The effect of parametric excitation on ship rolling motion in random waves

National Maritime Institute

lab. v.

Scheepshootikuntlt

Technische

Hogeschool

Delft

The Effect of Parametric Excitation

on

Ship Rolling Motion in Random

_Waves

by

J B Roberts*

*School of Engineering and Applied

Sciences, University of Sussex,

Falmer, Brighton, Sussex, BN 1 9QT

NMI R 100

October 1980

National Maritime-Institute

Feltham

Middlesex TW14 OLQ

0

Crown Copyright 1980

This report is Crown Copyright but may be freely reproduced for all purposes other than advertising providing the source is acknowledged.

National Maritime Institute

The Effect of Parametric Excitation on Ship Rolling Motion in Random Waves

by

J.B. Roberts*

* School of Engineering & Applied Sciences

University of Sussex, Palmer, Brighton, Sussex, BN1 9QT.

NMI Report R 100,

List of Contents

page number Introduction.

2 The equation of motion ₂

3 Averaging the equationS if

The FPK equations

6

5 Linear. damping

. 5.1 The envelope process

8

5,2 The roll angle response ₁₀

5.3 White noise excitation ₁₂ Linear-plus-quadratic damping ₁₃ 0.1 Purely parametric excitation ₁₄ 6.2 Purely non-parametric excitation ₁₆

6.3 Combined excitation ₁₆ 7 Linear-plUS-cubic damping ₁₇ Conclusions ₁₈ Acknowledgements ₁₉ 'References Appendix Figures

Summary

An equation of motion for uncoupled ship rolling motion in

irregular seas is considered in which parametric excitation, arising from the time dependence of the restoring moment, is included. It is

shown that, when the damping is light, the method of stochastic averaging can be applied to yield an approximate Markov model for the rolling motion. AppropriateFokkar-Planck equations are given from which some simple

expressions for the stationary response distribution are derived. For the case of linear damping it is found that the motion is unstable if the damping factor is sufficiently small. The inclusion of a quadratic damping term in the equation of motion is shown to limit the unstable motion predicted by the linear theory. Explicit expressions for the

dis-tribution of the roll response are derived for the case of combined linear

and quadrat c damping. The case of purely parametric excitation is

Notation

non-dimensional amplitude (see equation (46)) non-dimensional. amplitude (see equation (70))

A(t) amplitude process associated With 0(t)

normalisation constant damping function

strength of White noise process (see equation (59))

E{ }

K()

M(t)

p(t)

expectation operator

f2(A) functions defined by equations functions defined by equations

functions defined by equations functions defined by equations

restoring moment coefficient

normalisation constant in equation (48)'(F 1/2n)

restoring moment

modified Besgel function

scaled wave excitation moment .(see equation (4))

wave excitation moment

scaled parametric excitation process (see equation. (4).) p(AIA ; ) transition density function for A(t).

p(A,e1A ,e ;t) transition density function for A(t) and ecty

0

0

Ps(A) stationary density function for A(t)

Ps(A'e) stationary density function for A(t) and e(t)

Pg0)

stationary density function for 0(t)

P5(cj)) stationary density function for

0(t)

and (1)t)

Plt) parametric excitation process

q(t) process defined by equation (7)

Sm(w);Sp(W) spectral density functions for M(t) and

No

respectively

t- time

(20) and (21) respectively (11) and (12) respectively (13) and (14) respectively (15) and (16) respectively

period of oscillation (= 21T/W)

wm

(T)orp.(T). correlation functions for M(t) and p(t), respectively

W

X(X) Whittaker d function

11,

parameters defined by equations (24) to (27)

.roo

qaMMI1 Junction .

T(x,Y)

incomplete Gamma function

6( )

Dirac's delta function

2

Stall scaling parameter (E2 << 1)

linear damping factor

phase angle associated with 0(t) X parameter defined by equation (36)

parameter defined

by

equation ,(83)

parameter defined by equation (67): V cubic damping factor

parameter defined by equation (82)

quadratic damping factor

R standard deviation of roll

R in the linear case

. time lag

0 angle of toll

wt

+ 8

X non-dimensional roll angle

c= Vail)

1 Introduction

It is well:knOWn that ships Can, under suitable conditions, eXhibit d large amplitude resonant response to wave excitation, in the r011 mode. Many theoretical studies of this problem have been'Undertaken since the pioneering work of Froude (1861). Tor-a ship in beam waves an appropriate non-linear equation for rolling motion can

be

formulated fairly easily,

by incorporating the static restoring moment characteristic. A brief

survey of work in this area has been recently given by the author (Roberts

(1980)).

