Springing of ships - Considerations and computations for the development of a forecasting procedure

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HEINRICH SÖDING

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SPRJNGJNG OF SHIPS

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CONSIDERATIONS AND COMPUTATiONS FOR THE DE\'EI_OPMENT OF A FORECASTING PROCEDURE

Th

-ri

Hoe,chooI

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f i i-Ç7' H1 --r: NR, 7 ---17 SEP. 1982

SPRINGING OF SHIPS

Considerations and Computations for the Development of a Forecasting Procedure

1y H. Södirig

Publication of the Sonderforschungsbereich 98

The paper relies on contributions by G. Fietz, D. Hachnann, J. Isensee, K.-Y. Lee, G. Soproni arid the Germanischer Lloyd

Introduction and Summary

The term springing refers to bending vibrations of the main hull girder which are induced. by waves and which are not caused by slams.

In the following, only vertical vibrations, preferentially of the lowest mode, are treated; measurements have shown that these

vi-brations dominate the springing behaviour.

Springing is governed by the following kinds of forces (and moments):

.1. Restoring forces associated 'with elastic deformations. For the lower vertical modes, ìt is completely sufficient to treat thes2

forces by elementary beam theory including shear deformations and

a certain amount of shear lag, a is usual in hull vibration cal-culations. Hydrostatic forces should be included, too.

Mass forces caused 'by acceleration of real and added masses. Ad-ded masses may be calculated in the well-known way for

two-dimen-sional flow at the limit of high frequency and muplied by a

reduction factor for three-dimensional flow.

Da.mping forces. It seems that all theoretical investigations of

springing damping mechanisms have yielded values which are

con-siderably smaller than measured damping constants. In this paper,

the following kinds of dainping are considered:

a) Damping forces associated with wave generation. It has 'been

argued by some authors (f. i., i) that these forces are in-significant; calculations according-to a method similar to

(2) indicate, however, that this is true only if the ship sections are vertical at the waterline, whereas sections

in-- dined at the waterline turn out to have considerable damping

coefficients at springing frequencies. Calculations based on

normal or higher-order Lewis sections cannot show this

be-haviour as these sections have vertical sides at the waterline.

b) Damping forces from an assumed flow separation at the bilge

keels during vertical movements of the ship sections. These

forces which, presumably, are similar in nature to the roll

damping moments caused by the bilge keels may contribute substantially to the vibration damping.

-2-e) Flow separation at a transom stern, if the transom is sub-merged, at least for a certain percentage of tirne,by the ac-tion of waves. In many cases, the vortices generated by the oscillating transom may dissipate the mayor part of the

vi-bration energy.

Damping forces from vertical components of turbulent shear stresses caused by the combination of forward speed and

vibra-tion velocity These forces are shown to be insignificant.

Damping forces from roll damping fins. If the steering of the

fins i independent of springing motion or ancimetric on both ship sides, the springing damping effect of the fins is rela-tively small compared to thatof bilge keels.

Damping forces from a, band e together result in damping con-8tants of the same size range as that known from measurements.

A detailed comparison of measured and calculated damping constants, however, has not yet been made. It is supposed that non-hydro-dynamic damping forces (material and cargo damping) are

insigni-ficant in most cases.

4.

Exciting forces by waves. These forces may be developed into a Taylor series over the surface elevation to separate linear,

qua-dratic, cubic and so on forces. Here, only two components are

considered:

Linear forces. As their wave-length is in the range of io% of ship length only, these forces cannot adequately be handled

by strip theory. In this paper they are calculated using the Newman "reverse flow" theorem (3).

Quadratic forces resulting from the variation of the wetted

contour length of sections as explained in (4). These forces

are estìmated only by assuming a quasi-hydrostatic pressure

distribution near the water surface in the waves assumed to

be undisturbed by the presence of the hull. These results

should 'be considered only as an indication that quadratic springing exciting forces may be in the same size range as

* A recent paper by Grim shows that hydrodynamic springing exciting forces outweight these quasistatic forces by a factor of at least 10.

3

linear forces. The statistical method used for treating these

forces in a natural seaway, however, may be applied also in

cases of more elaborately calculated quadratic and higher-order forces.

Rough niinerical calculations have been performed for the

third-generation containership Tokyo Express on which extensive

mea-surements have been made. However, the records analysed until now show only whipping (slam-induced) and no essential springing vibrations. The calculations, too, result in very small springing

deflections of the hull in the millimeter range. So, it remains to be c-larified if a serious springing prediction method may he

developed along the lines indicated in this paper.

Relation between Forces and Motions

Due to the low damping of springing vibrations the modal approach is very suitable. This well-known method ist described here for

con-venience only for the case of springing motion.

