Bulging Modes of Circular Bottom Plates in Rigid Cylindrical Containers Filled with a Liquid
Shock and Vibration, Vol.
Bulging Modes of Circular Bottom Plates in Rigid Cylindrical Containers Filled with a Liquid
0 the liquid. The sloshing modes of circular cylindri
1 Department of Mechanics University of Ancona Ancona 60131 , Italy
In this article the free vibrations ofthe bottom plate ofan otherwise rigid circular cylindrical tank filled with liquid are studied, considering only the bulging modes (when the amplitude of the plate displacement is predominant with respect to that of the free surface). The tank axis is vertical, thus the free liquid surface is orthogonal to the tank axis. The liquid is assumed to be inviscid, and the contribution of the free surface waves to the dynamic pressure on the free liquid surface is neglected. Wet and dry mode shapes of the plate are assumed to be the same, so that the natural frequencies are obtained by using the nondimensionalized added virtual mass incremental (NA VMI) factors and the modal properties ofdry plates. This simplifies computations compared to other existing theoretical approaches. NAVMI factors express the nondimensionalized ratio between the reference kinetic energy of the liquid and that of the plate and have the advantage that, due to their nondimensional form, they can be computed once and for all. Numerical results for simply supported and clamped bottom plates, as well as for supported plates with an elastic moment edge constraint are given. For more accurate results, and to exceed the limits of the assumed modes approach, the RayleighRitz method is applied and results are compared to those obtained by using the NAVMI factors and other existing methods in the literature. © 1997 John Wiley & Sons, Inc.

INTRODUCTION
The study ofthe free vibrations of circular cylindri
cal tanks has interested many researchers; this is
obviously due to the wide application of tanks
in mechanical, aeronautical, and civil engineering.
Cylindrical tanks are often composed of a shell and
a circular bottom plate. Many studies investigated
vibrations of the shell and the bottom plate of
these containers, and some are reported here. The
liquidfilled tanks have two families of modes: the
sloshing and the bulging ones. Sloshing modes are
mann and Chang (1968),
Bauer and Siekmann
(1969)
, and
Capodanno (1989)
; other references
are given in
Abramson (1966)
. Both sloshing and
bulging axisymmetric modes of the bottom plates
were experimentally and theoretically studied by
Chiba, who also investigated the effect of the static
deflection due to the fluid weight
(Chiba, 1992,
1993)
and the effect of a Winkler foundation
(Chiba, 1994) on the bottom plate vibrations.
Nagaia and Takeuchi (1984)
studied plates of arbi
trary shape in contact with a viscous fluid, and
Nagaia and Nagai (1986)
studied circular bottom
plates on Winkler foundations in containers filled
with viscous fluid. Nonlinear sloshing was studied
by
Bauer et al. (1971)
, and nonlinear sloshing and
bulging modes of the bottom circular plate were
experimentally investigated by Chiba (1992). A
circular plate as the surface cover of a rigid circular
cylindrical tank was studied by
Bauer (1995)
.
Regarding the shell vibrations, due to the great
amount of literature, we only remember some
works of
Berry and Reissner (1958)
,
Lindholm
et al. (1962)
,
Baron and Skalak (1962)
, and
Kondo
(1981)
, and that of
Haroun and Housner (1981)
on the earthquake response of storage tanks.
Bauer and Siekmann (1971)
and Bauer et al.
(1972) studied the sloshing modes of circular cylin
drical containers with both flexible bottom plates
and flexible shells.
It is worth mentioning that the first modem
day studies on the vibrations of circular plates in
contact with fluids can be attributed to
Rayleigh
(1877)
and
Lamb (1921)
. However, they were in
terested in plates vibrating in a circular aperture
of an infinite rigid wall, so that the fluid was unlim
ited; this is a different problem from that given by
a cylindrical tank. This is also the case in the works
of
Kwak (1991)
,
Kwak and Kim (1991)
,
Ginsberg
and Chu (1992)
, and Amabili et al. (1995a,b).
In this article attention is focused on the bulging
modes of the flexible bottom plate of an otherwise
rigid circular cylindrical container with a vertical
axis and filled with liquid, so that the free surface
of the liquid is orthogonal to the tank axis. The
volume occupied by the liquid is cylindrical and
the liquid velocity potential can be obtained by
using the variable separation. This technique was
used in the quoted studies to find the velocity
potential of the inviscid liquid for sloshing and
bUlging modes. All these studies which include the
effect of the free surface waves of the fluid and,
in some cases, also the effect of the superficial
tension of the liquid
(Bauer and Siekmann, 1971)
or the inplane stress of the plate (Chiba, 1993),
give quite complex solutions that must be numeri
cally solved for each specific case; therefore, few
numerical results are available, especially for bulg
ing and asymmetric modes. On the contrary, in
this work the effects of the free surface waves
on the dynamic pressure at the free surface, the
superficial tension, and the hydrostatic pressure
are neglected
(Morand and Ohayon, 1992)
, so that
the plate vibrations, only considering bulging
modes, are studied by using a simplified theory.
