A study of vertical vibration of ships

'p

(ist report)

By Katsuo Ohtaka, * Member.

Fukuden Hibino,

Member.

Masatomo Oji, * Member.

Summary

Natural frequencies of the vertical vibration of bulk carriers and destroyers were calculated by

means of a digital computer, with a special attention to the shear rigidity, and the results were

compared with those of the measurements on their sea trials.

Both results showed good agreements in the lower mode, while in the higher mode there are discrepancies which are attributed to the estimation of the physical properties of ships. The ratio of calculated frequencies are nearly equal in the same type of ship and the empirical coefficients can be made for the similar ships.

Contents

Introduction

Method of computation Preparation of input data

Results

Conclusion

i

Introduction

Recently the digital computer technique has been developed and many complicated problems were solved numerically. In the field of the naval architecture, the David Taylor Model Basin has been engaged in the study of hull vibration using the digital computer2)3) and the comprehensive report

was published in this regard4).

As a shipbuilder we have to know the vibration characteristics at the design stage and, in the past several years, estimated the frequencies and modes of the hull vibration by means of a digital

corn-Table I Particulars of Ships

Ift

396)2O

* _{Engineer, Nagasaki Technical Institute Mitsubishi Heavy Industries.}

Ltd.

**

Engineer, Nagasaki Shipyard & Engine Works, Mitsubishi Heavy Industries Ltd.

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iÌi

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r

r

f

r QAT4 k .TlA

-- y---1

X

Fig. i _{( a )} Input Data for Calculation (Ship A)

A Study of Vertical Vibration of Ships

'o T .0CC ZA 'o AP. AP

Fig. i (b)

Input Data for Calculation (Ship B)

f.P. 91

2 Method of computation

Fundamental equations of vibration were made based on the Timoshenko-beam theory. The shear deflection and the rotational inertia were taken into account but the effect of a local vibration was

neglected. The fundamental equations of lateral vibration of a beam, together with the boundary

conditions, are shown in appendix 1. We considered the ship hull, although its cross section varying along the ship length, can be divided into many lumps whose cross sections are constant, as shown in Fig. 1. The lengths of the lumped beams are not necessarily equal but can be arbitrary, as shown in Fig. 1. Numerical method of computation are described in the appendix 1.

3

Preparation of input data

For the computation the distributions of the bending rigidity, shear rigidity, rotational inertia and the weight are necessary. Here we describe briefly how we calculated input data.

Table i Comparison of Frequencies

8v

J c4 : iû

Fig. 2 (a) Comparison of shear rigidity factors of a hollow triangular

section.

Bending rigidity (EI)

Moment of inertia of the section was calculated by the ordinary method employed in the strength calculation. All the effective longitudinal members up to the uppermost continuous deck were included.

Shear rigidity (K'GA)

Shear rigidity is one of the ambiguous factors in

the hull vibration. _{There are three commonly-used} method to calculate the shear rigidity, as shown in appendix 2. Prof. Watanabe' and prof. _Kumai") used the strain energy method, while McGoldrick2,

Jasper') and Leibowitz4) suggested the neutral axis method and they actually calculated the shear rig-idity by the web area method.

A C D

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MEAS-UREO ChtPI PC/TED

NEGLECTED S.4

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I NERTIA TAKEN NE ECTED 7 TAKEN NEGLECTED NEGLEcrro WEB 44 Mf rHODMErHOD WEB A4 ENEcr /01.3 MFrHOD7FTHOO 101 WEB AIEA 95 wEB4A MFTHOP EN(R&Y MFTHDD 92$ 7WEd 95 AtA MEOD EMIY MFTHDD '3.3 5 MrmoO 2- NODE

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A Study of Vertical Vibration of Ships 93

'.7

a'

0.0

-

80 ERERGY sia, (i')

-- BY WEB AREA 1HCV (W..)

45 1.0 ¡5 tO

Fig.2 (c) Comparison of the shear

rigidity factors of a mu-lticellular section.

Generally speaking, the shear rigidity factors obtained by these three methods are different. Comparison of the shear rigidity factors by these-methods is shown in Figs. 2(a), 2(b) and 2(c) for typical sections. From these figures we find the

foliwing facts.

i ) The shear rigidity factors K', K'5 and K' differ with each other, and the differences change with the shape and the scantlings of the section.

ii) In any case K' takes the greatest value. And the discrepancy between K' and K' is in some

case considerable.

In view of these facts we calculated the shear rigidity of ships by two method, i. e. by energy and web area method.

Weight

Hull weight was obtained by the method of trapezoidal distribution, as is used in the ordinary

strength calculation. By adding the weight of engine and the equipments, ship' s light weights were obtained. Weights of ballast, fuel and cargo were uniformly distributed in the tank or hold.

The weight corresponding to the added mass of surrounding water was estimated by Lewis-Landwe-ber's method7), which is illustrated in the articles by Townsin5>. Added mass for 2-noded vibration

was used for the higher mode vibrations. Rotational inertia of the section

lt is very troublesome to estimate the rotational inertia of the ship. As an alternative we made the following approximate estimation.

