EngineeringStructures,Vol.
BUTTERWORTH 01410296(94)000042
17, No. 3, pp. 221230, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 01414)296/95 $10.00 + 0.00
Free vibrations of partially filled cylindrical tanks R. K. Gupta Department of Engineering, University of Tasmania, Launceston, Australia 7250 (Received October 1992; revised version accepted March 1994)
The coupled free vibration characteristics of partially or completely liquidfilled ground supported circular cylindrical tanks are investigated. Attention is given to partially filled broad tanks with a height to diameter ratio of less than one. For the vibrations of the container itself Flugge's exact equations of motion are applied and the radial liquid dynamic pressure is obtained in terms of the liquid velocity potential (~). The complex roots of the eighthorder auxiliary equation are obtained by using four different methods. The suitability of these methods is discussed based on the relative magnitude of the various coefficients in the auxiliary equation. The contained liquid is assumed to be incompressible and inviscid. The bottom of the tank is considered to be flat. The results are compared with those obtained by Haroun and Housner and found to be in good agreement. Coupled natural frequencies in bulging modes are obtained for various parameters, i.e., (i) height to diameter ratio (A); (ii) the liquid depth factor (al); and (iii) the tank wall thickness factor (a2). It has been shown that Flugge's exact theory is a versatile tool which can easily be used for such problems. The results can be obtained to any desired degree of accuracy in any frequency range. Keywords: cylindrical tanks, free vibrations The evaluation of the free vibration characteristics of partially or completely liquidfilled cylindrical tanks is of primary importance for their design and for examining their safety against earthquakes. The dynamic behaviour of groundsupported liquidfilled storage tanks has been extensively studied during the past five years. Theoretical and experimental investigations have been conducted for a better understanding of their behaviour during earthquakes. Recently Haroun and Housner 13 have determined the dynamic characteristics of a liquidshell system by means of a discretization scheme in which the elastic shell is modelled by finite elements and the liquid region is treated as a continuum by boundary solution techniques. Special attention is given to cos0type modes for which there is a single cosine wave of deflection in the circumferential direction. It has been shown that the coupling between liquid sloshing and shell vibrational modes is weak l. Nash e t al. 4 have studied the elastic behaviour of liquid storage tanks subject to horizontal seismically induced motions of the base. Flugge's equations are used for the tank and the motion of the liquid is obtained by using the Laplace equation. Natural frequencies of the coupled liquidtank system are obtained by using a computer program called EXLITANK. In the present investigations the free vibration character
istics of partiallyfilled groundsupported broad cylindrical tanks have been determined. For the vibrations of the container itself Flugge's original equations are used and the radial dynamic pressure is obtained in terms of the liquid velocity potential (~b). The arbitrary constants in the expressions for the assumed solution, and the roots of the auxiliary equations are, in general, complex quantities which lead eventually to the problem of evaluating an eighthorder complex determinant. The eight roots of the auxiliary equation, the arbitrary constants of the assumed solution, and the elements of the determinant resulting from the application of boundary conditions are all found to be complex and are treated as such. The roots of the auxiliary equation are obtained by four different methods, discussed in Appendix 1. It has been assumed that the contained liquid is incompressible and inviscid and that the bottom of the container is flat and rigid. Then for the vibrations relevant to the liquid, the solution for the velocity potential is assumed as the sum of two sets of linear combinations of suitable harmonic functions, the unknown parameters of which are required to satisfy the remaining two boundary conditions, i.e., the kinematic condition along the wetted shell surface and the free surface condition of the liquid. With this procedure, one can determine the natural vibrations of both sloshing and bulging types which are
221
Free vibrations of partially filled cylindrical tanks: R.K. Gupta
222
dominated by the liquid surface vibration and the flexural vibrations of the shell, respectively. In the present study, only the bulging type of vibrations are investigated. The free vibrations of thin, empty, groundsupported tanks, free at the top and simply supported at the base have been studied by the author~. The present work extends the application to the case of shells that are partially filled with liquid. Flugge's exact equations are applied and results compared with those obtained by Haroun and Housner ~ and are found to be in good agreement.
Formulation of problem The problem under consideration is that of the free vibrations of a groundsupported liquidfilled cylindrical tank of radius a, height H and thickness h, which is filled to a depth d with an inviscid and incompressible liquid of mass density Pl. The bottom of the tank is assumed to be fiat. The fixed coordinate axes (x,O,r) are attached to the bottom of the tank with the origin O at its centre. The equations of motion for a circular cylindrical shell are given a s 6 1v
l+v
u" + ~  (l+k)//+ ~  v '  l~w'" v + 1~
l+v
kw"
+ uw'  ¢ ~02u =0
8
u = ~ A~eAiX/"cosmOsinwt i= I 8
v= ~
lv
,
C~e~iXn'cosmOsinwt
i=1
where m denotes the number of circumferential waves and o~ denotes the natural frequency of the coupled liquidtank system. The artibrary constants Ai, Bi, Ci are evaluated by considering the boundary conditions at each end of the shell. Substitution of equation (2) into equation ( 1) would result in the corresponding set of auxiliary equations.
