Mechanical criteria of ship stability

Mechanical Criteria of Ship Stability

i1gr luz. J. \V i n i a w s k i, Gdansk

In his present report the author takes up tite idea expressed by dr. ing. Krappingerinhis article entitled: "Was ist Scizijjsstahilitiit" (s."Sdtifjstedznik" 1960, Febr. issue).

An attempt is made to establish criteria of sip sta!,iiity by Inolcing USe nf iiOtjOflS appro/iriale to Ifl('(IUifltCS

general.

Em playing titis way0/ reasoning criteria are /ormulated whose final /orms are knoten all i/trough tite n'orld bitic/tic/i werc obtained by other proceedings, including those presentedin 10511 by prof. Wendel (s. "hansa"

1950, p. 572: "Wann kentert th sdzif]:").

In. their who/v titeaboL'esaidcriteria encircle problems o! s/tip stability limited to tuai restricted range onl'yto

whtidi they (ire reduced by the current shipbuilding practice and what was due to the necessity o! intrumlucin.g numerous simplifications in u/tat concerns tite nzut/tenatica1 model of phaen.omena considared. Tite lust

result-ing mainly from the uttermost difficulty to express analytically hot/i external actresult-ing forces and forces of buoyancy appearing in far more complicated conditions 1/lati titis is tite case for an immobile vessel in still touter.

1. Together with many oiiiers problems of stability to be found treated by applied rnedianics. the problems of ship stability range closely in a wide joint group of problems of the scitflce of mechanics as a whole, 'I'hey are founded on basic theoretical principles common to them all, they have common notions, employ common definitions. At the sanie

tinte. in particular cases. they show a wide diversity of modes

of treatment. nietliods and criteria applied to, whidt seem to

contradict their community. The last statement appears with

particuiar distinctness in what is called theory of ship staltilitv.

By example:

N. D. Moisieeff [1] defines the stability of a body in general

"an ability to subsist in u space of disturbed state of motion or to return from it to such of undisturbed one, viz to immobility."

A. N. Krvloff [21* gave a typical definition of the stability

of a ship as follows: 'We call stability the ability of a ship to float in an upright position and, if inclined under action

of an external cause, to return to the above said position after the external cause has ceased acting."

When comparing these two definitions, it aun be easily 'tatcd that the last one is limiting the general problem to a particular case and covering a very restrained range of positions and motions possible.

There is no wonder that, when limiting the matter since vears and years to sudi narrow a range and completing it foni time to another by some peculiar practical cases oniy these proceedings result in finding particular solutions to particular problems. This is why there continue to exist such questions conccrning ship stability as: what is the right

measure of the stability of a ship, when does a ship capsize as well as the subdividing of the stability problems in to "initial stability", "stability at large angles of heel" a. s. o.

It is only very recently that G. Weinblum [3] in his funda. ¡tienta! report on the subject tried to define ship stability as a "feature which enables the ship to perform, when remain-ing in a determined position, the task she is constructed for." Titis indeed, represents an attinipt to formulate a de-finition of a character general. It points out that its author appropriates to the notion of ship stability a sensibly more

extended meaning than it was (lone in theoretical naval archi-tecture beforehand.

As well the work to be performed by ships, as the positions

ut which their stability should be assured, are different for different types of ships. They are different even for a single ship given in accordance with varying conditions of her exploatation. A a general rule it can be said that the funda-mental function and the fundafunda-mental position of a ship are: to remain afloat in an upright position when nt rest in still water and when under action of the force of gravitation and

the hydrostatic force of buoyancy only. However the condition

of keeping upright does not necessarily concern every type of ship; it is known that, for instance a timber carrier is allowed to sail with a certain initial angle of heel.

Functions of the same ship when sailing on undulating sea

become sensibly widened as under the action of compulsory dynamical forces of the waves, herposition of stability will be determined by repented periodical movements in report to its upright positioti iitbin the limits of some determined amplitudes. Sail boats under way will roll about cerfain an

inclined position.

Besides rolling and pitching there exist amongst periodical movements of a ship on waves, some ai:igular displacements of the ship in a horizontal plane called yawning, which influence its ability to keep a determined straightforward course.

The few examples specifled could easily be nxtendcd furthermore by a number of others one which would take

into consideration other kinds of movements and other forces which involve them. They all arc inducing tite idea that stability shall characterisc ship's motions of every nature and

that this, the problem of ship's stability makes itself a

pro-blern general iii what is called "ship's theory" on the Continent ("Theorie du navire". "Sdiffstheori&', "Teoria Korabhia").

