come

In 1993, Bently Nevada Corporation introduced the "full spectrum" plot, as contrasted to the "traditional (half) spectrum" plot, and pioneered its application to rotating machinery monitoring and diagnostics. Since then, full spectrum plots have been included in all major Bently Nevada machinery management software packages, such as ADRE ® for Windows and Data Manager ® 2000 for Windows NT. Several articles have been published in past Orbit magazines which explain the meaning and applications of full spectrum plots and their advantages over half spectrum plots [START_REF] Southwick | Using Full Spectrum Plots[END_REF][START_REF] Southwick | Using Full Spectrum Plots Part 2[END_REF][START_REF] Laws | When you use spectrum, don't use it halfway[END_REF]. Nevertheless, full spectrum plots remain unfamiliar to many Bently Nevada customers. The questions most often asked are:

1. How is the full spectrum generated? 2. What is the correlation of it with half spectrum and/or with filtered orbits?

3. What can it do for me, and why do I have to bother learning about it?

Benefits of full spectrum plots

Before we answer these questions, we'd like to start with the following observation:

The objective of data processing is to extract and display the maximum amount of significant diagnostic information from the original signals generated by the transducers.

One of the pieces of information inadvertently hidden and ignored is the correlation between the vibration patterns from different transducers. The full spectrum plot displays the cor-Paul Goldman, Ph.D.

Sr. Research Scientist

Bently Rotor Dynamics Research Corporation e-mail: paul.goldman@bently.com

Application of full spectrum to rotating machinery diagnostics

relation between the vibration data from the X and Y components of the rotor or casing lateral response. At a glance, the full spectrum plot allows us to determine whether the rotor orbit or machine casing motion frequency components are forward or backward in relation to the direction of rotor rotation. This information, which is characteristic of specific machinery malfunctions, makes the full spectrum plot a powerful tool for interpreting the vibration signals of rotating machinery. The process of creating either full or half spectrum plots starts from digitizing the vibration waveforms. In the case of XY probes measuring rotor lateral vibration, there is one waveform from each channel. Combined, they generate a direct orbit. Note that the orbit represents the magnified path of the actual motion of the rotor centerline.

How the full spectrum is generated

Half spectrums are independently calculated from each waveform (Figure 1). During this calculation, a part of the information contained in the waveform and orbit is not retained. In particular, the relative phase correlation between X spectrum and Y spectrum components is not displayed. Thus, filtered orbits cannot be reconstructed using corresponding frequency components from X and Y half spectrums. Also, the half spectrum information shows no relationship to the direction of the rotor rotation. One process for obtaining a full spectrum, which demonstrates the correlation between the orbit and full spectrum, includes an expansion of the direct orbit into a sum of filtered orbits (Figure 2). Each filtered orbit has, in general, an elliptical shape. An elliptical orbit can be presented as a sum of two circular orbits: one is the locus of the vector rotating in the direction of rotation (forward), and the other is the locus of the vector rotating in the opposite direction (reverse). Both vectors rotate at the same frequency (the frequency of the filtered orbit).

What is the correlation of full spectrum with filtered orbits and half spectrum?

Since such a presentation of the filtered orbit can be done in only one way, forward and reverse circles are completely determined by the filtered orbit. An instantaneous position of the rotor on its filtered orbit can be presented as a sum of vectors of the instantaneous positions on the forward and reverse orbits: R ω+ e j(ωt+α ω ) +R ω -e -j(ωt+β ω ) . Here R ω+ and R ω - are the radiuses of the forward and reverse orbits, ω is the frequency of filtering, and α ω and β ω are phases of forward and reverse responses. In Figure 2, ω = Ω. Since Ω is the rotative speed, ω can, therefore, be equal to Ω or to 2Ω. (1X or 2X). Note that the major axis of the filtered orbit ellipse is R ω+ +R ω -, while its minor axis is R ω+ -R ω -. Forward precession of the filtered elliptical orbit (in the direction of the rotor rotation) means that R ω+ > R ω -, while reverse precession means that R ω+ < R ω -. To completely define an ellipse, the major axis orientation is needed. The angle between the horizontal probe and the ellipse major axis, (β ω -α ω )/2, is determined by the relative phase of the forward and reverse components. In two important cases, the ellipse degenerates into a simpler form:

1. If the filtered orbit is circular and forward, then the reverse component does not exist. If the filtered orbit is circular and reverse, then the forward component does not exist. There is no relative phase, and the major axis equals the minor axis.

2. If the filtered orbit is a straight line, then the amplitude of the forward component is equal to the amplitude of the reverse component. Relative phase is still important to defining the orientation of the line.

A full spectrum is constructed from the radiuses of the

What can it do for me?

So, now that we know how a full spectrum is created, and what additional information it contains in comparison to a half spectrum, the question is, "how can it be used?" In order to perform reliable diagnostics, all possible information has to be extracted from the available data. Since full spectrum contains more information than the half spectrum, it has an advantage from that perspective. It can be used for steady state analysis (full spectrum, full spectrum waterfall) or for transient analysis (full spectrum cascade).