Dynamic pressure Let ¢h(x,O,r,t) be the velocity potential introduced in order to express the velocity and dynamic pressure of a perfect fluid as Od? O~b aOCh] v(x,O,r,t) = (Vx,Vo,Vr) = a 0~' ~ ' Or/
(la)
o6
pd(X,O,r,t) =  p ~ t
3  u k w " + ¢v  7 2 0 2 v ~~t2=O
(lb)
3  v ,, "" + vu'  ~  k v "
(2)
8
w =~
, 1v u + 9" +  ~  ( l + 3 k ) v "
ku'" + ~ku
B~e~i~/~cosmOsimot
i=1
(x,O,r,t)
(3) (4)
where Pd is the linearized dynamic pressure exerted by the liquid which is obtained by neglecting the static pressure and nonlinear terms of components of v. For the irrotational flow of an incompressible inviscid liquid, the velocity potential, dp(x,O,r,t), satisfies the Laplace equation
+ 9 + w + k[w"" + 2w ....
v24, = 0 +'w'+2~+w]
+ 7
a 2
(lc)
(d < x < H)
in the region occupied by the liquid (0 < r < a, 0 < 0 ~< 27r, 0 < x < d). Since the velocity vector of the liquid is the gradient of the velocity potential, the liquid container boundary conditions can be expressed as follows. (a)
where ()'=
a '0(1 
(5)
202W
O~ (O,O,r,t) = 0 Ox
__~20(), p(s'2( 1  ~ )
ox ' ( ) = ox  
E
At the rigid tank bottom, x = 0, the fluid velocity in the vertical direction is zero, i.e.
'
(b)
h 2 Eh k = _~; D 1v 2 12a':
The letters u, v and w denote the axial, tangential and radial displacement components on the middle suface of the shell thickness corresponding to the coordinates x, 0 and r, respectively. The notation Pd represents the dynamic pressure exerted by the liquid on the tank wall in the radial direction, p,., v and E denote the mass density of the tank wall, Poisson's ratio and Young's modulus of elasticity, respectively. For homogeneous boundary conditions the modal frequency will be a function of the single value of axial waves (n). The general solution for modal vibration can easily be written in the following form
(6)
The liquid adjacent to the wall of the elastic shell, r = a, must move radially with the same velocity as the shell ~x(X,O,a,t ) = ~Ow (x,O,t)
(71
where w(x,O,t) is the shell radial displacement. (c)
At the liquid free surface t(d,O,r,t) = 0
(8)
The solution, ch(x,O,r,t), which satisifes the boundary conditions at the rigid bottom plate (equation (6)), and at the quiescent liquid free surface (equation (8)), can be expressed as
Free vibrations of partially filled cylindrical tanks: R.K. Gupta Ai[(I)
qb(x,O,r,t) = ~ m~l
223
n~l
+ A 2  Pik2m2)(lI
+ Plk3 A2) + (P2mAi) 2] = C i [ ( P 3 k m A 2)
[Am.(t)lm ( a . r / a ) cos ( a.2c/a) cos (toO)]
in which a . and I . . ( a . r / a )
(9)
(P2mAi)  ( f l  m 2 + P~k3A2)(vAi kA3i  Pikm2Ai)]
n
; (n = 1,2,...)
