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VIBRATIONS OF FLUID-FILLED HERMETIC CANS
Journal of Fluids and Structures (2000) 14, 235}255 doi: 10.1006/j#s. 1999.0267, available online at http://www.idealibrary.com on
VIBRATIONS OF FLUID-FILLED HERMETIC CANS M. AMABILI Department of Industrial Engineering, University of Parma, Parco Area delle Scienze 181/A Parma, 43100, Italy (Received 3 February 1999, and in "nal form 21 September 1999)
Free, conservative vibrations of a hermetic can, simply supported at the base, are studied. The can is composed by a circular cylindrical shell and two identical circular plates connected to the shell at its ends. The arti"cial spring method, which is an extension of the classical Rayleigh}Ritz method, is used to solve the system by using substructuring. The can is studied empty and "lled with an inviscid and incompressible #uid. Fluid volume conservation is applied. The interaction between the plates and the shell via the #uid is considered, and exact expressions for the #uid velocity potential are used. The e!ect of #exibility of joints between plates and shell is investigated. Results for a #uid-"lled, simply supported shell closed by rigid ends are also obtained and compared to the classical open-end shell. ( 2000 Academic Press
1. INTRODUCTION HERMETIC CANS ARE LARGELY EMPLOYED as containers of liquids. They are composed of a circular cylindrical shell, joined to two circular end-plates. Their modal behaviour has been investigated in the empty case by several researchers. Hirano (1969) studied axisymmetric vibrations of thin drums (cans) by minimizing the Lagrangian; this approach was extended to other modes by Takahashi and Hirano (1970). Takavoli & Singh (1989, 1990) studied theoretically a hermetic can by using the state-space method, which is a transfermatrix-based substructuring method; they also investigated the same can experimentally and by using a FEM code. Huang & Soedel (1993) applied the receptance method to the study of free vibrations of circular plates welded to a cylindrical shell, including the hermetic can case. A similar problem has been investigated by Yuan & Dickinson (1994) by using the arti"cial spring method which is an extension of the classical Rayleigh}Ritz method, suitable for substructuring. This method had been developed by Yuan and Dickinson (1992) and Cheng & Nicolas (1992). The arti"cial spring method has been extended to study #uid}structure interaction by Cheng (1994) in the case of a light-#uid approximation and by Amabili (1997a, b) in the case of dense #uid (liquid). The only work related to vibrations of cans in contact with dense #uid is that of Harari et al. (1994), studying a sti!ened and submerged cylindrical shell closed by two end-plates and immersed in unbounded #uid. They neglected the interaction between the plates and the shell via the #uid and used the expression of the #uid load on the end-plates obtained for circular plates with a rigid extension (circular plate in an in"nite ba%e). No studies are available for free vibrations of liquid-"lled cans. Otherwise, liquid-"lled tanks composed of a circular cylindrical shell and a #exible or rigid bottom plate have been widely studied, e.g. by Amabili et al. (1998), Amabili (1997a), Chiba (1996), Gonialves & Ramos (1996), Yamaki et al. (1984); however, in these studies there exists a free liquid surface and hence they are only partially related to the present case. 0889}9746/00/020235#21 $35.00/0
( 2000 Academic Press
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The present study investigates free, conservative vibrations of a hermetic can, simply supported at the base. The can is composed of a circular cylindrical shell and two identical circular plates connected to the shell at its ends. The use of arti"cial springs at the junctions allows us to study vibrations of this relatively complex system. In particular, the joints between the components of the structure are replaced by arti"cial springs that are distributed along the entire joint length. Only rotation at the shell}plate joints is coupled by rotational springs. The can is "rst studied empty and then "lled with an inviscid and incompressible #uid. Fluid volume conservation is applied. The interaction between the plates and the shell via the #uid is considered and exact expressions of the #uid velocity potential are used. The e!ect of #exibility of joints between plates and shell is also investigated. 2. THE EMPTY CAN The system studied is composed of a circular cylindrical shell and two circular plates connected to the shell at its ends. The bottom plate is assumed to be simply supported, in order to prevent rigid-body modes of the system. When a plate is joined to a circular cylindrical shell, in general three displacements and two slope connections could be considered, according to classical thin shell theory. However, if one only investigates lower modes of the system, the plates can be assumed inelastic in their plane. Moreover, in#uences of connection de#ections in the tangential planes of the shell can be neglected with respect to transverse amplitudes. Actually, for very thick plates, the shell axial constraint can be important and it should be properly modelled, but this is very limited for thin plates. Therefore, only the radial slope at the plate boundary can be considered to be coupled to the axial slope of the shell at the ends. A similar approach was used by Cheng & Nicolas (1992), Cheng (1994), Amabili (1997a) and Amabili et al. (1998). Huang & Soedel (1993) used two connections when the plate was not connected to the shell simple support. In the present case, only one connection is used. In fact, the joints between the shell and the plates give a reciprocal constraint that can be modelled as a simple support. Therefore, their reciprocal rotation at the edges is joined by introducing arti"cial rotational springs of appropriate sti!ness. The sti!nesses of these springs are chosen to be very high with respect to the sti!ness of the #exible components of the can, in order to simulate a welded connection between the shell and the plates. Moreover, the e!ect of the elasticity of the joint is investigated by using the present model. The top plate gives to the top end of the shell a constraint that can be conveniently modelled with a simple support for all the modes of vibration without movement of the can longitudinal axis. On the other hand, for beam-bending modes of the can, the cross-section of the shell at the top end remains circular but it moves with respect to the base, so that the top plate does not constrain these modes. Hence, these modes may correctly be studied by considering a free top-end of the shell component. In the present study, the shell is considered to be simply supported at both ends, so that this model cannot be used to study the beam-bending modes of cans simply supported only at the base. However, for the usual geometry of cans, lower shell-modes present more than one circumferential wave and can be conveniently studied by using the present model. The hermetic can considered has radius a and height ¸, and is conveniently studied in the polar coordinates (x, r, h); the origin O of the coordinate system is placed on the can axis at half-distance from the end-plates (Figure 1). Due to the axial symmetry of the structure, only modes of the shell and the plates with the same number n of nodal diameters (or circumferential waves) are coupled. A nodal diameter is a diameter in the cross-section of the shell (or in the plane of the plate) connecting points which are immobile during vibration.
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
237
Figure 1. Geometry of the can and coordinate system.
Besides, it is interesting to note that, due to axial symmetry, for each asymmetric mode (n'0) there exists a second mode having the same frequency and shape but angularly rotated by n/2n. In the present study identical end-plates are considered, so that the system is symmetric with respect to the plane x"0; therefore, it displays only symmetric and antisymmetric modes, in a radial section of the can, with respect to this plane. The Rayleigh}Ritz method (Amabili 1997b) is applied to "nd natural frequencies and modes of the #uid-"lled can, the time variation being assumed to be harmonic. Therefore, the mode shapes w of the shell wall in the radial direction (Figure 1) can be expressed as follows:
G
= cos (nh) + q cos [(2s!1)nx/¸] symmetric modes s s/1 w(x,h)" = cos (nh) + q sin (2snx/¸) antisymmetric modes, s s/1
(1)
where n and s are the number of nodal diameters and of axial half-waves, respectively, and q are the parameters of the Ritz expansion. The eigenvectors of the empty simply supported s cylindrical shell are used as admissible functions. The mode shapes w of the bottom circular plate in the transverse direction of the plate B can be given as (Amabili 1997a) = w (r, h)"cos (nh) + b [A J (j r/a)#C [I (j r/a)] , (2) B i n,i n n,i n,i n n,i i/0 where n and i are the number of nodal diameters and circles, respectively, a is the plate radius, b are the parameters of the Ritz expansion, and j is the well-known frequency i n,i parameter that is related to the plate natural frequency; J and I are the Bessel function and n n
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modi"ed Bessel function of order n, respectively. In equation (2), the in vacuo eigenfunctions of the simply supported circular plate are assumed as admissible functions. The trial functions are linearly independent and constitute a complete set. Values of j for simply n,i supported plates are given, e.g., by Leissa & Narita (1980). Similarly, mode shapes w of the top circular plate can be given as T = w (r, h)"cos (nh) + h [A J (j r/a)#C I (j r/a)] , (3) T i n,i n n,i n,i n n,i i/0 where h are the parameters of the Ritz expansion. To simplify the computations, the mode i shape constants, A and C , are normalized in order to have (Wheelon 1968) n,i n,i 1 : [A J (j o)#C I (j o)]2 o do n,i n n,i n,i n n,i 0 A2 n2 C2 n2 " n,i (J@ (j ))2# 1! J2 (j ) ! n,i (I@ (j ))2! 1# I2 (j ) (4) n n,i n n,i n n,i n n,i 2 j2 2 j2 n,i n,i A C # n,i n,i [J (j )I (j )#I (j ) J (j )] " 1, n n,i n`1 n,i n n,i n`1 n,i j n,i where o " r/a and J@ and I@ indicate the derivatives of J and I with respect to the n n n n argument. The ratio of the mode shape constants for simply supported plates is
G C
A
B
D
C
H
A
B
D
A /C "!I (j )/J (j ). (5) n,i n,i n n,i n n,i It is interesting to note that, for the system of Figure 1, the following conditions can be imposed: w "w symmetric; T B By using equations (6) we get
w "!w antisymmetric modes. T B
(6a, b)
h "b symmetric; h "!b antisymmetric modes. (7a, b) i i i i In order to solve the problem, we evaluate the kinetic and potential energies of the shell, plates and #uid. The reference kinetic energy ¹* of the shell, neglecting the tangential and S rotary inertia, is given by 1 2n L@2 1 = ¹*" o h : : w2dx a dh" o h a (¸/2)t + q2 , S 2 S S s n 2 S S 0 ~L@2 s/1 where h is the shell thickness, o is the density of the shell material (kg/m3) and S s 2n for n"0 t" . n n for n'0
G
(8)
H
In equation (8) the orthogonality of the sine function in the interval [!¸/2, ¸/2] has been used. Similarly, the reference kinetic energy ¹* of the plates is given by p = 1 2n a (9) ¹*" o h : : (w2#w2 ) r dr dh " o h a2t + b2, i T P P n P 2 P P B i/0 0 0 where h is the plate thickness and o is the density of the plate material (kg/m3). In P P equation (9) the orthogonality of Bessel functions (plate mode shapes) in [0, a] has been used.
