Compum & Strucrures Vol. 54. No. 5. pp. 997-1001. 1995 Copyright c: 1995 Elsevier Science Ltd Printed in Great Britain. All rights rexrved 0045-7949/95-$9.50 + 0.00
TECHNICAL NOTE FREE VIBRATION OF CYLINDRICAL FILLED WITH LIQUID
L. T. Lee and J. C. Lu Department
Mathematics, National Chung Hsing Taiwan 40227, Republic of China
(Received 16 June 1993) Abstract-A theoretical analysis for the free vibration of vertical simply supported cylindrical shells filled with ideal liquid has been investigated. The vibrations of the shell are examined by using the Donnell shell theory. The fluid is considered to be non-viscous and incompressible. The shell dynamic response function is obtained by applying Hamilton’s principle. The Galerkin method is used to simplify the differential equations into algebraic equations in order to obtain the natural frequencies and corresponding modes. The method shows its great advantage in saving of computing time. The results are compared with those available in the literature and excellent agreement is found between them.
garding research into shell vibration, Leissa [I] has collected most of the results before 1973. Chung  and Greif and Chung used the Rayleigh-Ritz method, for different boundary conditions, to obtain the natural frequency. Sharma and Johns [4,5] and Goldman  evaluated the natural frequency and modes for free and fixed boundary conditions. Chung  applied Stoke’s transformation technique to solve the frequency for different boundary conditions. For the vibration of shells with the interaction of internal fluid, Mnev and Pertsev , Junger and Feit , and Brown [IO] have done some research on this problem. Chu et a/. [l I] used the energy method to obtain the frequency parameters. Recently, Goncalves  has investigated the non-linear vibrations of thin-walled cylinders with liquid interaction. The main purpose of this paper is to find a quick and efficient method to obtain the natural frequency of a thin-walled cylinder tilled with liquid. The liquid is assumed to be non-viscous, incompressible and irrotational, so that the velocity potential function can be used to describe the flow condition and simplify the interaction between the shells and the liquid. By using the Donnell shell theory, the three coupled partial differential equations can be simplified to one uncoupled partial differential equation and the computational effort can be significantly reduced. Galerkin’s method is applied to simplify a high order partial differential equation into an algebraic equation. The results have been compared with the results of other research and excellent agreement has been found between them.
radius of shell Young’s modulus extensional rigidity, Eh/( I - v2) thickness of shell = h2/12a2 length of shell half wave number along axial direction added mass wave number along circumferential direction radial pressure hydrodynamic pressure
=mxll arc length (afl) time kinetic energy strain energy work done by the liquid axial, circumferential and radial displacements potential function strains at the middle surface curvature changes at the middle surface vibratory stress rotational angles of the X-Z and Z-8 planes density of shell density of liquid Poisson’s ratio circular frequency frequency parameter frequency parameter of shell filled with liquid frequency parameter of empty shell
OF SHELL VIBRATION
The shell under consideration is shown in Fig. I. It is a simply supported, thin-walled, circular cylindrical shell of radius a, length I and thickness h. The axial, circumferential and radial coordinates are denoted by, X, S and 2 respectively and the corresponding displacement components of a point on the middle surface are in turn denoted by u, r and W. The shell filled with liquid is considered to be loaded with the static load of the liquid at the first stage with the initial displacements, u,, v, and w,. If vibration then occurs only in the elastic range, the vibratory displacements, u, o and W, can be superimposed on the initial displacements. In this
Thin-walled axially symmetrical structures have been widely used in industry, flight structures and shipping. The dynamic responses of thin-walled structures filled with liquid, such as oil tanks or cooling towers, due to earthquake, motor and other transportation loading are of great importance. Re997
if one ingores the rotational kinetic energy terms, and where p is the density of the shell. The strain energy U is “=f
(u,e, + c,e,+u,,+,,,)dV
JJ and the work done by the liquid W is Zn I prwa dx dfI.
After tedious manipulation, a set of three equations of motion with corresponding boundary conditions can be obtained. For convenience, non-dimensional variables arc used as follows: U
Fig. 1. Coordinate system and geometry of the shell. 2=X
paper, however, only the vibratory displacements are considered for the dynamic response of the shells. By using Love’s first approximation the displacements can be described as:
nfJ= u(x, 8, I) + z$,,(x, 8, I) u, = w(x, 0, I),
s “(T-(I-W)dr=O. ,l
2pa2(1 + v)
pa’(l - v*) E
3-v L’,,,- -F 2pa’(
l-v + -j-V%
z v u + VM’,,
2pa’(l + v) E
kV8w + (1 - ~~)a’,,,,, = -Epa2(1 -v’)
3-v u,,, - __ 2
V4u - (2 + v )w,, ,,I + w,blJll = -
In eqn (4), the kinetic energy T is
V% - VW’,,,,+ WvY,,,, = --___
where L:, 6: and c’$ are the strains at middle surface and K,, K,, and K,,, are curvature changes at the middle surface. Nonlinear terms in the strain-displacement relation are considered only for E:., 6: and c$, to improve the accuracy of the results. Hamilton’s principle is used to obtain the equation of motion for the complex system where only the kinetic energy, potential energy and work done was considered.
