Statistics for prediction of ship performance in a seaway

STATISTICS FOR PREDICTION OF SHIP PERFORNANCE IN A SEAWAY

by

Michel K. Ochi

and

Wendy E. Bolton

Prepared for publication in INTERNATIONAL SHIPBUILDING PROGRESS

AUThORS

Dr. Michel K, Ochi

Naval Ship Research &Development Center 1J0 S. Navy.

Miss Wendy E. Bolton Admiralty. Experiment Works

STATISTICS FOR PREDICTION OF SHIP PERFORNANCE IN A SEAWAY PREFACE Page 1 I SPECTRAL ANALYSIS io 1.1 Random Processes io

1.2 Spectral Density Function 25

1.3 Linear System and Superposition Principle 34

1.4 Transient Wave Test Technique 46

1.5 Cross Spectral Analysis 51

1.6 Time-Domain Analysis (Deterministic Approach) 61

II PROBABILITY DISTRIBUTION ASSOCIATED WITH SPECTRAL ANALYSIS 75

2.1 Threshold Crossing and Expected Frequency 75

2.2 Distribution of Eivelope of a Narrow Band Random Process 84

2.3 Distribution of the Maxima (Peaks) of a Random Process 92

III RAYLEIGH PROBABILITY DISTRIBUTION AND ITS PROPERTIES 105

.3.1 Rayleigh Probability Distribution 105

Page

3.3

Examples of the Distribution 118

3.4

Goodness-of-Fit Tests 120

x2 -Test 121

Kolmogorov-Smirnov Test ₁₂₅

IV RANDOM SAMPLE AND RAYLEIGH DISTRIBUTION ₁₂₉

4.1

The Maximum Likelihood Estimator

₁₂₉

4.2

Statistical Properties of the Estimator ₁₃₄

4.3

The Minimum x2 Estimator 140

4.4

Estimation of the Parameter from a Small Number of Observations ₁₄₀ V APPLICATION OF PARTICULAR PROBABILITY DISTRIBUTIONS

₁₅₄

5.1 Long-Terni Probability Distribution ₁₅₄

5.2

Extreme Valùe Statistics ₁₇₁

5.3

Two-Dimensional Rayleigh Distribution ₁₈₈

5.4

Generalized Rayleigh Distribution for the Maxima(Peaks) of a

Random Process

Page

VI APPLICATION OF COUNTING PROCESSES 213

6.1 Poisson Random Process 213

6.2 Frequency of Occurrence of Ship Slamming 221

6.3 Prediction of Slamming Severity 228

6.4 Frequency and Severity of Deck Wetness 232

VII STATISTICS AS APPLIED TO POWERING CHARACTERISTICS 237

7.1 Powering Characteristic in Waves 237

7.2 Mean Increase of Power in Waves 240

7.3 Thrust and Torque Variations due to Waves 244

ACKNOWLEDGMENT 247

PREFACE

This monograph has been prepared in an attempt to provide a thorough

understanding of the statistics supporting current prediction techniques

of ship performance in a seaway. It may, therefore, be an aid to further

applications of various statistical concepts in the field of naval

engi-neering.

Statistics, including the fields of probability and stochastic processes,

has, in the last twenty years, been extensively employed in the prediction

problems of waves, ship motions and stresses in a confused sea. Within

this general sphere a number of interesting papers, many of which are cited

in the bibliography at the end of this monograph, have been written on

isolated topics. In view of the recent comprehensive advances in applied

statistics, and in consideration of the growing neceásity for the

appli-cation of statistics in prediction techniques, it is believed advisable

to collect various information facilitating prediction methods

and present them within one referencé. With this aim in mind several

aspects of applied statistics which are immediately relevant to problems

in naval architecture are selected, and their principles and applications

discussed. .

The prime object here is to indicate, with the help of examples, the

'practical. application of statistics. It is therefore- beyond the scope

of this monograph to give rigorous mathematical proofs of the underlying

principles or basic concepts. Special effort has been made, however, to

provide sufficiently detailed infoation on each subject for adequate

comprehénsion to be possible. Those who are particularly interested, in the

rigorous mathematical background of stochastic processes are recommended

'tô consult References [i] to [6]. ' '' .

-Certain'topics in- statistics, - such. as analysis of variance, regression

analysis, etc., are 'Ömitted from, this monograph although they have been

emp1oyedit the-naval engineering field. This isbecause several.

- excellent 'texts- on these subjects are already available to.the.

profession. For example, References [7],

[81, [9],

present full

de-scriptiOns of the techniques and ample practical examples.

This monograph consists of seven sections.

The first three parts of Section 1 present the basic principles and

concepts of the spectral analysis of random processes which provides the

foundation for one of the very important prediction techniques discussed

in this monograph. Two other specific prediction techniques, the transient

wave test and the time-domain analysis, are outlined in Sections 1.4 and

1.6, respectively. These methods have demonstrated unique merits in

predicting ship performance in waves: namely, the transient wave test

method reduces, by a considerable amount, the time for the model testing

required to characterize a ship in waves; and the time domain analysis

has a potential use in the short time prediction of ship motions, bending

moments, etc., of the immediate future.

Section 2 considers the probability distribution and related subjects

which are specifically associated with spectral analysis. The threshold

crossing problem is discussed in detail, since its application enables us

to predict the peak values of a random process, and to evaluate both the

frequency of occurrence of ship slanming and the extreme values of ship

motions. The discussion is directed towards a clarification of the

assumptions involved in the current prediction technique. Most of the

explanation .is limited to narrow band random processes, however, some

results from investigations on the effect of spectral bandwidth in

connectioú with the probability distribution is outlined in Section 2.3.

This latter subject appears to be of great importance for the prediction

statistical quantities associated with random processes.

Section 3. reviews tOpics related to the Rayleigh probability

distr.i-butioñ.. Extensive use has been made. of this distribution in conjunction

with

spectral analysis in order topredict.waves andship motions ma

seaway. Many prope'rties of the distribution, including the characteristic

function, are presentèd to facilitate further application of thé Rayleigh

distribution to various practical problems. Included in this section are

iscussions on two statistical goodness of fit tests, thé x2 (Chi-square)

test and the Kolmogorov-Smirnov test, which play a significant role in the

analysis of data obtained from experiments. In particular, the

Kolmogorov-Smirnov test is introduced together with a practical example, since the

test has not yet been used in the naval engineering field in spite of its

unique feature of taking into consideration each observed value instead of

a group of values.

