A stochastic theory for non-linear ship rolling in irregular seas

17 SEP. 1282

NIVt

National Maritime Institute

A Stochastic Theory for Nonlinear

Ship rolling in Irregular Seas

by

J B Roberts*

*School of Engineering and Applied

Sciences, University of Sussex,

Falmer, Brighton, Sussex,BN 1 9QT

NMI R 99

September 1980

National Maritime Institute

Feltham

Middlesex TW14 OLQ

Tel: 01-977 0933 Telex:263118

Lab,

V.

_{Scheepsbouwkunde}

Technische Hogeschool

Delft

.

0

Crown Copyright 1980

This report is Crown Copyright but maybe freely reproduced for all purposes other than advertising providing the source is

National Maritime Institute

A Stochastic Theory for Nonlinear Ship Rolling in Irregular Seas

by

J.B. Roberts*

sic School of Engineering and Applied Sciences, University of Sussex, Falmer, Brighton, SusSex,BN1 9QT.

NMI Report R 9 9

By a combination of averaging techniques with the theory of Markov

processes, an approximate theory is developed for the rolling motion

of g ship in beam waves. A simple expression is obtained for the dist-ribution of the roll angle, and is tested by a comparison with a set of

digital simulation estimates, due to Dalzell. Good agreement is obtained

Litt of Contents

page number

Introduction 1

2

The General Theory 5

2.1

The equations of motion

5

2.2

Averaging the equations 8

2.3

The response as ,a Markov process ₉

2.4

The FPK equation for 'the response 11

2.5 The Stationary solution -12

2.6 Special Cases

14

2.6.1 Linear restoring moment

2.6.2, White noise excitation 15

3 A Particular Case 17

3.1

Solution for free, tndatped 17

oscillations

5.2 Calculation of the response 20 distribution

3.3

The wave input 22

3.4

Comparison with simulation

23

results - process 3

3,5 Comparison with simulation 26 results - process 2 Conclusions 26 Acknowledgements

27

References Appendix Figures

non-dimensional damping factor (linear)

a1'a2 damping functions (see equation (17))

A damping constant (see equation (74))

A ,A2 averages of al, a2, respectively, over one cycle

A matrix containing Al and A2 (see equation (27))

A(t) amplitude process for response (see equation (63)) non-dimensional damping factor (quadratic)

b1, b2 functions defined in equations (17)

matrix containing 1)1 and b2 (see equation (27))

'damping constant (see equation (74)) function defined in equations (17)

cn Fourier coefficient related to cos 00(t) (see equation cn( ) Jacobian elliptic function

C(V) function of energy level, defined by equation (68)

C($) damping moment

dn( ) ' Jacobian elliptic function

diffusion matrix (seeP equation (28))

D11,D12 elements of D

D(V)

functions of V, defined by equation (55)

'F{.} expectation operator

f(V,0) defined by equation (8)

F(0) scaled damping moment ( = C(0)/1)

g(V,O)

defined by equation (9)

G(0)

scaled restoring moment (= K(0)/1)

H2' H3 non-dimensional wave input spectra

roll inertia

"strength" of white noise (see equation. (64))

k2 K( ) K(m) ml, m2 M(t) p(VIV ; p(ZIZ t) p5(V) Fs(01(;)

restoring moment constants (see equation (74)) restoring moment

'K2,normalisation constants

1

complete elliptic integral

in parameter for Jacobian elliptic functions

drift matrix (see equation (28)) elements of m

-roll excitation moment

transition density function for V(t) transition density function for Z(t)

-stationary density function for V(t) joint stationary density function for

0,$

p(A)

stationary density function for A(t)

P(A) cumulative distribution function for A(t) defined by equation (91)

Q(V) function of energy level, defined by equation (72)

Fourier coefficients related to sin eo(t) (see equation (A6)) n

sn( ) Jacobian elliptic function Ss(w) power spectrum for process X(t)

time

T(V) period of free, undamped oscillations

U(0)

potential energy function total energy of oscillation correlation function for X(t)

vector of unit Wiener processes (see equation (28))

x(t) defined by equation (79)

X(t) excitation process (= M(t)/I)

Y(t) scaled excitation process (see equation (14)) vector (V,A)

scaling parameterfor the damping moment

.

6( )

'Dirac's delta-funCtion'

6 =

phase angle (see equation (5))

eo e solution for .free, undamped oscillations

Slowly

varying phaseangle

A damping function (linear) (see equation (100))

p(y) function of energy level (see equation (36))

v(v)

function Of energy level (See eqUation (37))

-damping function (quadratic) (see equation (100))

aR standard deviation of roll angle

a

standard deviation of wave input

non-dimensional time (= wt)

(I) roll angle

critical roll angle

non-dimensional roll angle (=

0/0*)

frequency

frequency of free, undamped, linear oscillations

wp frequency at which wave input spectrum peaks

w1 frequency defined by equation (82)

w* frequency ratio, defined by equation (109)

. Introduction

The problem of predicting the rolling motion of ships due to wave

action has been a matter of considerable interest to naval architects

for centuries. This is because, for many ships, the natural frequency of rolling motion is of similar magnitude to the frequencies at which

wave energy is dominant. Moreover, the hydrodynamic damping associated

with rolling motion is usually relatively small. Thus ships may exhibit a large resonant response in the roll mode, and may even capsize, in sea

states which are not necessarily very severe.

In general the rolling motion of a ship is undoubtedly coupled with other motions, such as sway and yaw, a fact which considerably complicates

a theoretical treatment. Fortunately, however, there are two special cases,. In which the rolling motion is likely to be excessive, and in. which

it is reasonable. to consider this motion as uncoupled.. The

first

of these is the case of a Ship at 10w, or Zero, speed encountering unidirectional, .

beam waves. This situation is prone to occur whenever a ship iS in the hove-to Condition - e.g. through engine failure. The second is the case of a ship

in

unidirectional head waves:. here it is Well known that there 1S the pOssibility of instability, under following-wave conditions.

In this paper the first of these two iMportant Cases ship rolling in

beam seas - is considered

in

some detail, Analytical studies of this problem

have

a

long history, and originate in the pioneering work of Fronde (1). Fronde clearly recognised that ship rolling Was essentially

a

dynamics

prob-lem which must involve the inertia of the Ship, the effect

of

damping and the hydrodynamic restoring moment. Moreover, he demonstrated that both the

damping and restoring moments Varied

in

a-distinctly non-linear tanner with roll angle. By solving an appropriatedifferential equation he was able to

calculations of this kind have subsequently been made by many workers (e.g.

see Refs. 2-4).

During the last two decades efforts have been made to develop a more

real-istic theory by treating the wave input as a stochastic process. Unfortunately,

due to the non-linear nature of ship rolling, the linear spectral theory

(intro-duced into ship motion studies by St. Denis and Pierson (5), and used

success-fully for other motions, such as pitch and heave), is inappropriate here. A

general theory for non-linear system response to stochastic processes, of the

same scope as the linear theory, is not yet available and thus progress in dev-eloping a satisfactory stochastic theory for ship rolling has been slow.

