Metacenter and ship stability

Transcription

1 etacenter an sip stability Jacques égel, Janis Kliava To cite tis version: Jacques égel, Janis Kliava. etacenter an sip stability. merican Journal of Pysics, merican ssociation of Pysics Teacers, 00, 78 (7), pp HL I: al ttps://al.arcives-ouvertes.fr/al Submitte on 4 Jun 009 HL is a multi-isciplinary open access arcive for te eposit an issemination of scientific researc ocuments, weter tey are publise or not. Te ocuments may come from teacing an researc institutions in rance or abroa, or from public or private researc centers. L arcive ouverte pluriisciplinaire HL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recerce, publiés ou non, émanant es établissements enseignement et e recerce français ou étrangers, es laboratoires publics ou privés.

2 n te buoyancy force an te metacentre Jacques égel an Janis Kliava UR e Pysique, Université e Boreau 5 cours e la Libération, 405 Talence cee, rance bstract We aress te point of application of te buoyancy force (also known as te rcimees force) by using two ifferent efinitions of te point of application of a force, erive one from te work-energy relation an anoter one from te equation of motion. We present a quantitative approac to tis issue base on te concept of te yrostatic energy, consiere for a general sape of te immerse cross-section of te floating boy. We sow tat te location of epens on te type of motion eperience by te boy. In particular, in vertical translation, from te work-energy viewpoint, tis point is fie wit respect to te centre of gravity G of te boy. In contrast, in rolling/pitcing motion tere is uality in te location of ; inee, te work-energy relation implies to be fie wit respect to te centre of buoyancy, wile from consierations involving te rotational moment it follows tat is locate at te metacentre. We obtain analytical epressions of te location of for a general sape of te immerse cross-section of te floating boy an for an arbitrary angle of eel. We sow tat tree ifferent efinitions of viz., te geometrical one, as te centre of curvature of te buoyancy curve, te Bouguer s one, involving te moment of inertia of te plane of flotation, an te ynamical one, involving te secon erivative of te yrostatic energy, refer to one an te same special point, an we emonstrate a close relation between te eigt of above te line of flotation an te stability of te floating boy. inally, we provie analytical epressions an graps of te buoyancy, flotation an metacentric curves as functions of te angle of eel, for some particular sapes of te floating boies, viz., a circular cyliner, a rectangular bo, a parabolic an an elliptic cyliner. I. Introuction If a soli ligter tan a flui be forcibly immerse in it, te soli will be riven upwars by a force equal to te ifference between its weigt an te weigt of te flui isplace., p. 57 ne of te most long-staning for centuries an, peraps, one of te best-known laws of pysics is te rcimees Law stating tat a boy immerse in a flui is subjecte to a force equal to te weigt of te isplace flui an acting in opposite irection to te force of gravity. Tis force is calle buoyancy force or rcimees force ; we enote it by yrostatic, an its point of application will be referre to as., te subscript staning for

3 learly, is a resultant of all elementary yrostatic forces applie to te surface of te immerse boy. force is fully escribe by (i) its norm, (ii) its irection an (iii) its point of application. Note tat te point (iii) is not inclue in te above efinition of te rcimees' Law. n te oter an, it woul be surprising if suc a funamental epistemological question as tat of te point of application of te buoyancy force were not at all iscusse in te literature. By efinition of te resultant force, te point of application of soul be efine by requiring tat if we apply suc a force, te (macroscopic) beaviour of te boy will be te same as tat cause by te ensemble of elementary yrostatic forces. Note, owever, tat tis efinition oes not guarantee te uniqueness of tis point; inee, one may quite well conceive it to be ifferent in ifferent pysical situations. n analysis of te abunant bibliograpy concerne wit te stability of immerse boies sows tat many autors prefer to elue te question of te eact location of te point of application of te rcimees force, e.g., speaking of te line of action of tis force., p. 7 Wat is really sown is only tat te line of application of te rcimees force passes troug te centre of buoyancy, so tat one may guess tat can be locate anywere on tis line. If tis statement is sufficient in statics, it is certainly not sufficient in ynamics. Inee, one migt argue tat a force applie to a rigi boy, as a sliing vector, obeys to te principle of transmissibility stating tat te conition of motion of a rigi boy remains uncange if a force of a given magnitue, irection an sense acts anywere along te same line of action. However, in tis relation, an important amenment a been mae alreay more tan a century ago, namely, tat we may imagine a force to be applie at any point in te line of its irection, provie tis point be rigily connecte wit te first point of application. 4 consiere as rigi, te point of application of Below, we will sow tat, wile a floating boy can be is not necessarily rigily connecte wit it because te isplace flui is eviently not rigi. Terefore te question of te point of application of tis force is quite legitimate an meaningful. In a quite natural way one is tempte to relate te point of application of following remarkable points of a floating boy. to one of te tree (i) Te centre of gravity of te boy, enote by G (te istinction between te centre of gravity an tat of mass/inertia is of no relevance for te present analysis). To our knowlege, suc assignment as never been suggeste before. (ii) Te centre of buoyancy (centre of gravity of te isplace flui) enote by (from careen) or by B (from buoyancy); we prefer using te former notation. Note tat sometimes is efine as te geometrical centre of te isplace flui; meanwile, for a omogeneous flui (te case consiere below) bot efinitions coincie. ost frequently, te point of application of centre of buoyancy, e.g., see Refs. [5 (p. 58), 6, 7]. is assigne to te