The rotational inertia was divided into two parts, i. e. that of hull and the others on board the ship.

The former was assumed to be uniformly distributed along the girth length of the section and the

latter distributed uniformly within the area of the cross section. We calculated the rotational inertia. for ship A and B. (see Fig.l).

In this report the calculations of natural frequencies are described for two destroyers and two bulk carriers. All the input data for these ships-are shown in Figs. 1(a).1(d).

cQ ¿5 1.0 /5 ?0 _8/o

Fig.2 (b) Comparison of the shear

rigidity factors of a hal-low rectangular section.

i, i, neglected

f, i, t,

B full load taken into account

't f, neglected i, f, f, C ballast i, f, i ft D t, i, D i, i, -. wrTHOUT TATIONAL IRTLA 2 .3 4 5

'

7

Fig. 3 () Comparison of computed

and measured frequencies of ship A.

NO OF NODES

ND D NOD(3

Z 3 4 5 4

Fig. 3 (c) Comparison of computed and measured frequencies of ship C.

i,

energy method web arer method

'F

energy method web area method energy method web area method energy method

NO. OF NOD(S

. 7

Fig.3 (b) Comparison of computed

and measured frequencies of ship B.

3 4 .3 ₇

Fig.3 (d) Comparison of computed

and measured frequencies of ship D.

2C4

A Study of Vertical Vibration of Ships 95 Computed frequencies are tabulated in Table II, together with the measured frequencies. For con-vnience the frequencies are plotted in Figs.3(a)-3(d) against number of nodes. From these figures we find the following facts.

The effect of the rotational inertia is small in the destroyers.

The calculated frequencies with the shear rigidity obtained by the energy method are in better agreement with the measured frequencies than the frequencies with the rigidity obtained by the web area method.

The agreements between calculated and measured frequencies are better in the case of

the destroyers than the bulk carriers, but, even DESTROYER

1n that case there can be seen discrepancies in IO

the higher modes. This tact seems to be attri-

-SHIP C J I/lJ( CAR1fR

buted to the estimation of the added virtual

-mass of water and the effect of local vibration, and should be studied in the future.

In the case of bulk carriers the discre-pancies are considerable in the higher modes.

At present we have few available observations '

and unable to draw any definite conclusion. Fig. 4 Ratio of measured/calculated frequency.

But it is supposed that the discrepancies are

partly attributed to the effect of the rotational inertia of the bridge house, because these two bulk

carriers have tall bridges at the stern. In the future report we discuss in this regard.

To examine the relation between measured and calculated frequencies the ratio of them are shown in Fig. 4. In this figure we find that the ratio is equal in the same type of ship.

5 Conclusions

Comparison of the frequencies obtained by calculations and measurements of destroyers and bulk carriers yields the following conclusions.

(a ) Effect of tne rotational inertia on the frequency is small in the case of destroyer.

For the calculation of vertical vibration the use of the shear rigidity obtained by the energy method (eq. A2. 1 in the appendix 2) gives better approximation.

In the higher modes there are still discrepancies between calculations and measurements. It

is supposed that this is partly attributed to the physical properties of ships such as rigidity and partly to the effect of the local vibration. In the future report the present ambiguities should be clarified.

At the end the authors should express hearty thanks to the staffs of the laboratory and design

section who gave them sincere cooperation in calculations and measurements. References

S.Timoshenko. "Strength of Materials" 2nd ed. D.Van Nostrand, 1941 Part I.

R. T. McGoldrick and V. L. Russo, "Hull Vibration Investigation on SS Gopher Mariner". Trans. S.N.A.M.E. 1955.

N. H. Jasper. "Structural Vibration Problems of Ships-A Study of the DD692 class of Destroyers". D. W. Taylor Model Basin Report, C-36, Feb. 1960.

1) R. C. Leibowitz and E. H. Kennerd, "Theory of Freely Vibrating Non uniform Beams, Including

Methods of Solution and Application to Ships". D. W. Taylor Model Basin Report, 1317, May.

1961.

5) R.L.Townsin, "Estimates of Virtual Mass for Ship Vibration Calculations". The Shipbuilder

and M. E. Builder, Mar. 1962.

Vol.37, 1929.

Y. Watanabe. "On the causes of the stern vibration". Journal of Soc. of Nay. Arch. West Japan. Vol. 105, Aug. 1954.

I. Suetsugu. "A contribution to the vibration at the stern of single screw vessels". Journal of Soc. of N.A. Japan, Vol. 105, Aug. 1954.

Y. Watanabe. 'SOn the effects of the shear deflection upon the flexural hull vibration. Journal of Kyushu Zosenkai. May. 1934.

T. Kumai. "Vibration of ships under special consideration to the shearing vibration". Journal of Soc. of N. A. Japan. Vol. 99, July, 1956.

