Amplitude - frequency characteristics of wave bending moments

AMPLITUDE-FREQUENCY CHARLCTERISTICS OF WAVE BENDING MOMENTS

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Methods of calculation of atnplitude-freQuency characteristics.

Wave loads acting on ship hulls change in time and their alteration is random. In this connection the investigations _of regularities to which wave loads are subjected should be carried out by methods of probability theory and mathematical statistics. For such investigations in case of stationary wave processes the method of spectral transformations is widely used. in world _practice.

In order to use this method it is necessary to have the

cha.racte-ristics of a ship as a d.ynRmic system converting the wave process into the output processes,for instance, respoise-amplitude _operators of bending moments in the vertical plane.

At present two ways of determining such characteristics are

used: model experiment and calculations based on the equations of

heaving and. pitching motion.

The first method provides for testing of ship models on a serieE of waves of different lengths. _{However in a number of cases test} results obtained by different authors even at almost equal values of main model characteristics ( block coefficient, still water bending moment, longitudinal inertia radius) are essentially

different which is connected with scatter in model characteristics, particularities of model basins and experimental errors. Test

an exanpie the values of the coefficients

obtained ïn /1,2,3/ for models with almost equal main characte-ristics are considered. It is easy to find considerablc discro-pancies in results obtained, by different authors. Por instance the

data in /1/ and

/3/

which are in a good agreement at=O,7 and O8 differ almost by O% at =O6 in the whole considered

range

o±'2-/L and of speed ( fig.î). There is a marked di'ference

in the results of /1/ and /2/.

In this connection for checking the results o± calculatìo tecbiques it is expedient to use model experiment, repeating it in this case marr times statistically averaging the data obtained. Such investigations have shown that values 0±' wave bendin cm2ntS

obtained by way of using the equations of heaving and th: e.:

motion, written in the form offered by KorvinKroukovs1 ra

on an average in a good agreement with model test resuit

This conclusion in particular can be nado on the c'.

classical for the considered purpose

model

tests crriec out T i.:oor /2/. However even these rather extensive and re:'ole tests

the osuits o± which are presented as general nalytii. i

dences do not meet completely present reqairements, The ytical dependences obtained by Moor are based on the results of sts

carried out

with 1 actual ship models. The paraneters o is

ships cover a wide range but the factthat they change i rary combinations. This slightly reduc5 the value

_t

: because the arbitrary change of the paran-or it to determine with sufficient ac.suracy thi

So the regularities obtained in the process of work may be con-sidered Ofli as a result of first approximation. Besides, Moor carried out model tests only at comparatively high speeds ( Fr O,1) and this considerably reduces the reliability of

analyti-cal dependences for low speeds suggested by him. Finally, as an example, characterizing load distribution along the model length, Moor used only the longitudinal inertia radius. The choice of this parameter carnot be considered as an apt one, because the wave bending moment value to an equal and in some cases to a

greater extent is determined by the statical moment of weight forces ( bending moment value on still water).

Sufficiently comprehensive tests on the series of systematic models,taking into account this fact, were carried out by

Ju.A.Netsvetaev /4/. Ju.A.Netsvetaev treated originally the re-suits obtained and developed analytical dependences similar to the formulae, suggested by Moor for the definition of amplitude-frequency characteristics of the wave bending moment in seven sections over the ship length.

However analysing the influence of weight load distributn over the ship length on the wave moment value he proceeded from the availability o± single-valued connection between still water bending moment and longitudinal inertia radius, stating that this question requires an additional investigation.

It is worth noting among the model tests devoted to the point considered tests, carried out by Lotveit and others

/5/,

where the change of wave momnts depending on load distribution

over the

length

of a model of taer T-2 is thoroughly investi-gated. Attention should be drawn as well to tests, carried out by Vossers, Swaan and others

/3/,

who investigated

the if

lu-ence of block coefficient, ratio of nain dimensions and load

distribution OLL the bending moment on models of series 60. Tests,

carried out in Holland by Joosen and others should. be also marked //. They investigated the influence of a wide range of

te' odq,

parameters - configuration o± fore- an1'ratio o± main dimensions, block coefficient- on the wave ben.ding moment and compared. test

data with the results of the analytical calculations,mad.e

according to the nithods ol' Korvin-Kroukovsr and Vosser.