.For, the case of a ship in oblique or following waves, however,. attention

must be paid to the relationship between the restoring moment andthe Wave

profile relative to the ship. As waveS ImsS by the ship there is a

Con-tinuoUs variation in the effective Metacentric height = ie. the restoring

Moment must be treated as a time .dependent quantity in which the dependence

is

related to the wave motion,. When this effect is included in the equation

of motion for roll it appears in the form of 4 parametric excitation Which, in some circumstances, may give rise to instability.

The first study of-the influence' of parametric excitation on roll motion

Was undertaken by Grim (1952), Who.conSidered the Case of sinusoidal waves.

He showed that, when a periodic variation of metacentric, height is

intro-duced into a linearised, single degree of freedom equation for roll, the

equation can be reduced to the standard form, of, the Mathieu equation, Thus

a Mathieu Instability chart can be Used to determine appropriate stability criteria. Related work on the stability of roll motion in periodic waves

has subsequently been described by Kerwin (1955), VoSsers .(1962), Grim (1962),

Wahab and SWaan (1964), Roese (1969), Price (1972, 1975) and Rlocki (1980), Recently some attempts have been made to analyse the problem of roll,.

theory.

Here the parametric roll excitation term must be treated

as

a random process. One approach

to

this problem has been described by Price (1975), Which involves taking expectations of an integral solution

to the equation of motion. The resulting infinite hierarchy of equations was truncated to yield approximate stability Criteria for the mean.and

mean Square response. A rather different approach :has been pursued by

Eaddara (1975) and recently by Muhuri (1980). They treated the

para-metric component of the -restoring moment, and the excitation moment, as

white noise prodesses and assumed that the damping was linear. Under

these.ConditiOns the

'Till

displacement and velocity form a joint, two-dimensiOnal MakkOr process and it is possible to deduce moment Stability

criteria by using standard techniques.

The use of Markov process theory for ship

roll

stability has

the.attrad-tion

that a_powerful body of theory becomes aVailable. In this paper .a

different approach to the uSe ofMarkOv MethOds.is'described, which

is

based on the method of stochaStic averaging due to StratonoVitch (1964)

and Magminskii (1966). It will be Shown that this approach avoids the

need. to idealise the excitation processes as white noises. Furthermore,. it enables the 'effect of non,-linear damping to be incorporated easily into the theory.

2 The equation of_motion

In common with the authors ,previously cited in this paper, it will

be assumed here that the roll motion can be considered as uncoupled from the other motions. A suitable equation Of motion is then Of the .form,

see Block* (1980))

v+

2C(_.(1))

Here 0 is the roll angle

moment and the restoring

+ K((p) +(P(t)cp = M(t)

,r

(1)

and

C(0)

and

K(0)

are proportional to the damping

the parametric contribution to the roll restoring moment and M(t) is proportional to the wave excitation moment. The scaling parameter C2

is such that 62 << 1 and is introduced to emphasise the fact that the

damping is lighti.e. that the response is a narrow-band process.

The excitation moments, P(t) and M(t),

will

be modelled as

station-ary random processes, with zero mean., l'hespectru0 ,

can

where

be defined

by

the relationship

, r

Sp(w) = j coswT dT

(T)

= E{P(t)P(i+T)1 P

is the correlation function for P(t). The spectrum Of M(t), %,()), can

be defined similarly. It is noted that the processes P(t) and M(t) will be Correlated - i.e. their cross-sectruM will be non-zero. As it happens,

however, a knowledge of this cross-spectrum is not required here.

The theoretical approach which will be developed in this paper is based on the assumption that E i8 stall, and in fact is.asymptotically valid as

E 0. It is thus

of

importance to bring out the order of magnitude of

the _terms in equation (1), With respect to the parameter E.,. With M(t) and

-P(t) of order 6o, as implied in equation (1), the Standard deviation of the

roll response, OR, (assuming conditions are such that stationarity is

achiev-able)

turns Out to be of order c . To clarify the developmentof the theory it is convenient to scale P(t) and M(t) so that Gis of order E° This can

be achieved by introducing the processes p(t) and m(i) (of order

6°),

Where

P(t)

= E p(t);

M(t)

= E

m(t) (4)

In addition, the analysis here will be restricted to. those situations

Where the probability of the roll angle reaching very large values (i.e. clOse to the critical roll angle) is negligibly small. K() can then be Approximated

as

(7)

In terms of A(t) and e(t), equation (6) can be expressed as the following pair of exact equatiOns:

A = E2_Fl(A,) GA,(1),t)

6 . E2F2(A,o)

G2(4,(1,t)

(9) (10)

KO) =

_w2o(1)(1 + ke22 ) (5)

where w is the undamped natural frequency of roll and k is

a

restoring

coefficient (which may be positive or negative). In fact, this

approximation is not absolutely necessary, since it is Possible to use

here the general theoretical approach developed recently by the author

(Roberts (1980)) In that paper it was applied to the case of a

Ship

rolling in beam waves, in the absence Of parametric excitation, and

arbitrary non-linear damping and restoring moments were Considered..