The ship is assumed to be excited to vertical vibrations by regular

waves ol' amplitude h and frequency of encounter The ship sec-tions are assumed to be rigid. The vertical displacement Z(x,t) (positive downward) of the ship section at the longitudinal coor-dinate x (positive forward) at time t is described as a

superpo-sition of eigenforms

(x)Re(a

)et).

()

k

ak is the complex transfer function for the k-th mode. The real eigenforms'r(x) (mode shapes) describe the vibration profiles of the free undamped. vibrations of the different modes, including

heave and pitch motions.

The elastical properties of the hull girder and the buoyancy

for-ces are described by the influence function

(x,) designating the

vertical displacement of the section at location x under the

sta-tic action of a unit force at location and the accompanying hy-drostatic forces due to sinkage and trim caused by the unit force.

If the actual forces (excluding hydrostatic forces) on the hull per unit length are Q(,t) (positive upward), the vertical displacement

is

4-Z:(1:)= J)c()d

(2)

L

where 5 means integration over the whole ship length. The forces

Q(,t) are splitted into three components:

The forces m()-Z(,t) proportional to the vertical acceleration

of the vibrating section.

The forces d().Z(,t) proportional to the vertical velocity

of the section.

i)t

-The forces Re(f()he

- ) exertea by the wave on the ship

being assumed not to vibrate, f is the complex transfer function of wave forces per unit length.

Combining these definitions with (i), one obtains:

cL())

f()Je

L

(3)

V

_J.

Inserting this ints (2) and considering (i) results in

f(](x,)

d

a(x)tRe

{f

hett

t

This relation holds for all t values

Oflly

if

r j

As this relation is applicable for arbitrary exciting forces f, damping forces à and frequencies cJe, it may by applied also for the case f=O, d=O and the eigenfrequency of the k-th mode.

Under these conditions, the k-th vibratìàn mode will result:

ak4k(x) [k/s)

from which follows:

(c)

'This relation will be used to eliminate the influence function from

equation ₄ in the case of forced vibrations, To that purpose,

₍₄₎

SS

C')

It may be shown that the eigenforms'(x) are orthogonal to each other:

S O

C)

Equation

(6)

may now be simplified by changing the order of

inte-grtion, employing

(5)

and

₍₇₎

and the Naxwell relation Then, one obtains:

a £1

J4i()

Jf().

-one obtains from (8):

)QMj

From this equation follows the complex transfer function

5

(8)

-As the damping of the bending vibrations of ships is very small, substantial bending amplitudes ak result only if the frequency of

encounter differs only slightly from the eigenfrequency

So, for each frequency of encounter LJat most one mode k is exci-ted substantially. The sum over k in (8) may therefore be replaced by one term j if only the interesting regions of Ce near -. are

considered. Then, with the shorthands

$(x)()c,

(9)

Fig. i

mean velocity of the ship. If is the potential of the steady and

the periodical flow around the hull, the fluid pressure is, accor-ding to the Bernoulli equation,

2

f I

-+

J-s

is composed of a steady component due to the forward speed y

of

the ship and a periodical component due tó the vibration:

-s_ P

e

_(4g)

By inserting (15) into (14) and omitting steady terms, terms due to

the variation cf the steady potentialat the vibrating surface,

nonlinear oscillating terms and the periodical hydrostatic term gZ due to thefribration of the hull (as this term has been exclu-ded from Q(x)), one obtains an expression for the linearized hydro-dynamic periodical pressure on the hull:

4s

F

+

t

J

where D/Dt indicats time differentiation for a point moving

accor-ding to the steady flow potential 45:áiong the hull surface. As the

6

It is stated frequently that this and similar equations hold only if the distribution of damping forces d(x) over the ship length is

proportional to the mass distribution m(x). Actual damping force distribuions strongly contradict to this assumption. As is shown

above, however, this assumption is not required in case of low damping and exciting frequencies nea a resonance frequency (J

Forces due to the Vibration in Potential Flow

The hyclrodynamic potential flow forces due to the section velocity and acceleration will be determined by means of a strip method

ana-logous to that developed in (5). The coordinate system is directed as shown in fig. 1. It moves steadily with thé

e

x-component of the steady velocity is approximately -y, one may

write:

7

where ' indicates differentiation to x along the streamlines of

the steady flow on the ship surface. Therefore,

p

-

7iX4

It is assumed that 4 may be expressed in the form

RUe

H ere, is the complex amplitude of the potential function of the two-dimensional periodical flow around the ship section oscillating

with velocity amplitude i and frequency i . U is the complex am-plitude of the relative vertical velocity between the steady flow

and the vibrating ection (positive, if the water moves downward):

=

(

which with (i) gives

-From (.8) and i) follows

c(kx

A(x))e

Inserting that into (17) results in

(49)

Integration along the section contour gives the periodical hydro-. dynamic potential force Q. per unit length,

If

p R

with

where, by definition, m1 is the added mass per length and N is the

damping constant per lengthof the respectivè ship section.

