A stability analysis of the roll motion of a ship in an regular seaway

A STABILITY ANALYSIS OF THE ROLL MOTION

OF A SHIP IN AN IRREGULAR SEAWAY

by WiG. PRICE').

The restoring moment coefficient in the differential equation describing the roll motion of a ship

is assumed time dependent - being either sinusoidal or random in form. For a sinusoidal coefficient the motion equation reduces to the Mathieu type and the stability of motion discussed. For a ship in an irregular seaway, the coefficient will be a random time dependent function. The concepts of sta-bility from a probabilistic viewpoint are defined and a method summarised how the stasta-bility of mo-tion in this situamo-tion may be investigated.

1. Introduction

Although the motion and stability of .a ship in the horizontal plane in calm water have been ex-tensively studied and documented (e. g. see Mandel (1967)) the stability of a ship in a seaway has received less attention. This latter problem

has been investigated by Davidson (1918). Suther

-land and Korvin-Kroukovsky (1948), Weinblum and St. Denis (1950), Weinblurn (1951). Rydill

(1959), Lewis and Numata (1960), and Tasai (1968).

In a deterministic analysis, the seaway is usu-ally describedby a sinusoidal wave of known am-plitude, frequency and direction of travel such that the resultant ship motion may be written in a deterministic form. The mathematical model of the ship-wave system usually involves linear differential equations with constant coefficients.

However, Grim (1932), when investigating the stability of the roll motion of a ship in sinusoidal

head waves deemed it necessary to include a periodic restoring moment coefficient in the

mathematical model. s regular sinusoidal waves of frequency w pass along the ship there occurs

a variation in the metacentric height which is greater or less than the static value vhen the

wave trough and wave crest are respectively a--midships. Vossers (1962) described the rolling mode by the differential equation.

(t) + 2k (,), ci)(t) +2(1 +3 cos wt)

=K

cost

where i)(t)=dc (t)/dt etc.. k is the damping

fac-) Department of Mechanical Engineering, University College London.

tor. wo is the natural frequency of roll motion,

Kocip is the wave rolling moment amplitude and 3 is the amplitude of the variation of the

meta-centric height.

Time dependent coefficients have also been in-corporated into the mathematical models of Grim (1962), Wahab and Swaan (1964), Boese (1969) and Price (1972) when investigating the stability and control of a ship in following waves. Such co-efficients result from the interaction between the ship-sinusoidal wave disturbance and max' have complicated forms involving the characteristics of the sinusoidal wave and ship parameters.

Unfortunately, in nature, regular sinusoidal waves rarely, if ever, exist. In general, for a

random seaway, the resultant wave force or mo-ment and the time dependent coefficients are also random in form. In such a situation, the equa-tion of roll moequa-tion may be assumed expressable as,

103

Ci)(t) + 2kw0 cb(t) + T.(t`, +c(t) c.^:(t) =K(t) 0

where, analogous to the previous deterministic

equation, the coefficient w20;_, cosczt and wave mo -mentK coswt are replaced.by the random pro-cesses a(t) and K(t) respectively.

The two random processes ca(t) and K(t) are pro

-duced by the same irregular seaway and are there-fore probably inter -related. It is assumed that a(t) and K(t) are both stationary random processes though this assumption is invalid for the roll

re-sponse random process 7.,,(t.) when the ter m a(t) ap(t)

exists in the differential equation.

Due to the complexity of the random process Delft University of _Technology Ship Hydromecimnics Laboratori

Library Mekelweg 2- 2628 CD Delft The Netherlands Phone: 31 15 786373 - Fax: 31 15 781836

-104

coefficients, ageneral solution of the differential equation describing the roll motion is most un-likely. However, in this investigation. ways of defining the 'stochastic stability of the motion of the ship are discussed and the resulting stability criteria determined. In order to simply illustrate the concepts and method used, the roll model is initially assumed described by a linear

differ-ential equation with constant coefficients (a(t) = 0)

and the conditions for the stability of motion of a ship in a sinusoidal and random seaway are de-termined.

2. Concepts of stability

Since the coefficients of the differential roll

motion equation may be random functions or

para-metric excitations the determination of the stability of the motion becomes a probabilistic prob -lem. However, the stability concepts of any

de-terministic system are usually based on the Lyapunov stability criteria (for example, see

Cesari (1959)) and need be generalised in order to discuss the stability of the random system.