However, it was found that the non-linearity in the festorink_moment had a very Week effect on the res Wise distribution except at very high angles

of zol

. This provides some theoretical justification for approximating

the restoring moment, as indicated by equation (5). With this.

representa-tion it is possible to Apply here a Much simplified version of the general theoretical method (Roberts (1980)).

On. combining equations. (I), (4) and (5) one obtains the following

equatiOn

(CI; + 2C(ci)) + w + kE2cF

+ E

q-(t) = Em(t) (6)

\_-where

q(t) = p(t)/w

-3.Averaging the equations

Associated with the roll response process, 4)(0, is an amplitude

process, A(t) and a phase process, 6(t): these are uniquely defined by the equations

.= A doS (wt e) ₍₈₎

where

F (A,0) = 1(A,0) sin0 F2 (A,) = H1 (A,0) cos 0/A G1(A,0,t) = H2 (A,O,t) sin0

G2(A,0,t) = H2(A'0,t) cos 0/A

HI(A,0) = C(-(0 sin0)/w kw A3 cos3

o.

H2(A,0) = - in(t)/wo + w A cps' q (t)

and

ut + 0

(17)

Equations (9) and (10) will now be simplified by averaging over A

period of oscillation, T = 27r/w. This is essentially the method of

Bogoliubov and Mitropoisky (1961), but here the extension tO the stochastic case, as described by Stratonovitch (1964), is applied. The effect of the

averaging is to eliminate the small, rapid oscillations Which are

super-imposed on the relatively large, but slowly varying, components. of A(t) and e(t).

The terms F]. and F2. will be averaged first - this is quite

straight-forward, since these terms do not contain the excitation processes, m(t)-and q(t), explicitly. On averaging, equations (9) m(t)-and (10) become

A

=

E21(A) + c G1(A,O,t)

e =

E2f2(A) + 6 G2.-(A,0,t) where 1 r f1(A) = 27woj loA sin) sine 3kwoA3 kwoA3 r27 f2 (A) ₂₇ j cos40

d0=

-. 5

It is noted that the contribution to f1(A) arises solely from the damping term, since the non-linear stiffness term vanishes on averaging. Conversely,

the Only contribution to f2(A) is from the stiffness term - the damping term

has

a Zero average value.

de = E Where

a = 76 (2w

_{p o}

)/4u2,

_o = ffSM- w_{p o} s

I 7'

f

w (T) sin

2T

dT o o

(0)/2w 2

4 The FPK equations 6

As a final Stage

in

the averaging procedure, the StratonoVitCh KhaSminskii (SK) limit theorem

Will

be applied (see Appendix)., The

effect of this is essentially

to

average the terms Gi and

02.

Application

of the limit theorem shows that the joint process (11,0) convergeS-

weakly,.

as

c4-

0.,.t0 a two-dimensional MarkoV process, governed by the following

Ito equations (see Arnold

(l973)):-dA = (E

72f (A)

at.

A dt + (f3 + aA2

.2A dwi (A) t y)dt + + + (5) dW2 A2

(22)

(23)

(24)

'

(25)

The Markov process (A,13), defined by equations (22) and (23) has a transition probability. density function

p(MIA

,00;t), defined such that

0

p(A,SIAo' eo' .0dAde is the probability that 'A lies in the range A, A + OA,

-atideliesihratige'6,6+a,attimet,giventhatA=Aand 6 = eo

0

time t.

