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A rational approach to intact ship stability assessment
Ocean Engng. Vol. 6, pp. 493-516
Pergamon Press Ltd. 1979. Printed in Great Britain
A RATIONAL APPROACH TO INTACT SHIP STABILITY ASSESSMENT i. R. 0ZKAN Istanbul Technical University, Faculty of Naval Architecture, Teknik Oniversite, Istanbul, Turkey A b s t r a c t - - I n this study, asymptotic and total stability of the non-linear free a n d forced pure rolling motions o f a ship are investigated. A ship performing a rolling m o t i o n is taken as a dynamical system, Lyapunov's direct m e t h o d is employed in the analysis. By generating a timeinvariant Lyapunov function, conditions and the d o m a i n o f asymptotic stability are obtained for free rolling motion. Results o f the work o n " b o u n d e d n e s s " a n d " u n i f o r m b o u n d e d n e s s " o f the solutions o f the equation o f forced rolling motion, done by O z k a n (1977), that is, conditions o f total (practical) stability a n d its domain in the phase-plane are given a n d illustrated. SYMBOLS
E~+Ix, t
= = = = = = = = = = = = = = = = =
U
=
s(A) S(R) S(r) S R,
u(t)
= = = = = = = = =
N x II
= Euclidean n o r m o f x,
sgn inf sup W
= = = = = = = = =
A1
A A
a(x) a
b C1
d,e E E"x e(t) WM X
~
X1
xa = Y t I
fl
p+ M
E
L...) eB
Absolute value o f excitations N o r m o f a region Matrix o f constant coefficients Matrix o f variable coefficients Linear d a m p i n g coefficient Non-linear d a m p i n g coefficient Set o f continuous functions Coefficients o f restoring m o m e n t function max I e(t) I, Absolute value o f wave excitation n-dimensional Euclidean space Wave heeling m o m e n t Wind heeling m o m e n t Rolling angle .... Roiling velocity time Time interval; t E [t 0, + o0) Ex ~ × I, (n + 1)-dimensional inner product space Definition region o f i = f (t, x ) , U c_ E~+tx, t Definition region in a u t o n o m o u s systems Stability region Region o f initial states A n n u l a r closed region Stability region in a u t o n o m o u s systems Stability region in n o n - a u t o n o m o u s systems Positive limiting set Invariant set m-dimensional excitation vector
II x II = ( ~ x',)t i=1
Signum function Greatest Lower b o u n d Lowest upper b o u n d F o r every belongs to There exists C l o s e d - o p e n interval Set intersection Set inclusion 493
494
i.R. (~ZKAN
.)
=::= Derivative with respect to time = Time-dependent Lyapunov function = Time-invafiant Lyapunov function = Bound for the state vector, x A* E"x × I E"x = "xll[xll>~ b(ll x [I) : P.C.I. function a(tl x II) = N.N.C.I. function P.C.I. = Positive continuously increasing N.N.C.I. = Non-negative continuously increasing ( )~" = Transpose of a vector
x) V(x) V(t,
1. INTRODUCTION EXISTING stability criteria including those suggested by the "Intergovernmental Maritime Consultative Organisation', IMCO, are essentially based on the work done by Rahola (1939), and this work was performed by using data relating to 34 cases o f capsizing dating back to 1870. Values of the areas under the righting curves for the different angles o f statical inclinations and the initial "metacentric height" are used as the measures of stability. According to the current understanding of classical ship stability, a ship is assumed to be in an equilibrium state, say an upright position, when she floats on a free suffaceof water, under the effect of two opposite forces, weight and buoyancy (Fig. 1). w:
A;
gx(x)
:
w GZ.
(1)
L 'G
V
F I G . 1.
Forces acting on a ship floating at the uptight position.
When a ship is inclined to some degree, it is assumed that the ship keeps her new position as another equilibrium state and the heeling moment is taken equal to the restoring moment occurring due to the change of the underwater geometry of the ship (Fig. 2). The concept o f dynamical stability was first discussed by Moseley (Wisniewski, 1961), and is defined by the potential energy of the inclined ship with respect to the upright position. This criterion neither takes into account the hydrodynamical forces nor any other possible perturbations.
f 0
{gx(x)
-- M e
(x)} dx >..- 0.
(2)
A rational approach t o intact ship stability assessment
495
V
GZ ( x )
x:O v
X
57.3 °
Fig. 2. Staticheel and righting moraem.
GZ(x) Righting
moment arm
Me(x) W
\
X
FIG. 3. Dynamic stability.
As can be easily seen from the explanations hitherto given, existing stability criteria mostly rely on the geometrical and statical properties of a ship and rolling motion respectively. The equation of a forced non-linear pure rolling motion of a ship is given in the following form:
(I -F J)~ -F fz(;¢) -F gz(x) = e~(t) + WM.
(3)
By comparison of this equation with Equation (1), the following are deduced: (a) Equation (1) contains no velocity and acceleration terms. All inertial and damping moments are neglected, (b) excitations acting on the ship are not taken into account, (c) finally, the most important drawback of Equation (1) is to find a solution for a dynamical system by using statical equilibrium methods.
496
i.R. OZKAN
The stability of motion is not the case under consideration, and the various positions of inclined ship assumed to be stationary positions, are investigated by using the geometrical properties of the ship, in a statical way. However, such an approach assumes that every position, while the ship rolls, is an equilibrium point. But, if the rolling motion is a forced one, then it is not possible to talk about an equilibrium position. Even in free rolling motion there are no other equilibrium points except those defined by the non-linear restoring moment.
2. STABILITY OF FREE ROLLING MOTION In this part of the paper asymptotic stability of the non-linear free rolling motion of a ship will be investigated. Further information and details may be found in Ozkan (1977). Some theoretical considerations are given in Appendix 1 of this paper. Let us imagine a ship freely floating in an equilibrium position, say upright position, on the free surface of water. Due to the perturbations, the ship heels to a degree x, and begins to perform either an oscillatory or a divergent motion or reaches another equilibrium position and keeps it. The matter under discussion is to investigate whether the ship returns to its original position or not, after the perturbations stop disturbing the ship. In other words what we are looking for is the asymptotic stability of the equilibrium position. Such a rolling motion is governed by the following differential equation. (I + J) Yc + f~ (~c) + g~(x) -----O.
(4)
As can be easily seen, the problem is generally defined in the 2-dimensional phase space whose coordinates are rolling angle and velocity. Since I + J ~ 0, then Equation (4) becomes
+
+
g ( x ) = o.
(5)
By using the state variables, we obtain the following first order simultaneous equations: i q = A (xl,
=
•;c2 = f~ (xl, x ~ = - - f (xz) -- g(xO,
(6)
in matrix notation:
(7)
k a21(x) or simply = A (x) x; A(x) = ad(xi, x)); i = 1, 2 ; ) = 1,2. To obtain the conditions of asymptotic stability, let us prove the following theorem, in which the variable-gradient method is employed, to generate a Lyapunov function.
A
497
rational approach to intact ship stability assessment
Theorem: If for ~z =f(x), f ( x ) = O, there exists a bounded domain f~ and a real vector OV coxa
aal(x)xl + als(X)X2 + . . .
+ al,,(x)x,,
a2x(x)xl + a22(x)x2 + . . .
+ a2,,(x)x,,
OV c9x2
v
v = { V v,} =
OV
a,a(x)xl + a , a ( x ) x 2 + . . . + a,m(x)x,
Oxn
such that (1)
aVv, _ aVV/,
8xj
8x~
(2)
VV:/:o vx:/:o,
(3)
dV --vvr'i dt
(4)
dV. m is not identically zero for any other solution except dt
<0
Vx~O,
x = 0 in ~,
(5) V(x)-- / VVrdx 0
(a) V(x) > 0
V x :/: 0 and V(O) ----O,
(b) One of the curves of V = K = const, bounds the domain in the phase-plane, then x = 0 of ~ = f (x), that is, the equilibrium position of the free rolling motion is asymptotically stable in ~ which is the domain of asymptotic stability.
498
I.R.OZKAN
Proof: The equation of the non-linear free rolling motion is of the following form (I
+
J)3c 4- a,.~ 4- b~l.~l.x 4- 4 x
--
el x 3
--
O,
(8)
and since I + J :/: O, then
(9)
Yc + a X + b[Jc[Yc + d x - - e x ~ = O.
By using the state variables, this equation may be defined by the following first order simultaneous differential equations: )¢1 = X2
(10)
X, = - - dxl + exal - - ax~ - - b Ix2 [x~
[ vii VVl=
VV=
,,,x, +
V V , = ~V
,
]
dV - [(~nx~ + ~1~x2) (~slx~ + ~r~x~)] dt .
By substituting the values of x~ and x2 from Equation (10), we obtain: dV - [(**nxl + ~ x l ) (~sxl + o~x,)] [ dt
x2
[ - - dxl 4- ex81 - - ax2 - - b] x , [x~
(11)
or
dV d'--t- = °qlXlX~ + °cI~¢S2 - - ~ l d x 2 1
-~- ° ~ l e X l l - - a ~ l x l ' x ' ~ - -
boc211x~[xlx~
- - d o ~ q x ~ + ~q~exaax~ - - ao~,dc2s - - bat,,2[x21 x~2 •
According to the conditions d V ] d t < 0; equations should have no indefinite terms. For this reason by taking ~12 = 0hx = 0, the above equation takes the following form: dV dt
- - ag.nx', + (=n - ~ n d ) x l x , + o~nex'lx, - - ~ [
x,[ x ' , .
