} = {Q}.
(7)
Figure 3. Vibrational modes of the lateral surface of the structure shown in Figure 1. Driving line load P0 cos 2u is applied at the middle of segment 7. Curve 1 represents vibrations in vacuo, V 2 = 0·4; curve 2 represents vibrations in the fluid at the same frequency; curve 3 represents the static deflection (V 2 = 0).
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217
Figure 4. Dependence of attenuation V (dB) of the composite structure upon the frequency f (Hz) of excitation.
The vector {Y} consists of two parts: amplitudes of structural displacements and forces at the interfacial points and amplitudes of contact acoustic pressures at the collocation points. Correspondingly the matrix [B] is decomposed into four blocks Bnk , n, k = 1, 2. The structural–acoustic coupling is governed by all the matrix, but primarily by the block B22 . The latter is relevant to the second level equation (5). Its elements are obtained by integration over elementary parts Dj of the lateral surface of the structure: (nj) B22 =
g$ Dj
F(Xn , Y) +
g
%
G(Xn , Y)W(Y, Z) dSz dSy .
(l)
(8)
Here F and G are defined by equation (6); W is the Green function (1) or (2). The order of this SLAE is comparatively small: if one introduces 25 elements for the surfaces of each end cap and 50 elements for the surfaces of each cylindrical segment, then one has 250 unknown acoustic pressure amplitudes. There are also 84 unknown amplitudes of generalized displacements and generalized forces at the interfacial points. Then there are 334 algebraic unknowns in the two-level system. It should be emphasized that, since the kernels of the integral equations have closed forms (1), (2) and (6), it is possible to evaluate the elements of the SLAE analytically in some ranges of the parameters. If the vibrations of the structure shown in Figure 1 are considered in vacuo, then the boundary integral equations method provides one with the exact solution of the problem. 3. NUMERICAL ANALYSIS
Numerical analysis has been performed for reinforced shells of two types. A longitudinal section of the shell of the first type is shown in Figure 1. The lengths of all the segments are equal to the radius of the shell, which is taken as unity, and the thicknesses of all the segments are the same: hc = hp = 0·1. In Figure 3 curve 1 represents the amplitudes of vibrations excited in vacuo by a driving load P0 cos 2u, which is concentrated in the radial direction at the middle of segment 7 (see Figure 1), while curve 2 represents the amplitudes of vibrations in water (r/r0 = 0·128, c/c0 = 0·307; r and r0 are the densities and c and c0 are the speeds of sound in water and in the shell material, respectively). The dimensionless frequency of excitation is V 2 = 0·4 (V 2 = (1 − n 2)v 2R 2/c02). The dimensionless amplitude of vibrations is w* = ERw/P0 . The presence of the acoustic medium in this particular case increases the amplitudes of the displacements by 15–20%. To explain this effect one should note that the first resonance frequency of the ‘‘dry’’ structure given by V02 = 0·765 (this eigenfrequency corresponds to a predominantly radial mode of vibration of the shell). The added masses of water reduce this value and the frequency of the driving load becomes closer to the
. .
218
resonance one. Curve 3 in Figure 3 represents the static displacements caused by the same force when V 2 = 0. The shell of the second type is also reinforced by the bulkheads, as shown in Figure 1. Besides that, the shell is additionally reinforced by T-shape ring stiffeners; of which there are 17, uniformly spaced on each cylindrical segment of the shell. Parameters of the shell are as follows: the material is steel (E = 2 × 105 MPa, n = 0·3); the thickness is h = 0·014 mm, the radius is R = 1·5 m; the length of each segment is l = 4 m. Low frequency vibrations of the composite structure are examined so that a structural–orthotropic shell model [4] is used. The geometrical parameters of the stiffeners are as follows: the effective cross-sectional area is A = 980 mm2; the effective moment of inertia is I = 1600 mm4. The driving load is a point force applied at the middle of segment 7. Vibrations in the first eight circumferential modes (circumferential wavenumbers from 0 to 7) were determined and the velocity at the point of excitation was evaluated as a sum of the components obtained for each mode. The damping coefficient was taken as h = 0·05. The imaginary parts of the fundamental solutions (1) and (2) are retained (see reference [1]). The dependence of the attenuation V (dB) of the system calculated via the amplitudes of vibration at the point of excitation at the frequency f (Hz) is represented by the solid line in Figure 4. The dashed line represents the experimental data reported in references [5, 6]. The agreement between theoretical and experimental results appears to be good. 4. LOW FREQUENCY ASYMPTOTIC EXPANSIONS FOR THE FUNDAMENTAL SOLUTIONS
The numerical analysis described in section 3 was based on the use of fundamental solutions given by formulae (1), (2) and (6). The low frequency range of excitation is of special interest in many technical applications for reinforced cylindrical shells. In this range some important simplifications may be made by the use of asymptotic expansions of the Green functions. Then the inner integral in equation (8) (convolution of the Green function) may be solved analytically. For all the frequency ranges the fundamental solutions for the problem of the vibrations of the cylindrical shell are exponentials in the axial co-ordinate (see equations (2)) and thus these functions do not need any transformations. In the low frequency range power series expansions of the Bessel functions may be substituted in the fundamental solution (1) for the vibrations of a circular plate: W (p)(r, R) =
g
1 2pD
2p
cos mu{14 (r 2 + R 2 − 2rR cos u)[1 − g
0
− ln(s/2) − 12 ln (r 2 + R 2 − 2rR cos u)] +
s4 2 (r + R 2 64
− 2rR cos u)3[1 + 12 + 13 − g − ln (s/2) − 121 ln (r 2 + R 2 − 2rR cos u) + · · · } du,
g = 0·5772.
(9)
The fundamental solution (5) may also be simplified by the use of a power series expansion: exp(ivA/c) = 1 + (ivA/c) + (1/2!)(ivA/c)2 + (1/3!)(ivA/c)3 + · · · . Then, instead of equation (6) one obtains: G(=X − Y=) =
R0 2p
g$ p
0
01
01
1 iv 1 v i v + − A− A2 + · · · A c 2! c 3! c 2
3
%
cos mu du.
(10)
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219
The subscript m is omitted. The function (10) may be expressed as the following sum: G(=X − Y=) = G1 (=X − Y=) + G2 (=X − Y=) + G3 (=X − Y=). Then it is possible to compare the contributions of the components of the fundamental solution G(=X − Y=) corresponding to an incompressible fluid with the ones corresponding to the real and the imaginary parts of the acoustical correction to its value in various ranges of the parameters. These components are: G1 (x, j, r, R0 ) = G2 (x, j, r, R0 ) =
R0 2p
R0 2p
g
p
0
cos mu du, A
g$ 01 01 g$ 01 p
(11a)
1 v 1 v A+ A3 − · · · 2! c 4! c 2
4
−
0
G3 (x, j, r, R0 ) =
p
iR0 2p
v 1 v − A2 + · · · c 3! c 3
0
%
%
cos mu du,
cos mu du.
(11b) (11c)
The function G1 is that for an incompressible fluid (evidently, this is the Green function for Laplace’s equation), the function G2 is a correction due to acoustically added mass, and the function G3 is a correction due to energy radiation effects. Each term in the integral (11c),
g
p
[(x − j)2 + R02 + r 2 + 2R0 r cos u]2k cos mu du,
k = 1, 2, 3, . . . ,
0
may be easily solved analytically. Each term in the series (11b) is integrated by parts:
g
p
A 2k − 1 cos mu du =
0
2k − 1 m
g
$g
p
A 2k − 3 cos (m − 1)u du
0
%
p
−
A 2k − 3 cos (m + 1)u du R0 r,
0
k = 2, 3, 4, . . . .
(12)
Then, for k E m, all the terms in equation (11b) may be expressed via the fundamental solutions (11a) corresponded to various wavenumbers m. After a function T(m, M) is introduced as T(m, M) =
g
p/2
0
cos 2mu du z1 − M cos u
,
M=
2
4rR0 , (x − j)2 + (r + R0 )2
(13)
formula (11a) may be rewritten as G1 (x, j, r, R0 ) =
R0 pzx − j)2 + (r + R0 )2
T(m, M).
(14)
The following cases can then be distinguished. (i) A fundamental solution corresponding to axisymmetrical pulsations of the structure (m = 0). Then formula (14) has the form G1 (x, j, r, R0 ) =
R0 pz(x − j)2 + (r + R0 )2
(K(M) is the elliptic integral of the first kind).
K(M)
(15)
. .
220
Figure 5. Dependence of the potential G1 produced by a ring source in an incompressible fluid on the distance z. ——, exact formula (11(a)); ––––, approximate formula (19). m is the number of circumferential waves.