Inser-

A-=

tcfv

j, j.

Ct'for

(L)

('1 (20)

(24)

ting

(24)

into

(20)

gives the hydrodynamic potential force in terms of sectional added mass and damping constants:

Q

_[

k

r I

± (z2)

If the section area approaches zero at both ends of the ship,

there will be no concentrated forces at the ship ends. If, however, a transom stern is submerged, the values m and N' approach

infi-nit at the transom, which corresponds to a concentrated force at that location. Just as argued for the case of sway and yaw motions by Grim and Schenzle (6), it is assumed here that these concentra-ted forces do not really occur; the is assumed, instead, a

cir-culation as in case of an aerofoil, which causes a smooth flow over the transom and eliminates the concentrated forces. To take account

of this, integrations of Q(x) over ship length to yield total for-ces and moments will be extended only until ir'mediately before the transom. At the stem, no flow separation is expected; therefore,

integrations are extended at the forwa'd end of the ship to the point

- where the section area vanishes.

Other Forces due to the Vibration

As the damping forces contained in Q. will be shown to be very

small at least if no transom is immersed, it is necessary to inve-stigate also other possibly more effective damping forces.

'It may be assumed that the relative vertical velocity between shio section and surrounding water (positive, if the water moves down-ward) causes vertical resistive forces per unit -length of magnitude

- (23)

where b(x) is the breadth of the section and e a resistance coef-ficient supposed to be of the order of magnitude i for sections with effective bilge keels causing flow separation during upwcrd

move-ment of the section. Because of the nonlinear dependence on u, it is not possible to treat the _{motion of the ship in a natural}

sea-way as a superposition of regular oscillations of different

-9-Therefore, the relative vertical velocity u will be considered to be composed of a component u1 caused, in the first line, by the low-frequency motions of the ship and the waves, and a much

smal-ler component u2 due to the regular wave under investigation with a frequency of encounter near +he eigenfrequency of the respective bending mode:

Because /» ju2(1 we.may write:

(2g)

For vibration damping, only the second term oscillating with the

frequency C is effective. As

«

R (')

where U is given by equation (19), we obtain:

The component of the force caused by the real part of the amplitude of u2, being in phase with. the mass forces in which

are much greater, will be omitted. Then

,lhz

and with (23) and (24) one obtains

c()

In most cases it will be sufficient to use the temporal mean value of 1u1(x,t)t which may be calculated by usual strip theory for rigid-body heaving and pitching and by applying the short-term statistics of the motions in a natural seaway. The resistance co-efficient O for sections with bilge keels could perhaps be

deter-mined by model experiments or by theoretical calculations similar

- to those of Blendermann (7).The resistance of sections without

bilge keels or knuckles is supposed to be insignificant.

The speed of the ship causes a turbulent shear stress _{at the ship}

sJface)

tz SVCF

i

where the order of magnitude of CF

IS

The mean direction of is longitudinally backward. By the vertical velocity of the vibra-tIng ship sections, however, the direction of the relative velocity

lo

--R4

₃

/

where si is the ship mass per unit length:

(31e)

between water and ship is changed by an angle being about u2/v. This causes a corresponding change of the mean direction of c in the respective section of equal order of magnitude and, therefore, a vertical periodical component of the shear stress,

t

yvtCF

(29)

and a vertical force per unit length of about

Qi

(3 o)

T being the draft of the ship. The ratto between

BK and Q,

there-fore, is

(3î)

BK

1V)

which, apparently, is a small quantIty. So, Q may be disregarded.

If the ship employs two roll stabilizing fins each of area A, the periodical vertcal velocity u2 of the ship sections causes a peri-odical change of the angle of attack by u2/v, Therefore, periperi-odical

vertical forces

2:

(32)

are excited. The ratio between this force and the effects of bilge

keels of length 1 is

33)

-Q<

íLC

which, too, is much smaller than i in most cases. However, effective vibration damping could be accomplished if the fins would be steered

depending on the midship vertical acceleration. This is suspected to be possible without essential degradation of the roll damping effect; however, it would require a much greater turning rate of the fins.

Determination of Effective Mass and Dancing Constants

The total force Q per unit length due to the ship vibrations is the

sum of Q(equation 22),

Rct

i

L

ií

N'2t

A

lkL (

Comparison with equation gives the quantities in and d introduced

at the beginning: -

/144"i0-f,j4i

_t\J

-4--:-;---

& C k

'k

_________ i

By partial integration of the underlined terms one obtains

2.