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VIBRATIONS OF FLUID-FILLED HERMETIC CANS
It should be noted that the maximum potential energy of each mode of the empty shell (without the end plates) is equal to the reference kinetic energy of the same mode multiplied by the squared circular frequency u 2 of this mode. Moreover, due to the series expansion of s the mode shape, the potential energy is the sum of the energies of each single component mode. As a consequence, the maximum potential energy of the shell may be expressed as = (10) < "1 o h a(¸/2) t + q2u2 , s s n S 2 S S s/1 where u are the circular frequencies of the symmetric and antisymmetric #exural modes of s the simply supported shell that can be computed by using, for example, the FluK gge theory of shells (Leissa 1973). Similarly, the maximum potential energy of the two end-plates is the sum of the reference kinetic energies of the eigenfunctions of the plates in vacuum multiplied by uJ 2 : n,i D = = < "o h a2t + b2uJ 2 " t + b2 j4 , (11) i n,i a2 n P P P n i n,i i/0 i/0 where the plate circular frequency uJ 2 is related to the frequency parameter j by n,i n,i uJ "(j2 /a2) [D/(o h )]1@2 and D"E h3/[12(1!l2)] is the #exural rigidity of the plates; P n,i P P P P n,i l and E are the Poisson ratio and Young's modulus of the plates, respectively. P P In a can section containing the can axis, the rotations at the plate edges are considered joined to the rotations at the shell edges, as previously discussed. The shell and the plates are connected together by arti"cial rotational springs of appropriate sti!ness, continuously distributed around the circumference. The maximum potential energy of the two distributed rotational springs connecting the two end-plates and the shell is 1 < " C 2
2n
P GCA B Lw Lx
c
A B D CA B
2 Lw B # # Lr x/~L@2 r/a
Lw Lx
A B DH
2 Lw T ! a dh , (12) Lr x/L@2 r/a
0 where c is the spring sti!ness (N m/m). It should be noted that only the variation of the angle between the shell and the plates contributes to the potential energy stored by the rotational spring. Simple calculations for symmetric modes give
G
j j n2 = = = = + + q q (!1)s(!1)j(2s!1)(2j!1)# + + b b n,i n,h < "cat s j i h a a C n ¸2 s/1 j/1 i/0 h/0 ][A J@ (j )#C I@ (j )]][A J@ (j )#C I@ (j )] n,h / n,h n,h / n,h n,i / n,i n,i / n,i n = = j !2 + + q b (!1)s (2s!1) n,i [A J@ (j )#C I@ (j )] , n,i / n,i s i n,i / n,i ¸ a s/1 i/0 and for antisymmetric modes
H
G
(13a)
4n2 = = = = j j + + q q (!1)s(!1)jsj# + + b b n,i n,h [A J@ (j )#C I@ (j )] n,i / n,i s j i h a a n,i / n,i ¸2 s/1 j/1 i/0 h/0 4n = = j ][A J@ (j )#C I@ (j )]# + + q b (!1)ss n,i [A J@ (j )#C I@ (j )] . n,h n n,h n,h n n,h n,i n n,i s i n,i n n,i ¸ a s/1 i/0 (13b)
< "cat C n
H
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M. AMABILI
3. DYNAMIC BEHAVIOUR OF THE FLUID-STRUCTURE INTERACTION The can is considered completely "lled with an inviscid and incompressible #uid. Hydrostatic pressure e!ects and damping are neglected in the present study. For an incompressible and inviscid #uid, the deformation potential satis"es the Laplace equation
+ 2/ (r, h, x)"0.
(14)
The deformation potential / is related to the #uid velocity potential /I by /I (x, h, r, t)"iu/e*ut,
(15)
which is assumed to be harmonic; i is the imaginary unit and u is the natural circular frequency of vibration. The velocity of the #uid v is related to /I by v"!grad /I . The #uid deformation potential, using the principle of superposition, can be divided into /"/(1)#/(2),
(16)
where /(1) describes the velocity potential of the #uid associated with the #exible shell while considering the plates as rigid, and /(2) describes the #uid velocity potential for the #exible plates while considering the shell as rigid. The boundary conditions imposed to the #uid for the two complementary boundary value problems are (Amabili 1997b)
A B L/(1) Lx
A B L/(2) Lx
A B A B
"0, x/$L@2
"!w (r, h), B x/~L@2
L/(2) Lx
L/(1) Lr
"!w(x, h), r/a
"w (r,h), T x/L@2
(17a, b)
A B L/(2) Lr
"0.
(18a}c)
r/a
Equations (17) and (18) express the contact condition between a #uid (without cavitation) and a #exible or rigid wall. The reference kinetic energy ¹* of the #uid, by using the Greens F theorem (e.g. Amabili 1997b), is L/ 1 L/ 1 L/ 1 L/ :: / dS" o :: / dS! o :: / dS# o :: / dS F F F Lz 2 Lr 2 Lx 2 Lx SS`SB`ST SS SB ST 1 1 "! o :: (/(1)#/(2)) w dS# o :: (/(1)#/(2)) w dS P 2 F 2 F SS SB`ST "¹* #¹* #¹* #¹* , (19) F1 F1~2 F2~1 F2
1 ¹*" o F 2 F
where o is the mass density of the #uid, w indicates w when integrating on S and F P B B w when integrating on S , z is the direction normal at any point on the boundary surface of T T the liquid domain and is pointed outwards, S is the shell lateral surface, S is the bottom S B plate surface and S is the surface of the top plate. T 3.1. FLUID-SHELL INTERACTION In this section, the #uid}structure interaction of a simply supported, circular cylindrical and #exible shell of a can with rigid end-plates is considered. A large number of papers on the vibrations of partially #uid-"lled shells have been published [e.g. Berry & Reissner (1958), Lakis & PamK doussis (1971), Bauer & Siekmann (1971), Yamaki et al. (1984), Amabili & Dalpiaz (1995), Gonialves & Ramos (1996), and Chiba (1996)].
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VIBRATIONS OF FLUID-FILLED HERMETIC CANS
The liquid deformation potential /(1) is assumed to be of the form (Amabili 1997a) = /(1)" + q U(1) . s s s/1
(20)
The functions U(1) are given by s
G
C
D
= cos (nh) A rn# + A I (2mnr/¸)cos(2mnx/¸) symmetric, n,s,0 n,s,m / m/1 U(1)(x,r,h)" (21) s = cos (nh) + A I ((2m!1)nr/¸) sin ((2m!1)nx/¸) antisymmetric, n,s,m / m/1 where A are coe$cients depending on the integers n, s and m. For n"0, A will be n,s,m /,4,0 assumed to be zero, because two di!erent terms must be added to the deformation potential /, as will be discussed in Section 3.4. The functions U(1) satisfy the Laplace equation and the s two boundary conditions given in equations (17a); the condition given in equation (17b) is used to compute the coe$cients A . Di!erent solutions are obtained for symmetric and n,s,m antisymmetric modes. For symmetric modes, equation (17b) gives = 2mn A nan~1# + A I@ (2mna/¸) cos (2mnx/¸)"!cos ((2s!1)nx/¸). (22) n,s,0 n,s,m ¸ n m/1 Equation (22) must be satis"ed for all values !¸/24x4¸/2. If this equation is multiplied by cos (2njx/¸) and then integrated between !¸/2 and ¸/2, using the well-known orthogonality properties of the trigonometric functions, the equation becomes
C
D
1 !2(!1)s = p(S) I (2mnr/¸) U(1)"! cos (nh) rn# + s,m n cos(2mnx/¸) , s n (2s!1)nan~1 m I (2mna/¸) n m/1
(23)
where
C
D
L@2 ¸ (!1)m~s (!1)m`s p(S) " : cos (2mnx/¸) cos ((2s!1) nx/¸) dx" ! . s,m n 1#2m!2s !1#2m#2s ~L@2 (24) Therefore, the reference kinetic energy ¹* of the #uid associated with symmetric modes of F1 the shell is 1 2n L@2 ¹* "! o : : (/(1)) wa dx dh 1 F r/a 2 F 0 ~L@2
C
D
= = 4(!1)s(!1)ja¸ a = p(S) p(S) I (2mna/¸) " to + + qq # + s,m j,m n . s j (2s!1) (2j!1) nn m I (2mna/¸) 2n n F n s/1 j/1 m/1 For antisymmetric modes, equation (17b) gives = (2m!1)n I@ ((2m!1) na/¸) sin ((2m!1)nx/¸)"!sin (2snx/¸). + A n n,s,m ¸ m/1 Similar to the symmetric case, it gives 2 = p(A) I ((2m!1)nr/¸) s,m n U(1)"! cos (nh) + sin((2m!1)nx/¸), s n 2m!1 I@ ((2m!1) na/¸) n m/1
(25)
(26)
(27)
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M. AMABILI
where
C
D
L@2 ¸ (!1)m~s (!1)m`s p(A)" : sin ((2m!1) nx/¸) sin (2snx/¸) dx" ! # . sm n !1#2m!2s !1#2m#2s ~L@2 (28) Therefore, the reference kinetic energy ¹* of the #uid associated with antisymmetric modes F1 of the shell is a = = = p(A) p(A) I ((2m!1)na/¸) ¹* " t o + + q q + sm jm n . F1 n n F s j (2m!1)I@ ((2m!1)na/¸) n s/1 j/1 m/1
(29)
3.2. FLUID-PLATE INTERACTION In this section, the #uid}structure interaction of the simply supported, #exible end-plates of a can with a rigid shell is considered. Flexible bottom plates in rigid tanks were studied, e.g., by Bhuta and Koval (1964), Bauer and Siekmann (1971), Chiba (1993) and Amabili (1997c); Bauer (1995) studied the vibration of a circular top plate in a liquid-"lled rigid container. The liquid deformation potential /(2) is assumed of the form (Amabili 1997c) = (30) /(2)" + b U(2) . i i i/0 The functions U(2), for symmetric modes, are expressed as i = (31) U(2) (x, r, h)"cos (nh) + K J (e r/a) cosh (e x/a), i n,i,k n n,k n,k k/1 where e are solutions of the following equation: n,k (32) J@ (e )"0, k"1,2,R, n n,k rejecting the "rst solution e for n"0. The functions U(2) satisfy equations (14) and (18c). i n,k The constants K are calculated in order to satisfy equations (18a, b) n,i,k = + K J (e r/a) (e /a) sinh (e ¸/(2a))"[A J (j r/a)#C I (j r/a)]. (33) n,i,k n n,k n,k n,k n,i n n,i n,i n n,i k/1 Equation (33) must be satis"ed for all values 04r4a. If this equation is multiplied by (1/a2) J (e r/a) r and integrated between 0 and a, then using the orthogonality of the Bessel n n,k functions this becomes a (A b #C c ) n,i n,i,k n,i n,i,k , K " n,i,k (a e ) sinh (e ¸/(2a)) n,k n,k n,k where, by using equation (32), one obtains (Wheelon 1968)
C A BD
1 n 2 1 a [J (e )]2, a " : J2 (e r/a)r dr" 1! n n,k n,k a2 n n,k 2 e n,k 0 1 a j n,i J@ (j )J (e ), b " : J (e r/a)J (j r/a) r dr" n,i,k a2 n n,k n n,i e2 !j2 n n,i n n,k n,k n,i 0 1 a j n,i I@ (j )J (e ). c " : J (e r/a) I (j r/a)r dr" n,i,k a2 n n,k n n,i e2 #j2 n n,i n n,k n,i n,k 0
(34)
(35) (36) (37)
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VIBRATIONS OF FLUID-FILLED HERMETIC CANS
Hence, the reference kinetic energy ¹* of the #uid associated with symmetric modes of the F2 end-plates is given by
C
D
2n a 2n a 1 : : (/(2)) w r drdh# : : (/(2)) w r drdh ¹* " o F2 2 F x/L@2 T x/~L@2 B 0 0 0 0 = = = (A b #C c ) (A b #C c ) n,h n,h,k n,h n,h,k . n,i n,i,k n,i n,i,k "o a3t + + b b + F n i h tanh (e ¸/(2a)) a e n,k n,k n,k i/0 h/0 k/1 For antisymmetric modes, equation (31) is replaced by
G
= K x# + K J (e r/a) sinh (e x/a) 0,i,0 0,i,k 0 0,k 0,k k/1 U(2) (x,r,h)" i = cos (nh) + K J (e r/a) sinh (e x/a) n,i,k n n,k n,k k/1 The constants K in this case are given by n,i,k a (A b #C c ) n,i n,i,k n,i n,i,k . K "! n,i,k (a e ) cosh (e ¸/(2a)) n,k n,k n,k In addition, the constant K is given by 0,i,0 K "!2q , 0,i,0 0,i where (Wheelon 1968)
C
(38)
for n"0, (39) for nO0.