Using standard methods (Arnold

(1973))

a Fokker-Planck-Kolmogiorov (FPK)

equation for the transition probability density can be deduced directly from the Ito equations ((22) and (23)). The result is

7

[ 2 3a f3

I

2

- 7 7,7= ' (6

f1(A) + A + )p

[6

(A) + Y)Pi

at 2A ae

r(

-

2 2 1. a2 r, 1 r 13 +

z

L(f3 + a A2 )131 + + a, +d)pl DA 2 9e2

This equation is of the diffusion type, and the solution must satisfy the

initial condition'

p(A,e[A0,eclit)

(S(A -

A)a(e - 00) as t -0- 0 (29)

An inspection of equation

(32)

reveals that the Ito equation for A(t) is uncoupled from the phase equation. It follows that A(t) is a

one-dimensional MarkoV process, with a transition density p(AIA ;t) governed by the FPK equation

2f

3a a

1 2 2A

/

2

Ltc (A) + A + + = 2 [( + A2)pl

4.1 The stationary solution.,

/

If conditions are such that the response achieves stationarity then the stationary, joint probability density, ps(A,e) is given by

lim

p5 (A,0)

= p(A.,,e1A ,0 it)

t->03 o o

It can be obtained by setting ap/at = 0 in equation (28) and then solving

for

ps(A,e)-The conditions under which stationarity may be attained are dependent on the form of the damping function, fl(A). Here the case of 'linear damping

will be considered initially and later the analysis

will

be extended to

include the effect of non-linear damping.

5 Linear damping

This is the case discussed by Price (1975), Haddara (1975) and MUhuri

(1980), using different. approaches.

On setting

-2

E C(q) = 24%4)

where

4

is the

Usual,

non-dimensional,

lineardampingfactor,-Is

found

from equation (20) that

-E2f1 (A) =

-W

A 5.1 The envelope process

On substitUting equation (33) into equation (30), and setting

= 0, 4- a evident that if ps(A), the stationary density function

ap/at

of A(t), exists it must satisfy the equation

d2

_{(a 4.}

, d r, 3a _a

aA2)p j -

-W

A + + p

=o

2 s dA o 2 2A s

dA

The solution is found to be

2a4X_A

p8(A) ' 2 1+X

.,(a

aA )

'

where -.Q\- 60/a (36) 0-d

This solution will

only

exist provided that A > 0- i.e.

Tr6p(2410)/4wi3a (37)

-

\

Equations (35) and (37) have been obtained recently by Ariatatnam and

Tap. (1979), using: the same approach. They have shown that the inequality Of equation (37) gives the condition for sample stability for the approximate

amplitude procett governed by equation (22), and have given some justification for the assumption that it also gives the condition-for sample Stability of

the original system, described by equatiOn (6).

It is observed that the stability criterion of equation (37) shoWt that the onset Of instability it not affected by the presence

of

the

non-parametric excitation process, M(t). In fact the critical damping factor

depends

only

on the natural frequency of oscillation, wo, and the value of the spectral density of P(t), at twice the natural frequency. This is not unexpected since it is Well known that the stAbility, boundary for a linear

1

(33)

system excited by a periodic, parametric input hag tminimum when the excita-tion frequency is twice that of the natural frequency. For the case

of

purely

9

parametric random excitation the stability criterion of equation (37) was first deduced by Stratonovitch and Romanovskii (1950).

From equation (35) it is a simple matter to evaluate the moments

CO

E{An}

_{=1 Ps(A).AndA}

(38)

On carrying out the integration one finds that

F(1 +

11)

r(x

-E{An} =

a

1'(A + 1) The

nth

Moment exists provided that

X > n/2

Hence, for stability of the first moment

C > 3nS (2W )/8w 3 (41)

p

o o

and for stability of the second moment

3

> nS (w )/2w

p o

Returning now to equation (35) it is noted that, when stability exists, the amplitude distribution is not of the Rayleigh type- However, using the

standard formula

1

(1 + vx)v = exp(x) (43)

v±o

one finds, from equation (35), that as a 0 the Rayleigh form

A r A2 i p(A) = T-2expl - ,21 aR 2aR is approached, Where nSm(wo) 3 2Cw (42)

This result, for purely non-parametric excitation, can be obtained from the

a'

2

10

well-known linear spectral theory (assuming the restoring moment is linear). The equation for Ci, the standard deviation of the roll response, corresponds to the result obtained by using a white noise approximation for the

excita-tion process 1W(t), with spectral levels matching at wo.

If one introduces the non-dimensional amplitude

a = A/Cr

then, from equation (35), the probability density Of a is given by

p(a) =

Fig.1 showS the variation of p5(a) with a few various values of A,

including A 4- 00 (the Rayleigh case).

5.2 The 17611 angle response

It is evident from equation (28) that, just as in the case of non-parametric excitation (Roberts (1976)), the stationSry solution is of the

form

ps(A.,0) = K. ps(A) (48)

where p5(A) is given by equation.