(12)
A rational approach to intact ~hip stability assessment
499
Due to the condition V x V V = 0 we obtain:
which yields 0 ----- 6~11
t~x~
X1 •
To satisfy this equation, ~ax should be only the function of xl. Let us take: (13)
oqa = d~2 -- ex~xo~. Substitution of this value into Equation (13) results in the following equation: dV = -- ~q~x'2 (a + b Ix=I). d-T
(14)
Having determined dV/dt X
Xt
0
0
d x 21
V(x) = ~q,
X|
2"--
0
ex41
~%' T
x22
+ ~t'' --~ .
There is no point in choosing ~-n as a constant value; let ~2
=
4.
Then the Lyapunov function: V(x) = x~l (2d -- exZl) + 2x~2,
(15)
I~'(x) = -- 4x~ (a + b [x~ I)"
(16)
and its Eulerian derivative:
Asymptotic stability conditions: x~x (2d -- exat) q-- 2x~2 > O, - - 4x22 (a +
blx~ 1) < 0;
a +
bix, I >
o.
(17)
500
i.R. (}ZKAN
Equilibrium positions." According to i = f (x); f (0) = 0, i == 0 yields equations 0 ==xl : x2,
(118)
0 = -- ax2 -- b Ix2[x2 -- dxl + exal, which yield the following equilibrium points:
0 (0, 0),
(A --x/d/e, O) B (x/d/e, 0).
By using the characteristic equations it is found that:
0(0,0):
(a) ifa ~ - 4 d > f (b)
O,
stable node.
ifa s-4d
stable focus.
A (-- x/d/e, 0), (B x/d/e, 0),
unstable saddle points.
~,ol
A (-A F~,o)
0(0,0) Fig. 4. Equilibrium points of rolling motion in phase-plane.
Domain of asymptotic stability: As the rolling motion is taken as a non-linear one, it is not enough to know only the conditions for which the motion will be asymptotically stable. As far as the non-linear systems are concerned, stability is a local concept and one should design a domain of attraction around the asymptotically stable equilibrium position. Lyapunov's Direct Method gives us the possibility of taking into account the nonlinearities of the system and hence to evaluate the domain of stability. This property of the method used, demonstrates its basic advantage, bearing in mind that stability research
A rational approach to intact ship stability assessment
501
cannot be complete and satisfactory, without taking into account the non-linear behaviour of the system. Before going deep into the subject, it is suggested the reader studies Appendix 2, in which some definitions and a theorem on the extent of asymptotic stability may be found. In the following, a domain of attraction for the asymptotically stable equilibrium position of the free rolling motion will be constructed. Since a, b ~ 0, then it follows that
~._OV -
Ot
4x'~(a+blx,I)=o,
x.,=O,
R = {x [ x 2 = 0, x 1 X 0} is a s u b s e t o f the axis xv
Let us write the slope of the curve in the phase plane: tgX2
0X_..__22= Oxx
Ot
-- dx I +
exZl -- ax2 -- b [xz ]xz X2
"OXl
Ot As dxJdt ~ oo at every crossing point of the xx axis except xx ~ 0, then the possible invariant sets are found to be the equilibrium points. 0(0, 0), A (-- v/die, 0), B(x/d/e, 0). As a matter of fact, the points A and B were found to be unstable points which should be excluded from the domain of asymptotic stability. In conclusion, f~ : V(x) < K should be chosen by taking A and B as the boundary points. g(x) = V(x1, x2) = X~l(2d -- ex21) + 2x~2 = Const.
(19)
These curves cross the axis xl, at x = 4-x/die for x~ = 0 then V(x) = (+~d/e) ~ (2d -- e(+Vd/e)Z), d2
V(x) = K ---- - - .
(20)
e
The ordinate values of the points belonging to the curve V(x) = d2/e are as follows:
x2=4The other point of the curve:
c(0
x / ( - ~- -
x~l(2d-
(21)
I. R. OZKAN
502
extremum points: E(O, d/s/2e)
C
=
F(O, -- d/s/2e) G( s/d/e,
O)
D
=
= B
H(-- Vale, O) = A. According to the theorem given in Appendix 2, the interior of the domain bounded by the curve V(x) = an/e is d2 f2 ={x I V (x) < - - ,
I~'(x) < O}
(22)
e
and the set R in which d V/dt becomes zero is R = {x [ -- s/d/e < xl < s/d/e, x2 -:- O, d V/dt = 0}. The largest invariant set in R is M = {0}. This, also, is the positive limiting set F +. The domain already obtained (Fig. 5) is the interior of curves V(x) = an/e such that any rolling motion initiating within this domain never leaves it and in addition tends to the origin as the time increases.
•
E(O,d/~)~
stable equilibrium position ( I n v a r i a n t ~
~ /
x2= Y
/
~
/ V ~
~ ~- / /
stability
• v.
(
(laddie point)
~
~
(saddle point) ~F(O,-
FIG. 5.
xl'Vrx x ; d2
~-~ ~ ~ J
J ~ 1
'°'
. Asyrnp.stable trajectory
d/2eta)
Domain of asymptotic stability of the free non-linear pure rolling-motion of a ship.
A rational approach to intact ship stabilitya sessmcnt 3.
503
STABILITY OF FORCED ROLLING MOTION
Total (practical) stability: In the previous part of this paper we investigated the asymptotic stability of the equilibrium position and the behaviour of the trajectories of the free rolling motion of a ship under the assumption that the perturbations, which are small in magnitude, act on the ship for a finite duration. Stability, in fact, possesses the meaning that the moderate perturbations acting on the system do not turn the system's behaviour to an unstable feature and asymptotic stability implies that the effects caused by the perturbations disappear in time, after the perturbations cease. But, in practice, perturbations and excitations act on the system for longer periods. Under these circumstances it is necessary that one should mention total stability for the systems subjected to such continuous excitations. Due to the continuously acting perturbations and excitations, stability and asymptotic stability may not guarantee practical stability. At this point we need to ask the question whether asymptotic stability is a necessary condition for practical stability. It is not necessary at all. Due to the system's characteristics, running conditions, and mostly the mutual system-environment relations, the system may be practically stable while it is mathematically unstable i.e. oscillatory rolling motion under the effect of continuously acting excitations. Practical stability is to confine the solutions of the equation of motion to a domain which is safe enough, with respect to the initial conditions and the magnitude of excitations. In this part of the paper the results of two theorems on "boundedness" and "uniform boundedness", ("Lagrange Stability"), and especially a third theorem giving the conditions for, and the domain of practical stability of the solutions studied and proved by the author, will be given. For further information on the subject and details of proofs the reader is referred to Ozkan (1974, 1977). A theoretical background knowledge and the comments on total stability, in naval architectural terms, together with some basic definitions for non-autonomous systems are given in Appendix 3. A ship performing a forced rolling motion, under the effects of continuously acting excitations, belongs to a class of dynamical systems which are defined by the following differential equations :x = f (x) + u(t). Assuming that the ship is subjected to a time-dependent wave excitation and a constant wind heeling moment, the equation of motion may be written as follows: ( I + J ) .~ + fl(.x) + gl(x) =
ea(t) + WM,
or
"~ + f(J¢) +g(x) = e(t) + P.
(23)
Boundedness of solutions: Theorem (boundedness). By using the first integrals of the equation of motion of a rolling ship, it has been proved that the rolling angle and velocity are bounded, that is,
504
i.R. (~ZKAN
f
e(t)dt < 0o;
[e(t) [ ~ E, {[e(t) [ + P} ~ A1
(24)
0
and lim x
- + oo
[ f
(g(x)--A1)dx
[-+ d- c~
0
Theorem (uniform boundedness). By choosing a time-dependent Lyapunov function, it has been proved that the solutions are uniformly bounded in to, that is, bounds do not depend on the initial time. A region, which is the intersection of three sets each of which is obtained for the different features of the non-linear restoring moment, has been obtained. A* =
ADBDC
In addition to the conditions already given for the boundedness of the solution some further conditions on restoring moment are as follows:
g(x) ~>0, V x E (O,x/d/e), g ( x ) = O,
x = O, V d / e ,
G(x) ---- f g(x)dx > O, '¢ x E (0, v/2d/e), 0
G(x) = O,
g(x) ; G(x)
~
V x >~ V Z a / e ; x <~ -
/-g
V2a/e.
(x)
/ (x)= J'g(x)dx
X
(Zd~'aT~,O)
FIG. 6.
Restoring moment characteristics.
(25)
A rational approach to intact ship stability assessment
505
Apart from the results given above, one should construct a domain of total stability in the phase-plane. Theorem (domain of practical stability). By using the concept of a "Jordan curve" the domain of practical stability has been constructed as in Fig. 7.
bx2:Y;g(x) b,:Xt.°t
Trajectory of a 'Totally Stable' rolling motion
o. tara,
..abi,ity
FIG. 7. Domain of total stability. Curves:
PaPa = ½y~ q- G (x) q- Axx = Const, PxP2 ~- ½y2 + G (x) -- Axx = Const, P2Pa = ½Y~ q- G(x) = Const. Points: Po (0, b, + 6al A1 + 1) /)1 (xx, bl) 3¢|
f P~(xl+
xl
Ps (~/d/e, 0).
g(x)dx -- b21 2Ax
, y~),
(26)
506
]. R. OZKAN
The conditions under which the domain of practical stability is obtained, are as follows. (1)
.~f(.~) > O, V .x # O, x sgn ~ > 0
(2)
[e(t) [ <~ E;
(3/
lira [G(x) I = lim t f X
le(t) l + p <~ A1,
--~oC~
); -.+o0
g(xldx I--> or,
0
(4) righting moment and excitations: q(x) :dx-ex s
2A,: 2 mox{le(')l+ P)
FIG. 8. Righting moment and excitations.
