I4.r
-By
K. Fujii K. Tanida Y. YOkokura
Ship Srengch Department Research Institute
IshikawajimaHarima Heavy Industries Co., Ltd.
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Vibration o Elastic Body in Contact ich qatar
,iS-Vibration of Elastic Body in ontact with Water
1. Introduction
The vibration of local ship structures (web frame in
the tank (1), (2), (3) , ship bottom, engine room, etc.) has become a serious problem in conjunction with cracking or the vibration of the main hull
con-struction, and to solve this problem, it is necessary to calculate natural frequencies accurately.
Most ship structures are in contact with water, and the calculation accuracy of natural frequencies of them is fairly influenced by the calculation accuracy of the effects of water cbtact; aM nevertheless, the coupled vibration of a fluid and an elastic
structures is very complex and an exact solution for
it is difficult to obtain.. In fact, available calculation methods formulated for less complex structures have been applied as an alternative, but these require correction and the results are not
always satisfactory.
With the recent advancement of large-scale computers,
the Finite Elemeit Method is..nów beginning to he
applied not only to structural analysis but also
It has alr?dy been used in approximating added
virtual mass for a body vibrating on the boundary Of a fluid in a pattern of a rigid body or in a certain mode, and has reportedly givn satisfactorily
accurate results on comparison with solutions of
analysis. (8), (15)
This is an e*ample of the application of fluid analysis performed by this method to actual ships.
In the case of: the coupled vibration of a
lud and
an elastic body, calculation are further complicated because the vibration mode of an elastic body itself
is unknown, and both the ana-lysis of the vibration
of an elastic body and the afOrementioned f.uid
analysis mus be calculated imuJ.taneously..
FormularizatiOn concerning this probiem has been made
by Zienk-iewicz and others.
The Finite Element Method is versatile,as i.t allows any desired cpnfigurations and boundaries for both
a structure and a fluid region.
Because of this merit, it is considered to be a very
powerful method for solving this kind of problertt
This- paper will present a process and program of
solving the poblem of coupled vibration of a fl'id and an elastic body, specifically of the plate bending
vibration in contact with a 3-dimensional finite
fluid region. Since this kind of problem involves
tedious calculations, several techniques will be
employed to reduce calculation treatment--the
un-wanted variables of the matrix in elgenvalue analysis
are reduced by applying the calculation rtióde in air to those calculations in contact with water, for the purpose of simplifying the matrix operation.
As an example of the application of the Finite
Element Method to an actual ship vibration problem,
the vibration of web frames and perforated plates
in contact with water will be discusâed.
The effectiveness of this method is confirmed by comparison with experimental measurements of natural
frequencies, coupled vibration. made and fluctuation
of pressure distribution tken on acrylite models of simple configulations, by having them vibrate in air
and in water..
This paper will also discuss.the adequacy of, the
approximation by means of. the Rayleigh-Ritz's energy
method which has Once teen considered in a
paper by the auhors, for the vibration of web
frames in contact. with water.
also proved to give practically acceptable accuracy by comparison with the model experimental results ad
also the calbi4ated values by the Finite Element
Method. As an example of the application of the energy method to actual ships, a method of
calculat-ing added virtual mass for coupled vibration of a
web frame. and a. longitudinal in the tank iill be
presented in. this paper.
On comparison with measured values, it is fOund that
his method is. effective to represent the effect of
the coupled vibration in question.
2. Basic rincipies of Analysis by Energy ethod for Vibration in Water
When a body vibrates in a fluid, it reeives a forôe
from the fluid. This effect can often be explained by an attachment of an apparent mass, called added virtual mass, to the surface of the body.
If. a body vibrates in infinitesimally small amplitude
in an ideal fluid, the kinetic energy of the fluid, Tw; can be obtained from the following equation by
integrating te surface area of the vibrating body,
letting a velocity potential that satisfies. Laplace's
formula and boundary conditions of the fluid region be 5:
-Un, p=pw
Fluid density
v : Velocity of body
p : Fluid pressure
n : Normal Component to the body surface
t:
TimeAdded virtual mass defined b kinetic energy is
determined by Eq.. (1), assuning that the fluid kinetic
energy is equal to the k±netic energy which is yielded when added virtual mass vibrates in a vacuum at the.
same velocity as the body.
When the vibrating body is elastic, its vibration mode In water is also affected by fluid pressure.
Since both of its mode and pressure are unknown, the problem becomes more complex.
Strictly speaking, the coupled vibration of a fluid and an elastic body should be determined by integro differential equation. (5)
In practice, however, ihstead o.f an exact solution of
the integro differential equation, an approximate
,pw ap
Tw = ds (1)
value is' obtained by using the Ray1eigh-RtZ'S energy method as follows:
The vibration mde for an elastic body, W, can be shown as a combination of individual mode, wi, that satisfies bounday conditions,
w= E c
w.j ...(2)1
c1 : Unknown constant
If the vibratipn anplitude is sufficiently smail, the equation for the kinetic energy of an elastic body, Tm, will become a function of w. to form the follow-ing equation accordfollow-ing to a general vibration theory:
Tm = c. c M.. w. w.
1 J 1J 1 J
Mij : Vibration mass
Now, let the velocity
potential that atiie
boundary conditions of a fluid regiqn, when the elastic body is vibrating with = 1,.
0 (ij),
be
.
The velocity potentional in the mode defined in E. (2) is expressed by
-(3)
The kinetic energy of fluid, Tw, will then be yielded from Eq. (1): 1
Tw=-pff
- ds an = c c Mwij w. w. (5) WhereMwij = -w1f ds : added virtual mass.
It will be noted that the kinetic ezlergy of fluid can
be expressed in the same form as in Eq. (3)..
Consequently, the Ritz's method is appricable to vibrations in water as in air by altering the total energy of vibration in water to T = Tm + Tw.
As noted in Eq. (5), when more than two vibration modes are combined, the effect of a fluid is no
longer expressed by the scalar quantity of added
virtual mass. It should rather be expressed in the form of matrix using added virtual mass as an influ-ence coefficient of a fluid for each mode.
Calculation by Finit Eleiient Method of Vibra.tion p
Water
This section deals with how the Finite Element Method
is applied tO the calculations for natural
frequen-cies, vibration modes, and fluctuation of pressure distribution when coupled vibration of an elastic body and a fluid takes place in a 3-dimensional
finite fluid region. It also lists several examples of calculatiors for the purpose of assessing the accuracy of the ca]culated values.
Although only plate. bending vibration is studied,
it i..s obvious that the process is applicable to more
complex structures.
In this kind of problem to be solved by the Finite
Element Method,. matrix elements would become
con-siderably great. In order to reduce calculations,
the following techn.ques are used:
-(1) Since significant frequencies are often around.
the lower degrees of mode, structure stiffness and mass matrix are reduced to the degree that
the resulting accuracy is acceptably high. from
(2) A fluid region is divided into continuous blocks arranged in a series, and a serial elimination
is performed.
(3) VibratIon modes in water are expressed by linear
combinations of vibration modes in air in order
to simplify equations.
All calculations are made on a computer of. UNIVAC-lbS type. Procedures for the calculations will be
ex-plained below.
3.1 Vibration nalysis of Structure
A free vibration of plate bending is chosen as a subject for our study, giving consideration to the
following points:
In. this report, only bea e1irent and
Zienkiewicz's plate bending element. are.
taken into
consideration.-The power method is used for calculating eigen
values.
