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Distorted cylindrical shell response to internal acoustic excitation below the cut-off frequency
S.
442
N. YOUSRI AND F. J. FAHY
the generalized force acting is zero. Hence Firth’s theory does not predict quantitatively the response amplitude. The theory presented in the next section shows that if there is a small variation in the shell thickness, radius and/or Young’s modulus, these variations will couple the acting axisymmetric acoustic pressure to a non-axisymmetric vibrational shell mode. Rosen and Singer [7] calculated the effect of distortion on the vibration modes of cylindrical shell; they found that the influence of the initial geometrical imperfections (in radius) depends strongly on the mode of the initial imperfection and its relation to the mode of vibration, and that the imperfection may lower or raise the frequency. Ideally one should consider the shell thickness, radius and modulus of elasticity as variables [8]; this yields equations wherein terms containing these quantities must be differentiated once or twice with respect to shell co-ordinates. The resulting set of differential equations is essentially intractable. For small variations in thickness, radius and/or Young’s modulus constant average values can be used in the final perfect circular shell equations and then the coupling term (generalized force term) between excitation plane wave sound and the shell modal response can be calculated by taking into account these small variations. The theory presented in the next section is based on the above approach.
2. SHELL EQUATIONS The motion of a geometrically Reissner-Naghdi-Berry equations
l+Yu 2a
I
‘@
OF MOTION perfect circular [8] as;
WITH SMALL DISTORTION cylindrical
by the
.
1-y(]+p2)v .zZ+‘+BZ 2
shell can be described
-$=O,
(lb)
a2
where a is the shell outer radius, C, is the speed of longitudinal waves in the shell material, Ci = E/p,(l - r’), E is the Young’s modulus, h is the shell thickness, U, V, Ware the axial, circumferential and radial displacements, respectively. ii, v, ware the corresponding accelery is Poisson’s ratio, /I = ho/2/TZao, ation components, Z, r, cf, are the clyindrical co-ordinates, and the suffixes z and @ refer to differentiation in the axial and circumferential directions, respectively. (A list of symbols for reference is given in the Appendix.) P is the total pressure acting in the shell surfaces and can be expressed as P = Pi + Prt where Pi is the excitation which is given by
plane wave (blocked)
(2)
pressure below the acoustic cut-off frequency,
Pi = P,(z) e-‘O’
(3)
DISTORTED
and P, is the pressure radiated form have been assumed :
from the vibrating
U=
443
SHELL RESPONSE
shell. Modal
solutions
of the following
2 Un,,cos(k,z)cos(n@)e-‘“‘, n. s
V = 2 V,, s sin (k,z)sin n. 5 W=
(4a)
(n@) eC’O’,
(4b)
2 W,,,sin(k,z)cos(n@)e-‘“‘. n. s
(4c)
The pressure radiated, P,, is of the same form as the radial displacement, of vibration given by equation (4~). P, is given by (see section 3)
W. For the form
P, = 2 W,,,sin(k,z)cos(n@)o’pade-‘“‘, n. s where A is the fluid loading term (see section 3). Substitution into equations (1) gives 1-Y
Q2_k2az-s
2
]+Y n2 U,, + - 2
Q2-(1
+BZ)nz-
(5) of equations
(2), (3), (4) and (5)
k,anV,,,+yk,aW,,,=O,
(T)(l
(6a)
+8’)k$a2]V”,s
- [n + /?‘(kf a2 n + n3)] W,, s = 0,
(6b)
2 {-yk, a(U.,A W,,J + b + B’(kZa2n + n”)l(V, ,I W,, A
n.
s
+
[1+
B’(kz a2 + n2)2 - a’( 1 - pad/p, h)]} W,, s sin (k, z) cos (n@) = {( 1 - r2) a2/Eh} P,, (z), (6~)
where k, is the structural wave number, k, = m/L, L is the shell length, n is the ferential mode number, s is the shell axial mode number, p is the fluid density, density and Sz = ma/C,,. By eliminating U,,, J W,, sand V,,, J W,, 5from equation (6c), by using equations the solution for the modal displacement W,, s can be obtained by multiplying equation (6~) by the shell mode shape [sin(k,z)cos(n@)] and integrating over the with use being made of the modes’ orthogonality properties: this gives
shell circumps is the shell (6a) and (6b), both sides of shell surface,
-yk, a
2n
L (1
=
-
YZ)
----a’P,,(z) 0 0
Eh
sin(k,z)cos(n@)dzd@.
(7)
The right-hand side of equation (7) shows that a plane wave acoustic mode (PO is a function of z only) induces zero response in a structural mode having n # 0. The shell breathing mode (n = 0) occurs above the plane wave excitation frequency: therefore there is no frequency
S.