2)No

_cb

_{T_NT}

_T'iT

Linear Exciting Forces

In case of wave exciting forces as well as ship-motion-induced

(3G))

forces, potential-flow forces dominate over other forces. As, how-ever, for motion-induced forces the phase angle between motion and

force is very important, and as potential-flow forces are nearly

in phase with deflections, non-potential-flow forces being nearly

90 degrees phase-shifted against deflections have to be considered. For exciting forces where phase angles are unimportant, non-potential

- N. (37-)

As before, the sums over k may be substituted by one term J only in case of springing.

The mass and damping constants M. and D. follow from equations (o

and (10):

r1 i i "- 2. I j

vt'-

i -y- i;

ir

M-I

_JL

0j ¡AAQ

_LL

-1

which, by partial integration of theunderlined. terms,yields

(f

2. _'t'-

i

_{i -} I

2-t

()

the index T indicates the values at the transom stern°'(zero, if no transom is immersed).

12

-forces need not to be considered.

Another difference refers to the variation of forces with X:

Wheroas forces associated with the lowest bending mode have only one hump over the ship length, waves whose frequency of encounter is high encìugh to excite such vibrations have about 10 humps and 10

hollows over the ship length. That necessitates not only special care during the numerical integration of the forces over the ship length; it contradicts also to the fundamental assumption of the strip

method that the potential flow of the wave diffraction by the ship

is -confined essentially to the transverse sections of the ship. .Therefore, the "reverse flow" theorem of Newman

(3)

is used to

re-duce the problem to that of determining the radiation potential from ship bending vibrations, which, in turn, is approximated by the

strip method. Some uncertainties will remain, however, for ships with an immersed transom stern.

According to the Newman theorem, the exciting quantity defined by equation 11,

'

L')

may be calculated from the formula

'S

-(40)

The quantities occurring in this formula have the following meaning:

S designates the mean wetted surface of the ship.

(f,is

the complex amplitude of the potential of the incomming wave:

- -

(f0-

w-e

(L)

with = circular frequency of the wave,

Z k = wave number = -3

s

= acceleration of gravity, h = wave amplitude, k(y.sin _-

_x.cos»)

u.= angle of encounter between ship and wave.

is the complex amplitude of a potential which satisfies the Laplace

condition and the following boundary conditions:

a)

(L)

on the ship surface with

= unit normal vector into the body,

(112)

'1 y. = velocity amplitude of the bending motion of the ship

with profile 4'

Re(et)

displacement vector of the surface due to the saine bending motion,

steady velocity field for an assumed backward motion of

the ship with speed y.

b)

on the mean free surface z = O outside of the body.

e) The usual radiation condition, that is, outgoing waves at great

distances from the ship at the water surface.

d) The bottom condition ['i

T-J

The condition 4 on the body surface will be investigated more closely: For vertical bending vibrations, the displacement vector& is directed

-

13

-Ex+?4s,

gI

For slender ships, the square root is

L (v%_. -, I

Inserting the expressions

₍₄₇₎

until (51) into

(44)

gives:

(so)

vertically; that is,

( ?)

related to the velocity amplitudeftl-:

(1t Y)

The ship surface is assumed to be described by the functions y(x,$)

and z(x,$), where s istthe contour length of the ship section at the longitudinal coordinate x, measured from the waterline on port side

to the point (y(x,$),z(x,$)), divided by the whole contour length

A

of the wetted part of the section. The steady velocity field V on

the ship surface is approximated by the vector whose x-component is

equal to the ship speed and which is tangential to the lines s =

constant on the body surface:

(

y

)(

),

(Ltg)

where the index x designates partial derivation. The inward normal may be expressed by the functions y(x,$) and z(x,$):

a

'1COS-f

:Z_'ïCQ ¡3

=

-

k

(r1.b)']

-

14

--

_(oo)+V

(OO4)»(

v?1+s,rc)/A

J

--'t

z

j1-) /A.

A ,t L)

If B is the angle between the section contour and the horizontal

axis y (positive starboard), there follows

I)

COp cucL

-ft1.

J 'J (

.cix

tC A tL

H-

Acss

(tAco)' Jco3.

C)

As the second term is relatively small, it will be approximated by using a mean value for the whole section. is defined to

coin-cide with the correct value for straight V-shaped sections:

Acos

b waterline breadth of the section. Then, one obtains:

The condition is satisfied by the potential

where is the complex amplitude of the two-dimensional heaving

po-tential for the respective section defined on page

_7.