In the deterministic case, let 0 E 0 be the librium solution of the roll motion. Let the equi-librium be disturbed to an initial state To at time to so that for all t> to the ensuing roll motion is 0(t). The equilibrium solution is deemed stable

if, given E >0, there exists a (E, to) such that for all t >to

110(t)11 < E

provided that 11 00L <1 initially and where the

norm

! I 0(t) I I = 0(01

can be chosen independent of to, the equi-librium roll motion solution 0E0 is referred to as uniformly stable.

If 0 a ais stable, and if, in addition Lim I 0(t) I = 0

t

equilibrium is said to be asymptotically stable. Further, if these stability conditions hold for any initial roll motion 0_o the solution is asymptotic -ally stable in the large.

In other words, the roll motion of the ship is considered to be asymptotically stable if, after the ship has been disturbed from its initial

equi-librium reference condition the resultant roll

motion dies out with time. If the roll motion in-creases with time then the motion is considered unstable.

A generalisation of such deterministic stability criterion encompasses the definition of route or motion stability as proposed by the I. T. T. C. (1972). i.e. The stability of route or motion

sta-bility is that property of a body or ship which

causes it, when disturbed, to damp out extrane-ous motions set up by the disturbance and to re-duce them progressively to zero.

In the stochastic case, the norm 1! 0(011 is a random process and the conditions of stability of a deterministic system must be reformulated.

The stability concepts in the deterministic roll motion case are defined in terms of convergence relative to parameters such as the initial

condi-tion, 00. or the time parameter, t, in the

limit-ing process. It follows that there are at least as

many stability concepts for the study of the

stochastic motion as there are for the study of the deterministic roll motion. The deterministic con-cepts of stability have their counterparts in each of the common modes of convergence of prob-ability theory, i.e. convergence in probprob-ability,

expectations, almost sure converence etc.. In

fact, there are at least three times as many con-cepts of stability for the stochastic roll motion as for the usual deterministic motion.

Kozin (1969) redefined the previous determ-inistic stability criterion in terms of the stability of the expectations of the stochastic motion as

follows:

a. Stability of the mean: The equilibrium roll

motion solution has stability in the mean if the expectation exists and if, given E > 0, there ex-ists a n( E. to) such that for all t > to, the mean value of the norm,

E[ I I 0 (t) II 1< 6

provided that I 00 II < I initially. If,

Lim E[11 (t)11 1=0 t

the equilibrium motion solution is said to have asymptotic stability of the mean.

In other words, the stochastic roll motion of the ship is considered to have asymptotic stability of the mean if after the initial roll making dis-turbance, the mean roll motion dies out with II

-time. lithe mean roil motion increases with time then the stochastic motion is considered unstable in the mean.

b. Stability of the mean square: The equilibrium roll motion solution has mean square stability if, given E >9, there exists an n(E , to) such that for

all t > to the mean square value of the norm,.

El I 0(0'1 I 21 <

provided that I 001 < initially and where the square of the norm is by

1110(0112-11 crqt)ii 2

If,

Lim Ef !Hat)112 _{= 0} t

the equilibrium roll motion solution is said to,

have asymptotic stability of the mean square. This definition may be expressed similarly to (a): but with 'mean roll motion' replaced by the 'mean square roll motion', It is on these criteria that the stochastic stability of the roll motion is now defined. They are the counterpart concepts: usedindefining the stability of the deterministic roll motion..

3. Stochastic differentiation

The derivative of the stochastic roll 'motion

cD(t) may be defined as a limit i.e.

ocb(t) _d 0(0_ 10(t.5) - cD(t)

dt _.0 5

If the limit exists for all functions of the motion process (1:)(t), then 6(0 has the usual meaning of aderivative. Furthermore, the mean yalue of the derivative is given by,

E[cb(t)]=E[Lim /_8.0

cb"

8) -"c13(t)} 1{EI cp(t + 8)]i

(t)}

11 '=Lim

8.0

1 5

=E

(t

-Thus, the expected value of the stochastic

pro-cess cD(t) is differentiable and its derivative. equals E[ (t)11. Hence it follows that if P denotes a differential operator we have,

cpE[(t)], = El Pi 0(01T

The auto -correlation function of the roll motion process cD(t) at time t =ti and t=t2 is defined by