444
N. YOUSRI AND F. J. FAHY
matching and the response of this mode is negligible compared with the response of an asymmetric distorted shell mode where frequency matching occurs. Therefore an n # 0 shell mode may respond if a, h and/or E are functions of the shell co-ordinates 2, CD. Let a = a,[1 + 6,(Z, @)I,
h = h,[l + S,(=, @)I,
where a,, 6, and SE < 1. The quantity a2/Eh appearing series as
02, $?,(I 0 0
E = EJ
on the right side of equation
4
+2a,+6,-6,)=E
0 0
(7) can be expressed
n. S
_ 2(1 - Y2)u; L 2n pa(z) A,(z) cos2 (n@) sin (k,z)dzd4 nEohoLD ss
the internal
in Fourier
(9)
[I + ,c A”(Z) cos (n@)]. n
The coefficients A, in equation (9) can be estimated by measuring radii of the shell and fitting the results to the above formulae. Substituting equation (9) into equation (7) gives w
+ a,(~, @>I, (8a, b, c)
and external
2(1 - r2) = nE LD G,
(10)
0
00
where G is the generalized
force and
D = -yk, a(Un, ,/ W,,, ,) + [n + B”(k: a2 n +
+ [l
+ fi2(k:a2
n3)1CVn.s/Wn,s!
+ n2)2 - Q2(1 - pad/p,h)].
The solutions of D = Oare the dimensionless resonance frequencies, R, for the fluid loaded shell for the mode numbers n and S. D is a complicated function of a, h and E and the effects of 6,, dh and 6, in D have been neglected since these will only shift the resonance frequencies. Damping can be introduced into the analysis by replacing E. by the complex modulus E,(l + iqToT), where nToT is the total loss factor. Some possible forms of distortions are as follows. 1. One might have a tolerance in thickness h and/or radius a of the forms h = h,(l 2 6,) and/or a = a,(1 f: S,), respectively. 2. A line weld parallel to the shell axis at Q, = Q. can contribute to all modes; this can be analysed as follows. For the thickness, h, h = ho(l + 6,) + B6( @ - Qo), where the first term represents the tolerance in thickness and the second represents the line weld; 6 is the Dirac delta function, and B is the unknown amplitude. B can be obtained by multiplying both sides by a, and integrating, to give 277
ah d@ N 2na, ho + Ba,, 5 0
B = [weld area (perpendicular The term B6(@ - Qo) can be expressed
in Fourier
to shell axis)]/ao.
series as
B6( @ - Go) = 2 b, cos @I@), ” 2n
s
Bcos(n@)6(@
6),.,
- Go)d@ =+,
= 2B = [twice the weld arealla,.
DISTORTED
SHELL RESPONSE
445
to all modes. Similarly, variation of E at the Therefore (b,),,, is constant and contributes weld can be analysed. 3. The shell end boundary constraints, if not circumferentially uniform, can cause certain non-axisymmetric modes to be excited: i.e., if the shell is supported by four stiff arms it is likely that the n = 4, 8, 12 circumferential modes can be excited with plane wave sound, provided that frequency matching occurs. 3. THE FLUID LOADING
Cd)
For a single shell, A in equation (5) can be obtained by solving the Helmholtz equation and satisfying the boundary conditions at the shell surfaces and at infinity. The radiated pressure is given by [9] P,=
5 ~~paW,,,sin(k,a)cos(n~)e-‘“‘n. s
H,(v)
---
u74
(II) I’
where q2 = k2 - kz, J, and J,, are the Bessel function of order n and its first derivative, respectively, H, and H, are the Hankel function of order n and its first derivative, respectively. Hence A is given by d, as
This is a complex quantity; the real part is the reactive component (mass or stiffness loading) and the imaginary part is the radiation damping component; the imaginary part of A, is given by 4 =
(13)
nq’u2/Fi.(qu)l”’
Figures 1 and 2 give the vales of the real and imaginary parts of A,, respectively. It can be seen that lower order circumferential modes (small n) give higher radiation damping. Also for a steel shell vibrating in air, where pips = 1*2/7700, the fluid loading can be neglected unlike the case of a steel shell vibrating in water where p/pb = 998/7700. 16 14 12 10 e6‘I-
1 4 0
I
20
3
3
2 0 IO
b
0
1
3
20
3
“ii
-16’
Figure 1. Real part of A, (equation (12)).
v
446
S. N. YOUSRI
AND F. J. FAHY
Figure 2. Radiation loss factor qrad for an infinite circular shell vibrating in an n circumferential with an axial wave number k,.
mode and
For two concentric shells, on the assumption that the outer one is rigid and the inner one is vibrating with a radial displacement, W, as given in equation (4), then the value of A = A2 is a real quantity and is given by
J&4 7.W) - !tW) Y&4
A
z
Jn(q4 T”in(44 - uqw
Uqa)
1 ’
16 n=
I
21
14 -
Figure 3. Fluid loading function for two concentric shells (equation (14)).