Nearly all the theoretical work which has been undertaken so far is derived

from linear concepts. One approach involves replacing the original non-linear

equation of motion by an equivalent linear equation, with the damping and

restor-ing moments so chosen that the difference between the two equations is minimised.

This method, due to Booton (6) and Caughey (7),was first applied to rolling

motion by GoOdMan and Sargent(8) and subsequently by Kaplan (9) and Vassilopohlos

1

(10).

A variation of this technique has been described recently by Flower and

1

Mackerdichian (11). Another approach is to develop the solution of the

ton-linear equation in a perturbation .series (12). This method has been employed

1

by Yamanouchi (13) and Flower (14) to estimate the spectrum of the roll

res-1 ponse. Yet another extrapolation of linear theory ii the functional

represent-ation method described by Hasselman (15) and Vassilopoulos (16). This has been

applied to the ship roll problem by Dalzell (17) and Yamanouchi (18).

'These approaches are capable of yielding only limited information on the

roll

response statistics, Such as the mean square of the roll angle, or, 1.7'

some cases, the effect of nonlinearities on the response spectrum. HOweVer,

they can not yield useful' information on the probability distribution of the roll Angle - i.e. the deviatioh of this distribution from the Gaussian form

due to non-linear effects. It is precisely this information which is of vital importance for design purposes since it enables the probability of the roll angle

exceeding specified critical valUes to be quantified.

.Fortunately, there is one other method of attack,,

which is

not derived frOt linear theory, and which,

in

principle, At least, is Capable Of pre-dicting the form of the response distribution for 'non=linear system res-ponse. This is the so-called POIcker-Planck4olmogorOV (FPK) method (e.g. see (19)), which is related to the general theory of NArkov processes. the normal application Of this Method it is necessary to model the wave excitation as a white noise proceSs. For the rolling problem one can then. show that the joint response prOcess 0,$),' where 0 Is the angle of

toll

and 0 is its time derivative, is a two'-dimengional Markov process. This process

is characterised by a transitiOn density function, which is governed by an

. appropriate diffusion, or FPK,equation. The stationary solution of this

equation gives the' joint .probability distribution of

4

and (20). ThiA_

method

canbe

applied for any kind, of non-linearity but an analytical

solu-tion is only available 'for the case offnOn-llnear damping When this is Of a Very particular forth (21). For the roll response problem non-;linearities in both damping and restoring moments need to be considered and here the

approp--rate FPK equation must be solved numerically. Suitable solution techniqges

are available in the literature (22,23) but are complicated and time consuming to implement - so far they have not been applied to the study of ship rolling.

However, some results relating to the moments of the response have been

ex-tracted from the FPK equation by Haddara (20) and shOwn to agree with results obtained by other methods.

Partly because of the apparent necessity of adopting the rather drastic simplification of a white noise model for the WaVe input, and partly because of the difficulty of obtaining results in the case

of

non-linear damping, the FPK approach has not been enthusiastically pursued by those concerned With '

non-linear ship motions (e.g. see (24)). It is the object of this paper to show that, by combining the FPK method with an averaging approximation,

both these stumbling blocks can be surmounted.

'The proposed method hinges on the assumption that the damping is fairly

light, so that the roll response has an Oscillatory, or narrow-band,

char-1 acter. This appears to be a reasonable assumption in the Majority of cases

1.

of practical concern. In these Circumstances the total energy envelope, V of the roll response (potential plus kinetic energy) will be slowly :varying

with respect to time: it is then possible to approximate the equations of I

motion; expressed

in

terms of V and an appropriate phase angle, X, by averag-ing over one cycle of oscillation, assumaverag-ing that V and X are sensibly constant

over that period of time. The averaging which .is developed here for this purpose iS an extension of the method proposed by Stratonoviich (25), which

in turn is a generalisation, to the stochastic Case, of the deterministic

averaging technique of Bogoliubov and MitropoIsky (26). From the averaged equation for the energy envelope it is deduced that this process

is

a one-dimensional MarkoV process, uncoupled from the phase process. Hence a 'FPS

equation for V is obtained which can be Solved to obtain the stationary

dis-tribution.

The principal advantage of introducing the averaging approximation is that

the appropriate FPK equation is reduced from two dimensions to one dimension. The one-dimensional equation can be readily solved for arbitrary nom-linear damping and restoring moments - indeed, the stationary distribution of the

response can be expressed as a

simple

quadrature. Moreover, it is unnecessary to idealise the wave input ag. a white noise; the Method applies to a wave input

with an arbitrary spectrum. For the special .case of white noise excitation the

-general theory developed here reduces to results found previously by the author

, with the few known exact solutions.

To test the theory it is applied to the particular model studied by

Dalzell (32,33). By the use of a digital simulation technique, Dalzell

studied the amplitude of the roll response and obtained a comprehensive

set of estimated cumulative probability distributions. These results are very suitable for comparing with predictions obtained from the present

theory, and enable some assessment of the range of -Validity of the theory

to be made.

2 The General Theory

2.1 The_equations of motion

For a ship Undergoing rolling Motion, due to random beam waves, it

Will be assumed that the influence of

all

other, degrees of freedom can

be neglected. An appropriate equation of motion is then

+ a.C($) + Ic(0)

(1)

Here I is the roll inertia, 0 16 the roll angle, C($)

is

an arbitrary, non-linear damping moment, K(0) is

an

abitrary, non-linear restoring moment, and

mco

is the roll excitation moment:. (3 is

a

scaling parameter which, it will

be assumed at the outset, is Small. Initially the equation will be simplified

by divided throughout by thus

+

F6Y+ G(0) = X(t)

(2)

where. F = C/I, G = X/I and X .= M/I.

The total energy envelope, V(t), associated With the roll response, May

be defined by the relation

,L2 V

=2

* U(0) (3)

where

(t)

U(c)

= f

_{G(E) clE} (4)

.2

A /2 represents the kinetic energy of the ship, whereas

U(0)

represents its

potential energy. It is noted that when

a

0, and X = 0 the energy is a constant, independent of time. For the case of non-zero excitation, and

f(v,e)

F()

g(V,e) = G()

In the special case of free, to the expected results

V = 0 - i.e. V = a constant. and equation (7) reduces to

6 g-(11-19)

(2V) case

In the linear case the right hand side of equation (11) is a constant, and

hence e increases linearly with time. In the non-linear case the solution of 'equation (11), denoted e

co,

is no longer linear with time.

Small damping, the energy will be slowly varying with respect to time. It

is this basic property which enables an approximate theory to be developed.

As a first step the equation of motion will be rewritten in terms of V

and an associated phase process, e, defined such that

uoy

cose ; = -(210/

sia

(5).