The fact that just to present time the matters discussed have

been treated within a very restricted range, as well as the methods employed, result first of all from the particularities

of the development of scientific principles in shipbuilding.

More recent publications follow in general t?te tradition

repeating nearly word for word or with slight alternations ont y, the definitions established by their prectescssors. There also are in general no differences in what concerns the range and the ways of presentation of the "Ship stability" in adequate chapters of various treaties on Naval Architecture.

2. Following the principles of pure mechanics the

com-plete range of movements of a ship considered as a rigid body can be expressed by six equations, three of which refer to the components of displacement of the centre of gravity, e. i.

mx" = X; my" = Y; rnz" Z (1) and the other three arc expressing the values of rotation components with respect to the same centre of mass gravity

of the ship as above:

lip" + (EJ) x''

M

Jp" + (I - E)

' x' = M, (2)

Ex" + (JI)"q)

M,

in these equations the symbols X, Y, Z designate com-potients of the resultants of all forces acting an the ship's body reported to the three axes of rectangular coordinates passing through the ship's centre of gravity (fig. 1); the symbols M, M. M., mean the corresponding resultant moments of tile abovesaid forces, reported to the same axes;

the symbols I, J and E - moments of inertia of mass with respect of the three principal axes; rn designates the mass of

the ship considered.

Fig. I

As sonic of the component motions expressed by equations (i) and (2) are in shipbuilding of some greater practical importance, and as on the other hand the very complicated phenomena and forces acting on a ship in motion are extre-mely difficult to investigate and to be treated analytically, in the course of its historical development the so called "ship's theory" became subdivided into several distinct parts. It is obvious that in a more detailed way there were worked out those whidi notwithstanding some simplifying assumptions could by their results serve directly various praticai

con-structional problems.

The "ship's stability" in a form usually presented by typical

handbooks and monographies of ship's theory is presenting a case of the abovesaid. When studying its usual ranCe of problems treated one can state that those are limited to the investigation of motions defined by following simplified equations:

M=O

(3)

for transversal stability, and

M=O

(4)

for longitudinal stability, as well us

= 1"

(5)

the last only for the determination of the amplitude of heel under action of dynamic forces.

The simplification admitted in equations (3) and (4) will mean that the forces acting on a ship are considered as resulting in equilibrium, e. i. statically. Thus, the reaction of

the surrounding water becomes limited to the form of hydro-statical pressure only.

Similarly, when determining the maximal angle of heel by means of tite equation (5) all forces of dispersion arc omitted.

Contrary to that, when the phenomena of ship's motions are expressed by the equations general as abovesaid. in their

simplified form as exposed, the regarded stability criteria

can be established in a clear way by means of the energy method employing methods applied generally in mechanics.

Scintk ìd. S tS-1 - Her 41

88

-Criteria which are obtained when following the indicated way were already partly published under somewhat (liffering form however by Wendel [4). Horn 151 and Jens [61 the last making

use of them in a particularly interesting way. In this paper the author aims mainly at proposing a mnetilo(l of treating

tile problem.

3. After substitution of tite component moments functions

of all acting forces the equation (5) expressing tite motion of

ship can be written as follows:

lip" Ms,, (q)) - M1, (q)) (6) where:

M (q)) means the inchuing moment as a function of tite

angle of heel,

(q)) a similar function of tile righting moment of a couple of forces appearing when tile ship becomes inclined, e. i. the force of gravity and that of the

ship's buyoancy.

Multiplying 1)0th parts of tite equation (6) by dq) and inte-grating them we obtain:

r '

2

'M () thpjM1, ((p) d

(7) This equation expresses the kinetic energy of a ship inclined under action of the moment ; this energy equals to tite

sum of work performed by all tile forces which actually act

on the ship.

The maximal angle of dynamic heel can he determined by

putting:

as at the angle so determined the angular speed equals zero

and the ship is finding herself at a momentary immobility.

In order that tile rotation could take place in the opposite sense tite acceleration in this point should have a negative

value, e. i. the following condition must be fulfilled

(9)

From equation (7) there follows

(p' =

1/2/J [fM (q)) d_rM (p) dp

what means that to satisfy conditions (8) nd (9) there shall be

w w

j M,., (q)) d(p

=

M, ((p) dq) and

M,., <M1, . (11)

Tite conditions defined by (10) and (11) represent criteria of stability under a dynamic action of the inclining moments

by which the angle of heel does not transgress a given value. which would render impossible the coming hack to the initial position.