Section 4 discusses methods of estimating the Rayleigh parameter from

a random sample without carrying out spectral analysis. One advantage

of these methods is that they can be used to obtain the confidence limits

of the parameter for samples of only a small number of observations. Two

different estimation methods are introduced; one, by maximum likelihood

estimation, the other by minimum x estimation. The first portion of this

section describes the derivation of the maximum likelihood estimator and

its statistical properties. Estimation by the minimum x estimator is

then described together with examples of its practical application. By

this method the Rayleigh parameter can be estimated simply by counting the

number of observations greater or less than a preassigned constant. The

last part of this section explains in detail the methods of estimating

the confidence interval of the Rayleigh parameter from a small number of

observations.

Section 5 deals with the application of particular distributions to

four specific problems encountered in the field of ship performance

predictions. The first subject is the prediction of long-term ship

behavior which has been approached in a nutnber of different ways. Brief

descriptions of the problem are given here with examples. The most

appropriate approach to solve this problem is based on the concept that

the long-term is a marginal distribution of the short-term which can be

considered as a conditional probability distribution. However, since

the solution by this approach can only be obtained numerically, another

approach is to use the Weibull distribution so that the probability

distribution can be expressed by a mathematical formula. This is

considered in some detail so that its empirical nature can be understood

and the evaluation of its distribution parameters from test data can be

easily obtained. The second specific problem is the prediction of the

extreme values of ship performance. The application of order staUstics

is descr.ibed together with the asymptotic probability distributions of

extreme values for a large number of observations. Included in this

latter category are Gumbel's three asymptotic distributions which are.

given in sufficient detail to facilitate the evaluation of their

distri-bution parameters. The third subjèct is the theory of the two-dimensional

Rayleigh distribution. Although this. has not, as yet., been used in the

naval engineèring field, it has distinct potential in future applications.

The fourthand final subject discussed in this section is the generalized

Rayleigh probability distribution for the peak values of a random process

which is not necessarily narrow banded.

Section 6 outlines the counting processes with particular emphasis.

on the. application of the Poisson process to ship slaing and deck

wetness phenomena.. Two important properties of the Poisson process, one

in connection with the time interval between two successive events and the

forthe m-th time, lead to expressions för the probability density function

of the time interval between successive slams and of the time interval

between two severe slams, respectively. Frequency of occúrrence of ship

slaing is given by applying the techniques associated with the threshold

crossing problemand phase-plane diagram described in Section 3. The

probability of slam impact pressure is derived, from which the average

and-significant pressures can be evaluated. Of, perhaps evenmoreuse,

however, is the probability density function of extreme pressures obtained

by a use of order statistics. Although deck wetnessand slamming differ

in that the former depends only on relative motiOn while the latter depends

on relative velocity as well, statistically the two phenomena are similar

so that the Poisson process is again applicable. The probability density

Hfunction of the pressure due to the head of water flowing over the deck

is also determined.

-The final section, Section 7, outlinés the application of statistics

motion predictions can be made based on the linearity assumption for both

constant ship speed and constatit number f _{propeller revolutions, increase}

in powering characteristics such as added resistance, thrust and torque,

etc., in waves is rather sensitive to the propulsion System and linearity

does not always höld. _{Hence, the linear süperpösitïon technique for}

estimating ship response at speed in irregular seas may not be valid.

Statistical properties of powering quantities in irregular waves are

discussed both for mean increase and vatiation about the _{mean. A method}

of predicting the mean increase of powet in a séaway is given which

appears to agrée well with full scale triài results. _{The variation}

of the powering quantities about thé mean increase in a seaway appears to

follow the Rayleigh probability law; however, . prediction method for

,this variation based on results obtained in regular waves has yet to be

SPECTRAL ANALYSIS

1.1 Random Processés

Arandom process (or' stochastic process) is a family of rañdom

variables denotéd b X(t) T which can be described by means of

statisticl properties. The iñdex set T represents the observation time.

If T

= {

Ö', ±

1, ±2, -. - - } ,

thé stochastic process is called the discrete

time process; while if T

= { -

+

} , the process is rêferred to as a

continuous timé pröcess.

As an example, consider X(t) as the wave deviation from the zero-line

an observed wave record. x(t) varies randomly with time, but the histogram

f the observed data may show that X(t) follows the normal probability law.

If X(t) is takenas the peak-to-trough wave excursion, thén the Öbservation

ill shów that X(t) may quite likely follow the Rayleigh probability law.

The wave deviatiOn

froth

the zero-line as well as the peak-to-trough excursion,

both of which are function of time., are thee called thé randomprocesses.

The' example discussed above, hOwever, is based on the assumption

that thé process is stationary and ergodic(definitiorii will be given.

later) which has been generally accepted for short terti observation of

phenomena. which appear in the naval and ocçan engineering fields.

A.precise definition of randomness, on the other hand, needs the concept

that the statistical properties of a random processare established through

set of many simultaneous.obse:rvations instead of a single observation.

For better Ùnderstanding of this concçpt, let ús assimie that a large

úmber of wave-height measurement buoys of one kind are operating simultaneously

a certain sea area under identical conditions. A colléction of records,

äné for each buoy, representtng the variation of wave height as a function

of time is called an ensemble. The wave height will be characterized by a

certain random, function of time, 1x(t), 2x(t), - - 'x(t) for n buoys,

différirg from one another as shpwn inFigure.l.l. Then, the ensemble is

expresse4 by a set of n reòords ,{ 'x(t), 2x(t), where.

ñ must be a large number. Nöte that any given element record, "x(t), is

merely a special example out of an infinitely large number of possible

records that could be observd.

Characteristics of. the eisemble, {'x(t:), 2x(t), -

-expressed in terms of statistical values. For instancé, consider an -arbitrarily

chosen time t = t1 for the example shown in igure. 1.1. Instantaneous

amplitudes of all elethents. of the ensemble { 1x(t1), 2x(t1), - - - tj)

have different values, but the average value can be obtained

by'

1x(t1), k=l

which is cálled the ènsemble average, or ensemble mean. At a different time

t = t2, the ensemble average may be obtained by a similar manner but the value

may not necessarily be the same as that obtained at t = t. 'ortunately,

{this thconvenience des not apply if: the random process is ássumed tO be

statiotLary. There is still another difficulty in obtaining the statistical

Properties of a random process in practice, since, an ensemble. which consists

of a lage number of records is required for evaluation.

This

difficulty., hevet, can be removed by introducing an additional-assumption thatthe

random process is ergodico Under this condition, the statistical properties

of X(t) may be obtained from the time history of: a single record instead

12

n I

-

x(t)f are

is involved in the current technique for predicting waves and ship motiOns

L in a seaway. It is noted, howeer, that thé assumptio of ergodicity may

,tiot be applicable tO some particular problems. For example, a single wave

record cannot be used for predicting wave characteristics in the area ihere

fetch effect is evidenced.

Prior to discussing the spectral analysis, it may be well tb outline

the f ollowthg classifiéation of random processes in a sequence ranging

from general to specific so that thé basic concepts and assumptions

involved in a current prediction technique will be clearly grasped.