+ A 2  P,k2m2)(~~  ma+
(10) (11)
(Z/2)m+2p p!F(m+p+l)
pO
Using the boundary conditions of equation (7), we obtain Z
£
a
lll dO Ii
Do A8 + D1 A6 + D2 A4 + D3 A2 + D 4 = 0
(15c)
where
+ { p 2 _ k3Pl(P,k2 +
2)  1}] + {k3P,(~ + 2v) + a k , }
192 = m4[kP,{2  2P2P3  P~k3  P~k2} + {2 + P,k2  2P 2
A,..la)l'(a.) ~ = ~
+2p2k2k3 + P,k3}] + m2{2vP2P3  2u + 2vP~k3
# ( x, O,t )cos( a , x / a )dxcos( m O)
+ 2/'2  2P3  2Plk3 + l ) ( k P 2  2kP,  P,k2  3  2P,k3)}
2afi=dOfiW(x,O.t)cos(a.x/a)dxcos(mO) amn( t ) =
(15b)
D, = mZ[k{1 + p2 _ 2P2P3  2ksP2} (12)
= ¢v(x,O,t)
 ( ~  Pike m2 + Az)(P3kmA 2  m)]
Do = Plklk3
[Amn(t)°t"I"~(°t~)c°s(°6~c]a)c°s(mO)]
m~l n~l
Pik3 A2) + (P'21~tA/)2]
(A3 + P l m 2 A i ) (  P 2 m A i )
= Ci[ ( 1)1~i  k
Ira(Z) = ~
(15a)
are given by Bi[(O
a, =
m2

1
+ {P,k3 + O ( v + ~)} + ~ {P,k3(1  ~  ~)}
a,d.n.I'm( Ot,)
(13)
D3 = m6[kp~  2Plk2 + P~  P~k2k31 ] + m412P,P2  2VPl + 2kzP,P3  2P~ + 2PZk2k3 + 2 + ~ ( 3 + Plk3
The dynamic pressure can therefore be expressed as
+ 2P,k2  kP~)] + m2[{p22  P~(4 + 3k)  p2k2k 3  1 pe(x,O,a,t) =  P l ~b(x,O,a,t) + 2 O ( v P j  P3  ~  1  Plk3)}
 2ap~ ZZ
l
m>l n~l
fzdofi~
o
2
+ k_{p2 _ p2 _ v  P2fl + p2k2k3fl + ~} + If/(1 + P,k3)
('0.0,t)cos(a. rl/ a )dqcosz(m O)I.,( %)cos(a.ev/a)
+~{P,k3(1O)v
2+1~}]
a,£. (a°) D4 = m8p,k2 + m h {  2 P , k 2  1~(1 + P,k2)}
= _ 2apl ~~ d
(14) + m4
{
'
Plk2 + l~(f~ + 2 + 2Plk2)  ~ P l k 2 ~
f l ~( rb 0,t)cos( % n / a ) d n l m ( %)cos(o~.x/a)
f~
+ m2[f~(1 + 2 a + P~k2) + ~ { P , k 2 ( ~  1)+ 1~}]
aJ', (a.)
+ (~'~2+~k (1 +~'~~'~2)}
The analysis of a partiallyfilled tank is achieved in three steps: (i) Consider an empty tank to find the homogeneous solution of equations ( l a  lc). (ii) Consider the wet part of the tank to obtain the particular solution (iii) The boundary conditions at the top and bottom of the tank and at the junction of the wet and dry portions of the shell are then applied
Homogeneous solution The general solution for modal vibration is written in the form given in equation (2). The auxiliary equations can be written as
}
where 1 
Pl
2
v
1 + u
,Pz
k~= 1 k;kz=
2
3 
,P3
v
2
'~=Yzw2
1 + k; k3 = 1 + 3 k
The coefficients D j ( j = 0,1,2,3,4) are functions of E, m, h, a, Ps, v and w. Equations (15a and 15b) are linear functions of the complex constants Ai, Bi and C , while equation (15c) is a polynomial in the Ai. Equations (15a and 15b) can be solved for the constants A~ and Bg in terms of the Css, so that the displacement of equation (2) can be expressed in terms of just these complex constants and the roots of the auxiliary equation, the &. The eighthdegree polynomial in
Free vibrations of partially filled cylindrical tanks: R.K. Gupta
224
~t i (equation (15c)) is solved by four different methods, i.e. (i) ASCE manual method; (ii) Brown's method; (iii) Descarte's method; and (iv) Ferrari's method. Of these methods, Descarte's method is found to be more successful for the kind of tanks considered in the present study. By using all four methods the complex roots of the auxiliary equation for one such container have been obtained and compared in Reference 5.
where L2a4 plOJ21m( Otn)
dOot~/~ot.)[Z.oq(ko~.  P, km2 + n) + Z.m(1 + P3ka~) + YY ZZ]
YY= 1  I~ + k(ot~ + 2m2o? + m4  2m2 + 1)
ZZ
Complete solution The complete solution for the axial, circumferential and radial displacements can be expressed as
a3 pl tO2Im( Otn)

D Otnl~n(Otn)
hi eXd/%in( a.d/ a ) +   { eXd/acos( a.d/ a )  1}] Oln
T,.. =
\o#,/ / U:
Sn sin(a~/a) cosmOsincot (16a)
Ci Ai exex/a + Z 
v=
n~l
Ci Bie~/~ + ~ 
W=
Q~ c o s ( a ~ / a ) sinmOsimot (16b)
For a partiallyfilled tank, two sets of solutions are obtained, namely,
n~l
R. cos(a.x/a) cosmOsintot
Ci e'Xix/a + Z 
(16c) (i) For the wet part of the shell (0 < x < d) (ii) For the dry part of the shell (d < x < H)
n~l
where S., Q. and R. are the bounded coefficients and Ai and B i c a n be obtained from equations (15a and 15b). Substitution of equations (16a and 16c) into equations ( l a and lb) leads to the following relations
2 ~  ct.