Verbally they can be formulated afollows:

Under action of a determined inclining moment acting

dyna-mically, the maximal angle of heel attains a value by which

the work performed by the inclining mom-ent becomes equal

to that of the righting moment and the return to the original position will take place if at that angle of heel value of the

righting momenwill exceed the value of the inclining moment. Fig. 2 shows a diagram of the righting moment M1 (q) and three curves representing three inclining moments I, II and III of different characters.

M Mp ()

éA1

d1 _dmfdj, Fig. 2 /11 ¡1 q" = O (8)

Accordingly to the criteria as abovesaid the moments marked Il and 111 will cause the capsizing of the ship.

Under action of the rtloment 11 the ship heels to the limit

angle of dynamical inclination called dynamic angle of capsizing.

The same relationships are shown on fig. 3 by means of diagrams of work performed by the same moments, e. i. by

of diagrams of the dynamical stability.

E=

AL

4. The angle to which the ship will return after having been

inclined dynamically can be determined by recurring to the condition of tile maximum of the emetic energy which corres-ponds to the position of stable equilibrium.

The maximum value of the function

In'2 '

= M, (q) dq - j' M (p) dtp will appear at an angle cç determined

dE

---=0

and dq d2E

<0

&p2 'hat is to say, when

Taking into consideration that the expression dh/dq in equation (16) represents a generalisecl metacentric height at

a given angle of heel, it can be stated that, if the inclining

moment acts statically, the stability of a ship depends both of the value of tile righting couple and of the metacentric height at the given angle.

Fig. 4 shows a diagram of the righting moment M1, (tp) and

four curves of different shapes caracterising inclining

mo-ments JIV.

_f3 'fi

In accordance with equations (15) and (16) the positions

numbered 3 and 5 solely mark a stable equilibrium.

The angle q., which under action of a constant inclining

moment Il only corresponds to the maximumordinate of the

righting moment curve, is a limiting angle of tile statical stability and is called statical capsizing angle.

5. When in her initial position, e. i. under absence of an in-clining moment, the ship shall keep afloat upright, being sub-ject only to the force of gravity and that of buoyancy.

As in such case

M,0

an

d=

dq

the stability as by (15) and (16) will appear as follows:

dM

_>

-(18)

dcp

Thus, a stable equilibrium of a ship keeping freel' afloat in her upright position depends on the positive value of her

metacentric. height, e. i. in her inetacentre lying above the

centre of gravity.

6. Tile criteria of stability discussed in tile foregoing text are put together as shown in Table I.

These criteria refer to transversal stability. Similar criteria could also be computed in what concerns longitudinal

stability; the would however be of no importance a loss of stability by longitudinal heels becing practically out of

con-sideration.

Neutral equilibrium criteria require to be discussed sepa-rately. Following the energetical criterion of stability there

must exist a constant kinetic energy value within a dccermined range of angles of heel.

Therefore following conditions must be fulfilled: 1\'I,,. (q) = M,, (q)

dM,. (w) dM1, (ç)

dqi

-.

for an angle ç lying within a determined interval of

< PL' Wt) >

As an example illustrating theoretically the fulfilment of the

neutral equilibrium criteria may serve the case discussed by

Wendel [4] of a masted sail boat drawn under a bridge by the current.

Many handbooks illustrate the case of neutral equiiibrium a ship keeping afloat freely in an upright position, her

Centre of gravity co-inciding with her mctaccntre, e. i. when her MG = 0. Sudi example is faulty, as the equilibrium in

- 89 - Schiftstechnik Bd. t 19G1 - Heft 41

MM = o

(13) M,, = 0 (17) and dMv dM

<0 -

(14) dp dç

or, more distinctly, after suitable transformations,when

= M1) (15)

and

, dh dM, (16)

dq dç

The conditions expressed by equations (15) and (16) re-present the criteria which enable to verify the stability of a ship in a position of statical equilibrium, which equilibrium

shall be a stable one.

A ship inclined by a moment acting dynamically will return

to the abovesaid position. after the energy imparted to her undergoes dispersion.

The same criteria are equally valid for inclinations applied statically, in accordance with equation (3) and (4) which assume that independently of the shape of the curves of in-clining moments the transition to the position of equilibrium is effected statically and that thus the surplus of the work

effected by the inclining moment is lead away from tile system

during the ship heels and at every instant all acting forces

remain in equilibrium.