Consistent mass matrix is used as a mass
matrix. To calculate a mass matrix, Gaussz
(4) To reduce calculations for eigen. values,
stiffness and mass matrices are reduced as
explained below:
An equation for the free vibration of a
structure will be:
([K] - w2[M])
(o} = 0
(6)where [K] is stiffness intrix of tOtal
struc-ture. [N] is mass matrix, and the
displace-ment of nodal points (6} = {} sinwt.
Thus this problem becomes an ordinary eigen
value problem.
As the degree of freedom of the structure concerned increases, the time required to
calculate natural frequency will increase, and
it beäomes necessary to reduce rnati-x elements.
to the.degree that an accuracy sufficient. for
practical use can be obtained.
In this report, the calculation volume is
reduced as follows in accordance with
the-method described in reference (6).
The nodal-point displacements are separated in to the master and slave variables, assuming
that the slave variables satisfy only the
con-ditions in which, potential energy will be at its
minimum,
and that there exists no inertia forceat nodal points of the slave variables.
These assumptions mean that the vibration mode is expressed by linear combination of each static diplacement which takes place on application o unit weight to each variable.
Let nodal-point displacements be i (i = 1,
2 ...) and the slave-variables be s.
Now that potential energy must be at its mini-mum with respect to s, the elements in the
i-th row, the j-th column of a mass matrix [M] and a rigidity matrix [K] of a structure are finally expressed in the following equations:
Kij* = Kij - K,is (Kjs/Kss)
Mij* = Mij - Mis (Kjs/Kss) - M.js (Kis/Kss)
Mss (Kis/Kss) (Kjs/Kss)
...(7)
After the calculation of.. Eqs. (7), elements
in the s-th row, the s-th column of [K] and [MI
Illustrated in Fig. 1 are the results of
calculations done in accordance with the
bove-mentioned matrix reduction technique to obtain
natural frequency of a square. cantilever plate.
It is noted from the figure that natural frequency varies only slightly with the
decrease in master variables, which is seen -at
(a) through Cd) in the figure.
Considering that frequency of interest is normally around the lower mode, the reduction of matrices are shown to be very effective.
In the subsequent calculations, all rotational displacement exand Qy, is regarded as slave
variables, and el:iminated in the course of
calculations.
3.2. Calculation for Pressure Fluctuation of Fluid
Fluid pressure. fluctuation on an elastic body,
which occurs when the body vibrates in a cer.tain
mode on the boundary of a 3-dimensional if i±iite
fluid region, will be calculated here by the Finite
Element Method.
- In the calculation, the following assumptions are
The fluid is an ideal, non-viscous fluid
which is. imcothpreible and irrotational.
The vibration amplitude of a body is infini-tesimally small on the boundary of the fluid.
The fluid has a free surface. Pressure
fluctuation is: appróximäted to be zero on this
free surface, neglecting wave generation because this problem concerns only high
frequencies.
By the use of the Finite Element. Method, pressure
fluctuation is ultimately qiven by the following simultaneous simple equations based on the theory of variation method:
EH] {P} - [F} = 0 (8)
CF} .
(A][6}
[P1 : Fluid pressure vectorat nodal
points
C6} : Acceleration /ector of vibrating body at nodal points on boundary
In Ec. (8), [HI is a matrix representing the
con-dition of pressure. distribution, and coincides with
.Cp} is considered as the displacement, arid CF} the
external load. [Al is a matrix to covert the acceleration imparted to the boundry of the body into an equivalent external load in the direction perpendicular to the boundary.
onsequent1y, one an acceleration distribution on a body is known, pressure distribution over the
entire fluid. region can be calculated in the same
manner as In structural analysis.
'In this paper, a fluid region is divided basically into elements, each having four noda1 points an.
four faces. Pressure distribution inside the individual elements is expressed by P = cL1 + CL2X
+ + z, which is a function of linear
displace-ment between nodal points..'
Acceleration distribution on the boundary is also considered to be linear between nodal points.
In a 3-dimensional fluid region, nodal. points
increase immensely in numbers. In the author's
study, a simplified model as described.belOw is chosen, referring to literature (8):
(1) A 3-dimensional fluid region is divided into blocks as illustrated in Fig.. 2, so as to have
equalnuxnber of noda points at each plane.
Each block has an equal number of elements.
Only plane-N is in contact with a vibrating
body..
Under such conditions, Eq. (8) is converted into
Eq. (9). By repeatingthe calculation procedure of
Eq. (10) with a serial elimination technique, pressure. P} applied ona vibrating wall can be determined. H1 Ci. 0 0
c1T
Ca 0 c2.T H3 C3 0HN1 C_l
T 0 p2 (9) = [A] }[] = (HJ
[] = [Hi]
-[C]T. i1r1
[C1_11 = 2, 3, 4, N) (10)As
an
example of calculations, let us.consider a problem in which two opposing square side walls of a tank filled with water arenormally vibrating,
in such mode as illustrated in Fig 3 in antiphase.Pressure distribution on the vibratingwalls (z = 0,
s) can be obtained from the following
equations
using Fourier Series:
P(x,y) =
aopw2
AB1
cos±(Ys)
+ 1 i=O jij
sinh(Y1S)
2Trx
cosi-2.2ir/.z
+ .j +
=0.5, A1
=0.5, A1
= 0.0 (i > 2) (12) 4 2Bo =
-
1, B =(j
2) Fluid density :Angular frecuencv
In calculating by the Finite Element Method, only
part of, the fluid region consisting of 0 x 2/2
1 ir
and 0 z < s/2 is used taking advantage of syutetry.
The XY plane is divided into 7 x 7, and the area in the direction of z is divided into 6 blocks.
Fig. 4 shows
a comparison of calculated values of pressure distribution on a vibrating wall by Fourier series and Finite Element 4ethod.It will be noted that both results agree well. wth each other and that the inite Element Method can provide sufficient accuracy of calculation even when applied to such a case as above where a region is divided rather roughly.
3.3 Calculation. of Coupled Vibration of. Fluid and
Structure
If we apply Zienkiewicz!sxnethodof rapresentation4,
this problem,. whichis combination Of. the problems
presented in Sections 3.1 and 3.2, is boiled down
to a problern of solving the following simultaneous
equations:
- [A] (6} = 0 ...(13)
on wall of vibrating body (Plane-N)
:' Vector of displacement at
iodal
pQ.ntson vibrating body
In Eq. (14), [RI is a matrix to convert pressure
into an equivalent inertia force wOrking on a
vibrating body, and [RI is generally related to [A]
by the following equation.
[RI
If we eliminate CPN} from eqations (13) and (14),
and assume }
=
C.} sint, these equations are réarrange4 as follows:[[KI - w2([MI
(M,J)I C}
= 0Vector of pressure a nodal points
[AlT
(AlT[H7I [Al
[Mw] expresses the effect of interference of a fluid and an elastic body in terms of a matrix,
which may be called an added mass matrix..
Equation (16) indicates that natural frecuency in
water is calculable in the same way as that in air by merely adding [Mv] to [MI.
(15)
The calculation of equaion (16) can be reduced substantially by assuming that the vibration mode
of. a body in water is expressed by linear
combina-tion of vibracombina-tion modes in. air.
