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Free vibration of a clamped-free circular cylindrical shell partially filled with liquid—Part I: Theoretical analysis
Thin-Walled Structures 2 (1984) 265-284
Free Vibration of a Clamped-Free Circular Cylindrical Shell Partially Filled with Liquid---Part I: Theoretical Analysis M. C h i b a , N. Y a m a k i a n d J. T a n i Institute of High Speed Mechanics, Tohoku University, Sendai, Japan ABSTRACT Theoretical analyses are presented for the linear free vibration of a clamped-free cylindrical shell partially filled with an incompressible, inviscid liquid. For the vibration of the shell itself, the dynamic version of the Donnell equations was used and the problem was solved with the modified Galerkin procedure, taking the effect of the axisymmetric deformation due to the static liquid pressure into consideration. Concerning the vibration relevant to the liquid motion, the solution for the velocity potential was assumed as a sum of two sets of linear combinations of the suitable harmonic function, the unknown parameters of which were imposed to satisfy both boundary conditions along the wetted shell wall and the free liquid surface in a sense of appropriate series expansions. The procedure stated in the foregoing leads to a determinantal equation for the determination of the natural frequencies of the present shell-liquid system. To compare with the experimental results which will be stated in a companion paper, 1~detailed numerical results will be presented in another companion paper 13 on the free vibration characteristics of the two test cylinders partially filled with water.
NOTATION an, bin, A k, B j, B0 CI, C2, C3, C4
dnme D = Eh3/12(1 - v 2) E F(f)
Coefficients. Coefficients of eqn (32). Integral value defined by eqn (33). Flexural rigidity of the shell. Young's modulus. Stress function (nondimensional form).
265 Thin-Walled Structures 0263-8231/84/$03" 00 O Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
266
g@ h H i = x/-1 JN, IN
lo = H / L L N Nx, Ny, Nxr (nx, ny, nxy) P~(p,), Pal(pal) R
Sx (sx), i x (rex)
t(r) U, V, W (u, v, w) x(~), y(~), Z, r(o), O(n) Z = X/(1 - v2) • L2/Rh Otn, tan, Vn, Kn
= LN/TrR T = pcR/p~h
At(St) ENk V P~, P f
,I,( )
~(o~)
M. Chiba, N. Yamaki, J. Tani
Gravitational acceleration. Shell thickness. Liquid height. Unit of imaginary number. The first kind Bessei and modified Bessel function of the order N. Filling ratio of the liquid. Length of the shell. Circumferential wave number. Stress resultant components. Static and dynamic liquid pressure. Mean radius of the shell. Equivalent shearing force and bending moment. Time. Displacement components of the middle surface. Coordinate system. Geometric parameter of the shell. Parameters defined by eqns (25) and (27). Nondimensional parameter. Nondimensional parameter. Kronecker delta. Normal displacement of the liquid surface. Parameters satisfied by eqn (36). Poisson's ratio. Mass density of the shell and the liquid, respectively. Velocity potential of the liquid. Eigenfunction of a cantilever beam. Natural frequency.
Terms not shown here are defined in the text.
1 INTRODUCTION Recently, since the energy crisis, a large number of cylindrical oil-storage tanks with huge capacities have been built in the vicinity of industrial areas. To assess the safety of these structures against earthquakes, it is of
Free vibration o f a cylindrical tank - theoretical analysis
267
great technical importance to clarify the free vibration characteristics of circular cylindrical shells partially filled with liquid. Hence much research has been conducted on this subject through various analytical and numerical methods. 1-11In these studies, however, a variety of simplifying assumptions were used to overcome the inherent mathematical difficulty and the problem does not seem to have been thoroughly explored. With the object of obtaining practically accurate solutions of the problem when both edges of the shell are completely clamped, Yamaki e t al. 12 carried out a theoretical analysis on the linear free vibration of the cylindrical tank, considering the effects of the initial hoop stress as well as the free surface condition of the liquid which are usually ignored or approximated. The corresponding experimental studies were also conducted by using two polyester test cylinders containing water and employing a noncontacting proximeter utilizing fiberglass optics. Both theoretical and experimental results were confirmed to be almost coincident, revealing the validity of theoretical analyses as well as the degree of precision of the experimental data. As a continuation of these studies, similar research has been conducted under the different boundary conditions when the lower edge of the shell is completely clamped while the upper edge is free. In this report, we present results of theoretical analyses. For the vibration dominated by the shell response, the dynamic version of the Donnell equations was used and the problem was solved through the modified Galerkin method. As in the previous studies, it is assumed that the contained liquid is incompressible and inviscid, and that the bottom of the container is flat and rigid. Then, for the vibration relevant to the fluid motion, the solution for the velocity potential was assumed in the form of the linear combinations of the suitable harmonic functions, the unknown parameters of which were imposed to satisfy the remaining boundary conditions. Detailed calculations will be given in a companion paper,13 to clarify the whole aspect of the free vibration characteristics of the two test cylinders partially filled with water. The corresponding experimental results, together with a comparison with theoretical ones, will be given in another companion paper. 14
2 F O R M U L A T I O N OF THE PROBLEM We shall consider the free vibration of a clamped-free (cantilever) circular cylindrical shell with radius R, length L and thickness h, which is
268
M. Chiba, N. Yanu~ki, J. Tani
filled to a height H with an inviscid incompressible liquid. The bottom of the shell is assumed to be flat and rigid. The coordinate system is shown in Fig. 1. First, we will consider the nonlinear free vibration of the shell for convenience. Confining the problem to relatively low frequency ranges dominated by flexural motion of thin shells, we will apply the Donnell
o k.Z~2R
~
Fig. 1. Circular cylindrical shell partially filled with liquid.
nonlinear theory of the shell modified with the effect of the transverse inertia force. Then, by denoting the displacement components by U , V . W , the stress resultants by N~, Nv, Nx, the stress function by F, and the static and dynamic liquid pressures (positive when acting in the - z direction) by P~ and Pa, the governing equations may be expressed in the nondimensional form as ~ / 7 + ,~w ee + / 3 ~ ( w , w . , , - w2~,) = o
(t)
~2 r~ 4
W
--
O/ f,~cs¢ --
C
nx = 132 f . , , ,
(f.ee w m, - 2f.¢,~ w.en + f,n,~ w .~) + w,~ + p , + P d = 0
--
(2)
C
nv = f,e¢,
n,. = - / 3 f.e,
(3)
(4) By ,, - o~w + 5 B ~wT~ = f ~ - vB: f ~ [3u , + r e + [3w t w . , = - 2 ( I + u)/3f.e,
269
Free vibration o f a cylindrical t a n k - theoretical analysis
In the foregoing we have introduced the following notations: 7rx
zry
= -T-,
n = --F,
7rR t =
L ,
L
= T,
L2
= g - f i , o, = - r - g fi
F LSplg % = 1 L f = --E-~, Po = - ~ a - - ~ , t-" co, h2f~2° P~, (u, v) = ---~(U, V)
W
L2
H
(Nx, Ny, N~r), 10 - L
w = --~, (nx, ny, nxy) = ~
(5) L2
r = lq°t" f~° = --ffr
~h
1
=
~P
12(1- 2 ) ,
0,c =
~2 =
,Z
02
=
--~-r-,~/(1 /.(n
+~2
-
u2), D
Eh 3 =
12(1
/,,2)
02
= / 1:0 _---~:_--- rrlo [O:rr/o < (_--- rr
In these eqns, E, v and D are Young's modulus, Poisson's ratio and flexural rigidity of the shell respectively, while ps and P t are mass density of the shell and liquid, respectively. Further, t is time, g is the gravitational acceleration, f~ is the unknown natural frequency and N is the circumferential wave number of the vibration of the shell. In addition, subscripts following a comma stand for differentiation. The static liquid pressure Ps is P~ = p f g ( H -
(6a)
x) %
or
Ps =
£
1-
~p
(6b)
The dynamic liquid pressure Pa and the corresponding parameter pa will be explained later. It is to be noted that p0, I0 and Z are the nondimen-
270
M. Chiba, N. Yamaki, J. Tani
sional parameters related to the static liquid pressure, liquid height and shell geometry, respectively. Assuming that the lower end of the shell is clamped while the upper end is free, the boundary conditions are ~:= 0:w = w.~= u , = v = 0 ,
f
2rrN
(7a)
f.~d'0 = 0 0
(7b)
~:= 0 : s x = mx = n x = n x y = 0 where -MxL2/
sx = - S x L 3 / ( D h ' t r
3) = w . , e + (2 - v)/32w e~
and
m~ =
( D h T r 2) = w.ee + u f l 2 w . ~ correspond to the equivalent shearing
force and the bending moment, respectively. Now we shall derive the equations for the static deformation as well as the small amplitude free vibration, by specializing the preceding equations.
