Some aspects of ship motions in irregular beam and longitudinal waves

SOME ASPECTS OF SHIP MOTIONS

IN IRREGULAR BEAM AND LONGITUDINAL WAVES

STELLINGEN

De methode voor het beiekeneñ van de bewethgèn van een conventioneel schip

volgens de striptheorie kan geschlkt worden gemaakt voor catatharans.

Jong, B. de, "The hydrodynamic coeff1cient of two parallel identical cylinders oscillating in thé free.surfáce." Laboratorium voor Scheepsbouwkunde, report no; 268.

n

Door gebruik te maken van de resultaten van Grim betreffende

koersinstabili-teitvan een. schip in achterinkomende regelmatige goEven, kan m.b. y. de verdelingsdlchtheid voor de mómentane golfhoogÑ en golfiengte 'de kans

bere-kend worden, dat dit verschijnse[ zal optreden in een onregelmatige zee.

Grim, O., "Das Schiff in von achtem aufla,iiendèr See." Jandmch derSchifibautechnischen Gesellschaft 4S Band 1951.

m

De convergentle van de iteratiemethode van' Fil chakova -voor het afbeelden van een dwarsdoorsnede van een schip op eon ha1vecirkeI kan

worden.verbe-terd door de startwaarden van het procea te kiezen volgens de methode

beschre-ven door Kantorovich and KryLov.

Fi1chaJcova, V.P., "Conformal mappingof exteriorregions by the method of tiigonometric Interpolation." Prikiadnala Mekhanlka1 VoL 1, No 3, 1965. Tránslatlon by'R.D. Cooper, Report No, 143, Laboratorium voor. Scheepsbouwkode.

Gronin-Iv

De door Garrison verkregen resultaten voor de hydrodynamische

eigenschap-pen van een oneindlig lange àylinder met oneindig kleine diepgang in scheefin-komende golven kunnen, gezien de goêde resultaten van Ogawa, ook m. b. y. de

striptheorie worden verkregen. Laatst genoemde methode kan worden toege-past voor cylinders met een willekeurige vorm.

GarrIson, C. J., "On the Interaction of an Infinite shallow draft cylinder osclUating at the free surface with a train of oblique waves." J. Fluid Mecb. (1969), Vol. 39, part-2, pp. 227-255.

Ogawa, O. The drifting force andinoment on a ship In oblique regular waves." Laboratorium voor Scheepsbouwkunde, report No. 155.

V

¡let bepalen van de dynamisch eigenschappen van de ankerkabels van een schip in longitudinale gnlven met een langsscheepse onderstroom kan worden gefor-muleerd als een randwaarde probleem.

Jong, B. de, "Dynamisch gedrag van ankerkabels." Afsthdeerverslag. VI

Door gebruik te maken van een door Amiañtov en Tlkhoñov böschreven-methode kan bepaald worden in welke mate biJ het opwekken van onregelmatige golven door een golfopwekker met een begrensde flapuitwijking het ingangsspectrum van de golîopwekker wordt vervormd.

-Amiantov, 1.N. and Tikhonov, V.1., "The response of typical non-linear elements'to normally fluctuating inputs." Non-Linear Transformations of Stochastic Processes. Pergamon Press, 1965.

vn

Indien voor een schip in regelmatige golven de spanningen in de constructie als functie van de golilengte en golfhoogte bekend zijn, kan m. b. V. de verde-lingsdichtheid voor de momentane golflengte en golfhoogte het gemiddeld aan-tal malen per seconde- bepaald worden, - dat in een onregelmatige zee deze spanning een gegeven drempelwaarde overschrijdt.

Vm

Het woningaloop- en woningbouwbeleid van de gemeente Delft is meer gebaseerd op prestige en zakelijke motieven dan op de belangen van de toekomstige

be-woners. -

SOME ASPECFS OF SHIP MOTIONS

IN IRREGUIAR BEAM AND LONGITUDINAL WAVES

PROEFSCHRIFT

TER VERKRUGING VAN DE GRAAD VAN DOCTOR IN:DE TECHNISCHE WETENSCHAPPEN AAN DE -TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H.R. VAN NAUTA LEMKE, HOOG LERAAR IN DE AFDELING DER ELEKTROTECHNIEK

VOOR EEN 'COMMISSIE UIT DE SENAAT TE VERDEDIGEN

OPWOENSDAG7 OKTOBER 1970 TE 14.00UUR

DOOR BARTELE DE JONG wiskundig ingenieur geboren te Wieringermeer 1970 -DRUKKERIJ. BROÑDER.OFFSETNX.

Dit proefschrift is gòedgekeurddoor de promotoren PROE.DR;R.FIMMANEÑPRÓF.rR. J GERRITSMÄ

Aan m!jn ouders aan mijn vrouw

pag.

CHAPTER 1. INTRÓDUCTION. 9

CHAPTER 2. PRELÏMINARY RESULTS. ' 20

2. 1. Rolling In beam seas. 20

2. 2. HarmonIc solútion of the Duff ing equation., 26

2.3. Modél of random phases for a Gaussian procèss:

instantaneous amplitude and frequency;

quasi-stationary approximation method. 31

2. 4. Threshold crossing probability. 38

CONTENTS

SOME STATISTICAL CHARACTERISTICS WITH REGARD TO THE JUMP PHENOMENA IN THE OUTPUT AMPLITUDE.

3. 1. General results.

2. The average number per second of pward jumps

in the output when the ¡nput signal has a very narrow band width.

DETERMINATION OF THE PROBABILITY THAT THE

OUTPUT AMPLITUDE HAS VALUES ABOVE THE UN-STABLE REGION.,

1. Formulation of' the probtem

4. 2. Asymptotic approximation of the integral expres-sian for the probability that the output amplitude has values above the unstable region.

CHAPTER 5. PROBABILITY DENSITY FOR THE DURATION OF

EX-CURSIONS OF THE 'OUTPUT ENVELOPE ABOVE THE

UNSTABLE REGION. 103 CHAPTER 3. CHAPTER 4. 43 43 49 53 53 62

pag.-,

CHAPTER 6. SOME NOTES ON TITE TRANSVERSE STABILITY OF

SHIPS IN IRREGULAR LONGITUDINAL WAVES. 140.

6.1. The dynamicalLstabiiity of a ship in an irregular

foilôwing sea. 140

6. 2. Rofling, in irregular longitudinal waves. 148

6.3. Numerlcai example 152 APPENDICES. . 159 REFERENCES. . . . 192 SUMMARY. 194 SAMENVATTING. . 195' ACKNÓWEEDGEMENT. . . 196

CHAPTER 1

INTRODUCTION

In 1746 Pierre Bouguer expôuñded In his book !'Tralté. du Navire" a method

to calculate.the position of the metacenterof a ship before itwas launched*).

Earlier, already in the first part of the 17th century, the Dutchman_Simon

Stevin introduced the concept of: metacenter but, at that moment,: _nopractical

calculations were possible with respect to the position of this point.

The metacenter M is defined for asymmetrical ship in the upright_position, as the intersection point of the centerline plane with the vertical through the center of boyancy Bd when the ship has an infiniteslìnallysmaU heeling _angle d (Fig

i i )

_{This situation is considered in still water}

Fig. i.

The stability moment M8t for small heeling angles can be expressed In terms

of the metacentric height'M, which is the distance between M and the center

of gravity G:

5tPGGM8u11

PGM'I

P represents the weight of the ship, which Is equal to the weight V of the volume y of displaced fluid with specific gravity 'y. For a stable situation it is necessary that M is positioned above G. In this case the length Is defined as. -positive.

For a long time after this publication of Bouguer, the naval architects supposed that a positive metacentric height uf a certain degree was adequate to guarantee a safe operation under all conditions at sea. They did not realize

that the metacentric height does not define, the stability moment at finite angles

of heel; also the metacentric height can vary in certain wave conditions, such as encountered in following seas. With respect to the first mentioned item a serious discordarose in the British Admiralty in the 1860's between Navy of-ficers on one side and the director of the Naval Design Office, Sir Reed on

the other. Sir Reed, upholded theright conception that not the initial stability moment, I. e., the moment for small an l'es ofheel, which according to (1. 1)

is completely determined by the value of but that the magnitudes of the stability moment over a whole range ofheeling angles are. decisive for the safety with respect to capsizing. He argued, that for the values of these moments

the value of ÏÏ is not providing sufficient information. He stressed the

impor-tance of a high freeboard, which, according to him, both extends the range of heel for which a positive righting moment persists an4 also increases the

maximum value of the righting moment. The Navy Officers, however, under the

leadership of Captain Coles, were faithfull supporters of the existing conception

that only the valúe of the metacentric height Ï is important for the transverse

stability. Both parties stuck to their opinions very stubbornly and the only way

out of this deadlock appeared to be the building of tWo ships: one, the "Monarch",

according to the design of Sir Reed, while the other, the "Captain", was con-structed in correspondence with the views of Captain Coles. Both ships had the

same metacentric height and approximately the same main dimensions. However, the "Monarch" had a freeboard, which was more than twice the one of the

"Cap-tain". In 1870, under heavy weather circumstances, the "Captain", with Cap-. tain Coles commanding, capsized aid was lost iith all hands on board, While

Ldnàrch!',

'hich'wa sailìg at ónl

a'fè

1ldadiste

dd dut nfe1y.