Ho-wever in these experiments the step in wave frequencies is rather large and does not permit to bring to light a number of peculiarities of amplitude-frequency charactcristics,while the limits considered do not cover all the possible changes of para-meters. So in spite of a rather large volume of model tests,

carried out in different countries arid by different authors a nirmber of questions coDnected with the calculation of

amplitude-frequency characteristics of bending moments is still to be

settled.

Firstly it concerns the development of practically convenient arid simple calculation diagrams which take into account main parameters, characterizing individual peculIarities of each ship.

Secondiy it concerns systematical generalized data on regu-larities, which determine amplitude-frequency characteristics.

In order to olve these problems it seems reasonable to use the second method- the method of calculation. This method is used. in a number of papers of Soviet and foreign authors. Such invesbigation, for example, has been carried out by Japan

inve-stigator kuda /6/, who composed a calculation program for the definition of effective wave height, which included, as

an

bending moments was carried out in a number of papers of Ja.I.o-rotkin /7,8/. Using simplified equations

of heaving and pitching

motion of a symmetrical ship he investigated the qualitative effect

of the ship speed, longitudinal inertia radius and still water bending moment

on

the

wave

bending moment in a sufficiently wide range of changing ¡L.

An interesting work devoted to the solution o± the problem considered was carried out by Pavlinva and. Pbilippeo

/9/.

Representing ship lines in the form

of

a parabolic curves and

using simplified

equations of motion for a symmetrical ship the

authors derived regularities o± wave bending moment changes at systematic variations of different kinds of parameters ( _{ratio of} main dimentions,speed, still water bending moment, block

coeffi-cient).

The

_{regularities obtained are corrected according to the}

experimental data of Lotweit. It should be stated,however, that the authors al' this interesting work confined themselves to the consideration of comparatively narrow fi'equency band of regular waves

(X/L=

O,'5-'1,2) and. made a number of simplifying

-6-The use o± the computer permits to solvebbe problem of

plot-ting of amplitude-frequency characteristics of wave bending

mo-ment by a direct solution of' the equations of heaving and pitc1iin

motion not introducing into' the calculation simplifying

assump-tions of different kinds.

2, Initial data for computer programs.

Computer programs compiled especially for the computer

Llinsk-22 were used for the calculation of amplitude-frequency

characte-ristics.

Two programs were used for calculation:

- a program compiled for calculating a wave bending moment

in the middle section /10/ and

- a program compiled for calculating wave bending moments

for ten sections over the ship length and permitting as well to

establish correlation between load and bendingmoruents /11/.

Initial data varied within wide limits, covering the whole

practically possible range of their variation. The lines of ships

considered were characterized by curves of' sectional areas, the

form of which was given as a function of' block coefficient

according to the dependences derived in /12/.

In each case the weight load over the ship length was

distri-buted so as to satisfy the two given parameters: still water

moment and inertia moment about the midship section. At

ach

given value of still water bending moment three types of load

distribution corresponding to minimum, maximum and. some medium

value o± longitudinal inertia momentswere considered.

distribution along the ship length was attained when

- a ship was assumed to be symmetrical about the midship - light weight of a ship was assumed to be distributed in a

forza of a trapezoid

- deadweight load was distributed in ten compartments in _such a way that the centre of gravity of the forward or aft ends of the ship was from midship.

+ --Ch

-

2

(2)

where rn. - deadweight load in _{compartrnent,t}

- distance of the centre of gravity com-partment from the midship,m

- ship light weight,t

- distance of tie centre of gravity of the forward or after ends of ship light weight

from the niidship,m.

The value ZC _{at a given block coefficient and form of the}

curve of sectional areas, i.e. at a known location of a centre of buoyancy for half lengthCwas defined from the still water bending moment _{as a single-valued one. At introduction}

of

nondimensional values

the above mentioned dependence takes the following _form:

(4)

where _{- coefficient according to the diagram in}

fig.2.

-8--Two extreme values of the longitudinal moment of inertia (fig.3) may correspond to :

minimum- cargoes, being a part of a deadweight, are lo-cated in compartments closest to

maximum- cargoes,being a part of a deadweight, are lo-cated in the end compartments of the ship's half length.