Appendix i Fundamental Equation The following nomenclature is used in the paper

x : coordinate along the length of a ship

y vertical transverse deflection z slope duc to bending deflection

EI bending rigidity

k'AG shear rigidity

p weight per unit length Im : rotational inertia

¡ total length of a ship

length of i section

w frequency parameter of normal mode oscillation Bending Strain Energy

Vb=+f'

EI(_)2dx

(A 1.1)

Shearing Strain Energy

V5

=

--J

k'GA(-__ z)2dx (A1.2

Kinetic Energy of Translation

Kinetic Energy of Rotation

T5 J_fL

«P-Y)2dx

2 o g 8t T,.=1

_{2jo g}

at (A 1.3) (A 1.4) Total Strain Energy

21

V=1111E1( Oz 2

(-__z\1d

(A1.5)

2Jo

ax)+A\ax

J

Total Kinetic Energy

T=1f1-+- (

Oz

2jo lgYOt /

g

t. at )

jdx (A 1.6)

hence the Lagrangian function L=TV

can be writpen as follows.

--z' dx

(A 1.7)

i r55(

\

I, I Oz )2}dx _{1 fhIEJ( Oz )2+k'GA(} 2

LJ5\Ot)+_g_

2Jo1

Ox

I)

According to Hamilton's principal, _{J t}

f

taken between two fixed values of time,

j, and t,

is stationary for a dynamical trajectory.

'JI

g

V= _k'GA(_Z)

W=EIdz

dx

Hence the following set of equations is obtained

A Study of Vertical Vibration of Ships VT

(A 1.8>

equation.

Hence we lead two differential equations and a 3et of boundary condition at x=0, i

-

°_-8_k'GA(-

z)=o

g Ot dx

81z 8 dx

---8---EI0 _k'GA(-_ z)=O

EI-- = O

dx

at x=O,l

at x=O,l

\0x

/

JI (A 1.9) (A 1.10) To separate the two variables s and t, it is assumed that time variation of all dependent variable is sinusoidal.

Thus let

y= Yet,

x=zeic, etc. (A 1.11)

The Equations (A 1.8) to (A 1.11) then become a set of simultaneous ordinary differential equatiom in x, where the normal mode frequenoy w appears as an unknown parameter. A=w1

(A 1.12) EI

dZ+kPGA(dYz)...O

(A1.13) g dx dx \dX EI_d_=O _{at x=O,l}

k'GA(-_Z)=Oat x=O,l

J

The equation (A 1.11) to (A 1.13) are to be applied, in general, to the study of a non-uniforni

beam in which the physical characteristic change along the beam, will be convenient to difine the

following relations in a small distance of a beam.

d dY d2Y

dx

=EId2

(A 1.15)

-

' k'GA(

' _z)= _k'GA(_)

(A 1.16)

The formula (A 1.12) and (A 1.13) are transformed into

---Y+k'GA(._

Z)0

(A 1.17)

limA

Z+EI

+k'GA(4-_Z)=O

(A 1.18>

It will be convenient to define the following parameters

z

a g ¡Dv

b=

_g (A1.19) p =EI q =k'GA and let be (A 1.14) (A 1.20) (A 1.21)

(Z?AII?

MILS

:.44

CAL CAl. C end I3 7390 IBM 7040/7044 V' = Aa Y

V=W=0 at

x=0,1 Rewrite (A 1.22) to (A 1.23) into matrix form.

f,n I,n

-

I. PL A= o i

o

--j-o

o-i-o

o Ab o

..Aa O O O

lY

Iz

J V

X=1,I

(Ai.25)

Denote the ist and 2nd matrixes by X and

A respectively, eq. (A 1.19) is written as

X'=A.X

(Ai.26)

Let be H0

H0=I+--A

(A1.27)

ts_i

where I is eigen matrix.

XL+l=Ui.Xo (Ai.28)

X{Yo

o]

_{at x=0}

_(A1.29)

The unknown vector X can be determined from the continuity of U0 and boundary

con-dit ion s U1=H0_,.H2.H0_3. . .J.f0. U0 (A 1.30)

10

01

Uo=

00

To satisfy the boundary conditions at z=1 the value of the determinant must be zero. det U,1j =0

at x=0 (A 1.31)

(A 1.32) If we use middle class digital computor, it is easy to repeat these calculations dividing the physical elements at proper sections (in this report we used 50 sections) and varing frequency parameter

to satisy (A 1.32).

Appendix 2 Shear rigidity

Three commonly-used method in calculating the shear rigidity are as follows.

(a)

Strain energy method').

In this method the shear rigidity can be obtained by equating the work done by the external shearing force to the strain energy due to the shear stress in the beem'). Thus

f

IZAD

f

JS7EP PsIwr 'C 2,, , ¿, 4 (A 1.23) (A 1.24)

Af2dA

A Stude of Vertical Vibration of Ships 99

(A 2.1)

(A 2.2) Where V : shear force applied to a section.

A : area of the section.

shear stress in the section. Neutral axis methode).

In this method the shear rigidity factor is defined as the ratio of mean shear stress to the shear stress at the neutra' axis of the section. Thus

1.'

_n-

V

rNA ArNA

Where rNA shear stress at the neutral axis. Web area method").

In this method the effective shear K'A is approximated by A, where A is in the case of vertical vibration, the area of the vertical plating such as side shell and longitudinal bulkhead. Thus