2.1 Axisymmetric deformation due to static liquid premature With w0 and f0 denoting the axisymmetric static deflection and the corresponding stress function, respectively, we obtain the following equations from eqns (1)-(7). fo.ee = - a W o .
fo.,., = A~.e, = 0
(8)
Ol 2
-Ls(wo) =- Wo.eee~+ - - w o + p~ = 0
(9)
C
~: = 0:w0 = Wo,~ = 0 (10) ,~ = "n": Wo.~e = Wo.ee~ = 0
2.2 Asymmetric vibration with wave number N(A=0) First, we shall consider the small amplitude asymmetric free vibration of the shell around the aforementioned axisymmetric state. With u, v and w denoting newly the small incremental displacement components and f t h e corresponding stress function, the governing equations will be obtained from eqns (1), (2), (4), (7) as follows ~ 4 f + otw.£e + jO2 w.o£tw.n.~ = 0
(11)
271
Free vibration o f a cylindrical t a n k - theoretical analysis
L(w,D-- ¢4W-c Lu-cw0, uf
c
a~2wow~+
w.+pa
= 0
(12)
= 0 : w = w. ~ = f . ~ - v B z f . ~ = f. ~ + (2 + v)BZf. Or, = 0
(13a--d)
= 1 r : w. e e e + ( 2 - v ) 1 3 2 w
(13e-h)
e~=
w ee+v~2w~=f,~=f,e~=O
N o w we shall consider the associated free vibration of the liquid contained in the shell. Assuming the liquid to be incompressible and inviscid, and the m o t i o n to be irrotational, we introduce the velocity potential, ~ , which is to satisfy the following equation corresponding to the Laplace e q u a t i o n in the r, O ( = y / R ) and x coordinates
,bop+ ~,0+
~b,m+ ~
~bee = 0
(14)
Recalling that the b o t t o m of the tank is flat and rigid, and that the liquid is filled to height H , the boundary conditions are ~: = 0 : d , . , = 0
(15)
~: = *rl0 : 4'~,,+g6.~ = 0
(16)
w h e r e the last expression stands for the linearized flee surface condition. Further, along the wetted cylindrical surface p = l,O<~
(17)
while the dynamic liquid pressure Pa acting on the shell surface is Pd
-------py(tY~t)r = REp
(18a)
or Pd = --Y(6..),
(18b)
= ,~p
In the foregoing, the following nondimensional parameters have been introduced. 6 -
floRh'
r p = -'R" 'y -
prR ¢rg 1 p s h " ~o = - ~ o L " Pd -- p ~ n.-;-'~-TPa ~to
(19)
M. Chiba, N. Yamaki, J. Tani
272
2.3 Axisymmetric vibration with wave number N = 0 Specializing the preceding equation, the basic e q u a t i o n in this case becomes Ot 2
-La(w) ~ W eeae+--W+W.=+pd = 0 ¢
(20)
~ = O" w = w ' ~ = O ' = Tr " w , ~ = w . ~ =
(21)
6.p,+_l 6.0+
I O
4~ = 0
(22)
P T h e b o u n d a r y conditions for cb are given by eqns (15) through (17) as before.
3 METHOD OF SOLUTION
3.1 Axisymmetric deformation due to static liquid pressure C o n s i d e r i n g the b o u n d a r y conditions (10), the solution Wo(¢) is a s s u m e d in the form w0(() = ~ a . + . ( ~ )
(n = 1.2,3 . . . .
)
(23)
n
w h e r e the a. are u n k n o w n constants, and the tO. (¢) are eigen functions for a cantilever b e a m , that is (24)
t0.(() = /~.(coshan~: - cos~.() - v.(sinha.~: - s i n a . ( ) /~. = (coshan~" + COSa.Tr)/K.. v. = (sinha.Tr- sina.w)/K. I r . = X/(~) sinha.Trsina.Tr
J
(25)
F u r t h e r , the to. (() are normalized as
f ~+,(¢)t0,,(¢)d( = 8,,, ()
(26)
Free vibration of a cylindrical tank - theoretical analysis
273
while the a . are parameters satisfying the relation 1 + cosha.rr.cosa.zr = 0
(27)
For the determination of the unknown constants a., we apply the Galerkin m e t h o d to eqn (9) as
fro
(m = 1, 2, 3 . . . . )
(28)
(~b"(rr/0) + 2aZ.(rrl0a.v. -/z.)}
(29)
" L,(wo)" ~,.(sC)d~: = 0
which leads to a.
=
-/~o
4 4 a.rtc(ot. + 12Z2/'n "4)
A prime on the function 0. (~:) means differentiation with respect to ~:. Thus the static deflection Wo can be determined in terms of the shell g e o m e t r y Z, Poisson's ratio v, specific liquid pressure po and liquid filling ratio 10.
3.2 Asymmetric free vibration around the axisymmetric deformation 3 . 2 . 1 Stress f u n c t i o n
Assuming that the shell vibrates in N ( - 1 ) circumferential waves with circular frequency l'l, we put the deflection w and the stress function f i n the following form w(5, V, r) = e"~.cosv ~ bm0,,(~)
(m = 1,2, 3 . . . .
)
(30)
rn
(3l)
f(~, n, r) = ae/'~'cos'0f(~:)
where the b,, are unknown parameters. Substituting eqns (23), (30) and (31 ) into eqn (11) and integrating, we obtain the general solution of f as f ( ( ) = Clcosh/3~: + C2sinhfl~ + C3fl~:'cosh/~ + C4~" sinhfl~
_
b,. z
~1~4)i/l~ (~7) ]
a.b.,d,,,.e
(32)
274
M. Chiba, N. Yamaki, J. Tani
T h e u n k n o w n constants C1 to C4 can be determined with the boundary conditions (13c,d) and (13g,h), the result of which is given in A p p e n d i x 1. Further, d,,., is defined as d,,.e = rj0,~q/,(~), qJ=(~:)"qJe(~:)ds¢
(33)
T h u s we have obtained expressions of w and f satisfying both compatibility and b o u n d a r y conditions exactly. 3.2.2 Velocity potential We assume the velocity potential ~b, satisfying the governing eqn (14) and the b o u n d a r y condition (15), as
(k. j = 1, 2,3 . . . .
)
(34)
where Ak, B~ and B0 are u n k n o w n parameters and where O,vk(P, ~j) = J,v(e,Vk'O) cosh (~Nk'/~) ]
(35)
Further, the ~Nk are the roots of the equation Op
]0= I
2 (k = 1, 2, 3 . . . .
)
(36)
In the foregoing, JN and IN are the first kind Bessel and modified Bessel functions, of the order N, respectively. Now we shall determine the u n k n o w n parameters so that the remaining boundary conditions are satisfied in the form of the series expansions. First, from the boundary conditions (17), Bj and B0 can be expressed in terms of b,. as Bj = -(-I)J'Tj~,Kj,,,.b,..Bo = -To~Ko,,,b,,, " "
(m,j=
1,2.3 ....
) (37)
Freevibrationof a cylindricaltank-theoreticalanalysis
275
where
~'-- / (~)
4~
try
IN-1
/')/~0-N~10 1
(38)
+ IN+I ff~O
(39)
~,. = (-,)'Io" co~ (f0,) 0.,,,~,
In the foregoing, the following orthogonality conditions were taken into consideration. + 8~)-6,~j fo~C°S(~o ~) "c°S(To~)d~= ~rlo(1 1 ( m , j = O, 1, 2 . . . .