'This tragic lOss forced reater àttention to the dynamical aspects of transverse stability which, finally, resulted into the introduction of the con-cept of the dynamical stability of a ship, which can be defined as the amount of nergy, Which is available to resist, starting from the position of equilibrium, any external heeling energy to any inclined posItion We shall consider this quantity more closely.

Analogous to the metacenter M for the upright position, the metacenter M for some finite angle of heel' l Is defined as the point of intersection be-tween the lines through the centers of boyancy B

_{and Bd perpendicular to}

the waterlines W and (Fig.. 1. 2.). The index _{+.dl refers to quantities}

which relate to the angle' of heel i +di, where d Is, an infinitesimally small'

increase of i'.

i

The stability moment M8t for the heeling angle I' Is given by:

M5t PGZ (1.2)

Where is the righting arm for the angle . Plotting the" rÏgh'ting arm against the angle of heel' yields the well-known curve of static stability, (Fig. _{1.' 3.).} comparing the formuias(1. 1) and (1. 2), we obtain:

or

=M

when

=o

(1.3)

According to the lasí formula, we cän construct Mfrom the curve of static stability, (Fig. 1. 3. )'. Since fòrmula (i. i) is only valid for ships in a xiearly

upright poeltion, ?IMis called the initIai metacentric 'height. and Mt the -initiai

stability moment.

Fig: 1.3

The dynamical stability óf a ship, D5t, at a given inclination is defined as the energy, applied in-heeling the shop to that angle with infinitely small

velocity.

-So:

i

= S hl

d' =

i d'

(1.4)

P_

*'

We notice, that the 1ntegral GZ d represents the areabelow the curve of static stability in the interval (ò, ). It is further seén that, essentially; the

value of ii has no influence on the value of D5t. This fact was Very well

under-stood by Sir Reed and it was the fatal mistake of Captain Coles to overlook lt.

We observe, that the dynamical stability is completely dètermlnedby the

heeling moments for varying angles of inclination. -These moments refer to

the statical situation in still water and,, .therefore, the term "dynanical stability" Is rather deceptive. Since the quantity relates to the statics of the problem the

dynamics is only included in the name.

Already W. Froude and-Sir Reed recognizec, that under extreme conditions

the problem of the stability of a shipin a seaway shouid.be considered from the point of view of dynamics rather than statics. This is clear from the fact, that for a ship in waves the righting arms will vary continuously. Therefore, fOr a ship In an irregular sea, the lengths of these arms are random functions, the statistical properties of these functions being determined by:the- statistical characteristIcs of the seaway and the geometry of the ship.

In order to be able to predict the safety of -a ship with respect to its

trans-verse stabiUty in an analytical way, we should-have sufficient knowledge about

both the behaviour of a ship in an irregular sea and the external moments which might work on a ship, e g., duO to waves, wind, etc However1 since up to the present both the behavIour of a ship undér xtreme conditions and

sufficient knowledge about the nature of the sea on various ship routes are still lacking, the present standards of stébility re to a high degree based on a statistical investigation of the experiences, which' have beeñ- éncountered by

ships in the past This investigatioñ haé been initiáted by J. Rahola ("The

Judging of the Stability of Ships and the Determination of thé Minimum -Amount

of Stability", Helsinki1 1-939). -.

By analysing the transverse stability of both a great -numbér of casualties in-volving capsizings and a great number-Of vesséls, which operated in' a satis-factory way, he devised a histogram- for the righting arms of the stability moments, (Fig. 1-. 4.), which yielded the curve of the minimum amount of

static stability for safety at sea in undamaged condition.

-This method has' proved to be rather successfully. - - -.

OZ 0.2 m 01m '7 03m 0° - 15° 20°. _30°

FROM ;THE JUDGING OF THE STABILITY OF SHIPS -Sufficient values fär 2 AND THE DETERMINATION OF THE MINIMUM AMOUNT Critical

OF STABILITY -. BY J. RAMOLA - Insufficient

-Fig; 1.4 - -- -

-By analysing log-books- concerning dangerous situations of ships at sea

due to phenomena where the stability of the ship is-involved, _{itis-conéluded}

- that some pecial circumstances can -be indicated, which are in particular

beam waves of when it is in following, seas with 'a speed which is about equal to

the celerity of the dominant waves. In this thesis we will consider these two

cases more closòly. .. .- ' .. ' . ,,

Although the running of ships in beam seas has numerous disadvantages and, therefore, this situation will be. avoided, when resonance phenomena are

causing excessive rolling motions, there are occasions when this is necessary, e.g., In shallow restricted waters or when the ship has lost,propulsionpower

or rudder control. This situation also occurs when the controllability of a ship,

which is attempting to run, downsea, is overpowered by the wave action,

where-upon lt swings broadside, beamto the sea. This phenomenon is known as broaching.

A quite hazardoüs manner of rolling, which may occur to a ship in beam seas, is known as lurching, which.may be described asa staggeringmotion,

characterized by sudden high amplitudes and high accelerations. The predica-ment, which may arise,for ' ship in 'beam seas is very well described by the following twp passages concerning the famous steamship "Great Eastern",

quoted from D. B. Taylor's book "Steam Conquers the: Atlantic" and from For-. wood's "Reminiscences of a Liverpool Shipowner".

"The next voyage was disastrous. Leaving 'Liverpool on September 10, (1861),,

commanded by James Walker, she ran into a gale when three days out. . . On

the following day, a dull' grey one, awindof Increasing velocity veered to the

north4 bringing a heavy sea abeam and causing the vessel to slow down. When

one of thelife-boatsgot kose, orders were given to steer intothe' wind so that

lt might be cut away; butthe helm' refused to answer".;

"The big ship began to lurch and roll, heavily, taking heavy spray over-all.

Some of the movements were significant -she. hung when thrown over by a sea,

and recovered very slowly (..). The greàt.ship féll:lnto' the through of the sea

and became unmanageablé, lurching and rolling heavily and deeply . . .

Obviously, it may 'be very difficult to bring a ship back head into the sear once it là in a beam sea. '

Itwill be shown in the next chapter, .that the rolling motion In. a beam sea

+ 2k 3 -'--a + b-.3 1(t) (1.5)

where 1(t) is the rolling moment function, which is random In _{an irregular sea.}

It will be fùrthershown that.the relation between the waves and the-wave

ex-citing moments is linear.. Consequently, since the elevations of the sea surface are normally dietrlbuted the rolling moment has alBo that distrlbution.

In case the Input-function Isharmonic, we obtain

+ 2k 3 + a + b = R cos (ut 1-. 0) (1.6)

This equation is known as the Duffing equation and has been thoroughly

investi-gated, (see e.g. [3, 4, 5,6:1). One of the remarkable properties of this, equation

is .the occurrence f discontinuities. in the input-output relation. In the ampli-tude-frequency diagrams of -above oscillator this property manifests ItseLf_{as a} jump in the output amplitude when the amplitude and/pr the frequency of the

input passes through some critical value In systems, which_{are governed by} (1.6), this phenomenon is, observed as a-sudden Increase or decreaseln

ampli-tude, which corresponds with -the. descriptions of the lurching motlonin log-books.

With respect to the dynamical phenomena concerning the_tranverses

stabili-ty of a ship in a following sea, it is reported, that ships exhibited_sometimes signs of a temporary loss of tranverse stability when the ship_{speed was about} the same as the wave celerity. These phenomena are in particular observed

for fast container ships with a low prismatic coefficient. -It happens that, In

case the ship has at some moment an angle of Inclination due' to an external

-moment, it remains- for a long while in that position and appearsnot to be In a hurry to restore the upright position. Only, by making.a' drastic -course change,

a quick restoration of the uprlgth position can be effected.