From this assumption the vibration mode in water
(} equals:
c{30' } c2C50' } +
i-th order eigen mode of vibrating body in air
Unknown constant
Thus, Eq. (16) is rearranged as follows:
[.[F]T(K] (Fl - 2(F]T([MJ
[My]) [F]] (Z} =
(18)
[F]
= [{6o'
},
cS' },
6' }i]=
{cl, C2, C3,
ci}There exist the following relatioriships due to the
orthogonality of elgen modes in air:
+
J0'
co'
}rT[M]:{o=0
(r s) (l.9)
(o,}T[Ki(t}
= 0Therefore, matrices of [K] and [N] can be converted
into diagonal matrices.
Equation (18) is further reduced to an eigén value problem. of i x i. Consequently, the amount of calculations for vibration in water can remarkably
be reduced by utilizing the calculated. results
obtained in air. In our subsequent calculations by
Finite Element Method, in-air vibration modes up to.
5th order are used; namely, i 5, because. the authors' study aims at determining in-water vibra-tions in around the lower degree.
For a vibrating body in contact with water on both sides, the àuthois used a technique of doubling the
added mass matrix (MW], obtained by. the ca3.culaion'
of a fluid region on one side of the body.
As an example of the problem of the coupled.
vibration of a fluid and an elastic body, a problem of the vibration of aw elastic circular plate
subtherged in water. is considered; this plate is
length. And the natural frequency of the plate vibrating in the first and secondary modes (Fig. 5) are calculated by Lamb's method for comparison with those by the Finite Element Method.
Lamb has expressed the bending displacement of the circular plate vibrating, for instance, in the
first mode as follows:
w(r) = (1 -
L_)z
2 a2(approximation by one. term)
r2
r2
w(r) c ((1 -
_)2 + x(l -
(21)a2 a2
(approximation by two terms)
c, A: Unknown constants.
Using an exact solution of kinetic energy of a.fluid for a corresponding mode to that of Eq. (20) or (21),
he has proved ha.t the Rayleigh-Ritz's approximation
by one term orthe first two teris at the most. of the above equations is accurate enough for practical ase.
Table 1 compare natural frequencies caiculated
by the Finite Elemen.t MethOd and the energy method
of a vibrating steel circular pate of a/t = 50 with
both sides submerged in water.
For äalulation by the Finite Element Method,
model shown in Fig. 6 is contrived considering its
syrmrtetry. It.will be seen from the table that when
an iron circular plate of that size vibrates in water, its natural frequency is reduced to about
30% of that. in air in the fundamental mode and
about 50% in the secondary mode.
It. will also be noted that the discr.eancy. between
the calculated values of. the two methods is small:
1%. in air and 2.5% in water..
4. Tests on Web Frame Models (Examples of Calculatiors
b.y Finite Element Method and Energy Method).
This section discusses the problem of 'the vibration
of a web frame in the tank submerged in water,
study of. which has previously been made by the
authors et al. (1), as an example of the calculation for the coupled vibration of a fluid and an elastic body undertaken by the Finite Element MethOd and the
simple web frame models, the construction of which depends on selected basic parameters so that they can embody necessary vibration properties of actual web frames, are shown as well as the results of comparison of accuracies between the model tests and calculations by the Finite Element Method and the energy method.
4.1 Model Experiments
Web frame models as shown in Fig. 7 are prepared. Each modelis made of 3mm thick acrylite plates.
A rectangular-shaped water tank. of rigid
construc-tion measuring .630 x 420 x. 330mm is used to contain
the model, and measurements of natural frequencies
in water and in air as well, as vibration modes in
water and water pressure distribution are taken while the model is fastened to the tank at its
support frames with bolts.
For the purpose of comparison with the results of the experiments and calculations previously per-formed in reference (I), the models are subjected to the same boundary conditions; the three sides of
the web plate are fixed, while the remaining side
with a face plate is free..
All fixed sides of the web plate are reinforced by
systematically in depth d and flange width d, with a constant span Z, are prepared as shown in Table 2.
Since acrylite tends to creep1 - this tende,ncy was
pointed out in reference (10) - the Young'.s modulus
E obtained from static tensile tests cannOt be relied on in a dynamic situation.
Furthermore, E changes, though very, slightly, with
changing tnperature. Therefore, as suggested by
literature (10), the natural frequency of an.
acrylite dantilever is. measured each time a model
test is .perfbrmed, by forcing the cantilever to
vibrate. With the natu±a1 frequenc thus measured,
E is determined by reverse operation. The Poisson's ratio here is 0.3.
The web frame models are excited as they are.
fastened in the ,water tank which is fastened to a
shaking' platform with bolts, as illustrated in
Fig. 8.. Then the entire tank is forced to vibrate
in the direction parpendicular to the surface of
the web plate.
The water level in. the tank is set to the height h
where the flange is positidned for all models, considering the subsequent comparison with the results of calculations.
The measurements are taken in the following procedure: first, natural frequency is determined from resonarce curves by a XY recorder on receiving a signal
detected by a strain gauge attached to the web
plate; next, vibration modes fortheprimary resonance in water are determined by bringing a rod-shaped
probe from' a piezo-accelornter in contact witi
measuring points one at a time, and finally pressu±e distribution is measured by means of a pressure
gauge of 2.5mm thick with a diameter of 10mm attached
to the web plate.
All signals ued for the measurements are recorded on an electromagnetic oscillograph.
4.2 Calculation Model by Finite Element Method
The model used for calulatioby the. Fii,te Element Method is shown in Fig. 9.. .
With: this model, it is assumed that the flange in
contact with water is a stationary rigid wall,
that the web plate is in contact with water on both .sides and that the height of the water is y. = Ii.
Taking advantaqé of. the symmetry of x = Q/2,
one-half of the liqi region is considered.
The plane of x by y is divided into 7 x 7 of equal areas, and in the direction of z, the model is
divided into 8 blocks. A total of 576 nodal points exist in the model fluid region.
As fOr the web frame, the web plate is considered
as a pla-te-bending element and the face plate as
a beam element.
4..3 Simplified Calculation by Energy Method
Vibrations of the. prescribed model can roughly be
divided into the face plate predo±ninaiit vibration
and the web plate predominant vibratiOn, and both
a-re considered to be combined into a system of
vibration. Assuming a function of displacement satisfing the geometric boundary conditiçns that
the former (face plate predominant. vibration) has
three fixed sides and one free side (as shown at
(a.) in Fig. 10), and the latter (web plate pre
dominant viration) has three fixed sides and one ppdted.ide (as. shown at (b). in Fig. 10),
replace it, for the purpose of calculation, by. a
mass-spring system having 2 degrees of freedom, which is energetically equivalent.
() is hereinafter called F type mode, and (b) P type mode.
4.3.1 Calculation of Natural Frequency in Air
If the vibration mode of the entire web, w, is represented as
'- iT
w = A C sin (- x) (1 - cos y)
+ AY sin2 (j. x) sin -.y sin y} sinwt
(22)
A, A : Unknown constants
then the following equation will be derived by making the maximum kinetic energy and the maximum
strain energy to be equal:
+ + A 2k
CL)
2 (23)
mF 2Arn + A2 7
pmLt
in,. k : To be listed in Table 3.