A graphical representation of the relationShip between

0, $,

V an e is

shown in Fig. 1 This Shows a modified phase planeWith 45/V2 and

aS Coordinates. In this plane free undamped motion appears as a cir6Ular

orbit: position on this orbit may be specified by the radius V and angular

e

position

e.

In terms of V and e equation (2) can be recast as the following pair of

coupled, first-order equations:

V = Fif(r,e)(2v) sine (2)1 sine. X(t) (6)

gO_

am

Es)

coo--I

_{OV) COS}

6

72:8)

l't

e (21.0 where (7)

The excitation, X(t) will be modelled as a stationary random process,

with zero mean. In contrast with many other theories it is not necessary to assume here that it is a white noise or even that it is Gaussian. The spectrum of X(t), Sxw), may be defined 'by the relation

1

:S(w)

= mw x (T) cosWT dT 27_{_} where

w(T) =

E{X(t) X (t

+ T)}

(13)

is the correlation for X(t).

When stationary response conditions are achieved the standard deviation

of the roll angle, OR, is directly dependent on the degree of damping present: the smaller the damping the higher, the level of the response. In the linear

case it is well-known that d is of order .(3 and in fact this also holds in

the general, non-linear case E31]. To clarify the subsequent analysis it is convenient to scale the excitation

so

that CR is of order (3(3. Hence the process

-1

Y(t)

= e

X(t) (E2 = (3) (14)

is introduced into equations (6) and (7). They can then be wiitten as

2

= -E

a1(v,e) - Eb1(V,e,Y) (15)

where

e

-E2a2 (V,e) - Eb2,(V,e,Y) +

c(lv,e)

-a1 = -I(V,e)(2V) sine ; = (2V) sine Y(t)

,-f(V,e)cose (2V) Pose Y(t) (2V)* (12) (16) (17) (2V)Icose

It is emphasised that the substitution of Y(t) for X(t) does not imply that the analysis which follows is restricted to weak levels of excitation.

This step is made simply to bring out the order of magnitude

of

the various

2.2 Averaging the equations

It is evident from equation (15) that V is small if

E is

small - i.e. V is slowly varying. Similarly the phase process A(t), where

X(t) = c2a2(V,O)) -

c

b2(V,O,Y) (18)

will be slowly varying: On integrating equation (16) one has

e

=A +

fc(v,e)dt

In the case of a linear restoring moment the above integral term is simply

W t, but in the general case it is considerably more complicated. Here the

-o

simplifying assumption is made that, over periods of time in which V and X

are sensibly constant,. the integral term can be replaced by 00(t), the

so1U-tion for free, undamped oscillaso1U-tion. Then

e

A + 6 (18)

To approximate equations (15) and (16) the terms not containing Y(t)

explicitly (i.e.

a1, a2' and c) will first be replaced by their average value

over one cycle, assuming that if and A remain effectively constant over such

a time interVal_ This is esaentially the averaging concept proposed by

Bogoliubov and Mitropolsky [261. However, in Ref. 26 the method is restricted

to systems with only a small degree of non-linearity in the restoring moment. The technique developed here is a generalisation to the case of an arbitrary, non-linear restoring moment.

Considering the term al initially, its averaged value will be denoted Ai. Hence T(V) A1 (V)(V) - 1

f

a1(V,e) dt where T(V)

= 2/if

0 - Lq4))

and b is such that

U(b) = V (22)

T(V) is the period of free, .undamped oscillation, and depends on the energy level V, as indicated. Since $dt = dc, one has, from equations (5), (8),

(17) and (19),

4

A (V) = TiTvTio

F(V2[V - U(0)] )

The term a2 can be similarly averaged, to yield A2. In summary

then, after averaging has been carried out, equations (15) and (16)

are replaced by:

2

= -Z-A1 (V) - E b1(V'e'Y)

-= -62

A2(V)

-e

b2 (V"

- e Y)

2.3 The response as a Markov process

As a final step in the averaging procedure it is necessary to average

the terms b1 and b2. This is not a simple matter since the correlation be

tween V(t), e(t) and Y(t) must be considered. The "stochastic averaging'

method which is needed here was developed by Stratonovitch [251 and later

proved rigorously by Khasminskii [34]. The basic results are contained in

the so-called Stratonovitch-Khasminskii (SK) limit theorem.

To apply the SK limit theorem to the present problem it is convenient to

first cast equations(24) and (25) into the following matrix form:

2 Z

= e

A(Z) + Eb(Z,Y) where [ Z = 1 = V ; A = -Ai ; b = -lb, ' Z A -A2 -b2

The limit theorem state$ that, for equations of, the form (27); as

C -* 0 the vector process Z(t) converges weakly to a Markov process, governed

by the It& equation [35].

9

2

dZ = E m dt + ED dW

Here W is a vector of unit Wiener (or Brownian) processes and m

respectively, the drift and diffusion matrices of the Markov process. m

and D are given by the following' relationShips;

0 3b m = A

+ f<

E{

(-1 (3' )

> di aZ -+m t+T 03 present, two-dimensional,case

D D'

= f

< Ef(b) (bo) 1>

di-t .t+T

In the above < > denotes a time average, b' is the transpose of b and i

ab1 ab1

hr

3.A

ab 'ab

2 2 aV.

The SUbSeript t (tit t'+

T)

denotes that the quantity is to be evaluated at t (or t

+ T).

On performing the expectations- the elements Of 2 are treated as fixed quantities.

In the present case m and D D' are of the form

-m

[11a21

[D21

221

= D

ml ; D D' =

11

The evaluation of the elements of these two matrices is described in the

Appendix. There it is shown that

,A1(V)

+(V)

2 .+

'(V)1;

.)(V)1.

m2 - (V) (33) 2E where 2111 (V) . V. D - - -II 2 = D = 0 21

11(V) =

a 2

s

Enw(V)1 n=1

.4

x r v(V) =

CS

ow

(V

)1

ar-.71 .7\)(V) 2 2VE2 (28) and D are, (29) (30) the ( 31) (32) ,(37)

11

The coefficients

Sn and cn relate, respectively, to.Fourier expressions of

sin eo(t) and cos 00(t) (see Appendix).

2.4 The FPK equation for the response

From the above application of the SK limit theorem it is found that the

joint process Z = (V,A) converges to a Markov process as E_- O. For small but finite E, equation (28) will be a good model for Z.

-The process Z is completely defined by its transition density function

p(ZIZ

o;t), such that p(ZIZ ;t) dZ is the probability that the process lies in

- _-o

the range Z to Z + dZ at time t, given that it was at Zo at time t = O. From

-equation (28) it follows that p(ZIZ ;t) is governed by the Folder-Planck-- Folder-Planck--o

Kolmogorov (FPK)

equation f351 2 ' 2 ,2

4)

_ 2 r a E2

azaz

2 . (DilP) ' ' c- Li

-5y7 (m.P)

+=-- 1 1

-°

-1 1=1 j=1 I j --."