(35).

and, from the normalisation con-.

dltion, K =

1/27.,

Of course, such a solution only exists if IVA) exists

-i.e.,

if A> 0. It follows from equation (48) that, when Stationarity is

readied, the phase e is uniformly distributed between 0 and 27.

By a transformation from A, e variables to 0,,equations (48) and (35) yield an expression for the joint density, ps(0,0) of 0 and

0.

The result

is P5(4),c1)) = a2 11 2(1

+A)

otAR TW0 cr3

c(2

(i)2/co 2)/1+A

From this expression a wide variety of roll response statistics can be

(46)

(47)

11

calculated - e.g. the average number of crossings per unit time of a

critical amplitude level. As a 0 equation (49) reduces to the familiar bivariate Gaussian result for non-parametric excitation, i.e.

2 1 P (c1",$) Z71: exPf -121). '2

a'2V

21 7R wo

On integrating equation (50) With respect to ary density function for

0;

thus

a1

Ar(A + i)

Ps"))

4 -

- 2 i+X

Ta+1)

aci) )

As a 4. 0 this reduces to the Gaussian form

,2

P

ps(0)

exp ,

(27)

aR 2c2'

Equation (51) can be dimensionalised by introducing the non-dimensional amplitude

X = (i)/e (53)

The probability density for x is then, from equation(51),

i)

Ps(x)

=

_r

_{2 i+A}

1)

L2u(1

a

ri

x

3

2(1 + X)

Fig. 2 shows the

variationofps(x)

with

x

for various values of A including the Gaussian case. (X

m).

From equation (51) one can readily evaluate the moments of the roll

r ni

angle response, E{}. result is

l+n

n a r( 2 )r(x-Eio n/2 1/2

a

7

r(x

+ 1) 0 n odd (50)

one obtains the

station-(54)

n even

12

and the nth moment exists provided that

X > 0/2 (56)

It is noted that the condition for the nth moment of cli(t) to exist is identical to the condition for. the nth- moment of A(t) to exist (see

equation (40)).

For

n = 2 equation (65) gives the mean square roll angle

Ei(f)2}

= a

= a . _ m o -R 2a(X - 1) [2cwia3 - ffSp(2w6)] .7rS -(63 ) (52)

which reduces to equation (45), for non-parametric excitation, as Sp(2w0) +0.

2

As S (2W ) increases from zero

a

increases continuously, becoming infinite

P o

as X 4- 1.

5.3 White noise excitation:

.If P(t) and M(t) are treated

as

White noiSe processes then it

is

Un-necessary to use the stOchastic averaging method, in the linear case. It

is possible to derive results directly from the Ito equations for the joint

process (1),$. With this approach Muhuri (1980) has shown that mean square

stability exists if

4W3

> D (58)

Where. p i the "Strength" of the-white noise processj3C0

EIP(t) P(t + T)I

ms(t -

T) .(58)

From eqUation (2) the spectral level of P(t) is

Sp(w) = -17

and henCe, from equation (58)

> TrSP(0)/2w3₀

According to the present theory, derived from the stochastic averaging method, Mean square stability exists if

X > 1,

or

> TrSP (2w0 )/2w03 (62)

S (0) (a constant) (60)

Evidently equation (62) agrees with the exact result for the case of

White noise parametric excitation. It is worth noting, however, that the present theory takes the shape of the spectrum of P(t) into account

-this is not possible in the approach of Muhuri.

6 Linear plus .quadratic damping

A number of authors (e.g. palzell (1978)) have shown that, for many

ship forms, the damping function C($) is distinctly nonlinear. It is

worthwhile, therefore, to examine the influence of non-linearities in

damp-ing on the distribution of the roll response.

Initially it

will

be assumed that the damping is a combination

of

linear

and quadratic forma. _{This has been shown to be a good model in many cases,}

(e.g.. see Dalzell (1973)). _{(It is noted, however, that the present method}

can be applied to any form of non-linearity). Hence it is assumed that

p5(A) = 2 6

CO)

= 2C1.0.04) 13 pici)1(1) r 1 (3 -1r a- ,1 exp 1- _{LA -} tan

t(-4

a

Where c is a normalisation constant such that

CO

f p

(A)dA = 1

o s

(63)

Where

0 is

the quadratic damping /actor. On averaging over one cycle it is

found that (from equations (20) and (63))

E

f1(A)

= - co:Ak +

o pAl

By substituting this result into equations (28) and (30) one can obtain

appropriate FPK equations for the case of non-linear damping. As in the _linear case, the stationary solution can be found by setting

ap/at 7

O. From

equation (30) one finds that the stationary density function for A(t) is

given by (64) (65) (66) c A (13 + otA 1+

and

14

(67)

For the speClar-dase of linear damping ( p 0) equation (65) reduceS to the resUlt-found earlier (equation (35)).