)
(40)
Similarly, with expressions (34) and (37), the free liquid surface condition (16) leads to the equations
[to2--g'eNk'~'tanh (~Nk'~rrlo)]Ak
where
and where the following orthogonality conditions were utilized.
y01JN(eNk'P)'JN(~Nm'p)pdp = ~1[ 1 - ( N ) 2] .j2(~Nk).Skm (k, m = 1, 2, 3 . . . . )
(44)
276
M. Chiba, N. Yaraaki, J. Tani
Equations (41) represent a set of homogeneous linear equations in A kand
bin. Introducing the notations A = {A~, A2. . . . .
Ak} T, b = {bb b2. . . . .
(45)
b,,} r
eqns (41) can be expressed in the double size matrix form as
(O O){hAl =o92(IS){ Atbj
(46)
w h e r e | and O are unit and null matrices, respectively, and where Q and S are k × k and k × m matrices with the elements Qkt = g, " Euk" ~ tanh (~Uk" ~Irlo)" 8kt
(47) (k,l,m=
1,2,3 ....
)
3.2.3 Modified Galerkin method Now we shall seek for the conditions in the b,, for the approximate satisfaction of the remaining basic eqn (12). Noting that the coordinate functions we have employed, tO,,(~¢), do not satisfy the boundary conditions (13a,b) and (13e,f) completely, we apply the modified Galerkin m e t h o d proposed by Leipholz,~5 which leads to the following conditions
f2rrN-L(w,f)qJt(~)'cos~d~d~
f~" 0 ,,r 0
- (2 - v)B2 fl 2~rN[w ~,,. q,t]~ = ~"cosn dn )
-~- /.)~t~2
[w n,tb}]¢=~cos'0drt
(l = 1, 2, 3 . . . .
)
(48)
Expressing the dynamic pressure pd, in eqn (12), in terms of Ak and b,., and performing the integration, one finally obtains the equations in the following form
(0 E) {A b} = ~°2(GF) {A/ bl
(49)
where E and F are the l × m matrices while G is the l × I matrix. Actual expressions for the elements of these matrices are given in Appendix 2.
277
Free vibrationof a cylindricaltank - theoreticalanalysis
Equations (46) and (49) represent a system of homogeneous linear equations in Ak and bm. In order that one has nontrivial solutions, the d e t e r m i n a n t of the coefficients of these equations, A, should vanish. W h e n the values for v, Z, R/h, y, poand 10are given, A depends on N and to only, that is A(v, Z, R/h, y, P0, 10: N, to) = 0
(50)
Thus, one can determine for each wave n u m b e r N, the natural frequencies tot,., m -- l'l~m,m/II0, m, N = 1, 2, 3, . . . , and the corresponding values of the parameters b,, and Ak, representing the modes of vibration of the shell and the liquid, respectively. It is to be noted that w h e n the vibration of the liquid surface is expressed by A1or 81 = (AtL)/ (RhTr), one has 8f.~ =
[6, e]e - ,~,0
(51)
which leads to
~t(P, ~q, r) = ei~'cosr~ ~
k
Ak'eSk'f3Ju(~uk'p)'sinh(euk'f3"~'lo)
(52)
Further, it is to be added that when the shell wall is rigid one can put all of a., b,., Bj and B0 equal to zero in the present analysis. Then, the natural frequencies, fit,, o, of the liquid in a rigid circular cylindrical shell are given by fl(k.U) = to(k,u)'l-10 =
[
(")]
g'ENk'tanh ~-EN,
(53)
In the foregoing, subscript N indicates the n u m b e r of circumferential waves while k denotes the order of the radial mode implying the n u m b e r of the nodal circles.
3.3 Axisymmetric free vibration In this case, the solutions w and ~b are assumed as w(~, r) = e ;~ ~, bm~bm(~:)
(m = l, 2, 3 . . . .
)
(54)
m
t~(~,p,r)=/toe/w[
~k ZkOok+ 2i B]~°i+B°(p2-2f~2'2) ] (k, y = l, 2, 3 . . . .
)
(55)
M. Chiba,N. Yamaki,J. Tani
278
w h e r e bm, Ak, Bj and B0 are unknown parameters, and where 00k and ~0j are given by eqn (35) with N replaced by zero. Further Eok (k -- 1, 2, 3 . . . . " ¢01 = 0) are given by eqn (36) with N replaced by zero. Then, with the same procedure as before, the boundary condition (17) gives the relations
Bj = -(-1)JT]'~Kj.,.b.,,Bo=-T~Ko,,,b,. m
(m = 1 , 2 , 3 . . . .
)
m
(56) 2fl
1
T~ = 7rj'l' ( ~o] ) , T ~ - 2rcl°
(57)
while the free surface condition of the liquid leads to the equations
(Q* P) { A1 b/ = oj2(l S*) I bAl
(58)
where Q~'t = ~'Eok"fl" tanh (EOk'/~hrlo)"8k,, P~',. = 2~/32KoraSkl S'm =-
{ ~ H~fKj..+T~[R~-2(~rlo)e'8~,]Ko..}
t/,j=4
/( ~1o[ EL+ ( ~o] )2] "J°(e°')c°sh(e°k'~'t°)
(59)
(6o) 1/2
"k = 1
R~ = 4/[ 2okcosh(~ok.flarlo).Jo(~ok)]
" k :~ 1
Finally, we apply the Galerkin method to the remaining basic eqn (20)
f
o"La(W).q,,(Od6
=
0
(1=1,2.3 .... )
(61)
from which we have (O E*)
AI b~
= of(G* F*)
(62)
Free vibration of a cylindricaltank - theoreticalanalysis
279
In the foregoing, the matrices E~,, G I and ~ depend on v, Z, R/h, ~ and 10, the expressions of which are given in Appendix 3. Equations (58) and (62) represent a system of homogeneous linear equations in Ak and b,,, from which the natural frequencies and the corresponding modes of vibration can be determined through usual procedures. Further, the vibration of the liquid surface, 8r(0, z), is given by H
4
LH \
(63)
while the natural frequencies of the axisymmetric vibration of the liquid in the rigid circular tank will be given by eqn (53) with Nreplaced by zero.
4 CONCLUSIONS Theoretical analyses are presented for the free vibration characteristics of a clamped-free cylindrical shell partially filled with an incompressible, inviscid liquid. A generalized Galerkin method is used in which the solutions were approximated by linear combinations of the appropriate functions satisfying the boundary conditions and the governing equations, respectively, for the portions of the problem mainly associated with the shell wall vibration and the liquid motion. The initial axisymmetric deflection of the shell due to the static liquid pressure as well as the boundary condition along the free surface of the liquid are fully taken into consideration. Detailed calculations were carded out for the two test cylinders partially filled with liquid which clarify the whole aspect of the free vibration characteristics of the shell-liquid system. These will be given in a companion paper. 13The validity of the present theory will be assessed in another companion paper, 14 through comparison with the corresponding experimental ones. REFERENCES 1. Baron, M. L. and Shalak, R., Free vibrations of fluid-filled cylindrical shells, Proc. Am. Soc. Cir. Eng., J. Eng. Mech. Div., 88 (1962) 17--43. 2. Chu, W.-H., Breathing vibrations of a partially filled cylinder tank linear theory, J. AppL Mech., Trans. ASME, E-30 (1963) 532--6.