(a) Randon Process with Independent Increments

A random process X(t) is said to have independent increments if,

fr non-overlapping times t1 < t2

< t3 <- - -

< t, the n random

variables X(t2)X(t1), X(t3).X(t2),

- - X(t5) X(t_1) are

:Lstatistically independent. Nöte that the random variables X(t) and

X(t) may not necessarily be independent, but the increments X(t3)X(t1)

áre indèpendént. The difference between these two may be clarified in an

The random process with independent increments is sc;metimes called

additive. This is because the properties of a random process X(t) which

depend on its increments are unaffected by the addition or subtraction of

a process .Y(t), since the: increments .f the. new random process X(t)±Y(t)

are identical to the increments of the old process, X(t).

(b) Markov Process

Markov process is a random process whose subsequent state for .any

given time depends only on the state at the given time and does not

depend on the states at any preceding time. Simply speaking, if the

"present" of the process is known, the "future" behavior of the process

is independent of its "past". Let t1 < t2

< -

-

.-

<

t, and consider the probability that the random process X(t) will not exceed a certain

value x at a time t, given that its values at some 'arlier times are

Xx i Xt1==;,

(t)=x2

- -

-X(t1)

15

Note that the conditional distribution depends only on the most

recent value X(t) = at t = t_1 The conditional probability

function on the right-hand side of Equation (1.1) is called the transition

probability function. It can be shown that if the increment X(t) X(t1)

of the random process is independent of X(t) for all t < t <t3, then

the process is Markov process.

(c) Stationary Process

A random process is said to be stationary if all the statistjcal.

properties of the ensemble are invariant with respect to time shifts:

that is, all statistical functions which have partcu1ar values _averaged

over the whole ensemble at time t must have the same values at time

(t + r), for all T. Hence, the statistical properties of the random

variable X(t2) in Figure 1.1 are the same as those of X(t1) _{for arbitrarily}

A relaxing of the definition of stationarity from the strict to the

wide sense results in the so called weakly (or covariance) stationary

condition, which holds when the probability distribution representing

the ensemble is time dependent but when ensemble mean and covariance

have the following properties:

ensemble mean is constant

E[Xt)

=

= 'r'yi

for all t (1.2) auto-covariance function dependens only on time difference

Cov[X(t, xLttÎ

=

E

[t)_

E[(t)J}xa)-

E[xj]

x(t)

-

x(t + 'r)

-=

for all t (1.3)

It is noted that if the random process x(t) is strictly stationary then it

is weakly stationary also but the converse is not true. However, if x(t)

is weakly stationary and is a normal random process (defined by Equation

(1.10) then x(t) is also stationary in the strict sense. Waves and ship

motions are considered to beloug in this category.

This definition of weakly stationary may be extended tO two random

processes, thät the processes are said to be joi.ntlSr weakly stationary

their cross-covariance depends only on time difference T:

f each sàtisfies the conditions given in Equations (1.2) and (l3), and

Cv

[x(t),

(t)]

(d) Process with StatiOna Increments

if the increments of the random process, X(t) X(t.), are independent

ànd its probability distribuion depends Only on (t.- L), then the process

.1

x(t) is said to have stationary, increments. Note that the process X(t)

itself may not be stationary, but the increments are. stationa. In order

to clarify this difference, the following example is given:

-

(L+

T)-=

(.t)

fcirallt

9

17

Consider a random process N(t), t 20 which represents the number

occurrences of slamming in rough seas. The random process N(t) is

not stationary since its probability distribution is dependent on time, t.

Furthermore, N(t1) and N(t2) are not independent, since it is obvious that

N(t2) > N(t1) for t2 t1. No let X(t) be another random process defined

by,

x(t) = N( t +

ç) -

N(t) (1.5) Here, X(t) represents the number of slams occurring in a time

.inte.rvàl of length t beginning at The random process X(t) then becomes

independent of t, and depends oily on ç; Lé., X(t) is a randomprocess

with stationary increments although N(t) is not a stationary process.

Aswill be discussed later in Section 6, the prediòtion of the number of

occurrencesof slamming in a given period of time ts based on this

property.

(e) ErgodicRandoin Process

Suppose a stationary random process sátisfies the following

two conditions:

(i). each record of the ensemblé is statistically equivalent to

every other record,

all statistical properties for the ensemble at an arbitrary

instant of time hold true for a single record taken for a

sufficiently long interval of time; that is

=

T-'-°°

19

L

dt (1.6)

T3

ere p[kx(t)] is any statistical property Of the ensemble. If these

conditions are satisfied the process is said to be ergodic and significant

simplification is possible in the statistical analysis of data, since the

aibitrarily selected single record, shown in Figure 1.1., provides ali, the

statistical information which could otherwise be obtained only by analyzing

In the case when a random process döes hot satisfy the condition

given in Equation (1.6) for all functions

F&U,

but only for the mean

--.F[z.(t]

and covar.iance

*E

. 20 (1.7) (1.8)

then the process is said to be weakly ergodic. It should be noted that

for a random process to be ergodic it must first be stationary.

Table 1.1 shows a comparison of various statistical values such as

average, mean square, etc., for a stationary random process.and those for

stationary and ergodic random process. As can be seen in the table,

the. former is the ensemble averâge of a set of n records, while the latter is

a time average of a single record.

The treatment of waves and ship motions assumes that the ergodic

property .is satisfied. Even for waves generated ma towing tank the

characteristics are determined by analyzing a time history measured from a

single wave probe rather than measurements obtained from many probes

distributed throughout the tank. The random processes discussed in the

remaining portion of this monograph, therefore, will be treated under the

assumption that the ergodic property is satisfied.

(f) Random Process with Narrow-Band Spectrum

A narrow-band random process is defined to be one which contains only

a small range of frequencies, such that the significant portion of the

spectral density (or the average power over all frequencies) of the

process is confined to a narrow frequency band whose width is small

compared with the central frequency of this band. A sample function x(t)

from this narrow band random process exhibits the behavior of a sinusoidal

wave of slowly varying amplitude (or envelope ) A(t) and phase (t). This

situation, shown in Figure 1.2, may be represented by the equation

x(t = Act). CckL

/)Ö, OE(O<°°

(1.9)

where A(t) and (t) are time dependent, and u is the mean frequency of

the Spectral band.

A typIcal example. of a narrow band process is a random noise wave

when the incoming message is fed into a narrow band filter. This situation

is equivalent to a responsé in a strongly resonant vibratory system when

the input is, a wide band process. Assuming the energy spectrum of

processes representing ship motions to be narrow banded, it may be shown

that the envelope .atid peaks of these processes follow the Rayleigh

proba-bility law, as will be discussed in Sections 2.2 and 2.3. Although the

validity öf this assumption has not yet ben confirmed, the prediction

technique based ön this assumption has been widely used in the naval

engineering field.