This potential satisfies also the bottom and radiation conditions. The free surface

condition (46), however, is satisfied only approximately: numerical

estimations indicate that the second and third terms, which vanish for the limit of slender ships, are sufficiently small under the

pre-(sz)

(s3)

(cs)

C

(sv)

- co

r'

=-

ff1.-L) _L

-

15

-vailing conditions to be omitted. The remaining terms are satisfied by the function of equation 57 if the two-dimensional potential

is calculated with a free surface condition for frequency c That

is in contrastto strip method exciting force potentials, which are composed of two-dimensiona) potentials calculated for the wave

fre-quency c>, not

Insertion of (41),

₍₅₆₎

and

₍₅₇₎

into

ti ty

ss

H

bI?

¿2ií,

_bUj

_b)'1

r

J L

where T is the draft of the section at x and

C)

For ships with a submerged transom stern this result has to be cor-rected: In the derivation of the Newnian theorem there is assumed that

-4

(V)O,

(o)

where i ist the unit vector parallel to the intersection of the mean free surface and the ship. This relation holds if this intersection is a streamline of the steady velocity field. In the case of an im-mersed transom stern of breadth bT, however,

(40)

gives the exciting

quan-J cos

)7

(oo4)x(,0(

).(cs

/ L

,

j-.

Jr

The derivation

of

the theorem shows that in this case, the correction

.4

corr

) c

_{_yt 1(OT}

) b

(2)

has to be added to F. of formula 58. As -is the diffraction

po-tential amplitude for forward motion of the ship at the transom,

+

ts)Lb

this quantity cannot be calculated without solving explicitely the

diffraction problem.

To obtain an idea of the magnitude of this correction, we write

(c3)

with an unknown complex function a. Then the correction is

Corr

--ai

T'°

br

-

16

-5I1TT

_TF

e

Rough estimates of a are:

-O for head waves;

i for vaves withp&. between O and TV/2;

and a complex factor derived for an infinitely long cylinder of the

transom cross section forp values between 11/2 and rr. This diffrac-tion problem has been solved by Grim (8) and Urseli.

Practical Computation of Linear Snringing Motions

To investigate the magnitudes of the different terms, a computer

program for the calculation of linear springing motions has been

written. It recevesas input, besides other quantities, for each

of a number of equidistant sections the values m, N, the corresponding quantities for the calculation of exciting forces

appearing in equation

58,

b, 'bt _{,vj' c} and a temporal mean of

h-'11

Frequency-dependent terms, as,

e.

g., added masses an4damping

constants, are given for three different frequencies; the program interpolates actual values for a number of frequencies. M. and D. are integrated by Simpson's rule from equations 38 and

_39.

Prom these quantities follows the ratio r of successive amplitudes in

the case of free damped vibrations:

.,

D5--t-=

The exciting term F. is calculated from equation

58,

including the correction of equation ₆₄ with a given value of a. As F. is the in-tegral of a product of the rapidly oscillating function e1 with a more smooth function. the usual numerical integration rules would

number of

17

-special numerical integration rule for this case was developed: It may be shown that, if f(x) is approximated by a polynomial of second

degree between x1 and x1+2h, the following relation holds for

arbi-trary non-zero values of k:

JfkK

QtX

_{O.S_2f+4.f3}

4.

_M

2-f.-1-O.f3

7

_(Gb)

hk

j1

where f1 f(x1), f2 = f(x1+h), f3 = f(x1+2h) and

L2f

= f3-2f2+f1. This rule is applied for each double-spacing of sections to calculate

F..

J

From F., N. and D., ;thècomplex transfer function a. of the

sprin-J J J j

ging vibration follows (equation 13).

In a natural seaway of the spectrum of wave encounter S(c), the

variance of the vertical displacement at location x is

As 11a.j2 has significant values only in a small frequency region

near , where SB, F., M. and D. may be approximated by their values

at the resonance frequency .y , G may be exrressed as

t

With equation 13 follows

___

- .

a (c,)-' )

which after performing the integration gives

I

r

Quadratic Exciting Forces

Fig. S shows that the magnitude of the linear exciting force decreases strongly with increasing frequency of encounter. The

u-sual spectra of wave encounter decrease with L, too, in the

in-teresting frequency region. As in case of nonlinear wave excitation

longer waves of lower frequency of encounter can excite the ship

with the resonance frequency

ct-,

it seems necessary to investigate the order of magnitude of the nonlinear excitation. To that purpose, a highly simplified caseis examined whose importance was shown

recently by Grim and Schenzle (4): the second-order excitation due to the varying wetted length of the sections (see fig. 2). This variation of the wetted part of the section causes - on one ship

side under the assumption of hydrostatic pressure distribution,-a horizontdistribution,-al force per unit length of mdistribution,-agnitude

4

p2

Î3

t,uav'

* Compare footnote on page 2.