On substituting from equations (33) to (35) into equation (38) one obtains the equation . 2 a r

a-= 4{[liaAl(V)

- Tra(V)]Fil +

2 .5.7,AA2(V)p} +Tr ---2{vp(v)0 av a2 ,v(v)

v

.PJ

ax where

4(v) _

P(v)

vorl

2

Returning to equation (28) it is noted that, on combination with _equations

(33) to (35), it becomes

dV = [-A1(V) + Tra(V)1dt +12M1-1(V).V1 dW _(4l) 7v(V)

a =

[-aA2(V)]dt

[ 2V I cl-W (42)

The first of these equations shows that the energy process V is governed by an

Ito equation which is uncoupled from the phase equation - i.e. V(t) converges to a one-dimensional Markov process. _{The appropriate FPK equation for the}

trans-ition density function p(VIV0;-t) is given by

a 2

= g-4

113A1 (V) - Tra (v) p}

{vp(v)P}

at V aV2 (38) (43)

2.5 The Stationary Solution

In a realistic model of ship rolling the restoring moment will

reduce to zero at some critical angle of ro11,0*

say.

When driven by

random excitation the roll angle will eventually exceed

0*;

i.e. capsize

will result. Thus, strictly speaking, it is improper to discuss the

stationary response distribution in the roll mode, since stationarity

will never actually be achieved. However, from a practical point of view,

if the excitation is not too severe the probabilitythat will exceed

0*

will be negligibly Small, over periods of time in which it is reasonable

to consider the excitation as stationary.

Assuming that the excitation is such that stationarity is effectively

achieved, then, for the energy process, V(t),

p(VIV0;t) p5(V) (44)

as t becomes large, where p(V) is the stationary joint probability density

function for V(t). This can be found from equation (43) by setting ap/at = 0.

Ps(V) * VU(V) PIE') 71- 11(E)

r

V. -

Tra(01

0

(45)

Where K is a normalisation constant, chosen so that

foPs

(v) dV = 1 (46)

'Equation (45) enables the distribution of V(t) to be computed fairly easily

from the Original system parameters. The result incorporates the free,'

damped Solution, through the Fourier coefficients ss and

C.

An expression for the joint density function, ps(V,A), of the processes

V(t) and A(t) can also

be

found from the FPK equation tor this joint Matkov

process (equation (39)). ThUt

p(ZIZ

0'

-t) p (Z) = p5 ' (V A)

- -

-s

as t becomes large, and on setting ap/at 0 in equation (39), it is found

that the solution is of the form

13

Ps(V,A)

K2.P5(V)

(48)

is another normalisation constant, Such that

27°D

f f

_ps(V4X)

dVdA = 1 (49)

00$

Equation (48) shows that, for a given value of V,A is Uniformly

distrib-uted between 0 and

27.

Thus the distribution of the angle

e . 00-*

Xis

identical to the distribution of e It follows that the probability of the roll angle lying in the range dO, at a given energy level V, is equal to the proportion of the time spent in that range, during a free, undamped

oscilla-tion - i.e.

pa(01V)

d0 = dt/T(V) (50)

where

_Ps(011)

is the conditional density function for 0 and dt is the time

spent in the interval dO, during a free, Undamped oscillation. Now dt

2d0/$ , where

0

[2(V711(0))]i.

Heade

ps(V)

Ps(V,O)

P V)P(V) = (51)

(V-U(0)) T(V)

is the stationary

joint

density

fUnction

for V(t) and 0(t). Since

p5(V,0)

dV = p5(0,$)d3 (52)

and dV

= VitiTUTTY

d$, for a fixed

0,

it follows that

2 Ps(V)

7 T(V) (53)

is the appropriate expression for the joint density of 0 and $.

Equation (53) (or alternatively equation(51)) enables a wide variety

of statistics for the roll response to be evaluated. As a simple example,

the

nth

moment of the roll angle, E{On} , is given by

co

m

E{011} =

f

I

Onp.(V,O)

dOcIV

p (V) D (V)dV (54)

D(V) =

Here equation (51) has been used for ps(V,O) and Dn(V) is defined by

r-

n

y2 f x dx

T(V)

[V-U(0

For n = 2, equations (54) and (55) enable the mean Square roll angle,

GR2, to be computed.

2.6 Special cases

As a partial check on the validity of the fore-going theory it can

be compared with a variety of known results, obtained for special cases.

2.6.1 Linear restoring moment

The analysis is very considerably simplified if the restoring moment

is linear - i.e.

G(0)

= wo2 where wo is the undamped natural frequency.

The free, undamped solution is then simply harmonic - i.e. 00(t) = wot. Equations (36), (37) and (40) reduce to

p(V) = V(V)

= a(v)

= S(w)

(56)

and the FPK equations ((39) and (43)) are equivalent to those found earlier

by the author [28]. Solutions of the FIIK equations for this case have been

found to agree well with simulation results, even when the damping is

moder-ately high [28].

Naturally, further simplications ensue if the damping moment is also

linear - i.e. F(6) = . Then, from equation (22),

A1(V) = V (57)

and equation (45) becomes

V [P.E

- Isx(wo

p5(V) - exp { -

f

s

.dE}

vs (w

Tr

(w )E

x 0 0 x

0

= exp.(

"

}

(W )

x o (55)

Also, from equation (53),

15

p(0,0 =

K exp( - -

1w2

2 + (I) 1}-27Sx(wo) o Here

K2 and K3 are normalisation constants. Equation (59) may be recognised

as a joint Gaussian distribution for (I) and (1). From it the mean square roll

angle is readily found to be

7S (w )

2 x o_

a

= (60)

2

13(0

Equations (58) to (60) can be obtained from the standard linear theory

by using a white noise approximation for the excitation process - i.e.

re-ss.

placing

_S(w)

by a flat spectrum, of height Sx(w0). Thus, in the linear case, the application of the SK limit theorem is equivalent to a white noise

approximation. However, in the general case this equivalence does not hold,

since the shape of the spectrum plays an important role in the computation of

p(V) and )(V).

It is noted that, if an amplitude process A(t) is defined by

V(t) =

wo A2(t)/2 (61)

then, from equations (58) and (60), the stationarytjoint density function

for A(t) is given by

A.1

p5(A) = exp{. - A2

f

-aR2

2R

This is the well-known Rayleigh distribution. In the general case an ampli-tude may be defined by

V(t) = U[A(01 (63)

and the distribution of A(t) can be found from equation (45).