In Eq. (23), rnF/ kF and mp, kp represent equivalent
concentrated mass and spring constants when mass is concentrated at points A and B in Fig. 10,
re-spectively. mFl kFp represent coupled terms,
respectively. Both the mass and spring constants
ir'Dh
Spring constant: K = k
0, so as to make w2 in Eq. (23) a minimufli,
by letting the first term on the right side of the equation be u, the minimum value u will be
the root of the following quadratic aqua.tion:
(m . m, - mFP2) u2 - (k . m + kPntF - 2 k . m) u
+ (kk - k2) = 0
(25)By letting the minimum root. of Eq. (25) be u01.
dimensionless fundamental frequency c5 resulting
from the coupling of tie two modes equals:
(24)
/
/D
P Zt
In.
Neglecting the coupled effect of the two modes allows independent F type and P type frequeni.es,
and respectively to be expressed by
Vibration mass: M = hiP t X m
= =
P
4.3.2 Treatment of Added Virtual Mass
(1) Added Virtual Mass for F Type and P Type
Vibration Modes
In calling WF F type vibration mode, P type
vibration mode, and and the velocity
potentials that satisfy the boundary conditions of respective modes; when the web plate is in contact with water on both sides, the kinetic
energy of fluid, T, equals
T = T
+ 2TFP + XT
(28) where-0
w00F F
T = TFP pw hZ
- f
j_ (.q
2 aa F F
dx dy z=O dx dyz0
w)
dx dyz0
(27) (29) (30)TF and. T represent the kinetic energies of F
type predominant mode and. .P type predominant
mode respectively, while TFP shows the coupled energy of the two modes.
If the effect of the flange is neglected, TF
and T (Eq. 29) will be obtained
from
thefollowing equations: (1) coth(y. .$) T = p 2.hw2 (A.B.)2 1.1. cos2wt i=Oj=0 '.. 1. J (31) =
/j.
Cj +
1)Z.(Z)
1 (i = 0) , 1/2(I
0) I = 1/2For F type. mode,
0.5, A1
= -0.5,
A.. = 0 (1 > 2)(32).
B is determined by the same
equation as (12).
Ao =0.5, A1
= -0.5,
A. = 0 (i > 2)B0 = , B1
= -
, B. = 0(j
>- 2)(33)
Then, TF and T are replaced by equivalent masses, M and respectively, which
concentrate at points®andin Fig. 10.
M and can be expressed by the following
equations: M_ = e x FW F p h2.2 w
2rrul
)2(F type added virtual mass)
p h2,2
w
= e
Pw p
211/1 )
(P. type added virtual mass)
(34)
The second terms in the right side of the above first and second ecuations are added virtual masses for continuous square plates with
-type predominant mode, Where vi-bration
ampli-tude becomes larger. near the flange.
Therefore, it is necessary to correct. in
Eq. (34) fOr the added virtual mass affected by the flange (Eq. (35)).
C F
p hi2
w
2r" 1 ±
( i:i: .)Increasing ratio in added, virtual
mass by flange
(3.5')
When the web with a flange (width d),vibrates in F type mode, let the kinetic energy of a fluid be TFdI and when the web withOut a flange 2h x i and h x Z sizes, respectively. (11)
and e represent correction factors.
When S/i = 1.0, calculated e,. and, e will be
as shown in Fig. 11 where h/i is a parameter.
(2) Effect, of Flange on Added Virtual Mass
The effect Of the increase of added virtual mass due,to the presence of the flange, which
locally restricts ,flid flow is more
vibrates in the same mode, let it be TFO.
Then CF is determined from the equation,
CF = TFd/TFO. TFO is determined from Eqs. (31) and (32). A detailed explanation on the method of calculation for TFd is found in
reference (1) , in which a flange is regarded as a rigid wall. This paper therefore omits
such an explanation.
The results of the calculation for CF are shown
in Fig. 12, where d/2. and h/Z are parameters.
This figure indicates that CF linearly increases nearly in proportion to the flange width, and on the contrary, reduces as h/Z increases.
(3) Coupled Term of Added Virtual Mass for F Type
and P Type Modes
Calculations of Eq. (30) are simplified by
replacing wF and w by modes WA and w3 respec-tively of 2h: 9 and hx L size continuous
square plates supported at their edges which are submerged in an infinite fluid region, and by substituting velocity potentials and corresponding to WA and w respectively in Eq.
fluid; T, cánbe expressed in
a. simple equation MAW = W2COS2Ut X h22. p hi2 w. .. w , W (36) 2,ii/l + ()2
As previously explained, the added. virtual
masses, M and for F type mode and P type mode respectively are ultimately expressed by MAW and 4BW (Eq. (34)) multiplied by a
coef-ficient. Therefore, the kixati coupled mass can be calculated from the follOwing
equation, using the solution of Eq.. (36) :
Mpp
:;
FW+ Mp)
4.3.3 Calculation for Natural Frequency in Water.
In order to obtain natural frequency in water, the same calculation process as in air applies except that a term of the added virtual mass ia added to
(CF.eF ... ... (.37)
27r/
2h
the denominator in Eq. (23); namely, in Eqs. (23)
through (27), m, is substituted by (mF +
= Np/h2Pt
4.4 Comparison between Experimental and Calculated Values
The following are the findings from the comparison between measurements in model experiments and
calculated values:
(1) Table 4 compares the natural frequencies
measurements of web frame models in air and water with those calcilated by the Finite Element Method and the energy method.
As YOung's modulus E by dynamic test varies
for each model due' to temperature correction,
natural frequencies used iere are dimension-less.
From. ti table it wIll be seen that the
dif-férence between the calculated values by the
Finite Eleztent Method and the energy method is
5% at the most both in air and water. The measurements also agree with those
calculated values to the same degree except
errors in the measurements, both dalcutation
methods are accurate enough, for practical use.
Therefore, in the case of structures which à.re
as simple as these.models, it is presumed. that
considerably good. approximate values will be
obtained from simplified calculation with 2 degrees of freedom.
(2) In the case. of vibrations in water of models
No. 3 and No. 4, where h/p is small and d/h is extremely large, the calculated values are 15 to 20% higher than the measurements regardless of :the methods of calculations.
Probable reasons are that whereas the
contact-ing surfàc,e of the flange with water was
considered as a fixed wall for both calculation methods, in experiments the entire web frame vibrated horizontally and the flange itself vibrated in a torsional manner with farely large amplitude under a large d/h conditiOn. Presumably, larger values of added virtual mass measurements have resulted from
thi
effect. Considering that in actual web frames
d/h = .0.1 to 0.2 at the most, the calculations
on the assumption that a flange is aS fixed wall give enough accuracy for practial use.
In calculating natural frequency in water by
the energy methOd, as has been. stated
previOus-ly, the kinetic coupled term of a fluid
must be taken into account along with. the
coupled terms and kFp of ati elastic body in F type mode and P type mode.. Natural
frequencies obtained when this effect is
dis-regarded; namely, mF = 0, are also shown in Table 4 in the parentheses.
The resulting natural frequencies are
over-valued by 5 to 35%,; this error increases as
and become closer to each other, indicat-ing that the effect of the kinetic coupled term
of a fluid must. not be disregarded.
As for the Finite Element Method, the cal-culate4 values are further compared with the web model measurements of vibration mode in the fundamental frequency, pressure
distribu-tion, and natura]. frequency in higher modes.
A portion of the comparison is shown in Fig.
13 and Table 5. Degree N indicated in Table 5 represents an order which has the number of
N-i nodes in the direction of span 9.. of a web
calculated values and the measurements
indi-cates that. the Finite Element, Method is an
effective means for estimating precisely natural frequencies and vi±ration modes in water even in higher modes.