(14)
DISTORTED
SHELL RESPONSE
447
where b is the radius of the outer rigid shell, and Y, and P, are the Neumann function of order n and its first derivative, respectively. Figure 3 gives the value of A, for b/a = 1.2. Equations (1 I), (12), (13) and (14)arefortheinfiniteshellcase. Forafinitevibratinglength of a shell the radiation equation is in an integral form (see reference [9]), but above the axially coincident wave numbers (k > k,), the radiation from an infinite shell approaches that of the finite case [lo].
4. EXPERIMENTAL
INVESTIGATION
Experiments were performed to illustrate that a plane wave acoustic mode can excite an n # 0 shell mode directly and not through the supports, and also to confirm quantitatively the theory developed for the shell modal response with first order shell distortion affect. 4.1.
THE TEST SPECIMEN
The steel shell tested is shown in Figure 4. It has the following dimensions: shell average diameter 0.076 m, shell average thickness 0.000607 m, shell length 0.198 m. The shell has flanged ends welded to it and was supported at both ends independently of the excitation system. The distortion in radius and thickness were measured at four axial stations and eight circumferential stations. The results for aZih were expressed in the Fourier series in the form a2 4 -=-
h
ho
1+ i
A,(z) cos(J@)
(1%
.I=0
where the coefficients A,(z) are shown in Figure 5; only four terms have been considered since higher terms correspond to acoustic modes above the cut-off frequency. The first term with J = 0 will couple to the shell breathing mode n = 0 which is at about 120 kHz: i.e., above the cut-off frequency. The second term J = 1 will couple to the shell bending mode n = 1 which is at about 9.3 kHz; i.e., also above the cut-off frequency. Other modes with J = 2, 3 and 4 have been excited and analysed. The variation of Young’s modulus has not been investigated since the test shell was made of steel without any weld.
Figure
4. Test shell and supported
stand.
448
S. N. YOUSRI
AND F. J. FAHY
Z
Figure 5. Distortion coefficients (equation (15)).
4.2.
TEST ARRANGEMENT
Figure 6 shows a schematic diagram of the test arrangement. The Briiel and Kjaer standing wave apparatus was used, which consists of a stand, a loudspeaker, and travelling probe microphone, and a steel duct of 0.36 m length and 0.00138 m thickness which is fixed to the loudspeaker box at one end and loosely connected structurally to the test section with a ring of plasticine as seal. The transmitted vibration was damped out by housing the duct in a sand box. The test shell was also loosely connected to a termination duct which had a nearly rigid termination, A sine-random generator was used to drive the loudspeaker with a pure tone signal below the acoustic cut-off frequency of the test shell (2.6 kHz). The shell response was measured by a 0.2 gram Briiel and Kjaer accelerometer. K standing wove opporotus
0 ?I’ ’ Sme -rordom
generator
SCOp.2
Figure 6. Experimental set-up.
DISTORTED SHELL RESPONSE
449
4.3. TEST RESULTS The recorded acceleration for a slow frequency sweep of pure-tone excitation is shown in Figure 7, for frequencies up to 3 kHz. This shows that three modes were strongly excited below the cut-off frequency, which is about 2.6 kHz (the small peak at about 900 Hz is close to then = 0, s = 1 torsional mode). Each ofthese modes were identified by tuning the generator at the mode frequency and measuring the acceleration at 36 circumferential positions at midspan of the shell and at I 1 axial positions. The measured mode shapes are sho Nn in Figures 8.9 and IO. For each mode the axial centre-line pressure has been measured with the travelling probe microphone. The microphone was also moved radially to check that only a purely plane wave sound field was present. The amplitude of vibration of the standing wave duct which carried the sound from the loudspeaker to the test shell was also measured and it was found to be about 2 % in amplitude of that of the test shell. The generalized force was calculated for each mode from the measured pressure, mode shape and distortion, and the radial displacement was calculated from equation (IO). The total damping was measured by recording the decay rate of the acceleration response. The experiment was repeated with an outer concentric perspex shell of inner diameter of 0.1015 m and 0.00635 m thickness. The experitheoretical results are shown in Table 1. mental, and corresponding
Frequency
kizl
Figure 7. Acceleration response to pure-tone sweep acoustic excitation.
Figure 8. Mode shape for mode (2.1) at 2050 Hz.
450
S. N. YOUSRI AND F. J. FAHY
A
hal
Figure 9. Mode shape for mode (3,l) at 1655 Hz.
Ctrcumferenllol
3
Axlai
Figure 10. Mode shape for mode (4.1) at 1825 Hz.
