Application of finite element method to added virtual mass of ship hull vibration

(1)

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I

I Summary

This paper presents a general method for the calculation of the three-dimensional hydrodynamic

pressure by means of the finite element method and its application to the added virtual mass to

the ship vibration. In applying this method to the ship vibration, the added virtual mass is

obtained by assembling the pressures over the ship's vetted surface.

Calculations were carried out on prisms of various cross-section, and comparisons were made between the values obtained by this method and those by the conventional analytical method.

They were in good agreement for practical use.

The paper also presents a calculation method of the added virtual mass to the vertical vibration of higher mode.

i

Introduction

With the recent remarkable development of digital computers, the application of the finite element

method has been developed in the research field of static structural analyses.

An application of the finite element method to field problems such as heat conduction, hydrodynamics

etc. was originally introduced by Zienkiewicz and Cheung.(') They have shown that problems the physical behavior of which is governed by a general "quasi-harmonic" differential equation can be

treated as those in which a certain quadratic functional has to be minimized over a region, and therefore be solved by the finite element method, utilizing all the concepts of statical structural

analysis.

The hydrodynamic pressure acting on an immersed body in vibration can be obtained by the use of this formulation, too, and the so-called added virtual mass is supposed to be an integrated value of

the hydrodynamic pressure over the body surface.

Matsuura and Kawakami(2> have first applied this method to the caluculation of added virtual mass

to ship vibration by two-dimensional treatment. They showed that the virtual inertia coefficient obtained by this method for a bar of rectangular and circular cross-section agreed well with those by

the conventional method. This fact suggests that the application of the finite element method should

he a powerful tool in this research field.

Hirowatari and the author(8) have published a paper last year on an extended application of this

method to the three-dimensional problem.

In this paper, the author presents a general method of the calculation of three-dimensional hydra-dynamic pressure by means of the finite element method and its application to the calculation of the

added virtual mass to ship vibration.

The method was applied to an ellipsoid of revolution, an infinitely long cylinder etc. in three-dimensional motion, and the calculated hydrodynamic pressure acting on each body surface and the

* Technical Research Laboratory, Hitachi Shipbuilding & Engineering Co., Ltd.

Kouhei Matsumoto*, Member

83

Lab.

y. Scheepsbouwkunde

Technische Hogeschool

_t7

(t5ti 45f5

Application of Finite Element Method to Added

Virtual Mass of Ship Hull Vibration

(2)

s

84 127

so-called "correction factors for three-dimensional motion" were compared with the alues obtaine

by the old analitica! method. Further, the author presents a practical calculation method of the

correction factors for three-dimensional motion" to the added irtual mass of the vertical hull

vibration of higher mode.

2

Calculation of Hydrodynamic Pressure in Three-dimensional

Motion by the Finite Element Method

The finite element formulation is made as fol1on-. First, the fluid region under consideration is.

divided into finite number of elements. The hvdrodvnamic pressure distribution within each finite

element is assumed as a function of nodal values of the element, and the contributions of each element

are evaluated and assembled over the whole fluid, region. Thus, the calculation of hydrodynamic

pressure reduces to solve the following simultaneous equalions

(1)

where, matrix [S] corresponds to the well known stiffness matrix in the structural analysis.

When the acceleration vector (V) of a body in motion is given, the hydrodynamic pressure vector jp} will be calculated by equation (1). And the added virtual mass can be obtained by using the vector (p) acting on the body over its whole surface.

Matrices [S]C and [A]C for one finite element and their characteristics are given in Appendix.

In treating the problem as a three-dimensional, there are two difficul .es come into question. One is

that there is visually some confusion in dividing the three-dimensional fluid region into space finite elements, and the other is that the size of simultaneous equations becomes to large for practical purpose. To overcome these difficulties, the author introduced a "Layer Division Method". The method

is to divide the fluid region under consideration into finite

space elements by the following three kinds of curvedsurfaces.

That is,

A group of curved surfaces parallel to the boundary

surface of the body.

A group of curved surfaces which divide the surfaces

(1) above-mentioned in the longitudinal direction.

A group of curved surfaces which divide the surfaces

(1) above-mentioned in the transverse direction.

These groups of curved surfaces are shown in Fig. 1. The

surfaces (1) above-described are called as "Planes 1, 2, 3 N" in order from outside, and a third space surrounded by the Plane i-th and the (i+1)-th as "Layer i".

According to the "Layer Division Method", we can divide the fluid region into finite space elements. holding the following relationship.

Each "Plane" has the same number of nodal points.