BefOre discussing this solution

in

general terms it is useful to examine

two particular, important

cases-6A Purely parametric excitation

It was shown earlier that, in the Case of linear damping, the motion is

stable if X <

0.1When this condition lA satisfied the response level is

-determined by the level of the excitation process M(t)'. When M(t) is absent

the roll response is zero.

When non-linear, qUadratic.damping is included, in the theoretical model the, Of Course, the motion is still Stable -11,X > 0. However, now if A .< 0

it is possible for the nonlinear damping to limit the amplitude of the

Motion

in

other words it is possible for the roll motion to reach a station-ary condition, with a non-zero standard deviatibb. This is analogous to the phenomenon of limit cycling, which is well known in deterministic non-linear

vibrations-. In physical terms the Motion, which is unstable when A <

(according to the linear theory) will build

up until the mean rate of energy

dissipation due to damping is equal to the Mean rate of energy ,input from the

parametricexcitation

Since M(t) is assumed here to be zero, it follows that .=0

equation (65) reduces to

-(A) - ,exp{

p

_{i+x 1+2A}

_-

-

_/

a

A

where, from equation (66), the normalisation constant is given by

c

= p

a

/1(-2A)

15

the

non-dimensional amplitude.

4,

where .

= A/11 (70)

The probability density for a is then

psi

-r(-2X)-1+2Aa exP(-a)

< 0) (71)

The variation of p(a) with i, for various values of A, is shown in Fig.3. From equation (70) it is

a

simple matter to evaluate the Moments,

of

A(t)..One finds that

r(n - 2A)

EiAn} (A < 0)

r1,2x)

These moments will approach zer45 as A '÷ 0.

The stationary joint distribution of A(t) and e(t) is again of the form

Of equation (48). Hence., by a transfOrmatiOn from A,k) variables to

variables one finds that the joint density of 0 and 0 is given by

2 2 i 2A

P$

-/,f(o0

1

Ps(0,i0)

''=' ,2 1+X eXp{. [/° _P 27woN-2X).[CP2 co (A < 0)

From this various StatiStics of the response can be calculated. The simplest, and most important, of these is the standard deviation of the roll angle, aR, which is found tO be given by

2

a = A.(2X - 1)p2 _{< 0)} (74)

-Assuming that the linear cramping coefficient,

C, is

positive then the

minimum possible value of X is -1, and it assumes this Value in the special

case *here 0 (i.e. pure quadratic 'damping).. For this case equation (73) reduces

to

ps(15,4) 1 .271-w o1.12 exp{ 43,2 :2 2 wo 1 _ } (75) (i) ,(I) (73)

CW A2 .A

Ps(A) = eXp

{7.

16

A

and on idtegrating withcrespect to $, one finds that' the density function for 0(t)

ia

given by

Ps0) =

K

(I)

2 -1 p

np

Here K1( ) is the Modified Bessel function. The Corresponding standard deviation of the roll angle is given by

ak

= = 1.603 S (2w o)/pw3

p o

The above analysis clearly shows that the linear and quadratic compon-ents of damping have different effects, so,far as Stability is concerned.

In the absence of the excitation process, M(t), there is a critical value of

C, above which the roll motion is stable. However, for purely quadratic

damping there is no critical value of the coefficient p. In this case

- is in,f act non-zero for all finite Values of p and only approaches zero as

6,2 Purely non-parametric exCitation

When the 'strength' Of the parametric excitation approaches zero,

CL -4- 0 and eqUation (65) reduces to

8Pw A. co

9n8

(76 ) (77) (78)

It is clear from this expression that the nOn-linear component of the damping

has the effect, of increasing the rate of decay.of p5(A) with increasing A,

and that this effect will be most pronounced

in

the 'tails' of the distrib-ution.

It is shown

in

a recent paper by the author (Roberts (1980)) that equation (78) gives results in good agreement with a set of digital simulation estimates fOr the distribution of A(t), obtained by Dalzell (1973).

6.3 Combined excitation

17

where both M(t) and P(t) are non-zero. By combining this resUlt,With

equation (48),

and

transforming to

0,0

coordinates, one can obtain.

an

explicit expression for

p5(0,0).