5. Vibration of Perforated Plate in Water
In- an actual. ship, there are sections where perforated
structural materials are mainly used - the 'non-tight
floor in an aft peak tank,, for instance. Added virtual
decreases more rapidly for these perforated
-members than for non-perforated ones when they vibrate
in water. ' In calculating frequencies in water, there-. fore, it is important'to determine precisely the effects
Of. perforation on aded. virtual màss. ' .
Added virtual mass on perforated plates has been shown
experimentally using models, '
'but,, to the
authors' knowledge, 'no theoretical calculation of it
has ever been published. This report provides the
calculated. values of added virtuaL mass by, the Finite
Element Method on. perforated circular and square plates,
5.1 Model Experiments
A series of perforated circular an& square plates of 2mm-thick acrylite plates (Table 6) are. prepared, and their natural frequencies in. air and water are measured in the same manner as described in 4.1 of this paper while they are excited in the lateral
direction.
Each of the circular plates is fixed and the square plates supported at its edge and fitted with a
rectangular water tank by means of fit-up jigs as
shown in Fig. 14. The tank is filled with water to
an equal level with the top of the respective jigs for tests on vibration in water.
5..2 Calculation MOdels by Finite Element Method
All of the model experiments are performed in a
finite water region which contain one free surface -in the rectangular water tank.
In the case of a perforated plate,. water distributes
on both sides of the plate continuously and as a result the plate allows the fluid to pass through. .
its holes. Therefore, the pressure at its holes are unknown. In order to calculate the natural frequency
as the model experiments, it. is necessary to divide
the water regiOn. into very Small finite elements.
Therefore, to reduce the calculation, the following calculation model is conidered:
As to boundary conditions for a perforated square plate, Kito's continous square plate supported at its edge is cons.idered (the calculated value of a. non-perforated plate of
such condition is already shown), and for a
perforated circular plate, Lainb's circular plate
clamped at its edge with a rigid wall of in-finite length is considered. Thus the problem is reduced to a problem of the vir.ation in an
infinite fluid region.
Erbm the assumption that a fluid flow line in a hole is perpendicular to the plate surface in
contact: with water, a very lOw pressure P -is
estimated at. such a perforated.part.
Therefore, the authors considered the pressure
to be zero.
Fig.. 15 shows an example of division into finite.
elements of a perforated square plate in water. For a perforated circular plate, the same nthod of division as illustrated in Fig. 6 is used.
5.3 Comparison between Measured and Calculated Values
Natural frequencies measured and calculated by the Finite Element MethOd of perforated circular and.
square plates in air and. water are shown in Table 7 in terms of, dimensionless natural frequencies 2.
The table indicates satisfactory agreement. of the calculated values with the measurements with only about a 5% of error as for in air.
In calculations for natural frequencies in water by
the Finite. Element Method, an infinite fluid region
is considered as previously stated.
The resulting calculated valued agree with those of non-perforated circular and square plates calculated by LambTs method and Kit&s method with an accuracy
of 3% of higher. On the other hnd, experimental
values defy direct comparison with the calculated
values because the. experiments
are performed ma
finite fluid region, which leads to different added. virtual mass. However, the changes in natural
frequencies in water and in. added virtual mas of
perforated plates from thoe of non-perforated plates shown in Table 8 Indicate rather good agreement
between the calculated values and experimental values. Those calculated values and experimental values shown
in Table 8 are obtained in the saine manner as used
in reference (13) as follows:
In c2a and c2w are the natural frequencies of a plate
in air and in water, respectively, M is the ef-fective vibration mass of a:'plate, and Mw is added
virtual mats.,. the coefficient of added irtual mass.,
c, will be:
Mw a 2
-1
(38)When subscript 0 denotes a nonperforated plate, the
decreasing ratio. . of added virtual mass for a
per-forated plate is given by
MwI. C Mq
Nw0 £0 Mqo
In calculating vibration masses for plates, Mqo and M,. these are. regarded as energetically equivalent .
vibration masses for plate vibration mode; for the.
square plate, the mode of the square plate supported at its edqe in air is used, and for the circular plate Eq (.20) is used. The perforated area of
each plate is excluded from the region of
integra-tion of kInetic energy, and for the square plate its
perforated part is regarded as a rectangular one of
the same area for the conven-iencé of calculation.
The decreasing ratio of added virtual mass due to
the effect of: the perforation is shown in Fig. 16,
where a ratio of the perforated area to the area of
nonperf orated plate is used as a parameter,
The figure also shows experimental values of per-forated square plates obtained in reference (13) as well as an experimental fOrmula. This experimental formula was derived from model experiments on square plates of j/s = 2 and 4, and a/s = 1/2.
These experimental values agree well with the authors'
experimental values and -calculated values of the
authors' model of £/s = 2 and a/s =1/2.
However, with a perforated square plate of a/s = 1/3, the discrepancy between the authors' experimental and
calculated values- and. the values obtained from the
experimental formula referred to become great in a region of small perforated areas (y <0.1).
This indicates that the decreasing ratio of added virtual mass ot a perforated square plate must b
shown by using parameters both a/s and b/2,- instead of
perfOrated area ratio y. Thi also indicates a com-paratively small effect of £/s. From the above
plate with a small perforated area, the following equation can approximate more precisely the qualita-tive tendency of the plate (see Fig. 17):
The decreasing ratio of added virtual mass (Mw/Mw0)
=:1 - 26.5ci(1 - 2.lci
l.64cL2) (1 -
2.l.+ 1.,642)(40)
a
cwhere, ct.
= -,
= ,and c
must be such that arectangular area of ax c becomes equal
to
the areaof the perforation.
The decreasing ratio of a circular plate, on the
other
hand, is greater than that of .a square plate with a
corresponding ratio of perforated area (Fig. 16)
-Probable reaons are .not only the difference in edge conditions but also that the perforation exerts a
greater influence. on added virtual mass for
the
circular plate beause a radial flow of water becomes
predominant as compared with a peripheral flow when the vibration mode is syirunetric with respect to the.
6. Vibration of Web Frame in Water with Effect of Longitudinal
This section discusses the effect of the vibration of a
longitudinal, on the vibration of a web frame In the tank,
as an example of, the application to anàctual ship. As
typical vibration modes involved in the problem of web frame's vibration, the face plate and bracket pedominant vibration mode (F-B type) and the panel. and stiffener
pre-dominant vibration mode (P-S type) were mainly dealt with in reference (I) as mentiOned in ectidn. 3. It has been
revealed in si.thsequent tests on actual ships.that
longi-tudinals also. vibrate considerably with web frames. The
coupled effect of the two vibrating members is by no means negligible..
In the simplified equation for F-B type frequency presented in reference (1), the longitudinal, was treated just as an elastic spring, and the effect of mass' (mainly added virtual mass) was neglected in it. Then natural frequencies were overvalued where the longitudinal vibrated greatly, making
it difficult to obtain necessary accuracy. ':Thisdrawback
has already been pointed: out and corrected calculation
method suggeste4 by
researchers.,2'3
'However1 the effect of the. lonqitudinal. whidh .vibrate coupled with the webframe, specifically in water, mostly remain.s unsolved.
have applied the energy method, whiôh has been verified to give good approximate values in this kind of problem in
Section 3, and compared the calculated values. with measured
values on actual ships. Direct. calculations by means of
the Finite Element: Method will be made at some other
opportunity.when a general program is worked out.