(40)
(41)
D
A C 1 a q " : [A J (j r/a)#C I (j r/a)] r dr" 0,i J (j )# 0,i I (j ) . (42) 0,i 0 0,i 0,i 0 0,i 1 0,i 1 0,i 0,i a2 j j 0,i 0,i 0 Hence, the reference kinetic energy ¹* of the #uid associated with antisymmetric modes of F2 the end plates is given by
G
A B
e ¸ = = = (A b #C c ) n,i n,i,k n,i n,i,k (A b #C c ) tanh n,k , o a3t + + b b + F n i h n,h n,h,k n,h n,h,k 2a a e n,k n,k i/0 h/0 k/1 nO0
C
¸ = = = (A b #C c ) 0,i 0,i,k 0,i 0,i,k ¹* " o a3 t + + b b q q #+ 2 F F 0 i h a 0,i 0,h a e 0,k 0,k i/0 h/0 k/1 e ¸ ](A b #C c ) tanh 0,k , 0,h 0,h,k 0,h 0,h,k 2a
A BD
n "0. (43)
3.3. SHELL-PLATE INTERACTION VIA THE FLUID Equation (19) shows that in the reference kinetic energy of the #uid ¹*, two terms appear F because of the coupling e!ect of the #uid. In fact, even if the coupling springs between the plates and the shell were eliminated, the vibration of the two-end plates and the shell would remain coupled by the #uid inside the can. In particular, the quantity ¹* , for symmetric F1~2
244
M. AMABILI
modes, is given by 1 2n L@2 1 = = = ¹* "! o : : (/(2)) wa dx dh"! o at + + q b + K J (e ) f(S) , F1~2 r/a s i n,i,k n n,k n,s,k 2 F 2 F n 0 ~L@2 s/1 i/0 k/1 (44) where the constants K are given by equation (34) and n,i,k L@2 !2(2s!1)(!1)sna2¸ cosh (e ¸/(2a)) n,k f(S) " : cos((2s!1)nx/¸)cosh (e x/a) dx" . n,s,k n,k (2s!1)2n2a2#¸2 e2 n,k ~L@2 (45) The quantity ¹* , for antisymmetric modes, is given by F1~2
G
1 = = = ! o at + + q b + K J (e ) f(A) , s i n,i,k n n,k n,s,k 2 F n s/1 i/0 k/1 ¹* " F1~2 1 = = = ! o at + + q b !2q f(A)# + K J (e ) f(A) , 0,i,k 0 0,k 0,s,k s i 0,i 0,s 2 F 0 k/1 s/1 i/0
C
D
nO0, (46) n"0,
where the constants K are given by equation (40) and n,i,k L@2 !(!1)s ¸2 f(A)" : x sin (2snx/¸) dx" , (47) 0,s 2sn ~L@2 L@2 !4s(!1)sna2¸ sinh (e ¸/(2a)) n,k f(A) " : sin (2snx/¸) sinh (e x/a) dx" . (48) n,s,k n,k 4s2n2a2#¸2e2 n,k ~L@2 The component ¹* of the reference kinetic energy of the #uid has the following expression F2~1 for symmetric modes:
C
D
2n a 2n a 1 : : (/(1)) w r dr dh# : : (/(1)) w r dr dh ¹* " o F2~1 2 F x/~L@2 B x/L@2 T 0 0 0 0 !2(!1)s a3 1 = = "! o t + + q b (A J (j )#C I (j )) s i (2s!1)nj n,i n`1 n,i n,i n`1 n,i n F n n,i s/1 i/0 = (!1)m p(S) s,m (A m(S) #C k(S) ) , #+ n,i n,i,m mI@ (2mna/¸) n,i n,i,m m/1 n where the constants p(S) are given in equation (24), and (Wheelon 1968) s,m a m(S) ": I (2mnr/¸)J (j r/a)r dr n,i,m n n n,i 0 2mna3¸I (2mna/¸)J (j )#a2¸2j I (2mna/¸)J (j ) n`1 n n,i n,i n n`1 n,i , " 4m2n2a2#¸2j2 n,i a k(S) ": I (2mnr/¸)I (j r/a)r dr n,i,m n n n,i 0 2mna3¸I (2mna/¸) I (j )!a2¸2j I (2mna/¸)I (j ) n`1 n n,i n,i n n`1 n,i . " 4m2n2a2!¸2j2 n,i
C
D
(49)
(50)
(51)
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
245
The term ¹* for antisymmetric modes is given by F2~1 2 = = = (!1)mp(A) s,m (A m(A) #C k(A) ) , (52) ¹* "! o t + + q b + 2~1 n,i n,i,m F s i (2m!1)I@ ((2m!1)na/¸) n,i n,i,m n F n n s/1 i/0 m/1 where the constants p(A) are given in equation (28); m(A) and k(A) are obtained replacing 2m n,i,m n,i,m s,m by (2m-1) in equations (50) and (51), respectively. 3.4. SYMMETRIC MODES FOR n"0 AND FLUID VOLUME CONSERVATION In the case of symmetric modes and n"0 the condition for the conservation of the volume of the contained incompressible #uid must be imposed; in all the other cases it is automatically satis"ed by the assumed mode shapes. It is given by = 2n L@2 = 2n a + q : : cos ((2s!1) nx/¸)a dx dh"2 + b : : [A J (j r/a) s i 0,i 0 0,i s/1 0 ~L@2 i/0 0 0 #C I (j r/a)] r dr dh. 0,i 0 0,i
(53)
After simple calculations, one obtains = = !(¸/n) + q (!1)s/(2s!1)"a + b q . (54) s i 0,i s/1 i/0 For symmetric modes and n"0 two additional terms must be added to the #uid deformation potential / previously calculated. The "rst term is the constant / that gives 0 an additional contribution to the reference kinetic energy of the #uid, = = o a(¸/n)t / + q (!1)s/(2s!1)#o a2t / + b q . i 0,i s F 0 0 F 0 0 i/0 s/1 By using equation (54) it is found that this contribution must be zero. The second additional term is / and is given by a / "c (r2!2x2), a 0
(55)
(56)
where c must satisfy the condition of contact with a #exible wall 0 = L@2 (L/ /Lr) "2ac "!(1/¸) + q : cos [(2s!1) nx/¸]dx a r/a 0 s s/1 ~L@2
(57)
and = a (L/ /Lx) "!2¸c "(2/a2) + b : [A J (j r/a)#C I (j r/a)]r dr. a x/L@2 0 i 0,i 0 0,i 0,i 0 0,i i/0 0 From equations (57) and (58) we obtain, respectively,
(58)
= = c "[1/(an)] + q (!1)s/(2s!1) , c "!(1/¸) + b q . (59a, b) 0 s 0 i 0,i s/1 i/0 By using equation (54), we "nd that the right-hand side of equations (59a) and (59b) are equal; this result guarantees that / is e!ectively a solution of our problem. The term / is a a
246
M. AMABILI
associated with an additional reference kinetic energy given by
G
C
D
C
1 = = (!1)s (!1)j a2¸ 1 = = ¸2a2 o t + + qq #aJ ! + + bb q bI ! q F 0 n s j 2s!1 (2j!1)n i h 0,i 0,j 0,h 2 0,h ¸ s/1 j/1 i/0 h/0
DH
,
(60) where L@2 ¸3 (8!n2#4n2j!4n2j2)(!1) j aJ " : x2 cos [(2j!1)nx/¸] dx" 0,j 2n3(2j!1)3 ~L@2
(61)
and a bI ": r3 [A J (j r/a)#C I (j r/a)] dr"(a4/j2 ) MA [2J (j )!j J (j )] 0,h 0,h 2 0,h 0,h 3 0,h 0,h 0,h 0 0,h 0,h 0 0,h 0 #C [2I (j )#j I (j )]N . (62) 0,h 2 0,h 0,h 3 0,h 4. THE RAYLEIGH}RITZ METHOD In order to solve the problem, it is useful to introduce the Rayleigh quotient (Amabili 1997b; Zhu 1994) for the systems considered, which is (63) u2"(< #< #< )/(¹*#¹*#¹*). F P S S P C For the numerical calculation of the natural frequencies and the parameters of the Ritz expansion of modes, only N terms in the expansion of w, equation (1), and NM in the expansion of w and w , equations (2) and (3), are considered, where N and NM must be B T chosen large enough to give the required accuracy to the solution. Thus, all the energies are given by "nite summations. Here it is convenient to introduce a vectorial notation. The vector q of the parameters of the Ritz expansions is de"ned by qT"MMqNT, MbNTN, MqNT"Mq ,2q N, MbNT"Mb ,2, bNM !1 N . (64a}c) 1 N 0 Maximum potential energies < , < and < and reference kinetic energies ¹*, ¹* and S P S P C ¹* can be written in the following form: F ¹*d"qTMdq,
(66)
where u is the circular frequency (rad/s) of the #uid-"lled hermetic can. Equation (66) gives a linear eigenvalue problem for a real, symmetric matrix; real eigenvalues are obtained. In the case of symmetric modes, for n"0 the Galerkin equation is
C
KS#KP#KC D 0 -------------------D-----0T D 0
D
C
MS#MP#MF D MA !u2 --------------------- -D- ---/ MTA D 0 0
G H q
D
G H q
/
"0,
0
(67)
247
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
TABLE 1 Comparison of natural frequencies (Hz) computed for the studied empty can and n"3; S and P indicate shell- and plate-dominant modes, respectively Mode Symmetric
Antisymmetric
Present study 1st (S) 2nd (P) 3rd (P) 4th (S) 1st (P) 2nd (S) 3rd (P) 4th (S)
1734 2384 5124 6003 2365 4223 5177 7100
Huang & Soedel (1993)
Di!erence (%)
1735 2343 5080 5997 2326 4224 5125 7102
0)06 1)7 0)86 0)1 1)6 0)02 1)0 0)03
where the vector M has dimension (N#NM ) and represents the additional term to the A reference kinetic energy of the #uid given by equation (55) that must be zero; the last row in equation (67) expresses the conservation of the #uid volume. Matrix M includes the F contribution of equation (60) in this case. The "rst eigenvalue of equation (67) is zero; it corresponds to q"0 and must be disregarded. In addition, the pressure exerted by the #uid at a point of the can wall can be computed by using the linearized Bernoulli equation ( p) "o (L/I /Lt) "!o u2 (/) e*ut . 10*/5 F 10*/5 F 10*/5
(68)
5. NUMERICAL RESULTS AND DISCUSSION Numerical results have been carried out by means of the Mathematica software (Wolfram 1996). The following dimensions and material properties have been chosen in order to compare the results with those obtained by Huang & Soedel (1993) in the case of an empty can: mean radius a "0)1 m, ¸"0)2 m, h "h "0)002 m, E "E "206]109 Pa, S P S P o "o "7850 kg/m3 and l "l "0)3. Twenty shell and plate modes have been conS P S P sidered in the Rayleigh}Ritz expansions, and 20 terms have been included in the expansion of U(1) and U(2). They are enough to give a good accuracy, as will be shown in Section 5.2. i s The FluK gge theory of shells has been used, unless otherwise speci"ed. The plate and the shell are assumed to be joined by rotational springs of very high sti!ness c simulating a rigid joint. In the present case, it was found that a value of c"108 N is enough to simulate a rigid rotational joint; in fact, an increment to this value does not a!ect the natural frequencies of the system. The e!ect of the joint #exibility on natural frequencies will be investigated in Section 5.2. 5.1. EMPTY CAN This case has been analysed in order to compare the present results with data available in the literature. The comparison is shown in Table 1 for modes having n"3 nodal diameters. A very good agreement between the natural frequencies obtained by Huang & Soedel (1993), who used the receptance method, and the present results was found. It is interesting that, even though a completely #exible structure is studied, it is still possible to recognize modes dominated by the plate displacement (P mode) or by the shell displacement (S mode). Discussion on symmetric and antisymmetric modes with respect to
248
M. AMABILI
TABLE 2 Convergence of natural frequencies (Hz) with the number of terms N"NM in the expansion of mode shapes for the studied water-"lled can and n"3; S and P indicate shell and plate-dominant modes, respectively Mode
N"10
N"15
N"20
N"25
N"30
Symmetric
1st (S) 2nd (P) 3rd (S) 4th (P)
1013 1685 3922 4074
1012 1653 3918 4005
1011 1638 3914 3977
1011 1629 3909 3964
1011 1623 3905 3956
Antisymmetric
1st (P) 2nd (S) 3rd (P) 4th (S)
1683 2607 4061 4981
1652 2603 4003 4964
1637 2601 3976 4957
1628 2600 3960 4952
1623 2600 3951 4950
Figure 2. First four symmetric modes of the can "lled with water; n"3: (a) 1st S mode, 1011 Hz; (b) 1st P mode, 1638 Hz; (c) 2nd S mode, 3914 Hz; (d) 2nd P mode, 3977 Hz.