2.6.2 White noise excitation

Suppose that the excitation is a white noise of "strength" I - i.e.

w

(T)

= I 6(T) (64)

(59)

00,$)

= Ks e

Equations (36) and (37) then simplify to

I r 2

"V) =

irr

L sn

V(V) =

c2

(66)

n

From ParseValls theorem it follOWs that

(see

equation (A6) T(V) I

s

2 2 r T(V) j

si2

n8o(t)dt =

CI(IV) n=1 n 0 Where

vi

r C(V) = T(V) '

V-U(4))]kicP

(68)

Also

_r

_{2 2 :r("}

n1

n

1-Lo+cn2)

= T(V)

join2e(t)

+ cos20

0O

(ld = 2

= 0 . o and hence a(V) = 1/27 (69)

On substituting these results into equation (43) one obtains the FPS

equation

T 2

=

A4DA1(V)

-

ilpl + --=--2[C(V)pj (70)

9v

Which agrees with the result found earlier by Stratonovitch D51 and the

author [27,31] using different approaches. The stationary solution to equation (70) is

Ps(V) = K4T(V) exp{- .Q(V)1 (71) where Q(V)

j

r 1 clE -0

V A

(E)

C(E)

(72)

Hence, from equation (53)

U()3}

.Here S4 and K5 are further normalisation constants.

In the case of linear damping Q(V) = V and equation (73) agrees exactly

with the stationary solution of the two-dimensional Fokker-Planck equation

for the joint process (0,$). It also agrees with Caughey's exact SOlution

for damping of the form 4f(V) E211.,

(65)

(67)

3 A_Part.i.culat Case

To test the fore-going theory the following particular form of

equation (2) is chosen

0.4.441-101$4-k1cii-k-2

4) = X(t)

3

4

(74)

Thus the damping is assumed to be a combination of linear and quadratic

forms. Froude [11 and many others have shown that this is a reasonable

model in most cases. The linear plus cubic form chosen for the restoring moment is the simplest model which represents the basic features of the

measured restoring moment vs. roll angle curves. Equation (74) is the basis of the comprehensive simulation study by Dalzell [32, 33].

Following Dalzell, equation (74) will first be scaled and

non-dimen-sionalised, to reduce the number of disposable parameters. From equation

(74), the critical roll angle,(1) , at which the restoring moment is zero, is

= (k1/k2) (75)

and it is convenient to introduce a scaled roll angle, IP, 'defined by

= (1)/(1)* (76)

A non-dimensional time,T, can also be defined as

T = W

t where W = _{(k1 )} (77)

o

Equation (74) can now be recast as follows:

blidli)

+IP 43 = x(t) (78) where a =

A/wo;

b = BO*; x(t) -0 1. X(t) (79)

and differentiation is now with respect to

T.

3.1 Solution for free, undamped oscillations

To evaluate the functions

am

and

poo,

which appear in equation (45)

for ps(V), it is necessary to solve for free, undamped oscillations - i.e.

to find the solution of the equation

On integration the solution 18 found to be

11) = A

sn(wiT

+ K(W) Im)

where A is the amplitude of the oscillation (i.e. value of 4) at

= o) and sn is the Jacobian elliptic function

2 A w = (1 -1 2 T - _ 4K(m) (1 A2/2) o 63. Further 2

m-

(83) - A2

and K(m) is the complete elliptic integral , defined by

7/2

dE K(m) =

f

r

0

Ll -

msin2E1

It is noted that this solution is only valid for A < 1. For greater

amplitudes the motion is unstable. This is illustrated in Fig.2, which is a phase-plane portrait of the solutions of equation (80). Equation (81)

corresponds to orbital paths in the shaded area, which is the stable regime.

The natural period of oscillation, T, is given by equation (20), which

in the present case can be expressed as

(84)

As A -> 0, K(m) 7/2 and T 27. If w is the frequency of oscillation, corresponding to the periodic time T, (i.e.w = 27/T) then

= 7(1 - A3/2)

wo .2K(m)

Where w w = 1/27 as A -> O.

(85)

(86)

Fig. 3 shows the variation, of (.4/Wo.with A, computed from equation (86) This shows that w is fairly insensitive to the amplitude level, when this is

low. As A -> 1 w falls in value very rapidly and becomes zero at A = 1.

where

= Awl

cn(wit + KW* dn(wit

K(m)lm) (87)

where cn and dn are further Jacobian elliptic functions

I351.

Also

A2 A4

= 2

- 4

It follows that (see equation (5))

sino

- cn(w1T + K(m)Im)dn (W 1T + K(m)Im) (89)

and from this one can obtain cos 00(1.). Figs. 4(a) and (b) show, respectively, the variations of sin

e0(T)

and cos

eo(T)

with time over one complete cycle of

Oscillation, for various values of m As A 0, m 0 and sin 00(T) sin

T,

cos

00(T) ->

cos

T.

At the opposite extreme, as A 1, the deviation from

harmonic motion becomes very pronounced and, in the limit,e 0(t) becomes

dis-continuous, jumping between the values 0 and Ti. This reflects the fact that, for amplitude close to unity, the roll angle is very close to unity for a large

proportion of the periodic time - this is the well-known phenomenon of "creeping

motion" which occurs in the vicinity of singtlar points in the phase plane.

From equation (89) it is possible to calculate the coefficients sn and cn

which relate, respectively, to the Fourier expansions of sin

00(T)

and cos

eo(T)

(see equations (A6) and (A7). An expression for sn can be found analytically: the result is q exp { "P-7111) 1 It(m) 19 (n odd) . (90) (91) (88)

Unfortunately, it does not seem possible to obtain cn analytically - however

these coefficients are easy to compute numerically, using standard subroutines for the evaluation of the Jacobian elliptic functions. Again only the odd co-efficients are non-zero.

Fig.5 shows the variation of siand

c1 with m. For small m one finds, from

equation (90), that 3 K1

13_00

Vs 4S Cw(V)1

1 x

exP{

(92)

with an error of order m2 ci also appears to Vary linearly with m, at

Stall amplitudes and a good approximation is

c

1=

. +

_ld

m (93)

Figs. 6(a) and (b) show, respectively, the variation of s and c

(n = 1, 3 and 5) with m, on a logarithmic plot. 3.2 Calculation of the response distribution

and

e

are known, because it is then possible to determine

a(v)

and p(V).

In'view of the behaviour of the Fourier coefficients (see Fig.6) and

the fact that w(V) will usually correspond roughly with the frequency at

which S(w) peaks, it is evident that the first terms in equations (36) and

(37) will usually be dominant'-

i.e.

2

11(V) s

S(V)]

(94)

2

v(v)

s

_x5,)(1.)] (95)

1

One can expect the above approximations to be very accurate except at very

/i

large values of V. Moreover, in view of the fact that [(e12 +si2 )/21 is close to unity, except for large values of V (see Fig.5) one also has the

approximation

a(V) Sx5D(V)1 (96)

On substitution of these approximations into equation (45) one obtains

.f3A dE

I.