280
M. Chiba, N. Yamaki, J. Tani
3. Arya, A., Thakkar, S. and Goyal, A., Vibration analysis of thin cylindrical containers, Proc. Am. Soc. Cir. Eng., J. Eng. Mech. Div., 97 (1971)317-31. 4. Bauer, H. F. and Siekmann, J., Dynamic interaction of a liquid with the elastic structure of a circular cylindrical container, lng. Arch, 4tl (1971) 266--80. 5. Lakis, A. A. and Paidoussis, M. P., Free vibration of cylindrical shells partially filled with liquid, J. Sound Vib., 19 (1971) 1-15. 6. Stillman, W. E., Free vibration of cylinders containing liquid, J. Sound Vib., 3t} (1973) 509-29. 7. Komatsu, K., Vibration analysis of axisymmetric containers filled with liquid. (Finite element analysis using axisymmetric fluid finite elements), Trans. Japan Soc. Mech. Eng., 45 (1979) 295-303 (in Japanese). 8. Kondo, H., Axisymmetric free vibration analysis of a circular cylindrical tank, Bulletin. Japan Soc. Mech. Eng., 24 (1981) 215-21. 9. Haroun, M. A. and Housner, G. W., Dynamic characteristics of liquid storage tanks, Proc. Am. Soc. Cir. Eng., J. Eng. Mech. Div., 106 (1982) 783-800. 10. Haroun, M. A. and Housner, G. W., Complications in free vibration analysis of tanks, Proc. Am. Soc. Cir. Eng., J. Eng. Mech. Div., 106 (1982) 801-18. 11. Parkus, H., Modes and frequencies of vibrating liquid-filled cylindrical tanks, Int. J. Eng. Sci., 2ll (1982) 31%26. 12. Yamaki, N., Tani, J. and Yamaji, T., Free vibration of a clamped-clamped circular cylindrical shell partially filled with liquid, J. Sound Vib., 94 (1984) in press. 13. Chiba, M., Yamaki, N. and Tani, J., Free vibration of a clamped-free circular cylindrical shell partially filled with liquid--Part II: Numerical results, Thin-Walled Structures, 2 (1984) 305-22. 14. Chiba, M., Yamaki, N. and Tani, J., Free vibration of a clamped-free circular cylindrical shell partially filled with liquid---(Part III): Experimental results, Thin-Walled Structures, 3 (1985) in press. 15. Leipholz, H., Uber die Befreiung der Anstyfunktionen des Ritzschen und Galerkischen Verfahrens von den Randbedingungen, Ing. Arch., 36 (1976) 251-61.
A P P E N D I X 1: E X P R E S S I O N S F O R T H E C O E F F I C I E N T S IN E Q N (32) Ci = ~ C,,. = ~ {[,827r2(1 - T 2) + T2(2v,- l)]l~m m
m
--21-', [~'h'(1 -- T 2) -
T]12m- 21.',[]~'n-+T ( 2 v , -
+ 2Vl(2Vl +~TrT)I4~}/D,
1)]13.,
Free vibrationof a cylindricaltank- theoreticalanalysis C2 = 2 C2rn = 2 { - ( 2vl - 1 ) ~ - ( 1 - T 2) + T]Ilm m
m
- ~2m'2(1 - T 2) + 2v,]12..,- ( 2 v , - (2v, - 1 ) [ T ( 2 v ,
- 1) -
1)(,8.tr T - 2 v 0
I3,.,
#~]I4.,}/D,
1 ~ (C2., + 12,.) C3 = ~m C3m - 2vl-1 1
C4:
Zm C4" : "~'~Vl~m (llrrl--Clrtl)
T = tanhOTr, vt
- --, -
l+v
D, = 4v~- T2(2v,- 1)~+O=~r~(1- T 2)
I,., = I.,#~,.[(I + v , ) o ~ + (1 -
v,)J4]b..
~ ~ J.,,~a2lz<[v,ot~+ ( 2 - v,)~4]a.b,.
-
n
e
I2,. = {I,.~,.v,.[(1 - ~,),~..4 + (1 + ~,)~14]b..
+ ~
~ J . , . . a 3 v . [ v , a ~ - (2 +v,)-Ct4]a.b"}D8
n e
13 m
I,"
I,.
--
-
lfl~/(rr) [ 2l.,°13C*b,.- 2 2 J.meCe°te(°te+* -
cosh B~"
=t82
X/(~.)cosh Bzr
.
[2l,.a~b,.-
2o~ (a~ -/74) 2` J'"<
.
~
4
~4)anbm]
J.,.e(a~ + B4)a.b,. ]
2d.,,~ a(a4<_/74) 2 , C,. = (cot amrr + coth
ot,.Tr)/rt
281
282
M. Chiba, N. Y ~ k i ,
J. Tani
A P P E N D I X 2: EXPRESSIONS FOR THE ELEMENTS OF MATRICES E, G, F IN EQN (49) -I I
20tmCtl t "* "'" 4 ~ z { ' - T - - ' ~--- 4Oft 'A'~I--AN)--V(OI"C~+oqC*t)}
J
Elm
2B2~m(4(1 - v)C* - V.,) + rm
°t[32
"m = l
an'hnml- SP12 + P2 - --~ ~
~
n
j
C
2°[
q (a~-/3')
[
2a~
+
(
/33
sinhflTr + a3vt
/33
2
ol,,,V.,+ r,,,) : m =
~e oterean~l,,.~Yet
: e ~- I
otea,,d,,,,~(V~'r~+ 2fl2Ot3e)
V,,, = C*(1-ot,,'C*zr2)+l~,,,v,,,,
Y,,,l = C*
A *, = t~,~(amCt 2 . - atum/z/), rm = a ,4, +
dime =
e
(C,,,+2C4,.)/3:CT'ottX/(~').cosh/3~r-
+ (C2., + 2C3.,) (~2CTottX,/('n'). sinh/3~ -
°t "°t2I,, ., ~ ( 2 f l
~ a,,aj~lj,,~P,,,,
"e = 1 _
Oll
[A*
a---~.'~,,,i
-- . 4 .
~4
2 d j,,,e 1 o/(a4 __ ~ 4 ) 2 , hnml = hnml q'- (Ot4m __ ~4)2 (2f12 0t4 dnm! q- Fro" k,,,.l)
P,,.t = r e d . a + 2~2 k,,d, h,,,,,t =
f," q'. (~) ~,..(~) q',(¢)
dlj
~t
~l,,,)
)
Free vibration of a cylindrical tank - theoretical analysis k~m, =
~b" (se) • 0,~, (¢)" ~,(~) d~
SP12 = ~ a~
O;'(~)(C,m'coshB~+ C ~ s i n h f l ~ : + C3mB~:'coshfl~:
n
+ C4m~1~"sinh fls~) • ~bt(s¢) d~:
P2 = a B
Gik =
(C3,,,~coshfl~+ C~:sinhfl~:)41t(~:)d~:
-- ~ ' J N ( ~ N k ) ' X I k
Xi, = L~k'Ottcosh (eNk" f~rtlo) - L 2"eNk" f3"sinh (eNk'f~arlo) + L~,'cttvt
L]k = S3[/xtsinh (ott'n'lo) - vtcosh (ott'n'/o)] - S4[/ztsin (ottzrlo) + I"tCOS(~/~10)] L 2 = S 3 [ / . / . / c o s h (alrrlo) - v/sinh (atrdo) ] + S4[P.t" cos ( a t r r l o ) -
vlsin (alrrlo)]
1
1
L3k = S3"Ju S4, S3 = 012_ (ENkOfl)2, S4 = Ol21dg" (ENk,~)2
Ft,,, = Y ( ~ TRj" Kj,-,," K/t + ToKo,,," Kot ) + St,,,
TRj =
283
284
M. Chiba, N. Yamaki, J. Tani APPENDIX
3: E X P R E S S I O N S MATRICES
/
4
FOR THE ELEMENTS E * , G * , F* I N E Q N (62)
12Z2~
F,*m= ~'[ ~ TRTK,I"K,m+ Ta(K°,-2B2M,'K°..]+8,m
TR~ =
M, = l [ s i n h O/I
(atrrlo){/zt[2
+ (at'n'to) z] + 2v,(oqrr/o)}
- sin (ottTrlo) {/.*t[(ottTrlo) 2 - 2] - 2~,t(atrr/o) } - cosh
(atrrlo) {vt[2 +
(otlrr/o) 2] +
2tZl(ottrrlo)}
- cos (oqrrlo){ u,[(a,rr/o) 2 - 2] + 2/,t(a,rr/o) } ]
OF
Free Vibration of a Clamped-Free Circular Cylindrical Shell Partially Filled with Liquid---Part I: Theoretical Analysis M. C h i b a , N. Y a m a k i a n d J. T a n i Institute of High Speed Mechanics, Tohoku University, Sendai, Japan ABSTRACT Theoretical analyses are presented for the linear free vibration of a clamped-free cylindrical shell partially filled with an incompressible, inviscid liquid. For the vibration of the shell itself, the dynamic version of the Donnell equations was used and the problem was solved with the modified Galerkin procedure, taking the effect of the axisymmetric deformation due to the static liquid pressure into consideration. Concerning the vibration relevant to the liquid motion, the solution for the velocity potential was assumed as a sum of two sets of linear combinations of the suitable harmonic function, the unknown parameters of which were imposed to satisfy both boundary conditions along the wetted shell wall and the free liquid surface in a sense of appropriate series expansions. The procedure stated in the foregoing leads to a determinantal equation for the determination of the natural frequencies of the present shell-liquid system. To compare with the experimental results which will be stated in a companion paper, 1~detailed numerical results will be presented in another companion paper 13 on the free vibration characteristics of the two test cylinders partially filled with water.