.s it was already mentioned besides the two extreme possible values of moments the third intermediate value, corresponding tc a relative uniform distribution of a deadweight in compartments for a given still water bending moment was considered. At the development ol' the intermediate type of loading additional re-quirements were put to the deadweight distribution, which drew this distribution as near as possibl to the ypical one for

ships of general "standard" arrangement.

The values of inertia moments were characterized by relative longitudinal inertia radii 9./L. The values of the radii versus the values of block coefficients and parameters are

shown in fig. 4. It follows from the data obtained that mean correlations between and /L depend on. and for ships of standard types are well presented by the dependence :

(p/L)=c

₍₅₎

The scheme oÍ' load and supporting forces distribution for all the cases considered is given in fig.

5,6,7,8.

Calculation of motion and wave bending moments on a series of head regular waves at different ship speeds was made for every type ¿'f loading. The considered combinations of initial data are shown in. fig.9. Systematic data on :

In diagrams

- values and. distribution of loads over the ship length

- values and distribution of shearing forces over the

ship length

- values and distribution of wave bending moments over the

ship length

- correlation of load and wave bending moment arn.idships

were obtained from calculations made on the computer.

3. Results and their treatment

Some examples of calculated amplitude-frequency

characteris-tics of motion and wave bending moments are given in diagrams

o± fig.10. The total number of such diagrams amounted to more

than 120. Each diagram presents curves of non-dimentional

am-plitude-frequency characteristics in function of paraineterL/J\.

at seven values Fr.

--_a

cph

- non-dimensional peak-to-peak

value of wave bending moment

- non-dimeniona1 parameter of

he aviiag

- non-dimenSional parameter of

pitching

- Froude xiaber

L

- ship length,m

- ship

readth,m

h

_{- wave height,m}

- peak-to-peak value of a wave

moment ,tia

-lo--to which they correspond.

speeici Comparison 0±' calculation data with the results of the model experiment _{-fig.1l and with test results of other} invesiga-tors /4/-fig.12; /5/-fig.13;/2/-fig.14;/3/--fig0l5 shows a good

agreement.

Good. coincidence with the data of all the considered experi-ments has place at Fr' O.l _a1adt 1; a somewhat worse

coinc-dence has place on relatively short waves at Fr near zero. It may be explained either by errors connected with lack of sufficiently reliable data for the definition of damping coefficients _{at low}

speeds or by mistakes in. the experimental data,connected with the

effect of waves,reflected from tank walls at low speeds _{of models.} Taking into account the abovesaid these materials may be di-rectly used for practical calculations. However in this case it

is necessary to use the method of interpolation for _several para-meters, which is not always convenient. Therefore an attempt was made to generalize the obtained data in order to give them a more

convenient form for practical use.

Two methods of treatment of data were considered:

-Deduction of analytical dependences determining peak-to-peak values of wave bending moments at each wave frequency in function of parameters considered. In particular this method was used in /2/,/L./. It allows to get sufficiently accurate approximation for

curves of amplitude-frequency characteristics by means of analy-tical dependences. However, the dependences themselves are so complex that their practical application is connected with rather difficult calculations and does not give any advantages in compa-rison with direct interpolation by graphs.

- Definition of standard forms of amplitude-frequency charac-teristics in a twice normalized form as it suggested. in /13/. While this method is less accurate it has a number of advantages

because it allows to reduce considerably the number of calcu-lations and to use generalizations obtained in analytical inves-tigations. Thanks to these advantages the second method seems to be more preferable and will be used here later.

A. Midship section. Fig.16,17,18 show some typical examples of amplitude-frequency characteristics of wave bending moment, presented in a twice normalized form and plotted on the basis of

calculations,carried out on the computer for midship section. Values on the ordinate axis yPAm , corresponding to the

maxi-mum value of amplitude characteristic and values on the abscissa

axis , corresponding to J rn on the

axis (Ai , were taken as normalization factors. So values

were on the ordinate axis, and. values on the abs-cissa axis. In a usual case the form of amplitude-frequency characteristics depends on many parameters _r,

the most important of which are Fr and

The combinations of three parameters _{may be devided. into} three typical groups:

Group 1. Small and mean values of Prude number (Fr 0.15).