Random proce9ses may be classified aécording tò their probability

distributions,. There are two random processes (normal and Poisson),

which are frequently encountered in the study of naval engineering

problems. The behavior of waves, ship motions, wave-induced hull stresses,

thrust and torque variations in à seaway, etc., are considered to be normal

processes, while that of ship slamming and deck wetness can be treated as

a Poisson process. Since the Poisson process will be discussed in

detail in Section 6, the definition will be given later. The formal

definition of a nOrmal process is given as follows:

A random process is said to be a normal (or Gaussian) random process

if for every integer n and every set [t1, t of time instants

the random variables X(t1), X(t2), , X(t) shown in Figure 1.1. are

jointly distributed with

a

normal probability density fuction given by,

e

1=

(xt,).

Ct)

23

'z'

(1.10) M.

(m,

_{2) - - - -}

_mn

iit

¿II

determinant of z

= covariance matrix of X(t,), XJt) -Il

c0[

(t),

--

T

-6,12.

-

_61fl

By. assuming that the process is weakly stationary and ergodic,

= m = constant, m n

=

E [xct

X(L)]

-

₌

1ft)

zhere, R 1(T) = autocprrelationfunction .f X(t)

T = t.

J

Thùs, it can be seen that the. mean value and autocorrelation

function completelydefine the probability.law of the process, and the

joint probability function becomes a funcdon only of the time difference

T = t, and not of t. and t. separately.

One of the mportant properties of the normal process is that any

random process derived from a normal process by linear operations,

1, 2, 3, - n

ìricludiitg differentiation and intègration,is itself a normal. process. This

implies that if the input random process (waveS for example) is normal,

then the output random process (ship motion) is also normal provided that

:a linear relatiOnsh-p is maintained between them. Another example

6f the normal process is the ship bow motion relative to waves,1

since t is a lineár combination of waves, pitch: and heave motions all

of whichare considered to be normal random processes. Similarly, the

relative velocity between waves and ship bow motion can be considered to

be a normal process.

-1.2 Spectral Density Function

In general, two different approaches may be used to estimate the

statisticál properties of random pocegses, such as ship motions, etc.

i One is to take a random sample in the time domain, the other is to carry

out the spectral (or harmon-ic) analysis in the frequency dornain of the

process. The former approach will be discussed later in Section 4, whilst

the latter approach is the main subject of this section.

The significant advantages f spectral analysis are that it

facili-tates (1) the clarification of the physical mechanism (response

charac-teristics) of a random process such as ship motion, (2) the estimation og

motions in an arbitrafily given randöm sea assumn.g the thotions to belong

to a linear system, and (3) the possibility of simutating the' random

motions in the time domain.

The essence of spectral analysis lies in the following two theorems:

The Wiener-Khintchine theorem which states thàt the

autocorre-lation fUnction and the spectral density function are related to each other

by Fourier transformation provided that the random process is weakly

L stationary.

The ratio of the spectral density function of the output to that

of the thput ofa. linear system is equal to the square,.of the Fourier

transfom of the input response function Of the System.

Since t 1. beyond the scope of this note to give rigorous

mathemati-cal derivations 'of the formulae encountered in spectral analysis, the

theory of spectral analysis is limited, in the subsequent sections, to an

outline of its application to those problems frequently appearing in the

mechanical or naval engineering fields. Details of the mathematical

theory involved in spectral analysis may be found in References _[41 to [71.

First the relationship between the mean square value (see Table 1.1)

and the spectral density function will be discussed. Consider the random

process x(t), and define ,ç(t), as shown in Figure 1.3, such that

(x(t) XT(t)

o

Its Fourier Transform X,(w) is given by

T

(t).e cit

for

TTT

for

27 with the inverse transformation yielding

00 r

(t)=----

Xw).e

da).

ï

2rr

r

J 00 (1. 12) (1.13) (1. 14)

The concept of a truncated function xT(t) defined in equation (1.12) is

necessary for the existence f the Fourjer transform Of x(t), since

X(t)ldt

thay not necessarily converge to a f init value without the

-00

concept of

Let be the mean square value of the random process x(t). It

is giveii by,

(

00

1'x(t)CLt

Note that

xr()

has _a finite mean square for a finite value of T.

By using Parseval's theorem on Fourier transforms, the mean square can

be expressed in terms of frequency to be

However, the spectral density function S (w) of the. ràndom process

) is defined as } 28 (1.16) (1.15)

dt

'I

T

c

which can be seen to be a real, positive, function, sytietrical about the.

origin if x.)'is real. Thus, frOth equatiOns (1.16) and (1.17) the

following relationship between the mean square and the spectral density

functioti is derived, namely

2.

s

°) =

I

T- 2.TrT

:'-(Lu) dw

_{_.\ .5(w)}

_c4L)

Thus, the mean square is equal to the areà under the spectral density

function taken over the frequency rangé of zero tO infïnity.

It is noted that in some literature S(w) 'is termed the power spectral

density function. This is because if x(t) represents the current for a

one-ohm 1oad, then

Ç

becomes the average power. Hencé, from Equation (1.18), S(w) can be considered as the contribution to the average power

of the pOrtion of x(t) associated with a small frequency band 5w.

29

(1.17)

Suppose, on the other hand, x(t) is the random displacement of a mechanical

system, then P represents the average potential ene:gy. In this case,

S(w) may be called the energy spectral density function.

Since S(w)

Jut

may represent any quantity such as displacement, angle, etc., it may be

appropriate to designate S(w) as the spectral density function, in general.

The mean square will next be expressed with reference to the

auto-correlation function. In the case of a random process XT(t) which is at

least weakly stationary, the covariance defined in Equation (1.3) is only

dependent on time interval, T. Assuming that the process satisfies the

ergodic property, the autocorrelation function may be w.ritten

(t)

-

E [)

(tfT)]

=

t) XttT)

_(1.19)

T+oo

J

From Equations (1.13), (1.17) and (1.19),

S (w)

Too2T

where t1 t

+ T

Thus, the spectral density function S (w) of a statiönary

random process xT(t) is the Fourier transförm of its autocorrelation,

although the transförmatiòn has a slightly different form from that

H defined in Eqaton (1.13). This is because the spectral density

functipn S(w) is defLned by Équation (1.17) to conform with the relation,

generally held by those in.the engineering field, between and

as.given in Equation (1.18). If alternatively S(w) is defined as

00 r

£wt 4)i,

I Ç

t)

a

dt dt

T-'°°

27TJ

¿

T I

- j

-('t

dv

z):.

=

£7,1

-T-oo (1.20) (1.21)

c,o

-00

function is not eqúal to the mean square s given by Eqüation (1.18)

but is equal to i-r times the mean square'.