4 (iii) Te metacentre enote by. Tis point is of a special interest, since, in orer to assert te stability of te boy against overturn,, efine in te vicinity of equilibrium, soul be locate above G. We ave foun only one tetbook suggesting, witout emonstration, as a caniate for te point of application of, see Ref. [8, footnote p. 6]. n te oter an, in a very interesting wile, unfortunately, not easily accessible paper, Herer an Scwab 9 suggest a istinction between statically equivalent an ynamically equivalent resultant forces. Te former are efine for a fie position of te boy (statics) in wic case only a line an not a point of application of te resultant can be efine. Te latter are relate to te stability of a nominal state wit respect to small variations about tis state. In tis case te point of application of te resultant force is essential, an tese autors efine it as te ynamically equivalent point of application. or te particular case of floating parallepipe ( soe-bo ) Herer an Scwab euce from consierations base on te yrostatic energy tat te ynamically equivalent point of application of te buoyancy force is te metacentre. It as seeme quite tempting to generalize tis approac to te general case of te floating boy of arbitrary sape, incline troug an arbitrary angle. Te concept of metacentre ates back to Bouguer s Traité u Navire, e sa construction et e ses mouvemens (746) 0. Using te prefi µετά = beyon, e a esignate a specific point of a floating boy, efine as te intersection of two vertical aes passing troug te centre of buoyancy (te centre of gravity of te isplace flui) at two sligtly ifferent angles of eel. Besies, Bouguer formulate te well-known teorem relating te istance between an to te ratio of te moment of inertia of te plane of flotation an te volume of te isplace flui. Tree years later, Euler in Scientia navalis (749) gave a general criterion of te sip stability, base on te restoring moment: te sip remains stable as far as te couple weigt (applie at G) an te buoyancy force (wose line of application passes troug ) creates a restoring moment. cange of sign of te latter results in capsizing, an its vanising in incline position (at equilibrium it vanises by efinition) correspons to te overturn angle. Te problem of stability of te floating boies, wic can be trace back to rcimees imself, see [, n floating boies, Book I, pp. 5-6; Book II, pp. 6-00], as never cease to interest scientists an engineers -6, in particular, in te relation to te metacentre, e.g., see 7, 7-0 an as become an important part of acaemic stuies, 5, 6, 8,. eanwile, tere as been no significant progress in tis fiel since te original Bouguer s Treatise 0 an its reformulation in terms of a novel geometry in te Dupin s tetbook 7, relating te metacentric curve, i.e., te loci of, wit te buoyancy curve forme by te loci of, te former being te evolute of te latter. Te aim of te present work is to eluciate te question of te point of application of te buoyancy force in relation wit tat of sip stability. We consier two ifferent approaces to te efinition of te point of application of a resultant force base, first, on te work-energy relation an,

5 secon, on te equation of motion. We obtain appropriate epressions of te yrostatic energy an te location of te caracteristic points for a floating boy of rater general sape. n tis basis we sow tat te location of te point of application of te buoyancy force epens on te type of motion eperience by te boy. In particular, in pure translation tis point is fie wit respect to te centre of gravity G wile in rolling or pitcing motion it is fie wit respect to te centre of buoyancy (from te viewpoint of te work-energy relation) or locate at te metacentre (from te viewpoint of te rotational moment). In te framework of te formalism evelope in tis work, we present a quantitative analysis of te concept of te metacentre. We calculate te location of te geometrical, Bouguer s an ynamical metacentres for an arbitrary angle of eel an sow tat all tree efinitions, in fact, result in one an te same metacentric curve. inally, in te ppeni we apply te general relationsips to etermine te location of an for several particularly simple sapes of te floating boy. II. Buoyancy force an yrostatic energy onsier a floating, i.e., partially immerse in a flui, rigi boy, e.g., a vessel. In suc situation, strictly speaking, one soul also take into account te atmosperic pressure eperience by te part of te boy situate above te waterline. However, as far as te specific ensity of te air remains muc lower tan tat of te flui, as we ave assume in te analysis given below, te atmosperic pressure can be neglecte. In te formulae given below te surface S an te volume V concern only te submerge part of te boy, V being equal to te volume of te isplace flui. Te yrostatic force eerte on an element S of te immerse surface of te boy is = p S were p is te yrostatic pressure. Te minus sign in tis epression is ue to te fact tat, by convention, S is irecte outwars a close surface wile te yrostatic force is irecte inwars te boy. Integrating over te immerse surface an applying te graient teorem to pass from a surface integral to an integral over te immerse volume V yiels te buoyancy force as: = p S= p V. (II.) S Eq. (II.) sows tat ensity per unit volume of te flui V can also be consiere as a resultant of fictitious volume forces of energy) of ensity p. Te total yrostatic energy is calculate as V= p, eriving from a potential energy (yrostatic E = p V. V (II.) 4

6 rom te general relation between a force an te corresponing potential energy it follows tat = E. igure. Two ifferent representations of te coorinate systems efine in te tet. Left: Te floating boy turne anticlockwise wit respect to te orizontal line of flotation. Rigt: Te line of flotation turne clockwise wit respect wit te vertical floating boy. Te profile of te immerse cross-section is escribe by te function +. Te 4 iamons, circles an triangles inicate, respectively, calculate locations of te, an points. ne efines te plane of flotation as te plane in wic te boy is intersecte by te surface of te liqui, p. 67. Te line of flotation is efine as te intersection of te plane of flotation wit a vertical cross-section of te boy; te centre of te former is referre to as te centre of flotation an enote by. Below we will consier two types of motion of a partially immerse boy: a vertical translation an an oscillation about one of te orizontal aes of a floating boy. In most cases a floating vessel is muc longer in one of te principal orizontal irections tan in te perpenicular one, an te oscillations about te longituinal an transversal aes are respectively calle rolling an pitcing. We will not eplicitly consier te pitcing; inee, its caracteristics can be reaily obtaine from tose of te rolling motion. Te latter will be systematically escribe in two interrelate coorinate systems, see igure. Te main aes of te first one, for brevity referre to as te Eart frame, are enote by majuscules an cosen as follows: te X an Y aes are respectively irecte along te wit an te lengt of te boy an te Z ais is vertical ascening; te corresponing unit vectors are enote by ux, uy, u Z. Te origin of coorinates is cosen at te instantaneous position of te centre of flotation, an te orizontal XY plane coincies wit te plane of flotation. Te secon coorinate system, referre to as te boy frame, is rigily relate to te boy. Its aes are enote by minuscules, wit te y ais along te longituinal ais of te boy an te an z aes turne troug an angle ϑ (angle of eel) wit respect to te X an Z aes of te Eart frame; te corresponing unit vectors are enote by u, u y, u z. Te origin of coorinates is cosen at te location of at equilibrium, terefore, at equilibrium bot frames coincie an for an arbitrary angle of eel te relation between te respective coorinates is 5