As a consequence, the nondimensionalized added
virtual mass incremental (NAVMI) factor ap
proach, already successfully used for circular
plates by
Kwak and Kim (1991)
,
Kwak (1991)
,
Amabili et al. (1995a,b), and
Amabili and Dalpiaz
(1995)
, is applied so that all numerical computa
tions can be made nondimensional and the natural
frequencies of the plate in contact with the liquid
can be obtained directly from those in a vacuum,
considering the same plate boundary condition.
This is a computational simplification with respect
to other existing theoretical approaches; more
over, due to their nondimensional form, NAVMI
factors can be computed once and for all. The
proposed approach is based on the Rayleigh quo
tient for coupled vibrations
(Zhu, 1994)
and on
the hypothesis that the dry (in vacuum) and wet
(in liquid) mode shapes of the plate are unchanged
(assumed modes approach); the accuracy of this
approach is checked by using the RayleighRitz
method
(Zhu, 1995)
that removes the restrictive
hypothesis on the wet mode shapes. In particular,
the wet mode shapes are developed in a series by
using the dry mode shapes as admissible functions.
The RayleighRitz approach allows us to ex
ceed the limits of the assumed modes approach,
but it retains a remarkable simplicity of computa
tion, with respect to other analytical techniques
already applied to this problem. Moreover, nondi
mensional results are very useful for engineering
applications. It is also shown that the results of
the RayleighRitz method match very well with
numerical data available in the literature
(Chiba,
1993)
for bulging modes, obtained by using more
complex theories, and thus proving that the very
simple NAVMI factor approach is accurate
enough for many engineering applications.
THEORETICAL BACKGROUND
A thin circular plate having the thickness, h, and
the mass density, {)p, vibrating in a vacuum, is con
sidered. The plate material is assumed to be
linearly elastic, homogeneous, and isotropic; the ef
fects of shear deformation, rotary inertia, and
damping are neglected. The equation of motion
for the transverse displacement, w, of the plate is
governed by
(Leissa, 1969)
where D = Eh3/[12(1  zl)] is the flexural rigidity
of the plate and 1) and E are Poisson's ratio and
Young's modulus, respectively. In addition,
is the Laplace operator in the polar coordinates r
and e. The solution of Eq. (1) is obtained by using
the variable separation. In the case where bound
ary conditions at the edge are uniform, the solution
takes the following form
(Leissa, 1969)
:
w(r, e, t) = L L WmnCr) cos(me)f(t),
m~O n~O
where
f(t) = eiwt,
in which m and n represent the number of nodal
diameters and circles; Amn and emn are the mode
shape constants, whose ratio is determined by the
boundary conditions; Jmand 1m are the Bessel func
tion and the modified Bessel function of the first
kind; Amn is the frequency parameter, which is also
determined by the boundary conditions; a is the
plate's radius; and i is the imaginary unit. The
frequency parameter Amn is related to the circular
frequency w of the plate by
The values of the modal parameters Amn , A mn , and
Cnm are given by
Leissa (1969)
for clamped circular
plates, by
Leissa and Narita (1980)
for simply sup
ported circular plates, and by Amabili and col
leagues (1995b) for freeedge circular plates; Amn
for circular plates with elastic edge supports are
computed in
Azimi (1988)
; and the mode shape
(1)
(3)
(5)
(6)
constants can be obtained by using the bound
ary conditions.
The Rayleigh quotient for coupled vibrations
(Zhu, 1994)
is used to evaluate natural frequencies
of the plate in contact with the liquid. The original
idea is attributed to
Rayleigh (1877)
and
Lamb
(1921)
. Hence, we may write the following for
each case:
where V p is the maximum potential energy of the
plate and Tt and Tt are the reference kinetic
energies of the plate and the liquid, respectively.
Based on the hypothesis that the wet mode shapes
are equal to the dry mode shapes, natural frequen
cies of free vibration in a liquid, A, can be related
to natural frequencies in a vacuum, fv. Thus, the
following formula is obtained:
where gmn is the added virtual mass incremental
(AVMI) factor. This factor is given by the ratio
between the reference kinetic energy of the liquid,
whose movement is induced by the vibration of
the structure, and that of the plate
gmn =
Tt = f mn PLha ,
Tt Pr
where fmn is the NAVMI factor and PL is the mass
density of the liquid.
NAVMI FACTORS
A circular plate is considered to be the flexible
bottom of a rigid circular cylindrical tank filled
with an incompressible and inviscid liquid (see
Fig. 1); the liquid movement, considered as only
caused by the plate vibration, is assumed to be
irrotational. Then, the liquid movement associated
with each mode shape can be described by the
spatial velocity potential, c[>mn, that satisfies the
Laplace equation
The spatial velocity potential can be written as
V2 c[>mn = O.