CrUE

18

-ç

vJ

,Z

frch

2

This force is directed both at the wave crest and in the wave trough to the centre-line of the ship. If the section is inclined against

the, vertical coordinate in the mean waterline by the angle

,

the

total force of this pressure triangle has a direction normal to the

section contour. As the horizontal component remains unchanged, there

exists a vertical component, directed upward, of magnitude

'1 2

=

In the following, the case of head waves is treated only, and the

pitch and heave motions are disregarded. (The effect of ship motions could easily be included if the linear ship motions were calculated.)

The wave elevation in a natural seaway may be expressed as

-I

I2()

R(eL)

o)

e

fJ4i

where Seis the spectrum of wave encounter, are frequencies of encounter, km are the wave numbers associated to the frequencies

of encounter L , and are random phase angles distributed

-independently from each other with equal density in the range from

.0 to

2r.

The quantity 2 occurring in (69) follows from (70):

I

/

_j'

Li)

As

R)- RL)

for arbitrary complex figures z

conjugate, one obtains:

Ç)

E

I1

(ktIzr)

.P

J

The complex amplitude of the component of E which oscillates with

the frequency

Cis-L 1-2)

z2, where * designates the coaplex

(?Lj)

Por the calculation of the vibrations caused by q the exciting quantity

has to be calculated. The factor 2 is necessary as q is the force on one ship side only. From

₍₆₉₎

and.

(73)

follows:

e

20

-)]. L

xp{x(Li-!Q-e

where and are the phase angles and wave numbers, respec-tively, of the wave with frequency of encounter

C.)1=

whereas

41and

belong to the frequency of encounter

A

Sis

40

for positive frequencies only, the first

sum refers to C.i4 -values and the second to C,-values

If k is defined as

k

Fcr<

_(-)

fc

r

and if Eri is defined correspondingly, one obtains:

(Rc)

where the summation has to be performed over all positive values of

The vibration amplitude a is related to F. by equation 13:

J( (

Lj )M3

(g)

(The definition of a and F. is changed here from that of pages ₃

i J

and 5 by multiplying both values with h.) If the absolute value of

a is interpreted as being that cor'esponding to a spectrum S(2 ) the vibration displacement Z(x,t)/() , one obtains from

(13)

and (76):

J (- )M-A

i i

)L

_xLAi2

c)]'

L

From this equation, the spectrum S follows:

a s

/(M)

2

i ±Lb/)

j

1

with _'s1

(Lk)

L-('13)

C7

21

-and 'being the phase angle of the complex integral over L. The second factor of S is

V a

-i))

.(>

I

(

=

r

l'vi-i /t-%1

-

_{t 2}

T

, 4

-¿V "i /VV

f)

ll'\ .vl (141 ,t41

Without probf it is assumed that the cond term in

(so)

which may 'be positive or negative may be omitted for calculating the variance of the vibration, as during the subsequent integration over L)the different terms cancel each other in a statistical sense. Then,

2.

Jr

) ¿

-

) ) e- j 2.2

--

t:XV;

-lDt i(M- i

. t

(io)

/ L' ¡j

I

i

(:j)

L '-'

L1

(°zS )+DiM

o

Further, it is assumed that for a response spectrum confined essentially to a small frequency region compared with the 'breadth of the exciting wave spectrum, the phase angles of the complex amplitudes a.(e) may be taken to be uncorrelated inspite of the

fact that they depend on the same set of values for all

frequen-cies These assumptions will have to 'be verified by a closer examination. Under these assumptions, the variance c251 of vibration

deflections Z(x,t)/1() is obtained by integrating the vibration

spectrum:

G

5ca-As Sa is a narrow-band spectrum around the resonance frequency (J '

it seems possible to use not the exact values of M., D. and the

in-3 3

tegral over in

(ai)

at the frequency of encounter but to use the values for the frequency L. Then,

from (78),

₍₇₉₎

and (80) follows

S()

/(2N) A

r 22

-t(k

z

GJ

Mt) L o ¿)) + (c D' / '-l)

The integration over (Jem be performed explicitely:

2Z

)

(8i)

fs

J e. '

)S(i)/í1

e

4M- Ûi

. ) )

Numerical calculations have shown that the part of the integral over

from &7 to gives a small contribution only. Because of the

symmetry of the integrand to u/2 the formula may then be

simpli-) 1.

fied for numerical integration as follows:

¿(k-tk)v.