7 X= ( ) cosϑ+ ( z z) sinϑ. (II.) Z= ( ) sinϑ+ ( z z ) cosϑ We suppose tat, as is most often te case, te flui can be consiere as incompressible an omogeneous, so tat te yrostatic pressure epens only on te immersion ept: p gz = µ f were f (II.), (II.) become: an µ is te specific ensity of te flui an g is te acceleration of gravity. Ten eqs. = µ g Vu = µ gvu (II.4) f Z f Z V E = µ g Z V. (II.5) f V III. Two efinitions of te point of application of a force By efinition of te centre of buoyancy, its coorinates are X= X V ; Y= Y V ; Z= Z V V V V, (III.) terefore, from eqs. (II.5) an (III.), V V V E = µ gvz = Z. (III.) f Te concept of te yrostatic energy allows efining te point of application of te buoyancy force from te usual work-energy relation, e.g., see ref. []. Te elementary work is efine as te scalar prouct of a force wit an elementary isplacement of its point of application; on te oter an, for a force eriving from a potential energy it equals te iminution of tis energy. In te present case, δ W = r = Z = E. (III.) If te isplacement of te point matces tat of a efinite point of te boy, te latter point can be ientifie as te point of application of. s far as te above ientification is base on te workenergy relation, te corresponing point will be referre to as te energetical point of application of te buoyancy force. Tis efinition applies for any type of motion. noter possibility of efining te point of application of is base on te principle tat te rotational moment of a force vanises in its point of application. Tis efinition is applicable to types of motion were a rotational component is present, in particular, to rolling/pitcing. floating boy 6

8 eperiences bot te buoyancy force an te force of gravity g, an its total potential energy is te sum of te yrostatic an gravitational energies, Etotal= E+ Eg. Te couple + g prouces a rotational moment (torque) total= r + rg g were r an r G are position vectors of an G wit respect to an arbitrary coorinate system; obviously, origin of coorinates. total oes not epen on te coice of te Te strategy usually aopte in mecanics is to separate a general motion of te boy into translation of an arbitrary point, cosen insie or outsie te boy, an rotation about tis point. ost often te centre of rotation is cosen at G, in wic case r G= 0, but tis coice is not compulsory. or a floating boy te location of G epens not only of its sape but also of te istribution of weigts insie te ull, wic canges wit te sip loaing. Terefore, we ave cosen to focus on te buoyancy force an te relate yrostatic energy witout systematically calling to min te force of gravity an its potential energy. However, te rotational moment of te single force oes epen on te origin of r, terefore, in consiering only te buoyancy force an te yrostatic energy, te coice of te origin of coorinates is pysically meaningful. Te famous Euler s teorem states te following. of flotation. Te oscillatory movement of a floating boy (rolling or pitcing ing) ) can be escribe as a rotation about te centre oosing as te origin of r, te rotational moment of te buoyancy force, is calculate from a variation of E : E = r = ( Θ r) = Θ ( r ) = Θ (III.4) were te elements of te angle vector Θ= ( ϕ, ϑ, 0) are angles of rotation about te X an Y aes. In eriving eq. (III.4) we ave use te epression of variation of a position vector r in an infinitesimal rotation, r= Θ r. rom te latter equation one gets = Θ E were elements of te graient Θ are erivatives wit respect to te angles of rotation. In te particular case of rolling, te only non-vanising element of E = ϑ is u Y. (III.5) n te oter an, omogeneous flui it can be calculate as = r = µ g Z r S f S is te sum of moments of te elementary yrostatic forces, so, for a. (III.6) S 7

9 pplying te curl teorem an taking into account tat te curl of te raius vector is ientically null, r 0, we get ( ) = µ f g Z r V= µ f g ( XuY+ Yu X) V. (III.7) V By efinition of te moment of te buoyancy force, V = r = r u = µ gv( X u + Yu ). (III.8) Z f Y X rom eqs. (III.7) an (III.8) we obtain te coorinates of in te orizontal plane as te first moments of te immerse volume about te X an Y aes. Tese coorinates coincie wit te corresponing coorinates of, cf. eqs. (III.): X = X V Y Y V V = ; V. (III.9) V V X an Y give te lengts of te lever arms of for rolling an pitcing, respectively. Terefore, is applie along te vertical line passing troug. eanwile, as one migt epect, te vertical coorinate Z of te point of application of remains unefine. Interestingly, if we attempt to etermine te centre of gravity of a boy from vanising of te corresponing rotational moment in a given position, its vertical coorinate Z G will also remain unefine. eanwile, tis ifficulty is reaily overcome by rotating te boy troug an arbitrary angle about any non-vertical ais. rom te viewpoint of suc eperience, te point of application of te weigt can be efine as te intersection of all vertical lines passing troug te centre of gravity for ifferent angular positions of te boy. Te same proceure can be use to specify Z, an as will be iscusse later, te intersection of all vertical lines passing troug te centre of gravity of te isplace flui (i.e. te centre of buoyancy) for ifferent angular positions of te boy, by efinition, inicates te metacentre. Terefore, we nee an equation of motion involving an containing Z. In eq. (III.8) Z is eliminate because of te properties of te vector prouct r, so, in a somewat intuitive way, one may suggest an equation satisfie by te corresponing scalar prouct r. It can be obtaine consiering te equation of rotation of a soli: I Θ = were I is te tensor of inertia about an te elements of te angle vector Θ, te secon erivative of Θ, are angular accelerations aroun te X an Y aes. In orer to account for a small rotation about te state of te boy escribe by tis equation, we take te ifferentials of its bot sies: I Θ= = ( Θ r) = ( r Θ ). (III.0) 8