Roo t2) =-E[ cD(t1), cD(t2):1

whilst it can easily be shown, (e.g. see Price, and Bishop (1974)) that the auto correlation func -tion of the stochastic roll mo-tion derivatives

satisfy the relationships,. d2Roo ( ti, t2)1

Rci)cb(ti

-d ti -dt2

and if the process is stationary then,

-d2111)01N R

(T)1-0cD

where T t

1"

'I, Stability of simple deterministie and stochastic systems,

For simplicity, we shall Iconsider initially the, following linear differential equation with con-stant coefficients which is assumed to describe the roll motion of the ship to a known bounded de-terministic roll moment i. a.

pi (Kt) c, t) +2,1; 030q)(t) 4-t020 9(0 = K(t)

where the differential operator d2"

PI 11 + 2kca +c,)2.

°.dt 3

dt2

-The determination of stability criteria may be ob-tained from the impulsive equation,

P ci)(t)l 5(t)1

where am is the Dirac function. Multiplying the equation by e-iwt and integrating over the time interval produces the Fourier transform

1 1 1

)=H)= =

{

+ 2ilccoo

with an inverse relationship

1 e'cot 9(t,) -2Tr _{-r.42 +Liii.c40,c,)} +wo2 1 .2Tr H(w) eiwt Q. h(t) Coo,

where the receptanceII(ca) and impulsive function 2 T I t2) =t2 = -) E[ I

106

h(t) satisfy the relationship. from the initial conditions the solution of the. stochastic differential equation is given by,

h(t) =

27 5 H(w) eiwt du, and

00

H(w) 5 h(t) e dt.

By the residue theory, the roll response is given

by,

2

?(t)

2

_Are

r =1

where Ar is a constant and wr is a characteristic root of the characteristic equation,

-(42 +2ikw0 w+(42 = 0_o

Thus,

=ik L'CF- 1-1c2)

and the impulse roll response function -kw V(1 -k2)it

p(t) =Ale1 o o

+A2 e -k2)1t

which in the limit as t-.- becomes zero provided that 0<k <1, u> 0 indicating that the system is

asymptotically stable.

If the roll moment, K(t), is a bounded station-ary random process in the mean square, the re-sulting roll response is also a random process and the above procedure must be modified. We may obtain a deterministic differential equation by taking the ensemble average of the stochastic differential equation. This results in an equation in the deterministic variable E[(t)] and

determ-inistic moment E[K(t)] is given by,

P1E[c(t)] =E[K(t)].

By now repeating the above procedure we obtain.

E[?(t)] =131

el-i"o+I01-

-k21k t

2)t

+B2e/-1-")o -1'01/0- -1<

and the system is asymptotically stable in the mean provided that 0< k <1 and wo >O.

Apart from the deterministic terms obtained

p(t) = f h(t --r) K(7) d

where the impulse function, -kwot

h(t) - sinc0:V1 -k2)t t >0

V(1 -k2)

=0 t<0

and h(t -T) = 0 for 7 >t.

By multiplying the response 9(t) at times t1, t2 and averaging we have.

R( t1,

=E[c(ti) o(t2)1= f

f h(ti

-h(t2 -T2) E[K(T1)K(72)] d Ti d

= I

5 h(ti - h(t2 -T2)

-RKK(71 72) dTid 72.

By modifying the method used by Samuels (1061)

we may transform this integral relationship into the differential equation,

P21R99(ti, t2)=RKK(ti -t2) where

p.+ 2k w

dt. 9 2 ° - ° d i =1, 2. and -Ti)1 -5(ti -70. The change of variable,

T =t1 t2 t

transforms the differential equation to,

M211=199(t,.t - T)i =RI<K(T)

where the differential operators,

2 a a _{a4.° N., 2}

=(--+)

2k cz

° at

art 0 and 9 r

,

_{a , 2} . M₂₌ 9 " oT 0 87' t p-kco + 0 Ti) P1 -at

In order to determine criteria for the stability of the mean square when the roll moment

fu,nc--ion bounded in the mean square, it is

suffi-cient to consider the impulsive equation,

M2.1R9p . t - 7)1 = 5 (t) 5(7).

The solution of this equation obtained by using the double Fourier transform is given by.

.R9p(t, t -7 2 2 RcpT (t, t -7)

2

2 A

e M T =1 M eilwt + AO. - dcz,d

2.