lV(l+k)m 2
= kct3 + va.
v a.Q. 2}S. + ~l + m
1v } ~  ka.m 2 R.
l+v ~manS.+ 3v = m+~kmo~
Boundary conditions
{
1v ~rn2(12
The two sets of solutions together include 16 unknown coefficients. Enforcing the boundary conditions at the base and the top of the shell, and the compatibility equations at the junction of the lower and upper parts of the shell yields 16 simultaneous algebraic equations which can be written in a matrix form as [/)]{Ci} = {0}; i = 1,2,...,16
}
+3k)~x2 Qn
} R,,
(18)
(22)
The frequency equation is obtained by setting the determinant of the complex coefficient matrix, [/31, equal to zero. The various boundary conditions at the base, the top and the junction of the lower and upper parts of the tank are given in Appendix 2.
System's axisymmetric fundamental frequency
From equations (17) and (18), we obtain
S.=Z.R.
(19)
Q. = ~ . R .
(20)
where
Ogn =
/'/
23,
7r d
A simplified formula for natural coupled frequencies in bulging modes 7 is written as
[ot.{kot 2 + v  P,km2]{lI  m 2  P,k3~}  {1 + P3koc~.}{P2mZot.}] f l ( t~Jt~)lra a d 0 d x
A
to2 =
(24)
[m{P3kc~ 2 + l}{fI  P , k z m 2  c~} + { v + k a 2  P, krnaHP2maZ.}]
A A = { ~  P,k2 m2  oc~}{~  m 2  P,k30t]} +
From equation (9)
{P2mo&}2
Similarly,the substitutionof equations ( 16a 16c) into equation (Ic) yields the followingexpression for Rn in terms of the unknowncoefficients Ci.
=Amn'#rlcos(o:)comO 6 , r = amnl~)l~n ~OtnaJ COS otn
cosm0
(26)
8
gn = Zn Z i=1
CiTin
(21)
Assuming the internal dynamic pressure (Pal) to be uni
Free vibrations o f partially filled cylindrical tanks: R.K. Gupta form over the walls of the cylinder, the radial displacement (~) can be expressed as
225
Table2
Parametric study, natural frequencies (Hz) Circumferential mode (m)
a2~
(27)
w = Pl Eh ~

Substituting ~b,
A
n
0
1
2
3
4
5
1.0
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
6.25 13.98 18.75 6.75 14.26 18.95 7.31 14.53 19.13 7.89 14.77 19.48 8.39 15.46 23.48
4.14 10.11 14.02 6.00 9.88 14.36 6.86 11.86 16.07 6.72 11.12 16.15 5.90 11.76 17.70
4.01 9.71 13.97 5.28 9.80 14.11 6.57 11.97 16.03 6.30 11.46 16.85 5.60 11.73 17.54
3.69 9.45 12.76 4.86 9.31 14.17 6.43 12.35 15.88 6.24 11.67 16.73 5.10 11.40 17.82
3.72 10.12 13.18 4.57 9.24 14.61 6.37 12.06 15.87 6.79 11.20 16.12 5.13 11.14 17.57
3.66 10.31 12.97 5.22 9.49 14.38 6.29 11.90 15.23 6.88 11.38 15.63 5.54 11.81 17.60
0.8
Eh ll( a.) to2 = ct, pla3 [o(Oln)
(33) 0.6
Equation (33) is an approximate formula which is used to estimate the system's axisymmetric fundamental frequency and has proved to be useful in evaluating the frequency determinant, Iz l, requiring a trial value of to.