(ii) A fundamental solution corresponding to vibrations of a thin-walled structure without distortion of its cross-section (m = 1). Then formula (14) has the form G1 (x, j, r, R0 ) =
R0 pz(x − j) + (r + R0 ) 2
2
$
2−M 2 K(M) − E(M) M M
%
(16)
(E(M) is the elliptic integral of the second kind). (iii) Fundamental solutions corresponding to shell modes of vibrations of the structure (m e 2). Then, based on the identity T(m, M) = T(m − 1, M) − 2
g
p/2
sin u sin (2m − 1)u
du,
z1 − M cos2 u
0
(17)
the following formula is obtained: G1 (x, j, r, R0 ) =
$
R0 2−M K(M) M pz(x − j)2 + (r + R0 )2 m 2 E(M) − 2 s M k=2
−
g
p/2
%
sin u sin (2k − 1)u
du .
z1 − M cos2 u
0
(18)
The sum in equation (18) is regular for all values of M: when M = 0, it is equal to zero, and when M = 1, it is equal to smk= 1 (2k − 1)−1. After some quite cumbersome transformation one obtains G1 (x, j, r, R0 ) =
R0 pz(x − j)2 + (r + R0 )2
&
8
m
K(M) − 2 s (2k − 1) k=1
k 1 1−M (−1)p p E(M) − K(M) + s t [(2k − 1)2 × M M (2p + 1)! q = 1 p=1
− (2q − 1)2]
&
2
pG(2p + 2)
60
2p + 2
2p + 3 G 2
17
Here M1 = 1 − M and G( ) is the Euler G-function.
− 2
G(p)G(3/2)
0
2p + 3 2G 2
1
M1
''9
.
(19)
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221
Figure 6. The length Zm of an ‘‘influence zone’’ versus the number of circumferential waves m.
In Figure 5 the values of the functions G1 (x, j, r, R0 ) are presented versus the distance z = =x − j= when r = R0 = 1. Continuous curves correspond to the exact formula (11a) and dashed ones to the approximate formula (19). The latter gives negative values for the pressure at a certain distance from the ring; these values should be replaced by zero. It is convenient to introduce a so-called ‘‘influence zone’’ for analysis of the acoustic pressure generated by a ring source. For each m the length of this zone is equal to a maximum distance zm , which gives positive values for the formula (19). In Figure 6 the sizes of the influence zone zm versus the number of circumferential waves m are presented. It is clearly seen that the rate of decay of acoustic pressure sharply increases with an increase in the number m. This effect is explained by the fact that large values of m correspond to an alternating sequence of sources and sinks on the ring. The motions of the particles of the acoustic medium in this case are localized in the neighborhood of the ring. If the cylindrical surface of the radius R0 is considered, then only in cases of axisymmetric and beam-type excitations is there significant pressure at a distance larger than R0 . This effect remains the same if the function G2 (x, j, r, R0 ) is analyzed, which is in fact the difference between the functions G1 (x, j, r, R0 ) with different wavenumbers m (see equation (12)). The function F is the first derivative of G with respect to the co-ordinate which is normal to the lateral surface of the shell. In particular, for a cylindrical shell, F(x, j, r, R0 ) =
R0 4p
g
$
2p
(1 − cos u) 1 −
0
%
ivA exp(ivA/c) cos mu du. A3 c
(20)
It is useful to represent this function similarly to G: F(X, Y) = F1 (X, Y) + F2 (X, Y) + F3 (X, Y). Here one has F1 (x, j, r, R0 ) = F2 (x, j, r, R0 ) = F3 (x, j, r, R0 ) =
R0 4p
R0 4p
g
2p
(1 − cos u)
0
g $0 1 0 1 g $0 1 0 1
iR0 4p
2p
2p
0
% %
(21a)
1 1 1 1 + − A + · · · (1 − cos u) cos mu du, 2! A 3! 4!
(21b)
1 1 1 1 2 − − + A + · · · (1 − cos u) cos mu du. 2! 3! 4! 5!
(21c)
1−
0
cos mu du, A3
The functions Fk (x, j, r, R0 ), k = 1, 2, are expressed via the functions T(m, M) similarly to the functions Gk (x, j, r, R0 ), k = 1, 2. The first term in equation (21a) contains a singular part which has already been taken into account in equation (5) by the non-integral term (see reference [7]). The function (21c) is evaluated analytically.
. .
222
5. CLASSIFICATION OF TYPES OF STRUCTURAL–ACOUSTIC INTERACTION
Elements of the matrix B22 relevant to the boundary equation of the second level (see equation (8)) may be represented in the following form: (nj) = B22
g
[F1 (Xn , Y) + F2 (Xn , Y) + F3 (Xn , Y)] dSy +
Dj
gg Dj
W(Y, Z)[G1 (Xn , Y)
(l)
+ G2 (Xn , Y) + G3 (Xn , Y)] dSz dSy .
(22)
The first three terms on the right side of equation (22) may be evaluated by using formulae (21) for Fk , k = 1, 2, 3. The integration is performed analytically for each combination of positions of the observation point and the source point. All the double integrals in equation (22) may be transformed in the same manner: for the end caps,
gg Dj
1
W (p)(h, r1 )[G1 (x, nl, r, h) + G2 (x, nl, r, h) + G3 (x, nl, r, h)]r1 dr1 h dh
0
g
=
1
[G1 (x, nl, r, h) + G2 (x, nl, r, h) + G3 (x, nl, r, h)]
0
×
6g
7
W (p)(h, r1 )r1 dr1 h dh;
Dj
(23)
and for the cylindrical surface,
gg Dj
(c) W33 (j, x1 )[G1 (x, j, r, 1) + G2 (x, j, r, 1) + G3 (x, j, r, 1)] dj dx1
(l)
=
g
[G1 (x, j, r, 1) + G2 (x, j, r, 1) + G3 (x, j, r, 1)]
(l)
×
6g
Dj
7
(c) W33 (j, x1 ) dx1 dj.
(24)
The use of expansion (9) for the fundamental solution W (p)(h, r) permits one to evaluate the integral in square brackets of equation (23) analytically. Then the outer integral in equation (23) is computed by using the fundamental solution G(x, j, r, h) in the form (11). The integral (24) may be computed similarly because the fundamental solution has the simple exponential form (2). For the lateral surface of the cylindrical shell the convolution of the Green functions for the acoustic medium and the cylindrical shell may also be computed in another way. It has been shown in section 4 that the pressure field generated by a ring source is localized in the influence zone (see Figure 2). In this zone, for m e 2, the axial co-ordinates of an observation point and a source point satisfy the inequality =x − j= Q R = 1. This permits one to represent the argument of the elliptic integrals in the form M = (1 + (x − j)2/4)−1 = 1 − (x − j)2/4. Then the functions (11a) and (11b) may be rewritten as Gb (x, j, 1, 1) = d1b + d2b ln =x − j= + d3b (x − j)2 + d4b (x − j)2 ln =x − j=.
(25)
The coefficients dab , a = 1, 2, 3, 4, b = 1, 2, are quite cumbersome and they are not presented here. In fact, for m e 2 the integration in equation (24) should be performed only within the influence zone.
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223
The evaluation of the convolutions (24) for axisymmetric and beam-type modes of vibrations is more complicated. In these cases this procedure depends upon the values of the roots of the characteristic polynomials sk , k = 1, 2, 3, 4 (see equation (2)). Those roots which are large correspond to the boundary effect in the shell theory (see reference [3]). Then the integral (24) may be evaluated in asymptotic series of negative powers of sk . The integration of exponents with small roots should be performed numerically. A number of terms in equation (22) may be disregarded under certain conditions. The simplifications introduced by this disregard correspond to neglect of some features of the interaction. The basic features of the interaction are as follows: (i) structural–acoustic coupling, defined by the integrals
gg Dj
W(Y, Z)[G1 (Xn , Y) + G2 (Xn , Y) + G3 (Xn , Y)] dSz dSy
(l)
(the first term corresponds to an incompressible fluid model and the remaining two terms an acoustic medium model); (ii) compressibility of the medium, defined by the integrals
g
[F2 (Xn , Y) + F3 (Xn , Y)] dSy +
Dj
gg Dj
W(Y, Z)[G2 (Xn , Y) + G3 (Xn , Y)] dSz dSy ;
(l)
(iii) curvature of the structure, defined by the integrals
g
[F1 (Xn , Y) + F2 (Xn , Y) + F3 (Xn , Y)] dSy ,
Dj
(the first term corresponds to an incompressible fluid model, and the remaining two terms correspond to an acoustic medium model). These features are governed by the parameters of rigidity and geometry of a structure and also by the frequency of excitation. One may say that they influence the left side of equation (5), while the effect of the configuration of the driving load influences the right side of equation (5). For each specific configuration of the driving load the analysis of the interaction should be continued, based on the analysis already carried out for the homogeneous form of equation (5). The following simplified models for equation (5) can be considered: (1) incompressible medium, curvature of the structure and structural–acoustic coupling disregarded; (2) incompressible medium, curvature of the structure taken into account, no structural–acoustic coupling; (3) acoustic medium, curvature of the structure taken into account, no structural–acoustic coupling; (4) structural–acoustic coupling, incompressible medium, curvature of the structure disregarded; (5) structural–acoustic coupling, acoustic medium, curvature disregarded. An estimation of the validity of these models may be carried out in several ways. In particular, the comparison of values for energy radiated to the far field, as obtained from the exact formulation of the problem and from the simplified one should be performed. Then it is necessary to solve the two-level system and to calculate the amplitudes of vibrations of the structure. A solution for the two-level system is obtained numerically. Then all the further steps in the evaluation of the energy radiated from the structure must also be performed numerically. It would be more convenient to estimate the validity of models (1)–(5) by comparison (nj) of the matrix components B22 (see equation (22)). In this case model (1) corresponds to a pure diagonal matrix B22 with all the diagonal elements equal to 1/2. Each feature of the interaction gives a certain contribution to the matrix and transforms it from the pure diagonal form.