6.1 Calculation for Added Virtual Mass
6.1.1 Added Virtual Mass. for Longitudinal Predominant
Vibration
For studying the longitudinal predominant. vibration
mode (hereinafter called L-type) some assumptions. have
been given: the longitudinal is supported on the po-sition of web frames; and the inclination of web frames,
when they .vibrate as rigid bodies, is proportional to
that of the longitudinal supported at the web. frame,
consequen.t1y, opposing webframes vibrate in antiphasei
As a calcu].atiOn model tha.t meets all, of the above.
assumptions, a rectangular. water tank asshorn in Fig. 18 is considered, where y h .is a free Surface,.
and x = is a rigid wall.
When y = 0, it is assumed that the vibration mode
ii
-cos. .-
x.. sin .- z .sinwt...(41)
.5
The vibration mode Wf of web frames, when z = 0, can be expressed by
2TT
(h\ y
Wf = COS X
()
.sint
(42)The kinetic energy of a fluid under theabovementioned conditions can be calculated as a combination of the two different vibration modes illustrated in Fig. 19. For method of calculation, see Appendix-l.
Ifafluidinarangeof4<z<and 4<x<
takes part in the vibration, of one web frame wall,
with the web frame
and
the longitudinal are submerged in water.on both sides and on one side respectively, the kinetic energy TL of a fluid can be expressed inthe following form: . .
TL . = 2 PwCL x
w2cos2cLt .... (43)
V = £.h.s (Volume of the tank) : Fluid density
rw
CL = C + CL CL Coefficient for L type mode
where is
equivalent added Fig. 18 into the the longitudinal
the conversion of the Qnergetially
virtual mass for the mOde shown. in
concentrated mass at the center of
area 9xS. C
-L1 the. case where the longitudinal
the manner shown at (a). in Fig.. 19, CL2 is a coefficient
for the case where the web frame alone vibrates as shown at (b), and CL is a coupled term, which is expressed as a negative value.
Results of. numerIcal calculation for CL are shown in
Fig. 20. The following are evidenced by the figure.:
h/s governs the added virtual mass for L type mode whereas the effectof h/i. is small.
CL is almost constant independent Of h/s. and C a.nd C , in contrast, . increase rapidly with
L12
h/S. Thus in the range h/s < 0.6 the èffec of
the vibration of longitudinal predominates, while
in h/s > 0.6. the added virtual mass predominates
due to the vibration of web frame as a rigid body
predominates.
The added virtual mass on the continuous square plates of 9xs with supported edge becOmes
is a coefficient for.
+ (. .& ) 2
S
(submerged. on one side)
Co values, when h/L = 0.5, ae also shown in Fig. 20. The figure indicates that because
CL starts:
increasing significantly as h/s is increased from around h/s = 0.4,but co shows an inverse trend, the difference between the two magnifies itself as h/s is increased.
Con-sequently, the effect of the vibration of the web frame as a rigid body is considered too great to be neglected when an actual ship has h/s 0.4.
6.1.2 Coupled Term of Added Virtual Mass for F-B Type and L Type MOdes
When the web frame is deformed elastically, the F-B
type vibration mode is assumed to. be
2
WF = cos . (1 - COS
y). Sint
(45)If the web frames and longitudinal vibrate as shown in Fig. 21 in a combined mode of F-B type and L type, w = WF + XWL (A.: amplitude ratio), the tótal:kjfletjc
p.;.Z2
energy of fluid, Tw, will be
Tw = 2ATFL + .X2T (46)
(47)
p.V.cFL
2cos2wtwhere PW.V.CF is the concentrated mass at the center
of the face plate span, 9, , which corresponds to
equivalent added virtual mass for F-B type vibration mode when both sides of the web frame and 'longitudinal
are sumered in water. PW.VcFL is the coupled term
TF : Kinetic energy of fluid for F-B type mode
Kinetic energy of fluid for L type mode
TFL : Coupled energy of fluid for F-B type and L type mode
has &iready been obtained from Eq. ', and T
can be obtained by the same method of. calculation as
TL. For the method,. see Appendix-2'. As in the case
of TL, TF and TFL can be expressed by.
of added virtual, mass for F-B type and L type
vibra-tion modes.
Results of numerical calculations for CF and CFL are
shown in Fig. 22. The following are evidenced by this
figure:
Like CL, CF and CFL increase markedly as h/s is increased, and the effect of h/Z is relatively
small. ..
The. value of CL is ten times higher than that of
C, at the same indicated h/s. In calculation for natural frequencies (to be discussed later in this section) the same tendency has also been notedl the vibration mass of a structure for L type mode is also ten times higher than that for F-B type vibration mode. From these findings it may be
said that C, and. CL have almost equal effect on
natural frequency.
. .6.2 Coupled Vibration of F-B Type and L Type MOdes
In this section the energetically equivalent mass and spring terms Of the structure, M and K, are respectively determined for F-B type mode WF and L type mode
WL which
previous section.
In calculating them it is assumedthat.bothen.s of the.
web frame s:span. is limited, to bulkheads or to horiontal
'girders, nd that the depth of the web
frairte forthe F-B
type moe is from the face plate of it totie
face plateof the longitudinal (h), whereas. an overall depth of the
web frame, H, is used for the L type mode and coupled terms.
By using symbols shown in Fig. 23
and
the calculationmethods presented in reference (1) , they can be expressed as fol1ows
(a)
F-B Type
Mode5 =
E(f).{2.If
()3
i 151X2 (x)} MF W 8 zW51.X2(xJ
(b)L Type 'Mode
KL =f.).{2.()2rr2.If
+..
I
X2 (x1) y .h.t)}KFL = 2E.
). W.X2(x:.)
+ + E WLIX2 Cx.)
16
Cc) Coupled Term of F-B Type and L. Type Modes
(49)
MFL Ct). Wf +(1
T! .2)(50)
(.
W.X2
Cx.) + .-y.t.E)}
Where X(x.) sin2 (- x.) g Gravity acceleration E : Young's ModulusCoupled frequency f(cps) in air for F-B type and L type modes can be. obtained from the following quadratic equation in the same manner as described in section 43.
(1. - K)f
-
F2 +
2fFL.K)f2
+ F2L2 - fFL'K)
0 (51)5 =
(cps), -4/T (cbs).ML2
FL = (cps), K-
MFM
5
andL are natural. frequencies for the F-B type
mode and for L type mode, respectively.
umnng up
the vibration mass of a structure,
M,,
MFL, or ML, and the corresponding added virtual mass al1ws the natural frequency in water to be obtained. Let the corresponding added virtual mass beF' FL' and
respectively. These are calculated. with respect to.
an actual ship as follows:
As the coefficients of added virtual, mass, CF, CFL
and CL, definded in 6.1, do not allow for the effect of an increase of added virtual mass due to the
presence of the face plate that will take place when
underestimation of added virtual mass will result. However, approximately,
FL :
CF : CFL : CL (52)
For
F' an approximate equation taking the effects
of the face plate and water level into account, which is presented in reeence (1), is used and expressed by (3rr 8) x 0 .55 )( = 16rr p 9.2h 0.047 x W - (54)
The first. term inEq. (53) isa factor toconvert equally distributed mass into concentrated mass at. a center of the face plate.
and
L can be calculated from Eq. (52)
CFL
CF
(55)
(one side stthmergence) =
(53)
L
expressed by Eq.