0.4
0.2
Example problem In order to investigate an optimum tank geometry, a circular cylindrical tank of capacity Q(=15000 m 3) partially filled with a fluid of volume V(=10 000 m 3) has been studied. The coupled natural frequencies and mode shapes are obtained by varying height (H) to diameter (D) ratio (A) for the same quantity of liquid in the tank. The fluid is taken as water with mass density (Pl) 1000kg/m 3. The various data for the tank are as follows: E = 206 GPa; v = 0.3; Ps = 7850 Kg/m3; h = 30 mm Q = (~/4)D3(A); D = ( 4 Q / ~ A ) l / 3 ; H = DA; d = 4 V / ~ D 2 Based on the above equations, the various dimensions are listed in tabular form below Tank
h (m)
D (m)
H (m)
d (m)
A B C D E
1.0 0.8 0.6 0.4 0.2
26.7301 28.7941 31.6920 36.2783 45.7078
26.7301 23.0353 19.0152 14.5113 09.1415
17.8201 15.3569 12.6768 09.6742 06.0944
are made on the VAX 780 system available at the University of Melboume. All the real and complex numbers are obtained in REAL*16 and COMPLEX*16 precision, respectively. In the computations, the first five terms of the series in equation (16) are considered. From the results it is observed that (i) There is a good agreement of results obtained by using Flugge's theory and the results obtained in Reference 1 (ii) The natural frequencies in Reference 1 corresponding to m = 1 and 2 are not available for comparison. It is, therefore, doubtful whether the method used by Haroun and HousneP is applicable to estimate the lower order frequencies which are normally important in response calculations (iii) The results obtained in Reference 1 estimate the frequencies on the higher side The results of the example problem are presented in Tables 2  4 and Figures 1  4 . It is observed that
Results To illustrate the effectiveness of the analysis under consideration, a comparison of natural frequencies is made between the values computed using the present analysis and the values computed by Haroun and HousnerL In the present analysis the results are obtained by using Flugge's exact equations of motion 6. The comparison of results of natural frequencies in circumferential modes corresponding to first radial mode is shown in Table 1. The computations
Table 1 Natural frequencies in circumferential modes corresponding to first radial mode (Hz)
(i) Table 2 shows that in lower axial modes (n = 1,2) the higher natural frequencies for tanks lie in the lower order circumferential wave (m) zone while the lower natural frequencies are obtained in the higher order circumferential wave zone (ii) The natural frequency increases as the ratio A decreases. However, the trend does not follow for extremely shallow tanks (A = 0.2) (iii) The fundamental natural frequencies in various axial modes are sensitive to the parameter A (iv) Figure 3 indicates an increase in fundamental frequency as the depth factor ( a l ) decreases
Circumferential mode (m)
Table3 Type of analysis
1
2
3
4
5
6
7
8
Flugge's theory Reference 1
31,58 19.00 10.92 7.77 6.59 6.46 7.54 10.54 * * 11.85 8.06 6.57 6.77 8.28 10.60
(m=n=
A oq = d i D
*values not available Data: H = 12.5" (0.318 m); d = 11" (0.279 m); D = 8" (0.2032 m); h = 0.05" (0.0013 m); E= 0.735E06 psi (50.6E05 kPa); p , = 0.133E03 Ib.s2/in 4 (1.42E03 kg/m3); ~ = 0.3; pl = 0.93662E4 Ib.s2/in 4 (1000 kg/m 3)
Effect of liquiddepth factor (c~1) on natural frequency 1), Hz
1.0 0.8 0.6 0.4
1.0 3.95 4.05 4.17 4.61
0.8 4.69 5.16 6.15 6.49
0.6 6.36 6.49 6.90 6.91
Remarks 0.4 5.73 6.02 7.18 7.22
h = O.03m
Free vibrations of partially filled cylindrical tanks: R.K. Gupta
226
Table 4 Effect of liquiddepth factor (al = d/D) and tank thickness f a c t o r (~2h) on natural f r e q u e n c y (m = n = 1), Hz a~
%
Frequency (Hz)
Remarks
1.0 0.8 0.6 0.4 1.0 0.8 0.6 0.4 1.0 0.8 0.6 0.4
1.0
3.95 4.05 4.17 4.61 5.50 5.94 6.84 7.29 7.08 7.74 8.43 8.75
A= 1
2.0
3.0
9
6 ~5 4
•X = 1.0 I
3
0.4
0~6
0.8
1.0
a 1
Figure 3 Effect of factor al on f u n d a m e n t a l f r e q u e n c y ( m = n =1)
~X
= 0.4
10  
••........•
"
Z=

0.6
__X=0.2
~
~,= 0.8
L
.X= 1.0
fJ
C~
I
1
I
I
I
I
I
I
I
2 3 4 CIRCUMFERENTIALWAVENUMBER,m
I
5
=2.0
Figure 1 Effect of p a r a m e t e r s A and m on natural frequencies in first axial m o d e (n = 1)
_c~2 = 1.0
18 I
0.4
161
14
~
n
=
3
016
018
1.0
Figure 4 Effect of parameters a~ and a2 on f u n d a m e n t a l fiequency (m = n = 1)
u:
12 ~
j
n
=
2
f ~
o!~
o.',
o.'~
n
L~
=
l
,'.o
PARA.METER, ~. Figure2 Variation of f r e q u e n c y (m = 1) with p a r a m e t e r h in first three axial m o d e s (n = 1, 2, 3)
An approximate formula, developed for the estimation of the fundamental axisymmetric frequency produces reasonably good results. The solution of the eighthorder polynomial is sometimes troublesome as it produces eight complex roots and the method of solution depends upon the relative magnitudes of the coefficients a, /3, % and 6. The application of the four methods in solving quartic equation is discussed in Appendix 1. Descarte's method is found to be more successful for the kinds of tanks considered in the present analysis.