TABLE 1
Normalized maximum radial displacement of the shell
Mode
I Frequency
Experimental h W n,
n, s
(Hz)
2, 1
2050
Single shell Two coaxial shells
1655
Single shell Two coaxial shells
3, 1
4, 1
1825
\
PO
Single shell Two coaxial shells
b3/N)
0.16 x lo-’
C Frequency N4
0.43 x 10-l
\ (m3/N)
2058
Single shell Two coaxial shells
0.26 x lo-’
1639
Single shell Two coaxial shells
0.24 x lo-’ 0.43 x 10-7
W __!!G PO
0.21 x 10-7 0.24 x lo-’
Theoretical A
1802
Single shell Two coaxial shells
0.38 x lo-’ 0.19 x 10-7 0.19 x lo-’ 0.46 x lo-’ 0.46 x 1O-7
DISTORTEDSHELL RESPONSE
451
4.4. CONCLUSIONS A plane wave acoustic mode can excite a non-axisymmetric shell mode directly, and not only through supports. The response is directly proportional to the distortion in a’/hE. Classical modal analysis can be used provided that the small distortion is taken into account in the generalized force term. An agreement within a factor of less than 2 between theoretical and experimental results for the ratio of the radial displacement to the excitation pressure has been obtained. The effect of an outer rigid coaxial shell is to increase the response since it eliminates radiation damping from the inner vibrating shell (see section 3), but for a shell vibrating in air the radiation damping is very small compared to the total damping.
5. SUMMARY
AND CONCLUSIONS
An axisymmetric acoustic mode can excite a non-axisymmetric shell mode due to the geometrical distortion of the shell in radius a and/or thickness h, and also due to variation of the Young’s modulus E. The theory outlined in section 2 explains this mechanism of coupling. The experimental results presented in section 4 support the theory. It is found that the amplitude of vibration, and therefore the induced stresses, varies linearly with small variations in a, h and E in proportion to the variation of a’/Eh. It is probable that circumferential variations in cylinder and constraint conditions would have similar influences to, and possibly greater than, those investigated herein.
6. FUTURE
WORK
The theory developed in section 2 shows that for small distortions in radius a, thickness h and small variation in Young’s modulus E, for a circular cylindrical shell, the response is directly proportional to the resulting variation in a”/hE. This result has been obtained by using the undistorted cylindrical shell equations and introducing the distortion into the generalized force term. The authors suggest that the static deformation of the shell in radius, thickness and straightness of the shell axis should be considered as variables (functions of shell co-ordinates) when deriving the shell equations; also variation of Young’s modulus, i.e., due to a weld, can be considered. This yields shell equations which can be solved numerically for specific cases. Circumferential variation in end constraints should also be investigated. ACKNOWLEDGMENTS Support of the research acknowledged.
by Babcock
& Wilcox
(Operations)
Limited,
is gratefully
REFERENCES 1. D. FIRTH 1975 Transactions
of the 3rd International Conference on Structural Mechanics in Reactor Technology, Paper F2/10. The vibration of a distorted circular cylinder containing fluid. 2. P. G. BENTLEYand D. FIRTH 1971 Journal of Sound and Vibration 19, 179-191. Acoustically
excited vibrations in a liquid-filled cylindrical tank. 3. P. G. BENTLEYand D. FIRTH 1973 International Symposium, Vibration problems in industry, Keswick, England, lo-12 April 1973, Paper No. 614. Some background studies of acoustic vibration in fast reactors. 4. R. F. DURRANS 1975 Personal communication. 5. M. HECKL 1958 Acustica 8, 259-265. Experimentelle
Zylindern.
untersuchungen
zur schalldammung
von
S. N.
452
YOUSRI AND F. J. FAHY
6. C. L. MORFEY1971 Seventh International Congress on Acoustics. Budapest. 24A9. Transmission through duct walls of internally propagated sound. 7. A. ROSEN and J. SINGER 1975 Israel Institute qf Technology, Department sf Aeronautical Engirleering AFOSR, TR 73-2394. Influence of asymmetric imperfections on the vibrations of axially compressed cylindrical shell. 8. A. W. LEISSA1973 NASA SP-288. Vibration of shells. 9. S. N. YOUSRI and F. J. FAHY 1976 Institute qf Sound and Vibration Research Technical Report No. 83. Circular cylindrical shell response to an internal sound field above the cut-off frequency. 10. S. N. YOUSRIand F. J. FAHY 1975 JortrnalofSoundand Vibration 40,299-306. Acoustic radiation by unbaffled cylindrical beams in multi-modal transverse vibration.