Z

c

_e

_t'

)5(ci))

e L

Numerical Evaluations

To show the order of magnitude of the different terms, rmrerical calculations have been performed for the coiitaner vessel Tokyo

Exress (Loa = 287m; B

= 32,3

in) for drafts of 10.9 in aft and 10.5

in forward at 14 rn/s speed. 11 stations distributed eaually over

a length of 285 in were used. The hydrodynamic coefficients of the sections in two-dimensional flow have been calculated by

Dipl.-Ing. K.-Y. Lee by means of a program which is an extension

of that described in (2). The ftöw for each section was

aporoxi-mated by a superposition of typically 5 periodical sources and about

50 quadrupoles. Fig. 3 shows some of the geometric and hydrodyna-mic quantities, plotted over ship length. The variation of damping constants N similar to the variation of the angle of inclination

of the sections at the waterline is evident. Some of the

calcula-tions have been made for the Ship with an immersed transom stern

by assuming a waterline which was elevated 3.1 m above the actual

mean waterline at the last station, whereas at all other stations

only the mean waterline was used. For realistic calculations, it would at least be necessary to determine first of all the

low-fre-quency ship motions and then to estimate mean values of the hydro-dynamic quantities with the aid of these results.

i:

ì3)

t

30 , (i) 40 300 200 400 N

w

.10

t /Th,

zs

23 V f û

/

4'.14 fc'r

'N

c&c1tc4.

t)=

2.-t/-Ftc.

'3

'cicrt

brct L1

-z g

.F St(c

'Io g 9 '10 5

,Sctov

'Jr. rc o.

4.0

O,95

o. S

24

-Fig. 4 shows, as a measure of the vibration damping, the ratio r of successive amplitudes of free vibrations determined from equation

65 for different assumed eigenfrecuericies. It shows t10-t for rela-tively low eigenfrequencies the wave damping dominates, whereas for higher frequencies the damping by bilge keels and by an

immer-sed transom stern may be important. The damping of the transom is

approximately proportional to the square of it's breadth. - The ac-tual resonance frequency of the ship was

5.0

rad/s; the damping ratio r as determined from measurements made 'byDipl.-Ing. G. So-proni and Ing. grad. G. Fietz was typically

0.94.

The eigenform of the lowest bending mode for the loading condition

during the measurements was determined by the Germanischer Lloyd with the aid of mass and stiffness data prepared by Dipl.-Ing. J.

Isensee and Dipi.-Ing. D Hachmann (see Fig.

_3).

r-=

ra10

of

f

vLbr-o

k c___

I

.'

i L ..j I t 2.0. .

Fig. 4

q. o s s-o

I

t

L4oF

1F1/h

<N

2-4 4Q4

Fig. 5

shows the exciting quantity IF.( according to equation 58;

one curve contains also the correction of equation ₆₄ with values a decreasing linearly from 0.4 to 0.2 in the frequency range between

2.5

and _{5 rad/s. It is remarkable that the small modification of}

the waterline near the transom has such a profound effect on the

exciting quantity F.. That shows that variations of the immersed part of the ship due to low-frequency waves and associated motions will have to be considered for realistic predictions.

3.0

-Fig. 6 shows the absolute value of the bending deflection response function with it's sharp peak at resonance frequencyaS an indi-cation that the various simplifiindi-cations made in view of the narrow

frequency band in which the response has significant values are

justified. 13e

Dec1OA

cL r'JcV AL-IpUfucte

26

-its

E

ss

0,40

O.o5

Fig. 7 shows the ratio of the root mean square value of the bending deflection, divided

_bY\ie(L\37

for diffeent assumed resonance fre-quencies . The ship with immersed transom shows greater

vibra-tions inspite of it's higher damping constant because of an even greater increase of the exciting quantity F.. For comparison: The

application of an. _{ISSC- spectrum with a significant}

wave height of 6m and period of 11 s results - after trans-formation to the spectrum of wave encounter for head waves - in

\J Se C(C// _{= 0.082}

in.S'.

The oscillations of all the quanities shown with frequency and

the corresponding oscillations with ship speed are not assumed to have any practical relevance as, due to low-frequency waves and mo-tions, the positions of humps and hollows will change with time.

R ccïL Mc

Str

\Jcku

of ßci

V-cJ cL

27

-'iCI-i

ra:

-Q. 40

28

-ç

t

_/

/

/

I

L1. _C.)

{rct/.3

Fig. 8 was prepared to show the nature of the integration kernel

over C. in equation 82. The seaway spectrum S corresponding to the spectrum of encounter S6 was an ISSC-spectrum with H _6m and T = lis. The figure shows that the part of the integral frani

Wto o contributes

7% in case of the ship with immersed transom and less than 1%cUieyws.It shows further that the whole frequency region between the low-frequency limit of the wave spectrum (in this case at _{OcL/' ) andCj contributes substantially to the} integral; the main contribution, however, stems from those

combi-nations of waves where one component is near the range of maximal spectral energy (wave length 150m) whereas the other wave is short

(about 35m wave length); The suiii of the wave numbers remains nearly constant over the whole range of frequencies, corresponding to a