NOTATION an, bin, A k, B j, B0 CI, C2, C3, C4
dnme D = Eh3/12(1 - v 2) E F(f)
Coefficients. Coefficients of eqn (32). Integral value defined by eqn (33). Flexural rigidity of the shell. Young's modulus. Stress function (nondimensional form).
265 Thin-Walled Structures 0263-8231/84/$03" 00 O Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
266
g@ h H i = x/-1 JN, IN
lo = H / L L N Nx, Ny, Nxr (nx, ny, nxy) P~(p,), Pal(pal) R
Sx (sx), i x (rex)
t(r) U, V, W (u, v, w) x(~), y(~), Z, r(o), O(n) Z = X/(1 - v2) • L2/Rh Otn, tan, Vn, Kn
= LN/TrR T = pcR/p~h
At(St) ENk V P~, P f
,I,( )
~(o~)
M. Chiba, N. Yamaki, J. Tani
Gravitational acceleration. Shell thickness. Liquid height. Unit of imaginary number. The first kind Bessei and modified Bessel function of the order N. Filling ratio of the liquid. Length of the shell. Circumferential wave number. Stress resultant components. Static and dynamic liquid pressure. Mean radius of the shell. Equivalent shearing force and bending moment. Time. Displacement components of the middle surface. Coordinate system. Geometric parameter of the shell. Parameters defined by eqns (25) and (27). Nondimensional parameter. Nondimensional parameter. Kronecker delta. Normal displacement of the liquid surface. Parameters satisfied by eqn (36). Poisson's ratio. Mass density of the shell and the liquid, respectively. Velocity potential of the liquid. Eigenfunction of a cantilever beam. Natural frequency.
Terms not shown here are defined in the text.
1 INTRODUCTION Recently, since the energy crisis, a large number of cylindrical oil-storage tanks with huge capacities have been built in the vicinity of industrial areas. To assess the safety of these structures against earthquakes, it is of
Free vibration o f a cylindrical tank - theoretical analysis
267
great technical importance to clarify the free vibration characteristics of circular cylindrical shells partially filled with liquid. Hence much research has been conducted on this subject through various analytical and numerical methods. 1-11In these studies, however, a variety of simplifying assumptions were used to overcome the inherent mathematical difficulty and the problem does not seem to have been thoroughly explored. With the object of obtaining practically accurate solutions of the problem when both edges of the shell are completely clamped, Yamaki e t al. 12 carried out a theoretical analysis on the linear free vibration of the cylindrical tank, considering the effects of the initial hoop stress as well as the free surface condition of the liquid which are usually ignored or approximated. The corresponding experimental studies were also conducted by using two polyester test cylinders containing water and employing a noncontacting proximeter utilizing fiberglass optics. Both theoretical and experimental results were confirmed to be almost coincident, revealing the validity of theoretical analyses as well as the degree of precision of the experimental data. As a continuation of these studies, similar research has been conducted under the different boundary conditions when the lower edge of the shell is completely clamped while the upper edge is free. In this report, we present results of theoretical analyses. For the vibration dominated by the shell response, the dynamic version of the Donnell equations was used and the problem was solved through the modified Galerkin method. As in the previous studies, it is assumed that the contained liquid is incompressible and inviscid, and that the bottom of the container is flat and rigid. Then, for the vibration relevant to the fluid motion, the solution for the velocity potential was assumed in the form of the linear combinations of the suitable harmonic functions, the unknown parameters of which were imposed to satisfy the remaining boundary conditions. Detailed calculations will be given in a companion paper,13 to clarify the whole aspect of the free vibration characteristics of the two test cylinders partially filled with water. The corresponding experimental results, together with a comparison with theoretical ones, will be given in another companion paper. 14
2 F O R M U L A T I O N OF THE PROBLEM We shall consider the free vibration of a clamped-free (cantilever) circular cylindrical shell with radius R, length L and thickness h, which is
268
M. Chiba, N. Yanu~ki, J. Tani
filled to a height H with an inviscid incompressible liquid. The bottom of the shell is assumed to be flat and rigid. The coordinate system is shown in Fig. 1. First, we will consider the nonlinear free vibration of the shell for convenience. Confining the problem to relatively low frequency ranges dominated by flexural motion of thin shells, we will apply the Donnell
o k.Z~2R
~
Fig. 1. Circular cylindrical shell partially filled with liquid.
nonlinear theory of the shell modified with the effect of the transverse inertia force. Then, by denoting the displacement components by U , V . W , the stress resultants by N~, Nv, Nx, the stress function by F, and the static and dynamic liquid pressures (positive when acting in the - z direction) by P~ and Pa, the governing equations may be expressed in the nondimensional form as ~ / 7 + ,~w ee + / 3 ~ ( w , w . , , - w2~,) = o
(t)
~2 r~ 4
W
--
O/ f,~cs¢ --
C
nx = 132 f . , , ,
(f.ee w m, - 2f.¢,~ w.en + f,n,~ w .~) + w,~ + p , + P d = 0
--
(2)
C
nv = f,e¢,
n,. = - / 3 f.e,
(3)
(4) By ,, - o~w + 5 B ~wT~ = f ~ - vB: f ~ [3u , + r e + [3w t w . , = - 2 ( I + u)/3f.e,
269
Free vibration o f a cylindrical t a n k - theoretical analysis
In the foregoing we have introduced the following notations: 7rx
zry
= -T-,
n = --F,
7rR t =
L ,
L
= T,
L2
= g - f i , o, = - r - g fi
F LSplg % = 1 L f = --E-~, Po = - ~ a - - ~ , t-" co, h2f~2° P~, (u, v) = ---~(U, V)
W
L2
H
(Nx, Ny, N~r), 10 - L
w = --~, (nx, ny, nxy) = ~
(5) L2
r = lq°t" f~° = --ffr
~h
1
=
~P
12(1- 2 ) ,
0,c =
~2 =
,Z
02
=
--~-r-,~/(1 /.(n
+~2
-
u2), D
Eh 3 =
12(1
/,,2)
02
= / 1:0 _---~:_--- rrlo [O:rr/o < (_--- rr
In these eqns, E, v and D are Young's modulus, Poisson's ratio and flexural rigidity of the shell respectively, while ps and P t are mass density of the shell and liquid, respectively. Further, t is time, g is the gravitational acceleration, f~ is the unknown natural frequency and N is the circumferential wave number of the vibration of the shell. In addition, subscripts following a comma stand for differentiation. The static liquid pressure Ps is P~ = p f g ( H -
(6a)
x) %
or
Ps =
£
1-
~p
(6b)
The dynamic liquid pressure Pa and the corresponding parameter pa will be explained later. It is to be noted that p0, I0 and Z are the nondimen-
270
M. Chiba, N. Yamaki, J. Tani
sional parameters related to the static liquid pressure, liquid height and shell geometry, respectively. Assuming that the lower end of the shell is clamped while the upper end is free, the boundary conditions are ~:= 0:w = w.~= u , = v = 0 ,
f
2rrN
(7a)
f.~d'0 = 0 0
(7b)
~:= 0 : s x = mx = n x = n x y = 0 where -MxL2/
sx = - S x L 3 / ( D h ' t r
3) = w . , e + (2 - v)/32w e~
and
m~ =
( D h T r 2) = w.ee + u f l 2 w . ~ correspond to the equivalent shearing
force and the bending moment, respectively. Now we shall derive the equations for the static deformation as well as the small amplitude free vibration, by specializing the preceding equations.
