c
Ocean Engng. Vol. 6, pp. 493-516
Pergamon Press Ltd. 1979. Printed in Great Britain
A1 A A 4(x) a b
cl
d,eE.
E' e(i) WM X = X1 xs = y I U S(A) S(R) Sfr) SE,n
M u(r) 11x11 sgo ¡nf sup Y E 3[.)
n.
CA RATIONAL APPROACH TO INTACT
SHIP STABILITY ASSESSMENT
I. R. ÖZKAN
Istanbul Technical University, Faculty of Naval Architecture, Teknik Universite, Istanbul, Turkey AbstractIii this study, asymptotic and total stability of the non-linear free and foroedpure rolling motions of a ship are investigated. A ship performing a rolling motion is taken as a dynamical system. Lyapunov's direct method is employed in the analysis. By generating a time-invariant Lyapunov function., conditions and the domain of asymptotic stabilityare obtained for free rolling motion. Results of the work on "boundedness" and "uniform boundedness" of the solutions of the equation of forced rolling motion, done by Özkart (1977), that is, conditions of total (practical) stability and its domain in the phase-planeare given and illustrated.
SYMBOLS
= Absolute value of excitations = Norm of a region
= Matrix of constant coefficients = Matrix of variable coefficients = Linear damping coefficient = Non-linear damping coefficient = Set of continuous functions
= Coefficients of restoring moment function max I e(i) I, Absolute value of wave excitation
= n-dimensional Euclidean spa
= Wave heeling moment = Wind heeling moment = Rolling angle
-Rolling velocity ' = time
= Time interval; r E [t0, + )
= E,. X I, (n + 1)-dimensional inner product space Definition region of ± = f (r, x)., U
Definition region in autonomous systems = Stability region
Region of initial states = Annular closed region
= Stability region in autonomous systems = Stability region in non-autonomous systems = Positive limiting set
= Invariant set
rn-dimensional excitation vector
493
Deift UniversitY of TechflOIOY
'
Ship
l4yrom6Cfl1CS
LaboratOrYlibrary
MekeiWeg 2.2628 CD Deift
The Netherlands
phone: 3115 786373-Fax: 31 15781836
= Signum function Greatest Lower bound = Lowest upper bound = For every = belon to = There exists = Closed-open interval Set intersection = Set inclusion n = Euclidean norm of x, I x j = ( x)1 1=1
2 .. ..
.-::2'
.?.:
I'ç' --
."}
-''c'.4
'.
--Z
-T
494 C.) V(t,X) V(x) bIIxII) 0(11 z H) P.C.'. N.N.C.I. ( )T - . ':i -I. INTRODUCTIONEXISTING stability criteria includingthose suggested by the
"Intergovernmen1 Maritime
Consultative Organisation", IMCO, are essentially based on the work done
by Rahola
(1939). an.d this work was performed by using data relating to 34 cases of capsizing dating
back to 1870. Values of the areas under the righting curves for the different angles of
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 surface of water,
under the effect of two opposite forces, weight and buoyancy (Fig. 1).
w =
; g1(x) = w GZ. (I)Fio. 1. Forces acting on a ship floating at the upright position.
When a ship is inclined to some degree, it is assumed that the ship keeps hernew position
as another equilibrium state and theheeling moment is taken equal to the restoring moment occurring due to the change of the underwater geometry of the ship (Fig.2).
The concept of dynamical stabilitywas 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.
I{gi(x)M(x)}dx. 0.
(2)..r - '= -m =',
-Î. R. ÖZJCAN
Derivative with respect to time = Time-dependent Lyapunov function = Time-invariant Lyapunov function = Bound for the state vector, z
= E" x I
= - z Ixli>
= P.CJ. function N.N.C.L function
= Positive continuously increasing Non-negative continuously increasing Transpose of a vector
4G
!
B
.. 'w
.L'
t..4'
-s..':
&r
. 'pi-
-. - . - -'-- n
---4':L % -_,.. .' .-1.r..
.. ... . -. -.
I.-
-r4r
- n- --:,.z-.