"wave length of superposition" 21F/(k+k) varying only between 26 and 30 m. o.3 0. 8

o. ;

Q, Lf

0.2

I o_2 o

2

3

References

F. F. van Gunsteren: Some Further Calculatins of

Wave-in-duces Ship Hull Vibrations. Intern. Symp. on the Dynamics of

Marine Vehicles and Structures in waves,

_{p. 331}

(1974)

2

H. Söding, The Flow Around Ship Sections in Waves.

Schiffs--technik 20,

9 - i5(197)

3

J. N. Newman, The Exciting Forces on a Moving Body in Waves.

Journal of Ship Research 9,

190-z-199 (1965)

4

0. Grim and P

Schenzle, A Second-Order Effect on Wave Bending

Moments. Contribution to the 14. ITTC-Conference.

5

H. Söding, Eine Modifikation der Streiferirnethode.

Schiffstech-nik 16, 15-18 (1969)

6

0. Grim und P. Schenzle, Der Einfluß der Fahrgeschwindigkeit auf

die Torsionsbelastung eines Schiffes im Seegang. Report 7/1969

of the Forschungszentrum des Deutschen Schiffbaus (1968)

7

W. Blendermann, Der Spiralwirbel arm translatorisch bewegten

Kreisbogenprofil, Struktur, Bewegung und Reaktion.

Schiffs-technik 16, 3-14 (1969)

8

_{0. Grim and U. Kirsch, Forces on}

_{a Two-Dimensional Body Excited}

by an Oblique Wave. Jubilee Memorial W. P. A. van Lammeren.

Wageningen 1970

- 29

The seaway spectrum defined above causes the following root mean

sqare deflections d (see fig. 6 for definition):

Without transom

immersion

With transom

immersion

Linear Wave

Excitation

Quadratic Wave

Excitation

1.3 mm

0.5 mm

2.5 mm

2.5 mm

LEHRSTUHL UND INSTITUT FUR ENTWERFEN VON SCHIFFEN UND SCHIFFSTHEORIE

TECHNISCHE UNIVERSIThT HANNOVER

CALLINSTRASSE 15 / 3

HANNOVER / DEUTSCHLAND / TELEFON 0511 762 2442

Bisherige ESS Berichte

Nr. Titel Seitenzahl

i

Gebrauchsanleitung für die Programme

CHWA-RISMI und DECIS.

Zur Umwandlung von Gleichungssystemen,

Opti-rnierungsaufgaben und Entscheidungstaheilen

in Fortranprogramme.

H. Söding, Dezember 1974

₁₇

2

_{Long-term and Short-term Stability Criteria}

in a Random Seaway.

For Presentation at the International

Confe-rence "On Stability of Ships and Ocean

Vehic-lest? Glasgow 25 - 27 March 1975.

S. Kastner, Dezember 1974

₂₇

Das Procjramrn ARCHIMEDES 74.

Zur Durchführung von hydrostatischen Berech-nungen.

H. Söding u. I. Poulsen, Dezember 1974

₉₉

4

_{Ein Ansatz zur Festlegung zulässiger}

Spannun-gen bei mechanisch beanspruchten

Konstruktio-nen.

H. Söding, Dezember 1974

₈

5

_{On the Statistical Precision of Determining}

the Probability of Capsizing in Random

_Seas.

For Presentation at the International

Confe-rence "On Stability of Ships and Ocean

Vehic-les" Glasgow 25 - 27 March 1975.

i

S. Kastner, Januar 1975

₃₃

6

_{Das Programm TIDES zur Berechnung der}

Wasser-bewegung beim Stapellauf

von Schiffen.

H. Söding, Januar 1975

₆

7

Springing of Ships.

Considerations and Computations for the

Deve-lopment of a Forecasting Procedure.

H. Söding, März 1975

₂₉

Zum Entwurf von Schiffen.

i. Gebräuchliche Begriffe und Ma1e.

2. Vergleichende Betrachtung von zwei

Stück-guts chiffen.

Nr.

_Seitenzahl

9 _{Rechnergesttitzter Schiffsentwurf.}

Vorlesungsmanuskript.

H. Södinq, August 1975

₁₀₈

10

_{Fortran-Programmierkurs.}

Zur Vorlesung Rechnergestützter

Schi ffsentwurf.

H. Söding, August 1975

₆₈

li

_{Schwimmfähigkeit und Stabilität}

von Schiffen.

Vorlesungsmanuskript.

H. Söding, August 1975

12 _{TheTwo-Dimensional PotentiaJi Flow}

Excited by

a Body Oscillating at a Free Surface.

K.-Y. Lee

Description of the accompanying Program

ASYM1.

H. _{Söding, K.-Y. Lee, Sept. 1975}