1V4E_

1 (97) 2 J

0

WEe 2Sx[W(V)1

0 S.1 E

The second integral in equation (97) has an integrand which is singular at

= O. This difficulty can be overcome by writing

V A r ( 1 J0 s

Os

2E 21 ) +

f

E

o E

(98)

On combining equations (97)

and (98)

one then has

K1

eXp{

f

1

1

V 13A

(E)

clE V

.S.1 1 =

_I

_{( 12} - ) , (99) ps(V) ...

-r

'

43 LW(V)1 0 TiCs

S LW(V)j

x: 12 x 0 s1

where K11 is another normalisation constant. The integrand in the second

integral now approaches the finite limit 0.375, as E O.

The damping function Al(V) is given by equation (19) which, in the

present case, reduces to

Al(V) = ViaA(V) + bE(V)] (100)

where 2 A(V) T(V) Pin o(t) dt 2 V r r

E(V) - -

- Lsin3

00(01

dt T(V)

Fig. 7 shows the variation of A and E with the amplitude of oscillation, A.

It is noted that, in the case of a linear restoring moment, A(V) = 1 and

= 1.20Vi - this case is shown by the broken lines in Fig.7. It is evident

that the non-linearity in the restoring moment has a significant effect on A

and E at high amplitudes: ultimately, as A -> 1, both A and E approach zero in

value.

The foregoing results suggest that, at low amplitudes of oscillation, the

non-linearity in the restoring moment will be much less significant than the

non-linearity in the damping moment. Thus, for low amplitudes, one can

approx-imate equation (99) still further by setting si = 1,

Sx[w(V)1=

S(w) and

3/2

1 A2 [a + 0.565 b

4p-p (A) = K A expf

-2

_27S(w)

case a = 0 this is of the form

Ps- (A) = Ci A exp (= C2 A3)

where C and C2 are constant. This differs from the form

p5(A) = C _{A2 exp (-C2} A3)

S(w) = S (w ) H (w*) x

x p

2 1 Ps(V) = K2 exp {-

7S(w

V + 0.8 b V j/ x o) (103) where

K2 is another normalisation constant. For the case a= 0 it is - shown in ref. 28 that this agrees *ith a result due to Kirk [371.

When cast into a density function for A(t), equation (103) becomes

(104)

where again the non-linear-restoring moment hag been neglected. For the

suggested by Bell and Galef [38], although the exponential term is the same.

3.3 The wave input

Dalzell E32, 331 has used equation (78) as the basis for a simulation

study of the distribution of roll angle. In this work three different wave input processes (numbered 1, 2 and 3) were considered. Process 1 is

repres-entative of swell and has a very small bandwidth, which

is

similar in magni-tude to the bandwidth of typical roll response spectra. In this situation the

present theory will not yield accurate results: as StraionoVitch has shown

[251, results derived from the SK limit theorem will only be accurate when

the input spectrum bandwidth is appreciably larger than the response spectrum

bandwidth. Thus, for the purposes of comparing the present theory with Dalzell's

results, only Process 2 (intermediate bandwidth) and Process 3 (largest

band-width) are considered here.

The spectrum of Process 2 is given by

Where

e5/4

H2(w*) = exp( )

*5

4(1)21'4 and where 1 r

ve

H (w*) = expt - 1 - + Tr w*2 - - - -1 5 w* 4w*4 16 4

2

Here W is the frequency at which S(w) peaks and H23(w) are both scaled

such that H2,3(1) = 1. The spectral forms H2(w*) and H3(W*) are plotted

in Fig.8. Nearly all the simulation results available in Ref. 32 relate to

Process 3.

Both spectra can be specified by two parameters: firstly the standard

deviation,of x(T), given by

W' = w/wp For Process 3 2

f

s

(o) dw x 23 Sx(w) = Sx(wp) H3(w*) (110) (112)

and secondly the frequency ratio

= (113)

Wo

With the scaling in equation (78), wo = 1. aw can be directly related to

Sx(wp), by carrying out the necessary integration. It is found that, for Process 2

S (w ) = 0.7163 la 2/w (114)

x P W p

whereas, for Process 3

Sx(wP) = 0.3959

a

_W2/wP ' (115) 3.4 Comparison with simulation results - Process 3

There are four parameters involved in a comparison betwen the present

Fig. 9(a) shows the density function, p(a), for the amplitude A(t) (see equation (63)), in the case of pure quadratic damping (a = 0)

andwith

b = 1.0,

04

= 0.036 and Q = 0.90. The solid curve was computed from

the unsimplified theoretical solution given by equation (45). Also shown Is the result of using the approximation given by equation (103), which was

obtained by neglecting the effect of non-linearity

in

the restoring moment.

The difference between the two curves is very small, indicating that, of the

two non-linear terms (in damping and restoring moment), the damping term has

a dominant effect at low amplitudes. The standard deviation of the roll response, GR, was computed as 0.107, which compares favourably with the

simu-lation estimate of 0.102, obtained by Dalzeil, for this case. For comparison purposes, the Rayleigh density function for A(t) (see equation (62)) is also

shown in Fig. 9(a), where GR, as computed by the present theory, has been

used. It is noted that use of the Rayleigh distribution leads to a serious

over-estimation of the probability of A(t) reaching large values.

Fig. 9(b) shows the cumulative distribution A

P(A)

= f

p(E) dE (116)

0

of A(t), for the same set of parameters as in Fig. 9(a), plotted on normal

probability paper. Also shown in this figure are some simulation estimates

obtained by Dalzell [321. To give some idea of the statistical uncertainty inherent in these simulation estimates, approximate 95% confidence limits are

given for the three estimates at the highest amplitudes. Bearing in mind this uncertainty, it is clear that there is good agreement between the simulation

estimates and the present theory. As in Fig . 9(a), the Rayleigh distribution

is seen to overestimate the probability of reaching high amplitudes.

Figs 10(a) and 10(b) show similar comparisons between P(A) vs. A, as

computed from the present theory, and Dalzell's simulation estimates. Here a = 0.1 and b = 1: in Fig. 10(a) = 0.036 and = 0.55, whereas in Fig.10 b)

25

Cw = 0.054 and Q = 1.40. It is observed that very good agreement is

obtained for the case where Q = 0.55; here the response spectrum peak is

to the right of the wave spectrum peak - i.e. the wave spectrum is slowly

varying at frequencies close to the predominant roll frequency. For the case Q = 1.40 the agreement between theory and simulation is much poorer;

here the response spectrum peak is to the left of the wave spectrum peak

-i.e. the steeply sloping flank of the wave spectrum gives the main

contrib-ution to the roll response. In this latter situation, small errors in

simulation methods can be significant. To illustrate this, the theory was

recomputed by using the wave spectrum obtained by Dalzell by processing the

simulated wave input, rather than by using the theoretical wave spectrum

given by equation (111). This correction results in a substantially improved

agreement between theory and simulation. Similar corrections were tried for

the other comparisons reported here, but found to be negligible.