-It will be shown in chapter 6, that these phenomena are essentially -due to

a change in the transverse stability of the ship. It has been shown by B. Arndt and S. Roden, [7], that in a harmonic following_{wave the curve of righting arms}

varies periodically between two extreme positions w-ith a frequency equal- to the

frequency of encounter. These extreme positions are determined by the geometry of the ship the wave length and -the -wave height. Further, for every wave height,

the amplitude of oscillation of the stability_{curve has Its Largest value when the}

wave length is about equal to the ship length, attaining Its lowest value when

re-gion. For Increasing or decreasing válues of the wave length, the Influence of the waves on the transverse stability will diminish. If now the lower limit of

the curve of righting arms' Is below the minimum prescribed'stabilitY curve;

then, during each period of encounter, there is a- time Interval Inwhich the

stability of the ship does not fulfil the minimum requirements of transverse stability. This time intervAl will become longer when the ship speed is varied

in such a way that the frequency of encounter 'has a smaller value.

In connection with this Issue two papers of Grim [s, 9] should be mentioned.

Starting from the fact, that the largest change in the transverse stability occurs

when the wave length is equal to the ship length, he Investigated in which mea-sure 'the random following sea on the sides 'of the ship can be approximated by a regular wave with aiwâve length 'eqùal to the ship 'length [8 ] .'- Then' the am-plItude of thiS so-called effective wave Is a random time function. The-power

spectrum of'this function-is used-for the determination-of statisticalquantities with respect to the amplitude of' this wave; whIch give an insight into the-changes of the transverse Stability in' that particular random sea. In -his other paper

[9]

Grim points out, 'that the periodic 'changing of the transverse stability in a

regu-lar longitudinal wave can' give -rise to a spontanéous rolling motion.- The initial

metacenter height M is assumed to vary harmonically about some average value MØ with amplitude Ai and,frequencyw*), equal to the frequency of

encounter: ' '

-+ 'A GM' cos wt ' -' (1'. 7)

Consequently-, according to' (1.1-),- the initial stabilIty moment isa harmonic functiOn. Neglecting. the damping, ,the following equation is obtained forthe

rolling' motion' in a' longltudi'nal sea:

I +'yV

+ AG cos'wt) =o -

-'

(1.8)

'TMs'equatioá'is known as Math,Ieu'-s- differential equation, which can be written -in the-following canonical form:' ' - .- '

+ (a - 2q cos 2t) ' o - ' '(1. 9)

In the literâture this equation has been studied very extensively and the stable and unstable regions in the a, q-plane of the solutions of this equation have been determined. Of course, these regions reverto stationary ituations, where w

has a constant value.

In the literature about nonlinear systems with a stochastic Input, various methods are known to characterize the statistical properties of theoutput of

the system governed by (1.5).

Ariaratnam E 10] shows, that, when 1(t) represents a stationary Gaussian, white random process with zero mean,, the joint probability density p(0,3) Is

governed by the Fokker-Planck equation, which in this case has the form:

-?-

-

.-fr

(3P) +

¿-[(2k

+ a +

b3)p)

.= o (1.10)

where W0 is the white spectrum density of the excitation. The solution of this equation as obtained by Chuang and Käzda [ii] is expressed by:

C exp [ (42 +

a2

+ ,

b) I

Caughey [12] extended the method of equivalent linearization of Krylóff

and Bogolluboff [13] to nonlinear systems with random input. The nonlinear

equatIon (1. 5) is replaóed by a linear equation:

+ 2k $ + a = 1(t) (1.12)

The equivalent coefficients k and a are determined in such a way that the eq. eq

mean suare of the error term:

e(4,3) = 2 (k - k 3

(a-a)

+

b3

(1. 13)

has a minimum.

Crandall [14] considered the case, that the parameter b in (1. 5) is so small, that the nonlinear term in the restoring force Is small In comparison with the linear term. He obtained expressions for the statistical quantities of

the output by. applying the classical perturbation method, where a solution of

(t) + b-1-(t) 'i-b2 2(t) + -- (1. 14)

In the -method, which orginate8 from Norbert Wiener and which has been

adapted to all kinds of nonlinear problems, (see, e. g., Deutsch [15], van Trees [16], N. Wiener [17], Kuznetsov, Stratono-lch and Tlkhonov [18]),

the output function (t) is represented in terms of functionals which Involve the inputfûnctionl(t) andfuñctions h(T1, T2, Ta)', which are often óa[led

the Volterra Kernels of the nonlinear system:

. - . .

(t)=

_{j' h1t1(t -T)dT+ f}

_j' h(T, T2). I(t r) I(-t - T2) dT1, dT2 +

-+

f.. .j' -h(r1...,T) I(t - T1.)

(t- T)d'r1.,. .. dr+

(1.15) For a linear system only the first term of the series is maintained, This term

represents the well-known convolution integrai in which -h.1(T) is the Impulse

response function. Consequently, for -a nonlinear system, this -series may be

considered as .a generalization- of the linear convolution theorem. -When the

ker-nels h('r1, T2,- T).of a system are known then-the statistical properties

of thé output can be determined. .

-Although these methods are very-useful- for the determination of the

-sta-tistical properties of the output-signal, they are not suited' for'finding the statis-tical quantities, which relate to the discontInuous behaviour of the system,

governed by equation -(1.5).

-In this- thesis some statistical characteristics concerning these nonlinear phenomena are derived by using methods which.are essentially ali based on the Fourier series method of Rice, (see Rice [19 ] or Middleton [20]).

We will assume in our calculations, that the occurrence- of jumps in the Output

amplitude is- a rare event, i. e., lurching of the ship is a rather uncommon phenomenon. We further assume, that the wave exciting moment has a

rela-,tively narrow spectrum:

A-v- « :v_. . - . (1.16)

where is the characteristic bandwidth and Vm a representative midband frequency, (Fig. 1. 5.).

w(v)

Fig. 1.5.

The input of equation (L 5) is then a quasi-harrnonicsignal, which can be r. presented In the form:

1(t) R(t) cos _mt + 9(t))

where R(t) and e (t) are random function, vhich vary slowly as'tcompared with

cosw_mt.

For this reason we will apply the quasi-stationary approximation method for the determinationof statistical quantities with respect to. the jump phenom-ena in theoutput. Consequently,, We assuthe, that the output. attime tis only determined by the values, which the instantaneous.arnpltude R(t) and the

instan-tdneous frequency wm é(t). have atthat very moment. .

The quasi-stationary method will also be appUed for caiculationsith...

respect to the unstable phenomena of a ship in longitudinal waves. .. -.

V,,, V

CHAPTER 2

PRELIMINARY RESULTS

In this chapter we discuss some subjects which are needed for our

in-vestigations In the following chapters.

2.1. ROLLING IN BEAM SEAS

Weconsider a vessel' in a regular wave with its plane.ôf symmetry directed parallel to the 'crests of the approaóhing waves For the present we assume

that the transversO dimensional of The vessel are very small in comparison with the length of the wave. In addition'we assume the height of the wave tO be"very

small. 'For this reason, in 'the first orderapproximation, we may coñsider the moment, whiáh is expériencethby the vessel due to the presence of 'the fluid, to be the sum of the moment due to the motion of' the 'vessel In still water andthe

moment, which the vessel experiences in waves, when it is restrained in the

upright position.

1n this thesis only the case of pure roiling Is condklçred, i. e., the influence

of the heaving and swaying motion on the rolling motion is neglected.

The rolling moment on a vessel, executing in still water equivolumetric, transverse inclinations from the equilibrium position may be considered as the

sum of a hydroètatic moment and a moment, which is due 'to the hydrodynamic -reactive forces of the fluid on the vessel.

The hydrostatic moment is equal to the negative value of the stability

moment, since it tends 'to' decrease the value of the heeling 'angle:

Fig. 2..1

P is the weight of the vessel and ii the Initial metacentric height.

The hydrodynamic part of the moment consists of two components, one Is

proportional to the velocity, the other to the acceleration of the motion of thè vessel:

M = -N . (2.1.2)

(2.1.3)

The coefficients of proportionality N and m represent, respectively, the

damping and the added moment of inertia of themotion.

The total moirent, experienced by the vessel, oscillating In still water,

becomes: . .

M =-m

_st

-N4-PGMI

-2TTz/X.

p=p+pgz-pgl)e

(2.1.4)

Consider now the vessel restrained in upright position In a regular progressive

beam wave with frequency '», represented by: . . . . .