plane x"0 has been made in the previous sections. However, it can still be observed that, when n is odd and one observes a can section along its axis, one gets antisymmetric mode shapes with respect to the can axis; while, for n even, one gets symmetric shapes. 5.2. WATER-FILLED CAN The can is assumed to be completely "lled with water having a mass density o "1000 kg/m3. Table 2 shows the convergence of the method as the number of terms in F
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
249
Figure 3. First four antisymmetric modes of the can "lled with water; n"3. (a) 1st P mode, 1637 Hz; (b) 1st S mode, 2601 Hz; (c) 2nd P mode, 3976 Hz; (d) 2nd S mode, 4957 Hz.
Figure 4. E!ect of spring sti!ness c on natural frequencies of the water-"lled can; n"3: (symmetric); , 1st P mode (symmetric); , 1st P mode (antisymmetric:)
, 1st S mode
250
M. AMABILI
Figure 5. Natural frequencies of symmetric modes versus the number of nodal diameters; water-"lled can: P mode; , 1st S mode; , 2nd P mode; , 2nd S mode.
, 1st
Figure 6. Natural frequencies of antisymmetric modes versus the number of nodal diameters; water-"lled can: , 1st P mode; , 1st S mode; , 2nd P mode; , 2nd S mode.
the series expansions is increased; it is seen that 20 terms give good accuracy for this application. It was also observed that the "rst shell-dominant mode converges very quickly for both symmetric and antisymmetric cases. The third symmetric mode (shell-dominant) converges slowly as a consequence of its mode shape involving many plate modes; however, the di!erence between the natural frequency evaluated with 10 terms and the one evaluated with 30 terms is of 0)4% only. In general, more terms are necessary to estimate with good accuracy the natural frequencies of plate-dominant modes. Figure 2 shows symmetric modes of the water-"lled can for n"3. Natural frequencies are given in the caption and show a large reduction with respect to the empty case. The "rst and third modes are shell-dominant, while the second and fourth modes are plate-dominant. Figure 3 shows antisymmetric modes for the same case. In this case, the "rst and third
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
251
Figure 7. Natural frequencies of symmetric modes versus the number of nodal diameters; empty can: P mode; , 1st S mode; , 2nd P mode; , 2nd S mode.
, 1st
Figure 8. Natural frequencies of antisymmetric modes versus the number of nodal diameters; empty can: P mode; , 1st S mode; , 2nd P mode; , 2nd S mode.
, 1st
modes are plate-dominant, while the second and fourth are shell-dominant. The fourth mode is particularly interesting because it shows a large interaction of shell and plates in the dynamic behaviour of the system. The e!ect of the sti!ness of the joint at the shell}plate junctions is investigated in Figure 4. Here it is seen that all curves become horizontal for c"107 N; therefore a further increase of the joint sti!ness over this value does not a!ect natural frequencies. For the case studied, the values of joint sti!ness having the largest in#uence on natural modes lie between 103 and 105 N. Natural frequencies of symmetric and antisymmetric modes computed for di!erent numbers of nodal diameters n are given in Figures 5 and 6, respectively. Natural frequencies of plate-dominant modes increase with n, excluding the case n"0 of symmetric modes that has a particular behaviour, as already described in previous sections. In fact, this case is
252
M. AMABILI
Figure 9. Natural frequencies of the "rst four symmetric modes of the can versus the #uid density; n"3: , 1st P mode; 2nd S mode; 2nd P mode. S mode;
1st
governed by volume conservation, so that the plate mode without nodal circles is impossible and the "rst plate mode has one nodal circle. Natural frequencies of shell-dominant modes initially decrease with n, and then increase. In particular, the "rst shell-dominant symmetric mode has its minimum natural frequency for n"4. Figures 5 and 6 can be compared to the analogous Figures 7 and 8 obtained for the same empty can (with vacuum inside; in case of air inside, it is necessary to include the acoustic modes of the cavity) in order to evaluate the e!ect of the water contained. The e!ect of the density of the contained #uid on the natural frequencies of the "rst two shell- and plate-dominant symmetric modes with n"3 is shown in Figure 9. In particular, the 2nd shell-dominant mode is more sensible to the increment of the #uid density than the 2nd plate-dominant mode; therefore, for o :1000 kg/m3 the 2nd shell-dominant mode is F associated with a frequency lower than that of the 2nd plate-dominant mode. The relative edge sti!ness of the coupled shell and plates is fundamental to having a strong interaction between shell and plate modes. Figure 10 shows the e!ect of the thickness of the end-plates on the natural frequencies of the system for symmetric modes with n"3. In particular, the e!ect of the plate thickness on the "rst shell-dominant mode is given in Figure 10(a) and the e!ect on the "rst two plate-dominant modes is presented in Figure 10(b). Figure 10(a) shows that the natural frequency of the "rst shell-dominant mode changes signi"cantly for a plate thickness lying between 1 and 8 mm; outside this interval, its change is much less signi"cant. In fact, very thin end-plates cannot constrain shell rotation at the edge and give a constraint to the shell that is very close to a simple support; on the other hand, very thick end-plates give a constraint to the shell that is very close to clamped edges. The proposed method is suitable for the study of two plates in a rigid cylindrical container and of a shell closed by two rigid ends in the presence of dense #uid. The last case is particularly interesting because it allows comparison with the classical, and much simpler, problem of a simply supported shell with open ends (/"0 for x"$¸/2) completely "lled with water. Comparison is presented in Table 3 for a simply supported shell having the same material properties as the previously studied can and dimensions: a"0)175 m, ¸"0)6 m and h "0)001 m. Natural frequencies of a shell with closed ends are always S smaller than those of a shell with open ends. This is due to the larger added mass of the contained water that increases when the liquid is more constrained, as observed in other #uid}structure systems (Amabili & Kwak 1996). Di!erences for the shell studied with open
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
253
Figure 10. Natural frequencies of symmetric modes of the water-"lled can versus the plate thickness; n"3. (a) 1st S mode; (b) *r*, 1st P mode; *e*, 2nd P mode.