'Now, for the definition o. S. (w) gien in Eqúation (1.17), the

L autocorrelation functiOn Rer) can be.expressed inversely as,

Then, S (w) 'can be expressed

in

a form comparable to EquatiOn (1.13), that is

00

I,

__-5 (w)

R (T) Q

dT

(1.22)

_'G

In this case, however, the arèa under the spectral density

:R(t)

Sc)e

dw

The relationship between the spectral density function and the

autocorrelation fúnction given in Equations (1.20) and Ç1.23) is the

formulation of the Wiener-Khintchine theOrem of stochastic processes..

If the 'process x(t) is real, then

R(T)

and S(w) are real',

posÎtive functions, and are syú*etrical about the örigin. Equati6ns (12O)

.32

and (1.23) are therefore expressed by the following reciprocal forms, respect.ively

S

(W)==_

_ir

R 21 00 Cos

W

d

()

C,os

With r=o, Equations (1.18), (1.19) and (1.25) yield,

R0)= E.[(t)]

=

1

o

Thus, the area under the curve of the spectral density function

provides the mean square value, E[x2(t)1, which is equal to the variance

if the random process has zero mean.

It may be well to summarize the properties of the autocorrelation

function R.

(T)

here: xx

(1) R(T)

is a real-valued even function. That is,

R

t-T)

= R

(T) (1.27) xx.

Sod

zz 33 (1 .24) (1.25) (1.26)

R(T)

has a maximum value at T=0 0)

(1.28)

(0) is equal to the variance of a random process having zero

mean. That is,

(1.29)

34.

(4)

_(tco)==o

(1.30)

1.3 Linear System and Superposition Principle

Mechanical and physical systems may be interpreted as a transducer

which transmits energy of an input x(t,) to that of an output or response

y(t). Suppose the output is uniquely determined in terms of the input.,

then the system is completely defined if the nature of the dependence

of the output on the input is known. This nature of the response may be

A system is called linear if the response characteristics are

additive and homogeneous. The additive property is expressed as,

L[,ctt)]

= LkLt)]

L[xt)1 =

1(t) + (t) (1.31)

and the homogeneous property is expressed as,

where L is designated an operator which transforms x(t) to y(t), and a

is a constant. By combining these two properties, the linear system can be

defined as a system which satisfies the following relationship.

=

c L[Z1Ct] +

L [XaLt)]

=

c;

_{1(t).+ a2}

2(t)

(1.33) Consider the situation where a unit impulse is applied to a linear

system at t =- t0, and let h(t, t0) be a response of the system which is

called the weighting function or -impulse response function. It is Written

a s,

35

36

ck6 (1.36)

L.[(-t0)

te-j (1.34)

where ô( ) is a Dirac delta function, shown in Figure 1.4, and is

called the unit impulse function.

It can be shown that the response y(t) of a system to an arbitrary

input x(t) may be obtained in terms of h(t, t). This is because an

arbitrary input x(t) can be expressed as a sum of impulses; that is,

00

X(t)

=

Xt0)

t) dt0

(1.35)

In which

case, the response to an arbitrary input can be obtained from Equations (1.34), and (1 35) as,

(t)==

LLxt)]

=

r

Thus, the response y(t) can be obtained in terms of the impulse

response function h(t, t0).

Suppose properties of the linear system are invariant with respect

to time, then the system is called a time invariant linear system or a

constant parameter linear system, and it is expressed as,

= (t-t0)

(1.37)

The above equation implies that a time shift of the input merely

results in a time shift of the output. In this case, Equation (1.34)

becomes, .

Then, from Equation (1.36) the. response to an arbitràry input x(t)

can be expressed in terms of h(t) as,

37

(1.38)

Equation (1.39) is in the form of a convolution integral, and it

can be said that the output (ship motion, for example) is obtained as a

we.ghted infinite sum of the input (waves) over the entire time history.

In connection with Equation (1.39): it should be noted that

A necessary condition for the linear system to be physically

realizable is that,

for

O

A necessary and sufficient condition for the linearsystem to be

stable is that the input and output should be bounded for allS

t, that is,

38

Equation (1.39) will now be expressed in terms of the Fourier

trans-forms of the individual functions. 'For the existence of this transformation

to be possible, truncated functions XT(t) and YT(t) are introduced to

Le t

(w) =

(t)

T

be the Fourier transform of YT(t). Then,

wt

'

e

[

X(t-)

)

L)

x.Ct-4

(T)

dt

T

By the time shift theorem,

cù(u+)

e:

Ct-t)dt

dtL

39

jwt

X1.(w)- e. where, L(w) = Fourier transform of x(t)

From Equations (1.41) and (1.42), YT(w) can be expressed by,

(1.40) (1.42) 00 1' -00

[

t) L -I

Öo

-00

.Wt

-I

XT(T)

df]

(1.41)

00

t

wr

=

_{(-e). X) e.}

dT

X1(w) H cw)

(1.43)

where, .H(w) = Fourier transform of h(t), i.e., _{frequency response function.}

The convolution integral given in Equation (1.40) is thus reduced

to the simple relationship shown in Equation (1.43) by taking the Fourier

transforms of input, output, and the response to a unit, impulse.

Next, using Equation (1.43), the relationship between input and

output spectral density functions wIll be derived. The specti-al density

function for output is given In the same f

àth

as Equation (1.17).

s

=

_{I YTI}

(1.44)

From Equation (1.43),

s

'(Lo)=

_iw

(w)

_IH(w)r

=

_5(w)

(1.45)

.T-oc2T

T

Thus, the important conclusion is derived that the output spectral

density function 'is equal to the product of the input spectral density.

function and the squàre of the frequency response function. In navâl

engineering the latter, IH.(w)12, is often called thé esponse amplitude

operator of the system.

It may be well to nöte the follöwing with regard to the definition

of the spectral density function S(w) given earlier in Equation (1.17).

It wàs mentioned that if x(t) represents the random displacement of a

mechanicäl system, then S(w) may be called the énergy spécträl density

functiOn. The physical meaning of this definition is as follows:

Consider the random displacement of a linéar system having the

frequency response function

for

(1.46)

o

otherwise

This represénts the situation of a system which responds unifortily

in a particular frequency domain limited by w1 and w2, where (w2 - w)

could be very small. Then, the average potential energy of the output is,

jr

T o

00 'L

== L:m

T-+oO

2ltT

27rT j

LA) -00 2. 2.. XT(t)I

IHw)i

1W -00 42 (1.47)

Thus, the average potential energy of x(t) associated with the frequency

band u< w <W is given by the integral Of S(w) over the frequency domain.