= CCOS (Icy - w t) . .. (2.1.5)

2n

where k = is the wave number and X the wave length.

The pressure distribution in suäha wave is given by:

(2A.6)

where

con-eider the wave length to be large In comparison with thé beam of the ship. Then the wave surface may be taken plane for an area with the dimensions of the water-line areai (Fig 2. 1. 2). Further, assuming the Froudé-Kryloff hypo-thesis to be valid, It is seen from (2. 1.6), that for a ship with a draft, which Is. very small In comparison withthe wave length, the pressure distribution

on thé hull of the ship can be taken hydrostatic..

FiÙ. 21.2

The Qctual regular wave motion of thefluid Is' now .replaced'by the oscillatory

movement o the free surface according to:'

= c cos (wt. + 1» (2.1.7)

where Is the' maxlmum»angle of 'the actuâl' slope of the Wave surface. which,,

according to (2.1.5), is foundto'be:.'

2rï

=1cC a

o a X (2.1.8)

The moments' experienced by the vessel, are easily found by imagining the

mo-tion to be reversed, i. e., one considers the water level fixed and the vessel

rolling according to

_-

or, cos (wt + 4). The exciting moment is then found to

be:

Me m & + N,& + P GM (2.1.9)

Then, on the basis of d'Alembert's principle:

We find for the equation of motion:

+N3 vPl =P

+N&.+m'

(2.1.11) in which I I + m where I Is the moment of inertia of the vessel itself.

When the. draft of the ship is not very small In comparisoñ with the wave length,, then the hydrostatic approximation of the pressure distributiOn around

the ship is notaccurate enough anymore.

We shall consider now the hydrodynamic boyant force for thecase that. the

draft of the ship is not very small as compared with the wave length. However, the. assumption, that the beam is very small as compared with the wave length

Is. maintained. . .

We consider the time instant, that the .point of inflection ofthe wave j situated in the origin of the coordinates Oxyz, attached to the vessel, so that the starboard side is submerged, (see FIg. 2.1.3.), The wave profile at this moment can be written in the formt

sin ky (2.1.12)

or, since ky is small within the limits of the vessel's beam.

_kÇy = -

(2. 1. 13)

where the last equality is obtained by applying ( 2. 1.8).

Fig. 2.1.3

Substituting (2.1.13) in (2. 1.6), we findfor the pressure distribution on the hull

-24

T

_dl

r -kz

=-Pg0 j e

_dz

o

we find for the total' moment on the ship:

M = a-pg

L

STd

y2

tgß = -gpg a0

e

2dy

where I, is the moment of inertia of -the water-line areaat the depth.z abàut. its line of intersection with the symmetry plane of the vessel. Expression

(2. 1. 17-) represents the hydrodynamic boyant moment when the draft of the

vessel is not very small as compared with the wave length. Representing this

moment in the form:

M=PgVBMxaØ

- (2.i.18)

and comparing this to '(2.1. 17), we find for the coefficient XT the expression:

dû

_{- cos}

dxdz dz- (2.1.17) T

dI

i

r _e-kz xz dz

xo

(2. 1. 16) (2. 1-. 19)

where I, = V BM represents the moment of Inertia of-the waterline-area, y the

+ pgz + pg a0 ye

(2.1. 14)

-in-which the function y y(z) represents the contour of the ship.

Consider t*o equal surfacê elements dcr and d on the hull of the ship, which have -symmetrIc positions--with respect to the z-axis. Restricting oùr calculations to the-moments due -to. the vertical components of the pressure force on the'-hull-,. we obtain for the resulting moment due to the pressure force on the

+

-elements dû and dû

dIvi = 2pg a0-y2e1 sin dû - (2.-1.15)

where is the angle between the positive normal on- the hull of the ship and the

displaced volume.of. water and the distance from the center of boyancy B in

the upright posltion to the initial metacenter M.

Comparing expression (2. 1. 18) for the moment of the hydrodynamic boyant force to the moment of the hydrostatic boyant force, i. e., to the moment of'

the form stability, given by:

M0 = pg V BM (2. 1.20)

we see, that in' (2. 1. 18) the actual Wave slope a0is replaced by areduced wave

slope:

(2.1.21)'

So,. finally, for awave length, which Is not iery large in comparison with the draft of the vessel, we should replace in equation (2. 1. 11) the term P I1a, which refers to the hydrostatic component, the actual wave slope by Its reduced

value XT.4, a. In an analogous way the velocity & and the acceleration should be replaced by thé corrésponding reduced values & and x ?, where the correction coefficients and x, take the hydrodynamlc pressüre distribution in the wave into account. Then the equation of motion (2.1. 11) be-comes

+ N3 + PÏ 4'

P XrCY + N x & + m4, x, ¿ (2.1.22)

or

4-N6 + PÏYÏ 4, Mg.cos (wt +8) . (2.1.23)

where the wave, exciting moment Is given by

Ì4gC0s((al+.,0)=PGMXTa+NX& +m4,x4,d

(2. 1. 24) For large rolling amplitudes the boyant force will not be proportional to ?i anymorè.. It i's usual practice In naval architecture to replace aimply the factor - 4 by the. function l(.'l'), representing the righting arm of thé boyant

force. This function, which is' an odd one, is represented by the following power series:

l()

e1 i + C3 + 05 +

....

(2. I. 25) We will restrict in our calculations the power.serles to the first two terms. After: divldling (2. 1. 23) by I , the equation for pure rolÍing obtains the following form:

+ 2k

+al

-i- b3=.Rcos('wt+O)

(2. 1. 26)

It will be. seen in the fòllowln section, that the nonlinear term In the restoring

force of equation (2. 1. 26) is responsible for the occurrence of Jump phenomena. In the response curve of a system, governed by (2. 1. 26), these phenomena manifest Itselves as jumps. In the outputamplitude.

Finally, lt is easily derived from.(2. 1. 5), (2. 1'. 7), (2. 1. 8) and (2. 1. 24),

that the relatlon.between the wave exciting moment. and the incoming wave is

linear. Therefore an Irregular sea with a Gaussian distribution function

gener-:ates a wave exciting moment function, which has also.a normal distribution.

2.2. HARMONIC' SOLUTION OF THE DUFFING EQUATION

Ww consider the differential equation

+ 2k + a + b =

Rcos (ot+ e)

(2. 2. 1) which Is often called Duffing's equation since the first results concerning the solution of this equation were obtained by Duffing [21]. Hi's results together with results from other authors can, for example, be found: In [3, 4, 5, 6]

Except fór a harmonic oscillation, which we dénote by F1 cas wt+ F2 sinWL

the output can also contain a subharmonic oscillation F113 cOs' wt. It Is shown in [4; ch. IV], that a subharmonic oscillation can only occur when the

parameters of equatIon (2. 2. 1) satIsfy the inequality:

w2

9 [a

+(21bR2/1024 a2)1/[i-k2a2 [2o48/9bR2)- (1/2 a3))] (2. 2. 2) if b ' O and k O are not too 'large. This relation Is derived by. substituting asa first approxlmationofthesolutlon'of (2.2.1) the following exprésslon:

'I= F1 Cos ut + F2 sin wt + F1,,3 cosi wt _(2.2.3)

Equating the coefficients of cos wt, sin wt, cos w t and sin - wt we obtain four relations for the unknown variables F., F2, F113 and e. The quantity

O is here also an unknown variable since the phase angle of the subharmonic is taken zero. Further, the frequency w in these equations is found In the

corn-binations w2 and w. Eliminating now F1, F2 and iw from these four relations, we obtain a quadratic expression in the variable F1,,3 and the restriction, that F1,3 can only have real values yields above inequality.

We assume now that the system parameters and the excIting force are

such that the solution of (2. 2. 1) contains only the harmonic solútion, which we give the shape

F cos ut,.

For the determination of the amplitude-frequency relation for (2. 2.1)we sub-stutite 4= F cos w t. in, the left- hand side of (2. 2. 1). Neglecting the term In

cos 3wt which appears in the factorcos3 ut = cos 3 at cos u1.and equa-ting coefficients of COB ut and sin ut on each side of the resulequa-ting equation, we

obtain the two equations:

(a- w2i--bF2)F=RcosO

(2.2.4)

2wkF=R sin e

Except for the quantity F, the phase angle e Is in these quatlons also an unknown variable, since the phase of the harmonic solution Is taken zero. By

squaring and adding these relations we obtain the desired amplitude-frequency relation:

[(a- w2+-bF2)2+4 w2k2]F2=

(2.2.5)

The amplitude-frequency curves, which are represented' In Fig.; 2. 2; 1.,, are

obtained by plotting the relationship (2. 2. 5) foE w> o wheie we assigneda

po-sitive alue to the parameter b.

