A computation procedure for determination of ship responses to irregular seas

NAVAL

ARCHITECTURE

GLEN COVE

LONG ISLAND

NEW YORK

N

F

w

I.

for

DETR1I4INATION OF SHIP RESPONSES

to

by

Robert B. 2ubaly

and

Roger H. Compton

i4iti

inntitut1! nf Nanal Arrtifrrtixrr

CRESCENT BEACH ROAD GLEN COVE. L. I.. NEW YORK

A Ccz, utat ion Procedure_for_Determi nei or o

Ship Resoonses to reuiar Seas

by

Robert B. Zubaly

ar4

Roser H. Copton

As presented at Webb

Spring Seúnar on Ship Behavior at Sea

Introduction

A method of predicting the _{response cf a ship to an irregular}

storm sea was presented _{many years ago by St. Denia and Pierson (i).}

The method involves the _{representation of the sea by}

a spectrum def

in-ing the infinity of regular _{wave components making up the visible}

pattern. Lt is then assumed that the ship response can be obtained by the linear

superposition of its responses to all of the wave components. _This

leads to a response spectrum that provides a complete description _in

statistical terms of the ship' _{response to that particular sea.}

The application of the above _{principle to practical problems}

of ship design has been impeded _{by the apparent complexity of the}

calcu-lations involved. _{However, experience at Webb Institute}

in carrying out

many routine calculations of thIs _{type, based on available model test}

results in regular waves, has _{resulted in a convenient calculation}

form

for use with a desk calculator or slide rule. _{The procedure is}

read-ily adapted to electronic _{computer use.* This work had been done}

mainly in

It is the aim of this memorandum _{to describe the calculation}

iorm and to give detailed _{instructions as to its}

use. For convenience

in plotting _{nd interpreting the results of}

the calculations, sea

spectra are expressed in log-slope fors, _{(2). However, log}

ò instead

of lo

_.X

non-dimensional. _{It will ha found therefore that the calculation form can}

be used with any units (ice. metric, ft.- lb. sec., etc.)

Ob ect of Calculations

The calculations ae intendeo to predict the

_{response of any}

shipf or which

_{medal wave}

response data are available, at any heading

to any short-crested sea.

_{It is eumd that the}

sea ear' be

represented

by a point spectrum multiplied _{by a suitable ?spreading}

funtion. It

*

Numbers in parentheses refer _{to list of references at the end}

of this memorandum.

Such a computer program has been

_{written at Webb in Fortran language}

is thus necessary te perform a double integration with respect to

frequency and direction.

The calculation form included hereinAintended to integrate

nmericaiiy the following expression for any desired maan square ship

resoon.se, R:

[ Point

J

(SPeCtr

Sea

Freq.

Angle

Spread i ng

Function

The integrations are accomplished using Simpsons rule.

The

form also provides for the tabulation of values for the response

spec-trum components end for the integrated rescnse specspec-trum so that these

items may be plotted graphically to illustrate the composition of the

sMp rsponsa.

(See Fig. I.)

Sea Spectra

The point sea spectre obtained by ocean weather ships as given

in (3) are presented in a form suitable to the oceanographers who

de-rived th. However, they are no

in their most useful form for the

naval architect.

After considerable mv tiarcion and experimentation at Webb

Institute, it was decided that the "slopelog" spectral forni (2) was

most meaningful for application to ship response probleis.

The form

of the spectrum, which is non-dimensional and can thus be used in any

measurement system, is as follows:

Spectral orth nate:

[r('o

where

_[rloge

) )]

is the conventional amplitude spectral o:dinate

plotted on lOSe

)

_{instead of}

c)

In order to convert spectra given in various references to

this form, the following relationships apply:

r

-

-

-jr(/4c»

L/LJ

80

where

_j

(..4i

- Spectral oroinate of

(i)

- Spectral ordinate of

(3)

op.

/

J

2 7'f

_(sec)

_{spectralabscissain (1)}

H

hg

spectral abscissa in (3)

(i)

11/180

Resid9oertors

Ship responses such _{as those obtained from sources like (4)}

and (5) should be _{cross-plotted, non-dímensiottalized,}

divided by wave slope, and the resulting

fractIon squared to be compatible with the sea

spectrum by which it is to be multiplied. _{This squared fraction is the}

response amplitude operator, RAO.

By dividing the non-dimensional

response by wave slope, the resulting

_{response spectrum is in}

_{a form}

which is ittediateiy

_useful.

The following _{are exemples of response amplitude}

operators

for various ship responses assuming the sea _{spectrum to be expressed in}

slope-log form:

L Pitch

nc1

L/

where _{pitch angle,}

amplitude wave amplitude

7...

wave length

Heave

-

r-12.

where _{heave amplitude}

L _{ship length}

Vertical _{cceleratior at a given point}

r

"q

Lf.-_

Lz

2-)Z/X

4. Relative bow motion

T

Is

L'

where S = ampittude of relative bow motion

(i.e. difference between motion at bow and wave surfaça)

5. Wave bending moment

[,4/A

1

L2/ j

where H /L bending moment coefficient and

e

is an "effective wave height"

When plotting results of model tests run in regular waves at various headings to the model and with several wave lengths, it is rare

for the entire RAO curve to be clearly defined. lt is therefore neces

sary to extrapolate both ends of the RAO curves. Such extrapolation is

'accomplished by relying on some theoretical considerations, perhaps at zero forward speed, cr by íntuittve re3oning.