Fig.11 shows the variation of the standard deviation of the roll response,

UR with the standard deviation of the wave input, aw, for 0 = 0.55 and

= 1.40. In both cases a = 0.1 and b = 1.0. Very good agreement is obtained

for the case Q = 0.55. For Q = 1.40 the agreement is again considerably im-proved by using the same correction as in Fig. 10(b).

A range of comparisons between the cumulative distribution, P(A),as

pre-dicted by the present theory, and Dalzell's simulation estimates, are given in Figs. 12, 13, and 14. Related comparisons for the standard deviation of

the roll response, aR, are shown in Figs. 15, 16 and 17. _{In general the}

agree-ment is best for the lowest damped cases, as one would expect from the nature

of the approximations inherent in the theory. For the highest damping values

there is a tendency for the theory to overestimate the probability of the

amp-litude reaching high values - in other words the theory gives conservative

response, CR, the agreement between theory and simulation is quite good

even at the highest values of damping. Again the agreement tends to

deteriorate as the damping and amplitude level increase. 3.5 Comparison with simulation results - Process 2

Fig. 18 shows a comparison between the cumulative distribution, P(A),

as predicted by the present theory at Dalzell's simulation estimates, for

Process 2 wave input. Here a = 04, b = 1.0, Ow = 0.027 and I = 0.95. Asp

expected the agreement is here somewhat poorer than for the results for

Process 3, due to the narrower bandwidth of the excitation. However, it iS

noted that here the damping is relatively high. For lower damping values

one can anticipate a much better agreement. This is suppbrted by the

comp-arison for

0

shown in Fig. 19 which shows that for a = 0.01 and b = 0.1 very good agreement with the simulation estimates is .obtained. The tendency

for the theory to overestimate UR at high damping values, and high amplitude levels, is evidently a little more pronounced here than it was for the results

pertaining to Process 3. 4 Conclusions

The main conclusions are summarised as follows:

By combining averaging techniques with Markov process theory it is

possible to obtain a fairly simple expression for the distribution of

the roll response of a ship In irregular waves. The theory is valid

for arbitrary non-linear damping and,restorihg moment. The main restric-tion to the applicarestric-tion of the theory is that the damping must be fairly

light.

A comparison between predictions from the present theory and simulation

estimates of the roll distribution show that, for small to moderate angles

of roll, there is reasonably good agreement, for a realistic range of

damping values. The agreement is best at the lower damping values and for wide-band wave spectra; this trend is in accord with theoretical

27

For the particular model used for a comparison betweeh theory and

simulation, the non-linear damping effect dominated over the effect

of the non-linear restoring moment, except at very large angles of

roll. For ships with a stiffening restoring moment characteristic

at low roll angles, the non-linearity in the restoring moment may be much more significant. The present theory enables the relative

import-ance of the two non-linearities to be assessed, for a variety of ship

forts.

In principle the theory is applicable to very large angleS- of roll, provided that the damping is not exceSsive. Thus it may well be poss-ible to predict capsize probabilities, by a suitable extension of the

theory. In any event, further work is required to test the validity of the theory for very severe rolling conditions.

Acknowledgements

This work was undertaken during a period when the author was a consultant

at the National Maritime Institute (NMI), Feltham, Middlesex. It is a pleasure to acknowledge the help and advice received from NMI staff. In particular stimulating discussions with Dr. N.Hogben and Dr. R. Standing

were of considerable assistance. The author would also like to thank Mr.

H.H. Pearcey, Head of Research at NMI, for supporting this work, and for his

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Robb, A.M. "A Note on the Rolling of Ships", Trans. INA, Vol.100, 1958,

pp. 396=403.

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Caughey, T.K. "Equivalent Linearisation Techniques", J. Acoust. Soc.Am., Vol.35, No.11, 1963, pp.1706-1711.

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19711

PP.289-294.

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Technique to Nonlinear Rolling in Random Waves", Int. Shipbuilding _Progress,

Vol125, No. 281, 1978, pp.14-18.

Crandall, S.H. "Perturbation Techniques for Random Vibrations of Nonlinear

Systems", J. Acoust. Soc.Am., Vol.35, No.11, 1963, pp.1700=1705.

Yamanouchi, Y. "On the Effect of Non-Linearity of Response _{on Calculation}

of the Spectrum", Trans. 11th. Towing Tank Conf. Tokyo, Japan, 1966. _pp.387-390.

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Sea", Int. Shipbuilding Progress, Vol.23, 1976, pp.209-212.

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Systems to Ship Motion Performance in Random Seas", International

Ship-building Progress, Vol.14, No.150, 1967, pp.54-65.

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Functional Series Approach", Report. No. SIT-DL-76-1874. Stevens _Institute

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Caughey, T.K. "Derivation and Application of the Fokker-Planck Equation, to

Discrete Nonlinear Dynamic Systems subjected to White Random Excitation", J. Acdust. Soc.Am., Vol.35, No.11, 1963, pp.1683-1692.

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Equation to Nonlinear Ship Motions in Random Waves", Int. Shipbuilding

Progress, Vol.21, No.242, 1974, pp.283-288.

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Stochastic Excitation", Proc. Colloq. Int. Cent. Nat. Rech.Sci. Vol.148,

1964.

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and

From equation (30) one obtains the equation

o ab, , ab

-1

m = A i <B{(---) (b ) + (---) ( > dT (Al)

1 1 t 1,.. t+T a?, t

t+T

-m

and a similar expression for m2.

By referring to the definitions of 131 and b2 (See equation (17)) it is

seen that

ab11

abl

av sine. Y(t); = (2V)icose, Y(t) (A2)

(2V)-On substitution of these results into equation (Al) one finds that.

m = A1

+ f

< sine(t) Sine(t+T) + cose(t) cose(t+T)

> w (T)

dT. (A3)

1

Where

w (T)

= EIY(t) Y(t+T)} (A4)

is the correlation function for Y(t). This is related tow

(T)

as follows

(see equation (14)):

wx(T) = E2 w (T) (A5)

To evaluate the time average of the quantity in the < > brackets in

equation (A3) it is convenient to expand sine(t) and cos0(t) as Fourier series. If T is the periodic time then one can write

sine(t) = 1

sn

sin

arrnt

n=

-co

cose(t) = C cos 2ITnt

n

n=1

The coefficients s and

cn, and also T, are functions of the energy level, V. Since X is assumed fixed, and the time origin is arbitrary, the above Fourier

coefficients actually relate to the free, undamped phase solution 00(t).

On combining equations (A6) and (A7) one obtains <sine(t) sine(t+T) + dbse(t) cos (t+T)>

co 1 2 2

L "in 4.n

)

CosLnW(V)] n=1 . where 27 w(V) =

is the frequency of oscillation, at energy level V.

When equation (AS) Is substituted into equation: (A3) it is found that!