TABLE 3 Natural frequencies (Hz) of the simply supported shell with closed and open ends "lled with water (Donnell's theory of shells) for n"4 m
Closed ends
Open ends
Di!erence (%)
1 2 3 4 5 6 7 8
102)3 318)8 602)7 892)3 1160)2 1398)8 1609)9 1798)2
105)3 330)0 625)9 927)3 1205)5 1452)5 1669)7 1861)4
2)9 3)5 3)8 3)9 3)9 3)8 3)7 3)5
and closed ends lie between 2)9 and 3)9% for the modes considered. It is of interest to observe that the condition of rigid end-plates can be obtained by the present theory by giving a very high value to Young's modulus of the plates (E PR). P 6. CONCLUSIONS Natural modes of cans are largely a!ected by the presence of a dense #uid inside. In fact, for lower modes, the #uid pressure on the wetted surface is in phase with the structural
254
M. AMABILI
acceleration, thus the #uid appears as an added mass. However, for symmetric modes and n"0, the #uid volume conservation plays an important role, so that plate modes without nodal circles are no longer possible. Shell- and plate-dominant modes are still recognizable for the "rst modes and di!erent n values; however, already the "rst modes present an important interaction of shell and plate modes. This phenomenon is magni"ed when the #exural rigidity of shell and plates is similar (for the usual geometry, when the shell is thinner than the plates). The e!ect of the joint #exibility at the shell-plate links can be very large and the full range of #exibility has been investigated. The special case of a #exible, water-"lled shell closed by rigid ends displays natural frequencies larger by a few present than the same water-"lled shell with open ends; in fact, constraints to the #uid movement enhance the added mass of the contained #uid. ACKNOWLEDGEMENTS This research was partially supported by the Italian Space Agency (ASI). REFERENCES AMABILI, M. 1997a Shell-plate interaction in the free vibrations of circular cylindrical tanks partially "lled with a liquid: the arti"cial spring method. Journal of Sound and
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
255
LAKIS, A. A. & PAIG DOUSSIS, M. P. 1971 Free vibration of cylindrical shells partially "lled with liquid. Journal of Sound and
VIBRATIONS OF FLUID-FILLED HERMETIC CANS M. AMABILI Department of Industrial Engineering, University of Parma, Parco Area delle Scienze 181/A Parma, 43100, Italy (Received 3 February 1999, and in "nal form 21 September 1999)
Free, conservative vibrations of a hermetic can, simply supported at the base, are studied. The can is composed by a circular cylindrical shell and two identical circular plates connected to the shell at its ends. The arti"cial spring method, which is an extension of the classical Rayleigh}Ritz method, is used to solve the system by using substructuring. The can is studied empty and "lled with an inviscid and incompressible #uid. Fluid volume conservation is applied. The interaction between the plates and the shell via the #uid is considered, and exact expressions for the #uid velocity potential are used. The e!ect of #exibility of joints between plates and shell is investigated. Results for a #uid-"lled, simply supported shell closed by rigid ends are also obtained and compared to the classical open-end shell. ( 2000 Academic Press
1. INTRODUCTION HERMETIC CANS ARE LARGELY EMPLOYED as containers of liquids. They are composed of a circular cylindrical shell, joined to two circular end-plates. Their modal behaviour has been investigated in the empty case by several researchers. Hirano (1969) studied axisymmetric vibrations of thin drums (cans) by minimizing the Lagrangian; this approach was extended to other modes by Takahashi and Hirano (1970). Takavoli & Singh (1989, 1990) studied theoretically a hermetic can by using the state-space method, which is a transfermatrix-based substructuring method; they also investigated the same can experimentally and by using a FEM code. Huang & Soedel (1993) applied the receptance method to the study of free vibrations of circular plates welded to a cylindrical shell, including the hermetic can case. A similar problem has been investigated by Yuan & Dickinson (1994) by using the arti"cial spring method which is an extension of the classical Rayleigh}Ritz method, suitable for substructuring. This method had been developed by Yuan and Dickinson (1992) and Cheng & Nicolas (1992). The arti"cial spring method has been extended to study #uid}structure interaction by Cheng (1994) in the case of a light-#uid approximation and by Amabili (1997a, b) in the case of dense #uid (liquid). The only work related to vibrations of cans in contact with dense #uid is that of Harari et al. (1994), studying a sti!ened and submerged cylindrical shell closed by two end-plates and immersed in unbounded #uid. They neglected the interaction between the plates and the shell via the #uid and used the expression of the #uid load on the end-plates obtained for circular plates with a rigid extension (circular plate in an in"nite ba%e). No studies are available for free vibrations of liquid-"lled cans. Otherwise, liquid-"lled tanks composed of a circular cylindrical shell and a #exible or rigid bottom plate have been widely studied, e.g. by Amabili et al. (1998), Amabili (1997a), Chiba (1996), Gonialves & Ramos (1996), Yamaki et al. (1984); however, in these studies there exists a free liquid surface and hence they are only partially related to the present case. 0889}9746/00/020235#21 $35.00/0
( 2000 Academic Press
236
M. AMABILI
The present study investigates free, conservative vibrations of a hermetic can, simply supported at the base. The can is composed of a circular cylindrical shell and two identical circular plates connected to the shell at its ends. The use of arti"cial springs at the junctions allows us to study vibrations of this relatively complex system. In particular, the joints between the components of the structure are replaced by arti"cial springs that are distributed along the entire joint length. Only rotation at the shell}plate joints is coupled by rotational springs. The can is "rst studied empty and then "lled with an inviscid and incompressible #uid. Fluid volume conservation is applied. The interaction between the plates and the shell via the #uid is considered and exact expressions of the #uid velocity potential are used. The e!ect of #exibility of joints between plates and shell is also investigated. 2. THE EMPTY CAN The system studied is composed of a circular cylindrical shell and two circular plates connected to the shell at its ends. The bottom plate is assumed to be simply supported, in order to prevent rigid-body modes of the system. When a plate is joined to a circular cylindrical shell, in general three displacements and two slope connections could be considered, according to classical thin shell theory. However, if one only investigates lower modes of the system, the plates can be assumed inelastic in their plane. Moreover, in#uences of connection de#ections in the tangential planes of the shell can be neglected with respect to transverse amplitudes. Actually, for very thick plates, the shell axial constraint can be important and it should be properly modelled, but this is very limited for thin plates. Therefore, only the radial slope at the plate boundary can be considered to be coupled to the axial slope of the shell at the ends. A similar approach was used by Cheng & Nicolas (1992), Cheng (1994), Amabili (1997a) and Amabili et al. (1998). Huang & Soedel (1993) used two connections when the plate was not connected to the shell simple support. In the present case, only one connection is used. In fact, the joints between the shell and the plates give a reciprocal constraint that can be modelled as a simple support. Therefore, their reciprocal rotation at the edges is joined by introducing arti"cial rotational springs of appropriate sti!ness. The sti!nesses of these springs are chosen to be very high with respect to the sti!ness of the #exible components of the can, in order to simulate a welded connection between the shell and the plates. Moreover, the e!ect of the elasticity of the joint is investigated by using the present model. The top plate gives to the top end of the shell a constraint that can be conveniently modelled with a simple support for all the modes of vibration without movement of the can longitudinal axis. On the other hand, for beam-bending modes of the can, the cross-section of the shell at the top end remains circular but it moves with respect to the base, so that the top plate does not constrain these modes. Hence, these modes may correctly be studied by considering a free top-end of the shell component. In the present study, the shell is considered to be simply supported at both ends, so that this model cannot be used to study the beam-bending modes of cans simply supported only at the base. However, for the usual geometry of cans, lower shell-modes present more than one circumferential wave and can be conveniently studied by using the present model. The hermetic can considered has radius a and height ¸, and is conveniently studied in the polar coordinates (x, r, h); the origin O of the coordinate system is placed on the can axis at half-distance from the end-plates (Figure 1). Due to the axial symmetry of the structure, only modes of the shell and the plates with the same number n of nodal diameters (or circumferential waves) are coupled. A nodal diameter is a diameter in the cross-section of the shell (or in the plane of the plate) connecting points which are immobile during vibration.
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
237
Figure 1. Geometry of the can and coordinate system.
Besides, it is interesting to note that, due to axial symmetry, for each asymmetric mode (n'0) there exists a second mode having the same frequency and shape but angularly rotated by n/2n. In the present study identical end-plates are considered, so that the system is symmetric with respect to the plane x"0; therefore, it displays only symmetric and antisymmetric modes, in a radial section of the can, with respect to this plane. The Rayleigh}Ritz method (Amabili 1997b) is applied to "nd natural frequencies and modes of the #uid-"lled can, the time variation being assumed to be harmonic. Therefore, the mode shapes w of the shell wall in the radial direction (Figure 1) can be expressed as follows:
G
= cos (nh) + q cos [(2s!1)nx/¸] symmetric modes s s/1 w(x,h)" = cos (nh) + q sin (2snx/¸) antisymmetric modes, s s/1
(1)
where n and s are the number of nodal diameters and of axial half-waves, respectively, and q are the parameters of the Ritz expansion. The eigenvectors of the empty simply supported s cylindrical shell are used as admissible functions. The mode shapes w of the bottom circular plate in the transverse direction of the plate B can be given as (Amabili 1997a) = w (r, h)"cos (nh) + b [A J (j r/a)#C [I (j r/a)] , (2) B i n,i n n,i n,i n n,i i/0 where n and i are the number of nodal diameters and circles, respectively, a is the plate radius, b are the parameters of the Ritz expansion, and j is the well-known frequency i n,i parameter that is related to the plate natural frequency; J and I are the Bessel function and n n
238
M. AMABILI
modi"ed Bessel function of order n, respectively. In equation (2), the in vacuo eigenfunctions of the simply supported circular plate are assumed as admissible functions. The trial functions are linearly independent and constitute a complete set. Values of j for simply n,i supported plates are given, e.g., by Leissa & Narita (1980). Similarly, mode shapes w of the top circular plate can be given as T = w (r, h)"cos (nh) + h [A J (j r/a)#C I (j r/a)] , (3) T i n,i n n,i n,i n n,i i/0 where h are the parameters of the Ritz expansion. To simplify the computations, the mode i shape constants, A and C , are normalized in order to have (Wheelon 1968) n,i n,i 1 : [A J (j o)#C I (j o)]2 o do n,i n n,i n,i n n,i 0 A2 n2 C2 n2 " n,i (J@ (j ))2# 1! J2 (j ) ! n,i (I@ (j ))2! 1# I2 (j ) (4) n n,i n n,i n n,i n n,i 2 j2 2 j2 n,i n,i A C # n,i n,i [J (j )I (j )#I (j ) J (j )] " 1, n n,i n`1 n,i n n,i n`1 n,i j n,i where o " r/a and J@ and I@ indicate the derivatives of J and I with respect to the n n n n argument. The ratio of the mode shape constants for simply supported plates is
G C
A
B
D
C
H
A
B
D
A /C "!I (j )/J (j ). (5) n,i n,i n n,i n n,i It is interesting to note that, for the system of Figure 1, the following conditions can be imposed: w "w symmetric; T B By using equations (6) we get
w "!w antisymmetric modes. T B
(6a, b)
h "b symmetric; h "!b antisymmetric modes. (7a, b) i i i i In order to solve the problem, we evaluate the kinetic and potential energies of the shell, plates and #uid. The reference kinetic energy ¹* of the shell, neglecting the tangential and S rotary inertia, is given by 1 2n L@2 1 = ¹*" o h : : w2dx a dh" o h a (¸/2)t + q2 , S 2 S S s n 2 S S 0 ~L@2 s/1 where h is the shell thickness, o is the density of the shell material (kg/m3) and S s 2n for n"0 t" . n n for n'0
G
(8)
H
In equation (8) the orthogonality of the sine function in the interval [!¸/2, ¸/2] has been used. Similarly, the reference kinetic energy ¹* of the plates is given by p = 1 2n a (9) ¹*" o h : : (w2#w2 ) r dr dh " o h a2t + b2, i T P P n P 2 P P B i/0 0 0 where h is the plate thickness and o is the density of the plate material (kg/m3). In P P equation (9) the orthogonality of Bessel functions (plate mode shapes) in [0, a] has been used.