In this thesis we shall, derive all results for the case b> O. For b < O the calculations can be carried out along the samelines.

-IFil

Fig. 2.2.1

In this case the amplitude-frequency curves

óf to thé right, as shown in Fig. 2. 2.1. We curves areonly deflned'for w >0.

We obtain an expression for the lihes F2(w).

the amplitude-frequency curves are vertical respectto F and putting = o.

This results In the equation:

are leaned over tò the left Instead

ñotice that the amplitude-frequency

and F1(w) on which the tangents of by differentiating (2.. 2.5) with

a)2- 3k2 2 11/2]1/2

a)2 -3 k2w2}1/2]12

From these expressions we see, that the amplitude-fréquency curves have (2. 2.8)

- b2F + 3bF2 (a- w2) (a2 + w4- 2aw2 + k2w2) = o (2.2.6)

with solutions:

F

8 (2_

a) t _91b1 (w - a)2- 3k2 w2 1/2

(2. 2. 7

= _{(wa- a)+}

_9b

(2_

a)-

3k2w2)h/2

Since We considér only thé absolute values of F we obtain as fiñal result:

= [(w -a)±-j--

r8

2 8

((w -

2

((w2-only. vertical tangents when w satisflés the Inequality:

Thus for some arbitrary frequency w ₌ which satisfies the inequality

(2. 2. 9), the tangeñts of the amplitude-frequency curves. arevertical wheñ'the

absolute value of the output amplitude F satisfies the equations 1F1

= I Fi(wm)I

or JFj

_{= IF"2 (w)I' ,} where the right-hand sidé of these eqüatlòns is given by (2. 2. 8). The corresponding values for. the input amplitude, which We denote

by Rfl

'm and'Ri(Wm) (seeFig. .2.2. 1.), are found by subàtltuting succes-sively

I = Fi (um)I and FI. = F3 (W» In equatión (2.2.5) and solving

the résúlting two equations In R;

Replacing again Wm by w we find:

Í8

2 3 32 2,2 2 64

rl

R1 (w) = [ j- (w - a) w K (w- - a) - 81 bi r 8 2 3 32 2.2 2 64 R11 (w) =L 8lb (w - a) + w k

(w - a)

_811b1 We rewrite quation (2.2.5) in the form:

-- b2F6- - b (w2 -, a) F4 + ((w2 - a)2 + .2 1

iF -R =0

2, 2 (2.2.9) (2.2. 10)

-a)2- 3k22j32]'2

(2. 2. 11) -

a)'2- 3k2w2i3]1'2

- (2.2-I2)

which is an 'equation of the third degree in F,,. in which the variable R isa

-parameter. We see from FIg. 2. 2. 1., that for every w₌ wm, which satisfies (2. 2. 9) the equation (2. 2. 12) has for suitable values of R three real roòt8. We

further see,, that this equation has two equal roots F2 =F (um) for R= -R11_(Wm)

Consequently we may write (2. 2. 12) In the form:

-1-b2 [F2- F (wm))2 EF2-

_{F (um)} (2.2.13)

From this equation the third root F (um) is easily determined. Replacing

again Wm by w we obtain:'

(w2-a)2-3'k2w2

o

For the value w = w0, for which the equality sign Is valid, we find:

=

[(W2_

_{a) +9T[L}

(w2_ a)2- 3k2uj1F"2]1'2

(2. 2. 14)

Along the same lines, by substituting 'w = Wm and R = R1 (Wm) in (2.2.12), we obtain an equation, which has, a double root F2 = F (wm) and a third root

F2 = F (w), which Is given by:

r

8 2 iii r . 2 2 . 2 [1/2 Ti/2

L 9 b (w - a) - 91 bi (w .- a) - 3 k w i

j

(2. 2. 15)

The nonlinear restoring, force .a + b of the system (2.2. 1) produces the well-known jump and hysteresus phenomena. These properties are very veIl

demonstrated by considering the following two, cases:

We assume the amplitude and frequency.of the input to. have the values Ra and

wm respectively; so we start at point i on the curve R = Ra in Fig. 2. 2. 1. When

we increase R then IF I increases in a continuous way till the point 2 is reached.. A further increase of. R causes ä jump in the output.amplitude from IFi (Wm)I

to F3 (w)l ,

after which _.1,1 increases again iñ a continùous way. When,we now decrease R then a jump will occur from the point 4 to the point 5.

Above mentioned phenomena occur also when the amplitude is kept constant,

e. g., R R11 and the frequency is decreased' or iñcreased We start from the

point P on the curve 'R =R11 (wm) A decrease, of frequency causes a jump

from point 2 to point 3 and after that an increase of the frequency will cause a

jump from point 6 to point 7. 'Consequently above mentioned Jump phenomena take always place from a' point on the curve F1 (w)I to a point on the curve

F3 (w)I or from a point on the curve F2 (w) to a point on' the curve F4(w)I.

Obviously the values of Fj 'and lutbetween the curves IF1 (w)I and IF2 (w)

never occur in a stationary olutlon. This property can be seen. more clearly.

by the following agrument: Assuming

= cos w0t and 2 = 0cos u + ô i to' be two solutions of the differential equation (2. 2. 1), in which Ô is an infinitesimal quantity, then we find, that in the first order approximation Ô has to satisfy the differential

equation of Mathieu:, ' . . ' . . /

[(a +. bF2) +.- b 2 cos 2wt j ô = o (2. 1. 16) It'is weu-known, thät this equation can have solutions, which remain finite

and solutions, which tend .to infinity.

2

andthe frequency w.

It turns out that &ti tends to Infinity for those values of _{I F ( and win the}

region between the curves

Fi

_{(w)I and j '2 (w)}

For this reason these values of: F and w are said to be unstable.

We notice; that the response-curves showing the vertical jumps refer to statio-nary situations. In reality, however,, there will be a transition stage, during

which the amplitude of the output changes at a finite rate when the unstable

region Is passed

2.3. MODEL OF RANDOM PHASES FOR A GAUSSIAN PROCESS; INSTANTA-NEOUS AMPLITUDE AND FREQUENCY; QUASI-STATIONARY APPRO-MATION METHOD

In this section some characteristics of the random input signai 1(t) of the nonlinear system (1.5) will bespecified. We assume, that 1(t),. whichextends

from

=-.

to t

= ',

_{is a strictly stationary Gaussian process with a} continu-ous spectral distribution function F(v). Then 1(1) Is an ergodI process, I. e., time and ensemble averages yield the samé result (see appendix F).

Without restricting the generality of the discussion we may assùme, that the average value of the signal has zero value:

- EI(t) = o

The variance ofthé signal is obtaiñedtby putting s Ft in (E. 17):

E 12(t) =

2$

d F(v) w(y) dv (2.3.2)

where w(v)

-2

is the spectral density function, which Is definedfor

y _{o only. In this thesis we will assume that w(v) is always a continuous}

function. In virtue of the ergodicity of the process we have

T

E 12(t)

=

T-=

12(t) dt (2.3.3)

which Is the total average power of the signal.

Before Investigating various properties of the random signal 1(t), we

represent It in an approximate form, which is suitable for all kinds of analytical

Several representations of a Gaussian process are' possible 119, 20]. In

our case ft Is convenient to use the random phase mode, which in the real case has the form:

N

1(t) = E c cos (wt - Wa)' (2. 3.4) n=1

The angles cpi, cpi,. .., cp are Independent ateohastic variables with a

uni-form distribution over the range (0, 2n]. Further:

= [2 w(v) Av]h/2,

w =2r

V

and v= nAy

(2.3.5)

We observe that the model (2. 3.4) is a special case of the general serles re-presentatien (F. 18), with which we can approximate every stationary process

with any accuracy.

-The continuous spectral density w(v)of the signal 1(t) Is inthis represen-tation replaced by a discrete spectrum, which has only values for v, "2' (see FIg. 2. 3.1).

N

w()

_{= n1}

C2

ô (y - \')

(2. 3.6)

This spectrum belongs to. a signal with a finite length T = -. The average

power of the harmonic oscillation with frequency vn Is equal to w(vn) Av. This

is just the area of the rectangle, which has' its base onthe v-axis between

v_- Avandv+AvasindicatediflFlg.2.3.1.