2,2

. 2

(w -21k wow -coo)( A -211: co6X -coo)

Which by the residue theorem takes the form,

ic)rt.4-Am(t -7)4

where Arm denotes a con and corstant XM are the

eigenvalues of the characteristic equation,

x,

(0o2 - 2ik coo 2 ik0 =

The roots of this algebraic equation are

=1"b+_`')o1417-7:2')

=ikca C+) 17.

and at 7 =0, the mean square value of the response 'is given by,

R (t t)--=E[cp2(t)II 99 =A e-21<w,ut +Ale2kczct+ico

1-17;2)

o + A2 e -k -icoolti(1 -k2)1't

The system is asymptotically stable in the mean square provided that OK lc,- 1 and coo >0 since in

this case,

Lim Rpp (t,, t) 0, t 4.=

5. Stability- of dotermini4r. time dependent system

A more realistic mathematical model describ-ing the rolldescrib-ing mode of a ship. in a deterministic seaway was considered by Grim (1952) in which the variation of the metacentric. height was in-. eluded. This introduced a periodic time dependent coefficient into the differential equation which

was expressed by Vossers. (1962) as

,6(t) +2k(oo $(t) +c 9(1 + p cos um. cDf 0 0

Koo, cos cot.

The homogeneous equation is given by

+2"kco0 $(t)+4;(1.. +Acoswt) $(t) =0

which by the transformation,

t

$(t) =y(t) e x =c,)t

reduces to the following equation, 9 +1(.6 + cos x) y = 0 dx2 where co2(1 -k2) 2 and

E

--The homogeneous equation has reduced to the standard form of the Mathieu equation. Although the general solution of the Mathieu equation is. not known, the stability of its solutions may be 'deduced from a stability chart in the (a, a) plane.. This chart consists of a set of curves of the form 5=5(a) which divides the ( E , a) plane into stable and unstable regions. For values of ( ), which

define ,a point embedded in a stable region, the. solutions of the Mathieu equation are bounded and stable. For values of (a /4-0, 5) defining a point on the boundary, or embeddedin an unstable region, the solutions of the corresponding Mathieu equa-tion are unbounded and unstable. The stability chart of a Mathieu equation is shown i.n Figure 1

where the shaded areas are the stable regions. Thecurves 5 =53( a) defining the boundary between stable and unstable regions

have been

ap-proximated by Stoker (195.0). as power expansions in a., and these approximations are satisfactory for moderate values. of E as shown in Figure 2.

The stability chart shows that the, unstable

regions approach the 8-axis in indefinitely .close in the neighbourhood of

m2

= for m =01, 4-1, -"2,

.

If = 0 and a :0 (k .1)., all solutions of the free motion.are bounded and stable_ However, for ar

-bitrary values of the parameters .a and 5, the stability condition may be obtained from the ( a , 5)1

chart. For a damping factor k <1 and parameter

1 4+(2 1 is = t = (t) y E -,

108

Figure 1.

Figure 2.

=1+

0 (i.e. 5 > 0 and E j= 0 respectively) values of

the wave frequency may exist such that solutions are unstable. These values of wave frequency are

given by

c)

Oim T. 2

(1 -k-) --=

(1 - k ) for m - 0,1 , 2, ...

2 _To

where T is the wave period and To is the natural wave period of roll. The above instability

cri-terion for k 0 was first determined by Grim

(1952) when analysing the roll behaviour of a ship

running in head waves and further demonstrated experimentally by Kerwin (1955).

The non-homogeneous equation may be ex-pressed as

9

d- y

2(0

COS x) y K4` cos x

dx

where K*_an =K_anA.2. This has again a Mathieu

equation form but now includes a roll moment term of the same frequency as the time dependent coefficient. Rosenberg (1954) has shown that such equations have instabilities for the same values of the parameters E and 5 as the homogeneous

Mathieu equation.

6. Stability of siochastie time dependent systems

a. Stability of the mean.

The stochastic differential equation describing the rolling mode of a ship in a random seaway

may be written as follows,

43(0 + 2k wo (t)+p(t) +(t) qo(t) =K(t)

or,

p(t)1 = K(t) - a(t) 9(0

where now K(t) is a random roll moment and a(t) is a random coefficient both related to the sea-way. Apart from the initial conditions at t = 0, the integral solution of the differential equation is, in terms of the impulsive function, given by,

DC

p(t)= J h(t - 71) -a(Ti) q)(71) I d Ti.

On averaging the solution we have

E[W)I - I h(t lE[K(71)]

-E[a(Ti)?(Ti)ii d-ri

which contains the additional unknown E[a(t) TM].