239
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
It should be noted that the maximum potential energy of each mode of the empty shell (without the end plates) is equal to the reference kinetic energy of the same mode multiplied by the squared circular frequency u 2 of this mode. Moreover, due to the series expansion of s the mode shape, the potential energy is the sum of the energies of each single component mode. As a consequence, the maximum potential energy of the shell may be expressed as = (10) < "1 o h a(¸/2) t + q2u2 , s s n S 2 S S s/1 where u are the circular frequencies of the symmetric and antisymmetric #exural modes of s the simply supported shell that can be computed by using, for example, the FluK gge theory of shells (Leissa 1973). Similarly, the maximum potential energy of the two end-plates is the sum of the reference kinetic energies of the eigenfunctions of the plates in vacuum multiplied by uJ 2 : n,i D = = < "o h a2t + b2uJ 2 " t + b2 j4 , (11) i n,i a2 n P P P n i n,i i/0 i/0 where the plate circular frequency uJ 2 is related to the frequency parameter j by n,i n,i uJ "(j2 /a2) [D/(o h )]1@2 and D"E h3/[12(1!l2)] is the #exural rigidity of the plates; P n,i P P P P n,i l and E are the Poisson ratio and Young's modulus of the plates, respectively. P P In a can section containing the can axis, the rotations at the plate edges are considered joined to the rotations at the shell edges, as previously discussed. The shell and the plates are connected together by arti"cial rotational springs of appropriate sti!ness, continuously distributed around the circumference. The maximum potential energy of the two distributed rotational springs connecting the two end-plates and the shell is 1 < " C 2
2n
P GCA B Lw Lx
c
A B D CA B
2 Lw B # # Lr x/~L@2 r/a
Lw Lx
A B DH
2 Lw T ! a dh , (12) Lr x/L@2 r/a
0 where c is the spring sti!ness (N m/m). It should be noted that only the variation of the angle between the shell and the plates contributes to the potential energy stored by the rotational spring. Simple calculations for symmetric modes give
G
j j n2 = = = = + + q q (!1)s(!1)j(2s!1)(2j!1)# + + b b n,i n,h < "cat s j i h a a C n ¸2 s/1 j/1 i/0 h/0 ][A J@ (j )#C I@ (j )]][A J@ (j )#C I@ (j )] n,h / n,h n,h / n,h n,i / n,i n,i / n,i n = = j !2 + + q b (!1)s (2s!1) n,i [A J@ (j )#C I@ (j )] , n,i / n,i s i n,i / n,i ¸ a s/1 i/0 and for antisymmetric modes
H
G
(13a)
4n2 = = = = j j + + q q (!1)s(!1)jsj# + + b b n,i n,h [A J@ (j )#C I@ (j )] n,i / n,i s j i h a a n,i / n,i ¸2 s/1 j/1 i/0 h/0 4n = = j ][A J@ (j )#C I@ (j )]# + + q b (!1)ss n,i [A J@ (j )#C I@ (j )] . n,h n n,h n,h n n,h n,i n n,i s i n,i n n,i ¸ a s/1 i/0 (13b)
< "cat C n
H
240
M. AMABILI
3. DYNAMIC BEHAVIOUR OF THE FLUID-STRUCTURE INTERACTION The can is considered completely "lled with an inviscid and incompressible #uid. Hydrostatic pressure e!ects and damping are neglected in the present study. For an incompressible and inviscid #uid, the deformation potential satis"es the Laplace equation
+ 2/ (r, h, x)"0.
(14)
The deformation potential / is related to the #uid velocity potential /I by /I (x, h, r, t)"iu/e*ut,
(15)
which is assumed to be harmonic; i is the imaginary unit and u is the natural circular frequency of vibration. The velocity of the #uid v is related to /I by v"!grad /I . The #uid deformation potential, using the principle of superposition, can be divided into /"/(1)#/(2),
(16)
where /(1) describes the velocity potential of the #uid associated with the #exible shell while considering the plates as rigid, and /(2) describes the #uid velocity potential for the #exible plates while considering the shell as rigid. The boundary conditions imposed to the #uid for the two complementary boundary value problems are (Amabili 1997b)
A B L/(1) Lx
A B L/(2) Lx
A B A B
"0, x/$L@2
"!w (r, h), B x/~L@2
L/(2) Lx
L/(1) Lr
"!w(x, h), r/a
"w (r,h), T x/L@2
(17a, b)
A B L/(2) Lr
"0.
(18a}c)
r/a
Equations (17) and (18) express the contact condition between a #uid (without cavitation) and a #exible or rigid wall. The reference kinetic energy ¹* of the #uid, by using the Greens F theorem (e.g. Amabili 1997b), is L/ 1 L/ 1 L/ 1 L/ :: / dS" o :: / dS! o :: / dS# o :: / dS F F F Lz 2 Lr 2 Lx 2 Lx SS`SB`ST SS SB ST 1 1 "! o :: (/(1)#/(2)) w dS# o :: (/(1)#/(2)) w dS P 2 F 2 F SS SB`ST "¹* #¹* #¹* #¹* , (19) F1 F1~2 F2~1 F2
1 ¹*" o F 2 F
where o is the mass density of the #uid, w indicates w when integrating on S and F P B B w when integrating on S , z is the direction normal at any point on the boundary surface of T T the liquid domain and is pointed outwards, S is the shell lateral surface, S is the bottom S B plate surface and S is the surface of the top plate. T 3.1. FLUID-SHELL INTERACTION In this section, the #uid}structure interaction of a simply supported, circular cylindrical and #exible shell of a can with rigid end-plates is considered. A large number of papers on the vibrations of partially #uid-"lled shells have been published [e.g. Berry & Reissner (1958), Lakis & PamK doussis (1971), Bauer & Siekmann (1971), Yamaki et al. (1984), Amabili & Dalpiaz (1995), Gonialves & Ramos (1996), and Chiba (1996)].
241
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
The liquid deformation potential /(1) is assumed to be of the form (Amabili 1997a) = /(1)" + q U(1) . s s s/1
(20)
The functions U(1) are given by s
G
C
D
= cos (nh) A rn# + A I (2mnr/¸)cos(2mnx/¸) symmetric, n,s,0 n,s,m / m/1 U(1)(x,r,h)" (21) s = cos (nh) + A I ((2m!1)nr/¸) sin ((2m!1)nx/¸) antisymmetric, n,s,m / m/1 where A are coe$cients depending on the integers n, s and m. For n"0, A will be n,s,m /,4,0 assumed to be zero, because two di!erent terms must be added to the deformation potential /, as will be discussed in Section 3.4. The functions U(1) satisfy the Laplace equation and the s two boundary conditions given in equations (17a); the condition given in equation (17b) is used to compute the coe$cients A . Di!erent solutions are obtained for symmetric and n,s,m antisymmetric modes. For symmetric modes, equation (17b) gives = 2mn A nan~1# + A I@ (2mna/¸) cos (2mnx/¸)"!cos ((2s!1)nx/¸). (22) n,s,0 n,s,m ¸ n m/1 Equation (22) must be satis"ed for all values !¸/24x4¸/2. If this equation is multiplied by cos (2njx/¸) and then integrated between !¸/2 and ¸/2, using the well-known orthogonality properties of the trigonometric functions, the equation becomes
C
D
1 !2(!1)s = p(S) I (2mnr/¸) U(1)"! cos (nh) rn# + s,m n cos(2mnx/¸) , s n (2s!1)nan~1 m I (2mna/¸) n m/1
(23)
where
C
D
L@2 ¸ (!1)m~s (!1)m`s p(S) " : cos (2mnx/¸) cos ((2s!1) nx/¸) dx" ! . s,m n 1#2m!2s !1#2m#2s ~L@2 (24) Therefore, the reference kinetic energy ¹* of the #uid associated with symmetric modes of F1 the shell is 1 2n L@2 ¹* "! o : : (/(1)) wa dx dh 1 F r/a 2 F 0 ~L@2
C
D
= = 4(!1)s(!1)ja¸ a = p(S) p(S) I (2mna/¸) " to + + qq # + s,m j,m n . s j (2s!1) (2j!1) nn m I (2mna/¸) 2n n F n s/1 j/1 m/1 For antisymmetric modes, equation (17b) gives = (2m!1)n I@ ((2m!1) na/¸) sin ((2m!1)nx/¸)"!sin (2snx/¸). + A n n,s,m ¸ m/1 Similar to the symmetric case, it gives 2 = p(A) I ((2m!1)nr/¸) s,m n U(1)"! cos (nh) + sin((2m!1)nx/¸), s n 2m!1 I@ ((2m!1) na/¸) n m/1
(25)
(26)
(27)
242
M. AMABILI
where
C
D
L@2 ¸ (!1)m~s (!1)m`s p(A)" : sin ((2m!1) nx/¸) sin (2snx/¸) dx" ! # . sm n !1#2m!2s !1#2m#2s ~L@2 (28) Therefore, the reference kinetic energy ¹* of the #uid associated with antisymmetric modes F1 of the shell is a = = = p(A) p(A) I ((2m!1)na/¸) ¹* " t o + + q q + sm jm n . F1 n n F s j (2m!1)I@ ((2m!1)na/¸) n s/1 j/1 m/1
(29)
3.2. FLUID-PLATE INTERACTION In this section, the #uid}structure interaction of the simply supported, #exible end-plates of a can with a rigid shell is considered. Flexible bottom plates in rigid tanks were studied, e.g., by Bhuta and Koval (1964), Bauer and Siekmann (1971), Chiba (1993) and Amabili (1997c); Bauer (1995) studied the vibration of a circular top plate in a liquid-"lled rigid container. The liquid deformation potential /(2) is assumed of the form (Amabili 1997c) = (30) /(2)" + b U(2) . i i i/0 The functions U(2), for symmetric modes, are expressed as i = (31) U(2) (x, r, h)"cos (nh) + K J (e r/a) cosh (e x/a), i n,i,k n n,k n,k k/1 where e are solutions of the following equation: n,k (32) J@ (e )"0, k"1,2,R, n n,k rejecting the "rst solution e for n"0. The functions U(2) satisfy equations (14) and (18c). i n,k The constants K are calculated in order to satisfy equations (18a, b) n,i,k = + K J (e r/a) (e /a) sinh (e ¸/(2a))"[A J (j r/a)#C I (j r/a)]. (33) n,i,k n n,k n,k n,k n,i n n,i n,i n n,i k/1 Equation (33) must be satis"ed for all values 04r4a. If this equation is multiplied by (1/a2) J (e r/a) r and integrated between 0 and a, then using the orthogonality of the Bessel n n,k functions this becomes a (A b #C c ) n,i n,i,k n,i n,i,k , K " n,i,k (a e ) sinh (e ¸/(2a)) n,k n,k n,k where, by using equation (32), one obtains (Wheelon 1968)
C A BD
1 n 2 1 a [J (e )]2, a " : J2 (e r/a)r dr" 1! n n,k n,k a2 n n,k 2 e n,k 0 1 a j n,i J@ (j )J (e ), b " : J (e r/a)J (j r/a) r dr" n,i,k a2 n n,k n n,i e2 !j2 n n,i n n,k n,k n,i 0 1 a j n,i I@ (j )J (e ). c " : J (e r/a) I (j r/a)r dr" n,i,k a2 n n,k n n,i e2 #j2 n n,i n n,k n,i n,k 0
(34)
(35) (36) (37)
243
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
Hence, the reference kinetic energy ¹* of the #uid associated with symmetric modes of the F2 end-plates is given by
C
D
2n a 2n a 1 : : (/(2)) w r drdh# : : (/(2)) w r drdh ¹* " o F2 2 F x/L@2 T x/~L@2 B 0 0 0 0 = = = (A b #C c ) (A b #C c ) n,h n,h,k n,h n,h,k . n,i n,i,k n,i n,i,k "o a3t + + b b + F n i h tanh (e ¸/(2a)) a e n,k n,k n,k i/0 h/0 k/1 For antisymmetric modes, equation (31) is replaced by
G
= K x# + K J (e r/a) sinh (e x/a) 0,i,0 0,i,k 0 0,k 0,k k/1 U(2) (x,r,h)" i = cos (nh) + K J (e r/a) sinh (e x/a) n,i,k n n,k n,k k/1 The constants K in this case are given by n,i,k a (A b #C c ) n,i n,i,k n,i n,i,k . K "! n,i,k (a e ) cosh (e ¸/(2a)) n,k n,k n,k In addition, the constant K is given by 0,i,0 K "!2q , 0,i,0 0,i where (Wheelon 1968)
C
(38)
for n"0, (39) for nO0.