<l>mll = 1>mll(r, z)cos(me),
(8)
(9)
(10)
Rigid shell ~
Liquid
H
l'
Elastic plate J
I
'(
a
>I
The free liquid surface condition is described by
the zero dynamic pressure condition at z = H
(Morand and Ohayon, 1992; Zhu, 1994, 1995)
,
where H is the level of the liquid in the container.
This boundary condition is obtained neglecting
the contribution of the free surface waves and
superficial tension to the dynamic pressure of the
liquid at z = H. This simplification does not give
significant errors for tanks when only bulging
modes are studied, as is shown in the discussion
and comparison of numerical results. Studying the
shell vibrations, Kondo (1981) discussed this phe
nomenon observing that wave heights of the free
surface for bulging modes of circular cylindrical
tanks are so small that they almost coincide with
the undisturbed liquid level. The introduced sim
plification is the same obtained considering zero
gravity (without superficial tension) and does not
constrain the vertical velocity of the liquid. As a
consequence of the hypotheses, the free surface
(13)
does not exhibit an intrinsic capability to oscillate;
thus, the liquid free surface is not subjected to a
restoring force once it has moved, and sloshing
modes cannot be studied.
The condition of impermeable walls at the
liquidrigid tank interface for a noncaviting liq
uid is
( acf>mn)
ar r=a
=0
'
and the liquidflexible plate interface is
The solution of Eq. (12) with the condition given
by Eq. (13) is, for axisymmetric modes (m = 0),
cPon(r, z) = Kono(z  H) + i KOnkJO (BOk!..)
k=l a
(14)
(15)
(16)
1
z) sinh (cmk~)
. [cosh (Cmk
, (17)
a
tanh ( Cmk ~)
where
where Kmnk are constants. Equations (16) and (17)
can be obtained by those of
Bauer and Siekmann
(1971)
considering g = (J = 0; these equations must
satisfy all the boundary conditions. Equations (16)
and (17) satisfy the boundary condition (14) if Cmk
are solutions of the following equation
where J ~, is the derivative of the Bessel function
in respect to its argument. The constants Kmnk are
calculated in order to satisfy Eq. (15). For asym
metric modes we have
Recalling the orthogonality condition of the
Bessel function
(Spiegel, 1974)
, we have
where Okn is the Kronecker delta and (Xmk is given by
Moreover, we have
(Wheelon, 1968)
where
and for asymmetric (m # 0) modes is
We also introduce the
(Wheelon, 1968)
following
integral
(18)
(19)
(21)
If we multiply Eq. (19) by (l/a 2) Jm[cmk(r/a)]r and
then we integrate, this yields
Therefore, the constants Kmnk are given by
If one applies the Green's theorem to the har
monic function <Pmn , the reference kinetic energy
of the liquid can be computed as a boundary inte
gral
(Lamb, 1945; Amabili, 1995)
:
where V is the liquid volume, aV is the boundary
of the volume V, and s is the direction normal to
the boundary oriented inward to the liquid region.
Due to Eqs. (13) and (14), only the integration
over the plate surface gives a nonzero result; there
fore, the reference kinetic energy of the liquid is
given by
(27)
(28)
4Jrnn(r, 0)cos2(mO)r dr dO
( a4Jrnn )
az
z=O
1 '"
= 2PLa3"'mL
k=O
= ~ PLa2"'rn I: WrnnCap )4Jmn(ap, O)p dp
For axisymmetric modes (m = 0), the spatial veloc
ity potential of the liquid is described by Eq. (16);
for these modes, the boundary condition at the
liquidplate interface, Eq. (15), is satisfied if
(29)
~ ( r )
Kono + LJ KonkJo
SOkk=! a
SO(k )H
a' tanh SOk
a
(34)
The constant Kono is given by
where "'rn = 27T for m = 0 and 7T for m > 0, and
Tt depends on m and n. The reference kinetic
energy of the plate is given by
Tt = 12 pph fa f21T W~(r)cos2(mO)r drdO
o 0
_14"'mPPha20pmn '
where
OPmn = A;n [J;'2(Arnn) + (1  ~~) J;(Amn) ]
 C;n [I;'2(Amn )  (1 + ~J I; (Amn) ]
and J;' and I;' are the derivatives of corresponding
Bessel functions in respect to their arguments.