APPENDIX:
LIST OF SYMBOLS
outer and inner radii of shell mean radius of thin shell c speed of sound CP longitudinal wave velocity in the shell material D shell frequency equation Young’s modulus frequency Bessel function and its first derivative respectively Hankel function and its first derivative respectively h, ho thickness and mean thickness of shell k acoustic wavenumber n. s circumferential and axial structural mode numbers P acoustic pressure Pi excitation pressure P, radiated pressure PO pressure amplitude 4 wavenumber (q2 = k* - k$ time u, v, c: axial, circumferential and radial displacement, respectively u n. 5, V.,.s, ..W”>.S axial, circumferential and radial displacement amplitude respectively axial circumferential and radial acceleration, respectively u, v, w =, r, @ cylindrical co-ordinates P h;/12a: Poisson’s ratio P fluid density Ps shell density A fluid loading term frequency (Q = wa/c,) Q non-dimensional radiation loss factor had total loss factor ROT Dirac delta function; distortion in a, h or E s a0
442
N. YOUSRI AND F. J. FAHY
the generalized force acting is zero. Hence Firth’s theory does not predict quantitatively the response amplitude. The theory presented in the next section shows that if there is a small variation in the shell thickness, radius and/or Young’s modulus, these variations will couple the acting axisymmetric acoustic pressure to a non-axisymmetric vibrational shell mode. Rosen and Singer [7] calculated the effect of distortion on the vibration modes of cylindrical shell; they found that the influence of the initial geometrical imperfections (in radius) depends strongly on the mode of the initial imperfection and its relation to the mode of vibration, and that the imperfection may lower or raise the frequency. Ideally one should consider the shell thickness, radius and modulus of elasticity as variables [8]; this yields equations wherein terms containing these quantities must be differentiated once or twice with respect to shell co-ordinates. The resulting set of differential equations is essentially intractable. For small variations in thickness, radius and/or Young’s modulus constant average values can be used in the final perfect circular shell equations and then the coupling term (generalized force term) between excitation plane wave sound and the shell modal response can be calculated by taking into account these small variations. The theory presented in the next section is based on the above approach.
2. SHELL EQUATIONS The motion of a geometrically Reissner-Naghdi-Berry equations
l+Yu 2a
I
‘@
OF MOTION perfect circular [8] as;
WITH SMALL DISTORTION cylindrical
by the
.
1-y(]+p2)v .zZ+‘+BZ 2
shell can be described
-$=O,
(lb)
a2
where a is the shell outer radius, C, is the speed of longitudinal waves in the shell material, Ci = E/p,(l - r’), E is the Young’s modulus, h is the shell thickness, U, V, Ware the axial, circumferential and radial displacements, respectively. ii, v, ware the corresponding accelery is Poisson’s ratio, /I = ho/2/TZao, ation components, Z, r, cf, are the clyindrical co-ordinates, and the suffixes z and @ refer to differentiation in the axial and circumferential directions, respectively. (A list of symbols for reference is given in the Appendix.) P is the total pressure acting in the shell surfaces and can be expressed as P = Pi + Prt where Pi is the excitation which is given by
plane wave (blocked)
(2)
pressure below the acoustic cut-off frequency,
Pi = P,(z) e-‘O’
(3)
DISTORTED
and P, is the pressure radiated form have been assumed :
from the vibrating
U=
443
SHELL RESPONSE
shell. Modal
solutions
of the following
2 Un,,cos(k,z)cos(n@)e-‘“‘, n. s
V = 2 V,, s sin (k,z)sin n. 5 W=
(4a)
(n@) eC’O’,
(4b)
2 W,,,sin(k,z)cos(n@)e-‘“‘. n. s
(4c)
The pressure radiated, P,, is of the same form as the radial displacement, of vibration given by equation (4~). P, is given by (see section 3)
W. For the form
P, = 2 W,,,sin(k,z)cos(n@)o’pade-‘“‘, n. s where A is the fluid loading term (see section 3). Substitution into equations (1) gives 1-Y
Q2_k2az-s
2
]+Y n2 U,, + - 2
Q2-(1
+BZ)nz-
(5) of equations
(2), (3), (4) and (5)
k,anV,,,+yk,aW,,,=O,
(T)(l
(6a)
+8’)k$a2]V”,s
- [n + /?‘(kf a2 n + n3)] W,, s = 0,
(6b)
2 {-yk, a(U.,A W,,J + b + B’(kZa2n + n”)l(V, ,I W,, A
n.
s
+
[1+
B’(kz a2 + n2)2 - a’( 1 - pad/p, h)]} W,, s sin (k, z) cos (n@) = {( 1 - r2) a2/Eh} P,, (z), (6~)
where k, is the structural wave number, k, = m/L, L is the shell length, n is the ferential mode number, s is the shell axial mode number, p is the fluid density, density and Sz = ma/C,,. By eliminating U,,, J W,, sand V,,, J W,, 5from equation (6c), by using equations the solution for the modal displacement W,, s can be obtained by multiplying equation (6~) by the shell mode shape [sin(k,z)cos(n@)] and integrating over the with use being made of the modes’ orthogonality properties: this gives
shell circumps is the shell (6a) and (6b), both sides of shell surface,
-yk, a
2n
L (1
=
-
YZ)
----a’P,,(z) 0 0
Eh
sin(k,z)cos(n@)dzd@.
(7)
The right-hand side of equation (7) shows that a plane wave acoustic mode (PO is a function of z only) induces zero response in a structural mode having n # 0. The shell breathing mode (n = 0) occurs above the plane wave excitation frequency: therefore there is no frequency
S.
