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(1)

I4.r

-By

K. Fujii K. Tanida Y. YOkokura

Ship Srengch Department Research Institute

IshikawajimaHarima Heavy Industries Co., Ltd.

LaboWfl

Or

Scheepahydmmech81ica ftrchief

Mekelweg 2. 2623 CD ft

Pej 015 Th837S Fa 01.3 7. ::3

Vibration o Elastic Body in Contact ich qatar

(2)

,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

(3)

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

(4)

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.

(5)

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:

(6)

-Un, p=pw

Fluid density

v : Velocity of body

p : Fluid pressure

n : Normal Component to the body surface

t:

Time

Added 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)

(7)

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)

(8)

The kinetic energy of fluid, Tw, will then be yielded from Eq. (1): 1

Tw=-pff

- ds an = c c Mwij w. w. (5) Where

Mwij = -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.

(9)

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

(10)

(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

(11)

(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

(12)

that the slave variables satisfy only the

con-ditions in which, potential energy will be at its

minimum,

and that there exists no inertia force

at 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

(13)

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

(14)

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

(15)

.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

(16)

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 0

HN1 C_l

T 0 p2 (9) = [A] }

[] = (HJ

[] = [Hi]

-

[C]T. i1r1

[C1_11 = 2, 3, 4, N) (10)

(17)

As

an

example of calculations, let us.consider a problem in which two opposing square side walls of a tank filled with water are

normally 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 j

ij

sinh(Y1S)

2Tr

x

cosi-2.

2ir/.z

+ .j +

=0.5, A1

=

0.5, A1

= 0.0 (i > 2) (12) 4 2

Bo =

-

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

(18)

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)

(19)

on wall of vibrating body (Plane-N)

:' Vector of displacement at

iodal

pQ.nts

on 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}

= 0

Vector 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)

(20)

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'

(21)

co'

}rT[M]:{o

=0

(r s) (l.9)

(o,}T[Ki(t}

= 0

Therefore, 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

(22)

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.

(23)

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

(24)

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

(25)

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.

(26)

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

(27)

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.

(28)

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

(29)

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

(30)

= =

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 p

w hZ

- f

j_ (

.q

2 a

a F F

dx dy z=O dx dy

z0

w)

dx dy

z0

(27) (29) (30)

(31)

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

the

following 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 = 1/2

For F type. mode,

0.5, A1

= -0.5,

A.. = 0 (1 > 2)

(32).

B is determined by the same

equation as (12).

(32)

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

(33)

-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

(34)

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.

(35)

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

(36)

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

(37)

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.

(38)

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

(39)

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,

(40)

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

(41)

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.

(42)

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

(43)

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

(44)

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

(45)

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

c

where, ct.

= -,

= ,

and c

must be such that a

rectangular area of ax c becomes equal

to

the area

of 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.

(46)

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 web

frame, specifically in water, mostly remain.s unsolved.

(47)

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

(48)

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 in

the 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

(49)

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.

(50)

+ (. .& ) 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

(51)

energy of fluid, Tw, will be

Tw = 2ATFL + .X2T (46)

(47)

p.V.cFL

2cos2wt

where 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.

(52)

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

(53)

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 to

tie

face plate

of 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 calculation

methods presented in reference (1) , they can be expressed as fol1ows

(a)

F-B Type

Mode

5 =

E(f).{2.If

()3

i 151X2 (x)} MF W 8 z

W51.X2(xJ

(b)

L Type 'Mode

KL =

f.).{2.()2rr2.If

+..

I

X2 (x1) y .h.t)}

(54)

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 Modulus

Coupled 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.

(55)

(1. - K)f

-

F2 +

2fFL.K)f2

+ F2L2 - fFL'K)

0 (51)

5 =

(cps), -4/T (cbs).

ML2

FL = (cps), K

-

MFM

5

and

L 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 be

F' 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

(56)

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)

(57)

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:

(58)

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

(59)

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.

(60)

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

(61)

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

(62)

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.

(63)

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. . h

(64)

where 2 + 2 x (.&)2, A. -Ci = 0)

I

4

L ¶(4j2 -

1) ( . 1 A.B. - coshy.

ij

ij

. (z -2 E

j=0

.S

1ij sinh(

2

: j

2 +

(j+

x (L)Z L

4{rr(2j+l)(-l)

S

ir2(2j+1.)2

.2w

COS1 - X

where o. 2w 2..

ij

D = L2 = E 1=0 r t 0.5

0.0

(1

Ci >.2)

= 0, 1)

x cos(j + Tn(coswt (A- 2)

(65)

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 ultimatély

L1 L2 L12

be expressed in. the form .of Eq..

(43)

as follows: (A.D.)2

CL1 = 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 J

I.

=

0.5

3 C L11

A.2D.B

2(h)

1 jk

-i=0j=Ok=0

{(

)2

(k +

)2}.rr2

A.2B.D

E E

1 jk

i=0 j=0 k=0 {(rij)2 + 4k2}2

I.

1

(A-6)

(66)

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 k

y..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)

(67)

C

L

= 2

A.2B.E.

coth(1J)

1.1.

i=Oj=O

y..s

2

i:j

1)

+ 2

(!)

r (!....)2+ 4}

h

i=O j0 k=O

7T yr':

ij

A.2 E .D 1.

j k

(68)

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,

(69)

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,

(70)

(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

(71)

rrrr

rrr4r

rrrr

INumber

of degrees of freedom =60

(b)

vir4r4r4

'4,4,4,4

N. D. F. = 20

rArA

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,

(72)

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

(73)

Fig. 4 .Pressure distribution on the vibrating wall of a water tank

(74)

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

(75)

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

(76)

(Side Wall)

Fig. 10 Assumed modes of web. frame ._--- - .---- -.- --____ _._ _4._(__ __._

.--

-

-

-

-

-

-. .- _,_,. -.

-

'-

-

-

-

'- .-> ,..-- '-..-

---

--.

--.-

...-

_,-_

- -

-'- __*._. --

...--I I

--V

--I _# I i i_

-L' _r

I i

-

I

-I

I'

- i_ I.- -I i._I t - I

_1

I_I I_1 I

-

I_ -1

-

-

-' I_- E

-

-_I .-I

-

-V i_MI

r'

L-I (_#T I-.0 1 -- i

I-

I

-i

:_--k

_y

I L-'i i

j-

1 I I (Side Wall) äz. -(Bottom Wall) ay

Total 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

(77)

3.0

CC, 0 to U I.. C.) cn U

< .1.0

1 - I- -I 0.1 - 0.2

s/11.0

(Both sides-of the plate

immersed. in fluid L12 (F Type

241±(i.)2.AddedMs

MpepX. .Pr..

. (P Type

03

04

07

09

ep e

li/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

0.20

MF = eF X 0 15

0.05

0.1

dli

Fig. 12 0.2 ' 0.15 I-0 0.1 0 0.05

Cytaty

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