lim G (x) = lim ~ X ---~- aO
X - - ~ aO
Ig(x)
[dx --~ or,
0
g(x) < 2A1,
0 < x < ax;
g(x) > 2A1,
al < x < a'2,
g(x) = 2A1,
x = al, a 1,
a'l < x < + o o ,
(27)
t
(5) damping moment and excitations:
f ( x ) sgn.x >2A~
'q" [ ~ ] >bl,
f ( x ) sgn:~ = 2 A x
I ~ I =bl,
f ( x ) sgn:~ <2A~,
V I j [
(28)
A rational approach to intact ship stability assessment
507
f (t):ot+bltlt
, 2A~
bl
/I
FIG. 9. Damping moment and excitations. In most of the engineering applications, the righting moment g(x), on the contrary to the physical realities, is taken as having a single feature, increasing or decreasing. But, in fact, the systems do not behave in that way. As far as the rolling motion is concerned, righting arm curve is, at the beginning, a monotonically increasing function and after an extremum point behaves as a monotonically decreasing one. Bearing in mind this reality, the domain of practical stability is constructed by taking into account these properties of the GZ curve. l
Comments on the domain of practical stability: The domain shown in Fig. 7 is the domain of practical stability in which the solutions of the equations are uniformly bounded in to with respect to the set of initial conditions and excitations. A forced rolling motion initiating within the bounded parts of the first and third quadrants will never leave them and results in an oscillatory motion around the equilibrium position. Motions initiating at the outside of the bounds will tend to the interior of the bounds in a finite time, that is, the trajectories cross the bounds from outside. Second and fourth quadrants are convergent regions such that any motion initiating within these quadrants enters to the first or third quadrant and converges, since it is not possible that while the ship is heeling to one side, rolling velocity increases at the opposite direction. It is also found that when the roiling angle takes the values such that xa > a't, then the motion shows the character to differ away from the domain of practical stability. This is the case illustrated by the curve PaPs. At this part of the domain, to converge the solutions to the domain total stability, one should design an additional damping mechanism which begins to operate when the magnitude of the state vector, that is,
tlx II =(x,1 + x,,),, reaches to a certain value. The desired additional damping moment is:
M > Aa - - f ( y ) >i l e(t) I + P --f(Y)"
(29)
508
i . R . OZKAN , x 2
gCx)
!P~Cx,, b,)
/
i
X
O
at
X=> a,
CJ(X)
P~(x,,b~) ~, P2(x2,yz) t
I
X
CII
'
~
Xj
01
g(x)
xz
~xz,yo) X O,
FIG. lO.
~
- -
w
Xl
P3(X3,0)
Evaluation of the bounds in accordance with the different features of g(x).
In this study only wind and wave excitations are considered. But since the calculations are carried out for the norm of the excitation terms, any other type of finite excitations can be considered as well. This does not change the general features of the results obtained. In this case, the restoring and damping moment are required to have the greater values. 4.
CONCLUSION
Finally, the findings of this study can be summarized in terms of the current understanding of the ship stability problem, as follows: (a) To analyse the stability (i.e. asymptotic stability) of both free and forced rolling motion of a ship, by using the statical technique, is insufficient and results obtained by such an approach are completely unsatisfactory. This can be easily seen by comparing the results we obtained and the current ship stability criteria. This is the most important drawback of the criteria used for the time being.
A rational approach to intact ship stability assessment
509
(b) T h e i n t e r r e l a t i o n s b e t w e e n the ship a n d t h e e n v i r o n m e n t a r e t h e i n d i s p e n s i b l e items o f every stability analysis. A s y m p t o t i c a n d t o t a l stability c o n d i t i o n s a n d d o m a i n s are o b t a i n e d as t h e f u n c t i o n s o f t h e e q u a t i o n o f the m o t i o n itself. (c) T h e results o f this s t u d y c o v e r a n y t y p e o f s e a - g o i n g vessel, since there has b e e n n o specification m a d e o f t h e type a n d t h e size o f t h e ships. F o r a k n o w n ship, the e n v i r o n m e n t a l c o n d i t i o n s , such as, t y p e a n d the m a g n i t u d e o f e x c i t a t i o n s at w h i c h the ship will o p e r a t e safely, c a n be e s t i m a t e d i n a d v a n c e . APPENDIX I Some theoretical considerations of the stability of motion: The theory of stability of motion is concerned with investigating the effects of perturbations and excitations on the motion of a physical system. The state of any physical system can be defined by some variables, say n, at any time, t. These variables are known as the state variables which make up an n-dimensional state vector, x (t). x(t) = ( x l (t), x2 (t) . . . . .
(A.I.1)
x~(t)).
Taking the time as the independent variable, t E I = [to, + Qo), the state of the system is given by the ordered pairs, (x (t), t), which are the members of a subset of a (n + D-dimensional inner product space. (x, t) E U c E'~+tx, t = E"x x I
(A.1.2.)
Let the system be given by the following vector-differential equation: dx = - - = f (x, t) dt
(A.I.3.)
Where f (x, t) is an n-vector function defined in Ex,t"+l; f : I x E "+1 x,t --+ E"x.
Let x (t) be a solution of Equation (A.I.3.) at a time t. If the Cauchy-Lipschitz theorem is satisfied in U, then there exists a solution of Equation (A.1.3.) and it is unique. This solution draws an integral curve, g, in U. The curve g is a continuous set of points and can be formulated as follows: g = ( (x, (t), t)} E U c E n+lx,t s.t. t E [to, + oo); W t >I to. Due to the external effects, let us assume that the system's state reaches to another integral curve, gp, (Fig. A.I.1.).
x(t}
x(t)/~.., gP
i~( t o) 0
Fig. A.I.1.
i =
t
to
Perturbed and unperturbed integral curves of a system.
510
I . R . OZKAN
Defining the amount of deviation by y(t) and a special solution by ~,(t), we obtain x (t) : ~(t) -~ y(t).
(A.I.4.)
Substituting this into Equation (A.I.3.) and after some manipulations, we finally reach to = F (y, t), F (0, t) = 0.
(A.I.5.)
Every solution of Equation (A.1.3.) defined by y = 0 or x(t) = ~(t) is called art "unperturbed motion" of the system or a n "equilibrium position" and, in fact, this is the solution whose stability is to be discussed.
Definition: Given a n arbitrarily small positive number e, our problem is to find a positive number "q such that if the initial values x (to) of the variables x,(to) corresponding to t = tl ~- to satisfy [I x (to) I[ < q,
(A.I.6.)
then [ 4 x ( t ) t] <
E Vt
>/ ta.
(A.I.7.)
If such a a m b e r rl exists--then unpexturbed motion is called "stable" with respect to x~, xl . . . . . x,. Otherwise it is "unstable". It may happen that condition (A.1.6.) implies that lim x (t) -~ 0, t-p+
co
then every perturbed motion, sufficiently close to the unperturbed one, approaches it asymptotically. In this case, we shall say that the unperturbed motion is "asymptotically stable".
Stability in sense of Lyapunov : Let us think of a physical system. If the total energy of the system decreases monotonically, then it follows that the system's state will tend to a n equilibrium position. This holds, because energy is a non. negative function of the system's state and reaches a minimum when the motion stops. Lyapunov's direct method is the generalization of this idea, and the question of stability is thus convexted to the problem of the e,xistence of a positive definite scalar function, called Lyapunov function, whose time derivative is negative definite. Autonomous system: Throughout this paper a ship performing a free rolling motion is taken as a dynamical system in which the independent variable, time, does not appear explicitly. Governing equations for a n n-dimensional system are as follows. = f (x)
(A.1.8.)
= A (x) x
(non-linear)
= A~ x
(linear).
(A.1.9.) (A.I.10.)
Let the system be given by Equation (A.I.8.) such that f (0) = 0. Let S(R) and S(r); 0 < r .~ R and S(r) c: S(R), t/
IIxll
Z x~)t
The closed annular region is shown by
Sn,={xlr
~llx[l ~R}.
We shall suppose that in a certain open spherical region, S(A) = {x [[I x II < A} the basic existence theorem holds for Equation (A.I.8.). The partial derivatives with respect to eaeh state variable, that is,
A rational approach to intact ship stability assessment
511
v(x)
Stabilitl/~
Instability
/'o
X)K2
(a}
X2 9(x) > 0
Xl
~i
Vlx)>C
lity
"]
~"
Instability region
/
Instobilit region L
[
/ I I
K2 ~/(X)=Const. KI
(b) Fig. A.1.2.
"Lyapunov function and stability".
all exists and continuous in S(A). We, also, recall that through each point x(t) of S(A), there goes a unique path, I'. We shall designate the positive Limiting set by I'+. , . If for every S(R) there exists a spherical region $(r) such that every solution initiating within this domain never leaves S(R) as the time goes to infinity, that is lim II • ( 0 [I < R
.....
X---~ OO
then the zero solution of ~ = f(x) is said to be "stable". Apart from this, if, for every R, > 0 there exists a region such that every solution initiating within this domain tends to the origin as t ~ + 0% then the origin is said to be "asymptotically stable", if V Ro > 0 9S(Ro) s.t. lira II • (t) II ~ 0. t--~ -I- oo
512
]. R. OZKAN
Lyapunov function :
In autonomous systems a Lyapunov function is a scalar function of the system's state x(t), satisfying the following conditions. (1)
V(x)>O Ilxff
(2)
<
0
lim V(x) ~ ÷ oo, II x II --*
(4)
C E x ~,
V(x) :- 0, II x II =
(3)
Vx~O;xEf~
h
co
generalized upper right hand derivative is of the form d V(x) dt
l lim sup -~- [ V(x + h f (x) -- V(x)]. h ~ 0+
If V(x) E C a then this equation becomes equal to the Eulerian derivative
I;'(x)
OV(x) -
Ot
~V dx~ OV dx~ OV d x . =Ox---~xd-i- + ~ - ~ + ' ' " + Ox. dt '
(a)
(b) Fig. A.1.3.