- .-. ._ -¿.-
;t-
-- ,g . --_-;-
:i:.:-s. - S -.-'
-
- .-
- --:IA
4
;:
-,.,
n--ifr
;-.
1t-4' .--
- 1.-
--r-. -
-A rational approach to intact ship stability assessment 495
- Ftc. 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 shii is given in the
following form:(I + J)x +i) + g1(x) = e(t) + WM.
(3) By comparison of this equation with Equation (1), the following are deduced:Equation (1) contains no velocity and acceleration terms. All inertial and damping moments are neglected,
excitations acting on the ship are not taken into account,
finally, the most important drawback of Equation (1) is to find a solution for a
dynamical system by using statical equilibrium methods.
Fig. 2. Static heel and righting moment.
GZ ( x)
Righting moment arm
Moment arm
X
-r
-Excitation Me (xl
- frt---. 2.
:";
-i. -'----'---.-.'.'--'.--- .. .-:. .-
_ -- -. --. .ì'- -'C' . -' .r---ti-
-.---'
ii
---.--;-&,.
'--- . - ', 496 i. R. Özv_&NThe 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 anapproach assumes that every
position, while the ship rolls, is an equilibrium point. But, if the roUing 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 nonlinear free rolling motion of a -ship will be investigated. Further information and details may be found in Özkan (1977). Some theoretical considerations are given in Appendix i 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 orreaches 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)
X + fi (x) + g1(x) = 0..
(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
O, then Equation (4) becomesx +f() + g(x) = 0.
(5)By using the state variables, we obtain the following first order simultaneous equations: .i
=fi (x1, xJ =
x2 X2 f2 (xi,X2) =f(x2) - g(x1),
(6) in matrix notation:a(x)
L --- - -
- --.s--
v'.'
-'1 - -a12(x) xi a22(x)] I_x2 (7)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. or simply
.p
A rational approach to intact ship stability asscssment
Theorem:
If for x =f(x),f(x)
=
O. there exists a bounded domain Q and a real vectorsuch that
V
V={V VJ
=
av
ax2a,avir',,
VVO yxO,
=VVT.x<O VxO,
is not identically zero for any other solution except
X
=
O in Q,V(x)
=
VVTdx
V(x) > O y x
O and V(0)=
O,One of the curves of V
=
K=
const. bounds the domain Qin the phase-plane, then x
=
O of x=
f (x), that is, the equilibrium position of the freerolling motion is asymptotically stable in Q which is the domain of asymptotic stability.
a11(x)x1 + au(x)xs +... ± a1(x)x
a21(x)x1 + a22(x)x2 + - . -
+ a,,(x)x
a1(x)x1 ± ;2(x)x2+... + a,.,(x)x
A
-
--.
...--
t_ç,cì
-.,-- : _.1 . _- ;-. -- A -1.:- -y---
-, - -; ¿-- - -- 4 -----
A ... - - ; 498t
Proof:The equation of the non-linear free rolling motion is of the following form
(I+J)+ai+bjlk+djx
and since I + J O, then
x
+
ax
+ bIiI + dx - ex3
= O.By using the state variables, this equation may be defined by the following first order
simultaneous differential equations:
By substituting the values of x and x2from Equation (lo), we obtaitt:
dV
=
[(X +
21x2)(x1 +
x2)1 or dV dt xi = X2 T i. R.OZcANX2 = -
dx +
ex31-
ax2 -
bIx2 Ix2r
V7V1= Lvv=.
Ç,7V2= x2 dV=
+
12X2) (21x1 + 22X2)]-
dxx1x2 4: exex31x2 az,.x22 b.2Ix21 x.X2
r
L xi X].
e1x3i
j,
- dx1 + ex31 - ax2 bi x.,Ix2
=
x1x2+
X22 -
dx21 + ex41 - a21x1x2-
b1I
x21 x1x2dV
= -
aax22 + ( -
d)x1x2 + c,.ex31x2 22b1x21 X22.= o,
i
j.
According to the conditions dv/di
<
O; equations should have no indefinite terms. For this reason by taking 12=
.