2.1 Axisymmetric deformation due to static liquid premature With w0 and f0 denoting the axisymmetric static deflection and the corresponding stress function, respectively, we obtain the following equations from eqns (1)-(7). fo.ee = - a W o .
fo.,., = A~.e, = 0
(8)
Ol 2
-Ls(wo) =- Wo.eee~+ - - w o + p~ = 0
(9)
C
~: = 0:w0 = Wo,~ = 0 (10) ,~ = "n": Wo.~e = Wo.ee~ = 0
2.2 Asymmetric vibration with wave number N(A=0) First, we shall consider the small amplitude asymmetric free vibration of the shell around the aforementioned axisymmetric state. With u, v and w denoting newly the small incremental displacement components and f t h e corresponding stress function, the governing equations will be obtained from eqns (1), (2), (4), (7) as follows ~ 4 f + otw.£e + jO2 w.o£tw.n.~ = 0
(11)
271
Free vibration o f a cylindrical t a n k - theoretical analysis
L(w,D-- ¢4W-c Lu-cw0, uf
c
a~2wow~+
w.+pa
= 0
(12)
= 0 : w = w. ~ = f . ~ - v B z f . ~ = f. ~ + (2 + v)BZf. Or, = 0
(13a--d)
= 1 r : w. e e e + ( 2 - v ) 1 3 2 w
(13e-h)
e~=
w ee+v~2w~=f,~=f,e~=O
N o w we shall consider the associated free vibration of the liquid contained in the shell. Assuming the liquid to be incompressible and inviscid, and the m o t i o n to be irrotational, we introduce the velocity potential, ~ , which is to satisfy the following equation corresponding to the Laplace e q u a t i o n in the r, O ( = y / R ) and x coordinates
,bop+ ~,0+
~b,m+ ~
~bee = 0
(14)
Recalling that the b o t t o m of the tank is flat and rigid, and that the liquid is filled to height H , the boundary conditions are ~: = 0 : d , . , = 0
(15)
~: = *rl0 : 4'~,,+g6.~ = 0
(16)
w h e r e the last expression stands for the linearized flee surface condition. Further, along the wetted cylindrical surface p = l,O<~
(17)
while the dynamic liquid pressure Pa acting on the shell surface is Pd
-------py(tY~t)r = REp
(18a)
or Pd = --Y(6..),
(18b)
= ,~p
In the foregoing, the following nondimensional parameters have been introduced. 6 -
floRh'
r p = -'R" 'y -
prR ¢rg 1 p s h " ~o = - ~ o L " Pd -- p ~ n.-;-'~-TPa ~to
(19)
M. Chiba, N. Yamaki, J. Tani
272
2.3 Axisymmetric vibration with wave number N = 0 Specializing the preceding equation, the basic e q u a t i o n in this case becomes Ot 2
-La(w) ~ W eeae+--W+W.=+pd = 0 ¢
(20)
~ = O" w = w ' ~ = O ' = Tr " w , ~ = w . ~ =
(21)
6.p,+_l 6.0+
I O
4~ = 0
(22)
P T h e b o u n d a r y conditions for cb are given by eqns (15) through (17) as before.
3 METHOD OF SOLUTION
3.1 Axisymmetric deformation due to static liquid pressure C o n s i d e r i n g the b o u n d a r y conditions (10), the solution Wo(¢) is a s s u m e d in the form w0(() = ~ a . + . ( ~ )
(n = 1.2,3 . . . .
)
(23)
n
w h e r e the a. are u n k n o w n constants, and the tO. (¢) are eigen functions for a cantilever b e a m , that is (24)
t0.(() = /~.(coshan~: - cos~.() - v.(sinha.~: - s i n a . ( ) /~. = (coshan~" + COSa.Tr)/K.. v. = (sinha.Tr- sina.w)/K. I r . = X/(~) sinha.Trsina.Tr
J
(25)
F u r t h e r , the to. (() are normalized as
f ~+,(¢)t0,,(¢)d( = 8,,, ()
(26)
Free vibration of a cylindrical tank - theoretical analysis
273
while the a . are parameters satisfying the relation 1 + cosha.rr.cosa.zr = 0
(27)
For the determination of the unknown constants a., we apply the Galerkin m e t h o d to eqn (9) as
fro
(m = 1, 2, 3 . . . . )
(28)
(~b"(rr/0) + 2aZ.(rrl0a.v. -/z.)}
(29)
" L,(wo)" ~,.(sC)d~: = 0
which leads to a.
=
-/~o
4 4 a.rtc(ot. + 12Z2/'n "4)
A prime on the function 0. (~:) means differentiation with respect to ~:. Thus the static deflection Wo can be determined in terms of the shell g e o m e t r y Z, Poisson's ratio v, specific liquid pressure po and liquid filling ratio 10.
3.2 Asymmetric free vibration around the axisymmetric deformation 3 . 2 . 1 Stress f u n c t i o n
Assuming that the shell vibrates in N ( - 1 ) circumferential waves with circular frequency l'l, we put the deflection w and the stress function f i n the following form w(5, V, r) = e"~.cosv ~ bm0,,(~)
(m = 1,2, 3 . . . .
)
(30)
rn
(3l)
f(~, n, r) = ae/'~'cos'0f(~:)
where the b,, are unknown parameters. Substituting eqns (23), (30) and (31 ) into eqn (11) and integrating, we obtain the general solution of f as f ( ( ) = Clcosh/3~: + C2sinhfl~ + C3fl~:'cosh/~ + C4~" sinhfl~
_
b,. z
~1~4)i/l~ (~7) ]
a.b.,d,,,.e
(32)
274
M. Chiba, N. Yamaki, J. Tani
T h e u n k n o w n constants C1 to C4 can be determined with the boundary conditions (13c,d) and (13g,h), the result of which is given in A p p e n d i x 1. Further, d,,., is defined as d,,.e = rj0,~q/,(~), qJ=(~:)"qJe(~:)ds¢
(33)
T h u s we have obtained expressions of w and f satisfying both compatibility and b o u n d a r y conditions exactly. 3.2.2 Velocity potential We assume the velocity potential ~b, satisfying the governing eqn (14) and the b o u n d a r y condition (15), as
(k. j = 1, 2,3 . . . .
)
(34)
where Ak, B~ and B0 are u n k n o w n parameters and where O,vk(P, ~j) = J,v(e,Vk'O) cosh (~Nk'/~) ]
(35)
Further, the ~Nk are the roots of the equation Op
]0= I
2 (k = 1, 2, 3 . . . .
)
(36)
In the foregoing, JN and IN are the first kind Bessel and modified Bessel functions, of the order N, respectively. Now we shall determine the u n k n o w n parameters so that the remaining boundary conditions are satisfied in the form of the series expansions. First, from the boundary conditions (17), Bj and B0 can be expressed in terms of b,. as Bj = -(-I)J'Tj~,Kj,,,.b,..Bo = -To~Ko,,,b,,, " "
(m,j=
1,2.3 ....
) (37)
Freevibrationof a cylindricaltank-theoreticalanalysis
275
where
~'-- / (~)
4~
try
IN-1
/')/~0-N~10 1
(38)
+ IN+I ff~O
(39)
~,. = (-,)'Io" co~ (f0,) 0.,,,~,
In the foregoing, the following orthogonality conditions were taken into consideration. + 8~)-6,~j fo~C°S(~o ~) "c°S(To~)d~= ~rlo(1 1 ( m , j = O, 1, 2 . . . .