Geometrical representation of the domains of stability and asymptotic stability.
A rauonal approach to intact ship stability assessment I2(x) = V V(x) r.~ (5)
l:'(x) ~ 0
513
(A.l.ll.)
VxEt~.
Lyapunov stability theorem: If there exists a Lyapunov function V(x) in a spherical region around the origin, then the "unperturbed motion" or the "equilibrium position" is stable.
Lyapunov asymptotic stability theorem: In addition to the conditions for stability if l:'(x) < 0, then the "unperturbed motion" or "equilibrium position" is "asymptotically stable".
APPENDIX
2
Definition: positive limiting set, F + Let x(t) be a solution of x = f (x). I f w ~ :> 0artd T > 0 y t ~, Ts.t. [I x ( t ) -- Pll < then
p E F +,
and if x(t) is bounded then I ~+ is a non-empty compact invariant set.
Definition: Invariant set, M. If every solution initiating within the set M remains in M for t E [to, + or), then M is a n invariant set. According to this definition: ifv~>0
yT>0s.t.
Vt>T
ypEMand[lx(t)--pl[<~,
then x(t) approaches to the invariant set M as t --, + oo. For example if V t > 0, x(t) is bounded, then x(t) ---, F+ as t --, + or.
Theorem: Let f~ be a dosed and bounded (compact) set such that every solution x(t) of x = f (x) initiating within this set will remain in ~ . Let us assume that there exists a scalar function V(x); V(x) s Ca; posse~lng the first partial derivatives and its Eulerian derivative is negative sere-definite; I;'(x) < 0; in t2. Let R be the set of all points satisfying V(x) = 0; R = {x I ~(x) = 0} and M i s the largest invariant set in R. Then every solution initiating in f~ approaches to M as the time goes to infinity. x(t) --, M
W x(to) [ [2, t --* or.
Proof: Let x(t), x(t) E ~ be a solution o f x = f ( ~ ) . Since I;'(x) < 0, V(x) is a nonincreasingfunction of tithe. V(x) is continuous in f~ due to V(x) ~ 6"1 a n d is bounded from below. Because [1 is a compact set, then for every x E f~ it follows that l[ x (t) 11 > inf f~. Because of that reason V(x) will have a limit value as the time goes to infinity. lim V[x (t)] = C. In addition, since f~ is a closed set, the positive limiting set I"+ of x(t) will be in f~ and V(x) takes the value C in I'+ since it is continuous in ~ . V(x) = C, V x E I'% V (x) E Ca, x E t'l. F + is a n invariant set and IY(x) = 0 in P+. Hence r + becomes a subset of M, F + K M. This result shows that, as it is already mentioned above, every solution initiating in t~ tends to M. That is, if x(t,) E f~,
514
L R. OZKAN
then x(t) s M
ast~
+ oo.
APPENDIX
3
Total stability: Let the system be given by:
~k = f(x) + u(t).
(A.3.1)
Total stability is the existence of two compact sets and a number 8 > 0 for this system. Let t'l*o he the set of all initial states, x(to) of the system, x(to) E fPo, and fl* be the set of all ultimate states at which the system will operate in safe, x (t) E f~* V t ~. to. f~o* is a subset of fl*. flo* ~ f~*.
(A.3.2)
u(t) is a n-dirnemional vector representing the excitations and perturbations. If the set including every possible excitation and perturbation is U, then u(t) E
u,,
and I[u(t) ll < 6
V t ~. to, V x ( t ) .
(A.3.3)
According to the explanations given above, if V u (t) E Up and x (to) E fP0 ~ x (xo; to, t) E ~, V t ;~ to then the origin of Equation (A.3.1) is totally stable. In other words, total stability is the investigation and the justification of the following items simultaneously. (a) What are the excitations and perturbations expected due to the environment in which the system operates ? 8. (b) How sufficiently could the initial state of the system be controlled ? Set ~*0. (c) What should he the acceptable safe behaviour of the system as far as the engineering practice is concerned ? Set fl*. A topological representation of total stability and its comment for rolling motion is given in (Fig. A.3.1).
Boundedness of solutions: Definition: If V x (xe; to, t) and t > to there exists a ~ > 0 such that
]1 x (xo; to t)]1 < fl
(A.3.4)
then the solutions x (Xo; to, t) of ~[ = f (x, t) are bounded.
Definition: If the bound [3given in the above definition is independent of the initial time, then the solutions are uniformly bounded. Theorem: If there exists a time-dependent Lyapunov function, V (x, t), defined in the product space A* ~ E - a × I ; E ~
= { x I[Ixll > a } a n d l f f i { t t t ~ Ire + oo)}
provided that: (a)
a ( l j x l l ) ~ V(t,x) ~ b ( [ I x N )
whgTc
a(r) E P.C.I, and b(r) s N.N.C.I, (b) IY(x, t) ~ 0 in A*, then the solutions of x = f (t, x ) a r e uniformly bounded.
A rational approach to intact ship stability assessment
515
t~'-'-'-W M
' J '
\
\
~0~I
\
\
~,~ el(t),WM...
WM •
t
~
~--
~m's bound . . . . . . . . . . . ,i"/,o's bound
Fie. A.3.1.
Topological and physical representation of total stability in the forced rolling motion.
REFERENCES ANTOSlEWICZ, H. A. 1958. A survey of Lyapunov's second method. Contributions to the theory of nonlinear oscillations. Ann. Math. Stud. 41, pp. 141-166. C"mrr^vEv, N. G. 1961. The Stability of Motion. Pergamon Press, Oxford. CODmNt~TON, E. A. and LEVINSON,N. 1963. Theory of Ordinary Differential Equations. McGraw-Hill. HAHN, W. 1963. Theory and Applications of Lyapunov's Direct Method. Prentice & Hall. DE HEEItV.SO-mLTEMA-BA~r.a~R,A. R. 1969. Buoyancy and stability of ships. Tech. Publs. H. Stam. J.S.N.A. 1960. Advances in research on stability and rolling of ships. 60th anniversary series. 6. KALMAN,R. E and BERTRAM,J. E. 1960. Control system analysis and design via the second method of Lyapunov. J. has. Engng. pp. 371-393. Kuo, C. and ODABASl,A. Y. 1974. Theoretical studies on intact stability of ships. Phase 2 Report, University of Strathclyde. Kuo, C. and ODABASI, A. Y. 1975. Application of dynamic systems approach to ship and ocean vehicle stability. International Conference on Stability of Ships and Ocean Vehicles, Glasgow, Scotland, pp. 1-22. Kuo, C. and OD^aASl, A. Y. 1974. Alternative approaches to ship and ocean vehicle stability criteria. J. nay. Arch. KRA~VSKII, N. N. 1963. Stability of Motion. Stanford University Press. LAaALLE, J. P. 1964. Recent advances in Lyapunov stability theory. SIAM Review 6, pp. 1-11. LAaALLE, J. P. 1968. Stability theory for ordinary differential equations. J. diff. Equat. 4, pp. 57-65. LYAPUNOV,A. M. 1949. General problem of stability of motion. Ann. Math. Stud. 17, pp. 203-474.
516
i . R . OZKAN
MASSERA,J. L. 1949. On Lyapunov's conditions of stability. Ann. Math. 50, pp. 705-721. MASSERA, J. L. 1960. On the existence of Lyapunov functions. Publnes Inst. Mat. Estaddst., Montev. 3, pp. 111-124. ODABASl, A. Y. 1973. Methods of analysis of non-linear ship oscillations and stability. Departmental Report No. 08/73, University of Strathclyde, Scotland. ODAaASI, A. Y. 1976. Ultimate stability of ships. Trans. R. lnstn nay. Archit. ODABASI, A. Y. 1978. Conceptual understanding of the stability theory of ships. Schiff~technik 119. OZKAN, I. R. 1977. On the general theory of the stability of ships via Lyapunov's direct method. Ph.D. Thesis, Istanbul Technical University. OZKAN, I. R. 1974. On the boundedness and stability of ordinary differential equations. Departmental Report, University of Strathclyde, Scotland. OZKAN, I. R. A boundedness theorem on a class of non-linear second order differential equations .To appear. OZKAN, I. R. Effects of external excitations on the stability of ships. To appear. I~RSIDSKt, S. 1961. On the second method of Lyapunov. P M M 25, pp. 17-23. RAHOLA, J. 1939. The judging of the stability of ships and the determination of the minimum amount of stability. Helsinki. RAZUMIKHIN,B. S. 1958. On the applications of Lyapunov's method to stability problems. P M M 22, pp. 466-480. ROBB, A. M. 1958. A note on rolling of ships. Trans. Insm nay. Archit. pp. 396-402. TEUFEL, H., JR. 1972. Forced second order non-linear oscillations. J. math. Analysis Applic. 40, 148-152. THOMSON, G. and TOPE, J. E. 1970. International considerations of intact ship stability standards. Trans. Instn nay. Arch#. WISraEWSKI, J. 1961. Mechanical criteria of ship stability. Schiffstechnik 8, pp. 87-90. YOSmZAWA, T. 1953. Note on the boundedness of solutions of a system of differential equations. Mem. Coll. Sci. Engng. Kyoto imp. Univ. A2,8, pp. 293-298. YOSHIZAWA, T. 1955. On the stability of solutions of a system of differential equations. Mem. Coll. Sci. Engng Kyoto imp. Univ. A29, pp. 27-33. YOSHIZAWA,T. 1957. On the necessary and sufficient condititlBs for the uniform boundedness of solutions of ~ = f(x, t). Mem. Coll. Sci. Engng Kyoto imp. Univ. A30, pp. 217-226. YOSHIZAWA,.T. 1960. Stability and boundedness of systems. Archs ration. Mech. Analysis 6, pp. 409-421.