The number of finite elements included in each layer is the same.

(e) Only "Plane N" does coincide with the boundary surface of the body, and all the nodal points.

except on "Plane N" become the so-called "freenodal points".

When the fluid region is divided into the finite elements by the use of "Layei Division Method". equation (l can be converted into the following expression.

BOUNDARY SURFACE

OF A BODY

Fig. 1. Division of fluid region into finite elements

i

where, each subscript re;

As only the vector jP\ needed for our present

finally the following equi

where,

3 Cor

In order to ascertain t

the calculated hydrodyr obtained analytically for Three examples chose

çl) Translation of Flexural sinus Translation of Fig. 2 shows, as an of example (1). In th soids, 7 hyperbOloids o

of the ellipsoid are ch the "Layer Division M

eighth of the fluid re

considered here. App

eighth of fluid regio'

8-noded and the rest The fluid region of

assumed to be finite This approximation

- si

ciT o o

(3)

the values obtaine

tion method of

the-the vertical hull

tisional

ade: ciansideration is.

within each

finite-tiuns of each element

)n of hydrodynamic

(1)

al analysis.

amic pressure vector

:ained by using the

o in Appendix.

caesrion. One is

n of fluid region

Lite elements

outside, and a thin

space

elements-I all the nodal points Division Method",.

Application of Finite Element Method to Added Virtual Mass of Ship Hull Vibration ₈₅

-where, each subscript represents the "Plane" number.

As only the vector [.PV), which means the pressure acting on the boundary surface of the body, is needed for our present purpose, {Pj, [P2},., (PNI} are eliminated in turn from equation _{(2), and}

finally the following equation concerned with {PN} can be derived.

(3)

avhere,

[j] =

[S1]

[]

= [Si] - [Ct_1] T [-] -1 [C_] (i=2, 3, 4, .., N)

3

_{Comparison between Hydrodynamic Pressure Obtained}

by the Finite Element Method and by the

Old Analytical Method

In order to ascertain the accuracy of the finite element method, comparisons were made between the calculated hydrodynamic pressure by this method and the solutions which _{have already been}

obtained analytically for the three-dimensional motion by the old theory.

Three examples chosen as analytical solutions are given with the velocity potentials as follows. Translation of an ellipsoid of revolution moving perpendicular to the axis.

Iog _}cos_w _{by Lamb(4>}

A{-log

O Ci

} kV

-

go'

co(Co--1)

Flexural sinusoidal vibration of an infinitely long circular cylinder.

VK1(ka)

kK'(ka)cos O-cos kZ by TayIorJ5>

Translation of a finite circular cylinder moving perpendicular to the _axis.

4V , K1(k&-)

sin o sin kxZ

r i(ima)

Fig. 2 shows, as an example, the divided fluid region in the _case

of example (1). In this case, curved surfaces of 10 confocal

ellip-soids, 7 hyperboloids of two sheets and 10 planes including the axis of the ellipsoid are chosen as the three kinds of dividing surfaces in the "Layer Division Method". And because of symmetry, only

one-eighth of the fluid region cut off at the three planes of symmetry is

considered here. Applying the "Layer Division Method", the

one-eighth of fluid region is divided into 486 finite elements; 405 _are

8-noded and the rest 6-noded.

The fluid region of the body is actually infinite, but here it _is assumed to be finite and in the interior of

(9b)23 + (9k)2 =1.

by Kuinai.(6)

DIRECTION OF VIBRaTION

Fig. 2. An example of an

This approximation gives somewhat higher value of hydrodynamic _{ellipsoid of revolution}

Sl

C1T O O C1 S7 C22' o O O C2 Sa o O Cs O O SN1 CNI 7' CN-i SN -. - Pl P8 PNI PN

-O-.

o o O AVN

(2)

(4)

s

86

/[Pw'bV)

(i)

Translation of an ellipsoid of revolution moving perpendicular to the axis

irit

l27j'

-

ANaLYTICaL SJWTION BY TYLO5

AL L :,N BY LAL3 FINITE EEBT SOWTION

- ELEME'

3b

/ \

..T,CAL 23WTAN B KUMaI

F, ELEMENT SCLJTJON

Gb

P/I _V)

0 2

(iii) Translation of a Imite circular cylinder moving perpendicular to the axis

Fig. 3. Comparison of hydrodynamic pressures between finite element and analytical solution

pressure, namely added virtual mass, than that of taking into account the actual infinite fluid region,

but the difference is negligible small. Fig. 3-(i) shows the results of calculation. From this figure. it is known that sufficient accuracy is obtained in spite of such a rough division.