Equations (9)-(11) are coupled and can be simplified to
t,,=C$--ZK,, 0 ~\-(I=t,(J-ZK,,,,
and the corresponding
where ui., I$, and u$ are the initial static membrane stresses and u,, uOand ur;”are the vibratory stresses. According to Love’s approximation, the strain components at a distance z from the middle surface can be written as
pha* vu?\+ ~,o - w-kV4w=yw+pr
u:,=ob+a, r c r,,= elo + %I
1fV pha2 ., u,,,”+ __ u,,, - VW’,,= _ U 2 K
where u,, u, and u, represent the displacement components of any points along the X, S and 2 directions. The vibratory displacements u, u and w represent the displacement components of points on the middle surface. $, and J/,, are the rotation angles of the X-Z and Z-8 planes. The internal stresses are
Replacing the new variables without a bar, the equations of motion can be written as l-v u,,, + __ 2
u, = u(x, 0, r) + z$,(x, 8,f)
d’ 3-v $-TV?
V% + w,,, (16) I ‘I,
I + v)
‘o,,, + pr + w + k V%
+ VW,,, + M’,,,#, - v4pr.
Technical Note Applying Reissner’s simplifying assumption [ 131, eqn (I 7) can be reduced to
For a thin-walled cylinder with a simply supported condition at the ends, the radial displacement can be assumed as ?WRX
kV% f (1 - v2)w,,,,,
w = C,, cos nfI sin -
+ pa2(1 - VZ)
+ v4pr = 0.
From eqns (21) and (23), the potential function can be written as
4 = Q(r) cos nt7 sin !E I
Substituting eqn (24) into eqn (19) leads to
The liquid considered here is non-viscous, incompressible and irrotational, therefore the flow properties can be described by the potential function, 4(x, r, 0, t), which must satisfy the Laplace equation
4,,, + ; +,, + f $,I?,,+ 4,,, = 0.
By the Bernoulli equation, the hydrodynamic pressure acting on the wall is
where I. and K, are the first and second n th modified Bessel functions. From eqn (22), one can obtain A, = 0, therefore
d,, = W+,rl,=,r
OS n0 sin M”x sin ok
where pr is the density of liquid. To satisfy the boundary condition along the contact surface, one obtains
Substitution of eqn (27) into eqn (20) and using eqns (8) and (21), a nondimensional equation can be obtained as
Also, due to axial symmetry one can obtain
~,,LO = 0.
x cos n0 sin qx cos CM. (28)
a c3 ,
Circumferential nodal pattern
m=2 Axial nodal pattern Fig. 2. Mode shapes of two different patterns.
E u. O .s
Fig. 3. Three-dimensional
mode shape with n = 2, m = 3.
Fig. 5. Influence Galerkin’s substituted
method is then applied with eqns (23) and (28) into eqn (18) and this leads to Zn I F(o) ssII
G, dx dt’ = 0,
-v*) VYW,,, ) + V4Pr
Using the orthogonality k(q?+d)+(l
one can obtain E
(32) Let the frequency
which is exactly the same as the natural frequency of thin-walled shells obtained by Reissner .
= kV*w + (I - v2)o,,,,,
and the weighting
of liquid on frequency
PP F =0.132
Some calculations to test the theory in the case of thin-walled shells filled with liquid are presented herein. The parameters used for these studies are v = 0.3, 7= I/a = 4, 7i = h/o = l/400, pF/p = 0.132, where pr is the density of sea water and p is the density of stainless steel. For the vibration modes of the cylindrical shell, different m gives a ‘beam-type mode’ and n gives a ‘lobar-type mode’, which are shown in Fig. 2. A three-dimensional view of the mode shape is also shown in Fig. 3. The frequency parameters of shells filled with liquid, R,, and of empty shells, R,, for lobar-type mode are shown in Fig. 4, where we can find that the lowest frequency does not occur at the lowest values of n; in this case the fundamental frequency occurs at the mode with m = 1 and n = 5. This is exactly the same as the result obtained by Arnold and Warburton and reported in Kraus . They explained this phenomenon by considering the strain energy of the middle surface under both bending and stretching. It can be seen that R, is much lower than C& because the liquid absorbs part of the vibratory energy which makes the frequency lower. Also
then with pr = 0 *2
Pdl - v2) (1 - +?q
s e I&
R,=(l + k(q2 + n’)‘,
ia ti v = 0.3,-i= Fig. 6. Frequency
Fig. 4. Frequency
m Pp l/a = 4, h = h/a = 11400, 7 = 0.132 parameter variation ence of liquid.
with m under influ-
shell filled with liquid. Previous research which has been done on this topic must solve at least a 3 x 3 complex eigenvalue problem or I2 coupled algebraic equations which need appropriate computing software to give a more precise solution. But for a draft design or an urgent analysis, this method shows some advantage over other methods.
z $ a
P @ 0.2 z $ .s
Fig. 7. Frequency parameter variation with thickness of shell.
known as ‘added mass’, is the same as the result obtained by Goncalves 1151.Therefore, the liquid can be considered as a mass added to the cylindrical shells. It is observed that the value of R,/Rv varies greatly for smaller n but approaches I and n increases to a sufficiently large number as shown in Fig. 5. For a fixed n, RL/Rv increases as m and also approaches 1, which is shown in Fig. 6. For a thicker shell the frequency parameter increases more rapidly and the value of R,/Rv becomes higher as m increases, as shown in Fig. 7. This can be explained as a thin shell absorbs less energy when it is filled with liquid. For higher density of the liquid, the frequency parameter increases more slowly because of the higher absorption rate of the vibration energy which also influences the natural frequency of the shell. CONCLUSION
The method illustrated in this paper gives a simpler and faster way to solve the natural frequency of a thin-walled
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