444
N. YOUSRI AND F. J. FAHY
matching and the response of this mode is negligible compared with the response of an asymmetric distorted shell mode where frequency matching occurs. Therefore an n # 0 shell mode may respond if a, h and/or E are functions of the shell co-ordinates 2, CD. Let a = a,[1 + 6,(Z, @)I,
h = h,[l + S,(=, @)I,
where a,, 6, and SE < 1. The quantity a2/Eh appearing series as
02, $?,(I 0 0
E = EJ
on the right side of equation
4
+2a,+6,-6,)=E
0 0
(7) can be expressed
n. S
_ 2(1 - Y2)u; L 2n pa(z) A,(z) cos2 (n@) sin (k,z)dzd4 nEohoLD ss
the internal
in Fourier
(9)
[I + ,c A”(Z) cos (n@)]. n
The coefficients A, in equation (9) can be estimated by measuring radii of the shell and fitting the results to the above formulae. Substituting equation (9) into equation (7) gives w
+ a,(~, @>I, (8a, b, c)
and external
2(1 - r2) = nE LD G,
(10)
0
00
where G is the generalized
force and
D = -yk, a(Un, ,/ W,,, ,) + [n + B”(k: a2 n +
+ [l
+ fi2(k:a2
n3)1CVn.s/Wn,s!
+ n2)2 - Q2(1 - pad/p,h)].
The solutions of D = Oare the dimensionless resonance frequencies, R, for the fluid loaded shell for the mode numbers n and S. D is a complicated function of a, h and E and the effects of 6,, dh and 6, in D have been neglected since these will only shift the resonance frequencies. Damping can be introduced into the analysis by replacing E. by the complex modulus E,(l + iqToT), where nToT is the total loss factor. Some possible forms of distortions are as follows. 1. One might have a tolerance in thickness h and/or radius a of the forms h = h,(l 2 6,) and/or a = a,(1 f: S,), respectively. 2. A line weld parallel to the shell axis at Q, = Q. can contribute to all modes; this can be analysed as follows. For the thickness, h, h = ho(l + 6,) + B6( @ - Qo), where the first term represents the tolerance in thickness and the second represents the line weld; 6 is the Dirac delta function, and B is the unknown amplitude. B can be obtained by multiplying both sides by a, and integrating, to give 277
ah d@ N 2na, ho + Ba,, 5 0
B = [weld area (perpendicular The term B6(@ - Qo) can be expressed
in Fourier
to shell axis)]/ao.
series as
B6( @ - Go) = 2 b, cos @I@), ” 2n
s
Bcos(n@)6(@
6),.,
- Go)d@ =+,
= 2B = [twice the weld arealla,.
DISTORTED
SHELL RESPONSE
445
to all modes. Similarly, variation of E at the Therefore (b,),,, is constant and contributes weld can be analysed. 3. The shell end boundary constraints, if not circumferentially uniform, can cause certain non-axisymmetric modes to be excited: i.e., if the shell is supported by four stiff arms it is likely that the n = 4, 8, 12 circumferential modes can be excited with plane wave sound, provided that frequency matching occurs. 3. THE FLUID LOADING
Cd)
For a single shell, A in equation (5) can be obtained by solving the Helmholtz equation and satisfying the boundary conditions at the shell surfaces and at infinity. The radiated pressure is given by [9] P,=
5 ~~paW,,,sin(k,a)cos(n~)e-‘“‘n. s
H,(v)
---
u74
(II) I’
where q2 = k2 - kz, J, and J,, are the Bessel function of order n and its first derivative, respectively, H, and H, are the Hankel function of order n and its first derivative, respectively. Hence A is given by d, as
This is a complex quantity; the real part is the reactive component (mass or stiffness loading) and the imaginary part is the radiation damping component; the imaginary part of A, is given by 4 =
(13)
nq’u2/Fi.(qu)l”’
Figures 1 and 2 give the vales of the real and imaginary parts of A,, respectively. It can be seen that lower order circumferential modes (small n) give higher radiation damping. Also for a steel shell vibrating in air, where pips = 1*2/7700, the fluid loading can be neglected unlike the case of a steel shell vibrating in water where p/pb = 998/7700. 16 14 12 10 e6‘I-
1 4 0
I
20
3
3
2 0 IO
b
0
1
3
20
3
“ii
-16’
Figure 1. Real part of A, (equation (12)).
v
446
S. N. YOUSRI
AND F. J. FAHY
Figure 2. Radiation loss factor qrad for an infinite circular shell vibrating in an n circumferential with an axial wave number k,.
mode and
For two concentric shells, on the assumption that the outer one is rigid and the inner one is vibrating with a radial displacement, W, as given in equation (4), then the value of A = A2 is a real quantity and is given by
J&4 7.W) - !tW) Y&4
A
z
Jn(q4 T”in(44 - uqw
Uqa)
1 ’
16 n=
I
21
14 -
Figure 3. Fluid loading function for two concentric shells (equation (14)).