From this,

on

Integration one-can Obtain

.statistics such asUhfortUnately.it is difficult to carry out the

R'

appropriate integrations analytically, in this general Case; but it is a simple matter to apply numerical integration procedures.

Other forts Of non-linear damping lead to simpler-analytical expressions for the stationary density functions,' In the remaining part of this paper, one such fort is examined briefly.

7..Linear-plua,eubic damping Here it

is

assUmed that

62 c($) 2;(00$ Nci)3/Wo

(79)

Dalaell

(1978) has shown that this model fits 'a variety of experimental

data about as well as (and, in some cases, better than) the linear-plus-quadratic model, if the coefficients and are suitably choSen

On averaging C (), according to equation (79), Over one cycle one

obtains

E fi(A)--w A(C + 3vA2 )

o 8

and, following the sate procedure as before, the stationary density function for A(t) is found to be

(80)

Where

and

A = (C

a

The nottaliaation constant C is .given A_A-1 exp(-a/aEl r(lrAA/a ) p (A) = -

CA

eXp{ + aA ) (81) (83) by (84)-= 8a/3W0 (82)

18

where

r(x,y) is

the incomplete Gamma function,

It is interesting to note that equation (81)'gives the Rayleigh'

form of density function

in

the special case where A = 0.

Again, various roll statistics can be deduced from equation (81), by

using equation (48) and transforming to (1),0 coordinates.

For

eXample the

mean square response is given by

2 ₌

13,A)./2

(3(1-A)/2

. exp(---) W

R 4 aE -1-A

2

where

WA

11,

is

Whittaker's function, which is related to the coefficient

hypergeometric function. For the special case of purely parametric

excita-tion this reduces to

GR2

4/2

(A < 0) (86)

8.Concluslons

The main conclusions are summarised as follOws:

1, The joint alplitude and phase process, (A,e)iassociated with the roll response approaches a two-dimensional Markov proceSs as the damping

approaches zero; 'Related analyses (e.g. Roberts (1976) and (1980)) show that the Markov model is likely

to

be good apprOxitation.for the

finite, but small, damping values Which are usually encountered in

prac-tice. For a given magnitude Of damping the accuracy

of

the Markov model

will improve

as

the bandwidths of the excitation processes inCrease.

-(Roberts (1980)).

2. In the Markov approximation, the amplitude process A(t) is uncoupled

from the phase equation. This enables the stationary distribution of,

the amplitude response to be found analytically (if it exists)._

I.

(85)

3.. In the case of linear damping, there is a critical Value of the damping factor, below Which the roll motion is unstable. The critical Value

19

is proportional to the spectral density of the parametric excitation

process-, evaluated at twice the natural frequency of roll.

When non-linear damping is included, the .motion is found to be Stable under

all

conditions - i.e. stationary conditiOns are achievable Under

all conditions.

When stationarity exists the joint distribution of the roll angle and

N Velocity

is

simply related to the distribution Of A(t), This enables the probability denSity of the roll angle to be found, by

integration,-and hence the Stintegration,-andard deviation Of the

roll

motion,

In the Special case of

purely

parametric excitation the analysis is

considerably simplified. Here roll Motion only exists

if

is less than its critical value. Particularly simple expressionS.for the

Statistics of the roll motion are found in, the case of pure quadratic

damping.-Acknowledgements

This work was begun

during

a. period When the author: wasa consultant

at the National Maritime Institute (NMI), Teltham, Middlesex. It

is

a.

pleasure to acknowledge the help and advice received from NMI. staff. In

particular, stimulating discussions with Dr. 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}

References

1, Ariaratnam, S.T. and Tam, D.S.F. "Random Vibration and Stability of a Linear Parametrically Excited Oscillator", ZAMM, Vol.59, pp.79-84,

1979.

Arnold, L. "Stochastic Differential Equations Theory and Applications", J,Wiley, New York, 1973_

Bpese, P. "Steering

of

a Ship in a Heavy Following Seaway", Jahbuch der SchiffbaUteChnisChenGessellschaft, 1969 (DRIC TransiationNo,..2807). 4..Bogoliubov, N.N. And Mitropolsky,y.A. "Asymptotic Methods in the Theory

of Non-Linear Oscillations", Gordon and Breach, New York, 1961.

Blocki,W. "Ship Safety in Connection with Parametric Resonance Of the

Roll"', International Shipbuilding Progress, Vol.27, pp.36-52, 1980.