By applying the central limit theorem It can be shown, that the

represen-tatIon (2. 3. 4) converges to a normal distributed process with averáge value zero andvarianceJ w(v) dv when the discrete spectrum (2. 3.6) approximates the corresponding continuous one. This proof is, for example, given by Rice in [19]. He shows that the distribution of the sum of the series (2. 3. 4)

ap-proaches the normal law when N and Av -. o such that N Av = F, where F

is any positive constant.

The model' (2. 3. 4) will be used to derive several results relating to the process 1(t). However one of the manipulations In these derivations will

al-ways be the limiting operation N -. and Av -. o so that the final resùlts al-ways relate to the original normal process with a continuous spectrum.

WIV)

Fig. 2.3.1

Analogous to Rice's method we choose a frequency um. which is a re-presentative midhand frequency, and write (2. 3.4) is the form:

N 1(t)

= n1 c

cos (wut - Wt -

+ wt)

I COS

Wt - I

sin Wmt (2.3.7)

where and N I =

c cos (w t -

t - q)

c n=1 n n m N I = E

c SIfl(Wtw t-p

n=1 n n m n R(t) = [12 + 12]h/2 C S (2.3.8)

It Is obvious, that In the limit N - and Av -. o the processes I(t) and 15(t)

have also a normal distribution and a continuous power spectrum and are there-fore ergodic.

Ve assume the power spectrum to be relatively narrow. Then the values

of c will decrease very fast to zero when n - ml Increases In value. Thus 1(t) and 1(t) are very slowly varying functions as compared with cos wmt. We define:

(2.3.9)

Then (2. 3. 4) can be written In the form:

1(t) = R(t) cos(s

t + e(t))

(2;3. 11)

In virttie of the relations (2.3. 9) and (2. 3. 10) the functions R(t) and e (t) are also slowly varying functlönsas compared with cos Wmt when the spectrum Is

relatively narrow. Therefore we may Interpret R(t) as the envelope of th signal 1(t). From the physical' pòlntof view this Is also clear by considering. an oscillogram of a Gaussian process with a relatively narrow spectrum, (FIg. 2.3.2.).

Fig. 2.3.2

It can be shown, that e (t) is uniformly distributed over the range (o, 21r] while the envelope R(t) has the Rayleigh' distribution

(2. 3. 12)

In which 4' represents the total average power of the sIgnal 1(t). From (2.3.11)

it Is seen, that the functions R(t) and w1(t) = wm + Ô can be Interpreted as the

instantaneous amplitude and frequency o the signal. ê(t) represents here the

tizne, derivative of the phase function O (t).

The choice of the nildband' frequency um is rather' arbitrary and It is not Immediately evident whether or not a different choice of thisfrequency leads to the same R(t) and w1(t). Also it Is not clear what becomes of our physical' concept of envelope when the process has a spectrum, which is not relatively narrow. In order to eliminate this arbitrariness we shall give a more direct

formulation of R(t) and w1(t).

By writing z(t) in the fòrm:

f Wt

i[(w = wm)t -z(t)= e n=1 c e N _{l(w t'- q :)}

z(t)= E

c e n n n=1 n

which Is the complex form of the random phase model (2. 3.4) for the process 1(t). We now define the onvelópe R(t).andthe Instantaneous frequency w(t) of the

process 1(t), respectively, as the absolute value and the time- derivative of the

argument 'of the complex function z(t)'. . Thus representing z(t by.

z(t) = _z(t)I

elt),

we obtain R(t) _{= I} w,(t) (2.3. 13) (2.3.14) '(2. 3, 15) (23. 16)

it can be shown, that the definitions (2. 3. 15) give the same results a the :earllèr

given definitions due to Rice, for any choice of themldband frequency., The définitions (2.3. 1:5) are a special case óf the definitioñs for the en-velope and Instantaneous frequency as given by Dugundji L 15, 22; '23] for an...

arbitrary élgnal 1(t). He started from a complex valiedfunction

z(t) = 1(t) + 1 1(t), (2.3. 17)

which he called the pre-envelopeof the waveform 1(t). The function 1(t) repre-señts :the Hilbert transform [.24] of 'the function: I(t).given by 'the principal

value of the' integral:. ' ' .

. .:

I(t)=.

_$ d (2. 3. 18)

The envelope and instaneousfróquency.areobtainedby substituting expression

-So in the general case:

R(t) = [12(t) + Î2(t)]1/2

(2. 3. 19) (

w1(t) = -- arctg _{. 1(t)}

The stochastic processes R(t) and w1(t) are defined oñ the process 1(t). Conse-quently, according to F. 3., if 1(t) Is a strictly stationary and ergodic process,

then so are the processes R(t) and w1(t).

- i(wt- cpa)

We note, that the function z(t) E e- e is the pre-envelope

n=1 n

of 1(t) = E1 c cos (Wnt

P) since sin

nt - w) is the Hilbert transform of

COB (wt- wn

In section 2. 2. we saw, that the stationary relation between the input and

the output variables displ'ay Jumps for those values of the amplitude and fre-quency of the Input, which satisfy the equations R = R1 (w) and R = R11 (w), where

the right-hand sides of these equations are given 'by (2. 2. 10). Consequently we

may expect, that under certain conditions the output.of the system (1.5), which has an Input 1(t) = R(t) coB (wmt + ,(t)), which is both amplitude and frequency modulated, will also exhibit discontinuous iihenomena In 'its amplitude. We now

make the decisive step'.by stating, that jumps will only occur for those values

--of the Instantaneous amplitude R(t) and,frequency w(t) =: + ê', ,which satisfy the equation R= R1 (wi) and.R

We come to uch'a staternònt 'by -assuming that. at time t = to, when the Input function has' the -instantaneous frequency 'W:i = .wm + O (t0) andinstantaneous amplitude R(t0). the harmonic output function is determined as if the system is excited by the harmoaic signal

R(t0) cos [(w + ê(t0))t + P

-- m

and may therefore be written as F(t0)'cos [.(wm+ ê(t0)) t 'j 'where the'thn-plitude F Is calculated by substituting R R(t0) and w = W + é(t0) in '(2. 2. 5).

Thus, In this so-called quasi-stationary method, which is applied for the

determination of the discontinuous points in the output amplitude' the output at time t = to Is assumed to depend only on the input át time t = to and not on the

time 'history of the input signal as is the case'ln reality. From an-Intuitive point of view It is easily understood, that this approximation will be'more

accurate when R(t)' and O (t) are more slowly varying functions as compared

with COS Wmt . . -,

it will: be showil now, for: a special case, that the rate of amplitude and frequency 'modulationof a Gaussian process with a relatively narrow bandwidth is easily derived from the probability density p(R ](, Ó) given by. (A. 14)

p(R,,*, _{,='. exp [-} --fb2R2 + b0(t2 + R2Ô2)

-

2b1R2 Ô i] (2.3.20)

Assume,, that the power spectrum has the form::

.4r.

)2/2a2

w(v)

-

e (2. 3. 21)

Here the: characteristic bandwidth and the midband frequency are given by c

and v, respectively. The parameter (r is again the-total average power of

the signal. Since we consider only signals with a relatively narrow bandwidth, i. e., «1, we may replace in'(A. 3) the integration -with respect to v over the rangé (o,, =) by an Integration over (- , -o'), thn we obtaln i

3n 2n-1/2

b

=2

ir

2n

b_2n-1 =0

We substitute (2.3.. 22) in (2.3. 20) and obtain::

R2 t R2 p(R,i3O)

= 8n2

exp L r(n+1/2) ;lr

n=-0,1,2...

(2.322)

n = 1,2...

It will be seen In the following chapters, that we impose for our approxi-mations there the additional ôonditiönR/4,/'2

» i,

i. e., we consider only peaks of the envelope, which are at high levels as compared with the average

power of 'the signal. -' -.

Henceforth we take _' 1, which reduces (2. 3. 23) to:

p(R R 8) R2

2 exp [

R: 1

2 (R2 + R28 2)1 (2 3 24)

8n o

. 2(2rra)

vhile the above mentioned condition for R is reduced to R» i. Consequently

l. =C(2na.) and Ô' =ìO:(2:TToR'). Thus,, in sense of the probability theory, ac-cording as the instantaneous amplitude of the signal has a larger value, the

value of the instantaneous frequency w + e will differ less from the mldband frequency Wm Consequently, for large values of R the frequency modulation is of a higher order than the amplitude modulation. This Is an important fea-ture for our further calculations.