)
(40)
Similarly, with expressions (34) and (37), the free liquid surface condition (16) leads to the equations
[to2--g'eNk'~'tanh (~Nk'~rrlo)]Ak
where
and where the following orthogonality conditions were utilized.
y01JN(eNk'P)'JN(~Nm'p)pdp = ~1[ 1 - ( N ) 2] .j2(~Nk).Skm (k, m = 1, 2, 3 . . . . )
(44)
276
M. Chiba, N. Yaraaki, J. Tani
Equations (41) represent a set of homogeneous linear equations in A kand
bin. Introducing the notations A = {A~, A2. . . . .
Ak} T, b = {bb b2. . . . .
(45)
b,,} r
eqns (41) can be expressed in the double size matrix form as
(O O){hAl =o92(IS){ Atbj
(46)
w h e r e | and O are unit and null matrices, respectively, and where Q and S are k × k and k × m matrices with the elements Qkt = g, " Euk" ~ tanh (~Uk" ~Irlo)" 8kt
(47) (k,l,m=
1,2,3 ....
)
3.2.3 Modified Galerkin method Now we shall seek for the conditions in the b,, for the approximate satisfaction of the remaining basic eqn (12). Noting that the coordinate functions we have employed, tO,,(~¢), do not satisfy the boundary conditions (13a,b) and (13e,f) completely, we apply the modified Galerkin m e t h o d proposed by Leipholz,~5 which leads to the following conditions
f2rrN-L(w,f)qJt(~)'cos~d~d~
f~" 0 ,,r 0
- (2 - v)B2 fl 2~rN[w ~,,. q,t]~ = ~"cosn dn )
-~- /.)~t~2
[w n,tb}]¢=~cos'0drt
(l = 1, 2, 3 . . . .
)
(48)
Expressing the dynamic pressure pd, in eqn (12), in terms of Ak and b,., and performing the integration, one finally obtains the equations in the following form
(0 E) {A b} = ~°2(GF) {A/ bl
(49)
where E and F are the l × m matrices while G is the l × I matrix. Actual expressions for the elements of these matrices are given in Appendix 2.
277
Free vibrationof a cylindricaltank - theoreticalanalysis
Equations (46) and (49) represent a system of homogeneous linear equations in Ak and bm. In order that one has nontrivial solutions, the d e t e r m i n a n t of the coefficients of these equations, A, should vanish. W h e n the values for v, Z, R/h, y, poand 10are given, A depends on N and to only, that is A(v, Z, R/h, y, P0, 10: N, to) = 0
(50)
Thus, one can determine for each wave n u m b e r N, the natural frequencies tot,., m -- l'l~m,m/II0, m, N = 1, 2, 3, . . . , and the corresponding values of the parameters b,, and Ak, representing the modes of vibration of the shell and the liquid, respectively. It is to be noted that w h e n the vibration of the liquid surface is expressed by A1or 81 = (AtL)/ (RhTr), one has 8f.~ =
[6, e]e - ,~,0
(51)
which leads to
~t(P, ~q, r) = ei~'cosr~ ~
k
Ak'eSk'f3Ju(~uk'p)'sinh(euk'f3"~'lo)
(52)
Further, it is to be added that when the shell wall is rigid one can put all of a., b,., Bj and B0 equal to zero in the present analysis. Then, the natural frequencies, fit,, o, of the liquid in a rigid circular cylindrical shell are given by fl(k.U) = to(k,u)'l-10 =
[
(")]
g'ENk'tanh ~-EN,
(53)
In the foregoing, subscript N indicates the n u m b e r of circumferential waves while k denotes the order of the radial mode implying the n u m b e r of the nodal circles.
3.3 Axisymmetric free vibration In this case, the solutions w and ~b are assumed as w(~, r) = e ;~ ~, bm~bm(~:)
(m = l, 2, 3 . . . .
)
(54)
m
t~(~,p,r)=/toe/w[
~k ZkOok+ 2i B]~°i+B°(p2-2f~2'2) ] (k, y = l, 2, 3 . . . .
)
(55)
M. Chiba,N. Yamaki,J. Tani
278
w h e r e bm, Ak, Bj and B0 are unknown parameters, and where 00k and ~0j are given by eqn (35) with N replaced by zero. Further Eok (k -- 1, 2, 3 . . . . " ¢01 = 0) are given by eqn (36) with N replaced by zero. Then, with the same procedure as before, the boundary condition (17) gives the relations
Bj = -(-1)JT]'~Kj.,.b.,,Bo=-T~Ko,,,b,. m
(m = 1 , 2 , 3 . . . .
)
m
(56) 2fl
1
T~ = 7rj'l' ( ~o] ) , T ~ - 2rcl°
(57)
while the free surface condition of the liquid leads to the equations
(Q* P) { A1 b/ = oj2(l S*) I bAl
(58)
where Q~'t = ~'Eok"fl" tanh (EOk'/~hrlo)"8k,, P~',. = 2~/32KoraSkl S'm =-
{ ~ H~fKj..+T~[R~-2(~rlo)e'8~,]Ko..}
t/,j=4
/( ~1o[ EL+ ( ~o] )2] "J°(e°')c°sh(e°k'~'t°)
(59)
(6o) 1/2
"k = 1
R~ = 4/[ 2okcosh(~ok.flarlo).Jo(~ok)]
" k :~ 1
Finally, we apply the Galerkin method to the remaining basic eqn (20)
f
o"La(W).q,,(Od6
=
0
(1=1,2.3 .... )
(61)
from which we have (O E*)
AI b~
= of(G* F*)
(62)
Free vibration of a cylindricaltank - theoreticalanalysis
279
In the foregoing, the matrices E~,, G I and ~ depend on v, Z, R/h, ~ and 10, the expressions of which are given in Appendix 3. Equations (58) and (62) represent a system of homogeneous linear equations in Ak and b,,, from which the natural frequencies and the corresponding modes of vibration can be determined through usual procedures. Further, the vibration of the liquid surface, 8r(0, z), is given by H
4
LH \
(63)
while the natural frequencies of the axisymmetric vibration of the liquid in the rigid circular tank will be given by eqn (53) with Nreplaced by zero.
4 CONCLUSIONS Theoretical analyses are presented for the free vibration characteristics of a clamped-free cylindrical shell partially filled with an incompressible, inviscid liquid. A generalized Galerkin method is used in which the solutions were approximated by linear combinations of the appropriate functions satisfying the boundary conditions and the governing equations, respectively, for the portions of the problem mainly associated with the shell wall vibration and the liquid motion. The initial axisymmetric deflection of the shell due to the static liquid pressure as well as the boundary condition along the free surface of the liquid are fully taken into consideration. Detailed calculations were carded out for the two test cylinders partially filled with liquid which clarify the whole aspect of the free vibration characteristics of the shell-liquid system. These will be given in a companion paper. 13The validity of the present theory will be assessed in another companion paper, 14 through comparison with the corresponding experimental ones. REFERENCES 1. Baron, M. L. and Shalak, R., Free vibrations of fluid-filled cylindrical shells, Proc. Am. Soc. Cir. Eng., J. Eng. Mech. Div., 88 (1962) 17--43. 2. Chu, W.-H., Breathing vibrations of a partially filled cylinder tank linear theory, J. AppL Mech., Trans. ASME, E-30 (1963) 532--6.