Dalzell,J.F. "A Note on the Distribution of Maxima of Ship Rolling", Journal of Ship Research, Vol.17, No.4, pp.217-226. 1973.

Dalzell,J.E. "A Note on the Form of Ship Roll Damping', Journal o Ship.Research, VO1.22, No.3, pp.178-185, 1978.,

8.- Froude,W. "On the Rolling ofShips", Transactions of the Institution of Naval Architects, Vo1.2, pp480-229. 1861...

9_ Grim,0.."RolisChwingungen Stabilitat ünd Sicherkeit im Seegang",.

Schiffstechnik, Band 1, pp-10.21, 1952.

10.Grim, O. "Surging Motion and Broaching Tendencies in a. Severe Irregular Sea",, Stevens Institute Of Technology, Report No.929, 1962.

Haddara,M.R. "A Study of the Stability of the Mean and Variance of Rolling Motion in Random Waves", Int.Conf. on Stability of Ships and Ocean Vehicles, March 1975. University of Strathclyde, Glasgow.

Kerwin,J.E. "Notes on Rolling in Longitudinal Waves", International Shipbuilding Progress, Vol.2, pp.597-614.-1955.

Khasminskii,.R.Z. "A Limit Theorem for the Solution of'Diffei'ential

Equations with Random Right Hand Sides", Theory of Probability and its

Applications, Vol_11, pp.390-405,' 1966.

14: Muhuri, P.K. "A Study of the Stability of the Rolling Motion of a Ship in an Irregular Seaway",International Shipbuilding Progress, Vol.27,

pp.139-142, 1980.

15. Price, LG. "The Stability of A Ship in a Simple Sinusoidal Wave", International Symposium on Directional Stability and Control of Bodies

Moving in Water-. (ed. R,E.D. Bishop, A.G. Parkinson, and R.. Eatock

Taylor), pp.167-179. I.Mech.E London. 1972.

16.Price,W.G. "A Stability Analysis of the Roll Motion of a Ship in an Irregular Seaway", International Shipbuilding Progress, Vol_22,

pp,103-112,.1975.

17. Roberts, J.B. "Stationary Response of Oscillators with Non-linear Damping to Random Excitation", Journal of Sound and Vibration, Vol.50, pp.145-156, 191.

Robets,J.B. "A

Irregular Sear. Stratonovitch,R. and Breach, New

21

Stochastic Theory for Non-linear Ship Rolling in

1980. (to be published).

Iv?

_{1.7-6 cych)}

L. "Topics in the Theory of Random Noise", Gordon

York, 1964.

Stratonovitch,R.L. and Romanovskii,Yu.M., "Parametric Effect of .a

Random Force on Linear and Non-Linear Oscillatory Systems". Nauchnye Doklady Vysshei Shkoly, Fiziko=Mat. Nauk., Vol.3, pp.221. 1958.

Translated in "Non-Linear Transformations of Stochastic Processes", ed. by P.I. Kuznetsov, R.L. Stratonovitch and V.I. Tikhonov,

Pergamon Press, 1965, pp.322-326.

VossersG. "Resistance, Propulsion and Steering of Ships", The Technical Publishing Company, The Netherlands, 1962.

Wahab,,R. and,Swaan, W.A. "Coursekeeping and Broaching of Ships in Following Seas", International Shipbuilding Progress, Vol_11, pp.

Appendix

-Equations

MO

and (19) can be east into the following Matrix fOrit:

Z 62-f(Z) + E q(z4t) (Al)

Where

A

The SK limit theorem (see Stratonovitch (1964) and Khasminskii (1966)) states that, for equations of the form (Al), as E 0 the vector prodess Z(t)

/S.

converges weakly to a MArkov process, governed by the Ito equation

dZ = c2 m dt + E D dW (AS)

Here W is'a vector of unitWiener processes and 'm and D are, respectively,

the drift And diffusion matrices, given by

Dp

<g{(-=)

(G')

} > dT OZ t4-t DD'

= f

<E {(G) (G) } > dT (A5) t+T

In the Above < > denotes a time average, which in this case is over one

period of oscillation. T = 27r/w0. The time subscript, t or t+T , indicates

that the quantities are to be evaluated at those times. In carrying out the expectation operation, A and e are treated as constants.

r

,(A2)

I 0

eit

2

The probability density funcion for

0(t)

fox Various

vAues of X. Linear damping_

FIG. 2

611

I0

01

-01 1 1 2 6

Thu probability density function for A(t) for various

values of -;_. COMbined linear and quadratic damping. Parametric excitation only.

3 4