. .
224
T 1 Elements of the matrix B22 for n = 1, m = 10 and (a) vR/c = 0·001 and (b) vR/c = 0·1 (a) bHE × 105
k 1 4 7 10 13 16 19 22 25
ZXXXXXCXXXXXV
h/R = 0·001
h/R = 0·01
−0·121 −0·090 −0·065 −0·038 −0·018 −0·007 0·0 0·005 0·004
0·170 0·140 0·092 −0·076 −0·052 −0·034 −0·024 −0·021 −0·003
(b) bHE
k 1 4 7 10 13 16 19 22 25
ZXXXXXCXXXXXV
h/R = 0·001
h/R = 0·01
−0·633 −0·471 −0·331 −0·231 −0·160 −0·083 0·0 0·054 0·041
−0·182 −0·151 −0·124 −0·100 −0·071 −0·050 −0·030 −0·010 −0·008
The matrix components (22) relevant to structural–acoustic coupling can be denoted by BHE , the components relevant to compressibility of the medium by BAC and the components relevant to effects induced by the curvature of the structure by BHD . A comparison of the values of the components of the matrix corresponding to these three basic features of the interaction is performed in section 6. 6. ANALYSIS OF THE PARAMETER REGIMES OF STRUCTURAL–ACOUSTIC COUPLING FOR VIBRATION OF CYLINDRICAL SHELL WITH END CAPS
The vibrations of the steel structure shown in Figure 1 in water are considered: c/c0 = 0·307 and r/r0 = 0·128. The vibrational modes with m circumferential waves are analyzed separately. Let an observation point be placed at the end cap. The integral containing the normal derivative F over the end cap is equal to zero (the singular part of this integral produces the non-integral term 1/2). Piecewise constant approximation of the contact acoustic pressure is introduced with NE = 25 (NE is the number of elements). Since the curvature of the contact surface is equal to zero, the comparison has been performed for the simplified models (1) and (4) of section 5. The values of the real parts of the elements of the matrix B22 are presented in Tables 1(a) and 1(b). These elements are denoted as BHE . A point of observation is set on the edge
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225
T 2 The natural frequencies of an isolated segment of a ‘‘dry’’ cylindrical shell h/R
m 1 2 3 4 5 6 7 8
ZXXXXXXXXCXXXXXXXXV
0·001
0·003
0·01
0·03
0·10
1·99 0·652 0·481 0·354 0·265 0·203 0·159 0·128
1·99 0·652 0·481 0·354 0·266 0·206 0·166 0·141
1·99 0·653 0·483 0·361 0·281 0·239 0·228 0·244
1·99 0·661 0·503 0·410 0·389 0·432 0·519 0·637
1·99 0·743 0·687 0·775 0·985 4·04 4·57 5·11
of the circular plate so that the first index n is equal to 1. The second index characterizes the position of a source. It is presented in the first row of each table. Table 1(a) corresponds to a circumferential wavenumber m = 10 and frequency parameter vR/c = 0·001. The second and the third rows present the elements BHE of B22 for h/R = 0·001 and h/R = 0·01, respectively. Table 1(b) corresponds to a circumferential wavenumber m = 10 and frequency parameter vR/c = 0·1. The second and third rows present elements BHE of B22 for h/R = 0·001 and h/R = 0·01, respectively. In both the cases considered the imaginary parts of the elements BHE are much smaller than the real ones and there is practically no radiation to the far field from the structure in this vibrational mode. It is clearly seen that in a case of low frequency excitation the influence of coupling effects upon the contact acoustic pressure may be neglected regardless of the position of the source and the simple model (1) is valid. When the driving load is at a comparatively high frequency (vR/c = 0·1), the influence of coupling is small near the center of the plate while in the neighborhood of the edge it becomes essential and model (4) is preferable. Let a point of observation be placed on the surface of a cylindrical segment. The natural frequencies of an isolated segment of a ‘‘dry’’ structure shown in Figure 1 are presented in Table 2. In Tables 3–6 the real and imaginary parts of the values for the elements of the SLAE BHD , BAC and BHE are presented. The number of boundary elements introduced is 50. The point of observation is set at the middle of a cylindrical segment and the first index is equal T 3 Real parts of elements of matrix B22 for n = 25, vR/c = 0·1, m = 8 k
BHD × 102
BAC × 106
BHE × 106 (h/R = 0·1)
BHE × 104 (h/R = 0·03)
BHE × 103 (h/R = 0·01)
BHE × 10 (h/R = 0·003)
BHE × 10 (h/R = 0·001)
25 22 19 16 13 10 7 4 1
0·633 −0·211 −0·231 −0·095 0·051 0·134 0·129 0·064 0·001
−0·260 −0·246 −0·209 −0·158 −0·104 −0·057 −0·026 −0·006 0·0
−0·503 −0·392 −0·184 −0·091 −0·042 −0·019 −0·008 −0·003 −0·001
−0·178 −0·147 −0·075 −0·041 −0·021 −0·010 −0·004 −0·001 0·0
−0·377 −0·354 −0·232 −0·167 −0·113 −0·072 −0·042 −0·006 0·0
−0·100 −0·094 −0·088 −0·079 −0·070 −0·060 −0·050 −0·042 −0·033
0·542 0·518 0·593 0·627 0·662 0·695 0·723 0·744 0·755
. .
226
T 4 Real parts of elements of a matrix B22 for n = 25, vR/c = 1, m = 8 k
BHD × 102
BAC × 104
BHE × 103 (h/R = 0·1)
25 22 19 16 13 10 7 4 1
0·633 −0·211 −0·231 −0·095 0·051 0·134 0·129 0·064 0·001
−0·266 −0·252 −0·214 −0·162 −0·106 −0·059 −0·027 −0·010 −0·003
−0·155 −0·117 −0·057 −0·028 −0·013 −0·006 −0·002 −0·001 0·0
BHE × 102 BHE × 10 BHE (h/R = 0·03) (h/R = 0·001) (h/R = 0·003) −0·685 −0·560 −0·331 −0·202 −0·118 −0·065 −0·034 −0·016 −0·012
0·297 0·454 0·839 0·929 0·829 0·552 0·155 −0·270 −0·624
0·063 0·143 0·261 0·241 0·133 −0·021 −0·163 −0·235 −0·209
BHE (h/R = 0·001) 0·166 −0·080 0·786 0·707 0·360 −0·120 −0·535 −0·711 −0·578
to 25. The second index characterizes the source position. Only values of the elements of a matrix with k = 1, . . . , 25 are included in the tables since values of elements with k = 26, . . . , 50 are the same. The values for BHD depend only upon the circumferential wave number m. The values for BAC depend both upon m and the dimensionless frequency of excitation vR/c. The values for BHE depend upon m, vR/c and also upon the thickness of the shell h/R. The analysis has been performed for the following values of thickness: 0·001; 0·003; 0·01; 0·03; 0·1. T 5 Elements of matrix B22 for h = 25, vR/c = 1, m = 3; (a) real parts; (b) imaginary parts (a)
k
BHD × 102
BAC × 103
25 22 19 16 13 10 7 4 1
0·957 0·063 −0·079 −0·092 −0·060 −0·024 −0·006 −0·009 −0·027
−0·249 −0·247 −0·242 −0·232 −0·217 −0·197 −0·174 −0·149 −0·127
BHE × 10 ZXXXXCXXXXV h/R = 0·03 h/R = 0·01 0·057 0·051 0·080 0·162 0·233 0·252 0·258 0·251 0·230
0·149 0·060 0·365 0·448 0·709 0·762 0·774 0·742 0·667
BHE ZXXXXXCXXXXXV h/R = 0·003 h/R = 0·001 0·049 0·101 0·119 0·102 0·241 0·259 0·263 0·253 0·225
0·179 −0·074 0·702 0·359 0·783 0·836 0·843 0·802 0·715
(b) 5
k
BAC × 104
BHE × 10 (h/R = 0·03)
BHE × 104 (h/R = 0·01)
BHE × 104 (h/R = 0·003)
BHE × 103 (h/R = 0·001)
25 22 19 16 13 10 7 4 1
−0·115 −0·115 −0·115 −0·114 −0·114 −0·113 −0·113 −0·112 −0·111
−0·865 −0·745 −0·664 −0·624 −0·614 −0·631 −0·677 −0·759 −0·877
−0·259 −0·214 −0·195 −0·186 −0·183 −0·188 −0·199 −0·219 −0·263
−0·879 −0·720 −0·666 −0·631 −0·621 −0·638 −0·680 −0·737 −0·892
−0·288 −0·241 −0·224 −0·213 −0·211 −0·216 −0·228 −0·247 −0·292
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227
T 6 As Table 5, but with m = 1 (a) k
BHD × 10
BAC × 102
25 22 19 16 13 10 7 4 1
0·131 0·039 0·018 0·007 0·002 −0·001 −0·003 −0·004 −0·006
−0·220 −0·219 −0·214 −0·207 −0·198 −0·187 −0·175 −0·161 −0·147
BHE × 10 (h/R = 0·01) 0·019 −0·008 0·114 −0·003 0·108 0·192 0·170 0·416 0·477
BHE (h/R = 0·001) 0·017 −0·252 0·393 −0·134 0·065 −0·037 0·606 −0·014 0·501
(b) k
BAC × 102
25 22 19 16 13 10 7 4 1
−0·278 −0·278 −0·277 −0·275 −0·273 −0·271 −0·268 −0·264 −0·260
BHE × 102 (h/R = 0·01) −0·160 0·059 0·085 0·097 0·096 0·983 0·056 0·017 −0·191
BHE × 10 (h/R = 0·001) −0·163 0·058 0·093 0·097 0·096 0·083 0·063 0·015 −0·192
The low frequency vibrations with the large number of circumferential waves (vR/c = 0·1; m = 8) have been considered first (see Table 3). In this case the influence of the curvature of the structure is negligibly small (BHD 0 10−3) and the influence of the compressibility of the medium is also small (BAC 0 10−7). The imaginary parts of BAC and BHE are of the same order (BAC 0 BHE 0 10−16) and also should be ignored. There is no radiation from a structure in this vibrational mode at this frequency. In this case there is the alternative to use either the simple model (1) or the model (4) when structural–acoustic coupling effects cannot be disregarded. For a thick shell (h/R = 0·1) structural–acoustic coupling is insignificant (BHE 0 10−7), but for a thin shell (h/R = 0·001) these effects become significant. Generally, in this case model (1) is acceptable if h/R q 0·01 and model 4 should be used for h/R Q 0·01. In Table 4 similar data are presented for vR/c = 1 and m = 8. The values for BHD remain the same as in the previous case and the effect of curvature should be ignored. Despite a certain increase in the values for BAC they are still small and therefore should be neglected. Structural–acoustic effects are significant in most of the range of thicknesses considered (0·001 Q h/R Q 0·03). Moreover, when the thickness is small (h/R = 0·001) the values of BHE are greater than 1/2 (the diagonal element of the matrix relevant to model (1)). Then model (4) should be used. It should also be noted that for a very thick shell (h/R = 0·1) contributions of each feature of the interaction are equivalent. The imaginary parts of BAC and BHE are of the same order (BAC 0 BHE 0 10−12) and should be omitted. In Table 5(a) similar data are presented for vR/c = 1, m = 3. When the thickness is large (h/R = 0·1) it is still possible to use the simple model (1). While for thin shells
228
. .