Then, by using Eq. (9), the AVMI factor is given by
(31)
(32)
and the NAVMI factor is
= TOn,
(30)
where
(Wheelon, 1968)
The constants KOnk' for k > 0, are obtained by
Eq. (27) computed for m = 0; therefore, for axi
symmetric modes, the reference kinetic energy of
the liquid is
and the NAVMI factor is given by
r  2 [2H 2
On  Opmn ;; TOn
NUMERICAL RESULTS
NAVMI factors are computed for circular plates
with different boundary conditions by using
Eqs. (33) and (38) and the software Mathematica
(Wolfram, 1991)
. Data for simply supported plates
is listed in Table 1 for m :S 4, n :S 3 and for
different liquid depth ratios H/a. NAVMI factors
(36)
(37)
(38)
increase with the ratio H/a and decrease with m
and n. In particular, the fundamental mode (m =
oand n = 0) presents a high value of the NAVMI
factor for H/a ?: 1, due to the movement of the
liquid center of mass during vibrations; on the
contrary, no change of the center of mass height
is verified for mode shapes having m > O. The
limit NAVMI factors, when H/a goes to infinity,
are given for asymmetric modes. These limit val
ues cannot be obtained for axisymmetric modes
by using the assumed modes approach but can be
obtained by the RayleighRitz method. A similar
behavior is shown in Table 2, where the data for
clamped plates is reported; however, the factor
for the fundamental mode is lower in this case in
respect to simply supported plates. NAVMI fac
tors for axisymmetric modes are also plotted in
Fig. 2. The case of supported plates with a constant
elastic moment constraint at the edge is studied
for different torsional distributed stiffness K t
[moment/unit length] values; this constraint simu
lates well the plate boundary condition when this
plate is welded to a circular cylindrical shell.
Resuits are reported in Tables 3 and 4 for axisymme
tric and asymmetric modes, respectively, and were
obtained by using the modal parameters given in
Azimi (1988)
. The full range 0 (simply supported
plate) + 00 (clamped plate) of the stiffness K t
was studied.
IMPROVED SOLUTION:
RAYLEIGHRITZ METHOD
The RayleighRitz method
(Meirovitch, 1986)
is
applied to eliminate the restrictive hypothesis that
dry and wet modes have the same shape. All the
other hypotheses, previously introduced, are re
tained. The wet mode shapes W, by using the un
known parameters qn and the admissible functions
Wmn , can be described by
00
W(r, 0) = cos(mO) 2: qnWmn(r),
n=O
(39)
where Wmn is given by Eq. (4). To simplify the
m = 1
functions; in fact, dry mode shapes are quite simi
lar to wet mode shapes. The trial functions Wmn
are linearly independent and constitute a com
plete set.
The spatial distribution of the velocity potential
of the liquid, <P, calculated at the liquidplate in
terface (z = 0), is given by
where
<P(r, e,O) = ¢(r,O)cos(me),
00
¢(r, 0) = L qn¢mn(r, 0).
11=0
(42)
(43)
Using the principle of superposition, considering
that the plate deflection is given by the sum of
Eq. (39), and Eq. (43), the function ¢ at the liquid
plate interface is given by the following sum:
computations, the mode shape constants, Amn and
emn , are normalized to have
The result of quadrature of Eq. (40) is [see
Wheelon eqs. 11.106, 33.10, and 31.101 (1968)]
In Eq. (39) the eigenfunctions of the plate vibrat
ing in a vacuum are assumed to be admissible
00 00
¢(ap, O) = L qn L KmllkJm(SmkP)'
n~O k~O
(44)
0.7 r                       ,
0.6
~
I:~::I
1:n =2\
.n =3
o ~~~~~
o 0.5 1.5 2
H/a
The reference kinetic energy of the liquid is
given by
expression for the reference kinetic energy of the
liquid for asymmetric modes is obtained:
Tt =  12 PL f2u fa (a<p)
a a az Fa
<P(r, 0) r dr d8. (45)
+ Cm/Ymjk).
(46)
Using the boundary condition at the liquidplate
interface (a<plaz)z=a =  W(r, 8), the following
The reference kinetic energy of the liquid for axi
symmetric modes (m = 0) is then given by
Tt = 21PLa2l/Jrn00~ ~00 qnqj [ 2H'Ton'TOj
+ ~ KOnk(Aoj/3 ojk + CO/YOjk) ].
The reference kinetic energy of the plate, using
the normalization introduced in Eq. (40) and the
orthogonality of the dry mode shapes, is given by
The maximum potential energy of the system, con
sidering an incompressible liquid, coincides with
that of the plate and can be computed as a sum
of the reference kinetic energies of the dry eigen
functions,
To perform numerical computations for each fixed
m value, only a finite number N of terms must
be considered in all the previous sums. To this
purpose, the vector q of the unknown parameters
is introduced
q = (
1
.
qqol
q~l
To make the formulas more compact, the follow
ing constant is also given:
The NAVMI matrix describes the inertial effect
of the liquid on the modes. Therefore, this is the
extension of the NAVMI factor to the Rayleigh
Ritz approach. Moreover, the NAVMI factors are
the diagonal elements of the NAVMI matrix.