The conditions as by (15) and (16) can l)e expressed verbally thus:

The position of stable equilibrium under action of a deter-mined inclining moment will take place at such an angle of heel at which tile inclining moment becomes equal to the rithting moment and the first derivative of the righting mo-nient will he greater than the first derivative of the inclining

T

(12)

'f5 o.

'f'

case exposed can be either stable or unstable, depending of

the shape of the righting moments curve in the neighbourhood of qi

=

0. Neutral equilibrium needs a complementary

con-dition to be satisfied, viz, that of M (qi) = O for a determined range of angles neighbouring closely qi

=

0.

The abovenamed conditions for a neutral equilibrium when in its initial position are fuluiled for example by a cylinder of a circular section. whose centre of gravity lies in the

longitu-dinal axis of that body, supposing the axis to be parallel to the surface of flotation. It satisfies for an arbitrary range of

l 1)0th the conditions M1, (cp)

=

O and M0' (ç) 0.

The above exposed criteria eml)race conditions of the stability of ships within that elementary range in which they appear in the common ship-building practice. This restried

way of treatment is due to the necessity of introducing nume-rous simplifications in the complete mathematical model of the phoenomena considered and particularlY owing to the difficul-ties of expressing analytically both external acting forces and the forces of buoyancy appearing in more or less complicated conditions presented by an undulating sca.

Table i

dv

SCHIFFSTECHNIK

Forschungshefte für Schiffbau und Schilismaschinenbau

Verlag: Schifrahrts-Verlag,,Hansa"C.Schroedter & Co.. Hamburg 11, Stubbenhuk 10. Tel. Sa.-Nr. 364981. - Schriftleitung: Prof. Dr.-Ing. Kurt Wendel, Hamburg. - Alle Zuschriften sind an den obigen Verlag zu richten. - Unaufgefordert eingesandte Manuskripte werden nur au! ausdrücklichen Wunsch zurückgesandt. - Nachdruck, auch auszugsweise, nur mit Genehmigung des Verlages. - Preis für das Einzelheit DM 6,1(1; Jahres-Abonnement DM 30, zuzüglich Postzustellgehühr, Abonnementskündlgun-gen müssen his splitestens einen Monat vor Ablauf des Jahres-Abonnements beim Verlag vorlieAbonnementskündlgun-gen. - AnzeiAbonnementskündlgun-genteitung: Irmgard Daht, Hamburg. - Anzeigcnprclsliste Nr.2. - Bankkonto: Vereinsbank, Abteilung Hafen. - Postscheckkonto: Hamburg Nr. 141 87. Höhere Gewalt entbindet den Verlag von jeder Lieferungsverpilichtung. - Erfüllungsort und Gerichtsstand Hamburg.

Druck: Schroedter & Hauer, Hamburg 1.

Notwithstanding this the author is quite persuaded that.

when having recourse to some more complicated methods of mechanics general, followed by laboratory research and

sup-ported by statements obtained on real shipsit will be

pos-sible to step UI) to sonic more extended forms of equations of. ship's movements and to the researdi of the stability of some.

lt could be expected that in this way a bridge will be erected letwcen the actually separated and simplified ship stability problems of to-day. and a special chapter of science em-bracing those of the rolling of ships pushed far more mathe. matica'llv. _{(Eingegangen ani 8. Juli 1960)}

References

[1] N. D. Mois ice f f : Ocerki razvitja teorii ustoicivosti.

A. N. Kryloff: Sobranie trudow. e. 1. Moskva.

G. Weinblum: Die künftige Entwicklung des Schiffes im Lichte der Schiffstheorie, HANSA" 1952- S. 1541.

K. W e n d e 1: Wann kentert ein Schiff?,, HANSA" 1950, S. 972.

F. H o r n : Wann kentert ein Schiff? HANSA" 1950. S. 1459. J. J e n s : Stabilitiitsgleichungen, Stabilitiitskriterien und

Kenterpunkte, Schiffstechnik, 3. Band 1956, S. 151.

1 Stability Criteria Unstability Criteria Critical angle o! capsising Dynamical 5M dr 5M1, d'I' 'dT 5M1, -.S Stability M, <M M M1 M M1, . M1, M M M1 Statical Stability dM,,, dM1, dM,,. dM, dM,,. dM1, dq dT dv - dv Initial Stabïlity M0 O p _> M1, O dM1 Sehiftstechnlk Bd. s - 1961 Heft 41 ₉₀