(h/R Q 0.01) model (4) is the most appropriate one. It should be noted that the case of moderate thickness (hR = 0·03) appears to be the most complicated one, when no simplified model is acceptable. Imaginary parts of BAC and BHE are of the same order and both of them should be taken into account. These imaginary parts are presented in Table 5(a). In Table 6 similar data are presented for the beam-type mode vR/C = 1, m = 1. When the thickness of the shell is small, only structural–acoustic coupling effects should be taken into consideration; for moderate thicknesses both the acoustic and structural–acoustic effects are essential (model (4) should be used). The imaginary parts are presented in Table 6(a). 7. CONCLUSIONS
The vibrations of a reinforced cylindrical shell of finite length have been analyzed by the two-level boundary integral equations method. Asymptotic formulae for the Green functions for vibrations of circular plates in an acoustic medium have been obtained for a low frequency range of excitation. Numerical analysis of vibrations of reinforced shells of two types has been performed along with analysis of the system of boundary equations. Based on these analyses several simplified models of structural–acoustic interaction have been suggested. The ranges of validity for these models have been estimated. REFERENCES 1. L. I. S and S. V. S 1995 Journal of Sound and Vibration 184, 195–211. Analysis of structural–acoustic coupling problems by a two-level boundary integral equation method, part 1: general formulation and test problems. 2. S. P. T and S. W-K 1959 Theory of Plates and Shells. New York: McGraw-Hill. 3. A. L. G, V. B. L and P. E. T 1979 Free Vibrations of Thin Elastic Shells. Moscow: Nauka (in Russian). 4. E. A. S, I. L. E and A. V. V 1988 Nekotorye Problemy Mekhaniki Sudovikh Konstruktsiy, 62–67. A simple estimation of the first natural frequencies of orthotropic cylindrical shells (in Russian). 5. I. K. P and S. A. R 1983 On Impedance of Cylindrical Shell Reinforced with Ring Stiffeners. DR 1905, Ts.N.I.I. Rumb (in Russian). 6. I. K. P and S. A. R 1984 Discretized Model to Evaluate Mechanical Impedance of Stiffened Cylindrical Shell. DR 1905, Ts.N.I.I. Rumb (in Russian). 7. P. K. B and R. B Boundary Element Methods in Engineering Science. New York: McGraw-Hill.
ANALYSIS OF STRUCTURAL–ACOUSTIC COUPLING PROBLEMS BY A TWO-LEVEL BOUNDARY INTEGRAL EQUATIONS METHOD, PART 2: VIBRATIONS OF A CYLINDRICAL SHELL OF FINITE LENGTH IN AN ACOUSTIC MEDIUM S. V. S Department of Engineering Mechanics, Marine Technical University, Lotsmanskaya 3, St. Petersburg 190008, Russia (Received 6 April 1992, and in final form 15 March 1994) In Part 1 a method was proposed to evaluate the contact acoustic pressure and displacements of the surface of a composite thin-walled structure vibrating in an acoustic medium. Here the full formulation of a problem and its numerical aspect are examined in some detail. As an example, the method is applied to the analysis of vibrations of a reinforced cylindrical shell of finite length. 7 1995 Academic Press Limited
1. INTRODUCTION
The companion paper [1] has been devoted to a formulation of a new, two-level system, boundary integral equations method for computing the contact acoustic pressure on a surface of a thin-walled composite structure immersed in an acoustic medium. The displacements of the structure and the far field radiated sound pressure can then be readily deduced. Several simple test problems have been analyzed to validate the efficiency of the proposed method. This paper is concerned with the corresponding numerical technique with special reference to vibrations of a reinforced circular cylindrical shell with end caps. Attention is focused on the phenomenon of vibrations of the composite structure considered; the subsequent computation of the far field radiation pressure presents no difficulty. The analysis of the two-level system is performed based on the low frequency asymptotic representation of the kernels of the boundary integral equations. A classification of the parameters regime for structural–acoustic coupling is suggested. The notation used here, particularly for the geometry of the problem and the characteristics of the shell and the acoustic medium, is the same as in Part 1 [1]. 2. FORMULATION OF THE PROBLEM
The structure considered is a cylindrical shell with two end caps and three intermediate bulkheads. The longitudinal section of this composite structure is presented in Figure 1, with the enumeration of its elementary segments. The driving load arbitrarily distributed on the surface of the structure is assumed to be expanded into a series of trigonometric functions of the angular co-ordinate. Then, due to the axial symmetry of the structure, there is no coupling of the vibrational modes with different circumferential wavenumbers m and so each of them may be analyzed separately. 213 0022–460X/95/270213 + 16 $12.00/0
7 1995 Academic Press Limited
. .
214
Figure 1. The longitudinal section of a composite structure consisting of circular plates (1–5) and segments of cylindrical shell (6–9).
Vibrations of circular plates are considered in the framework of Kirchhoff theory [2]. The positive directions for displacements and forces are shown in Figure 2. The equations of the first level for a circular plate vibrating on the mth mode are presented in the companion paper [1]. There are two boundary integral equations for each plate. A Green function for vibrations in the mth mode is: W (p)(X, Y) =
1 [Y (sA) + (2/p)K0 (sA) − iJ0 (sA)], 8Ds 2 0
A = =X − Y=,
s 4 = rhv 2/D
(1)
(all the notations are defined in Appendix B of reference [1]). Vibrations of segments of cylindrical shells are considered in the framework of the Goldenvejzer–Novozhilov shell theory [3]. The positive directions for displacements and forces are shown in Figure 2. There are eight boundary equations of the first level for each segment of a cylindrical shell, which are also presented in reference [1]. A set of Green functions has the following form: 4
Wj1(c)(x, j) = s cjk ajk exp sk =x − j=, k=1 4
Wj2(c)(x, j) = s cjk bjk exp sk =x − j=, k=1 4
Wj3(c)(x, j) = s cjk exp sk =x − j=,
j = 1, 2, 3, 4.
(2)
k=1
Figure 2. Positive directions for generalized displacements (Figure 2(a)) and generalized forces (Figure 2(b)) of segments of the composite structure.
– , 2
215
There are 42 boundary integral equations of the first level for the whole composite structure shown in Figure 1. These equations contain the unknown contact acoustic pressure p and also 84 algebraic unknowns: displacements and forces at the interfacial points (points A, B, C, D and E of Figure 1). The conditions of continuity of the displacements and the equilibrium conditions should be formulated for each point of an intersection. They are presented here for the point B (see Figure 1). Since the Kirchhoff plate model is used, which postulates the absence of tangential deformations of the plate, the circumferential and radial displacements of cylindrical segments 6 and 7 are put to zero: w62 = w63 = w72 = w73 = 0.