Similar to Eq. (52), the reference kinetic energy
of the plate is obtained
, 1 TT
T p = 2: apphq .q,
where I is the N X N identity matrix. Then, the
maximum potential energy of the plate takes the
following expression:
where P is the N X N diagonal matrix given by
5
15
20
and Oij is the Kronecker delta. In order to find
natural frequencies and wet mode shapes of the
plate vibrating in contact with the liquid, the Ray
leigh quotient for coupled vibration in an inviscid
and incompressible liquid
(Zhu, 1994)
is used.
Minimizing the Rayleigh quotient with respect to
the unknown parameters qn, one gets
~ Pq  A2(pphl + PLaMdq = 0,
a
(58)
where A is the circular frequency of the wet plate.
Equation (58) is a Galerkin equation and gives an
eigenvalue problem. It would be convenient to
introduce the following nondimensional constants:
fP=A2aD4pPh ,
nand JL are called the wet frequency parameter
and the densitythickness correction factor, re
spectively. Then, Eq. (58) can be written in the
following nondimensional form:
Pq  fP(1 + JLMdq = O.
(61)
It is interesting to see that, if the NAVMI matrix
ML is diagonal, the system ofEq. (61) is uncoupled;
in this case, the approximate solutions given in the
previous section become exact.
DISCUSSION AND COMPARISON OF
NUMERICAL RESULTS
The numerical solution to the Galerkin equation,
Eq. (61), is obtained by using the Mathematica
(55)
(56)
(57)
(59)
(60)
0
n=1
nQ
10
Jl
FIGURE 3 The wet frequency parameters n for axi
symmetric modes (m = 0) of clamped plates having a
ratio H/a = 0.5, as a function of the densitythickness
correction factor JL. Curves relative to the first three
axisymmetric modes.
(Wolfram, 1991)
computer program. The compu
tation of the NAVMI matrix ML is performed
by using Eq. (53) for asymmetric modes and
Eq. (54) for axisymmetric modes. Due to the
nondimensional form of these equations and the
number of nodal diameters m once fixed, ML
depends only on the plate boundary conditions
at the edge, on the ratio Hla, and on the Poisson
ratio v; clamped plates are independent from v.
The numerical results for three different cases
are reported here; these results are also used as
a testing bench for the assumed modes approach.
The NAVMI matrix for axisymmetric modes (m
= 0) of clamped plates having a ratio Hla =
0.5 is
o~==========~~====~
0.5
1
dP
0.38260
0.072123
0.072123
The NAVMI matrix for axisymmetric modes (m = 0) of clamped plates having a ratio H/a = 2 is
The NAVMI matrix for modes with a nodal diameter (m = 1) of clamped plates having a ratio H/a =
2 is
Six terms in the series expansion of mode shapes
are used in the cases presented; this choice allows
a good evaluation of the first three eigenvalues
(Meirovitch, 1986)
. Observing Eqs. (62)(64), one
can find that the NAVMI factors previously com
puted are the diagonal elements of the NAVMI
matrices. Moreover, the offdiagonal elements can
be positive or negative, but generally have a
smaller absolute value than the diagonal elements;
exceptions are obtained in Eq. (63), where the
axisymmetric modes of a clamped plate having a
ratio H/a = 2 are considered. It is clear that the
matrix in Eq. (62), relative to clamped circular
plates with a ratio H/a = 0.5, is more diagonal
than the others reported in Eqs. (63) and (64). In
fact, an increment of the offdiagonal elements
with the ratio H/a can be observed; on the
contrary, the importance of the offdiagonal elements
decrease with m, as verified comparing Eqs. (63)
and (64).
Figure 3 shows the wet frequency parameters
n for axisymmetric modes (m = 0) of clamped
plates for H/a = 0.5 as function of the density
thickness correction factor fL. Curves relative to
the first three axisymmetric modes are given. The
circular frequency of this plate is obtained by using
Eq. (59) and Fig. 3. The percentage errors that
one commits by using the NAVMI factor solution,
instead of the RayleighRitz solution, is plotted
in Fig. 4 for this plate as function of fL. It is very
interesting to note that the fundamental mode is
accurately estimated by the NAVMI factor solu
tion and that the error increases with the number
of nodal circles and with fL. In Figs. 5 and 6 the
(62)
(63)
(64)
' n=O
o~================~
o 5 10 15 20
FIGURE 5 The wet frequency parameters n for axi
symmetric modes (m = 0) of clamped plates having a
ratio H/a = 2, as a function of the densitythickness
correction factor /L. Curves relative to the first three
axisymmetric modes.