280
M. Chiba, N. Yamaki, J. Tani
3. Arya, A., Thakkar, S. and Goyal, A., Vibration analysis of thin cylindrical containers, Proc. Am. Soc. Cir. Eng., J. Eng. Mech. Div., 97 (1971)317-31. 4. Bauer, H. F. and Siekmann, J., Dynamic interaction of a liquid with the elastic structure of a circular cylindrical container, lng. Arch, 4tl (1971) 266--80. 5. Lakis, A. A. and Paidoussis, M. P., Free vibration of cylindrical shells partially filled with liquid, J. Sound Vib., 19 (1971) 1-15. 6. Stillman, W. E., Free vibration of cylinders containing liquid, J. Sound Vib., 3t} (1973) 509-29. 7. Komatsu, K., Vibration analysis of axisymmetric containers filled with liquid. (Finite element analysis using axisymmetric fluid finite elements), Trans. Japan Soc. Mech. Eng., 45 (1979) 295-303 (in Japanese). 8. Kondo, H., Axisymmetric free vibration analysis of a circular cylindrical tank, Bulletin. Japan Soc. Mech. Eng., 24 (1981) 215-21. 9. Haroun, M. A. and Housner, G. W., Dynamic characteristics of liquid storage tanks, Proc. Am. Soc. Cir. Eng., J. Eng. Mech. Div., 106 (1982) 783-800. 10. Haroun, M. A. and Housner, G. W., Complications in free vibration analysis of tanks, Proc. Am. Soc. Cir. Eng., J. Eng. Mech. Div., 106 (1982) 801-18. 11. Parkus, H., Modes and frequencies of vibrating liquid-filled cylindrical tanks, Int. J. Eng. Sci., 2ll (1982) 31%26. 12. Yamaki, N., Tani, J. and Yamaji, T., Free vibration of a clamped-clamped circular cylindrical shell partially filled with liquid, J. Sound Vib., 94 (1984) in press. 13. Chiba, M., Yamaki, N. and Tani, J., Free vibration of a clamped-free circular cylindrical shell partially filled with liquid--Part II: Numerical results, Thin-Walled Structures, 2 (1984) 305-22. 14. Chiba, M., Yamaki, N. and Tani, J., Free vibration of a clamped-free circular cylindrical shell partially filled with liquid---(Part III): Experimental results, Thin-Walled Structures, 3 (1985) in press. 15. Leipholz, H., Uber die Befreiung der Anstyfunktionen des Ritzschen und Galerkischen Verfahrens von den Randbedingungen, Ing. Arch., 36 (1976) 251-61.
A P P E N D I X 1: E X P R E S S I O N S F O R T H E C O E F F I C I E N T S IN E Q N (32) Ci = ~ C,,. = ~ {[,827r2(1 - T 2) + T2(2v,- l)]l~m m
m
--21-', [~'h'(1 -- T 2) -
T]12m- 21.',[]~'n-+T ( 2 v , -
+ 2Vl(2Vl +~TrT)I4~}/D,
1)]13.,
Free vibrationof a cylindricaltank- theoreticalanalysis C2 = 2 C2rn = 2 { - ( 2vl - 1 ) ~ - ( 1 - T 2) + T]Ilm m
m
- ~2m'2(1 - T 2) + 2v,]12..,- ( 2 v , - (2v, - 1 ) [ T ( 2 v ,
- 1) -
1)(,8.tr T - 2 v 0
I3,.,
#~]I4.,}/D,
1 ~ (C2., + 12,.) C3 = ~m C3m - 2vl-1 1
C4:
Zm C4" : "~'~Vl~m (llrrl--Clrtl)
T = tanhOTr, vt
- --, -
l+v
D, = 4v~- T2(2v,- 1)~+O=~r~(1- T 2)
I,., = I.,#~,.[(I + v , ) o ~ + (1 -
v,)J4]b..
~ ~ J.,,~a2lz<[v,ot~+ ( 2 - v,)~4]a.b,.
-
n
e
I2,. = {I,.~,.v,.[(1 - ~,),~..4 + (1 + ~,)~14]b..
+ ~
~ J . , . . a 3 v . [ v , a ~ - (2 +v,)-Ct4]a.b"}D8
n e
13 m
I,"
I,.
--
-
lfl~/(rr) [ 2l.,°13C*b,.- 2 2 J.meCe°te(°te+* -
cosh B~"
=t82
X/(~.)cosh Bzr
.
[2l,.a~b,.-
2o~ (a~ -/74) 2` J'"<
.
~
4
~4)anbm]
J.,.e(a~ + B4)a.b,. ]
2d.,,~ a(a4<_/74) 2 , C,. = (cot amrr + coth
ot,.Tr)/rt
281
282
M. Chiba, N. Y ~ k i ,
J. Tani
A P P E N D I X 2: EXPRESSIONS FOR THE ELEMENTS OF MATRICES E, G, F IN EQN (49) -I I
20tmCtl t "* "'" 4 ~ z { ' - T - - ' ~--- 4Oft 'A'~I--AN)--V(OI"C~+oqC*t)}
J
Elm
2B2~m(4(1 - v)C* - V.,) + rm
°t[32
"m = l
an'hnml- SP12 + P2 - --~ ~
~
n
j
C
2°[
q (a~-/3')
[
2a~
+
(
/33
sinhflTr + a3vt
/33
2
ol,,,V.,+ r,,,) : m =
~e oterean~l,,.~Yet
: e ~- I
otea,,d,,,,~(V~'r~+ 2fl2Ot3e)
V,,, = C*(1-ot,,'C*zr2)+l~,,,v,,,,
Y,,,l = C*
A *, = t~,~(amCt 2 . - atum/z/), rm = a ,4, +
dime =
e
(C,,,+2C4,.)/3:CT'ottX/(~').cosh/3~r-
+ (C2., + 2C3.,) (~2CTottX,/('n'). sinh/3~ -
°t "°t2I,, ., ~ ( 2 f l
~ a,,aj~lj,,~P,,,,
"e = 1 _
Oll
[A*
a---~.'~,,,i
-- . 4 .
~4
2 d j,,,e 1 o/(a4 __ ~ 4 ) 2 , hnml = hnml q'- (Ot4m __ ~4)2 (2f12 0t4 dnm! q- Fro" k,,,.l)
P,,.t = r e d . a + 2~2 k,,d, h,,,,,t =
f," q'. (~) ~,..(~) q',(¢)
dlj
~t
~l,,,)
)
Free vibration of a cylindrical tank - theoretical analysis k~m, =
~b" (se) • 0,~, (¢)" ~,(~) d~
SP12 = ~ a~
O;'(~)(C,m'coshB~+ C ~ s i n h f l ~ : + C3mB~:'coshfl~:
n
+ C4m~1~"sinh fls~) • ~bt(s¢) d~:
P2 = a B
Gik =
(C3,,,~coshfl~+ C~:sinhfl~:)41t(~:)d~:
-- ~ ' J N ( ~ N k ) ' X I k
Xi, = L~k'Ottcosh (eNk" f~rtlo) - L 2"eNk" f3"sinh (eNk'f~arlo) + L~,'cttvt
L]k = S3[/xtsinh (ott'n'lo) - vtcosh (ott'n'/o)] - S4[/ztsin (ottzrlo) + I"tCOS(~/~10)] L 2 = S 3 [ / . / . / c o s h (alrrlo) - v/sinh (atrdo) ] + S4[P.t" cos ( a t r r l o ) -
vlsin (alrrlo)]
1
1
L3k = S3"Ju S4, S3 = 012_ (ENkOfl)2, S4 = Ol21dg" (ENk,~)2
Ft,,, = Y ( ~ TRj" Kj,-,," K/t + ToKo,,," Kot ) + St,,,
TRj =
283
284
M. Chiba, N. Yamaki, J. Tani APPENDIX
3: E X P R E S S I O N S MATRICES
/
4
FOR THE ELEMENTS E * , G * , F* I N E Q N (62)
12Z2~
F,*m= ~'[ ~ TRTK,I"K,m+ Ta(K°,-2B2M,'K°..]+8,m
TR~ =
M, = l [ s i n h O/I
(atrrlo){/zt[2
+ (at'n'to) z] + 2v,(oqrr/o)}
- sin (ottTrlo) {/.*t[(ottTrlo) 2 - 2] - 2~,t(atrr/o) } - cosh
(atrrlo) {vt[2 +
(otlrr/o) 2] +
2tZl(ottrrlo)}
- cos (oqrrlo){ u,[(a,rr/o) 2 - 2] + 2/,t(a,rr/o) } ]
OF