Above property has been demonstrated for the power spectrum given by

n (2.3.21). It is however easily seen, that on account of the factor (y

-(A. 3) the coefficients b/2B and b1/2B in the exponent of (2. 3. 30) can be made

just as large as we want for every type of power spectrum by taking the

charac-teristic bandwidth of the spectrum smali,enough.

2.4. THRESHOLD CROSSING PROBABILITY

Let f = 1(w1) be some continuous and differentiable function of the instan-taneous frequency w1= W + Ö Sinde wm Is a constant, we consider f as a'

function of Ô 'and hence of t. We assume, that' the random envelope R(t) of the

Gaussian process intersects the graph of the function f(t) in some point t+ T

in the time interval (t, t + dt), where the function 1(t) and 'R(t) satisfy the

re-lation:

dR df

>

-ai

We further assume, that the length dt of the Interval is so' small, that In this interval the functions 1(t) and R(t) can be considered as llnear'wlth probability

almost one.

In the point of intersection we have:

f (t'-i- T), = 1(t) -i- fT

R(t+T) = R(t)+RT

where o <T <dt. Solving for T, we find:

T

f(t) - R(t)

which, since we assumed T to be in the inÑrval (t, t + dt), yields

(2.4.1)

(2.42)

while (2.4.4) càn be written in the from:

df..

f+---edt- Rdt'< R <f

d8

f+tdt-dt

< R < f (2.4.4) Sincö f =

- , we obtain from (2.4. i);

ft > -' ' '(2.4.5)

Assume, that the simultaneous 'probability distribution oíthevariables R, it, Ô 'and e Is represeñtedbyp(R,*, 'Ô, 8). Then theprobability, that'the envelope R(t) 'of the Gaussian process' '

1(t) = _{R(t) cos f Wmt}+ 8(t) 1

intersects the' functiòn f(t) ¡n the interval (t, 't+ dt) 'on the condition that (2.4. i)

Is valid, while at the same time 'the instantaneoüs freqùency of 1(t) has'a value

in the Interval

(W+

Ó W Ô + dÓ), is given by:'

't, f'

o $d

$dÑ dR p, it, 0, Ó) (2.4.7)

0

dO

-de

The 'integration intervals for R' and k result from the inequalities (2. 4. 6)and

'(2. 4. 5), respectIvely. " '

Since --

dt - t dt is very small in comparison with'f, We put lñ the

inte-grandR= f.

Integrating (2.4.7) with respectto R, we obtain:

t'

t

dtdO

$ dÒ (2.4.8)

dO.

When we assume, that the envelope R(t) intersects 'the function f(t) on' the

condition 'that, instead of ('2.4. '1),' the lnequality

dR < df

'dt

dt

is satisfied, then it cañ be shown, that the inequality (2.4. 6) has to be replaced by:

df.

f <R < f+--Òdt-Rdt

Condition (2. 4. 9) may be written in the form:

df..

:Êt <- e

The expression for the probability, which is similar to (2. 4.8) except for the

condition (2. 4. 1), whIch Is now replaced by (2. 4. 9), can be derive4 in the same

way andis givenby:

df..

ã9

-.dtdô $ d

$ dñ (Ñ.----) p(R=f(Ô),Ô ,ft,e)

(2.4.12) In case the function f(t) Is a constant: f = c then, by performiùg.an ad-ditional integration with respect to 9 over the range (- o), expression

(2;4. 8) can be reduced to:

dt

j Ü p(R=c, ñ)dâ

(2.4.13)

- o

which is Rice's resultfor the probability, that the envelope R(t) passes in the time interval (t, t + dt) through the constant level 1= e with apositive slope. In a similar way we obtain from (2.4.12) for the probability, that R(t) passes

in (t, t + dt) through the level f = c with a negative slope:

-dt j ft p(R=c, t)dÑ

(2.4.14)

According to (F. 34) the integrals of the expressions (2. 4. 13) and (2. 4. 14)

árè the intensities of the stream of upward and downward crossings, respecti-vely. Since the 1(t)-process Is strictly stationary and ergodic, itfolLows from

F. 3., that sols the envelope process R(t).

Further, R(t) has continuous sample functions and a continuous one-dimensional

distribution function. Therefore, in virtue of the diecüssions In F. 4., above (2.4.9)

(2.4. 10)

(2.4.11)

mentionedintensities are equal to thecorresponding average number of

cross-ing per second. Denotcross-ing the average number per second of, upward and down-c

ward crossing by N and Nd, respectively, we obtain:

N=

f°Ñp(R= c,Ñ)dÑ

(2.4. 15)

Nd

H

p(R= c, ñ) dÑ

Combining the arguments, which yielded the formulas (2. 4. 7) and (2. 4. Ï2)

we can derive an expression for the simultaneous probability, that R(t) passes

through the function f(t) in the time interval (t1, + dt1) and alter that through

the function g(t) in the time interval (t2, t2 dt2) on the condition, that in the

first Intersection point the slopes of R(t) and f(t) satisfy the inequality -î-->

-and the instantaneous frequency has a value in the intervál (wm+ O1, w + ê1 +

+ dO1), while in the second intersection point R(t) and g(t) satisfy

and the instantaneous frequency has a value In (wm+ 82 Wm + 82

+dO2).

The following result is obtained:

d02 2 - dt1dt2 dO1, dO2 _J dé1 $

S2 J

2 (R1 _d - (2.4.16) p(R1 =

f(O1,11,01'R2

=

g(Ô2),2.ñ2,2)

where p(R1, R1, 61

,R2, R2, 02 02) represents the joint dansity. of the variables

R1,

1' and R2, R2, 2 at time t= t1 and t= t2, respectively. When the functions f and g are constants:

f= c1 and g=c2 (2.4. 17)

then, alter performing .the Integrations with respect ot the variables O1

2

and over the same range (-

, w),

we obtain:,

- dt1 dt2

_{J '}

.1 2R1 Û2 p(R1 =c1,.ñ1, R2 = c2, R2).

m

which represents the probability, that R(t) passes in (t1, t1 i- dL1) throughthe

level f .= c, with a positive slope and through the level g.= o2 with a negative.

-CHAPTER 3

SOME STATISTICAL CHARACTERISTICS WITH REGARD TO THE JUMP PHENOMENA IN THE OUTPUT AMPLITUDE

In this chapter we derive first a general expression for the probability that in some elementary time interval (t, t + dt) an upward jump occurs in the

output amplitude of the system (2.2. 1). From this density we derive the

mathe-matical expectation of the number of upward jumps per second In thè output

amplitude. This statistical quantity provides us with a first insight in the dis-continuous properties of the system. Finally we derive In the first sectionan

expresslonJor the average number .per secondj of upward jumps with heights Which, exceed sáme prescribed value.

In the second section we derive asymptotic approximations for the Integral expressions for above mentioned statistical quantIties when the band width

becomes very narrow.

We consider. again a signal with a relatively narrow bandi width. So the instantaneous amplitude and frequency are slowly varying functions. We further assume- that the upward and downward. jump8 occur at levels which _{are so}

high, that the probability of occurrence of values of R higher than these levels are very, small., Consequently we may assume that the excursions of R(t) above these levels -are rare, and have a short duration which implies that the

time, intervals between successive upward -and' downward jumpé are very short.

3.1. GENERAL RESULTS

In the previous chapter we saw that an upward jump in thé outputoccurs either for the amplitude value R = R11 (w), when for some constant frequency

w = w8 > w0 the amplitude R of the Input R cos (wt+ O) is increased from some R <R.. (W8) until someR> R11 (w8), or for the frequency w = w,, when

for some constant amplitude R = Rt > 1(w0), where Rt = (we), the

fre-quency is decreased from some w> w until some w < we., For this reason we may expect, that a simultaneous variation of the frequency and the

ampli-tude of the input signal can cause jumps In the output ampliampli-tude. We shall

con-sider this more closely.

Wo WI W1 + e

Fig. 3.1.1

Werepresent at every time instant t the input R(t) cos (wmt - 3 (t)) as a

point In FIg. .3. 1. 1. wIth coordinates (R, (Um + O). Thus the behaviour of the input as a function of time can be studied by considering the movements of the

point (R, ù+ O) in the (R, w)-plane. Consequently, according to

thequasi-sta-tionary approximation method which we described. In sectIon 2. 3., the Input signal causes an upward jump in the output when the instantaneous amplitude R(t) and frequency w = Wm + ó (t) vary in such a way that the point (R, wm + passés through the curveR = R11(w) from region A into region B.