(3a)
The first index is the number of the segment considered; the second one is the number of the component of the generalized displacements vector (see Figure 2(a)). For plates there is a sole component of this vector: wk3 , k = 1, 2, 3, 4, 5; w'k3 is its first derivative. The corresponding generalized forces are denoted as Qk1 and Qk2 respectively (see Figure 2(b)). For segments of the shell there are four components of the vector of generalized displacements: wkn , k = 6, 7, 8, 9 and n = 1, 2, 3, 4; wk4 = w'k3 . The corresponding generalized forces are denoted as Qkn . The remaining conditions ensure continuity of the axial displacements, w61 = w71 = w23 ,
(3b)
w64 = w74 = w'23 .
(3c)
and continuity of the angles, Equilibrium conditions for point B (they are similar for points C and D) have the following form (see Figure 2(b)): equilibrium of forces acting upon the edges of cylindrical segments 6 and 7 and plate 2, Q61 = Q71 + Q21 ;
(3d)
Q64 = Q74 + Q22 .
(3e)
equilibrium of bending moments, There are ten conditions (3) for each intermediate point B, C or D. The conditions for the end caps, say, for the point A, are w62 = w63 = 0,
w61 = w13 ,
Q61 = Q11 ,
w64 = w'13 ,
Q64 = Q12 .
(4a–c) (4d, e)
Thus one has 42 conditions of compatibility for the generalized displacements and generalized forces at the interfacial points for all the segments of the structure. The equation of the second level has the following form (lengths of cylindrical segments are denoted as l; the radius of the cylindrical shell is taken as unity: R = 1): 1 2
p(x, r) +
g g
1
F(x, 0, r, h)p(0, h)h dh +
0
+
1
g
4l
F(x, j, r, 1)p(j, 1) dj
0
F(x, 4l, r, h)p(4l, h)h dh − rv 2
0
g
1
G(x, 0, r, h){r1 [Q11 (r1 )W (p)(h, r1 )
0
+ Q20(h, r1 )w'13 (r1 ) − Q12 (r1 )1W (p)(h, r1 )/1r1 − Q10(h, r1 )w13 (r1 )]=r 1 = 1 +
g
1
0
9
p(0, r1 )W (p)(h, r1 )r1 dr1 }h dh − rv 2 s k=6
g
(lk )
G(x, j, r, 1)
. .
216 ×
6g
(c) (c) W33 (j, x1 )p(x1 , 1) dx1 + [Qk1 (j)W31 (x1 , j)
(lk )
(c) (c) (c) (j, x1 ) + Qk3 (j)W33 (j, x1 ) − Qk4 (j)W34 (j, x1 ) + Qk2 (j)W32 0 0 0 (j, x1 ) − wk2 (j)Q32 (j, x1 ) − wk3 (j)Q33 (j, x1 ) − wk1 (j)Q31 0 (k − 5)l 2 (j, x1 )]=xx11 = + wk4 (j)Q34 = (k − 6)l } dj − rv
g
1
G(x, 4l, r, h){r1 [Q51 (r1 )
0
× W (p)(h, r1 ) + Q20(h, r1 )w'53 (r1 ) − Q52 (r1 )1W (p)(h, r1 )/1r1 − Q10(h, r1 )w53 (r1 )]=r 1 = 1 +
g
1
p(4l, r1 )W (p)(h, r1 )r1 dr1 }h dh
0
= rv 2
g
1
G(x, 0, r, h)
0
q(0, r1 )W (p)(h, r1 )h dh r1 dr1 + rv 2
0
g 6 g 1
×
g
1
9
k=6
0
7
3
j=1
G(x, 4l, r, h)
0
q(4l, r1 )W (p)(h, r1 )h dh r1 dr1 + rv 2 s
× s
g
1
g
G(x, j, r, 1)
(lk )
W3j(c) (j, x1 )qj (x1 , 1) dx1 dj.
(lk )
(5)
Here G (the index m is omitted) is the Green function corresponding to a ring source (see reference [1]): Gm (=X − Y=) =
R 4p
g
2p
0
exp(ivA/c) cos mu du; A
A = =X − Y= = z(x − j)2 + R 2 + r 2 − 2Rr cos u .
(6)
F is a derivative of G: F = 1G/1n+ (n+ is distance along the outward unit normal to the fluid domain at the surface of the structure). All the rest of the notation is the same as in reference [1]. A piecewise constant approximation of the acoustic pressure along the surface of the structure is introduced. Then the standard collocation procedure is used to solve the integral equation (5) (the points of collocations are at the centers of the elements) and the two-level system of boundary equations is transformed to a system of linear algebraic equations (SLAE): [B]{Y} = {Q}.
(7)
Figure 3. Vibrational modes of the lateral surface of the structure shown in Figure 1. Driving line load P0 cos 2u is applied at the middle of segment 7. Curve 1 represents vibrations in vacuo, V 2 = 0·4; curve 2 represents vibrations in the fluid at the same frequency; curve 3 represents the static deflection (V 2 = 0).
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217
Figure 4. Dependence of attenuation V (dB) of the composite structure upon the frequency f (Hz) of excitation.
The vector {Y} consists of two parts: amplitudes of structural displacements and forces at the interfacial points and amplitudes of contact acoustic pressures at the collocation points. Correspondingly the matrix [B] is decomposed into four blocks Bnk , n, k = 1, 2. The structural–acoustic coupling is governed by all the matrix, but primarily by the block B22 . The latter is relevant to the second level equation (5). Its elements are obtained by integration over elementary parts Dj of the lateral surface of the structure: (nj) B22 =
g$ Dj
F(Xn , Y) +
g
%
G(Xn , Y)W(Y, Z) dSz dSy .
(l)
(8)
Here F and G are defined by equation (6); W is the Green function (1) or (2). The order of this SLAE is comparatively small: if one introduces 25 elements for the surfaces of each end cap and 50 elements for the surfaces of each cylindrical segment, then one has 250 unknown acoustic pressure amplitudes. There are also 84 unknown amplitudes of generalized displacements and generalized forces at the interfacial points. Then there are 334 algebraic unknowns in the two-level system. It should be emphasized that, since the kernels of the integral equations have closed forms (1), (2) and (6), it is possible to evaluate the elements of the SLAE analytically in some ranges of the parameters. If the vibrations of the structure shown in Figure 1 are considered in vacuo, then the boundary integral equations method provides one with the exact solution of the problem. 3. NUMERICAL ANALYSIS
Numerical analysis has been performed for reinforced shells of two types. A longitudinal section of the shell of the first type is shown in Figure 1. The lengths of all the segments are equal to the radius of the shell, which is taken as unity, and the thicknesses of all the segments are the same: hc = hp = 0·1. In Figure 3 curve 1 represents the amplitudes of vibrations excited in vacuo by a driving load P0 cos 2u, which is concentrated in the radial direction at the middle of segment 7 (see Figure 1), while curve 2 represents the amplitudes of vibrations in water (r/r0 = 0·128, c/c0 = 0·307; r and r0 are the densities and c and c0 are the speeds of sound in water and in the shell material, respectively). The dimensionless frequency of excitation is V 2 = 0·4 (V 2 = (1 − n 2)v 2R 2/c02). The dimensionless amplitude of vibrations is w* = ERw/P0 . The presence of the acoustic medium in this particular case increases the amplitudes of the displacements by 15–20%. To explain this effect one should note that the first resonance frequency of the ‘‘dry’’ structure given by V02 = 0·765 (this eigenfrequency corresponds to a predominantly radial mode of vibration of the shell). The added masses of water reduce this value and the frequency of the driving load becomes closer to the
. .
218
resonance one. Curve 3 in Figure 3 represents the static displacements caused by the same force when V 2 = 0. The shell of the second type is also reinforced by the bulkheads, as shown in Figure 1. Besides that, the shell is additionally reinforced by T-shape ring stiffeners; of which there are 17, uniformly spaced on each cylindrical segment of the shell. Parameters of the shell are as follows: the material is steel (E = 2 × 105 MPa, n = 0·3); the thickness is h = 0·014 mm, the radius is R = 1·5 m; the length of each segment is l = 4 m. Low frequency vibrations of the composite structure are examined so that a structural–orthotropic shell model [4] is used. The geometrical parameters of the stiffeners are as follows: the effective cross-sectional area is A = 980 mm2; the effective moment of inertia is I = 1600 mm4. The driving load is a point force applied at the middle of segment 7. Vibrations in the first eight circumferential modes (circumferential wavenumbers from 0 to 7) were determined and the velocity at the point of excitation was evaluated as a sum of the components obtained for each mode. The damping coefficient was taken as h = 0·05. The imaginary parts of the fundamental solutions (1) and (2) are retained (see reference [1]). The dependence of the attenuation V (dB) of the system calculated via the amplitudes of vibration at the point of excitation at the frequency f (Hz) is represented by the solid line in Figure 4. The dashed line represents the experimental data reported in references [5, 6]. The agreement between theoretical and experimental results appears to be good. 4. LOW FREQUENCY ASYMPTOTIC EXPANSIONS FOR THE FUNDAMENTAL SOLUTIONS
The numerical analysis described in section 3 was based on the use of fundamental solutions given by formulae (1), (2) and (6). The low frequency range of excitation is of special interest in many technical applications for reinforced cylindrical shells. In this range some important simplifications may be made by the use of asymptotic expansions of the Green functions. Then the inner integral in equation (8) (convolution of the Green function) may be solved analytically. For all the frequency ranges the fundamental solutions for the problem of the vibrations of the cylindrical shell are exponentials in the axial co-ordinate (see equations (2)) and thus these functions do not need any transformations. In the low frequency range power series expansions of the Bessel functions may be substituted in the fundamental solution (1) for the vibrations of a circular plate: W (p)(r, R) =
g
1 2pD
2p
cos mu{14 (r 2 + R 2 − 2rR cos u)[1 − g
0
− ln(s/2) − 12 ln (r 2 + R 2 − 2rR cos u)] +
s4 2 (r + R 2 64
− 2rR cos u)3[1 + 12 + 13 − g − ln (s/2) − 121 ln (r 2 + R 2 − 2rR cos u) + · · · } du,
g = 0·5772.