Within these limi _{ts of Prude numbers the form of} amplitude-frequency characteristics is constant _{and little depends on}

parRmeters _{At a change of from}

positive (0.2) to negative _{ones (-0.1) the area occupied by} the curve of an amplitude-frequency characteristic somewhat increases. However in all cases the change in form is not

-.12-Group 2. Mean values o± Frude nuibers (0.15 < Fi-. '( _0.25).

For this range of speeds, especially at negative values of the parameter , the appearance of a poorly expressed second maximum is usual. At values _{0.10 the generalized form of} the amplitude-frequency characteristic eproaches the form typi-cal for group 1.

Group

3.

High values of Frude numbers. (Fr _{0.25). In this} range of speeds at _{ksw the form of the}

amplitude-frequency characteristic has two pronounced peaks. When the

values o±'

k _{increase these peaks gradually smooth away and. at}

0.2 they vanish at all and the curve aquires the _form

si-inilar to that at mean Prude numbers (group 1). _{With the incrase}

of block coefficient 8 the flattening of the second peaks occurs at smaller values of . Thus at 0.6 the second peaks smooth away only at and at _=0.9 at

k5=o.

Prude numbers less than 0.2 are typical for transport ships,

especially on waves. In this range of Prude numbers ( o F _0.2:

it is possible to use a unified form of amplitude-frequency cha-racteristic, the ordinates of which are given in table 1.

Table 1.

Cg/LA)

rrra% 0.4 _0.5 0.6 _0.7 0.8 _{0.9 1.0 1.1 1.2 1.3 1.4}

0.06 0.18 0.36 0.56

0.77

_0.95

1.0

_0.9

_{0.62 0.33}

_0.0

However at values of

k<o;

_0.75

_{and at 0.15 F.}

0.2 _{tabular values of non-dimenSional frequency}

_/shou1d

be multiplied by the factor

(8)

In order to determine J\ _and

t

_{it is necessary to} take into account the parameter _{J/L at}

r 0.15.

With the decrease of /L and at constant _the amp-litudes of pitching and. heaving reduce. In this case wave

bonding moment amplitudes do not change considerably (fig. 10 a,b). At constant /L and with th increase of \ the motion parameters on the contrary vary relatively little at Forms, siven in fig.16-18, should be used at Fr> 0.2. For practical usage of amplitude-frequency characteristics in a twice normalized form it is necessary to have information about the normalization factors i.e. values _JMm and loca-tions of maxima of amplitudefrequency characteristics _{on the} axis of frequency ( rno _{) versus values of different}

pa-rameters. With the accepted 11eine and with two maxima on the

curve it is sufficient to determine the largest of them ( the main one).

The analysis of calculation results has shown that the value of main maximum

J\Ç

_{and its location on the axis of} frequen-cy - rno depends on all the parameters considered.

This dependence is rather complex and. characterized by diagrams,given in fig.19,20,21,22. At low speeds (Fr < 0.15) the number of parameters considered may be reduced withoub a

serious error to three ( ,

k

) and values vrÇf'irncc

and -j-m are represented as approximate dependences.

= O.020 (8+o5)(o,s4

-

o6)[i+

(0.2

a considerable decrease of the amplitudes of wave

bending moment

(fig. 10 a,c). At possitive values of

0.15 the alteration of

motion parameters with the increase of speed does not practically

change the values of wave bending moments (fig.lOd). However at

sufficiently larger k

1

0.2 and

0.8 there i8 a

ten-dency to increase La a wave moment at large Fr.

B. Distribution over the ship length.

The analysis of data of the calculation program shows that

the form of distribution of peak-to-peak values of wave bending

moments over the ship length is steady. With accuracy sufficient

for practical calculations this distribution for ratios

within the limits of 1+3.5 may be considered as constant.