wet frequency parameters n and the percentage
errors are given for axisymmetric modes (m = 0)
of clamped plates for Hla = 2. These figures are
given in order to understand the changes due to
the different ratio of Hla; it is clearly an increment
of the percentage errors that are maximum for
modes having one nodal circle (n = 1). In Figs. 7
and 8 the wet frequency parameters n and the
percentage errors are plotted for modes with one
nodal diameter (m = 1) of clamped plates with a
ratio Hla = 2. It is interesting to observe that the
frequency error for all modes and for all the J1,
values are lower in this case with respect to the
one shown in Fig. 6; therefore, as well as the off
diagonal elements becoming less important for
0
5
10
H
0
H
H 15
[il
0
1
5
10
J.l
0
5
15
20
0
5
15
20
FIGURE 8 The percentage errors of the NAVMI fac
tor solution in respect to the RayleighRitz results as
a function of /L for clamped plates. Modes with a nodal
diameter (m = 1) and H/a = 2.
o
5
O~____~___~_~~~____~____~
15
20
FIGURE 7 The wet frequency parameters n for
modes with a nodal diameter (m = 1) of clamped plates
having a ratio H/a = 2, as a function of the density
thickness correction factor 11. Curves relative to the first
three modes.
modes having nodal diameters, the percentage er
rors decrease with m.
The wet mode shapes are investigated by using
Eq. (61); data relative to the three is reported. In
Fig. 9 the dry and wet mode shapes are plotted
along a radius for axisymmetric modes of clamped
plates with Hla = 0.5 and J1, = 10. Modes with up
to two nodal circles are considered. It is clear that,
as for the natural frequency, the mode shape of
the fundamental mode (m = 0; n = 0) shows
little change. Similar results are given in Fig. 10
for axisymmetric modes of clamped plates with
Hla = 2 and J1, = 10 and in Fig. 11 for modes
with one nodal diameter of clamped plates with
Hla = 2 and J1, = lO.
The results of the NAVMI factor approach
n=O
10
.\.l
>=:
~ 0.8
Ql
o
cO 0.6
rl
U0.l. 0.4
•.1
'0
'0 0.2
Ql
.~
rl
~ 0.2
So!
~ 0.4
O~~.~.~~~~~~
o
0.2
0.4
0.6
0.8
1
o~~~~~~~=~~
o
0.2
0.4
0.6
0.8
1
and RayleighRitz methods are compared to
those given by
Chiba (1993)
for bulging modes
of a clamped circular bottom plate, where both
the effects of the free surface waves on the
dynamic pressure and the inplane stress of the
plate are considered. Results of
Chiba (1993)
are dimensional and refer to a steel plate having
radius a = 0.144 m, thickness h = 0.002 m,
Young's modulus E = 206 Gpa, Poisson ratio
v = 0.25, mass density pp = 7850 kg m3 in
contact with water, having PL = 1000 kg m3•
In Fig. 12 the natural frequencies (Hz) of the
first three axisymmetric modes of the plate inves
tigated by
Chiba (1993)
are compared to the
results obtained by using both the simple NAVMI
factor approach and the RayleighRitz method.
It is clear that the results of the RayleighRitz
method always match very well with Chiba's
results (see Fig. 12), confirming that the applied
free surface condition (zero dynamic pressure at
Z = H) is correct when one is studying bulging
modes. Moreover, the simple NAVMI factor
approach gives quite good results, especially for
the fundamental mode and for small values of
r
r
FIGURE 10 Comparison of () dry and () wet modes for clamped plates; axisymmetric
modes (m = 0) and H/a = 2.
O~~~\~t~I
o
0.2
0.4
0.6
0.8
1
the ratio H/a; in fact, it was previously found
that, when the ratio H/a increases, the accuracy
of the NAVMI factor approach decreases.
CONCLUSIONS
The proposed approach to the problem has the
advantage that the natural frequencies are quickly
obtained by using the NAVMI factors and the
modal properties of dry plates. The NAVMI fac
tors are listed for application to engineering and
design; to this end, some common constraints at
the plate edge are studied. In spite of the computa
tional simplicity, the present approach gives good
results, especially for the fundamental mode
(m = 0 and n = 0) or modes without nodal circles
(n = 0), and for values ofthe ratio H/a ~ 1; with.in
the aforestated range, mode shapes havmg no CIr
cular nodes present nearly equal wet and dry mode
shapes, so that the assumed mode approach gives
nearly exact natural frequencies.
When more accurate results are necessary, the
RayleighRitz method, which retains a relative
computation simplicity and gives nondimensio?al
results, can be used. This method for bulgmg
modes gives results nearly equal with those ob
tained by using more complex theories.
NOMENCLATURE
plate radius
mode shape constants
r
D
E
Iv
A
h
H
m
n
r
t
Tt
T:
Vp
()
<Pmn
N'
:.1...:.
>(,,)
c:
Q)
::I
0e
....
~
::I 1000
1iS
z
.