However, by multiplying the initial stochastic

differential equation by a(t1),a(t1) a(t2) a(ti) a(t) and then averaging produces the set of hierarchy equations,

PIE[p(t)1} --E[K(t)] -Ek,-(t) p(t)]

PIE[a(t1)9(t)11-E[a(t1)K(t)] - E[a(ti) a(t) 9(t)] etc.

Since it is not possible in general to solve the =

00

infinite number of averaged hierarchy equations it becomes necessary to truncate the sequence

at a preferably low order. Richardson (1963)

showed that such a truncation may be achieved by writing,

E[a(ti)... a(t) TM] E[a(ti). a(t)] E[cp(t)]

which is equivalent to the local independence as-sumption of Bourret (1962). In the simple case.

E[a(ti) a(t) p(t)] =E[a(ti) a(t)] E[p(t)]

and the hierarchy equations reduce to a closed system of two equations given by,

PIE[p(t)] =E[K(t)] - E[a(t) cp(t)]

P/E[a(ti) p(t)]) E[a(ti) K(t)]

-E[a(ti) a(t)] E[l)(t)]. The solution of the last equation is given by,

E[o(t)] - 5 5 h(t - Ti) h(Ti - T2) Raa(T/ -T2)

00

E[o(T2)] dT1 dT, =j h(t - Ti) K(Ti) d

R (T - dT dT

ak 1 1 2

which transforms into the differential form.

PIE[o(t)]/ - f h(t -72) Ran(t - 72) E[(72)1 d T9

00

=E[K(t)] - f h(t -72) Pt

ltd

-72, _72

after applying the operator P/ and using the result that

Plh(t)/ =5(t)

Since the stationary random process a(t) is re -latedto the seaway which is described by a wide-band process (i. e. the wide-bandwidth parameter E. -.1 as determined from the PiersonMoskowitz spec -trum), then the correlation time of the process

a(t) is very short. Thus, R(T) decreases very

rapidly for increasing 1. and

!Rol »Ril»iR21>...

E[a(ti) p(t)] = 5 h(t - 72) lE[a(tir) K(72)]

where Rn= J

R(T)dT.

-E[a(t1)a(72)]E[?(72)11c172

which at t1 - T1 becomes, By letting s =t -T9 the series expansion of the

integral is given by, E[a(71),p(71)] =5 h(71 - ) (

RaK 71 72)

5 h(s)Rcza(s)E[cD(t -s)] ds

0.3

-Raa(Ti - E[1(T2)]; d T2 ₀₀

where the correlation functions of the stationary

- f

h(5)Ra(s)1E[0(t)] +sE[CD(t)]

random processes a(t), K(t) are given by,

RaK(T1 72) =E[a(71) K(72)] and

E[(t)].. ds

s2

2 Rac,(71 -72) =E[a(71) a(72)].

E[GD(t)] _miE[cio (t)]

The equation describing the mean roll motion is given by

where m= = 5 snh(s)R_an(s)ds, (n =o, 1, ...), and

the differential equation for the mean roll

response reduces to,

E[CD(t)] ±(2k,..)0 - m1) E[D(t)]C../20 - M0) E[cp(t)]

The complementary solution, obtained by

set--5 5

h(t - Ti) (Ti T2) =E[K(t)] - f h(t -12)RK(t -72) d 12. -=t -72) , 00 -a - EIG)(t)1

-il0

ting the right-hand zero, has the form

9

EICDMI

2 Ar ePr t

r =1

where Ar is a constant and pr are the eigenvalues of the characteristic equation,

p- -4- (2k wo p + c.)20 -m0) =0

given by,

Pl, 2 r -(k -12m1) + wo2(1 -k2) +k

2 -11111 mo

The asymptotic stability of the mean exists pro-vided that,

9 ₂

kwo> ;1-ri1 and w-0(1 -k2)> mo +%m -kW() n.11,

and a limited stability of the mean exists when,

2

m A

and }two>

o o o

-giving a zero valued root of p and a constant mean

value of the roll response as la. Stability of the mean square.

By multiplying the integral solution,

CD(t) = _I h(t -7) K(7) -a(7) cp(7)1 dT

of the stochastic differential equation at times t1, t2 and then averaging the product we obtain,

13$0(t1, t2)=E[cD(t1) -cp(t2)]

= 5 5 h(ti -71) h(t9

IRKK(71 -79) -E[K(71) a(72) $(72)] -E[K(72)a(71) (Ti)] +E[a(71)a(72) 0(71)

cl)(72)11 d T1 d72.