(9)
The fundamental solution (5) may also be simplified by the use of a power series expansion: exp(ivA/c) = 1 + (ivA/c) + (1/2!)(ivA/c)2 + (1/3!)(ivA/c)3 + · · · . Then, instead of equation (6) one obtains: G(=X − Y=) =
R0 2p
g$ p
0
01
01
1 iv 1 v i v + − A− A2 + · · · A c 2! c 3! c 2
3
%
cos mu du.
(10)
– , 2
219
The subscript m is omitted. The function (10) may be expressed as the following sum: G(=X − Y=) = G1 (=X − Y=) + G2 (=X − Y=) + G3 (=X − Y=). Then it is possible to compare the contributions of the components of the fundamental solution G(=X − Y=) corresponding to an incompressible fluid with the ones corresponding to the real and the imaginary parts of the acoustical correction to its value in various ranges of the parameters. These components are: G1 (x, j, r, R0 ) = G2 (x, j, r, R0 ) =
R0 2p
R0 2p
g
p
0
cos mu du, A
g$ 01 01 g$ 01 p
(11a)
1 v 1 v A+ A3 − · · · 2! c 4! c 2
4
−
0
G3 (x, j, r, R0 ) =
p
iR0 2p
v 1 v − A2 + · · · c 3! c 3
0
%
%
cos mu du,
cos mu du.
(11b) (11c)
The function G1 is that for an incompressible fluid (evidently, this is the Green function for Laplace’s equation), the function G2 is a correction due to acoustically added mass, and the function G3 is a correction due to energy radiation effects. Each term in the integral (11c),
g
p
[(x − j)2 + R02 + r 2 + 2R0 r cos u]2k cos mu du,
k = 1, 2, 3, . . . ,
0
may be easily solved analytically. Each term in the series (11b) is integrated by parts:
g
p
A 2k − 1 cos mu du =
0
2k − 1 m
g
$g
p
A 2k − 3 cos (m − 1)u du
0
%
p
−
A 2k − 3 cos (m + 1)u du R0 r,
0
k = 2, 3, 4, . . . .
(12)
Then, for k E m, all the terms in equation (11b) may be expressed via the fundamental solutions (11a) corresponded to various wavenumbers m. After a function T(m, M) is introduced as T(m, M) =
g
p/2
0
cos 2mu du z1 − M cos u
,
M=
2
4rR0 , (x − j)2 + (r + R0 )2
(13)
formula (11a) may be rewritten as G1 (x, j, r, R0 ) =
R0 pzx − j)2 + (r + R0 )2
T(m, M).
(14)
The following cases can then be distinguished. (i) A fundamental solution corresponding to axisymmetrical pulsations of the structure (m = 0). Then formula (14) has the form G1 (x, j, r, R0 ) =
R0 pz(x − j)2 + (r + R0 )2
(K(M) is the elliptic integral of the first kind).
K(M)
(15)
. .
220
Figure 5. Dependence of the potential G1 produced by a ring source in an incompressible fluid on the distance z. ——, exact formula (11(a)); ––––, approximate formula (19). m is the number of circumferential waves.
(ii) A fundamental solution corresponding to vibrations of a thin-walled structure without distortion of its cross-section (m = 1). Then formula (14) has the form G1 (x, j, r, R0 ) =
R0 pz(x − j) + (r + R0 ) 2
2
$
2−M 2 K(M) − E(M) M M
%
(16)
(E(M) is the elliptic integral of the second kind). (iii) Fundamental solutions corresponding to shell modes of vibrations of the structure (m e 2). Then, based on the identity T(m, M) = T(m − 1, M) − 2
g
p/2
sin u sin (2m − 1)u
du,
z1 − M cos2 u
0
(17)
the following formula is obtained: G1 (x, j, r, R0 ) =
$
R0 2−M K(M) M pz(x − j)2 + (r + R0 )2 m 2 E(M) − 2 s M k=2
−
g
p/2
%
sin u sin (2k − 1)u
du .
z1 − M cos2 u
0
(18)
The sum in equation (18) is regular for all values of M: when M = 0, it is equal to zero, and when M = 1, it is equal to smk= 1 (2k − 1)−1. After some quite cumbersome transformation one obtains G1 (x, j, r, R0 ) =
R0 pz(x − j)2 + (r + R0 )2
&
8
m
K(M) − 2 s (2k − 1) k=1
k 1 1−M (−1)p p E(M) − K(M) + s t [(2k − 1)2 × M M (2p + 1)! q = 1 p=1
− (2q − 1)2]
&
2
pG(2p + 2)
60
2p + 2
2p + 3 G 2
17
Here M1 = 1 − M and G( ) is the Euler G-function.
− 2
G(p)G(3/2)
0
2p + 3 2G 2
1
M1
''9
.
(19)
– , 2
221
Figure 6. The length Zm of an ‘‘influence zone’’ versus the number of circumferential waves m.
In Figure 5 the values of the functions G1 (x, j, r, R0 ) are presented versus the distance z = =x − j= when r = R0 = 1. Continuous curves correspond to the exact formula (11a) and dashed ones to the approximate formula (19). The latter gives negative values for the pressure at a certain distance from the ring; these values should be replaced by zero. It is convenient to introduce a so-called ‘‘influence zone’’ for analysis of the acoustic pressure generated by a ring source. For each m the length of this zone is equal to a maximum distance zm , which gives positive values for the formula (19). In Figure 6 the sizes of the influence zone zm versus the number of circumferential waves m are presented. It is clearly seen that the rate of decay of acoustic pressure sharply increases with an increase in the number m. This effect is explained by the fact that large values of m correspond to an alternating sequence of sources and sinks on the ring. The motions of the particles of the acoustic medium in this case are localized in the neighborhood of the ring. If the cylindrical surface of the radius R0 is considered, then only in cases of axisymmetric and beam-type excitations is there significant pressure at a distance larger than R0 . This effect remains the same if the function G2 (x, j, r, R0 ) is analyzed, which is in fact the difference between the functions G1 (x, j, r, R0 ) with different wavenumbers m (see equation (12)). The function F is the first derivative of G with respect to the co-ordinate which is normal to the lateral surface of the shell. In particular, for a cylindrical shell, F(x, j, r, R0 ) =
R0 4p
g
$
2p
(1 − cos u) 1 −
0
%
ivA exp(ivA/c) cos mu du. A3 c
(20)
It is useful to represent this function similarly to G: F(X, Y) = F1 (X, Y) + F2 (X, Y) + F3 (X, Y). Here one has F1 (x, j, r, R0 ) = F2 (x, j, r, R0 ) = F3 (x, j, r, R0 ) =
R0 4p
R0 4p
g
2p
(1 − cos u)
0
g $0 1 0 1 g $0 1 0 1
iR0 4p
2p
2p
0
% %
(21a)
1 1 1 1 + − A + · · · (1 − cos u) cos mu du, 2! A 3! 4!
(21b)
1 1 1 1 2 − − + A + · · · (1 − cos u) cos mu du. 2! 3! 4! 5!
(21c)
1−
0
cos mu du, A3
The functions Fk (x, j, r, R0 ), k = 1, 2, are expressed via the functions T(m, M) similarly to the functions Gk (x, j, r, R0 ), k = 1, 2. The first term in equation (21a) contains a singular part which has already been taken into account in equation (5) by the non-integral term (see reference [7]). The function (21c) is evaluated analytically.
. .
222
5. CLASSIFICATION OF TYPES OF STRUCTURAL–ACOUSTIC INTERACTION
Elements of the matrix B22 relevant to the boundary equation of the second level (see equation (8)) may be represented in the following form: (nj) = B22
g
[F1 (Xn , Y) + F2 (Xn , Y) + F3 (Xn , Y)] dSy +
Dj
gg Dj
W(Y, Z)[G1 (Xn , Y)
(l)
+ G2 (Xn , Y) + G3 (Xn , Y)] dSz dSy .