= A (Vi.+ a(V)

₍₄₁₉₎

ml

where,

a(17) = _79

(S2

+

C)

S [nw(V)] 267 n=1 n x

And use has been made of equations (12) and (A5).

can be evaluated by a similar procedure. The result is found to be

zero.

Evaluation of the elemente of the diffusion matrix,

From equation (31), it is seen that

co

D

= f

<Ei(b )

11

t)t+d

(Al2)

and Similarly for D12' D21 and P22'

On substitution of the expression for b1, given by equation(17) intO

equation (Al2) one has

-to

'D11 = 2V

f <sine(t)

sine(t+T)? w

(r) dT

(A13)

-00

Again the time averaged quantity in the < > brackets can be evaluated by

ex-panding sine(t) as a Fourier series, according to equation (A6). Hence co <sine(t) 008(t+T)> .

r

2

- sn cPs[nW(V)j n=1 ' (A14)

On combining this result with equatioh (A13) it is found that

D _ s n=1 2 S [do(Vil (A15) 11 2 n x ; E A2 (AS)

D12' D and D2 can be evaluated in a similar fashion. It is

found that

and

D12 = D21 =

0

CO 2 2 Tr

_c

₂

_s

nw(V)]

(A16)

,

n

2VE

n=1

- --RHA

0.2

0

0

01

02

03

0.4

0.5

0.6

0.7

08

0,9

1'O

A

VARIATION OF COSeo (-T) WITH TIME OVER ONE CYCLE

a'

CD

0

-VARIATION OF

SIN% (r) WITH 1-1,ME OVER ONE CYCLE

,111 t

Variation of the modulii of sl, s,3 and s5 with m Variation of the modulii of cl, C3 and c5 with m

t

FIG.6

-111111

11111

I I 1 I

11111

1 i 1 I I I

11-T IL

1

111j

I I I I 1 1 I

11111

I

o

o

,

o

_a

I 1111

1 -1

1111

11

III

1 1 1

A

1-2

1 - 13

0-6

4

2 (QUADRAT 1C;)

1/

.201) 2

VARIATION OF A AND

1:-71 WITH THE AMOLITUDE OF OSCILLATION A

A

I NEAR)

1'0

0-8

0.7

0.6

0.5

0.4

0.3

0,2

0-1

1

0

H3

07

-

H3

(ig)

0.9

08

06

0-5

0-4

0.3

0-2

01

0

THE NON-DIMENSIONAL SPECTRAL FORMS

H2 AND H3

FIG. 8

NA

101

10

-1

10

16

-3

10

0.1

RAYLEIGH

PRESENT THEORY

PRESENT THEORY

WITH LINEAR

RESTORING MOMENT

0.2

0.3

Probability density function for the amplitude A a = 0, b = 1, Ow = 0.036, Q = 0.90. Process 3.

0.4

A

=NI 411 GM/ 0.1

PRESENT

THEORY

RAY I_EIGH

DALZELLIS SIMULATION

ESTIMATES

-I

= 0

b 1.

0-1

02

0 3

0.4

0-S

0-6

A

Cumulative probability distribution for the amplitude A.

a = 0, b = 1, = 0.036, S2 = 0.90. Process 3.

FIG. 9. ( b)

p( A)

0.999

0.99

0.95

0.90

0430

0:70

0-60

050

0.40

0.30

0.20

0.10

005

0.02

-0.1

111=MI =NO .1111 ammll

DALZELLI S

SIMULATION

ESTIMATES

0-0.999

) 0 .991

095

09

0-8

0.7

06

0.5

0.4

_0.3

_{0. 2}

0.1

0.05

1=m,

PRESENTTHEORY--01

(CORRECTED)

Pr

PRESENT

dug,

THEORY

1111. o

0

Al 0) 11 . I-. 1-A II 11 Q Q 11 II

00

.0 "0 . 4). CI) 0 0 InM 0 031. 0 Cr P. H. tfl rt. Cr

rt-#

DALZELLIS

CD

SI MULAT ION

ESTIMATES

5:1: 140

=NO

01-b

O'2

DALZ ELL'S 5 I MU LATION

Q

055

O n

14O

Variation of the standard deviation of roll, GR, with the

standard deviation of the wave input,

qw.

a -,-- 0.1, b = 1.0, 2 = 0.55 and 1.40. Process 3.

FIG.11

0-50

0-40

0.30

0'20

PRESENT

THEORY

b= 1.0

PRESENT

THEORY

b = 0.1

DALZELCS SIMULATION

ESTI MATES

b = 0 -1

b =

1

0

= 0.01

0-02

001

0-005

1

-0-1

0

0-1

0-2

0-3

04

0-5

0-6

Cumulative probability distribution for the amplitude A.

a = 0.01, ow = 0.036, S2 = 0.90 b = 0.1 and 1.0. Process 3.

0-999

P (A)

099

0-95

0-90

0-80

0-70

0-60

0-10

0-05

0 .999

p(A)

0 . 99

95

0.90

-SO

-70

60

0-SO

40

0.30

.20

0.10

0-05

0-02

0-01

0-005

-01

DA LZELL'S SIMULATION

ESTI MATES

Cumulative probability distribution for the amplitude A.

a = 0.03, avi = 0.036, = 0.90 b = 0.1 and 3.0. Process 3.

FIG. 13

0-4

0.5

0-6

0.999

(A)

0-99

0-95

0.90

0.80

0-70

0.60

0 50

0.40

30

0.20

-10

0.05

0-02

001

0-005

DALZ ELL'S SIMULATION

EST I MATES

Cumulative probability diStributioii for the amplitude A. a = 0.1 , aw = 0.036, Q = 0.90

b = 0.3 and 1.0. Process 3.

0.6

0-3

0.4

0.5

DAL ZELI2S

SIMULATION

b:01

b = 0-3

b:10

Variation

of

the standard deviation of roll, CR, with the

standard deviation of the wave inpUt, Cw_

a = 0.01, = 0.90. b = 0.1, .1,0 and 3.0. Process 3,

FIG. 15

:

Variation of the standard deviation of roll, GR, with the

standard deviation of the wave input, GI'q.

0-15

6.10

005

FIG .17

001

0-02

003

crw

0-04

Variation of the standard deviation of roll, On, with the

standard deviation of the wave

input-, auc

= 0.10, 2 = 0.90. b =:0.3 And 1_0. Process 3.

P(A)

0.99

0-90

0-80

070

0-60

0.50

40

-30

0-20

0-10

DALZELLIS SIMULATION

ESTIMATES

Cumulative probability distribution for the amplitude A.

a.= 0.1, b Cw 0.027,

2 =

0.95. Process 2.

Variation of the standard deviation of roll, _{GR, With the} standard deviation of the Wave input, C

= 0.96

a = 0.01 and b = 0.1, a = 0.01 and b = 1.0, a = 0.10.and b = 1.0, Process 2.

FIG. 19