=
O, the above equation takes the following form:qo)
A rational approach to intact ship stability assessment
Due to the condition VxVV = O we obtain:
a a
- (x21x1 +
x22x2) = _(xx1 + ;2x2),
cx1 cx2 which yields Ex11 O = X1. cx2To satisfy this equation, x should be only the function of x1. Let us take:
= dx2 - ex2cc.
Substitution of this value into Equation (13) results in the following equation: dV
- - ccx22(a +bIx2).
ai
Having determined dV/dt V(xj=
f VVT.dx
=
f \7Vdx1 + f VV2dx2.
._-'-.;g
. -- --
,,. % C -499i4.
V(x) = :2
- x
+ x22There is no point in choosing x22 as a constant value; let
= 4.
Then the Lyapunov function:
V(x) = x21 (2d - ex21) + 2x22,
and its Eulerian derivative:
V(x) = - 4x22(a + bIx2).
(16)Asymptotic stability conditions:
x21 (2d - ex) + 2x22 > O,
4x22(a+bIxaI)<O; a+bjxaI>O.
(17) (15)_
-
r
- ---':'.-
'..
:ç- ;
s.--,. -. -;J-.---4I','
tn, .;.J--500 Ì.R.Özx Equilibrium positions:According to x = f (x); f (0) = 0,. = O yields equations 0 =x1 = x2,
0 =
ax2 b!x21x2 dx1 + ex31,which yield the following equilibrium points:
0(0,0), (As/0) B(\/7O).
By using the characteristic equations it is found that:
0(0, 0): (a)
ifa2-4d-?. 0,
(b) ifa24d <0,
A ( i/d/e, O), (B i/dIe, 0),
- ' '-_ - -c: ja
J-,) ::'
- - s! -- .,:'. c;r-t !Pcu r-'--A (-17,O) -.s___- ,. --r..
B (I7,0) 0(0,0)Fig. 4. Equilibrium points of rolling motion in phase-plane.
(18)
Domain of asymptotic stability:
As the rolling motion is taken as a non-linear one, it is not enouah 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
non-linearities of the system and hence to evaluate the domain of stability. This property of the method used, demonstrates its basic advantatze, bearing in mind that stability rcscarch
stable node.
stable focus.
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 3V
- 4x22(a + bIx2I) = O, x2 = O,
R
=
{x Ix2 = O, x O}is a subset of the axis x1.Let us write the slope of the curve in the phase plane: 3x2
3x2 dx1 + ex31 - ax2 - b Ix, Ix2
ÔX 3x X2
As d.x2Jdl
-
at every crossing point of the x1 axis except x1 0, then the possibleinvariant sets are found to be the equilibrium points 0(0, 0), A (i/die, 0), B(i/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, fl V(x) < K should be chosen
by taking A and B as the boundary points
V(x)
=
V(x1, x2)=
x21(2d ex21) + 2x22=
Const (19)These curves cross the axis x1, at x
=
± i/d/e for x2=
O thenThe ordinate values of the points belonging to the curve V(x)
=
d2/e are as follows:The other point of the curve:
V(x)
=
(±i/d/e)2 (2d
-V(x)=K=_.
(20) x2=
±
x21(2d - ex21)\ 2)
c
D(
Q' 0,=1'
\
i/2e/
(21)i
ç-- --- -- --j..
- -- .,.;
-i-4,*- -'--q - - -... ,__ - . -.
-- -
- - - ---
,..' .,. - - - -)._--,j_. -. - -.: ;:-.. .''
... ... -- ;-'3Ç( - ;. 4-__-'t';
-. - --
- ...- -- - --.- .,.,.- f-. - -.. -., - -- ---. __a__ --r. -
.Ç.'
-.--. h- - ._-
.'p!l-
..'. -.-
- -,-- -,--. . -i..-nt
. .. . -. ,--
- ; - .,
--.'-.;
: - -.-.. -.----
._-.--_ --
- -.- --
---.- ..--.