The calculation results for examples (2) and (3) are shown in Fig. 3-(ii) and (iii). In which the value obtained by this method for example (3) is somewhat higher than that by analytical method. This may be due to the difference of the boundary conditions taken at the end of the cylinder.

4 Three-dimensional Correction Factor to Added Virtual Mass

for Infinite Prisms of Various Cross-sections

The three-dimensional correction factor to added virtual mass, J, is generally defined as Kinetic energy of surrounding fluid in three-dimensional motion

Kinetic energy of surrounding fluid in two-dimensional motion

But in this paper, the following definition is taken for the factor J.

Hydrodvnamic pressure acting on the body surface in three-dimensional motion hydrodynamic pressure acting on the body surface in two-dimensional motion

Of course, the kinetic energy and the hvdrodynamic pressures are different in magnitude, but there will be no difference between the two Jvalues defined above.

J-values for an infinitely long prism of circular and rectangular croSs-section in sinusoidal vibration have been derived by Taylor) and Joosen et al.(7), respectively. These value arc compared with those obtained by the finite element method. Results of the comparison are shown in Fig. 4, in which 1/b is the ratio of 1/4 wave length of the vibration pattern to the half breadth of the prism. It canbe

seen from this figure that the values obtained by this method are good agreement with those by

P' [PW2bVJ

¿t

(ji) Flexural sinusoidal vibration of an infinitely long circular cylinder

Application of Fi

Taylor for practical use

J_values calculated at 1Jb

tion and a midship section sectional coefficients of the

0.957 and A=1 where eis ficient and the draft-bean corner of the midship se

seen in Fig. 4, thelatter equal to the value for rei,

5 ApplicatiOn

Element Method

of Added Viri

Ship Vil

In applying the finitet

is necessary to employ sufficient accuracy. Tht

number of finite element becomes large. Therefo

vibration of higher mo

enough capacity. In orc Taking advantage of th

the fluid region to be region. Now, if the sb

along her length. As t

on the vibration ampli

Therefore, there exists

respectively in the fo.

of p=O and (n+l) sur

Fig. 5. Divis

(5)

sub-..:tion of ari

r cylinder

.ytical solution

mite fluid regione From this figure.

ii',. In which the

tnalytical method.

e cylindei-.

Mass

ed as

Application of Finite Element Method to Added Virtual Mass of Ship Hull Vibration 87

Taylor for practical use. In this figure, the

J-values calculated at 1/b=3 for a Lewis

aec-tion and a midship Secaec-tion are also presented. The

sectional coefficients of the Lewis section aie0=

0.957 and 2=1 where o is the sectional area

coef-ficient and the draft-beam ratio, and the round corner of the midship section has r/b=0.12. As seen in Fig. 4, the latter twoJ-values are nearly 025

equal to the value for rectangular section.

5

Application of the Finite

Element Method to Calculation

of Added Virtual Mass to

Ship Vibration

In applying the finite element method to the calculation of added virtual mass to ship vibration, it is necessary to employ a division method corresponding to the vibration profile in order to keep.

sufficient accuracy. This means that the higher the natural frequency of ship is, the larger the number of finite elements becomes, and accordingly the size of simultaneous equtions to be solved

becomes large. Therefore, it seems to be very difficult to calculate the added virtual mass to ship. vibration of higher mode directly, because the digital computer available at present stage have not

enough capacity. In order to eliminate this difficulty, the author introduced the following device. Taking advantage of the fact that the vertical vibration of a ship is symmetry about her center line, the fluid region to be taken into account can be reduced to one half of the whole surrounding fluid

region. Now, if the ship excutes n node vertical vibration, she may have n nodes and (n+1) loops.

along her length. As the hydrodynamic pressure acting on the wetted surface of the ship depends. on the vibration amplitude, the pressure distribution should correspond to the mode of vibration. Therefore, there exists surfaces where p=0 and ß=maximum corresponding to each node and loop. respectively in the fluid region. That is to say, the fluid region under consideration has n surfaces

of p=0 and (n+1) surfaces of ( )=o and the whole fluid region can be divided into 2n sub-.