Pergamon Press Ltd. 1979. Printed in Great Britain
A RATIONAL APPROACH TO INTACT SHIP STABILITY ASSESSMENT i. R. 0ZKAN Istanbul Technical University, Faculty of Naval Architecture, Teknik Oniversite, Istanbul, Turkey A b s t r a c t - - I n this study, asymptotic and total stability of the non-linear free a n d forced pure rolling motions o f a ship are investigated. A ship performing a rolling m o t i o n is taken as a dynamical system, Lyapunov's direct m e t h o d is employed in the analysis. By generating a timeinvariant Lyapunov function, conditions and the d o m a i n o f asymptotic stability are obtained for free rolling motion. Results o f the work o n " b o u n d e d n e s s " a n d " u n i f o r m b o u n d e d n e s s " o f the solutions o f the equation o f forced rolling motion, done by O z k a n (1977), that is, conditions o f total (practical) stability a n d its domain in the phase-plane are given a n d illustrated. SYMBOLS
E~+Ix, t
= = = = = = = = = = = = = = = = =
U
=
s(A) S(R) S(r) S R,
u(t)
= = = = = = = = =
N x II
= Euclidean n o r m o f x,
sgn inf sup W
= = = = = = = = =
A1
A A
a(x) a
b C1
d,e E E"x e(t) WM X
~
X1
xa = Y t I
fl
p+ M
E
L...) eB
Absolute value o f excitations N o r m o f a region Matrix o f constant coefficients Matrix o f variable coefficients Linear d a m p i n g coefficient Non-linear d a m p i n g coefficient Set o f continuous functions Coefficients o f restoring m o m e n t function max I e(t) I, Absolute value o f wave excitation n-dimensional Euclidean space Wave heeling m o m e n t Wind heeling m o m e n t Rolling angle .... Roiling velocity time Time interval; t E [t 0, + o0) Ex ~ × I, (n + 1)-dimensional inner product space Definition region o f i = f (t, x ) , U c_ E~+tx, t Definition region in a u t o n o m o u s systems Stability region Region o f initial states A n n u l a r closed region Stability region in a u t o n o m o u s systems Stability region in n o n - a u t o n o m o u s systems Positive limiting set Invariant set m-dimensional excitation vector
II x II = ( ~ x',)t i=1
Signum function Greatest Lower b o u n d Lowest upper b o u n d F o r every belongs to There exists C l o s e d - o p e n interval Set intersection Set inclusion 493
494
i.R. (~ZKAN
.)
=::= Derivative with respect to time = Time-dependent Lyapunov function = Time-invafiant Lyapunov function = Bound for the state vector, x A* E"x × I E"x = "xll[xll>~ b(ll x [I) : P.C.I. function a(tl x II) = N.N.C.I. function P.C.I. = Positive continuously increasing N.N.C.I. = Non-negative continuously increasing ( )~" = Transpose of a vector
x) V(x) V(t,
1. INTRODUCTION EXISTING stability criteria including those suggested by the "Intergovernmental Maritime Consultative Organisation', IMCO, are essentially based on the work done by Rahola (1939), and this work was performed by using data relating to 34 cases o f capsizing dating back to 1870. Values of the areas under the righting curves for the different angles o f statical inclinations and the initial "metacentric height" are used as the measures of stability. According to the current understanding of classical ship stability, a ship is assumed to be in an equilibrium state, say an upright position, when she floats on a free suffaceof water, under the effect of two opposite forces, weight and buoyancy (Fig. 1). w:
A;
gx(x)
:
w GZ.
(1)
L 'G
V
F I G . 1.
Forces acting on a ship floating at the uptight position.
When a ship is inclined to some degree, it is assumed that the ship keeps her new position as another equilibrium state and the heeling moment is taken equal to the restoring moment occurring due to the change of the underwater geometry of the ship (Fig. 2). The concept o f dynamical stability was first discussed by Moseley (Wisniewski, 1961), and is defined by the potential energy of the inclined ship with respect to the upright position. This criterion neither takes into account the hydrodynamical forces nor any other possible perturbations.
f 0
{gx(x)
-- M e
(x)} dx >..- 0.
(2)
A rational approach t o intact ship stability assessment
495
V
GZ ( x )
x:O v
X
57.3 °
Fig. 2. Staticheel and righting moraem.
GZ(x) Righting
moment arm
Me(x) W
\
X
FIG. 3. Dynamic stability.
As can be easily seen from the explanations hitherto given, existing stability criteria mostly rely on the geometrical and statical properties of a ship and rolling motion respectively. The equation of a forced non-linear pure rolling motion of a ship is given in the following form:
(I -F J)~ -F fz(;¢) -F gz(x) = e~(t) + WM.
(3)
By comparison of this equation with Equation (1), the following are deduced: (a) Equation (1) contains no velocity and acceleration terms. All inertial and damping moments are neglected, (b) excitations acting on the ship are not taken into account, (c) finally, the most important drawback of Equation (1) is to find a solution for a dynamical system by using statical equilibrium methods.
496
i.R. OZKAN
The stability of motion is not the case under consideration, and the various positions of inclined ship assumed to be stationary positions, are investigated by using the geometrical properties of the ship, in a statical way. However, such an approach assumes that every position, while the ship rolls, is an equilibrium point. But, if the rolling motion is a forced one, then it is not possible to talk about an equilibrium position. Even in free rolling motion there are no other equilibrium points except those defined by the non-linear restoring moment.
2. STABILITY OF FREE ROLLING MOTION In this part of the paper asymptotic stability of the non-linear free rolling motion of a ship will be investigated. Further information and details may be found in Ozkan (1977). Some theoretical considerations are given in Appendix 1 of this paper. Let us imagine a ship freely floating in an equilibrium position, say upright position, on the free surface of water. Due to the perturbations, the ship heels to a degree x, and begins to perform either an oscillatory or a divergent motion or reaches another equilibrium position and keeps it. The matter under discussion is to investigate whether the ship returns to its original position or not, after the perturbations stop disturbing the ship. In other words what we are looking for is the asymptotic stability of the equilibrium position. Such a rolling motion is governed by the following differential equation. (I + J) Yc + f~ (~c) + g~(x) -----O.
(4)
As can be easily seen, the problem is generally defined in the 2-dimensional phase space whose coordinates are rolling angle and velocity. Since I + J ~ 0, then Equation (4) becomes
+
+
g ( x ) = o.
(5)
By using the state variables, we obtain the following first order simultaneous equations: i q = A (xl,
=
•;c2 = f~ (xl, x ~ = - - f (xz) -- g(xO,
(6)
in matrix notation:
(7)
k a21(x) or simply = A (x) x; A(x) = ad(xi, x)); i = 1, 2 ; ) = 1,2. To obtain the conditions of asymptotic stability, let us prove the following theorem, in which the variable-gradient method is employed, to generate a Lyapunov function.
A
497
rational approach to intact ship stability assessment
Theorem: If for ~z =f(x), f ( x ) = O, there exists a bounded domain f~ and a real vector OV coxa
aal(x)xl + als(X)X2 + . . .
+ al,,(x)x,,
a2x(x)xl + a22(x)x2 + . . .
+ a2,,(x)x,,
OV c9x2
v
v = { V v,} =
OV
a,a(x)xl + a , a ( x ) x 2 + . . . + a,m(x)x,
Oxn
such that (1)
aVv, _ aVV/,
8xj
8x~
(2)
VV:/:o vx:/:o,
(3)
dV --vvr'i dt
(4)
dV. m is not identically zero for any other solution except dt
<0
Vx~O,
x = 0 in ~,
(5) V(x)-- / VVrdx 0
(a) V(x) > 0
V x :/: 0 and V(O) ----O,
(b) One of the curves of V = K = const, bounds the domain in the phase-plane, then x = 0 of ~ = f (x), that is, the equilibrium position of the free rolling motion is asymptotically stable in ~ which is the domain of asymptotic stability.
498
I.R.OZKAN
Proof: The equation of the non-linear free rolling motion is of the following form (I
+
J)3c 4- a,.~ 4- b~l.~l.x 4- 4 x
--
el x 3
--
O,
(8)
and since I + J :/: O, then
(9)
Yc + a X + b[Jc[Yc + d x - - e x ~ = O.
By using the state variables, this equation may be defined by the following first order simultaneous differential equations: )¢1 = X2
(10)
X, = - - dxl + exal - - ax~ - - b Ix2 [x~
[ vii VVl=
VV=
,,,x, +
V V , = ~V
,
]
dV - [(~nx~ + ~1~x2) (~slx~ + ~r~x~)] dt .
By substituting the values of x~ and x2 from Equation (10), we obtain: dV - [(**nxl + ~ x l ) (~sxl + o~x,)] [ dt
x2
[ - - dxl 4- ex81 - - ax2 - - b] x , [x~
(11)
or
dV d'--t- = °qlXlX~ + °cI~¢S2 - - ~ l d x 2 1
-~- ° ~ l e X l l - - a ~ l x l ' x ' ~ - -
boc211x~[xlx~
- - d o ~ q x ~ + ~q~exaax~ - - ao~,dc2s - - bat,,2[x21 x~2 •
According to the conditions d V ] d t < 0; equations should have no indefinite terms. For this reason by taking ~12 = 0hx = 0, the above equation takes the following form: dV dt
- - ag.nx', + (=n - ~ n d ) x l x , + o~nex'lx, - - ~ [
x,[ x ' , .