(55) represents the added
virtual mass when a fluid exists only in the tank.
If the 1ongtudina1.is in contact with a fluid
out-side the tank, the value for a continuous square
plate 2xs with supported edge is added to
Hence,
(both side submergence)
2
=Q.___:
+.
F CF
4iT/i -- (;)
where h/sis used as a parantete
for C.F, and h/s
C
andC.
6.3
Comparison between Calculated and Measured; Natural
Frequencies on Actual Ships
Based on. the above-mentioned calculation methods, the.
authors have calculated the natural frequericies ofweb
frames in tanks, both in air and water, using the same
five 1HIships as used for previous researches (reference
(1),. (2) and (3)), and then compared the calculated values H
with measurement values.
The results of. the comparison
are shown in Table 9.
Here are some findings:
In air, the calculated values agree quite well with
the measured values.. In water, the calculated values are found to be about 20% higher than the measured ones, but the amount of deviation in both values is
as little as around ±10%. Qualitatively, therefore, they satisfactorily agree with each other. This
report has concentrated on the coupled effects of F-B type and L type vibration modes. Considering
that in actual ships the. atural frequency. in water
should be. less than the calculated figures due; to
the coupled effects with web panels., stiffeners, etc.,
it is presumed that conparatiely accurate natural frequencies will be obtained if the calculated values are corrected by the use of a factor of 1.2 or so.
The ratio of vibration anp1itude at the center of face plate, AF, to that of the center of longitudinal. span, AL, is about 0.2 to about 0.5 in both calculated and measured values, indicating a comparatively good
agreement. . .. . . .
Calculated values obtained by simplified equations in reference (1) are also listed in Table 9.. These values are relatively inaccurate, especially when. they concern the L type node predominant, because of
of longitudinals. This suggests that this effect
cannot be disregarded.
(4) Under submerged conditions, web frames on bulkheads,
where longitudinals are relatively weak, tend to
vibrate in the L type predominant mode, and web frames on the bottom plate in the F-B type predominant mode. In the latter case, the F-B type frequency is
corn-paratively close to the L type frequency, which results in a great coupled effect; when coupled, . their frequency becomes about 20% lower than an in-dependent F-B type frequency.
7. Conclusion.
This paper was intended to evaluate the. accuracy of te authors program worked out to analyze natural frequencies by combining techniques of fluid analysis and structure analysis by means of the Finite Element Method, which is considered to be effective in solving coupled vibration problem of fluids and structures. Several calculations were performed for evaluation purposes. As .a practical application,, 'a problem of vibrations of web frames in a
tank and perforated plates submerged in water,. often
en-countered as vibration problem of actual ship, was taken as a subject, and a comparison of the above calculated values with the results of model expeiments was
attempted. . '
In addition, the energy method, a. means extensively used for solving this kind of problem, was also applied to
the problem of coupled. Vi:ratiOflS of web frames in the
tank with., longitudinals. The resultmn calculated values
were compared with measured values on actual .ships. . This
findings are: , .
(1) The comparison between the results of web frame model
experiments and the caIcuations by Finite Element Method indicates that the calculated values of natural
vibration modes, and fluid pressure distribution 'under
submerged conditions are very accurate, and thus the ef-ficiency of the Finite Element Method is verified.
Fundamental frequencies were calculate4 also by simplified equations of the energy method by modelizing 2degrees"of
freedom system. The discrepancy between the results of
the sirrplifiéd calculations, the calculations by' the
Finite Element Method, and model experiments was only about 5%, which implies that the energy nethod an
pro-vide.satisfaqtory accuracy from a practical viewpoint.
(2) As to the vibration of perforated plates in water,
cacu-lation models of perforated squar plates with supported edge and perforated circular plates wjth clamped edge were used to determine the effect of perforations on
added virtual mass by the Finite Element Method assuxni,n
that the plates were in. an infinite liquid.region and
that pressure at the perforations was 0.. 'The calculated
values are proved, by comparison with results of model experiments, to have acceptable accuracy. An approxi-matidn formula has also been obtained.
'(3) As to the vibration of web frames in the tank in actual ships, the coupled vibration of longitudinals and web
equations for natural frequencies reflecting this effect were developed by the use of the energy method. The
re-sulting calculated natural frequencies in water are about 20% higher than the measurements in actual.ships. But.
the percentage of deviation in both are found to be.
com-paratively small, about ± 10%. Qualitatively, they agree well with each other.
In this paper calculations, by .the Finite Element Nethdd were
confined only to plate bending vibrations in water. It is
obvious that analyses of vibrations of shells or other
3-dimentional structures in water would involve immensely tedious
calculations. The paper demonstrated that such techniques used in analyzing natural frequencies as the reduction of the degree of freedoir and the substitution of a vibration mode in water by the combination of vibration modes in air; are of assistance
in reducing calculations. with practically, acceptable accuracy.
Despite the fact that ship constructions are mostly submerged within.water, effects of water on them have not accurately
been grasped. to date... It is expected that the vibration analysis.
program on submerged sttutures presented in this paper will, be a basis for a generalization of the program. for solving problems of ship vibrations.
Appendix-i Calculations for Coefficient CL
The velocity potential of a water. tank side wail which vibrates in the Mode as shown in Fig. 18 is given by coribining
and L2 as shoin in Fig. 19. .BOindary conditions to be
satisfied by L and L are: -
-1 2
(1)
At .y=
1(5W) At
-,
S =:j2
az az
and Lztt satisfy the above boundary conditionsan4
Laplace's equatiOn can be expressed by
A.D.
= {PoPo(y." h) +
1 J
Li
1J
6:.
1J
(except for both i = 0 and
j =
0)X cOsi
XCosj
z Lx
wcosu.t (A-i)(2)
At x=
I : ax 3x (4) Pt z.= 0,a:1
=0,a:z
a:L2 (3) At y = 0,. ay. = wz, - =sinh6(
- h) coshô. . hwhere 2 + 2 x (.&)2, A. -Ci = 0)
I
4L ¶(4j2 -
1) ( . 1 A.B. - coshy.ij
ij
. (z -2 Ej=0
.S1ij sinh(
2: j
2 +(j+
x (L)Z L4{rr(2j+l)(-l)
Sir2(2j+1.)2
.2wCOS1 - X
where o. 2w 2..ij
D = L2 = E 1=0 r t 0.50.0
(1Ci >.2)
= 0, 1)x cos(j + Tn(coswt (A- 2)
Kinetic energy of a fluid for is calculated assuming that a fluid in the range of - < z < is involved in the
vibration of one web frame wall. With this, dimensionless added virtual masses, c , C
ändc
.', will ultimatélyL1 L2 L12
be expressed in. the form .of Eq..
(43)
as follows: (A.D.)2CL1 = A02D02
i=0 j=O
(except for i = 0 and j = 0) (A-4.)
1.0 (i = 0)
1.0 (j = 0)
i - 0.5 Ci 0) '
- C
0.5 (j 4 0)
(A.B.)2 Y.
cL = 2 E E 1 coth(
1JIJ)1
(A-5)2
i=0 j=0 ij
2 1 JI.
=0.5
3 C L11A.2D.B
2(h)
1 jk
-i=0j=Ok=0
{(
)2(k +
)2}.rr2A.2B.D
E E1 jk
i=0 j=0 k=0 {(rij)2 + 4k2}2
I.