(14)
DISTORTED
SHELL RESPONSE
447
where b is the radius of the outer rigid shell, and Y, and P, are the Neumann function of order n and its first derivative, respectively. Figure 3 gives the value of A, for b/a = 1.2. Equations (1 I), (12), (13) and (14)arefortheinfiniteshellcase. Forafinitevibratinglength of a shell the radiation equation is in an integral form (see reference [9]), but above the axially coincident wave numbers (k > k,), the radiation from an infinite shell approaches that of the finite case [lo].
4. EXPERIMENTAL
INVESTIGATION
Experiments were performed to illustrate that a plane wave acoustic mode can excite an n # 0 shell mode directly and not through the supports, and also to confirm quantitatively the theory developed for the shell modal response with first order shell distortion affect. 4.1.
THE TEST SPECIMEN
The steel shell tested is shown in Figure 4. It has the following dimensions: shell average diameter 0.076 m, shell average thickness 0.000607 m, shell length 0.198 m. The shell has flanged ends welded to it and was supported at both ends independently of the excitation system. The distortion in radius and thickness were measured at four axial stations and eight circumferential stations. The results for aZih were expressed in the Fourier series in the form a2 4 -=-
h
ho
1+ i
A,(z) cos(J@)
(1%
.I=0
where the coefficients A,(z) are shown in Figure 5; only four terms have been considered since higher terms correspond to acoustic modes above the cut-off frequency. The first term with J = 0 will couple to the shell breathing mode n = 0 which is at about 120 kHz: i.e., above the cut-off frequency. The second term J = 1 will couple to the shell bending mode n = 1 which is at about 9.3 kHz; i.e., also above the cut-off frequency. Other modes with J = 2, 3 and 4 have been excited and analysed. The variation of Young’s modulus has not been investigated since the test shell was made of steel without any weld.
Figure
4. Test shell and supported
stand.
448
S. N. YOUSRI
AND F. J. FAHY
Z
Figure 5. Distortion coefficients (equation (15)).
4.2.
TEST ARRANGEMENT
Figure 6 shows a schematic diagram of the test arrangement. The Briiel and Kjaer standing wave apparatus was used, which consists of a stand, a loudspeaker, and travelling probe microphone, and a steel duct of 0.36 m length and 0.00138 m thickness which is fixed to the loudspeaker box at one end and loosely connected structurally to the test section with a ring of plasticine as seal. The transmitted vibration was damped out by housing the duct in a sand box. The test shell was also loosely connected to a termination duct which had a nearly rigid termination, A sine-random generator was used to drive the loudspeaker with a pure tone signal below the acoustic cut-off frequency of the test shell (2.6 kHz). The shell response was measured by a 0.2 gram Briiel and Kjaer accelerometer. K standing wove opporotus
0 ?I’ ’ Sme -rordom
generator
SCOp.2
Figure 6. Experimental set-up.
DISTORTED SHELL RESPONSE
449
4.3. TEST RESULTS The recorded acceleration for a slow frequency sweep of pure-tone excitation is shown in Figure 7, for frequencies up to 3 kHz. This shows that three modes were strongly excited below the cut-off frequency, which is about 2.6 kHz (the small peak at about 900 Hz is close to then = 0, s = 1 torsional mode). Each ofthese modes were identified by tuning the generator at the mode frequency and measuring the acceleration at 36 circumferential positions at midspan of the shell and at I 1 axial positions. The measured mode shapes are sho Nn in Figures 8.9 and IO. For each mode the axial centre-line pressure has been measured with the travelling probe microphone. The microphone was also moved radially to check that only a purely plane wave sound field was present. The amplitude of vibration of the standing wave duct which carried the sound from the loudspeaker to the test shell was also measured and it was found to be about 2 % in amplitude of that of the test shell. The generalized force was calculated for each mode from the measured pressure, mode shape and distortion, and the radial displacement was calculated from equation (IO). The total damping was measured by recording the decay rate of the acceleration response. The experiment was repeated with an outer concentric perspex shell of inner diameter of 0.1015 m and 0.00635 m thickness. The experitheoretical results are shown in Table 1. mental, and corresponding
Frequency
kizl
Figure 7. Acceleration response to pure-tone sweep acoustic excitation.
Figure 8. Mode shape for mode (2.1) at 2050 Hz.
450
S. N. YOUSRI AND F. J. FAHY
A
hal
Figure 9. Mode shape for mode (3,l) at 1655 Hz.
Ctrcumferenllol
3
Axlai
Figure 10. Mode shape for mode (4.1) at 1825 Hz.