(12)
A rational approach to intact ~hip stability assessment
499
Due to the condition V x V V = 0 we obtain:
which yields 0 ----- 6~11
t~x~
X1 •
To satisfy this equation, ~ax should be only the function of xl. Let us take: (13)
oqa = d~2 -- ex~xo~. Substitution of this value into Equation (13) results in the following equation: dV = -- ~q~x'2 (a + b Ix=I). d-T
(14)
Having determined dV/dt X
Xt
0
0
d x 21
V(x) = ~q,
X|
2"--
0
ex41
~%' T
x22
+ ~t'' --~ .
There is no point in choosing ~-n as a constant value; let ~2
=
4.
Then the Lyapunov function: V(x) = x~l (2d -- exZl) + 2x~2,
(15)
I~'(x) = -- 4x~ (a + b [x~ I)"
(16)
and its Eulerian derivative:
Asymptotic stability conditions: x~x (2d -- exat) q-- 2x~2 > O, - - 4x22 (a +
blx~ 1) < 0;
a +
bix, I >
o.
(17)
500
i.R. (}ZKAN
Equilibrium positions." According to i = f (x); f (0) = 0, i == 0 yields equations 0 ==xl : x2,
(118)
0 = -- ax2 -- b Ix2[x2 -- dxl + exal, which yield the following equilibrium points:
0 (0, 0),
(A --x/d/e, O) B (x/d/e, 0).
By using the characteristic equations it is found that:
0(0,0):
(a) ifa ~ - 4 d > f (b)
O,
stable node.
ifa s-4d
stable focus.
A (-- x/d/e, 0), (B x/d/e, 0),
unstable saddle points.
~,ol
A (-A F~,o)
0(0,0) Fig. 4. Equilibrium points of rolling motion in phase-plane.
Domain of asymptotic stability: As the rolling motion is taken as a non-linear one, it is not enough to know only the conditions for which the motion will be asymptotically stable. As far as the non-linear systems are concerned, stability is a local concept and one should design a domain of attraction around the asymptotically stable equilibrium position. Lyapunov's Direct Method gives us the possibility of taking into account the nonlinearities of the system and hence to evaluate the domain of stability. This property of the method used, demonstrates its basic advantage, bearing in mind that stability research
A rational approach to intact ship stability assessment
501
cannot be complete and satisfactory, without taking into account the non-linear behaviour of the system. Before going deep into the subject, it is suggested the reader studies Appendix 2, in which some definitions and a theorem on the extent of asymptotic stability may be found. In the following, a domain of attraction for the asymptotically stable equilibrium position of the free rolling motion will be constructed. Since a, b ~ 0, then it follows that
~._OV -
Ot
4x'~(a+blx,I)=o,
x.,=O,
R = {x [ x 2 = 0, x 1 X 0} is a s u b s e t o f the axis xv
Let us write the slope of the curve in the phase plane: tgX2
0X_..__22= Oxx
Ot
-- dx I +
exZl -- ax2 -- b [xz ]xz X2
"OXl
Ot As dxJdt ~ oo at every crossing point of the xx axis except xx ~ 0, then the possible invariant sets are found to be the equilibrium points. 0(0, 0), A (-- v/die, 0), B(x/d/e, 0). As a matter of fact, the points A and B were found to be unstable points which should be excluded from the domain of asymptotic stability. In conclusion, f~ : V(x) < K should be chosen by taking A and B as the boundary points. g(x) = V(x1, x2) = X~l(2d -- ex21) + 2x~2 = Const.
(19)
These curves cross the axis xl, at x = 4-x/die for x~ = 0 then V(x) = (+~d/e) ~ (2d -- e(+Vd/e)Z), d2
V(x) = K ---- - - .
(20)
e
The ordinate values of the points belonging to the curve V(x) = d2/e are as follows:
x2=4The other point of the curve:
c(0
x / ( - ~- -
x~l(2d-
(21)
I. R. OZKAN
502
extremum points: E(O, d/s/2e)
C
=
F(O, -- d/s/2e) G( s/d/e,
O)
D
=
= B
H(-- Vale, O) = A. According to the theorem given in Appendix 2, the interior of the domain bounded by the curve V(x) = an/e is d2 f2 ={x I V (x) < - - ,
I~'(x) < O}
(22)
e
and the set R in which d V/dt becomes zero is R = {x [ -- s/d/e < xl < s/d/e, x2 -:- O, d V/dt = 0}. The largest invariant set in R is M = {0}. This, also, is the positive limiting set F +. The domain already obtained (Fig. 5) is the interior of curves V(x) = an/e such that any rolling motion initiating within this domain never leaves it and in addition tends to the origin as the time increases.
•
E(O,d/~)~
stable equilibrium position ( I n v a r i a n t ~
~ /
x2= Y
/
~
/ V ~
~ ~- / /
stability
• v.
(
(laddie point)
~
~
(saddle point) ~F(O,-
FIG. 5.
xl'Vrx x ; d2
~-~ ~ ~ J
J ~ 1
'°'
. Asyrnp.stable trajectory
d/2eta)
Domain of asymptotic stability of the free non-linear pure rolling-motion of a ship.
A rational approach to intact ship stabilitya sessmcnt 3.
503
STABILITY OF FORCED ROLLING MOTION
Total (practical) stability: In the previous part of this paper we investigated the asymptotic stability of the equilibrium position and the behaviour of the trajectories of the free rolling motion of a ship under the assumption that the perturbations, which are small in magnitude, act on the ship for a finite duration. Stability, in fact, possesses the meaning that the moderate perturbations acting on the system do not turn the system's behaviour to an unstable feature and asymptotic stability implies that the effects caused by the perturbations disappear in time, after the perturbations cease. But, in practice, perturbations and excitations act on the system for longer periods. Under these circumstances it is necessary that one should mention total stability for the systems subjected to such continuous excitations. Due to the continuously acting perturbations and excitations, stability and asymptotic stability may not guarantee practical stability. At this point we need to ask the question whether asymptotic stability is a necessary condition for practical stability. It is not necessary at all. Due to the system's characteristics, running conditions, and mostly the mutual system-environment relations, the system may be practically stable while it is mathematically unstable i.e. oscillatory rolling motion under the effect of continuously acting excitations. Practical stability is to confine the solutions of the equation of motion to a domain which is safe enough, with respect to the initial conditions and the magnitude of excitations. In this part of the paper the results of two theorems on "boundedness" and "uniform boundedness", ("Lagrange Stability"), and especially a third theorem giving the conditions for, and the domain of practical stability of the solutions studied and proved by the author, will be given. For further information on the subject and details of proofs the reader is referred to Ozkan (1974, 1977). A theoretical background knowledge and the comments on total stability, in naval architectural terms, together with some basic definitions for non-autonomous systems are given in Appendix 3. A ship performing a forced rolling motion, under the effects of continuously acting excitations, belongs to a class of dynamical systems which are defined by the following differential equations :x = f (x) + u(t). Assuming that the ship is subjected to a time-dependent wave excitation and a constant wind heeling moment, the equation of motion may be written as follows: ( I + J ) .~ + fl(.x) + gl(x) =
ea(t) + WM,
or
"~ + f(J¢) +g(x) = e(t) + P.
(23)
Boundedness of solutions: Theorem (boundedness). By using the first integrals of the equation of motion of a rolling ship, it has been proved that the rolling angle and velocity are bounded, that is,
504
i.R. (~ZKAN
f
e(t)dt < 0o;
[e(t) [ ~ E, {[e(t) [ + P} ~ A1
(24)
0
and lim x
- + oo
[ f
(g(x)--A1)dx
[-+ d- c~
0
Theorem (uniform boundedness). By choosing a time-dependent Lyapunov function, it has been proved that the solutions are uniformly bounded in to, that is, bounds do not depend on the initial time. A region, which is the intersection of three sets each of which is obtained for the different features of the non-linear restoring moment, has been obtained. A* =
ADBDC
In addition to the conditions already given for the boundedness of the solution some further conditions on restoring moment are as follows:
g(x) ~>0, V x E (O,x/d/e), g ( x ) = O,
x = O, V d / e ,
G(x) ---- f g(x)dx > O, '¢ x E (0, v/2d/e), 0
G(x) = O,
g(x) ; G(x)
~
V x >~ V Z a / e ; x <~ -
/-g
V2a/e.
(x)
/ (x)= J'g(x)dx
X
(Zd~'aT~,O)
FIG. 6.
Restoring moment characteristics.
(25)
A rational approach to intact ship stability assessment
505
Apart from the results given above, one should construct a domain of total stability in the phase-plane. Theorem (domain of practical stability). By using the concept of a "Jordan curve" the domain of practical stability has been constructed as in Fig. 7.
bx2:Y;g(x) b,:Xt.°t
Trajectory of a 'Totally Stable' rolling motion
o. tara,
..abi,ity
FIG. 7. Domain of total stability. Curves:
PaPa = ½y~ q- G (x) q- Axx = Const, PxP2 ~- ½y2 + G (x) -- Axx = Const, P2Pa = ½Y~ q- G(x) = Const. Points: Po (0, b, + 6al A1 + 1) /)1 (xx, bl) 3¢|
f P~(xl+
xl
Ps (~/d/e, 0).
g(x)dx -- b21 2Ax
, y~),
(26)
506
]. R. OZKAN
The conditions under which the domain of practical stability is obtained, are as follows. (1)
.~f(.~) > O, V .x # O, x sgn ~ > 0
(2)
[e(t) [ <~ E;
(3/
lira [G(x) I = lim t f X
le(t) l + p <~ A1,
--~oC~
); -.+o0
g(xldx I--> or,
0
(4) righting moment and excitations: q(x) :dx-ex s
2A,: 2 mox{le(')l+ P)
FIG. 8. Righting moment and excitations.