_'_s_ --
s-.. .._'-i
- -.-i. __.__, - --- -------
--- --------:-
-_:_-'-. .--.;_s.__
-R. (xi';'(X) o E(0,d/1e) 0(0,0) AsympToTically - Ja?, 0) (saddle point) x: yVd/e <x1 < Vd/e,
-= O, dV/di = O).F(Q,- d/I') i Asymp. stable trajectory V(x):'V(x1,x5)--Extent of asymp. sta bility B(Ja7i3O) (soddle point)
Fio. 5. Domain of asymptotic stability of the free non-linear pure rolling-motion of a ship.
___*. .- -. _-.; -.-. ---
-502 Ì. R. ÖZKAN extremum points: E(O, d/-s,/2e) = CF(O, - d/1/= D
G(/d/e, O) = B
H( /d/O) = A.
According to the theorem given in Appendix 2, the interior of the domain bounded by the
ci.rve V(x) c!2/e is
fl={xIV(x)<,V(x)<O}
(22)and the set R in which dV/di becomes zero is
R = (x
I
-The largest invariant set in R is M = {O}. This, also, is the positive limiting set r. The domain already obtained (Fig. 5) is the interior of curves V(x) = d2/e such that, any rolling motion initiating within this domain never leaves it and in addition tends to the
.-. -
-,
--=-,..- r -, ?- ''-
--- - . L' 'P- - L' Ç-.- - --.: -- 4 ., --.---.
-- -_y
-- ---- -.-
-. --,--- -,---,---,---
,.- - - - -'.. - ,- ._,-._ - .- _4_-
-. - ----a. .,- --A rational approach to intact ship stabilitya sessmcnt 503
3. STABILITY OF FORCED ROLLING MOTION Total (practica!) stability:
-In the previous part of this paper we investigated the asymptotic stability of the
equili-brium 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
perturba-tions 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 asymp totic 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-environ-ment 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 coiífine the solutions of the equation of motion to a domain which is safe enough, with repect 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 Özkan (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 fu.ed rolling motion, under the effects of continuously acting
excitations, belongs to a class of dynamical systems which are defined by the following differential equations
* = f(x) + u(i).
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)
X +
f1() +
g1(x) = e1(r) + WM, orX + 1(x) ±g(x) = e(i) + 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,
xI<3i;IiI<a
Vt?- l,
_;
,-.---
__.z '"
-¿
--. _i- - - -- ,.-j;:-
. ' ...-..r
and o r-4. o Xf()i>O VO
fe(i)dt<; le(i)
IE, (le(i) +
F} A1 (24) limf (g(x)_A1)dxJ_*+
Theorem (uniform boundedness). By choosing a time-dependent Lyapunov function, it
has been proved that the solutions are uniformly bounded in t0, that s, 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.
= AflBflC
In addition to the conditions already given for the boundedness of the solution some
further conditions on restoring moment are as follows:
g(x) >
O, ' X E (O,g(x)=O x=O,i/,
X
;;
FIG. 6. Restoring moment characteristics.
-:
504 i.R. ÖZKAN (25) G(x)=
J g(x)dx >
O, V X E (O, -.75iie), G(x) = 0,y
x>
x- ñ77e.
S' . .. -
-J .:-- -.:;-.-;
Y.'.::; --
--j
-.:-'''?''-
:#- -z-'-,;
: :..-
- '-,, 4t.c.'r J-" .-
.. -. » --y
-r.-
4: iz-: z'
A rational approach to intact ship stabilityasscasrnent 505
Apart from the results given above, one should construct a domain of total stability in the phase-plane.
Theorem (do,nain of practical stability). By using the concept of a "Jordan curve" the domain of practical stability has been constructed as in Fig. 7.
x2 y,g(x)
Curves:
Points:
P1 (xi, b1)
P3 (Vd/e, O).
FIG. 7. Domain of total stability.
= y2 +
G (x) + A1x = Const,F2P3 = y2 + G(x) = Const.