References (v) Figure 4 indicates an increase in fundamental frequency as the thickness factor ( a l ) increases
1 2 3
Conclusions Flugge's exact theory is a versatile tool which can be applied to any vibration problem for circular cylindrical tanks. This is able to estimate the coupled natural frequencies of the system up to any desired degree of accuracy in any frequency range.
4
5
6
Haroun, M. A. and Housner, G. W. 'Complications in free vibration analysis of tanks', Proc. ASCE, Engng Mech. Div. 1982, 108, 801818 Haroun, M. A. and Housner, G. W. 'Dynamic characteristics of liquid storage tanks', Proc. ASCE, Engng Mech. Div. 1982, 108, 783800 Haroun, M. A. 'Vibration studies and tests of liquid storage tanks', J Earthquake Engng Struet. Dyn. 1983, 11, 179206 Nash, W. A,, Shaaban, S. H., Watawala, L. and Lee, S. C. Flexible shells (Eds E. L. Axelrad and F. A. Emmerling) SpringerVerlag, Berlin, 1984 Gupta, R. K. 'Free vibrations of cylindrical containers,' Report L002ENG101992 Department of Engineering, University of Tasmania, Launceston, Australia Flugge, W. Stresses in shells SpringerVerlag, Berlin, 1960
Free vibrations of partially filled cylindrical tanks: R.K. Gupta 7 8 9
Gupta, R. K. and Hutchinson, G. L. 'Free vibration analysis of liquid storage tanks', J. Sound Vib. 1988, 122 (3), 4 0 1  5 0 6 ASCE Manual of Engineering Practice 1951, ASCE Manual No 31, 'Design of cylindrical shell roofs', ASCE, New York, 1951 Bamard, S. and Child, J. M. Higher algebra, Macmillan, 1936
227 1
(A1.5)
1
Appendix I
1 X4 = 2 [   ~ 1 1  ( ~ 2 2  ~ 3 3 ) ]
Solution o f eighthdegree equation
The auxiliary algebraic equation (15c) contains only the even powers and has eight complex roots occurring in four conjugate pairs. The existing methods of determining the roots are tedious and involve several operations with complex numbers. The eighthdegree equation is first recast as a quartic equation by means of a suitable substitution as follows:
where 2a
Zl = Yl
2a Z2 = Y2
(A1.6)
~
2a Z3 = Y3  ~ 
(AI.1)
7/  A 2
3
Then equation (15c) may be written in the form where 7/4 q_ O~7/3 "1 /37/2 ..1_ ~//V "l 6 = 0
(
(A1.2)
1)
Yl = P cosA + ~
where a = D~/Do;/3 = D2[Do; "y = D3/Do; 6 = D4/D o
Y l = p ( c o s A + 1 sinA)
The biquadratic equation (A1.2) is solved by four different methods briefly described in the following: (a) A S C E M a n u a l method: ASCE Manual 318 essentially consists in reducing the eighthdegree equation in the first instance to a quartic equation (A1.2) whose resolvent cubic equation is solved by trigonometric methods. The quartic equation (A1.2) is transformed into a reduced quartic by letting
1 sinA)
where (a2312c)1/2
p =  
(A1.8)
/~i ~'~ "~ ~/7/ (i = 1,2,...,8) can easily be obtained. By substituting cos A = y / p on the righthand side of the
Thus
7/=x 4 which takes the form X4 + ax 2 + bx + c = 0
(A1.3)
where a, b and c can be dervied in terms of a, /3, y and & The four complex roots of equation (A1.2) are derived as follows: O~ 4
O/ 7/2 '~X2  4
7/3 = X3
(A1.7)
/
O/
7/1 Xl
sinA
(A1.4)
O/ 4
trigonometric identity, i.e.: cos3A = 4cos3A3cosA the lefthand side can be expressed in terms of a, b, and c. Now let A = 30°A, then: cos3A = cos(90°3A) = sin3A. From the above A can be expressed in terms of a, b and c. (b) B r o w n ' s method: Brown's method 9 is based on the theorem that the expression of any biquadratic form as a product of quadratic factors is dependent on the solution of a cubic equation only. The quartic equation (A1.2) is reduced to a pair of quadratic equations as follows 7/2 _{._
+
{o
"1 ( ( ~ 1 2 ) 2   ( f l " 0 ) )
1/2 7/
}
+ ( 0 / 2 ) 2  6) 1/2 = 0
Og 7/4 = X4  q~
where
(A1.9) 1
x,
(~
+
~)] where
Free vibrations of partially filled cylindrical tanks: R.K. Gupta
228 R
02+A
0=13
(AI.10)
A = aT48
(AI.ll)
R = a13 T  y z  a28
(Al.12)
The first approximation is 0 = C and successive approximations are obtained by the rule R
0,+l = C  ~ + ~
Corresponding to any root, q, of equation (A1.22), we can find values of l, m, l' and m' which will satisfy conditions. In the above equation I = ae  4bd + 3c 2
(A1.23)
J = ace + 2 b c d  ad 2  c 3  eb 2
(A1.24)
The roots of the reducing cubic equation (A1.22) are obtained as
(Al.13) tl = sl +
where 0, is the nth approximation.