10 Developing te ouble vector prouct an making use of te fact tat is orizontal, yiels prouct: is vertical wile Θ = r ( Θ) Θ( r ) = Θ ( r ). (III.) In te case of te rolling motion, from eqs. (III.5) an (III.) we get for te sougt-after scalar ² E Y = = r = Z. (III.) ϑ² ϑ Tus, efining te vertical coorinate of te point of application of following teorem:, we are le to te Te e scalar prouct of te buoyancy force an te position vector of its point of application from te origin at te centre of flotation is given by minus te erivative of te rotational moment wit respect to te angle of roll or pitc or by te secon erivative of te yrostatic energy wit respect to tis angle. Tis result can be consiere as a particular case of te general analysis of te ynamically equivalent point of application given by Herer an Scwab 9. Eqs. (III.9) an (III.) are necessary an sufficient to etermine all te tree coorinates of te point of application of. s tis etermination is base on ynamical equations involving te rotational moment, tis point will be referre to as te ynamical point of application of te buoyancy force. It is easy to sow tat te secon erivative of te total potential energy can be epresse as ² E total total Y = = ( r rg) = ( Z ZG), (III.) ϑ² ϑ irectly yieling te vertical istance between G an. In wat follows we present a moel calculation of te location of te ifferent points of application of te buoyancy force. Te calculations refer to te immerse part of te transversal crosssection of te floating boy for brevity enote by immerse cross-section, so, te results given below refer to a portion of te boy of unit lengt along its longituinal ais. Te caracteristics of te wole boy can be reaily obtaine by weigte integration along te y/y ais. 9

11 igure. Eample of an immerse cross-section, (i) an (ii) representing fully an partially immerse pieces. In te general case te sape of te immerse cross-section can be quite complicate. It is not necessarily suppose to be symmetric wit respect to te z ais, e.g., te case of a Venetian gonola; moreover, it can be elimite by a multi-value function, e.g., see igure. However, it can always be represente as a combination of pieces of two ifferent types: (i) tose fully immerse, elimite from te top an from te bottom by functions z= f ( ) an z= f ( ), respectively, an along te orizontal ais by te abscissas inf an sup ; (ii) tose partially immerse, elimite from te top by te line of flotation an from te bottom by a one-value function z= f( ) an along te orizontal aes by te abscissas P an Q. Te epressions of areas an coorinates of te centres of buoyancy for suc pieces are given in ppeni I. + IV.. Te point of application of te buoyancy force in translation or efiniteness, we consier a vertical ascening motion of te boy (a ry-ocking ). Tis problem is most conveniently treate in te boy frame at ϑ= 0. We assume tat te boy is in equilibrium; terefore, te force of gravity is suppose to be applie along te vertical line passing troug. irst we consier a fully immerse piece of type (i) elevate troug a small vertical istance b. Its area an ence remain constant, cf. eqs. (.), an from eq. (III.) we get E = z z = z. (IV.) 0

12 igure. Immerse cross-section elimite by te single-value function + for < 0 an for > 0. b is a small vertical translation. Te iamon an circle an triangle inicate, respectively, te locations of, an. rom comparison wit te work-energy relation, cf. eq. (III.), it follows tat z = z, (IV.) terefore, te point of application of is rigily connecte wit. eanwile, in tis case is rigily connecte wit te immerse piece itself, so, can be assigne to any point locate on te vertical line troug. ne can see tat for a fully immerse piece only te line an not te point of application of can be efine. Net, we treat a partially immerse piece of type (ii), see igure. In te starting position te line of flotation is locate at z = 0, so, te buoyancy force is, cf. eq. (II.4), Q 0 f f Q. (IV.) = µ g z = µ g f P f P an te yrostatic energy is, cf. eq. (II.5), Q 0 Q = µ f g = µ f g. (IV.4) P f P E z z f Wen te piece is elevate troug b> 0, te line of flotation moves own to z= b, so tat te cange in is Q b = µ f g z = µ f gl b (IV.5) P 0 were L= Q P is te lengt of te line of flotation forϑ= 0.Te corresponing cange in E in linear approimation is, cf. eq. (II.5),

13 Q 0 b Q E= µ f g ( z b) z + z µ f g f b f f rom eqs. (IV.) an (IV.6) one gets:. (IV.6) P E = b, (IV.7) so, a comparison wit eq. (III.) sows tat z = b. (IV.8) s in te case of a fully immerse piece, is rigily connecte wit te partially immerse piece in translational motion. or an immerse cross-section of a general sape te coorinates of are weigte sums of te corresponing coorinates of separate pieces, terefore remains rigily connecte wit suc a piece. s G is rigily connecte wit te immerse boy an, in translation, G is locate on te ais of application of, one can consier tat is applie at G. Tis assignment being base on te workenergy relation, te corresponing point of application of one. P will be referre to as an energetical n te oter an, is not rigily connecte wit a partially immerse piece; terefore, in tis case it woul be an error to consier tat is applie at. Inee, from eqs. (III.) an (IV.7), E = z z = b, (IV.9) an substituting, E, an E from te respective epressions, one gets Q Q z L f f b P P =. (IV.0) bviously, only for a fully immerse boy, wen L= 0, one gets z = b. In oter cases te relation between te isplacements of te boy an of its centre of buoyancy can be very ifferent. Te ifference between z an z is particularly obvious for a partially immerse parallelepipe, z= f( ) = const= b for P, Q, see igure 4. In tis case Q P f = bl an Q f = b L, so tat for a finite isplacement b P eq. (IV.0) yiels z = b. (IV.)

14 Tus, te isplacement of is only alf of tat of G (an of te point of application of te buoyancy force). igure 4. Square immerse to a ept b. b an z are, respectively, vertical translations of te square an of. It can be easily sown tat if te contour of te immerse cross-section f( ) is a singlevalue function, Q P z is always smaller tan or equal to b. Inee, from eq. (IV.0) one as Q f( ) L f ( ) P. (IV.) In te integral form tis inequality was first publise on p. 4 of te paper by Bouniakowsky, but in can be irectly euce from te aucy-scwarz inequality vali for any square-integrable functions. ne can see tat z b, te limiting case z = b occurring for te rectangle, as iscusse before. In te general case te relation between z an b can be very ifferent. or instance, consier te immerse cross-section sown in igure 5, looking like te bulb-sape keel of certain yacts participating in te merica s cup regatta 4. Surprisingly, in tis case one obtains so, wen te boy is elevate te centre of buoyancy is lowere over te same istance! z b, igure 5. ross-section of a bulb-sape keel. Te profile of te keel an te bulb are escribe, respectively, by 9 ( ) an ± Te iamons inicate te positions of.