This contains the unknown terms E[K(71)a(72)

p(72)], E[K(72) a(71) (71)] and E[(71) a(T2) cD(71)

$(79)]. From the previous section it was assumed that in order to obtain a closed set of hierarchy equations we have,

E[K(71) a(72) T(72)] = E[K(71)C1(72)1E[ 0(72)] =RKa(71 -T2) E[cD(72)]

E[K(-12) C1(71) (I)(71)1 _F1< T2 _Ti) E[(1-1)1

and we shall now assume that,

E[a (71) a(T2) cp(71) cD(79)1

=Era( Ti) a(72)1 E[°(71) °(72)]

=R(71 -79) Roo (Ti, 72). This latter condition has also been adopted by Samuels (1960) and the covariance integral equa-tion may be rewritten as,

cc. Rvi)

t2) -5

5 h(ti -71) h(t2 - 72) Raa (71 -79) Bcpcl) (T1 72) d 71 d 72 ,NO = 5 f h(ti - 72) h(t2 - 72) IREK(71 - 72) 00-00

RKa(Ti - 79) E[c)(72)] aK(71 -72)

E[0(71)]td71d72 since R_{al<S1 -72) rilKa(72 -71)}(

By the procedure described in section(a) we may transform this integral relationship into the differential equation,

Mi M2 Rizo (t, t -7) -Raa (7) R,D0 (t, t

=RKK(7) -RKa(7) E[$(t -7)] -RaK(7) E[a)(t)].

Provided that the system is stable in the mean

expectation such that, E[cD(t)] is bounded together with bounded covariance functions RKK, Ricci and

RaK , then the previous equation may be replaced by the impulsive equation,

Mi M2 Roo (t, t -7) -Raa(7) Ro(t, t -7)

=5(t) 8(7).

Papoulis (1965) has shown that if the

auto-cor-relation function R(T) is analytic then it may

be expressed by the Taylor series

9 d-Raa(0)

R(T)=R(0)+T

d 72

since the odd time derivatives of the stationary

random process are zero. The mean square

stability criteria are determined when T =0, and 2 -Raa(T)1=Raa(,0) +

equation,

M.2 Roo (t, t -T) -Raa(0) Roo (t. t -T) =5(t)8(T)

which has solution

1

R,Do(t, t -T) =

4.n.2 I

-eilwt +Mt -"*da)dA

(.2 _{-2ikco co-co 1(7'}

22

_{-21ka, -cc') -R (0)}

o o o o aa

This may be evaluated by the theory of residues and the mean square value of the roll response is given by,

zor

Roo (t, t) =E[cD2(t)I = 2 Are r)t

r=1

where A is a constant and co . A are the

eigen-r

values of the characteristic equations. Solutions

of co andAr are given by

/I

9 2 R00 (0)

(w+A)1,2=2ika,0_--v

(.(1k )

-o

-1(`)

The asymptotic stability of the mean square is valid, provided that,

q(1

-k2) (0), 0<k <1

although, limited stability of the mean square

exists when

.40(1 -k2)(1 3k2)_{=Raa (0)}

resulting in a zero valued root of A) and the

meansquare value of the roll response is a con-stant in the limit as t

7. Conclusions

The restoring moment coefficient in the dif-ferential equation describing the roll motion of a ship in a seaway is assumed either sinusoidal or

random time dependent in form.

For a

de-terministic seaway, the resulting Mathieu equa-tion describing the ship's response indicates many regions of stability and instability. In contrast. the analogous random situation is governed by conditions deter mined from a simplified analysis involving the mean and mean square expectation

statistics. Although criteria obtained from the

value of the response is bounded, it appears for practical application to assume that this is always

so.

Since the restoring coefficient is dependent on the form of the seaway, it appears beneficial to consider stability in the mean square expectation rather than the mean expectation. In this way, the stability criteria being a function of the auto -correlation function of the restoring coefficient, may be related to the auto-correlation function of the seaway and hence to the spectral density function of the irregular seaway. In other words, the criteria for the mean square expectation is dependent on energy content of the seaway.

From the simplified analysis of the problem discussed, it is evident that by considering the

expectations of the roll motion response then

theoretical stability criteria may be deduced even for a ship in a random irregular seaway.

References

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