By substitutingIn (2. 4.8) for the functiOn f(ê) the expression for R11 (ui) glvenby (2.2. 1Q), wherew Is replaced by w_m+ we obtain the probability

that Inthe-time Interval (t, t+dt)ajump occurs in the output aiiplitùde, whiLe the absolute value of theinstantaneous frequency has a value In the Interval

(w

+8 w

+d+dO):

m ' m

dtd

$cI

i(t-

_{) p(R=}

+ _ _dR11

iU

r (3.1.1) p(R=Rii(ô),èt= _2Wm_

In order to avoid confusion we denoted in the second term the variable'é by Ô'

which after that has to be replaced by -2w

-The first termin expression.(3. 1.1) refers to. thecontribution'of the positive values of the instantaneous frequency, while the second term takes its negative

values into account.

In deriving formula(3. 1.1;), it is, assumed that every passage throughthe

level R11 _{from region A into region B is attended by an upward jump. This is}

true with probability almost one, because we assumed In. the beginning of this

chapter that the probability of occurrence of-values of R higher than.the 1 veIs

R1, andR11 is very small.

From Fig. 3. 1. L it is seen that.the occurrence of upward jumps in the

outpüt amplitude is only possible when has values In. the interval (w0- w, .

By integrating (3. 1. 1) with respect to Ô over above mentioned rangé of values we obtain the probability that an upward jump occurs In the output amplitude

in the time interval (t, t dt) without regard to the' frequency 'of the signal'.

dt 'Jedé : WoWm N TWIU8 $ d f.dÑ

(

_-

_{' ) [p(R}

=R11(0), è,'t,è) dR dô p(R=Rj1(0),Ô'

Analogous to the argument used in section 2. 4., the average nümber of

times per second an upward jump occurs in the output amplitude is readily

derived from (3. 1.2).

-$d

J°dR (i

e)

[p(R=R11(),.,it,)-i--

_{u_nil}

dÔ (3.1.3)

p(R=R11.(Ô), Ô' =2W Ô,Ñ,) .1

The statistical quantity N has much practical interest. Therefore we shall express it in terms of the spectrum and system parameters'and reduce it'to a

simpler form.. ' .. .

We subsútute expression (A. .13) for the probability density p(R, R,9, 0) in formula (3. 1.3)., Next, .we introduce the change of variables:

dRi'

d'O

-(3.1.2)

(3, Ji. '4)

Then, we obtain:

N= Jd

$d

WOWfl.j

-M33 ÇR11 +

[I e e) +

Ii(,e'

= -

2wm - e 0))

(3 1 5)

where In the second term the variable O, which has to be replaced by 2Wm Ó

is again denoted by Ô', while

1'

3

r

3/2 1/2

RRnexpL

2 + 211 2 . 33 RJ (M11+2 M120) +M22

(2

+ +

2 ö2)

2M23 (R11 11 2 Ru + 2.e

(ó

)2)

+ M33

(R1

O2 + 4R1 2 dR11 +

4'R1 Ó,+ 4

e2.o2(dhlI I _dO. . de.

The Integrations with respect to R and O are now readily performed. This

yields: N=

jdê

2 - 2W

-

Ô) } (3. 1. 7) Wm In which: I ÇO) 'B3 R [M22

(ll)2

- 2M23 { R + 2ò

_}

₊ 2 1/2 . dR .dR1 dR11 2 2rr'M33 - {M22

d3 M23(R11 +

40 dO) +M 3(R11e ±202 dO.)}

+

F

dR1i2i1/2

. . dO-"J {M22 - 4M23 4M23ò2}

(M22() -.2M23.(R1i,

1 +20

(1I

)

F-R -(M13+M230)2

ecp- 2V.

_{2B M33} ê2ö (3.1.6) +2 M12,e +M22e2J] (3. 1'. 8) M33(R11 +

;I}

Since the relation (2.2. 10) between R11 and w _{= wm} e is very complicated,

the integration with respect to e _{has to be performed numerically.}

Since we assumed in the beginning of this chapter that the .probabllltyof

occurrence of values of R higher than the levels R.1 andR11, are very small, we

may expect for physical reasons that most of the time Intervals between an

upward' jump and a' successive downward jump are very small*).

We further assùmed that the power,spectrum of the input signal is'

relatIvely-narrow. Consequently, during such a, time Interval, the. Instantaneous values of the input amplitude and frequency remain approximately the, same.

Assume that. at time t an upward jump occurs in the output amplitude.

'Then it is seen from Fig. 2.2.1.. that during' such an intervalbetween an

upward jump and a successive downward jump the additional height of the out-put amplitude due to the discontinuity has values between

F3(.w +O) _{- F1 (um}+ e.) _{and 'F2 (w +8) - F4(u} .i-OE)

where + _{O represents the instantaneous frequency at time t.}

The height S(w + _{O) of the initial jump' in the output amplitude, when the}

point (R, Wm + e.) in Fig. 3. 1. 1. passes from region A into region B, is given

by:.

S(w O)

= »m -

Ilm

°)I (3. 1. 9)

where 'F1. and are given by the-formulas (2.2.7) and (2. 2.13). 'This quantity can be conceived as a measure foE the seriousness of the discontinuity.

SinceS('w + _{O) Is a' monotonie increasing function, an 'expression for}

the' average number per secánd N.', of discontiñuities with Initial' jump heights which exceed. some value. S =S(w) is., - analogous to (3. 1.. 3)', expressed by

co

N $dO

$d

$dñ (Ñ.

(.p(R.=.R1(Ó),O:,íl,O) +

W wm - dB11

dO

p(R='R11.(0), O'

_=-

2w- O,,ñ,ö) }

(3.1.10)

In chapter 5 it is shown, for a special case, that' this is indeed true.if the original signal R(t)'CoS (mt + e(t)). which is both amplitude and' frequency modulated, is approximated by the pure amplitude-modulated signal R(t) cos Wmt.

Applying the same procedure, which reduced 'forimila (3.1.3) to»(3. I. 7) ve obtain for (3. 1. 10)

=

'$o

_'2

2' ='

(3.1.11)

wherethe function '2 in the'Integrañdis given by (3. 1.8)...

By plotting S against N we can judge to a certain extent the seriousness

of the discontinuous properties of thé system. Since S( Wm + ). is, a monotonic increasing function, N(S) will' be a decreasing one. For 'example the graph in Fig. 3. 1. 2 may represent the relation between S and '

SI

Fig 3.1.2

It is obvious, 'that for S = o, N is equal to N, which Le the average total'

num-beriof'jumps per second in' the output amplitude 'as givèn by formula (3.1.. 7).

Rem ark '1:

For symmetric power spectra the expression (3. 1. 8) for12(Ô)

can be' considerably simplified, because, in this case the coefficients b1 n=1, 2, . .'..;, are all equal zero. Consequently the cofactors M12 and M23 vanish. Remark 2: The average number of jumps per second, performed by the

amplitude modulated cosine function R(t) COB wt, is, according. to formula

(2.4.13), givenby:

3.2. THE AVERAGE NUMBER PER SECOND OF UPWARD JUMPS'IN' THE OUTPUT WHEN THE INPUT SIGNAL HAS A VERY NARROW BAND WIDTH We consider the case, that the input signal has the power spectrum:

We shall determine an approximate expression for the integral (3. 1. 7) whenthe characteristic bandwith a has a very small value.

The coefficients b which are defined by (A. 3) for the powerspectrum (3. 2. 1)

are given by (2. 3. 22).

Substituting these values in (A. 7), we obtain for the cofactors which appear in (3. I.?):

12

= 6(21ra) ;.

We consider first the case:

2nv

_m = w

i

-(V-V )2/2a2

w(v=

e m 'M13 = 2(21ra)l0; > _w o (3. 2.1) (3.2.5) M33 = 2(21ra)8 = M23 '° (3.2. 2)'

The determinant, of the moment matrix, is calculated from (A. 8) and (A.9):

B2 = 4(2rra)12 (3.2.3)

We substItute (3. 2. 2) and (3. 2. 3) in (3. 1. 7). This yields:

=

dRJ1 2

, 2

dR11 _1/2

i-

$ción.1 {(

_dO ) (2(21TO)2'I-4Ö

4R1 -b-

o+4 }

W0 W

- (3. 2.4)'

e

We notice that for the rangew6- wm < Ô <

> R _o

w) >0

(3.2.6)