15 V. Duality of te point of application of te buoyancy force in rolling/pitcing motion Now we eamine te location of te point of application in rolling/pitcing. In te general case, in tis type of motion a rotation troug an angle ϑ is combine wit a vertical translation of te centre of buoyancy, see igure 6. igure 6. Two immerse cross-sections turne troug an angle ϑ. Left: tat of igure ; rigt: te one elimite by te functions ( ) 4 f+ = (upper branc) an ( ) 6 f = for < 0 an f ( ) = for > 0 (lower branc). Te full an empty iamons, circles an triangles inicate, respectively, te locations of, an before an after te rotation. t an arbitrary angle of eel te line of flotation is escribe by te equations z z z z Q P = = Q P tanϑ (V.) were te coorinates of are ( ) ( ) = + P Q z = z + z P Q. (V.) Te curve escribe by in te course of te rolling/pitcing motion is calle flotation curve. Te area of te immerse cross-section an te coorinates of te centre of buoyancy are given by eqs. (.), (.) an (.4). V.. Energetical consierations epresse as Denoting by flotation te ept of below te line of flotation, E in te boy frame is 4

16 flotation E= = ( ) sinϑ ( z z) cosϑ (V.) an in te Eart frame (remining tat by efinition of te Eart frame, Z = 0 ) cf. eq. (II.), E = Z. (V.4) In orer to calculate te erivative of E wit respect to ϑ, we consier a small variation of te angle of roll an make use of te fact tat te area of te immerse cross-section remains constant in te course of te rolling/pitcing motion: ϑ = 0 (V.5) (ere an below te primes enote erivatives wit respect to ϑ). Substituting = i+ ii from eqs. (.) an applying te Leibniz rule to calculate te erivatives of te parameter-epening integrals, we get ( ) + tanϑ z z = 0. (V.6) P Q P Q Taking te erivative of eq. (V.) yiels ( ) L P Q tanϑ z P + z Q =. (V.7) cosϑ Te angular erivatives of te couples of variables ( P, z P ) an ( Q, z Q ) are relate troug te equation of te function z= f( ). Using te cain rule for te erivatives of composite functions, f f z P = P ; z Q = Q, (V.8) P Q from eqs. (V.6) an (V.7) we get a system of equations incluing only two unknown erivatives, an Q. Resolving tis system yiels P L L P = ; Q =. (V.9) f f sin ϑ cos sin cos ϑ ϑ ϑ P Q rom tese epressions te erivatives of an z, respectively, an z are straigtforwar. By te way, te centre of flotation as a very particular role in tis motion. Uner a small variation of te angle of roll, ϑ ϑ+ ϑ, its coorinates cange as follows: + ϑ ; z z + z ϑ. (V.0) 5

17 Te initial an te new lines of flotation are escribe by, cf. eq. (V.), z ( ϑ) = z + ( ) tan ϑ ; flotation z z z ( ϑ+ ϑ) = + ϑ+ ( ϑ) tan( ϑ+ ϑ) flotation, (V.) z + ( ) tanϑ+ ϑ cos² ϑ were we ave mae use of te fact tat, accoring to eq. (V.6), tanϑ z = 0. Te centre of rotation (pivot point) of te line of flotation is te intersection of te initial an te new lines of flotation, terefore it verifies te system of equations (V.) yieling z pivot pivot = = z. (V.) Eqs. (V.), (V.) provie a simple but quite general proof of te Euler s teorem, see Section III. Note tat Euler imself a prove tis teorem only in te vicinity of equilibrium. Taking te erivatives of eqs. (.), (.4) wit respect to ϑ an transforming te results by means of eqs. (V.9) yiels z = = L cosϑ. (V.) L sinϑ Inserting an z in te erivative of eq. (V.) wit respect to ϑ, we get E = ( ) cosϑ+ ( z z) sinϑ. (V.4) Surprisingly, tis equation looks as if in eq. (V.) only te trigonometric functions were ϑ-epenent, wic is generally not te case. Transforming to te Eart frame, cf. eq. (II.), yiels E = X. (V.5) bviously, tis result is consistent wit te relation Z = X ϑ (V.6) arising from te fact tat te rotation occurs about an X = 0. eq. (III.5), Te rotational moment an te lever arm of te buoyancy force wit respect to are, cf. 6

18 = X ; a = X. (V.7) In te position of stable equilibrium at ϑ = 0 te total potential energy Etotal= E+ Eg of a floating boy is minimal an te rotational moment of te couple + g vanises. However, te yrostatic energy at equilibrium oes not necessarily take an etreme (minimal or maimal) value, so tat in te general case te moment of te buoyancy force wit respect to, oes not vanis. If at ϑ= 0 E as a minimum, a eviation from equilibrium prouces a restoring moment (of opposite sign to tat of ϑ). n te contrary, if at ϑ= 0 E as a maimum, an overturning moment (of te same sign as tat of ϑ) is prouce. eq. (III.), Te elementary work of te buoyancy force in a small rotation about is epresse as, cf. δw = Z = E ϑ. (V.8) omparing wit eqs. (V.5), (V.6) yiels Z = Z, (V.9) i.e., te vertical isplacement of te point of application of coincies wit tat of. Terefore, is rigily connecte wit, an, as far as an are locate on te same vertical line, one can consier tat is te energetical point of application of in rolling/pitcing motion. Tis result is ifferent from tat for te translational motion, in wic case one can consier tat eq. (IV.8). V.. Dynamical consierations is applie at G, cf. Taking te erivative of eq. (V.4) wit respect to ϑ, after a transformation yiels te secon erivative of E : L E = ( ) sinϑ+ ( z z) cosϑ cosϑ z sinϑ. (V.0) In te latter formula te terms in an z escribe te isplacement of te point wit respect to te boy frame. eanwile, eqs. (III.4) an (III.), escribing te angular erivatives of E, refer to a coorinate system wit origin at te centre of rotation. Terefore, eq. (V.0) soul be consiere in a coorinate system centere at, in wic case truncate secon erivative of E : an z vanis, an we get te 7