Fig i shows the sea spectrum in ziope-log form, the bending

moment RAO discussed previously, and the resuliug respouse spectrum. The'area of the latter gives the mean square waie bending moment coef f icient for the particular sea ccnditon represented by the sea spectrum.

B as Ic Da tuird

The LoLowing quantities are needed as basic information for the use of the computation form:

h Point sea spectrum (3) -- for the present farm the spectrum is chaflged to slope-log form and the spreading function

2/fr-

cos.

2. Response amplitude operators -- for the particular ship

form under investigation obtained by cross-plotting model test results (4, 5)(or by theoretical calculations).

An

Angular nomenclature can best be clarified by the following sketch:

-/

/,

'5,4/P

77qC/<

DfJfr/M/4N7 hvve ¿?'cr/úA/

(M/VD

¿,.qc,,a,v)

\

k

i4/iVñ

C-4''P24/-,IT

'cT/OA/

where

angle between ship course and dotiinant

wave (wind)

,8û1

- angle between dotninant wave

(wind) and

cotnponent wave

-9O

= angle between ship course and wave component

i8oi

a.

* t'

Rigorously speaking, this equation is not correct since the

sun of,a

and,/t1, with the Umits given can be greater than

180

However, due to the syiretry of t. ship about fts centerline,

the fo1lowng substitutions c'ay be made:

RAO f

2250

2700

etc.

157½°

\ 135°

RAO for Jt-

.l12½0

)900

(etc.

The enclosed tool, "SWAT," will be helpful in determining

the correct conibination of angles for any given ship to wave situation.

The a1igtxnent is accoap1ished by placing the e.al1 disk

(showing values of/..t

so that the required ship heading angle

is aligned with the large "wind" arrow on the larger disk (showing

values

The values of/L are then paired with the

appropri-ate values of

and they can be entered on the computation form.

For example, if a quartering seas case

heading

45°)

is required, the

450

arrow on the RAO disk is aligned with the

"wind"

arrow (JÇ - O) on the Sea Spectr

disk, and the paired angles

become;

These values

of /L, are

permanently

entered on

the

computa-Clon form.

These values of

,U. are to be

entered on

computation form

ln the

appropri-ate spaces.

Colus for/L,

are not provided ott the computation sheet

as the spreading function is identically zero at these anglas

1'omenci ature

ibg5 ¿)

natural iogarit

c

frequency in 1/sec. units.

This is the particular spectral abscissa in use

for the enclosed form.

-o..

-90

135

-67½

-45

90

-22½

45

45

90

45

RAO response amplitude operator for the particular response being investigated, in the foztii of a non-dimensional ratio or angle divided by wave slope, quantity squared.

R mean square response, non-dimensional.

r root mean square (rms) response. It s

from this quantity that various statistical predictions can be made as is indicated in the following table:

average response 0.866 r

aignificant response (average of the 1/3 highest

responses) 1.415 r

average of the 1/10 highest

responses 1.80 r

highest expected response in N cycles:

N 20

1.87r

2.12r

ÇptationFormDtai1s

Colttr

-- lists values of the spectral abeissa, iO

at which the sea spectrum and the RAO curves will be read. The entire

range of signIficant response energy must be covered. The form is set

up for en interval on log of 0.1.

Columu - records values of the point sea spectrum

ordinates at indIcated va1uc cf Lo )

Columns ® ®)®) J)

J) (2J

response amplitude operators for each log5 ) . Each o these columns

is for a particular angular ccmpontú (Values read f ron plotted RAO's.)

Columus ®

Ç1J

response spectrum component curve ordinates for each io

be platted

as respcnse speCtrum c

Columns

® ® 6J. G)

3 --

contributions to the total integrated response from the various angles

at each loge ) value. The constants are for the Simnson's integration

over angle.

Column - tabulates the results of the integration over

angle and is thus the listing of the integrated response spectrum

ordi-nates for each value of log ¿i.) . is the symbol used for such

ordinates. (They may be plottod as integrated response spectrum curve.)

Coluut

3

integration over spectral abscissa. iog

Column tabulates the functions of integrated

response for the determination of the area under the response spectrum

curve.

References

(i) St. Denis, Manley and Pierson, Willard J., "On the Motions of

Ships in Confused Seas, Transactions SNAME, Vol. 61, 1953,

pp. 28O358.

Le-is, Edward

in Irregular Seas" Webb Institute Report, October 1963.

Moskowltz, L., pierson, W. J., Mehr, E.,

Wave Spectra Estimated

From Wìve Records Obtained by the OWS WEATHER EXPLORER and the

OWS WEATHER REPORTER,f? New York University Research Division

Report, Part I, November 1962, and

Part it, March 1963.

Vossers, G., Swaan, W. A. and Rijken, H., "Vertical

and Lateral

Bending ?lonient Measurements on Series 60 Models," International

Shipbuilding Progress, Vol. 8, No. 83, July 1961.

Vossers, G., Swaar., W. A and Rljkn, H., "Experiments with Series

60 Models in Waves," Transactions SNAME, 'loi. 68, 1960, pp. 364.

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