(22)
The first three terms on the right side of equation (22) may be evaluated by using formulae (21) for Fk , k = 1, 2, 3. The integration is performed analytically for each combination of positions of the observation point and the source point. All the double integrals in equation (22) may be transformed in the same manner: for the end caps,
gg Dj
1
W (p)(h, r1 )[G1 (x, nl, r, h) + G2 (x, nl, r, h) + G3 (x, nl, r, h)]r1 dr1 h dh
0
g
=
1
[G1 (x, nl, r, h) + G2 (x, nl, r, h) + G3 (x, nl, r, h)]
0
×
6g
7
W (p)(h, r1 )r1 dr1 h dh;
Dj
(23)
and for the cylindrical surface,
gg Dj
(c) W33 (j, x1 )[G1 (x, j, r, 1) + G2 (x, j, r, 1) + G3 (x, j, r, 1)] dj dx1
(l)
=
g
[G1 (x, j, r, 1) + G2 (x, j, r, 1) + G3 (x, j, r, 1)]
(l)
×
6g
Dj
7
(c) W33 (j, x1 ) dx1 dj.
(24)
The use of expansion (9) for the fundamental solution W (p)(h, r) permits one to evaluate the integral in square brackets of equation (23) analytically. Then the outer integral in equation (23) is computed by using the fundamental solution G(x, j, r, h) in the form (11). The integral (24) may be computed similarly because the fundamental solution has the simple exponential form (2). For the lateral surface of the cylindrical shell the convolution of the Green functions for the acoustic medium and the cylindrical shell may also be computed in another way. It has been shown in section 4 that the pressure field generated by a ring source is localized in the influence zone (see Figure 2). In this zone, for m e 2, the axial co-ordinates of an observation point and a source point satisfy the inequality =x − j= Q R = 1. This permits one to represent the argument of the elliptic integrals in the form M = (1 + (x − j)2/4)−1 = 1 − (x − j)2/4. Then the functions (11a) and (11b) may be rewritten as Gb (x, j, 1, 1) = d1b + d2b ln =x − j= + d3b (x − j)2 + d4b (x − j)2 ln =x − j=.
(25)
The coefficients dab , a = 1, 2, 3, 4, b = 1, 2, are quite cumbersome and they are not presented here. In fact, for m e 2 the integration in equation (24) should be performed only within the influence zone.
– , 2
223
The evaluation of the convolutions (24) for axisymmetric and beam-type modes of vibrations is more complicated. In these cases this procedure depends upon the values of the roots of the characteristic polynomials sk , k = 1, 2, 3, 4 (see equation (2)). Those roots which are large correspond to the boundary effect in the shell theory (see reference [3]). Then the integral (24) may be evaluated in asymptotic series of negative powers of sk . The integration of exponents with small roots should be performed numerically. A number of terms in equation (22) may be disregarded under certain conditions. The simplifications introduced by this disregard correspond to neglect of some features of the interaction. The basic features of the interaction are as follows: (i) structural–acoustic coupling, defined by the integrals
gg Dj
W(Y, Z)[G1 (Xn , Y) + G2 (Xn , Y) + G3 (Xn , Y)] dSz dSy
(l)
(the first term corresponds to an incompressible fluid model and the remaining two terms an acoustic medium model); (ii) compressibility of the medium, defined by the integrals
g
[F2 (Xn , Y) + F3 (Xn , Y)] dSy +
Dj
gg Dj
W(Y, Z)[G2 (Xn , Y) + G3 (Xn , Y)] dSz dSy ;
(l)
(iii) curvature of the structure, defined by the integrals
g
[F1 (Xn , Y) + F2 (Xn , Y) + F3 (Xn , Y)] dSy ,
Dj
(the first term corresponds to an incompressible fluid model, and the remaining two terms correspond to an acoustic medium model). These features are governed by the parameters of rigidity and geometry of a structure and also by the frequency of excitation. One may say that they influence the left side of equation (5), while the effect of the configuration of the driving load influences the right side of equation (5). For each specific configuration of the driving load the analysis of the interaction should be continued, based on the analysis already carried out for the homogeneous form of equation (5). The following simplified models for equation (5) can be considered: (1) incompressible medium, curvature of the structure and structural–acoustic coupling disregarded; (2) incompressible medium, curvature of the structure taken into account, no structural–acoustic coupling; (3) acoustic medium, curvature of the structure taken into account, no structural–acoustic coupling; (4) structural–acoustic coupling, incompressible medium, curvature of the structure disregarded; (5) structural–acoustic coupling, acoustic medium, curvature disregarded. An estimation of the validity of these models may be carried out in several ways. In particular, the comparison of values for energy radiated to the far field, as obtained from the exact formulation of the problem and from the simplified one should be performed. Then it is necessary to solve the two-level system and to calculate the amplitudes of vibrations of the structure. A solution for the two-level system is obtained numerically. Then all the further steps in the evaluation of the energy radiated from the structure must also be performed numerically. It would be more convenient to estimate the validity of models (1)–(5) by comparison (nj) of the matrix components B22 (see equation (22)). In this case model (1) corresponds to a pure diagonal matrix B22 with all the diagonal elements equal to 1/2. Each feature of the interaction gives a certain contribution to the matrix and transforms it from the pure diagonal form.
. .
224
T 1 Elements of the matrix B22 for n = 1, m = 10 and (a) vR/c = 0·001 and (b) vR/c = 0·1 (a) bHE × 105
k 1 4 7 10 13 16 19 22 25
ZXXXXXCXXXXXV
h/R = 0·001
h/R = 0·01
−0·121 −0·090 −0·065 −0·038 −0·018 −0·007 0·0 0·005 0·004
0·170 0·140 0·092 −0·076 −0·052 −0·034 −0·024 −0·021 −0·003
(b) bHE
k 1 4 7 10 13 16 19 22 25
ZXXXXXCXXXXXV
h/R = 0·001
h/R = 0·01
−0·633 −0·471 −0·331 −0·231 −0·160 −0·083 0·0 0·054 0·041
−0·182 −0·151 −0·124 −0·100 −0·071 −0·050 −0·030 −0·010 −0·008
The matrix components (22) relevant to structural–acoustic coupling can be denoted by BHE , the components relevant to compressibility of the medium by BAC and the components relevant to effects induced by the curvature of the structure by BHD . A comparison of the values of the components of the matrix corresponding to these three basic features of the interaction is performed in section 6. 6. ANALYSIS OF THE PARAMETER REGIMES OF STRUCTURAL–ACOUSTIC COUPLING FOR VIBRATION OF CYLINDRICAL SHELL WITH END CAPS
The vibrations of the steel structure shown in Figure 1 in water are considered: c/c0 = 0·307 and r/r0 = 0·128. The vibrational modes with m circumferential waves are analyzed separately. Let an observation point be placed at the end cap. The integral containing the normal derivative F over the end cap is equal to zero (the singular part of this integral produces the non-integral term 1/2). Piecewise constant approximation of the contact acoustic pressure is introduced with NE = 25 (NE is the number of elements). Since the curvature of the contact surface is equal to zero, the comparison has been performed for the simplified models (1) and (4) of section 5. The values of the real parts of the elements of the matrix B22 are presented in Tables 1(a) and 1(b). These elements are denoted as BHE . A point of observation is set on the edge
– , 2
225
T 2 The natural frequencies of an isolated segment of a ‘‘dry’’ cylindrical shell h/R
m 1 2 3 4 5 6 7 8
ZXXXXXXXXCXXXXXXXXV
0·001
0·003
0·01
0·03
0·10
1·99 0·652 0·481 0·354 0·265 0·203 0·159 0·128
1·99 0·652 0·481 0·354 0·266 0·206 0·166 0·141
1·99 0·653 0·483 0·361 0·281 0·239 0·228 0·244
1·99 0·661 0·503 0·410 0·389 0·432 0·519 0·637
1·99 0·743 0·687 0·775 0·985 4·04 4·57 5·11
of the circular plate so that the first index n is equal to 1. The second index characterizes the position of a source. It is presented in the first row of each table. Table 1(a) corresponds to a circumferential wavenumber m = 10 and frequency parameter vR/c = 0·001. The second and the third rows present the elements BHE of B22 for h/R = 0·001 and h/R = 0·01, respectively. Table 1(b) corresponds to a circumferential wavenumber m = 10 and frequency parameter vR/c = 0·1. The second and third rows present elements BHE of B22 for h/R = 0·001 and h/R = 0·01, respectively. In both the cases considered the imaginary parts of the elements BHE are much smaller than the real ones and there is practically no radiation to the far field from the structure in this vibrational mode. It is clearly seen that in a case of low frequency excitation the influence of coupling effects upon the contact acoustic pressure may be neglected regardless of the position of the source and the simple model (1) is valid. When the driving load is at a comparatively high frequency (vR/c = 0·1), the influence of coupling is small near the center of the plate while in the neighborhood of the edge it becomes essential and model (4) is preferable. Let a point of observation be placed on the surface of a cylindrical segment. The natural frequencies of an isolated segment of a ‘‘dry’’ structure shown in Figure 1 are presented in Table 2. In Tables 3–6 the real and imaginary parts of the values for the elements of the SLAE BHD , BAC and BHE are presented. The number of boundary elements introduced is 50. The point of observation is set at the middle of a cylindrical segment and the first index is equal T 3 Real parts of elements of matrix B22 for n = 25, vR/c = 0·1, m = 8 k
BHD × 102
BAC × 106
BHE × 106 (h/R = 0·1)
BHE × 104 (h/R = 0·03)
BHE × 103 (h/R = 0·01)
BHE × 10 (h/R = 0·003)
BHE × 10 (h/R = 0·001)
25 22 19 16 13 10 7 4 1
0·633 −0·211 −0·231 −0·095 0·051 0·134 0·129 0·064 0·001
−0·260 −0·246 −0·209 −0·158 −0·104 −0·057 −0·026 −0·006 0·0
−0·503 −0·392 −0·184 −0·091 −0·042 −0·019 −0·008 −0·003 −0·001
−0·178 −0·147 −0·075 −0·041 −0·021 −0·010 −0·004 −0·001 0·0
−0·377 −0·354 −0·232 −0·167 −0·113 −0·072 −0·042 −0·006 0·0
−0·100 −0·094 −0·088 −0·079 −0·070 −0·060 −0·050 −0·042 −0·033
0·542 0·518 0·593 0·627 0·662 0·695 0·723 0·744 0·755
. .