19 L E = ( ) sinϑ+ ( z z ) cosϑ. (V.) In te Eart frame E is furter simplifie to L E = + Z, (V.) yieling te vertical istance from te line of flotation to, cf. eq. (III.),: E L flotation= Z = = + Z. (V.) n te oter an, te istance of from te line of flotation is Z (Z is negative in te system of coorinates cosen), terefore, L = Z Z =. (V.4) In accorance wit te Bouguer s teorem 0, tis epression correspons to te metacentric istance, vie infra. Terefore, one can conclue tat te ynamical point of application of te buoyancy force coincies wit te metacentre. Note tat Herer an Scwab 9 ave carrie out a similar calculation in te particular case of a rectangular immerse cross section in equilibrium position. However, in teir equations (5, 55), corresponing to te above eqs. (V.), (V.), incorrectly appears te istance between G an, in contraiction wit te Euler s teorem. In fact, te latter istance soul be calculate from te secon erivative of te total potential energy, cf. eq. (III.). Te eistence of two ifferent points of application of ( an ) euce from two ifferent efinitions ( energetical an ynamical ) for te same type of motion may seem paraoical. Wile tese two points result in ientical rotational moments, because te corresponing lever arm is te same, teir locations on te vertical ais are ifferent. In orer to unerstan te cause of tis uality, te reaer can consier te apparently trivial case of te point of application of te force of gravity rolling/pitcing motion. pplying to g te same formalism, as before for g eerte on te floating boy in, cf. eqs. (V.), (V.4), (V.4), (V.0), (V.), (III.), (V.), sows tat bot efinitions of te point of application of converge to te centre of gravity of te boy. g In fact, te ifferent beaviour of an g is ue to te fact tat te sape of te immerse volume oes not remain constant in te course of rolling/pitcing motion. Te corresponing variation gives rise to te aitional term in 8

20 eq. (V. ) an ff., epening on te lengt of te line of flotation L an responsible for te uality of te point of application of. or a fully immerse boy L= 0, so tat is merge wit, cf. eq. (V.4), an te above uality isappears. VI. etacentre an sip stability ctually, in te literature one can fin tree ifferent efinitions of te metacentre an te metacentric curve. We ave cosen to esignate tem as te geometrical one (te evolute of loci of te centre of buoyancy), te Bouguer s one (relating te metacentric istance to te moment of inertia of te plane of flotation) an te ynamical one (suggeste by te fining by Herer an Scwab tat te ynamically equivalent point of application of te buoyancy force is te metacentre). 9 Previously, te geometrical metacentre as been efine for any angle of eel wereas te Bouguer s metacentre was consiere only in te vicinity of equilibrium an te ynamical metacentre only in te particular case of a soe-bo in equilibrium 9. Besies, te eisting emonstrations of te Bouguer s teorem are employing a oc geometrical constructions an are far from being rigorous, see Refs. [8 (pp )], [ (pp. 8-8)]. In tis contet, we woul like to clarify te following two points: (i) Do te tree efinitions of te metacentre aress one an te same point of te floating boy an if yes, is tis true for any angle of eel? (ii) Wat is te pysical meaning of te metacentric curve for an arbitrary angle of eel? To begin wit, we epress te relation between te metacentric curve z ( ) an te buoyancy curve z ( ) by means of te usual efinition of te evolute of a curve as te locus of its centers of curvature: 5 were N N = z ; z = z +. (VI.) D D N= + z ; D= z z. (VI.) Te epressions of an z are calculate by taking te erivatives of eqs. (V.) wit respect to ϑ an inserting (VI.) yiels P an Q from eqs. (V.9), see ppeni I. Substituting tese epressions in eq. 9

21 6 6 L L N= 44 ; D= 44, (VI.) so tat eq. (VI.) reuces to = z ; z = z +, (VI.4) ence, te geometrical metacentric istance is z z L geometrical= ( z z) + ( ) = =. (VI.5) cosϑ Tis istance is eactly te same as te istance from to te ynamical point of application of te buoyancy force, cf. eq. (V.4), terefore te ynamical metacentre coincies wit te geometrical one. Now let us emonstrate te Bouguer s teorem for an arbitrary angle of eel. Te statement of tis teorem is as follows: Te metacentric istance is equal to te ratio of te moment of inertia I of te plane of flotation wit respect to a orizontal ais an te immerse volume V of te isplace flui : Bouguer I =. (VI.6) V s before, we consier te immerse cross-section of te floating boy. In tis case eq. (VI.6) becomes Bouguer= I (VI.7) were I is te moment of inertia of te line of flotation wit respect to, L J = λ λ= L, (VI.8) L ence, J L =, (VI.9) wic proves te teorem. s te rigt-an sie of tis equation coincies wit tat of eq. (VI.5), te Bouguer s metacentre coincies wit te geometrical one. We conclue tat for any angle of eel, all tree above-mentione efinitions of te metacentre are equivalent: 0

22 L = = =. (VI.0) ynamical geometrical Bouguer Generalizing eqs. (VI.0) to te tree-imensional case, one soul perform te integration along te longituinal ais of te boy to get te epression of te global rolling metacentric istance as D = L ma 0 L( ϑ, y) y ( y) Lma (VI.) were similar way. L ma is te total lengt of te boy. Te pitcing metacentric istance can be etermine in a rom eqs. (V.5), (V.7) an (V.) it is seen tat te metacentric eigt (vertical istance from te line of flotation to te metacentre) is irectly proportional to te angular erivative of te lever arm of te buoyancy force: flotation E = a =. (VI.) Tus, te caracter (restoring or overturning) of te rotational moment of te buoyancy force epens on te sign (resp. positive or negative) of rate of angular epenence of te former. flotation, an te absolute value of te latter etermines te VII. onclusion We ave sown tat te location of te point of application of te buoyancy force epens not only on te type of motion of te floating boy (translation or rolling/pitcing) but, in te latter case, also on te efinition of tis point. In translation tis point remains fie wit respect to te centre of gravity of te boy wile in rolling/pitcing it is subject to a uality. Namely, from te viewpoint of te work-energy relation it is fie wit respect to te centre of buoyancy wile from te viewpoint of te rotational moment it is locate at te metacentre. Tis peculiarity of te buoyancy force is ue to te fact tat, wereas te immerse boy can still be consiere as a rigi one, te sape of te isplace flui oes not remain constant in te course of te motion. Inee, in te case of a completely immerse boy, te sape of isplace flui remains fie, so tat metacentre an te centre of buoyancy coincie. Te concept of non-uniqueness of te point of application of a resultant force seems quite unusual; neverteless, as we ave sown, tis non-uniqueness is an inerent feature of te buoyancy