Fig. 5. Division of whole fluid region into sub-regions

075 to -J

LO

CIRCULAR SECTION BY TAYLOR

FINITE ELENTLU11ON;

-* CIRCULAR SECTION

CTANGULAR SECTION

LE'l5 FOI SECTION

(OO957, X.i)

° MIDSHIP SECTION

O 2 3

Fig. 4. Three-dimensional correction factors of infinite prisms with various uniform cross-sections

Jo BY LJoIB

'Jo

* BY LEWIS

A

FINITE BEIENT ETHI

TRANSLATION

2-NODE

° 3-NODE

4 6 6 IO 2

L/B

Fig. 6. Comparison of J-values of ellipsoids of revolution between

finite element and analytical

solution

,)fl

i I (Jn

nitude, but there

ìusoidal vibration ipared with those

4, in which 1/b

prism. It can be at with those by,

Lo 09 w Q8 - 07 0.6

(6)

88 127 '

and

regions by those surfaces. This means that the hydrodynamic pressure will be calculated by

assembl-ing the values independently calculated for each sub-region. Fig. 5 shows, as an example, the

.division of the fluid region for 2 node vertical vibration.

This method is applied to the J-value calculation for the vertical vibration of an ellipsoid of

revolution. In this case, the dividing surfaces are approximately considered as those of hyperboloids.

if we assume that the vibration modes are represented as

for 2 node vibration

for 3 node vibration,

where p: a variable adopted an elliptic coordinate, then the dividing surfaces, become ,u=O, ±/5 and =0, ±-J-.-, ±}/-- for 2 and 3 node vibration, respectively.

The J-values obtained by this method are plotted in Fig. 6 and compared with the analytical values obtained by Lewis(8) assuming the vibration as a shear vibration. This is because that the direction of acceleration vector taken in this calculation is normal to the free surface and this corresjonds to the shear vibration. In Fig. 6 the coefficients J for the motion of translation in a direction perpen-dicular to the axis are shown and compared with the value by Lamb. From Fig. 6 it is known that

these results are good agreement with the analytical ones.

In calculating added virtual mass to Ship hull vibration of higher mode, it is more reasonable to adopt the local three-dimensional correction factor for each Section instead of the overall J-value

which is Constant over the whole ship length(°). The distribution off-value can be calculated indepen. .dently for each sub-region by this method. But as the vibration mode becomes higher, the distance

between adjacent dividing surfaces along the ship's length, i (nearly 1/4 of wave length of vibration),

becomes shorter, and the dividing surfaces which exist on the parallel part of the ship body become

approximately planes perpendicular to the direction of ship length. Therefore the J-value for the

parallel part is considered to be Constant and nearly equals to the value represented in Fig. 4. This

means that only the J-value for the after and fore parts of the ship should be calculated by this method.

It is also supposed that the ship's form will affect a little upon the local J-value. Therefore if the

ship's form is not so particular, it will be unnecessary to carry out this calculation for every ship to

'evaluate the natural frequencies of ship.

fi

Conclusing Remarks

A general method for the calculation of three-dimensional hydrodynamic pressures acting on a body

in vibration is investigated by means of the finite element method. To apply this method to ship

vibration, the "Layer Division Method" is introduced for practical treatment. Some examples are

solved by this method and the results of calculations are compared with those derived analytically.

They indicate good agreement.

The added virtual mass to ship hull vibration, which is defined in this paper as the hydrodynamic pressure acting on the ship's wetted surface, is obtained by this method and compared with those

obtained by the old theory. Moreover, for higher mode vibration, a method of dividing the fluid region into sub-region is devised. Three-dimensional correction factors for ellipsoids of revolution are

obtained by this method and compared with analytical solutions by Lewis. These comparisons indicates good agreement for practical use. Application of this method to actual ship's form will be shown in

the next paper.

5/

V₌ V2 -3

/7

t

The author would like t

1jjrowatari of Technical Rest

Zienkie\viCZ, O. C. ai Engineer. Sept. 24, Matsuura, Y. and S Moment of Inertia c Ilirowatari, T. and I of J.S.S.C. on the i Lamb: IlydrodYnal Taylor, J. L.: Sam Kumai, T.: On th

the Vertical Vibrat Joosen, w.P.A. am Mass of vibrating Report No. 40 S, i Lewis, F. M.: The Vol. 37, 1929 AnderSon, G. and

Several Nodes and

i

Matrix [SV and [A We adopt the tetrahed

pressure distribution with

natriX se reduce to a I

where,

i=1,2,3,4, and the ot

1,2,3,4.