H/a
FIGURE 12 Comparison of natural frequencies obtained by using different theories as a
function of the ratio H/a. (D) NAVMI factors approach; (e) RayleighRitz method; ()
results of
Chiba (1993)
.
circular frequency of the plate (rad/s)
frequency parameter of the wet plate
Laplace operator
= V2~72, iterated Laplace operator
Journal of
Engineering
Volume 2014
Volume 2014
Sensors
Hindawi Publishing Corporation
ht p:/ www.hindawi.com
Volume 2014
International Journal of
Hindawi Publishing Corporation
ht p:/ www.hindawi.com
Active
and
Advances in
Civil Engineering
Hindawi Publishing Corporation
ht p:/ www.hindawi.com
Journal of
Robotics
Hindawi Publishing Corporation
ht p:/ www.hindawi.com
Advances
OptoEle
Submit
your manuscr ipts
VLSI
Design
Hindawi Publishing Corporation
ht p:/ www.hindawi.com
Hindawi Publishing Corporation
Navigation and
Observation
Engineering
International Journal of
and
Engineering
Electrical
and
Computer
International Journal of
Aerospace
Engineering
Abramson , H. N. , Editor , 1966 , "The Dynamic Behavior of Liquids in Moving Containers," NASA SP106 , Government Printing Office, Washington, DC.
Amabili , M. , 1995 , "Comments on 'Rayleigh Quotients for Coupled Free Vibrations'," Journal of Sound and Vibration , Vol. 180 , p. 526 .
Amabili , M. , and Dalpiaz , G. , 1995 , "Vibration of a FluidFilled Circular Cylindrical Tank: Axisymmetric Modes of the Elastic Bottom Plate," Proceedings of the International Forum on Aeroelasticity and Structural Dynamics , Vol. 1 , June 2628, Manchester, UK, pp. 38 . 1  38 .8.
Amabili , M. , Dalpiaz , G. , and Santolini, c., 1995a, "FreeEdge Circular Plates Vibrating in Water," Modal Analysis: The International Journal ofAnalytical and Experimental Modal Analysis , Vol. 10 , pp. 187  202 .
Amabili , M. , Pasqualini , A. , and Dalpiaz , G. , 1995b , "Natural Frequencies and Modes of FreeEdge Circular Plates Vibrating in Vacuum or in Contact with Liquid," Journal of Sound and Vibration , Vol. 188 , pp. 685  699 .
Azimi , S. , 1988 , "Free Vibration of Circular Plates with Elastic Edge Supports Using the Receptance Method," Journal of Sound and Vibration , Vol. 120 , pp. 19  35 .
Baron , M. L. , and Skalak , R. , 1962 , "Free Vibrations of FluidFilled Cylindrical Shells," Journal of the Engineering Mechanics Division , ASCE , Vol. 88 , No. EM3 , pp. 17  43 .
Bauer , H. F. , 1995 , "Coupled Frequencies of a Liquid in a Circular Cylindrical Container with Elastic Liquid Surface Cover," Journal of Sound and Vibration , Vol. 180 , pp. 689  704 .
Bauer , H. F. , Chang , S. S. , and Wang , J. T. S. , 1971 , "Nonlinear Liquid Motion in a Longitudinal Excited Container with Elastic Bottom," American Institute of Aeronautics and Astronautics Journal , Vol. 9 , pp. 2333  2339 .
Bauer , H. F. , and Siekmann , J. , 1969 , "Note on Linear Hydroelastic Sloshing," ZeitschriJt fUr Angewandte Mathematik und Mechanik , Vol. 49 , pp. 577  589 .
Bauer , H. F. , and Siekmann , J. , 1971 , "Dynamic Interaction of a Liquid with the Elastic Structure of a Circular Cylindrical Container," Ingenieur Archiv , Vol. 40 , pp. 266  280 .
Bauer , H. F. , Wang , J. T. S. , and Chen , P. Y., 1972 , "Axisymmetric Hydroelastic Sloshing in a Circular Cylindrical Container," AeronauticalJournal , Vol. 76 , pp. 704  712 .
Berry , J. G. , and Reissner , E. , 1958 , "The Effect of an Internal Compressible Fluid Column on the Breathing Vibrations of a Thin Pressurized Cylindrical Shell," Journal of Aeronautical Sciences , Vol. 28 , pp. 288  294 .
Bhuta , P. G. , and Koval , L. R. , 1964a , "Hydroelastic Solution of the Sloshing of a Liquid in a Cylindrical Tank," Journal of the Acoustical Society of America , Vol. 36 , pp. 2071  2079 .
Bhuta , P. G. , and Koval , L. R. , 1964b , "Coupled Oscillations of a Liquid with a Free Surface in a Tank," Zeitschrift fUr Angewandte Mathematik und Physik , Vol. 15 , pp. 466  480 .
Bleich , H. H. , 1956 , "Longitudinal Forced Vibrations of Cylindrical Fuel Tanks," Jet Propulsion , Vol. 26 , pp. 109  111 .
Capodanno , P. , 1989 , "Vibrations d'un Liquide dans un Vase Ferme par une Plaque Elastique," Revue Roumaine des Sciences TechniquesSerie de Mecanique Appliquee , Vol. 34 , pp. 241  266 .