Hénce., S(w) is considered as the contribution of x(t) to the average

poteñt-ial energy in a frequency element 6w, and hence S,(w) may be called

the energy spectral density function in this case.

me prediction of ship responses to waves such

as

motions, bending moment, etc., is based on Equation (1.45)., and pioneering work on this

subject wai carried out by St. Denis and PiersOn in 1953 (10]. Since

then,, the prediction method has been applied widely to various problems on

the behavior of ships and märine structúres.

through numerous examples, it maybe well to list. the assúùiptioñs iñvolved

in the current prediction tèchniqüe. These are,

waves and ship motions are considered to 'be stationary, normal

random processes with zero mean

spectral density function of waves and ship motions are. considered

to be narrow-banded

ship motions are considered to be a time invariant linear system,

and 'the superposition principle is applicable to the prediction

of ship motions in irregular seas.

Definitions regarding Items ( ) and () were. given eatlier in Section

1.1, and the definition of'linear system in Item (c) is given in this

Section. The Superposition principle states that the respoñse óf a

'linear system is the sum

of the

component responses to the respective input components acting individually. This may be interpreted to mean

that the response

of

a ship to irregular waves can be represented by the

i sum of ship responses to the component waves

In practical application of the prediction technique using

Equátion (1.45), the frequency response function H(w) must be obtained

either by theoretical calculation or by a series of model experiments in

regular waves. H(w) may also be obtained by a particular test technique

which will be discussed in the next section.

The mathematical model usually employed to represent ship motion

response is in the form of a second order linear differential equation

with a sinusoidal excitation force and frequency dependent coefficients.

Although this approach does give some measure of the relationship between

the cause (wave excitation) and effect (ship response), it may not be

adequate from a strict mathematical viewpoint. Tick (111 attacked this

problem with a consideration of the general characteristics of linear

systems. Cummins 112] gave two approaches; namely, the impulse response

àpproach and the hydrodynamic equation.approach. The former assumes a

H _{linear system and isa good representation.for calculatingthe response}

but poor, in assessing why or how the ship responds to varioûs conditjons.

FOr instance, any restraint involvéd in model experiments affects the

impulse response function in any coupled mode, and hence it may not be

correct to use the response function obtained thus for prediction of ship

motion. in a seaway.. The second approach does not have this disadvantage

and the hydrodynamic equations are similar to the currently applied

equations of motion except that the coefficients of the new equations

are independent of frequency and that convolution integrals over the past

history of the velocity are includéd. These two treatments of the ship

motion problem with linear coupling àre complementary; the one for response

calculation, the other for responsé analysis.

Two examples of the comparison between experimentally obtained and

predicted spectral density functions are shown in Figures 1 5 and 1 6

The respönse

amplitude

operators used in these predictions are those obtained from regular waves tests Included also in Figure 1 5 are

spectra resulting from the response amplitude operators calculated by

Korvin-Kroukovsky's method (3]. Figuré 1.6 shows àn example of the

conditiohs pertaining to sea state and ship daft that are considered to b

severe for navigation 114]. Good agreement between the predicted and

»measured motion spectrà can be seen in these examples.

:1.4 Transiént Wave Test. Technique

It vas mentioned in the preceding section that the frequency response

function H(w of hip motion is determined either by theoretical

calcu-lation or by a series f model experiments in regular waves over the

range of wave lengths of interést. However, if the set of regular wave

tests could be replaced by a single test experiment in a wave disturbance

'having energy distributed over all wave lengths, a considerable amount of

test time would be reduced. Such a wave disturbance is the so called

transient wave which is described by Davis and Zarnick (15] whô have

deve loped a test technique with this wave to determine the frequency _response

characteristics of a ship's motion.

A transient wave is produced by generating a wave train, the frequency

which decreases linearly with time from the highest to the lowest

frequency desired, in such a way that the fast moving (low _frequency)

waves catch up the slower (high frequency) ones to coalesce, at. _some

point in space and time, and so produce a very large wave which may be

thought of as a unit impulse.

The mathematical background to the transient wave system is _as

follows: Consider the time dependent wave height _{(x1, t) observed at a}

fixed point x1 shown in Figure 1.7. This wave, which is assumed to be one

of many travelling in the same direction and to have finite energy per unit

crest length, can be represented by its Fourier transform N(x1, w) thus,.

1

(x,t)=

_:::;;

_{4(x) e}

dw

Here N(x1, w) is regarded as being an infinitesimal wave _{componentj of}

positive.frequency. This same wavecomponent will be observed again at a

point x2 but with a phase lag of (w2x/g) defined over the frequency

interval (O, ) or (wwx/g) over

(_e,

),

where x is the distance between

two points. _{The time dependent wave height at this point is therefore} (1.48)

Comparing the above equation with Equation (1.43), the operator

exp {_. _wi Ixig} can be considered as the, frequency response function thich

relates wave heights measured at two pôintè separated by a distance x in the

direction of travel.

Take

fl(x1,

t) to be the wave height of the initially generated wave at

point x1 and T(x2, t) to be that of the coalescence of the waves at point

as shown in Figure 1.7. Then, since this wave at point x2 may be

thought of as a unit impulse., and since the Fourier transform of a unit

impulse is one., it follows that

48 00

V

r

_T

e

(1.49)

-00

By letting N(x2 ) be the Fourier transform of (x2, t), Equation

(1.49) yields

_iwIw!1

Thus, by taking the inverse transform, the initial wave at point x1 is shown to be given by e.

î

ov*)T

-oo

tw

r'ít

co

where, S(u) and C(u) are Fresnel sine and cosine integrals defined by

LOt) d..o

(1.52)

2,

w)

=

(1. 51)

Hence, Equation (1.50) gives the Fourier Transform of

_fl(x1,

t) to be

I O

Let OL.==I \\Z7TX

can approximately be exprecsed by

t)=J

cosÇ

Then, for lárge values of a, Equation (1.53)

50

(1.54)

The above equation , shown in Figure 1.8 in non-dimensional form with

I,

time scale reversed, defines the transient wave which would be generated

in a towing tark with constant amplitude and linearly decreasing frequency

to result in a very large wave, at some point in space and time, which

corresponds to a unit impulse created for a brief instant. Examples of

the application of.this transient wave tèst technique to ship motion

predictions in a seaway may be found in References (15], [16], and [17].

Examples of comparison of frequency response functions obtained from

transient wave tests, regular wave tests, and by Newman's theory are shown

in Figures 1.9 and 1.10. A good agreement between these three can be seen.

1.5 Cross Spectral Analysis

The relationship between the spectral density and the correlation

function obtained for a single rardom prdcess xT(t.) discussed in Section

1.2 will now be extended to two random processes x(t) and YT(t). Here,

both processes satisfy all the properties given in SectIon 1.2

indi-vidually. If, in addition, x1,(t) and yT(t) are the input and output of

a system, the cross spectral analysis will provide complete information

on the linear system, since it nOt only provides the relationship between

the input and output spectra but also thé phase between them, This will

be dIscussed later in this section.