(A1.25)
$2
1
ifi
1
ifi
t2 =  ~(Sl + s2) + ~  (Sl  s2) (c) Descarte's method: Descarte's method 9 can also be used by expressing the quartic equation as
t3 =  ~(s, + s2)  ~  (sl  s2) u = a~)4 + 4br/3 + 6cr)2+ 4 d ~ + e
(A1.26)
(A1.27)
(Al.14) where
where a = 1; a = 4b; 13 = 6c; 3 ' = 4d; 8 = e
(Al.15)
sl = [r + (q2 + r2)1/2]~/3
(AI.28a)
s2 = [r  (q2 + F2)1/2]1/3
(AI.28b)
Equation ( A l . 1 4 ) is expressed in biquadratic form as where u = a('o 2 + 21r/+ m)('q 2 + 21'r) + m ' )
(Al.16) I
a
d e lm' + l' m = 2  ; m m ' = a a
l m =
l' m'
0 l' 0 m'
2 l+l' m+m'
l m
(Al.17)
m+m' ] Im' +l' m 2mm'
Substituting the above values for l+l', m+m', etc, and multiplying each row by a/2, we have "1
b
b 3c2all'
all' d
3c2all' ] d e
=
l' 
(Al.18)
=0
a
m
(d +~)+
2c 4t m' =   +   a a
0 0
l+l' 211' lm' +l' m
1 =0
(A1.29)
The condition of equation ( A l . 1 7 ) evaluates the unknown coefficients as
l + l' = 2b; m + m' = 6c _ 4 ll' a
J
q =  ~; r =  g
Expanding and equating coefficients, we have
2 (d
({c+_~}z

e)1/2
m
bm')
a (mre')
2b l' l =  a
(A1.30)
(A1.31) (A1.32)
(A1.33)
where t represents any root of the reducing cubic equation (A1.22). The two quadratic equations in equation ( A l . 1 6 ) may be solved as usual to obtain four roots of the quartic equation ( A l . 1 4 ) .
(Al.19) (d) Ferrari's solution: In Ferrari's solution 9 the quartic is expressed as
If we write au
=
(ax 2 I 2hx + s) 2  (2mx + n) 2
(A1.34)
(A1.20)
t = c  all'
Expanding and equating coefficients, we have Equation (A1.19) becomes I
a b c+2t
b et d
c+2t ] d = e
2m 2 = as + 2b 2  3ac; mn = bs  ad; n 2 = s 2  ae (A1.35) (A1.21) Eliminating m and n, (s 2  ae )( as + 2b 2  3ac ) = 2( bs  ad) 2
which when expanded is 4t 3  It + J = 0
(A1.22)
which reduces to
Free vibrations of partially filled cylindrical tanks: R.K. Gupta s 3  3 c s z + ( 4 b d  a e ) s + ( 3 a c e  2 a d z  2eb 2) = 0
229
(i) S i m p l y  s u p p o r t e d e n d ( b a s e )
(A1.36) u = v = w Mxl~=o = 0
The second term can be removed by the substitution s = 2t + c
(a)
(A1.37)
u
8 E A i f W i e'
=
(A2.1)
i=1
and equation (A1.36) becomes 4t 3  It + J = 0
(A1.38)
i=1
n~l
i=1
n~l
i=l
&'
which is the reducing cubic. Equation (A1.35) becomes m 2 = at + b 2  ac mn = 2bt + bc  ad
K
(d)
(A1.39)
M.=~(w"
n 2 = (2t + c ) z  a e
i=1
+ vfCu'vV)
8
M. = ~Clsi a [(A 2  vm2  / ~ ~ i
Thus, if t~ is a root of equation (A1.38) and
 ~
ni)
i=1
~ Tg,,(La~+

m l = ( a t l + b2  ae)l/2; nl = (2btl + b c  a d ) / m l
l.q'rl2L + Z n L a n +/.q'n
~nL)]
= 0
the equation, u = 0, can be put into the form ( a x 2 + 2 b x + c + 2h) 2  (2ml + nl) 2 = 0
(A2.4) (A1.40)
(ii) F r e e e n d ( t o p )
gx= Tx= S
and its roots are the roots of the quadratics, i.e. a x 2 + 2 b x + c + 2tl = +(2mix + n l )
(A1.41)
=MxL.= 0
Expressions for the above quantitites in its exact form are as follows
It should be noted that the three roots of equation (A1.38) correspond to the three ways of expressing u as the product of two quadratic factors.