(40)
(41)
D
A C 1 a q " : [A J (j r/a)#C I (j r/a)] r dr" 0,i J (j )# 0,i I (j ) . (42) 0,i 0 0,i 0,i 0 0,i 1 0,i 1 0,i 0,i a2 j j 0,i 0,i 0 Hence, the reference kinetic energy ¹* of the #uid associated with antisymmetric modes of F2 the end plates is given by
G
A B
e ¸ = = = (A b #C c ) n,i n,i,k n,i n,i,k (A b #C c ) tanh n,k , o a3t + + b b + F n i h n,h n,h,k n,h n,h,k 2a a e n,k n,k i/0 h/0 k/1 nO0
C
¸ = = = (A b #C c ) 0,i 0,i,k 0,i 0,i,k ¹* " o a3 t + + b b q q #+ 2 F F 0 i h a 0,i 0,h a e 0,k 0,k i/0 h/0 k/1 e ¸ ](A b #C c ) tanh 0,k , 0,h 0,h,k 0,h 0,h,k 2a
A BD
n "0. (43)
3.3. SHELL-PLATE INTERACTION VIA THE FLUID Equation (19) shows that in the reference kinetic energy of the #uid ¹*, two terms appear F because of the coupling e!ect of the #uid. In fact, even if the coupling springs between the plates and the shell were eliminated, the vibration of the two-end plates and the shell would remain coupled by the #uid inside the can. In particular, the quantity ¹* , for symmetric F1~2
244
M. AMABILI
modes, is given by 1 2n L@2 1 = = = ¹* "! o : : (/(2)) wa dx dh"! o at + + q b + K J (e ) f(S) , F1~2 r/a s i n,i,k n n,k n,s,k 2 F 2 F n 0 ~L@2 s/1 i/0 k/1 (44) where the constants K are given by equation (34) and n,i,k L@2 !2(2s!1)(!1)sna2¸ cosh (e ¸/(2a)) n,k f(S) " : cos((2s!1)nx/¸)cosh (e x/a) dx" . n,s,k n,k (2s!1)2n2a2#¸2 e2 n,k ~L@2 (45) The quantity ¹* , for antisymmetric modes, is given by F1~2
G
1 = = = ! o at + + q b + K J (e ) f(A) , s i n,i,k n n,k n,s,k 2 F n s/1 i/0 k/1 ¹* " F1~2 1 = = = ! o at + + q b !2q f(A)# + K J (e ) f(A) , 0,i,k 0 0,k 0,s,k s i 0,i 0,s 2 F 0 k/1 s/1 i/0
C
D
nO0, (46) n"0,
where the constants K are given by equation (40) and n,i,k L@2 !(!1)s ¸2 f(A)" : x sin (2snx/¸) dx" , (47) 0,s 2sn ~L@2 L@2 !4s(!1)sna2¸ sinh (e ¸/(2a)) n,k f(A) " : sin (2snx/¸) sinh (e x/a) dx" . (48) n,s,k n,k 4s2n2a2#¸2e2 n,k ~L@2 The component ¹* of the reference kinetic energy of the #uid has the following expression F2~1 for symmetric modes:
C
D
2n a 2n a 1 : : (/(1)) w r dr dh# : : (/(1)) w r dr dh ¹* " o F2~1 2 F x/~L@2 B x/L@2 T 0 0 0 0 !2(!1)s a3 1 = = "! o t + + q b (A J (j )#C I (j )) s i (2s!1)nj n,i n`1 n,i n,i n`1 n,i n F n n,i s/1 i/0 = (!1)m p(S) s,m (A m(S) #C k(S) ) , #+ n,i n,i,m mI@ (2mna/¸) n,i n,i,m m/1 n where the constants p(S) are given in equation (24), and (Wheelon 1968) s,m a m(S) ": I (2mnr/¸)J (j r/a)r dr n,i,m n n n,i 0 2mna3¸I (2mna/¸)J (j )#a2¸2j I (2mna/¸)J (j ) n`1 n n,i n,i n n`1 n,i , " 4m2n2a2#¸2j2 n,i a k(S) ": I (2mnr/¸)I (j r/a)r dr n,i,m n n n,i 0 2mna3¸I (2mna/¸) I (j )!a2¸2j I (2mna/¸)I (j ) n`1 n n,i n,i n n`1 n,i . " 4m2n2a2!¸2j2 n,i
C
D
(49)
(50)
(51)
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
245
The term ¹* for antisymmetric modes is given by F2~1 2 = = = (!1)mp(A) s,m (A m(A) #C k(A) ) , (52) ¹* "! o t + + q b + 2~1 n,i n,i,m F s i (2m!1)I@ ((2m!1)na/¸) n,i n,i,m n F n n s/1 i/0 m/1 where the constants p(A) are given in equation (28); m(A) and k(A) are obtained replacing 2m n,i,m n,i,m s,m by (2m-1) in equations (50) and (51), respectively. 3.4. SYMMETRIC MODES FOR n"0 AND FLUID VOLUME CONSERVATION In the case of symmetric modes and n"0 the condition for the conservation of the volume of the contained incompressible #uid must be imposed; in all the other cases it is automatically satis"ed by the assumed mode shapes. It is given by = 2n L@2 = 2n a + q : : cos ((2s!1) nx/¸)a dx dh"2 + b : : [A J (j r/a) s i 0,i 0 0,i s/1 0 ~L@2 i/0 0 0 #C I (j r/a)] r dr dh. 0,i 0 0,i
(53)
After simple calculations, one obtains = = !(¸/n) + q (!1)s/(2s!1)"a + b q . (54) s i 0,i s/1 i/0 For symmetric modes and n"0 two additional terms must be added to the #uid deformation potential / previously calculated. The "rst term is the constant / that gives 0 an additional contribution to the reference kinetic energy of the #uid, = = o a(¸/n)t / + q (!1)s/(2s!1)#o a2t / + b q . i 0,i s F 0 0 F 0 0 i/0 s/1 By using equation (54) it is found that this contribution must be zero. The second additional term is / and is given by a / "c (r2!2x2), a 0
(55)
(56)
where c must satisfy the condition of contact with a #exible wall 0 = L@2 (L/ /Lr) "2ac "!(1/¸) + q : cos [(2s!1) nx/¸]dx a r/a 0 s s/1 ~L@2
(57)
and = a (L/ /Lx) "!2¸c "(2/a2) + b : [A J (j r/a)#C I (j r/a)]r dr. a x/L@2 0 i 0,i 0 0,i 0,i 0 0,i i/0 0 From equations (57) and (58) we obtain, respectively,
(58)
= = c "[1/(an)] + q (!1)s/(2s!1) , c "!(1/¸) + b q . (59a, b) 0 s 0 i 0,i s/1 i/0 By using equation (54), we "nd that the right-hand side of equations (59a) and (59b) are equal; this result guarantees that / is e!ectively a solution of our problem. The term / is a a
246
M. AMABILI
associated with an additional reference kinetic energy given by
G
C
D
C
1 = = (!1)s (!1)j a2¸ 1 = = ¸2a2 o t + + qq #aJ ! + + bb q bI ! q F 0 n s j 2s!1 (2j!1)n i h 0,i 0,j 0,h 2 0,h ¸ s/1 j/1 i/0 h/0
DH
,
(60) where L@2 ¸3 (8!n2#4n2j!4n2j2)(!1) j aJ " : x2 cos [(2j!1)nx/¸] dx" 0,j 2n3(2j!1)3 ~L@2
(61)
and a bI ": r3 [A J (j r/a)#C I (j r/a)] dr"(a4/j2 ) MA [2J (j )!j J (j )] 0,h 0,h 2 0,h 0,h 3 0,h 0,h 0,h 0 0,h 0,h 0 0,h 0 #C [2I (j )#j I (j )]N . (62) 0,h 2 0,h 0,h 3 0,h 4. THE RAYLEIGH}RITZ METHOD In order to solve the problem, it is useful to introduce the Rayleigh quotient (Amabili 1997b; Zhu 1994) for the systems considered, which is (63) u2"(< #< #< )/(¹*#¹*#¹*). F P S S P C For the numerical calculation of the natural frequencies and the parameters of the Ritz expansion of modes, only N terms in the expansion of w, equation (1), and NM in the expansion of w and w , equations (2) and (3), are considered, where N and NM must be B T chosen large enough to give the required accuracy to the solution. Thus, all the energies are given by "nite summations. Here it is convenient to introduce a vectorial notation. The vector q of the parameters of the Ritz expansions is de"ned by qT"MMqNT, MbNTN, MqNT"Mq ,2q N, MbNT"Mb ,2, bNM !1 N . (64a}c) 1 N 0 Maximum potential energies < , < and < and reference kinetic energies ¹*, ¹* and S P S P C ¹* can be written in the following form: F ¹*d"qTMdq,
(66)
where u is the circular frequency (rad/s) of the #uid-"lled hermetic can. Equation (66) gives a linear eigenvalue problem for a real, symmetric matrix; real eigenvalues are obtained. In the case of symmetric modes, for n"0 the Galerkin equation is
C
KS#KP#KC D 0 -------------------D-----0T D 0
D
C
MS#MP#MF D MA !u2 --------------------- -D- ---/ MTA D 0 0
G H q
D
G H q
/
"0,
0
(67)
247
VIBRATIONS OF FLUID-FILLED HERMETIC CANS
TABLE 1 Comparison of natural frequencies (Hz) computed for the studied empty can and n"3; S and P indicate shell- and plate-dominant modes, respectively Mode Symmetric
Antisymmetric
Present study 1st (S) 2nd (P) 3rd (P) 4th (S) 1st (P) 2nd (S) 3rd (P) 4th (S)
1734 2384 5124 6003 2365 4223 5177 7100
Huang & Soedel (1993)
Di!erence (%)
1735 2343 5080 5997 2326 4224 5125 7102
0)06 1)7 0)86 0)1 1)6 0)02 1)0 0)03
where the vector M has dimension (N#NM ) and represents the additional term to the A reference kinetic energy of the #uid given by equation (55) that must be zero; the last row in equation (67) expresses the conservation of the #uid volume. Matrix M includes the F contribution of equation (60) in this case. The "rst eigenvalue of equation (67) is zero; it corresponds to q"0 and must be disregarded. In addition, the pressure exerted by the #uid at a point of the can wall can be computed by using the linearized Bernoulli equation ( p) "o (L/I /Lt) "!o u2 (/) e*ut . 10*/5 F 10*/5 F 10*/5
(68)
5. NUMERICAL RESULTS AND DISCUSSION Numerical results have been carried out by means of the Mathematica software (Wolfram 1996). The following dimensions and material properties have been chosen in order to compare the results with those obtained by Huang & Soedel (1993) in the case of an empty can: mean radius a "0)1 m, ¸"0)2 m, h "h "0)002 m, E "E "206]109 Pa, S P S P o "o "7850 kg/m3 and l "l "0)3. Twenty shell and plate modes have been conS P S P sidered in the Rayleigh}Ritz expansions, and 20 terms have been included in the expansion of U(1) and U(2). They are enough to give a good accuracy, as will be shown in Section 5.2. i s The FluK gge theory of shells has been used, unless otherwise speci"ed. The plate and the shell are assumed to be joined by rotational springs of very high sti!ness c simulating a rigid joint. In the present case, it was found that a value of c"108 N is enough to simulate a rigid rotational joint; in fact, an increment to this value does not a!ect the natural frequencies of the system. The e!ect of the joint #exibility on natural frequencies will be investigated in Section 5.2. 5.1. EMPTY CAN This case has been analysed in order to compare the present results with data available in the literature. The comparison is shown in Table 1 for modes having n"3 nodal diameters. A very good agreement between the natural frequencies obtained by Huang & Soedel (1993), who used the receptance method, and the present results was found. It is interesting that, even though a completely #exible structure is studied, it is still possible to recognize modes dominated by the plate displacement (P mode) or by the shell displacement (S mode). Discussion on symmetric and antisymmetric modes with respect to
248
M. AMABILI
TABLE 2 Convergence of natural frequencies (Hz) with the number of terms N"NM in the expansion of mode shapes for the studied water-"lled can and n"3; S and P indicate shell and plate-dominant modes, respectively Mode
N"10
N"15
N"20
N"25
N"30
Symmetric
1st (S) 2nd (P) 3rd (S) 4th (P)
1013 1685 3922 4074
1012 1653 3918 4005
1011 1638 3914 3977
1011 1629 3909 3964
1011 1623 3905 3956
Antisymmetric
1st (P) 2nd (S) 3rd (P) 4th (S)
1683 2607 4061 4981
1652 2603 4003 4964
1637 2601 3976 4957
1628 2600 3960 4952
1623 2600 3951 4950
Figure 2. First four symmetric modes of the can "lled with water; n"3: (a) 1st S mode, 1011 Hz; (b) 1st P mode, 1638 Hz; (c) 2nd S mode, 3914 Hz; (d) 2nd P mode, 3977 Hz.