When any surface (i the boundary matrix

(7)

culated by assembl. s an example, the of an ellipsoid of ise of hyperboloids.

ie,a=0, ±7- and

e analytical values that the direction is Corresponds to a direction perpen-it _{is known that} ìore reasonable to verall J-value calculated

indepen-her, the distance

ngth of vibration),

ship body become J-value for the

d in Fig. 4. _This d by this method. Therefore if the for every ship to

ed analytically.

p

te hydrodynamic

red with those

the fluid region

revolution are

risons indicates ill be shown in

Application of Finite Element Method to Added Virtual Mass of Ship Hull Vibration 89

7 Acknowledgement

The author would like to acknowledge the continuing guidance and encouragement by Dr. T.

flirowatari of Technical Research Laboratory, Hitachi Shipbuilding & Engineering Co., Ltd. References

Zienkiewicz, O. C. and Cheung. Y. K.: Finite Elements in the Solution of Field Problems, The

Engineer. Sept. 24, 1965

Matsuura, Y. and Kawakami, H.: _{Calculation of Added Virtual Mass and Added Virtual Mass} Moment of Inertia of Ship Hull Vibration by the Finite Element Method, J.S.N.A.. Vol. 124 Hirowatari, T. and Matsumoto, K.: On the Vibration of an Elastic Body in a Fluid, Symposium of J.S.S.C. on the Matrix Structural Analiysis. 1969

Lamb: Hydrodynamics. 6th Edition.

Taylor, J. L.: Some Hydrodynamical Inernia Coefficients. Phil. Mag., 9, 1930.

Kumai, T.: _{On the Three-dimensional Correction Factor for the Virtual Inertia Coefficient in}

the Vertical Vibration of Ships. J.S.N.A. Vol. 112

Joosen, W.P.A. and Sparenberg, J. A.: _{On the Longitudinal Reduction Factor for the Added}

Mass of Vibrating Ships with Rectangular Cross-section. _{Netherlands Research Centers T.N.O.,}

Report No. 40S, 1961

Lewis, F. M.: The Inertia of the Water surrounding a Vibrating Ship. _{Trans. S.N.A.M.E.,}

Vol. 37, 1929

Anderson, G. and Norrand, K.: A Method for the Calculation of Vertical Vibration _with

Several Nodes and Some other Aspects of Ship Vibration, R.I.N.A., Vol. 111, 1969

Appendix i Matrix [S] and [A]'

We adopt the tetrahedral finite element as shown in Fig. A-1. If we assume the hydrodynamic pressure distribution within a finite element is _{P=1+û2x+t3y+a4z}

matrix [S]C reduce to a following expression,

b1= (_i)i _Yi+s z1+3 [S1jJC=

i 111

36V where,

V=i

6 xi 1/1 X2 1/2 X3 1/s X4 1/4 z' z2 z3 z3 acting on a body

i

1 1. s method to ship c1= (-1) --i Xi+1 X4+2 e examples are zi+1 z1,2 Zt+3

i

1

i

Xt+i rica Xj.f 3 yi+1 lIL+s

i=1,2,3, 4, and the other constants are defined by cyclic interchange of the _{subscripts in the order} 1,2,3,4.

When any surface (i, f k) of a tetrahedral finite element faces on the boundary surface of body,

the boundary matrix {A]C can be obtained as follows.

211

[A]t= P:I 1 2 1

(8)

p

Fig. A-1. A tetrahedral finite element

coMJsrrE ELE.ENT WiTH 5 NODES

--4

(2 TEl HEDL

/

ELEMENTS) 2 W)TH 6 NODES (3 E.EMENTS) WTH B NODES (5 ELEMENTS)

Fig. A-2. Composite finite element with 5,6, and 8 nodes

XL Yt

'

where, _c= _Xj _y

i

Xt y

i

Xj, _{Yi represents the coordinate of nodal point i, j, k of the plane perpendicular to the direction}

of

vibration.

2 Characteristics of Matrix [S]

The matrix [S] has the following important characteristics. _{This characteristics are very advan} tageous for practical computer programming.

The values of elements of matrix [SJ are not changed by

any clockwise (or anticlockwise) order of numbering when viewd from _{one apex.} translations to any direction,

and (3) rotations about any axis.

3 Basic Finite Elements used

The tetrahedral finite element gives very confusing division for practical case. Therefore composite

elements with 5, 6 and 8 nodes were used as shown in Fig. A-2. _{The average value of matrix [S]}

for two combination of tetrahedral elements was used to compose an element having 6 or 8 nodes.

(L

î,

Elastic-Plastic B

The recent developmei

approach, termed "the f. In the previous paper2

and the authors furtheri

most stiffened plates,

stiffened plate moves a' are induced in the jnst2 stiffening can be taken

As a basic example o

fur several kinds of sti

And compressive buckl

stiffener. The results the usefulness of the r