K
(a)
M x = a~(z w
,,
+ v¢~ 
u'
 v O)
8
Applications of above methods
2 cdirY[ }[2  lgn 2  1 ~ i  Fm2Bi]e AiH/a =
The applicability of the above methods in solving quartic equations depends upon the relative magnitudes of the coefficients a, /3, 7 and 6. The ASCE M a n u a l method is successful when the coefficient 6 is the largest among a,/3, y and 8, and a is relatively larger than /3 and % Brown's method is applicable when the coefficients T and 8 are small compared to a and/3. Descarte's method is normally successful when the coefficient/3 is larger than a, 1' or 6. Ferrari's solution is found to work when the coefficient T is larger than/3. Among all the above methods, Descarte's method is found to be more successful for the kinds of containers considered in this paper.
0 (A2.5)
i=1
(b)
Nx = D (
8
+ v¢ + v w )  ~ w "
[ ;cf
i= 1
+
h2A?] . . . ~ j eXiH/a = 0 ( 1 2 . 6 ) + , ,  .12a
t.
K T~ = ~ (1  v ) ( w '  v') a
(c)
8
E fdirYXi(m + Bi)eXl H/a
(A2.7)
i=1
Appendix
2
(d) S x 
Boundary conditions
Four boundary conditions at each end of the container, simplysupported at the base and free at the top, are applied to obtain a set of eight homogeneous equations in terms of eight unknown complex constants (Ci, i = 1, 2, ..., 8), i.e. [/5]{Ci} = {0}, i = 1, 2 ..., 8. Since the boundary conditions are homogeneous, the determinant of the complex coefficient matrix, 161, must be zero for a nontrivial solution. This results in the characteristic equation whose eigenvalues determine the natural frequencies of the container. The corresponding eigenvectors determine the mode shapes. The various bounary conditions for the simply supportedfree case are as follows
D ( 1  v) (a+v')+ a 2
K ( 1  v) a 3   2( V
8 h2Ai E ~iirY[mAi "1 Ain i + ~
(m + ni]e AiH/a
p   W ¢ ')
(A2.8)
i=1
(iii)
Compatibility conditions (x = d)
(a)
= Uwe, 8
8 =
i=1
Tin/ e x#/" ]
[A , + Z ZL
i=1
sin(,
°d/a)
n>l
(A2.9)
230
Free v i b r a t i o n s o f p a r t i a l l y f i l l e d c y l i n d r i c a l tanks: R.K. G u p t a
(f) Nx(dry)la= gx(we,)ld
Vd~yd : I)wet[d
(b)
~BiCf ~y:
~C~ii ~'
i=1
i=1
i+
cos(a.d/a) n~ 1
(A2.10)
Ti,,/eh,~m1
i=1
[
(AA~ + a
a3 J
= ~ C ~ ~' a(he4~+~nBi+ v )  a3 j i=1
(c)
W.r~b = Wwe,b 8
s
Efdi ry : E C Wet[1 + ~ 2 . c o s ( a . d / a ) i=1
i=1
n>l
2~cos( a,,d/a) T~./e~,~/~]
Ti,,le x,am]
(A2.14)
(A2.11)
(g) L(~,~,)b T~(weol~ :
w,~o,],~= W~e,la
(d)
EC}I~YAi(m + Bi) = EC~Yie' A,(m + B~) ECdirY Ai = ECwietk Ai  E a,,L sin(and~a) i=1
i=1
i=1
i=1
n~l
 ~ ( m + ~.)a.L sin(a.d/a) T.,/eX,~/"1 (A2.12)
Ti,,/e*,a/a1
(h) S*(dr~)ld = L(~,O]d
(e) Mx~.~b = Mx~weob 8
ECi ry ~ t I   (  m A l i=,
i= I
8 [
= Eft
et (1~ 2 
(A2.15)
J
]~FFI2  1 ~ i 
z
+ AiBi)+
/la
~T
(m + ni)
= _~C,w. . . 2. . .a( mA, + A~B~)+ ~3 (m + Bit
lIFnBi )
i= 1

~cos(a,d/a)2,(a 2 +
vm 2

][.a\
+ a,,~,, + Ka,, (m+ ~,,)} a~
n>_l
n~l
+ otnZ,, + wn ~nTiJeXid/"l]
X\(1
(A2.13)
Z, sin( a,,d/a ) Ti,,/e~'ia/a] 2
(A2.16)