Distri-bution curve has two peaks only in the region

0.5, where

the valuesÌthemselves

are small ( fig.23). Design speed and

still water bending moment have the most pronoimced influence

upon the bending moment distribution over the ship length. At

zero speed and rather long waves the maximum of the wave moment

coincides with the midship, whatever the sign of the still water

bending moment may be. With the growth of speed the maximum of"

the wave moment the more moves to the fore end the less the

parameter kis. At

sw=0.1 and r=°2 the maximum of the

wave moment moves forward from the midship about 0.IL and its

value exceeds the wave moment value amidships approximately by

1.1 times (fig.24). The wave bending moment values in the region

of the maximum may be found from the value

J\k

for midship

section by multiplying it by the factor

and values obtained froni this formula should not be assumed equal to more than 1.1.

/L

YY/L

CONCLUS IONS

The dependence between numerical pararneters,characte-rizing the weight load distribution by still water moment

and. relative longitudinal inertia radius /L,is not one-valued. At a given the values of

f

¡L may change su.fficiently.

The mean relations between and. 5)/L depend on the block

coefficient and. for ships of a standard type are well reflected by the dependence (

S ).

Maximum and minimum values of ¡L may differ from the mean ones within the limits of _15%.

At low speeds (Fr <0.2) parameter has a

deternii-ning influence on the wave bending moment value, at high speeds

(Fr > 0.2) the influence of

f

/L acquires the same significance. The form of the amplitude-frequency characteristic of a wave bending moment for midship section presented in a twice normalized form, should be chosen taking into account _ship speed on waves and, still water bending moment.

The form of wave moment distribution over the ship length is constant and depends mainly on the design speed and ratio

Forms,obtained in this work, may be recommended for prac-tical usage.

Parameters ,JM _max, and (

/L,X),

which are necessary for the application of these forzns,may be assumed in depen-deuce on ,

k,,

¡L, Fr of a particular ship.

-

16-REFERENC1S

1.. loosen P.A.,Wahab R. and. Woortman J.J. Vertical Motions

and Bending Moments in Regular Waves. Report of the Netherlands Ship Model Basin,Wageningeen.

Moor

D.J6

Longitudinal Bending Moments on Models in Head Seas. TRINA,1966.

Vossers G., Swaan NOA. and Rijken . Vertical and. Lateral

Bending Moment Measurement on Series 60 Models. International Shipbuilding Progress, vol.9, W83, 1961.

Netsvetaev Ju.A. Analysis of Model Test Results for In-vestigation of Wave Bending Moments of Transport Ship Hulls on Head Waves. Transaction of the Central Scientific Research Institute naxned after A.N.Kriloff. Issue 245.L.,Shipbuilding,

1968.

5,

Lotveit M. Wave Loads on a T-2 Tanker Model. European Sbipbuilding,v.10,N10,196l; v.13, N3,1961.

Fukuda. Results of the Computer Program for Amplitude-response-operators of Wave Bending Moment on Regular Waves.

Transactions of ISSC-III,1967.

Korotkin Ja.I. Investigation of Wave Moments in Midship Section of Ship Hull

in

_{Regular and Irregular Waves.} Transac-tions of the VI Scientific Session of Shipbuilders of Poland.

Gd.ansk, 1966.

Korotkin Ja.I. Analysis of Overall Strength of

Maine

Transport Ships0 A Promotion Work for a Doctorate of Techni-cal

Sciences. LEI, 1969.

Motion Speed and. Forward Load in Practical Calculations of Nave Bending Moments of Transport Ships. Transactions of the Central Scientific Research Institute named after Kriloff A.N. Issue 245, L., Shipbuilding, 1968.

Kondrikov D.W. On. Calculation of Heaving and Pitching Motion of Transport Ships. Transactions of Register of Shipping of the USSR. " Problems of Seakindliness of Ships". L.,

Tran-sport,l967.

Maximadji A.I.,Semikalenov V0N.,Markozov G.W.,Ch&tir kin N.Y. Computer Program for Amplitude-Frequency C haracteris-tics o± Iave Bending Moments on the Computer "Minsk", Transaction

of the Certral Scientific Research Institute of Merchant Marine N 59,1964.

Maximadji A.10 and others0 Low-Alloyed Steel in

Shipbuil-e

ding0

L0 "Shipbuilding", 1964.

Kozijakov LW. On Rational Structure of Formulae for Determining Statistical Characteristics of Wave Loads. "Ship-building", N 8, 1966.

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