23 force. Wile tis fining is not epecte to bring about canges in practical applications, it as a certain funamental an eucational interest for te mecanics of floating boies. It woul be interesting to fin out weter some oter pysical forces o possess a similar non-uniqueness of te point of application. Using te general approac base on te yrostatic energy formalism, we ave sown tat te various efinitions of te metacentre ( geometrical, Bouguer s an ynamical ), in fact, concern one an te same istinct point of te immerse boy. Tis fining ols (i) for any sape of te immerse boy an (ii) not only in te vicinity of equilibrium but also for any angle of eel. Besies, from te viewpoint of te rotational moment te metacentre proves to be te point of application of te buoyancy force in te rolling/pitcing motion. Tese finings se new ligt on te long-staning concept of te metacentre. noter, more practical, implication of tis stuy concerns te criterion of sip stability in relation to te location of te metacentre. Inee, te metacentric eigt is proportional to te angular erivative of te lever arm of te buoyancy force. Tus, te caracter (restoring or overturning) of te rotational moment of te buoyancy force epens on te sign (resp., positive or negative) of te eigt of te metacentre above te line of flotation, an te absolute value of tis eigt etermines te rate of its angular epenence. Te moel evelope in te present stuy allows one to get analytical epressions for te location of te metacentre for te floating boy of an arbitrary sape. Tus, it presents a certain interest for teacing an practice of naval mecanics an engineering, as well. ppeni I: I Some epressions use in te main tett or te areas of fully an partially immerse pieces of te cross section of a floating boy, respectively, type (i) an (ii), introuce in Section III, we get: ii i sup f+ sup inf f inf Q zflotation ( ) = z = f f Q = z = z L cosϑ f P f P + (.) were inf an sup are te etreme abscissas of te fully immerse pieces, an P ( P, z P), Q ( Q, z Q) an L are, respectively, enpoints an te lengt of te line of flotation for a boy incline troug an angle ϑ:

24 L Q P =. (.), cosϑ Te coorinates of te centre of buoyancy of te immerse cross-section are calculate by aapting to te two-imensional case te general formulae eqs. (III.), (III.9). Te orizontal coorinates of for te pieces of types (i) an (ii) are sup + sup = z = ( f ) i + f inf f inf. (.) Q z flotation Q = z = + L sinϑ cos ² ϑ+ ( ) f P f P ii Te corresponing vertical coorinates of are z z i f sup f + sup inf f inf ( ) = z z = f+ ² f ² z z z L( L sin ϑ z ) cosϑ ( z f f ).(.4) Q z flotation Q = = ii + + P f P Te epressions of te secon erivatives of te coorinates of te centre of buoyancy use in Section VI are as follows: z L 4 = 8 + sinϑ f f sinϑ cosϑ sinϑ cosϑ P Q. (.5) L sinϑ sinϑ = + + cosϑ f f sinϑ cosϑ sinϑ cosϑ P Q 4 8 cos ϑ ppeni II: Some special cases of immerse boies Here we consier te buoyancy, flotation an metacentric curves for several simple sapes of floating boies. ll subsequent results are reaily obtaine as particular cases of te general formulae erive above.

25 II.. ircular cyliner igure 7. loating circle of raius R an centre angle of te immerse part α. onsier te transversal cross-section of a floating long circular cyliner, its immerse part representing a circular segment of raius R an centre angle α, see igure 7. In tis instance, te natural coice of te origin of coorinates, ifferent from te previous one, is te centre of te circle. Te immerse area is = R ( α sin α), (.6) an te abscissas of te enpoints of te line of flotation are = R sin ( α ϑ) ; = R sin( α+ ϑ). (.7) P Q In te boy frame bot an escribe circular arcs of equations ( ϑ ) = R cosαsin ϑ ; z ( ϑ) = R cosα cosϑ (.8) an, cf. eqs. (.), (.4), ( ) 4 R sin α ( ) 4 sin sin ; sin z R α ϑ = ϑ ϑ = α α α sin α cos ϑ, (.9) see igure 8. In particular, at equilibrium = 0 an 4 z takes te values of R, Rπ an 0 respectively, for α= 0, π an π. rom eqs. (VI.4) it follows tat te metacentric curve is reuce to a single point, te centre of te circle: = 0 ; z = 0, (.0) an from eqs. (.9) an (.0) te metacentric istance is = sinα R α sin α. (.) 4 4

26 igure 8. Buoyancy an flotation curves for a floating circle wit R = 00 an α= 60. Te iamons an circles inicate, respectively, te locations of an for ϑ= 0, 0,60 an 90. Te metacentric curve is reuce to a point (triangle). In accorance wit eq. (V.), te ept of below te line of flotation is flotation 4 sin α = R cosα α sin α. (.) Bot istances an te yrostatic energy E = m g are inepenent of ϑ. s a consequence, flotation f te lever arm of te buoyancy force vanises. Te floating equilibrium is stable if G is locate below te centre of te circle (te ais of te cyliner). II... Rectangular bo igure 9. loating square of sie l. b is te immersion ept at equilibrium. onsier a floating boy of rectangular cross-section (a bo, see igure 9) an enote l its wit an b te immersion ept at equilibrium. Te origin of coorinates is cosen at. or efiniteness we consier te case were two angles of te rectangle are immerse. bviously, = bl, = P an = l. Te buoyancy an te metacentric curves as functions of te angle of eel are, cf. eqs. Q (.), (.4), (VI.4), ( ) l ² ( ) l ² ϑ = tan ; z b 4 tan ² b ϑ ϑ = + b ϑ (.) l 5