Chiba , M. , 1992 , "Nonlinear Hydroelastic Vibration of a Cylindrical Tank with an Elastic Bottom, Containing Liquid. Part I: Experiment," Journal of Fluids and Structures , Vol. 6 , pp. 181  206 .
Chiba , M. , 1993 , "Nonlinear Hydroelastic Vibration of a Cylindrical Tank with an Elastic Bottom, Containing Liquid. Part II: Linear Axisymmetric Vibration Analysis," Journal of Fluids and Structures , Vol. 7 , pp. 57  73 .
Chiba , M. , 1994 , "Axisymmetric Free Hydroelastic Vibration of a Flexural Bottom Plate in a Cylindrical Tank Supported on an Elastic Foundation," Journal of Sound and Vibration , Vol. 169 , pp. 387  394 .
Ginsberg , J. H. , and Chu , P. , 1992 , "Asymmetric Vibration of a Heavily FluidLoaded Circular Plate Using Variational Principles," Journal of the Acoustical Society of America , Vol. 91 , pp. 894  906 .
Haroun , M. A. , and Hausner , G. W. , 1981 , "Earthquake Response of Deformable Liquid Storage Tanks," Journal of Applied Mechanics , Vol. 48 , pp. 411  418 .
Kondo , H. , 1981 , "Axisymmetric Free Vibration Analysis of a Circular Cylindrical Tank," Bulletin of the Japan Society of Mechanical Engineers , Vol. 24 , No. 187 , pp. 215  221 .
Kwak , M. K. , 1991 , "Vibration of Circular Plates in Contact with Water," Transactions of the ASME , Journal of Applied Mechanics , Vol. 58 , pp. 480  483 .
Kwak , M. K. , and Kim , K. C. , 1991 , "Axisymmetric Vibration of Circular Plates in Contact with Fluid," Journal of Sound and Vibration , Vol. 146 , pp. 381  389 .
Lamb , H. , 1921 , "On the Vibrations of an Elastic Plate in Contact with Water," Proceedings of the Royal Society of London, Series A , Vol. 98 , pp. 205  216 .
Lamb , H. , 1945 , Hydrodynamics, Dover, New York, p. 46 .
Leissa , A. W. , 1969 , "Vibration of Plates," NASA SP160 . Government Printing Office, Washington, DC.
Leissa , A. W. , and Narita , Y. , 1980 , "Natural Frequencies of Simply Supported Circular Plates," Journal of Sound and Vibration , Vol. 70 , pp. 221  229 .
Lindholm , U. S. , Kana , D. D. , and Abramson , H. N. , 1962 , "Breathing Vibrations of a Circular Cylindrical Shell with an Internal Liquid," Journal of Aerospace Sciences , Vol. 29 , pp. 1052  1059 .
Meirovitch , L. , 1986 , Elements of Vibration Analysis, 2nd ed., McGrawHill , New York.
Morand , H. J.P. , and Ohayon , R. , 1992 , Interactions FluidesStructures , Masson, Paris.
Nagaia , K. , and Nagai , K. , 1986 , "Dynamic Response of Circular Plates in Contact with a Fluid Subjected to General Dynamic Pressures on a Fluid Surface," Journal ofSound and Vibration , Vol. 70 , pp. 333  345 .
Nagaia , K. , and Takeuchi , J. , 1984 , "Vibration of a Plate with Arbitrary Shape in Contact with a Fluid," Journal of the Acoustical Society of America , Vol. 75 , pp. 1511  1518 .
Rayleigh , Lord J. W. S. , 1877 , Theory of Sound, 2nd ed. (1945 reissue) , Dover, New York.
Siekmann , J. , and Chang , S. c. , 1968 , "On the Dynamics of Liquid in a Cylindrical Tank with a Flexible Bottom," Ingenieur Archiv , Vol. 37 , pp. 99  109 .
Spiegel , M. R. , 1974 , Fourier Analysis , McGrawHill , New York.
Tong , P. , 1967 , "Liquid Motion in a Circular Cylindrical Container with a Flexible Bottom," American Institute of Aeronautics and Astronautics Journal , Vol. 5 , pp. 1842  1848 .
Wheelon , A. D. , 1968 , Tables of Summable Series and Integrals Involving Bessel Functions, HoldenDay, San Francisco.
Wolfram , S. , 1991 , Mathematica: A System for Doing Mathematics by Computer , 2nd ed., AddisonWesley , Redwood, CA.
Zhu , F. , 1994 , "Rayleigh Quotients for Coupled Free Vibrations," Journal of Sound and Vibration , Vol. 171 , pp. 641  649 .
Zhu , F. , 1995 , "RayleighRitz Method in Coupled FluidStructure Interacting Systems and Its Applications," Journal of Sound and Vibration , Vol. 186 , pp. 543  550 .
Volume 2014