P0 (0, b1 + 6a1 A1 ± 1)fg(x)dx
b21P2(x1+
2A1 312), (x2,y2) z= y2 + G (x) -
A1x = Const, (26)Trajectory of a 'Totally P0(O,y0)
Extent of total stability
Stable' rolling motion
506
a
The conditions under which the domain of practical stability is obtained, are as follows.
if(x) > O, V X
O,xsgn
i.>
O e (t) E; e(i)+ P
A1, - --'s---- --- -- -
-- -L - ...-.-.-; .- -
-- .-
- .- a. 'a.a..
-.,;_"-'?"
---.-'--- ---.-'---
---.
-.-. ,_ ,. -. -. '.n1; :rr -. 1. R. ÖZKAN lit G(x) j=
J
J g(x)dx
j -+righting moment and excitations:
limG(x)=lim J
g(x)dx-c,
Of(x)
sgn x > 2A1f(i) sgn
x =
2A1V (i
f > b1,j i = b1,
f(x)
sgn i <2A1, V
j i
<b1.
g(x)
<2A1, O
<x <a1;
a'1 <x
<
g(x) > 2A1,a1 <x <a'2,
g(x) =2A1,
x =
a1, a'1,damping moment and excitations: Fio. B. Righting moment and excitations.
'
;;k;
f:
::
z
---
f_t - - - . ..., . - ..-... -
-. _1 . -..- .- - - -.--
-,-.. - -
-.. .-_, ,- - -...*-f.
--' -C -...-_:,? --_ .. -.r-
....
- . ..- .,.f-' - .. . --e- -,;--..
-
. . -.. .- .-- f¿... -'.
. - . --'--- . _._';.- . . - .-- .--t.L-J..L
A rational approach to intact ship stability assessment 507
f (i)sgni
f (i)ci+bI
lib,
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 curve.
Comments on ¡lie domain of practical srabiliiy:
The domain shown in Fig. 7 is the domain of practical stability in which the solutions of the equations are uniformly bounded in t, 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 rolling angle takes the vaiues such thât x1 > a'1, thei the
motion shows the character to differ away from the domain of practical stability. This is the case illustrated by the curve P2)'3. 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,
x = (x21 + x22),
reaches to a certain value.
The desired additional damping moment is:
M > A1
f(y)
e(i)+ P - f(y).
(29)2 A,
_ --- -. -,.-t __L
.,, ..- - .- - -
g(x) g(x) Ci o. o X2 .(O,y0) X2 P,(x1, b1) P, (x,,b,) X, P2 CX2 , y2 ) XI X' P2(x3 0) FIG. 10. Evaluation of the bounds in accordance with the different features of ¿'(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 under-standing 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.
508 Î. R. ÖZKAr4
o' Ci
A rational approach to intact ship stability assessment 509
The interrelations between the ship and the environment are the indispensible
items of every stability analysis. Asymptotic and total stability conditions and domains are obtained as the functions of the equation of the motion itself.
The results of this study cover any type of sea-going vessel, since there has been no specification made of the type and the size of the ships.
For a known ship, the environmental conditions, such as, type and the magnitude of
excitations at which the ship will operate safely, cn be estimated in advance. APPENDIX I
Sorne theoretical considerations of the stability of motion:
The theory of stability of motion is concerned with investigating the effectsofperturbations and excita-tions on the motion of a physical system.
The state ofany physical system can be defined by some variables, say n, at any time, t. These variables arc known as the state variables which make up an n-dimensional state vector, x (t).
x(t) = (x1 (z), x2 (z)...x(t)). (A.l.1) Taking the time as the independent variable,t E I = [t0, + ), the state of the system is given by the
ordered pairs, (x (t), 1), which are the members of a subset of a (n ± 1)-dimensional inner product space.
(x,t) E U C =E" X ¡ (A.1.2.)
Let the system be given by the following vector-differential equation:
x =
=f(x,z)
\Vhere f (x, z) is an n-vector function defined in Ex,n+l;
-
f: IxE"12
-. E.
Let x (z) be a solution of Equation (AJ.3.) at a tme t.
Ifthe CauchyLipschitz theoren. atisfled in U. then there exists a solutionofEquation (A.l.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 ={(z,(z),t)} E UCE11,1s.t.z E[z0,+
);yt.