1(A-6)
Appendix-2 Calculations for Coefficients CF and CFL
Velocity potential F of. a web frame for F-B type
vibration mode will be
A.E. - coshy... (z - ..)
1 J
1J .cosl. -r-x E ky..s...
i=0 j=0 ii sinh(x cos(j + ...) . y x wcost (A-7)
where
E..
J
(j
= 0)(j 0)
The coefficients, CF and CFL, in Eq. (47) can be calculated. in the same manner as above. Hence ultimately .
(A.B.)2 .y CF = 2 E E ..
coth(1') 1.1
(A-8) 1=0 j=0 ij 1 3 where {l.0 (i = 0) i. = 0.5 - 0.5 (i 0)C
L
= 2
A.2B.E.
coth(1J)
1.1.
i=Oj=O
y..s
2i:j
1)
+ 2
(!)
r (!....)2+ 4}
h
i=O j0 k=O
7T yr':ij
A.2 E .D 1.j k
References
Katsuya Fujii,Koji Tanida, Yoshio Ochi, andTeruyuki
Nonaka: Vibration of Web Frame in Tank, 1111 Engineering
Review Vol. 12 No. 4 (1972. 7) pp. 321-343
Ochi, Nagano, Hirai, Kimura, and Yoshida: Measurements
and Design Standard of Vibration of Web Frame in. Taflk.
1st Report Measurements on Actual Sbips: IHI Engineering Review Vol. 15 No. 2 (1975. 3) pp. 216-232
Ochi, Nagano, Yoshida: Measurements and Design Standard of Vibration of Web Frame in a Tank - 2nd Report Design Standard: IHI Engineering Review VOl. 15 No. 3 (1975. 5) pp. 390-407
O.C. Zienkiewicz: The Finite Element Method in Engineering Science: McGraw-Hill London (1971)
:.(5) Fumiki Kito:. Principles of HydrO-Elasticity: Yokendo
(1970)
(6) B.M. Irons: Structural Eigenvalue. Problems, Elimination of
Unwanted Variables: J.A.I.A.A. Vol. 3, No. 5, 1965
(7). M.V. Barton: Vibration of Rectangular and Skew Cantilever
Plates: Journal, of Applied Mechanics Vol. 18 No. 2 June,
Kouhei Matsumotb: Application Of Finite Element Method Of Added Virtual Mass of Ship Hull Vibration (1st Report), Journal of the Soc. of Naval Arch. of Japar No. 127 (1970. 6) pp. 83-90
H. Lamb: On the Vibrations of an Elastic Plate. in.Con.tact
with Water: Proc. Roy. Soc. London, Ser. A. i92l pp. 205-216
Yóshikazu Matsuura: Model Experiments of Vertical Vibra-.
tion of Ships., Journal of the Soc. of Naval Arch.. of Japan
No. 109 (1961. 6) pp. 433-441
Furniki Kito:Added Virtual Massof Plane Plates Vibrating in Contact with Water, Collection of Miscellaneous Articles by Zosen Kyokai No. 266 (1944. 5) pp. 1-10
Satoru Onurn:. Lateral Vibrationof Deep Girders in Water,
Journal of the Soc. of Naval Arch. of Japan No. 130 (1.971.
.12) pp. 285-296 . .
.
Ryuichi Nagarnoto, Masao Ushijima, Katsuo Ota]a and Satoru
Oriuma: VibratiOn of Local Structures Immersed
in
Water, Journal of the Soc. of Naval Arch. of West Japan No... 4.0(1970. 5) pp. 151-170
Masaki Matoba .and Tatsuro Kawamoto: Experimental .Study
on the Vibration of Stiffened Plate Contacted with Water,
(1968. 5) pp. 193-208
(15) Yoshikazu Matsuura and Hajime Kawakami: Calculation of Added Virtual Mass and Added Virtual Mass Moment of
inertia of Ship Hull Vibration by the Finite Element Method, Journal of the Soc. of Naval Arch. of Japan
rrrr
rrr4r
rrrr
INumber
of degrees of freedom =60(b)
vir4r4r4
'4,4,4,4
N. D. F. = 20rArA
13(d)
rAr4rJA
rArArAr4
N. D. F. =Fig. I Natural f1.quency of square cantilever plate (reduction
of matrix)
All degrees of. freedom eliminated except lateral deflection at ringed nodes.
N
(jr) (
) denotes the resultsby BartOn(7:N N 3.470 8.665 24.31 30.38 48.92 1 3.467 2 8.554 3 21.58 4 27.13 5 31.28 1 3.467 ( 3.494) Without Reduction. 2 8.549 ( 8.547) 3 21.53 (2L44 ) 4 27.00 (27.46 ) ox 5 31.07 (31.17 ) N I 3.467 2 8.565 3 22.16 4 27.26 5 33.43 1 2 3 .4 5 I,
t.
Vibrating Plane Fluid Region
Fig. 2. Division of dimensional fluid region into finite elements
Vibration.Mode wa0 cos2fx(1 cos-y)sin cut
Fig. 3 Vibration mode of side walls of water tank
Fig. 4 .Pressure distribution on the vibrating wall of a water tank
Infinite Rigid Plane
Fluid(p)
Elastic Circular Plate
Fig. S Vibration, of a circularplate clamped along the circumference in contact with. water
x-yplane(zO)
11 I I 1 I Circular Plate. Circular Plate Bldck Number of Fluid:7 - Boundary Cond. 1st.. 2nd -,Elastic Circular Plate
dP_
.f_
z
P=o
P=O
f=o
p=o
ai__
E_
öy (6)f=o
frP,0
Supporting Frame
418mm
Rectangular Tank (Breadth:420mm)
I
Fig. 7 Description of web frame models
/ Shaking Platform
4-
,-Fig. 8 Excitation method
® Sec. 120 m
Bolt hole to be fixed to tank
(Side Wall)
Fig. 10 Assumed modes of web. frame ._--- - .---- -.- --____ _._ _4._(__ __._
.--
-
-
--
-
-. .- _,_,. -.-
'--
-
-
'- .-> ,..-- '-..----
--.
--.-...-
_,-_- -
-'- __*._. --
...--I I
--V
--I _# I i i_-L' _r
I i-
I -II'
- i_ I.- -I i._I t - I_1
I_I I_1 I-
I_ -1-
-
-' I_- E-
-_I .-I-
-V i_MIr'
L-I (_#T I-.0 1 -- iI-
I-i
:_--k
_y
I L-'i ij-
1 I I (Side Wall) äz. -(Bottom Wall) ayTotal Joint Number of Fluid : 576
x.
Web Plate
Model caluculated by F.E.M.. of web frame vibrating in contact with water
(a) Mode for Face PlatePredominant (F TYPE) (b) Mode forPanel PredOminant (P TYPE>
3x.
(Sym. Plane)
P=0(Free Surface)
(Face Plate) ay
3.0
CC, 0 to U I.. C.) cn U< .1.0
1 - I- -I 0.1 - 0.2s/11.0
(Both sides-of the plate
immersed. in fluid L12 (F Type
241±(i.)2.AddedMs
MpepX. .Pr..
. (P Type03
04
07
09
ep eli/I
Fig. 11 Added virtual mass'for F and P type vibration mode (without flange)
L0
(Both sides- of the plate:
-immersed in fluid 1.0 - t