226
T 4 Real parts of elements of a matrix B22 for n = 25, vR/c = 1, m = 8 k
BHD × 102
BAC × 104
BHE × 103 (h/R = 0·1)
25 22 19 16 13 10 7 4 1
0·633 −0·211 −0·231 −0·095 0·051 0·134 0·129 0·064 0·001
−0·266 −0·252 −0·214 −0·162 −0·106 −0·059 −0·027 −0·010 −0·003
−0·155 −0·117 −0·057 −0·028 −0·013 −0·006 −0·002 −0·001 0·0
BHE × 102 BHE × 10 BHE (h/R = 0·03) (h/R = 0·001) (h/R = 0·003) −0·685 −0·560 −0·331 −0·202 −0·118 −0·065 −0·034 −0·016 −0·012
0·297 0·454 0·839 0·929 0·829 0·552 0·155 −0·270 −0·624
0·063 0·143 0·261 0·241 0·133 −0·021 −0·163 −0·235 −0·209
BHE (h/R = 0·001) 0·166 −0·080 0·786 0·707 0·360 −0·120 −0·535 −0·711 −0·578
to 25. The second index characterizes the source position. Only values of the elements of a matrix with k = 1, . . . , 25 are included in the tables since values of elements with k = 26, . . . , 50 are the same. The values for BHD depend only upon the circumferential wave number m. The values for BAC depend both upon m and the dimensionless frequency of excitation vR/c. The values for BHE depend upon m, vR/c and also upon the thickness of the shell h/R. The analysis has been performed for the following values of thickness: 0·001; 0·003; 0·01; 0·03; 0·1. T 5 Elements of matrix B22 for h = 25, vR/c = 1, m = 3; (a) real parts; (b) imaginary parts (a)
k
BHD × 102
BAC × 103
25 22 19 16 13 10 7 4 1
0·957 0·063 −0·079 −0·092 −0·060 −0·024 −0·006 −0·009 −0·027
−0·249 −0·247 −0·242 −0·232 −0·217 −0·197 −0·174 −0·149 −0·127
BHE × 10 ZXXXXCXXXXV h/R = 0·03 h/R = 0·01 0·057 0·051 0·080 0·162 0·233 0·252 0·258 0·251 0·230
0·149 0·060 0·365 0·448 0·709 0·762 0·774 0·742 0·667
BHE ZXXXXXCXXXXXV h/R = 0·003 h/R = 0·001 0·049 0·101 0·119 0·102 0·241 0·259 0·263 0·253 0·225
0·179 −0·074 0·702 0·359 0·783 0·836 0·843 0·802 0·715
(b) 5
k
BAC × 104
BHE × 10 (h/R = 0·03)
BHE × 104 (h/R = 0·01)
BHE × 104 (h/R = 0·003)
BHE × 103 (h/R = 0·001)
25 22 19 16 13 10 7 4 1
−0·115 −0·115 −0·115 −0·114 −0·114 −0·113 −0·113 −0·112 −0·111
−0·865 −0·745 −0·664 −0·624 −0·614 −0·631 −0·677 −0·759 −0·877
−0·259 −0·214 −0·195 −0·186 −0·183 −0·188 −0·199 −0·219 −0·263
−0·879 −0·720 −0·666 −0·631 −0·621 −0·638 −0·680 −0·737 −0·892
−0·288 −0·241 −0·224 −0·213 −0·211 −0·216 −0·228 −0·247 −0·292
– , 2
227
T 6 As Table 5, but with m = 1 (a) k
BHD × 10
BAC × 102
25 22 19 16 13 10 7 4 1
0·131 0·039 0·018 0·007 0·002 −0·001 −0·003 −0·004 −0·006
−0·220 −0·219 −0·214 −0·207 −0·198 −0·187 −0·175 −0·161 −0·147
BHE × 10 (h/R = 0·01) 0·019 −0·008 0·114 −0·003 0·108 0·192 0·170 0·416 0·477
BHE (h/R = 0·001) 0·017 −0·252 0·393 −0·134 0·065 −0·037 0·606 −0·014 0·501
(b) k
BAC × 102
25 22 19 16 13 10 7 4 1
−0·278 −0·278 −0·277 −0·275 −0·273 −0·271 −0·268 −0·264 −0·260
BHE × 102 (h/R = 0·01) −0·160 0·059 0·085 0·097 0·096 0·983 0·056 0·017 −0·191
BHE × 10 (h/R = 0·001) −0·163 0·058 0·093 0·097 0·096 0·083 0·063 0·015 −0·192
The low frequency vibrations with the large number of circumferential waves (vR/c = 0·1; m = 8) have been considered first (see Table 3). In this case the influence of the curvature of the structure is negligibly small (BHD 0 10−3) and the influence of the compressibility of the medium is also small (BAC 0 10−7). The imaginary parts of BAC and BHE are of the same order (BAC 0 BHE 0 10−16) and also should be ignored. There is no radiation from a structure in this vibrational mode at this frequency. In this case there is the alternative to use either the simple model (1) or the model (4) when structural–acoustic coupling effects cannot be disregarded. For a thick shell (h/R = 0·1) structural–acoustic coupling is insignificant (BHE 0 10−7), but for a thin shell (h/R = 0·001) these effects become significant. Generally, in this case model (1) is acceptable if h/R q 0·01 and model 4 should be used for h/R Q 0·01. In Table 4 similar data are presented for vR/c = 1 and m = 8. The values for BHD remain the same as in the previous case and the effect of curvature should be ignored. Despite a certain increase in the values for BAC they are still small and therefore should be neglected. Structural–acoustic effects are significant in most of the range of thicknesses considered (0·001 Q h/R Q 0·03). Moreover, when the thickness is small (h/R = 0·001) the values of BHE are greater than 1/2 (the diagonal element of the matrix relevant to model (1)). Then model (4) should be used. It should also be noted that for a very thick shell (h/R = 0·1) contributions of each feature of the interaction are equivalent. The imaginary parts of BAC and BHE are of the same order (BAC 0 BHE 0 10−12) and should be omitted. In Table 5(a) similar data are presented for vR/c = 1, m = 3. When the thickness is large (h/R = 0·1) it is still possible to use the simple model (1). While for thin shells
228
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(h/R Q 0.01) model (4) is the most appropriate one. It should be noted that the case of moderate thickness (hR = 0·03) appears to be the most complicated one, when no simplified model is acceptable. Imaginary parts of BAC and BHE are of the same order and both of them should be taken into account. These imaginary parts are presented in Table 5(a). In Table 6 similar data are presented for the beam-type mode vR/C = 1, m = 1. When the thickness of the shell is small, only structural–acoustic coupling effects should be taken into consideration; for moderate thicknesses both the acoustic and structural–acoustic effects are essential (model (4) should be used). The imaginary parts are presented in Table 6(a). 7. CONCLUSIONS
The vibrations of a reinforced cylindrical shell of finite length have been analyzed by the two-level boundary integral equations method. Asymptotic formulae for the Green functions for vibrations of circular plates in an acoustic medium have been obtained for a low frequency range of excitation. Numerical analysis of vibrations of reinforced shells of two types has been performed along with analysis of the system of boundary equations. Based on these analyses several simplified models of structural–acoustic interaction have been suggested. The ranges of validity for these models have been estimated. REFERENCES 1. L. I. S and S. V. S 1995 Journal of Sound and Vibration 184, 195–211. Analysis of structural–acoustic coupling problems by a two-level boundary integral equation method, part 1: general formulation and test problems. 2. S. P. T and S. W-K 1959 Theory of Plates and Shells. New York: McGraw-Hill. 3. A. L. G, V. B. L and P. E. T 1979 Free Vibrations of Thin Elastic Shells. Moscow: Nauka (in Russian). 4. E. A. S, I. L. E and A. V. V 1988 Nekotorye Problemy Mekhaniki Sudovikh Konstruktsiy, 62–67. A simple estimation of the first natural frequencies of orthotropic cylindrical shells (in Russian). 5. I. K. P and S. A. R 1983 On Impedance of Cylindrical Shell Reinforced with Ring Stiffeners. DR 1905, Ts.N.I.I. Rumb (in Russian). 6. I. K. P and S. A. R 1984 Discretized Model to Evaluate Mechanical Impedance of Stiffened Cylindrical Shell. DR 1905, Ts.N.I.I. Rumb (in Russian). 7. P. K. B and R. B Boundary Element Methods in Engineering Science. New York: McGraw-Hill.