TABLE 1
Normalized maximum radial displacement of the shell
Mode
I Frequency
Experimental h W n,
n, s
(Hz)
2, 1
2050
Single shell Two coaxial shells
1655
Single shell Two coaxial shells
3, 1
4, 1
1825
\
PO
Single shell Two coaxial shells
b3/N)
0.16 x lo-’
C Frequency N4
0.43 x 10-l
\ (m3/N)
2058
Single shell Two coaxial shells
0.26 x lo-’
1639
Single shell Two coaxial shells
0.24 x lo-’ 0.43 x 10-7
W __!!G PO
0.21 x 10-7 0.24 x lo-’
Theoretical A
1802
Single shell Two coaxial shells
0.38 x lo-’ 0.19 x 10-7 0.19 x lo-’ 0.46 x lo-’ 0.46 x 1O-7
DISTORTEDSHELL RESPONSE
451
4.4. CONCLUSIONS A plane wave acoustic mode can excite a non-axisymmetric shell mode directly, and not only through supports. The response is directly proportional to the distortion in a’/hE. Classical modal analysis can be used provided that the small distortion is taken into account in the generalized force term. An agreement within a factor of less than 2 between theoretical and experimental results for the ratio of the radial displacement to the excitation pressure has been obtained. The effect of an outer rigid coaxial shell is to increase the response since it eliminates radiation damping from the inner vibrating shell (see section 3), but for a shell vibrating in air the radiation damping is very small compared to the total damping.
5. SUMMARY
AND CONCLUSIONS
An axisymmetric acoustic mode can excite a non-axisymmetric shell mode due to the geometrical distortion of the shell in radius a and/or thickness h, and also due to variation of the Young’s modulus E. The theory outlined in section 2 explains this mechanism of coupling. The experimental results presented in section 4 support the theory. It is found that the amplitude of vibration, and therefore the induced stresses, varies linearly with small variations in a, h and E in proportion to the variation of a’/Eh. It is probable that circumferential variations in cylinder and constraint conditions would have similar influences to, and possibly greater than, those investigated herein.
6. FUTURE
WORK
The theory developed in section 2 shows that for small distortions in radius a, thickness h and small variation in Young’s modulus E, for a circular cylindrical shell, the response is directly proportional to the resulting variation in a”/hE. This result has been obtained by using the undistorted cylindrical shell equations and introducing the distortion into the generalized force term. The authors suggest that the static deformation of the shell in radius, thickness and straightness of the shell axis should be considered as variables (functions of shell co-ordinates) when deriving the shell equations; also variation of Young’s modulus, i.e., due to a weld, can be considered. This yields shell equations which can be solved numerically for specific cases. Circumferential variation in end constraints should also be investigated. ACKNOWLEDGMENTS Support of the research acknowledged.
by Babcock
& Wilcox
(Operations)
Limited,
is gratefully
REFERENCES 1. D. FIRTH 1975 Transactions
of the 3rd International Conference on Structural Mechanics in Reactor Technology, Paper F2/10. The vibration of a distorted circular cylinder containing fluid. 2. P. G. BENTLEYand D. FIRTH 1971 Journal of Sound and Vibration 19, 179-191. Acoustically
excited vibrations in a liquid-filled cylindrical tank. 3. P. G. BENTLEYand D. FIRTH 1973 International Symposium, Vibration problems in industry, Keswick, England, lo-12 April 1973, Paper No. 614. Some background studies of acoustic vibration in fast reactors. 4. R. F. DURRANS 1975 Personal communication. 5. M. HECKL 1958 Acustica 8, 259-265. Experimentelle
Zylindern.
untersuchungen
zur schalldammung
von
S. N.
452
YOUSRI AND F. J. FAHY
6. C. L. MORFEY1971 Seventh International Congress on Acoustics. Budapest. 24A9. Transmission through duct walls of internally propagated sound. 7. A. ROSEN and J. SINGER 1975 Israel Institute qf Technology, Department sf Aeronautical Engirleering AFOSR, TR 73-2394. Influence of asymmetric imperfections on the vibrations of axially compressed cylindrical shell. 8. A. W. LEISSA1973 NASA SP-288. Vibration of shells. 9. S. N. YOUSRI and F. J. FAHY 1976 Institute qf Sound and Vibration Research Technical Report No. 83. Circular cylindrical shell response to an internal sound field above the cut-off frequency. 10. S. N. YOUSRIand F. J. FAHY 1975 JortrnalofSoundand Vibration 40,299-306. Acoustic radiation by unbaffled cylindrical beams in multi-modal transverse vibration.
APPENDIX:
LIST OF SYMBOLS
outer and inner radii of shell mean radius of thin shell c speed of sound CP longitudinal wave velocity in the shell material D shell frequency equation Young’s modulus frequency Bessel function and its first derivative respectively Hankel function and its first derivative respectively h, ho thickness and mean thickness of shell k acoustic wavenumber n. s circumferential and axial structural mode numbers P acoustic pressure Pi excitation pressure P, radiated pressure PO pressure amplitude 4 wavenumber (q2 = k* - k$ time u, v, c: axial, circumferential and radial displacement, respectively u n. 5, V.,.s, ..W”>.S axial, circumferential and radial displacement amplitude respectively axial circumferential and radial acceleration, respectively u, v, w =, r, @ cylindrical co-ordinates P h;/12a: Poisson’s ratio P fluid density Ps shell density A fluid loading term frequency (Q = wa/c,) Q non-dimensional radiation loss factor had total loss factor ROT Dirac delta function; distortion in a, h or E s a0