Due to the external effects, let us assume that the system's state reaches to another integral curve, g,, (Fig. A.l.I.).
x(t)
¿(ti
o t
Fig. A.I .1. Perturbed and 'unperturbed integral curves of a system.
(A.1.3.)
e
;.
-:---'--i L
ç
a 510_E;
-'-.t-; -''- - - --tc&-
2- pfr - -i. R. ÖZKANDefining the amount of deviation by y(z) and a special solution by (t), we obtain x (z) = (:) + y(t).
Substituting this into Equation (A.l.3.) and after some manipulations, we finally reach to = F (y, z), F (0, z) 0.
(A.l .4.)
(A.l .5.) Every solution of Equation (A.l.3.) defined by y = O or x(z) = (t) is called an "unperturbed notion" of the system or an "equilibrium position" and, in fact, this is the solution whose stability is to be discussed.
Definition:
Given an arbitrarily small positive number e, our problem is to find a positive number 11 such that i the initial values x (t0) of the variables x.(t0) corresponding tot = t1 - t0 satisfy
fi iO0) II TI, (A.1.6.)
then
11x0) II< E Vt ti. (A.l .7.)
If such a number 11 existsthen unperturbed motion is called "stable" with respect to x1, x2,. . ., x. Other-wise it is "unstable". It may happen that condition (A.l.6.) implies that
lint x (t) 0,
t-' +
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 an 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 converted to the problem of the existence 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 an n-dimensional system are as follows.
x = 1(x) (A.l.8.)
i = A (x) x (non-linear) (A.l.9.)
i = A x (linear). (A.l.10.)
Let the system be given by Equation (A.I.8.) such that f (0) = 0. Let 5(R) and S(r); O < r R and
S(r) 5(R),
The closed annular region is shown by
xl <R; (
x) <R.
= {x r 11x11 4 R}.
We shall suppose that in a certain open spherical region, 5(A) = {x I Il x j < A) the basic existence
0 2. " s) >0 Instability région V(x) Stability 0 (o) VIa)K1 (b)
Fig. A.1 .2. "Lyapunov function and stability".
_(af11
-
2' a1' .aU exists and continuous in SCA). We, also, recall that through each point xO) of 5(A), there goes a unique path, r. We shall designate the positive Limiting set by I'.
If
for every S(R) there exists a spherical region Sfr) such that every solution initiating within this domain never leaves S(R) as the time goes to infinity, that isum x(t)II< R
x-.
then the zero solution of i f (x) is said to be "stable". Apart from this, if, for every R0 > O there exists a region such that every solution initiating within this domain tends to the origin as t
+ , then the
origin is said to be "asymptotically stable",if yR0>03S(R0)s.t.limIJx(t)l!-iO. t-. + Instability Instability region 4 K3 K2V() Cortst. K1 -
:-'
-;2
.xtt ka...:
.'... - L-:1 .
'-
:".--;-.':r
Lyapunov function:In autonomous systems a Lyapunov function is a scalar function of the-system's state x(z), satisfying the following conditions.
V(x)>O
VxO;xEÇZCE1',
II x Il c h V(x) =O, II x Il = Ohm V(x) - + ,
II x !'-generalized upper right hand derivative is of the form
- um sup -[V(x + hf(x) - V(x)].
dt h-.O Ji
If V(x) E C' then this equation becomes equal to the Euleriart derivative
& x1 dt ax2 dt ax,, dr
Fig. A.1.3. Geometrical representation of the domains of stability and asymptotic stability. (o)
(b)
.:..:.-- .:..:.--'
.-. - -- -.-.- -%-_-.._;_ - v.i5v -p Theorem:Let Q be a closed and bounded (compact) set such that every solution x(t) of x = 1(x) initiating within this set will remain in Q. Let us assume that there exists a scalar function V(x); V(x)E C1; possessing the
first partial derivatives and its Eulerian derivative is negative scm-definite; V(x) -E 0; infl. Let R be the set of all points satisfying V(x) = 0; R = (X I '(x) 0} and M is the largest invariant 5et in R.