plane x"0 has been made in the previous sections. However, it can still be observed that, when n is odd and one observes a can section along its axis, one gets antisymmetric mode shapes with respect to the can axis; while, for n even, one gets symmetric shapes. 5.2. WATER-FILLED CAN The can is assumed to be completely "lled with water having a mass density o "1000 kg/m3. Table 2 shows the convergence of the method as the number of terms in F
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Figure 3. First four antisymmetric modes of the can "lled with water; n"3. (a) 1st P mode, 1637 Hz; (b) 1st S mode, 2601 Hz; (c) 2nd P mode, 3976 Hz; (d) 2nd S mode, 4957 Hz.
Figure 4. E!ect of spring sti!ness c on natural frequencies of the water-"lled can; n"3: (symmetric); , 1st P mode (symmetric); , 1st P mode (antisymmetric:)
, 1st S mode
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Figure 5. Natural frequencies of symmetric modes versus the number of nodal diameters; water-"lled can: P mode; , 1st S mode; , 2nd P mode; , 2nd S mode.
, 1st
Figure 6. Natural frequencies of antisymmetric modes versus the number of nodal diameters; water-"lled can: , 1st P mode; , 1st S mode; , 2nd P mode; , 2nd S mode.
the series expansions is increased; it is seen that 20 terms give good accuracy for this application. It was also observed that the "rst shell-dominant mode converges very quickly for both symmetric and antisymmetric cases. The third symmetric mode (shell-dominant) converges slowly as a consequence of its mode shape involving many plate modes; however, the di!erence between the natural frequency evaluated with 10 terms and the one evaluated with 30 terms is of 0)4% only. In general, more terms are necessary to estimate with good accuracy the natural frequencies of plate-dominant modes. Figure 2 shows symmetric modes of the water-"lled can for n"3. Natural frequencies are given in the caption and show a large reduction with respect to the empty case. The "rst and third modes are shell-dominant, while the second and fourth modes are plate-dominant. Figure 3 shows antisymmetric modes for the same case. In this case, the "rst and third
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Figure 7. Natural frequencies of symmetric modes versus the number of nodal diameters; empty can: P mode; , 1st S mode; , 2nd P mode; , 2nd S mode.
, 1st
Figure 8. Natural frequencies of antisymmetric modes versus the number of nodal diameters; empty can: P mode; , 1st S mode; , 2nd P mode; , 2nd S mode.
, 1st
modes are plate-dominant, while the second and fourth are shell-dominant. The fourth mode is particularly interesting because it shows a large interaction of shell and plates in the dynamic behaviour of the system. The e!ect of the sti!ness of the joint at the shell}plate junctions is investigated in Figure 4. Here it is seen that all curves become horizontal for c"107 N; therefore a further increase of the joint sti!ness over this value does not a!ect natural frequencies. For the case studied, the values of joint sti!ness having the largest in#uence on natural modes lie between 103 and 105 N. Natural frequencies of symmetric and antisymmetric modes computed for di!erent numbers of nodal diameters n are given in Figures 5 and 6, respectively. Natural frequencies of plate-dominant modes increase with n, excluding the case n"0 of symmetric modes that has a particular behaviour, as already described in previous sections. In fact, this case is
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Figure 9. Natural frequencies of the "rst four symmetric modes of the can versus the #uid density; n"3: , 1st P mode; 2nd S mode; 2nd P mode. S mode;
1st
governed by volume conservation, so that the plate mode without nodal circles is impossible and the "rst plate mode has one nodal circle. Natural frequencies of shell-dominant modes initially decrease with n, and then increase. In particular, the "rst shell-dominant symmetric mode has its minimum natural frequency for n"4. Figures 5 and 6 can be compared to the analogous Figures 7 and 8 obtained for the same empty can (with vacuum inside; in case of air inside, it is necessary to include the acoustic modes of the cavity) in order to evaluate the e!ect of the water contained. The e!ect of the density of the contained #uid on the natural frequencies of the "rst two shell- and plate-dominant symmetric modes with n"3 is shown in Figure 9. In particular, the 2nd shell-dominant mode is more sensible to the increment of the #uid density than the 2nd plate-dominant mode; therefore, for o :1000 kg/m3 the 2nd shell-dominant mode is F associated with a frequency lower than that of the 2nd plate-dominant mode. The relative edge sti!ness of the coupled shell and plates is fundamental to having a strong interaction between shell and plate modes. Figure 10 shows the e!ect of the thickness of the end-plates on the natural frequencies of the system for symmetric modes with n"3. In particular, the e!ect of the plate thickness on the "rst shell-dominant mode is given in Figure 10(a) and the e!ect on the "rst two plate-dominant modes is presented in Figure 10(b). Figure 10(a) shows that the natural frequency of the "rst shell-dominant mode changes signi"cantly for a plate thickness lying between 1 and 8 mm; outside this interval, its change is much less signi"cant. In fact, very thin end-plates cannot constrain shell rotation at the edge and give a constraint to the shell that is very close to a simple support; on the other hand, very thick end-plates give a constraint to the shell that is very close to clamped edges. The proposed method is suitable for the study of two plates in a rigid cylindrical container and of a shell closed by two rigid ends in the presence of dense #uid. The last case is particularly interesting because it allows comparison with the classical, and much simpler, problem of a simply supported shell with open ends (/"0 for x"$¸/2) completely "lled with water. Comparison is presented in Table 3 for a simply supported shell having the same material properties as the previously studied can and dimensions: a"0)175 m, ¸"0)6 m and h "0)001 m. Natural frequencies of a shell with closed ends are always S smaller than those of a shell with open ends. This is due to the larger added mass of the contained water that increases when the liquid is more constrained, as observed in other #uid}structure systems (Amabili & Kwak 1996). Di!erences for the shell studied with open
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Figure 10. Natural frequencies of symmetric modes of the water-"lled can versus the plate thickness; n"3. (a) 1st S mode; (b) *r*, 1st P mode; *e*, 2nd P mode.
TABLE 3 Natural frequencies (Hz) of the simply supported shell with closed and open ends "lled with water (Donnell's theory of shells) for n"4 m
Closed ends
Open ends
Di!erence (%)
1 2 3 4 5 6 7 8
102)3 318)8 602)7 892)3 1160)2 1398)8 1609)9 1798)2
105)3 330)0 625)9 927)3 1205)5 1452)5 1669)7 1861)4
2)9 3)5 3)8 3)9 3)9 3)8 3)7 3)5
and closed ends lie between 2)9 and 3)9% for the modes considered. It is of interest to observe that the condition of rigid end-plates can be obtained by the present theory by giving a very high value to Young's modulus of the plates (E PR). P 6. CONCLUSIONS Natural modes of cans are largely a!ected by the presence of a dense #uid inside. In fact, for lower modes, the #uid pressure on the wetted surface is in phase with the structural
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M. AMABILI
acceleration, thus the #uid appears as an added mass. However, for symmetric modes and n"0, the #uid volume conservation plays an important role, so that plate modes without nodal circles are no longer possible. Shell- and plate-dominant modes are still recognizable for the "rst modes and di!erent n values; however, already the "rst modes present an important interaction of shell and plate modes. This phenomenon is magni"ed when the #exural rigidity of shell and plates is similar (for the usual geometry, when the shell is thinner than the plates). The e!ect of the joint #exibility at the shell-plate links can be very large and the full range of #exibility has been investigated. The special case of a #exible, water-"lled shell closed by rigid ends displays natural frequencies larger by a few present than the same water-"lled shell with open ends; in fact, constraints to the #uid movement enhance the added mass of the contained #uid. ACKNOWLEDGEMENTS This research was partially supported by the Italian Space Agency (ASI). REFERENCES AMABILI, M. 1997a Shell-plate interaction in the free vibrations of circular cylindrical tanks partially "lled with a liquid: the arti"cial spring method. Journal of Sound and
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LAKIS, A. A. & PAIG DOUSSIS, M. P. 1971 Free vibration of cylindrical shells partially "lled with liquid. Journal of Sound and