lim G (x) = lim ~ X ---~- aO
X - - ~ aO
Ig(x)
[dx --~ or,
0
g(x) < 2A1,
0 < x < ax;
g(x) > 2A1,
al < x < a'2,
g(x) = 2A1,
x = al, a 1,
a'l < x < + o o ,
(27)
t
(5) damping moment and excitations:
f ( x ) sgn.x >2A~
'q" [ ~ ] >bl,
f ( x ) sgn:~ = 2 A x
I ~ I =bl,
f ( x ) sgn:~ <2A~,
V I j [
(28)
A rational approach to intact ship stability assessment
507
f (t):ot+bltlt
, 2A~
bl
/I
FIG. 9. Damping moment and excitations. In most of the engineering applications, the righting moment g(x), on the contrary to the physical realities, is taken as having a single feature, increasing or decreasing. But, in fact, the systems do not behave in that way. As far as the rolling motion is concerned, righting arm curve is, at the beginning, a monotonically increasing function and after an extremum point behaves as a monotonically decreasing one. Bearing in mind this reality, the domain of practical stability is constructed by taking into account these properties of the GZ curve. l
Comments on the domain of practical stability: The domain shown in Fig. 7 is the domain of practical stability in which the solutions of the equations are uniformly bounded in to with respect to the set of initial conditions and excitations. A forced rolling motion initiating within the bounded parts of the first and third quadrants will never leave them and results in an oscillatory motion around the equilibrium position. Motions initiating at the outside of the bounds will tend to the interior of the bounds in a finite time, that is, the trajectories cross the bounds from outside. Second and fourth quadrants are convergent regions such that any motion initiating within these quadrants enters to the first or third quadrant and converges, since it is not possible that while the ship is heeling to one side, rolling velocity increases at the opposite direction. It is also found that when the roiling angle takes the values such that xa > a't, then the motion shows the character to differ away from the domain of practical stability. This is the case illustrated by the curve PaPs. At this part of the domain, to converge the solutions to the domain total stability, one should design an additional damping mechanism which begins to operate when the magnitude of the state vector, that is,
tlx II =(x,1 + x,,),, reaches to a certain value. The desired additional damping moment is:
M > Aa - - f ( y ) >i l e(t) I + P --f(Y)"
(29)
508
i . R . OZKAN , x 2
gCx)
!P~Cx,, b,)
/
i
X
O
at
X=> a,
CJ(X)
P~(x,,b~) ~, P2(x2,yz) t
I
X
CII
'
~
Xj
01
g(x)
xz
~xz,yo) X O,
FIG. lO.
~
- -
w
Xl
P3(X3,0)
Evaluation of the bounds in accordance with the different features of g(x).
In this study only wind and wave excitations are considered. But since the calculations are carried out for the norm of the excitation terms, any other type of finite excitations can be considered as well. This does not change the general features of the results obtained. In this case, the restoring and damping moment are required to have the greater values. 4.
CONCLUSION
Finally, the findings of this study can be summarized in terms of the current understanding of the ship stability problem, as follows: (a) To analyse the stability (i.e. asymptotic stability) of both free and forced rolling motion of a ship, by using the statical technique, is insufficient and results obtained by such an approach are completely unsatisfactory. This can be easily seen by comparing the results we obtained and the current ship stability criteria. This is the most important drawback of the criteria used for the time being.
A rational approach to intact ship stability assessment
509
(b) T h e i n t e r r e l a t i o n s b e t w e e n the ship a n d t h e e n v i r o n m e n t a r e t h e i n d i s p e n s i b l e items o f every stability analysis. A s y m p t o t i c a n d t o t a l stability c o n d i t i o n s a n d d o m a i n s are o b t a i n e d as t h e f u n c t i o n s o f t h e e q u a t i o n o f the m o t i o n itself. (c) T h e results o f this s t u d y c o v e r a n y t y p e o f s e a - g o i n g vessel, since there has b e e n n o specification m a d e o f t h e type a n d t h e size o f t h e ships. F o r a k n o w n ship, the e n v i r o n m e n t a l c o n d i t i o n s , such as, t y p e a n d the m a g n i t u d e o f e x c i t a t i o n s at w h i c h the ship will o p e r a t e safely, c a n be e s t i m a t e d i n a d v a n c e . APPENDIX I Some theoretical considerations of the stability of motion: The theory of stability of motion is concerned with investigating the effects of perturbations and excitations on the motion of a physical system. The state of any physical system can be defined by some variables, say n, at any time, t. These variables are known as the state variables which make up an n-dimensional state vector, x (t). x(t) = ( x l (t), x2 (t) . . . . .
(A.I.1)
x~(t)).
Taking the time as the independent variable, t E I = [to, + Qo), the state of the system is given by the ordered pairs, (x (t), t), which are the members of a subset of a (n + D-dimensional inner product space. (x, t) E U c E'~+tx, t = E"x x I
(A.1.2.)
Let the system be given by the following vector-differential equation: dx = - - = f (x, t) dt
(A.I.3.)
Where f (x, t) is an n-vector function defined in Ex,t"+l; f : I x E "+1 x,t --+ E"x.
Let x (t) be a solution of Equation (A.I.3.) at a time t. If the Cauchy-Lipschitz theorem is satisfied in U, then there exists a solution of Equation (A.1.3.) and it is unique. This solution draws an integral curve, g, in U. The curve g is a continuous set of points and can be formulated as follows: g = ( (x, (t), t)} E U c E n+lx,t s.t. t E [to, + oo); W t >I to. Due to the external effects, let us assume that the system's state reaches to another integral curve, gp, (Fig. A.I.1.).
x(t}
x(t)/~.., gP
i~( t o) 0
Fig. A.I.1.
i =
t
to
Perturbed and unperturbed integral curves of a system.
510
I . R . OZKAN
Defining the amount of deviation by y(t) and a special solution by ~,(t), we obtain x (t) : ~(t) -~ y(t).
(A.I.4.)
Substituting this into Equation (A.I.3.) and after some manipulations, we finally reach to = F (y, t), F (0, t) = 0.
(A.I.5.)
Every solution of Equation (A.1.3.) defined by y = 0 or x(t) = ~(t) is called art "unperturbed motion" of the system or a n "equilibrium position" and, in fact, this is the solution whose stability is to be discussed.
Definition: Given a n arbitrarily small positive number e, our problem is to find a positive number "q such that if the initial values x (to) of the variables x,(to) corresponding to t = tl ~- to satisfy [I x (to) I[ < q,
(A.I.6.)
then [ 4 x ( t ) t] <
E Vt
>/ ta.
(A.I.7.)
If such a a m b e r rl exists--then unpexturbed motion is called "stable" with respect to x~, xl . . . . . x,. Otherwise it is "unstable". It may happen that condition (A.1.6.) implies that lim x (t) -~ 0, t-p+
co
then every perturbed motion, sufficiently close to the unperturbed one, approaches it asymptotically. In this case, we shall say that the unperturbed motion is "asymptotically stable".
Stability in sense of Lyapunov : Let us think of a physical system. If the total energy of the system decreases monotonically, then it follows that the system's state will tend to a n equilibrium position. This holds, because energy is a non. negative function of the system's state and reaches a minimum when the motion stops. Lyapunov's direct method is the generalization of this idea, and the question of stability is thus convexted to the problem of the e,xistence of a positive definite scalar function, called Lyapunov function, whose time derivative is negative definite. Autonomous system: Throughout this paper a ship performing a free rolling motion is taken as a dynamical system in which the independent variable, time, does not appear explicitly. Governing equations for a n n-dimensional system are as follows. = f (x)
(A.1.8.)
= A (x) x
(non-linear)
= A~ x
(linear).
(A.1.9.) (A.I.10.)
Let the system be given by Equation (A.I.8.) such that f (0) = 0. Let S(R) and S(r); 0 < r .~ R and S(r) c: S(R), t/
IIxll
Z x~)t
The closed annular region is shown by
Sn,={xlr
~llx[l ~R}.
We shall suppose that in a certain open spherical region, S(A) = {x [[I x II < A} the basic existence theorem holds for Equation (A.I.8.). The partial derivatives with respect to eaeh state variable, that is,
A rational approach to intact ship stability assessment
511
v(x)
Stabilitl/~
Instability
/'o
X)K2
(a}
X2 9(x) > 0
Xl
~i
Vlx)>C
lity
"]
~"
Instability region
/
Instobilit region L
[
/ I I
K2 ~/(X)=Const. KI
(b) Fig. A.1.2.
"Lyapunov function and stability".
all exists and continuous in S(A). We, also, recall that through each point x(t) of S(A), there goes a unique path, I'. We shall designate the positive Limiting set by I'+. , . If for every S(R) there exists a spherical region $(r) such that every solution initiating within this domain never leaves S(R) as the time goes to infinity, that is lim II • ( 0 [I < R
.....
X---~ OO
then the zero solution of ~ = f(x) is said to be "stable". Apart from this, if, for every R, > 0 there exists a region such that every solution initiating within this domain tends to the origin as t ~ + 0% then the origin is said to be "asymptotically stable", if V Ro > 0 9S(Ro) s.t. lira II • (t) II ~ 0. t--~ -I- oo
512
]. R. OZKAN
Lyapunov function :
In autonomous systems a Lyapunov function is a scalar function of the system's state x(t), satisfying the following conditions. (1)
V(x)>O Ilxff
(2)
<
0
lim V(x) ~ ÷ oo, II x II --*
(4)
C E x ~,
V(x) :- 0, II x II =
(3)
Vx~O;xEf~
h
co
generalized upper right hand derivative is of the form d V(x) dt
l lim sup -~- [ V(x + h f (x) -- V(x)]. h ~ 0+
If V(x) E C a then this equation becomes equal to the Eulerian derivative
I;'(x)
OV(x) -
Ot
~V dx~ OV dx~ OV d x . =Ox---~xd-i- + ~ - ~ + ' ' " + Ox. dt '
(a)
(b) Fig. A.1.3.