Iñ the same way that the spectral deúsity function for a single

random process is he Fourier transform of the autocorrelation function

(seeEquation 1.20), so the crossspectra1 density function for two random

prOcesses is. the. Fourier transform of the cross-correlation function. The

proof of this statement is given in the following discussion:

Analoguous to the mean square value of a single random process, the

mean square value associated with two random processes is given by,

ni (L)

ct

T-2T1

T T

By Parsevalts theorem on Fourier transforms, the above equation may

be expressed in terms of frequency as,

To

On the other hand, the cross-spectral density function S('t) of

these two pròôessés is defined to be

x

=

Îj)

X(-w)

T-*cT

52 'I (1.55) (1.56) (1.57) (IT 21'

z'-Xtt)

dt

as,

By the inverse Fourier transform the above equation may be expressed

sx

-

XT

(-w y (w)

T aocO

r

₍

_tht

-.

£

J

cW

(t1) e

T-oo T T

-o

On

-oÖ 53 cLt1

dt

(1.58)

where, _.ti= and

_T)

-.

E

[xc

1(ki-í)J

=.

íY (ft)

(1.59)

c'o

Thus,.it hasbeen shown that the cross-spectral density function for

two random prôcesses is the Fourier transform of the cross-correlation

(1.56) ând (1.57), the mean square and the cross-spectral density

function has the following relationship:

The cross correlation function has the following properties:

(1)

R(T)

is an odd function and the mirrór image of R(T).

That is,

(2) It has an upper bound;

(p)

2.Z.

-

(°)+(°)ì

J (1,60) (1.61) 1.62) 54

R(±oo> = O

(1.63)

R,(0)

is not necessarily the maximum value of R (T), neither

does it have the special significance associated with the

Since the cross-correlation fünction is not an even function, its

Fourier transform, namely the cross-spectral density function has thé

complex form

(w)=

(w) + ¿

(w)

(1 64)

where., the real part C(w) is called the e spectrum which represents the

in-phase component, whilè the imaginary part Q(w) is called the

quadrature spéctrûm whIch represeflts the out-óf-phase cömponent of S(w).

It is of interest tO nöte that

C.,.(t)= C

C-w)

=

_c

(uì)

9

Q(w)= Qw)

S (-m)

= 3

()

= S)

9

₉

where

S*(w)

is the complex conjugate of

The rnagnitudè of the cross-spectrum is the amplitude.spectntm given

55

(1.67)

5

(w)

-

H(w)

zi

E(w)

e.

S

(w)

S (u))

H(w)

'dz.

where H*(w) = complex conjugate of H(w)

Suppose x(t) and y(t) are the input and output, respectively, of a

time invariant linear system. Then, from Equations (1.59) and (1.39),

the cross correlation R may be written thus

c'o

()

(t)(t)

dt

Tco2TJ

T

r

00

56 (1.68)

and the phase lag of process y(t) with respect to _{process x(t) is}

(w)

=i_

Q(w)

(1.69)

C o)

This phase spectrum, which is a measure of the relative phase of x(t)

and y-(t) may be written in terms of the spectral density functions as

follows:

( )?

S (w)

e

J

-00

--L Wt0 r

R ('tt) dt0

(1.71)

By taking the Fourier transform of the above equation, the following

relationship between the cross-spectral density function añd the spectral

density function of the input is derived:

).

(t) dt0

dt

57

Hcw)

S(o)

(1.72)

Here, H(u.,) is the complex frequency response function defined in

connection with Equation (1.43). Since the cross-spectral density

function is complex, Equations (1.64) and (1.72) yield the sqùare. of the

absolute value of the frequency response function to be

dt

(1.73)

This is the response amplitude operator defined in Equation (1.45);

hence, by comparing these two equations, the following quantity w)

termed coherency is derived:

+ Q

(w)

5

(u)}

(w)}

2.

(1.74)

Although the coherency as defined in the above equation was obtained

under the assumption that x(t) and y(t) were the input and output,

respectively, of a linear system, this assumption is not always required.

Coherency, .y(u), between two general random processes is defined to

be a measure of the linear dependence or correlation between the frequency

components of the two processes. Its interpretation is analogous to the

square of the correlatiOn coêfficient between two random variables in

probability theory. In the case of a time invariant, linear system, the

coherency theoretically becomes unity. Suppose, on the other hand, two

processes are statistically independent, then (w) = O at all

frequen-cies. In practice, however, coherency usually lies somewhere between

these two extremes as may be seen in Figure 1.11 from [18]. Very small

coherency, generally, indicates a lack of correlation between the two.

processes concerned. Although physically a strong linear correlation

appears to exist it does not necessarily follow that coherency will be

large and some other explanation is required.

The hypothesis that one normal process is linearly dependént on

another is usually accepted if the coherency between them is greater or

equal to 0.85 for all frequencies at which the power spectral density is

significant for thè processes. _A possible reason for the coherency not

satisfying this limit may be _{one of the following [18][l9]:}

insufficient spectral estimates especially for the cross spectra which

may have rapid fluctuation over the frequency range of interest.

the presence of noise in either or both of the records.

rapid change in phase with frequency

wave reflection, etc. from the tank sides and wave absorber

in the case of model experiments.

actual minor non-linear effects.

short crestedness of the normal _{wave process.}

In relation to this last point Pierson [20] _{has shown that, whereas the}

coherency between ship motion and wave is unity for a long crested sea,

in the case of a short crested sea it is usually less than one. _This

probably arises from the complexity of a situation where, for a particular

heading of the craft into the waves, different phases of the response _can

result from the same frequency of _{encounter. Pierson maintains that}

representing motion in a short crested seaway only _{as a function of time}

provides insufficient information and that if it were represented as a

function of time and space then a coherency _{close to one could be obtained.}

A complete set of curves from the _{cross spectral analyses of pitch}

and heave motions on a destroyer is _{given in Figure 1.12.}

This shows the

auto spectra each of the pitch and heave (a) _{(b), the real and imaginary}

parts of the cross spectra (c) Cd), and the coherency between the two

motions (è).

1.6 Time-Domain Analysis (Deterministic Approach)

The most usually employed method of evaluating ship response (motions,

accelerations, bending moments, etc.) in irregular seas is that resulting

from spectral analysis techniques discussed in the preceding sections.

By this approach, statistical characteristIcs (such as average, significant,

and extreme values) of the ship's response to a random excitation can be

evaluated. A different stochastic approach, which enables the time history

of ship motions; accèleration, etc. to be found in terms of the observed

time history of the wave profile, is called the time-domain analysis or

deterministic evaluation of ship motions, and is based on the convolution

integral with respect to the impulse response function shown earlier in

Equation (1.39).