Then every solution initiating in Q approaches to M as the time goes to infinity. x(z') - M y x(t0)E Q, r. co.
Proof:
Let x(t), x(t) EQ be a solution of x = f (ii). Since V(x) . 0, V() is a nonincreasing function of time. V(x) is continuous in Q due to V(x) EC1 and is bounded from below. Because Q.is a compact set, then for
everyX EQ it follows that x (t)ji > in!Q. Because of that reason V(x) will have a limit value as the
time goes to infinity.
11m V[x (r)] = C.
In addition, since Q is a closed set, the positive limiting set r of x(:) will be in Q and V(x) takes the value
C in r+ sin it is continuous in Q.
V(x) = C, y
x E r,
V(x) e C1, x Q.r is an invariant set and '(x) O in r. Hence r becomes a subset of M, r M. This result shows that, as it is already mentioned above, every solution initiating in Q tends to M. That is,
if x(0)
Eli,'
-'J'
-..-,-- .-..-,---..-,--;_
- '-_ ;----4
-)_
---A rac!onal approach to intact ship stability assessment 513
V(x) = 7V(x)T.1 (A.l.11.)
(5)
'(x)0 Vxfl.
L apunar stability theorem:1f 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 V(x) < 0, then the "unperturbed motion" or "equilibrium position" is "asymptotically stable";
APPENDIX 2 Definition: Positire limiting set, r
Let x(t) be a solution of ,t = f (x).
C> Oand T> 03 t> Ts.t. Ijx(r) - PI <
then
PE r,
and if x(r) is boundea then r 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 [r0, + co), then M is an invariant set. According to this definition:
ifvC>0
3'T>Qs.Ly:>T3pEMandIJx(t)pj<Ç,
then x(t) approaches to the invariant set M as: + co. For example if y t > 0, x(t) is bounded, then
x(t) - r as t -. + co.
then
Total stability:
Let the system be given by:
and provided that: where and x(:)EM
asl+.
a(i) E APPENDIX 3 a(UxII) V(t,x)b(IIxIl)
Theorem:If there exists a time-dependent Lyapunov function, V (x,t), defined in the product space
= E" XI; Ea
(x IIIxl
> a) and! = {t ¡t E[r0 ± )}i = f(x) + a(i). (A.3.l)
Total stability is the existence of two compact sets and a number 8 > O for this system.
Let Cl'Ø be the set of all initial states,x(t0)of the system, x(:0) E Q, and Q° be the set of all ultimate states at which the system will operate in safe, x (z) E fl
y t>
r0. flu' is a subset of Q'.Q'. (A.3.2)'
a(/) is a n-dimensional vector representing the excitations and perturbations. If the set includingevery possible excitation and perturbation is U,, then
I!u(:)ll
<8
yi.
t,
yx(t).
(A.3.3)According to the explanations given above, if
y
u(t) L U, ana x (r0) E 'O - x(x0; t, t) E Q, yt> t0thenthe origin of Equation (A.3.l) is totally stable.
In other words, total stability is the investigation and the justification of the following items simulta-neously.
What are the excitations and perturbations expected due to the environment in which the system operates? 8.
How sufficiently could the initial state of the system be controlled? Set Q',.
What should be the acceptable safe behaviour of the system as far as the engineering practice is concerned? Set Q'.
A topological representation of total stability and its comment for rolling motion is given in (Fig. A.3.I).
Boundednessofsolutions:
Definition:
If
y
a (x0;t, t)andt i,there exists a > O such thati a (x,;t, t) II < (A.3.4)
then the solutions x (x; t,, t)of x = f (x,t)are bounded.
Definition:
If the bound given in the above definition is independent of the initial time, then the solutions are uniformly bounded.
a(r) E P.CJ,
b(r)E N.N.C.I,
J' (x,t) Oin ¿s', then the solutions of X = f(t, X)are uniformly bounded.
r,
A rational approach to intact ship stability assessment 515
Fio. A.3.1. Topological and physical representation of total stability in the forced rolling motion.
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-,'-:.:7-' ::
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