Geometrical representation of the domains of stability and asymptotic stability.
A rauonal approach to intact ship stability assessment I2(x) = V V(x) r.~ (5)
l:'(x) ~ 0
513
(A.l.ll.)
VxEt~.
Lyapunov stability theorem: If there exists a Lyapunov function V(x) in a spherical region around the origin, then the "unperturbed motion" or the "equilibrium position" is stable.
Lyapunov asymptotic stability theorem: In addition to the conditions for stability if l:'(x) < 0, then the "unperturbed motion" or "equilibrium position" is "asymptotically stable".
APPENDIX
2
Definition: positive limiting set, F + Let x(t) be a solution of x = f (x). I f w ~ :> 0artd T > 0 y t ~, Ts.t. [I x ( t ) -- Pll < then
p E F +,
and if x(t) is bounded then I ~+ is a non-empty compact invariant set.
Definition: Invariant set, M. If every solution initiating within the set M remains in M for t E [to, + or), then M is a n invariant set. According to this definition: ifv~>0
yT>0s.t.
Vt>T
ypEMand[lx(t)--pl[<~,
then x(t) approaches to the invariant set M as t --, + oo. For example if V t > 0, x(t) is bounded, then x(t) ---, F+ as t --, + or.
Theorem: Let f~ be a dosed and bounded (compact) set such that every solution x(t) of x = f (x) initiating within this set will remain in ~ . Let us assume that there exists a scalar function V(x); V(x) s Ca; posse~lng the first partial derivatives and its Eulerian derivative is negative sere-definite; I;'(x) < 0; in t2. Let R be the set of all points satisfying V(x) = 0; R = {x I ~(x) = 0} and M i s the largest invariant set in R. Then every solution initiating in f~ approaches to M as the time goes to infinity. x(t) --, M
W x(to) [ [2, t --* or.
Proof: Let x(t), x(t) E ~ be a solution o f x = f ( ~ ) . Since I;'(x) < 0, V(x) is a nonincreasingfunction of tithe. V(x) is continuous in f~ due to V(x) ~ 6"1 a n d is bounded from below. Because [1 is a compact set, then for every x E f~ it follows that l[ x (t) 11 > inf f~. Because of that reason V(x) will have a limit value as the time goes to infinity. lim V[x (t)] = C. In addition, since f~ is a closed set, the positive limiting set I"+ of x(t) will be in f~ and V(x) takes the value C in I'+ since it is continuous in ~ . V(x) = C, V x E I'% V (x) E Ca, x E t'l. F + is a n invariant set and IY(x) = 0 in P+. Hence r + becomes a subset of M, F + K M. This result shows that, as it is already mentioned above, every solution initiating in t~ tends to M. That is, if x(t,) E f~,
514
L R. OZKAN
then x(t) s M
ast~
+ oo.
APPENDIX
3
Total stability: Let the system be given by:
~k = f(x) + u(t).
(A.3.1)
Total stability is the existence of two compact sets and a number 8 > 0 for this system. Let t'l*o he the set of all initial states, x(to) of the system, x(to) E fPo, and fl* be the set of all ultimate states at which the system will operate in safe, x (t) E f~* V t ~. to. f~o* is a subset of fl*. flo* ~ f~*.
(A.3.2)
u(t) is a n-dirnemional vector representing the excitations and perturbations. If the set including every possible excitation and perturbation is U, then u(t) E
u,,
and I[u(t) ll < 6
V t ~. to, V x ( t ) .
(A.3.3)
According to the explanations given above, if V u (t) E Up and x (to) E fP0 ~ x (xo; to, t) E ~, V t ;~ to then the origin of Equation (A.3.1) is totally stable. In other words, total stability is the investigation and the justification of the following items simultaneously. (a) What are the excitations and perturbations expected due to the environment in which the system operates ? 8. (b) How sufficiently could the initial state of the system be controlled ? Set ~*0. (c) What should he the acceptable safe behaviour of the system as far as the engineering practice is concerned ? Set fl*. A topological representation of total stability and its comment for rolling motion is given in (Fig. A.3.1).
Boundedness of solutions: Definition: If V x (xe; to, t) and t > to there exists a ~ > 0 such that
]1 x (xo; to t)]1 < fl
(A.3.4)
then the solutions x (Xo; to, t) of ~[ = f (x, t) are bounded.
Definition: If the bound [3given in the above definition is independent of the initial time, then the solutions are uniformly bounded. Theorem: If there exists a time-dependent Lyapunov function, V (x, t), defined in the product space A* ~ E - a × I ; E ~
= { x I[Ixll > a } a n d l f f i { t t t ~ Ire + oo)}
provided that: (a)
a ( l j x l l ) ~ V(t,x) ~ b ( [ I x N )
whgTc
a(r) E P.C.I, and b(r) s N.N.C.I, (b) IY(x, t) ~ 0 in A*, then the solutions of x = f (t, x ) a r e uniformly bounded.
A rational approach to intact ship stability assessment
515
t~'-'-'-W M
' J '
\
\
~0~I
\
\
~,~ el(t),WM...
WM •
t
~
~--
~m's bound . . . . . . . . . . . ,i"/,o's bound
Fie. A.3.1.
Topological and physical representation of total stability in the forced rolling motion.
REFERENCES ANTOSlEWICZ, H. A. 1958. A survey of Lyapunov's second method. Contributions to the theory of nonlinear oscillations. Ann. Math. Stud. 41, pp. 141-166. C"mrr^vEv, N. G. 1961. The Stability of Motion. Pergamon Press, Oxford. CODmNt~TON, E. A. and LEVINSON,N. 1963. Theory of Ordinary Differential Equations. McGraw-Hill. HAHN, W. 1963. Theory and Applications of Lyapunov's Direct Method. Prentice & Hall. DE HEEItV.SO-mLTEMA-BA~r.a~R,A. R. 1969. Buoyancy and stability of ships. Tech. Publs. H. Stam. J.S.N.A. 1960. Advances in research on stability and rolling of ships. 60th anniversary series. 6. KALMAN,R. E and BERTRAM,J. E. 1960. Control system analysis and design via the second method of Lyapunov. J. has. Engng. pp. 371-393. Kuo, C. and ODABASl,A. Y. 1974. Theoretical studies on intact stability of ships. Phase 2 Report, University of Strathclyde. Kuo, C. and ODABASI, A. Y. 1975. Application of dynamic systems approach to ship and ocean vehicle stability. International Conference on Stability of Ships and Ocean Vehicles, Glasgow, Scotland, pp. 1-22. Kuo, C. and OD^aASl, A. Y. 1974. Alternative approaches to ship and ocean vehicle stability criteria. J. nay. Arch. KRA~VSKII, N. N. 1963. Stability of Motion. Stanford University Press. LAaALLE, J. P. 1964. Recent advances in Lyapunov stability theory. SIAM Review 6, pp. 1-11. LAaALLE, J. P. 1968. Stability theory for ordinary differential equations. J. diff. Equat. 4, pp. 57-65. LYAPUNOV,A. M. 1949. General problem of stability of motion. Ann. Math. Stud. 17, pp. 203-474.
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i . R . OZKAN
MASSERA,J. L. 1949. On Lyapunov's conditions of stability. Ann. Math. 50, pp. 705-721. MASSERA, J. L. 1960. On the existence of Lyapunov functions. Publnes Inst. Mat. Estaddst., Montev. 3, pp. 111-124. ODABASl, A. Y. 1973. Methods of analysis of non-linear ship oscillations and stability. Departmental Report No. 08/73, University of Strathclyde, Scotland. ODAaASI, A. Y. 1976. Ultimate stability of ships. Trans. R. lnstn nay. Archit. ODABASI, A. Y. 1978. Conceptual understanding of the stability theory of ships. Schiff~technik 119. OZKAN, I. R. 1977. On the general theory of the stability of ships via Lyapunov's direct method. Ph.D. Thesis, Istanbul Technical University. OZKAN, I. R. 1974. On the boundedness and stability of ordinary differential equations. Departmental Report, University of Strathclyde, Scotland. OZKAN, I. R. A boundedness theorem on a class of non-linear second order differential equations .To appear. OZKAN, I. R. Effects of external excitations on the stability of ships. To appear. I~RSIDSKt, S. 1961. On the second method of Lyapunov. P M M 25, pp. 17-23. RAHOLA, J. 1939. The judging of the stability of ships and the determination of the minimum amount of stability. Helsinki. RAZUMIKHIN,B. S. 1958. On the applications of Lyapunov's method to stability problems. P M M 22, pp. 466-480. ROBB, A. M. 1958. A note on rolling of ships. Trans. Insm nay. Archit. pp. 396-402. TEUFEL, H., JR. 1972. Forced second order non-linear oscillations. J. math. Analysis Applic. 40, 148-152. THOMSON, G. and TOPE, J. E. 1970. International considerations of intact ship stability standards. Trans. Instn nay. Arch#. WISraEWSKI, J. 1961. Mechanical criteria of ship stability. Schiffstechnik 8, pp. 87-90. YOSmZAWA, T. 1953. Note on the boundedness of solutions of a system of differential equations. Mem. Coll. Sci. Engng. Kyoto imp. Univ. A2,8, pp. 293-298. YOSHIZAWA, T. 1955. On the stability of solutions of a system of differential equations. Mem. Coll. Sci. Engng Kyoto imp. Univ. A29, pp. 27-33. YOSHIZAWA,T. 1957. On the necessary and sufficient condititlBs for the uniform boundedness of solutions of ~ = f(x, t). Mem. Coll. Sci. Engng Kyoto imp. Univ. A30, pp. 217-226. YOSHIZAWA,.T. 1960. Stability and boundedness of systems. Archs ration. Mech. Analysis 6, pp. 409-421.