Since the impulse response function provides the time history of the

response of a system to a unitimpulse, the superposition of these unit

impulses to represent wave excitation yields the total response of the

system. This method of obtaining the time history of ship motions was

first applied by Fuchs and MacCamy to a simple floating body in irregular

waves (21] (22], and extensive application of the method was made at the

Davidson Laboratory on displacement ships, submerged bodies, and hydrofoil

craft (23]. Although this deterministic approach may, in practice, be

little used in its capacity of ship response evaluation, it does have a

potential use in the short time prediction of waves, ship motion and

bending moment, etc. in the time domain of the immediate future (241, (251,

(26].

The following discussion outlines the method of time-domain analysis

as applied to ship heaving motion in irregular seas:

First, consider the response in regular waves. If it is assumed that-the

heave displacement from equilibrium, z, is obtained from an analysis of

the coupled heave-pitch motion then z may be expressed as the sum of two

components proportional, to the exciting force, Fet, _{and the exciting}

iwt

moment M e _{respectively. Hence z may be expressed as} o

A:wt

Z(t)=7e

= ((4))

çe

-

M0e

(1. 75)

where the complex functions

_f(W)

_{and Ø(w) of frequency,}

w, are the

frequency response functions in heave _{per unit exciting force and moment,}

respectively.

The ship's response is dependent on the forcing functions F and M,

and the functions are obtained by considering the in-and out-of-phase

pressure distributions over the hull when restrained from moving in the

wave system.

Let

_11(t)

be the wave elevation at a location

_(,

_{) along the hull}

(which may be submerged) _{referred to its origin.}

fl(t) can be

written by,

where, g(

T

H5(w). v)

Ma)=

H(w)Yw)

o O

7'l

U 2

-H(tii)

u)L

- ()

-t- _C..,,, 64-1 (1.77)

) is the integrated value of g(w, , ) with respect to and c'

and Hf(W) and H(w) are the complex transfer functions for the regular

excitaU.on-wave system. In general, they take the förm [23],

(1.78)

A)t

),).

_e

_(1..76)

where, = wave amplitude

e.

By using EquatiOn (1. 76), the forcing functions for regular waves may

where,

Then, Equation (1.75) may be rewritten in the form

Z=

(w)

H(W)

)

o o

¡l.79)

H(w)=Co) H)

(w) H w)

)

»1

Equation (1.79) indicates that H(w) is the effective frequency

response function of a heaving ship, which is free to pitch, in waves.

The corresponding impulse response functionis the inverse Fourier transform

of 11(w), namely

64-2

o

iw

In contrast to the case of motions in regular waves, the case of

instantaneous motion in arbitrarily time-varying waves requires that the

forces and moments be functions explicitly of time and not functions of

discrete frequency as in the regular wave case. Hence., llelWt

and

Me1t

must be replaced by 1t), F0(t), and N(t), respectively , and

the time history of heave motion, z(t) can only be obtained by applying the

Fourier integral transformation technique. Thus the Fouriertransform of

the heave resulting from arbitrarily time-varying waves is,

r

LW?

r

_Lw'

Z() e.

d'C

=

_H(w)

_{t) e.}

(1.81)

00

where T is a dummy variable for t. With Equation (1.80) in mind, now

take the inverse transformation of Equation (1.81) to give

27f 'J

I

-òo

r

1S. H (w) z

J

tÖ

).e.

d}

6

d

t,

r

65 iwt e o

66

=

(T)

t-t)

(1. 82)

-00

This is a convolution integral of the wave and h(t) which may be

con-sidered as the effective impulse response function for heave in the ship-.

wave system, and has the same form as that given by Equation (1.39). Thus

from Equation (1.82) it may be seen that h(t), as defined in Equation

(1.80), is associated with the linear integral transformation of a pair

of functions z(t). and 1(t), and is called the kernel function.

In. order to evaluate Equation (1.82), it is necessary to calculate

h(t) from past ship response records and to measure the wave

sufficiently in advance of the ship to be able to predict its response in

the required time domain,

of the

immediate future. The impulse response function h(t) may be ascertained from model experiments either in regular

or irregular waves, or by full scale trials at sea. - The effective frequençy

response function defined in Equation (1.79) may equally well be written

-L E(w)

where A(w) is the response amplitude and [-e(u)3 is the phase lag of the

motion with respect to the surface wave when both the motion ansi wave are

measured at the same place and same instant of time. These quantities

can be obtained, in the usual way, from time domain records of the wave

and motion processes resulting from regular wave experiments. Thus

Equation (1.80) yields

r

_

Lj.wE)wwi-}

(t)-L

A(w)

zTr

In the case of evaluation under irregular wave conditions it is usual

to obtain the frequency recponse function from spectral arialysis.where,

according to Equation (1.45) the response amplitude operator A2(w) is

givenby . . . .

67

dw

1. 84)

(1.85)

with S(w) and S() the auto spectral ordinates of the heave motion and

with C(w) and Q(w) the co- and quadrature spectra of wave input and

ship response output. Using specific values of and , therefore, it is

possible to evaluate Equation (1.84) and, with a knowledge of fl0(t),

Equation (1.82) also.

The situation may arise where, given the wave (not necessarily regular)

elevation at some position A on the free surface, it is required to obtain

the time domain representation of the wave at some other position B, a

distance along and below the free surface (see Figure 1.13). For this,

the wave frequency response, H(w), is obtained from Equation (1.83) by

considering the ship to be shrunk to a point and indistinguishable from

a fluid element [231. That is,

_)!W

_T

CUJ

H(w)== e

w

L Q

(w

.(ui)= +

wz }

Cw

68 (1.86) (1.87)

The inverse Fourier transform of Equation (1.87) results in

-00

00 ,,

ü)--_f

p

'J -oo where, w2= s e.

tw

dw

Cs(wt

_ut-F)

dt)

±ìt

e

e..

4Z

For the wave at the surface ( _C O ), Equation (1.88) redüces to

t)==/[

_cos(.).Lj-±

_cÇi'Ç)}

+

s(1i).}]

(1.88)

1.89)

Noté. that the above equation is the same result obtained earlier in.

The Fresnel sine and cosine integrals S(u) and C(u) defined in

Equation (1.53) each tend to (½) as u tends to infinity. Thus as

(__/_i_)

_{tends to infinity or} tends to zero, that is when position

B differs little from position A

(t)

>

/i

7r

w

T)

Whereas for submerged craft when equals zero but is non zero,

(t /

\\Z/Z

becomes purely imaginary in which case

70

(1.90)

zf

=

I.

so that Equation (1.87) gives

w

₂₁₁

p

(1.91)

In summary, therefore, it may be seen that by using the appropriate value

of h(t), defined by Equations (1.88) to (1.91) inclusively, and replacing