27 an ( ) l ² ( ) l ² ϑ = tan ϑ ; z ϑ = b+ 4 b b cos² ϑ. (.4) Bot curves are illustrate in igure 0. Te metacentric curve is V -sape, an for te metacentric istance we get = l ² b cos ϑ. (.5) Te ept of below te line of flotation an te yrostatic energy are, respectively, cf. eq. (V.), flotation l ² sin ² ϑ = Z= 4 + b cosϑ b cosϑ. (.6) E =µ g f flotation igure 0. Buoyancy an metacentric curves for a floating square wit l = 00 an b = 00. Te iamons an triangles inicate, respectively, te locations of an for ϑ= 0,5, 0 an 45. Te flotation curve is reuce to a point (circle). Te lever arm of te buoyancy force, in accorance wit eq. (V.7), is a = l + sinϑ b cos ϑ. (.7) b Te angular epenence of a is sown in igure, left. or larger boies E is minimal at equilibrium, ϑ= 0 an increases wit te angle of eel, cf. eq. (.6); te sign of a is opposite to tat of ϑ, corresponing to a restoring moment. or narrower boy an opposite beaviour is observe: at equilibrium E is maimal, so tat a an ϑ ave te same sign, corresponing to an overturning moment. In te vicinity of equilibrium te cange in te sign of a takes place at a critical value l b= 6. 6

28 igure. Left: lever arm of te buoyancy force for a floating rectangle of wit l an immersion ept b vs. te angle of eel. Rigt: metacentric eigt. Te ϑ -values are limite by te requirement tat of two angles of te rectangle be immerse. or l b = one always as an overturning moment, for l b=.5 te moment is an overturning one for eel angles less tan ϑ an becomes a restoring one at larger ϑ. or l b= one always gets a restoring moment. rom eqs. (.5) an (.6), te metacentric eigt is, cf. eq. (VI.), l sin ϑ cos ϑ. (.8) b cos ϑ flotation= b cosϑ+ 4 igure, rigt sows te angular epenence of flotation. Te conition of stability of te vessel requires tat at equilibrium ( ϑ= 0 ) G lie below. rom eq. (.8), te limiting eigt of G wit respect to te plane of flotation as a function of te l b ratio is Z G l = + b b lim (.9) for l b, Te corresponing grap, sown in igure, illustrates te amazing stability of a raft. Inee, Z b! igure. Limiting values of wits l (in relative units). Z G for a rectangular bo (full line) an a parabolic cyliner (vie infra, ase line) for ifferent Te special case of a rolling ro can be obtaine from te above equations by assuming l 0. In tis case, coincies wit : 7

29 = = 0 ; z = z = b. (.0) II.. Parabolic cyliner igure. loating parabola escribe by eq. (.). onsier a floating cyliner of cross section escribe by te parabola b z= 6 b, (.) l ² see igure. Te parameters in eq. (.) are cosen so as to facilitate a comparison wit te rectangular bo; inee, l is te lengt of te line of flotation at equilibrium, an te immerse area is =bl. Te origin of coorinates is cosen at te centre of flotation at equilibrium, an te abscissas of te enpoints of te line of flotation are l l P= l+ tan ϑ ; Q= l+ tanϑ. (.) b b In te boy frame an escribe parabolas of equations an l b l b = tan ϑ ; z = 4 tan ϑ (.) l z b b l b = tan ϑ ; = tan ϑ (.4) (note tat an coincie). Te metacentric curve is given by, cf. eqs. (VI.4), l l = tan ϑ ; z = 5 b+ 4 b b cos ϑ, (.5) as in te case of floating bo, it is V -sape. Te metacentric istance is, cf. eqs. (VI.5), = l b cos ϑ. (.6) 8

30 Te ifferent caracteristic curves are sown in igure 4. Tere is muc similarity between te cases of te rectangular bo an te parabolic cyliner; inee, in bot cases te istance between te enpoints of te line of flotation remains constant, Q P= l, an te metacentric istance is te same, cf. eqs. (.5) an (.6). eanwile, te centre of flotation of te bo is fie wile tat of te parabolic cyliner is a function of te angle of eel, cf. eq. (.). Te epressions of te ept of below te line of flotation an te yrostatic energy for te parabolic cyliner are, cf. eq (V.), = b cos ϑ ; E = µ gb l cosϑ. (.7) flotation 5 5 f igure 4. Buoyancy, flotation an metacentric curves for a floating parabola wit l = 00 an b = 00. Te iamons, circles an triangles inicate, respectively, te locations of, an for ϑ= 0, 8 an 56. s a ifference from te rectangular bo, for te parabolic cyliner te lever arm of te buoyancy force, cf. eq. (V.7), for ϑ> 0 is always positive, resulting in an overturning moment a 5 b sin = ϑ. (.8) rom eqs. (.6) an (.7), te metacentric eigt is, cf. eq. (VI.), ϑ= 0, = l b cosϑ b cos ϑ. (.9) flotation 5 Te limiting eigt of G wit respect to te plane of flotation as a function of te l b ratio for Z G lim l = 5+ b b, (.0) is illustrate in igure, vie supra, in comparison wit te rectangular bo. In tis case also Z b for l b, so, G can be locate very ig above te plane of flotation, in spite of te above-mentione tenency of te buoyancy force to overturn te boy. 9

31 II. 4. Elliptic cyliner igure 5. loating ellipse of semi-aes a an b. onsier an elliptic cyliner, see igure 5, of cross-section escribe by z b a = ± a. (.) were a an b are, respectively, te semi-major an semi-minor aes. s in te case of te circular cyliner, te origin of coorinates is cosen at te centre of te transversal cross-section. To simplify te formulae, we limit ourselves to te case of alf-submerge cyliner, so tat te line of flotation always passes troug its centre an te area of its immerse alf is = πab. Below we use te notation R( ϑ) = a sin ϑ+ b cos ϑ (.) ( Rϑ ( ) is te raius of te same ellipse turne troug te angle ϑ+ π ). In a position incline troug an angle ϑte line of flotation meets te ellipse at te points P,Q ab cosϑ =. (.) R( ϑ) Te coorinates of are = 4 a sinϑ 4 b cosϑ ; z πr( ϑ) = πr( ϑ), (.4) an from eqs. (VI.4) one calculates te location of te metacentre: a ( a ) ( ) 4 b sin ϑ 4 cos ; b a b ϑ = z =. (.5) πr ( ϑ) πr ( ϑ) Te buoyancy an metacentric curves are sown in igure 6 for rolling about two equilibrium states, respectively, wit te major ais parallel an perpenicular to te line of flotation. In te first case te metacentric curve is Λ -sape wile in te secon case it is V -sape. rom eqs. (.4), (.5) te metacentric istance is, cf. eq. (VI.5), 0