One of the possible applications of full spectrum is for analysis of the rotor runout caused by mechanical, electrical, or magnetic irregularities. Depending on the periodicity of such irregularities observed by the XY proximity probes, different combinations of forward and reverse components are observed. The rules for such an analysis are summarized in Table 1. The amplitude and frequency components generated by the irregularities of the rotor do not change with rotative speed, unless there is a change in the rotor axial position. In that case, a new pattern will emerge, but it will follow the same rules.

forward and reverse components of the filtered orbits. The horizontal coordinate equals ± frequency (+ for the forward and -for the reverse components), and the vertical coordinate equals the peak to peak amplitude of the corresponding forward or reverse component.

In ADRE ® for Windows, the full spectrum is obtained as the result of an FFT transformation of the sampled signals. The X component is the direct input, and the Y component is the quadrature input (Figure 3). Half spectrums are obtained independently from X and Y sampled signals. Each of them is considered as the direct input to the FFT, while the quadrature input is zero.

It is important to note that the full spectrum forward and reverse component amplitudes can be used to recover the shape of the corresponding filtered orbit. Determining the orientation of the orbit is not possible in the full spectrum without the relative phase information, however. There is no way to make any judgment on the shape of the filtered orbit using half spectrums.

The full spectrum is unaffected by probe orientation or probe rotation, as is the orbit. The X and Y half spectrums are dependent on the actual probe locations and can change dramatically with changes in their orientation. These characteristics, along with the enhanced applications of the full spectrum, make it superior to the half spectrum.

Periodicity of irregularities

M Ma aj jo or r f fr re eq qu ue en nc cy y c co om mp po on ne en nt ts s Information regarding the full spectrum content generated by some rotating machinery malfunctions is contained in Table 2.

Once per cycle

1X forward (can include mechanical bow)

Twice per cycle

2X forward and 2X reverse, with the same magnitudes

Three times per cycle

3X reverse

Four times per cycle

4X forward and 4X reverse, with the same magnitudes

Conclusions

Table 2 is incomplete. Our knowledge of the application of full spectrum to rotating machinery monitoring and diagnostics is gradually evolving. However, even from what we know now, this new data presentation format is worth using. It allows assigning a direction to the rotor lateral response frequency analysis, and thus provides a better foundation for root cause analysis. Unlike individual half spectrums, full spectrum is independent of the particular orientation of probes. This independence, among other advantages, makes a comparison of different planes of lateral vibration measurements along the rotor train considerably easier.

Figures 1 &

 1 Figures 1 & 2 show the generation of the half and full spectrum plots.

Figure 1 .

 1 Figure 1. Half spectrum data-processing sequence: spectrums are obtained independently using the data from each probe.

Figure 2 .

 2 Figure 2. Full spectrum data-processing sequence: the spectrum obtained from the data from two probes.

Figure 3 .

 3 Figure 3. Mathematical procedure for obtaining a full spectrum.

Table 1 . Runout signature analysis.

 1

	Type of malfunction	Frequency component	Forward	Reverse	Comments

Table 2 . Rotating machinery malfunctions as displayed by a full spectrum plot. The minus (-) means the components are nonexistent or, typically, of very small amplitude.

 2


Lucas Riondet

Maud Rio

Véronique Perrot Bernardet

Peggy Zwolinski

Toward the eco-design of emerging technologies for energy and the industry of the future: how to integrate upscaling in life cycle assessment?

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

sCALe-IT

Toward the eco-design of emerging technologies for energy and the industry of the future: how to integrate upscaling in life cycle assessment (LCA)?

State(s) of the Art and research questions

The planetary boundaries framework shows the current unsustainability of our societies: 6 of the 9 limits guarantying a safe operating space (SOS) for humanity have been crossed in 2022. Facing this massive issue, institutions all over the world address the necessity to build collectively the transitions toward sustainability: IPCC/IPBES reports, European Commission reports, SNBC and RTE/ADEME scenarios in France. In engineering, scientific methods to support the anticipation capacity of product designers in an upscaling context are lacking.

Therefore, designers should be better equipped to follow sustainable strategies by assessing the sustainable potential of an emerging technology before its upscaling in an environmental perspective.

PhD subject (March 2021-2024)

The term "upscaling" is polysemic in the literature: A characterization requirement.

( 3 )•••x18

 3 related to massive production to reach a level of a cumulated service Deploying a technology, reaching a level(1) related to productivity and/or TRL From laboratory to industrial scale (4) Focused on interactions between different technologies and complex system integration Up and down zooming(2) related to producibility Industrialization, massification(5) related to the definition of an absolute (environmental) limit.Down-limitingContext: An environmental sustainability framework RQ 1 : What is an upscaling and how to environmentally assess it? RQ 2 : How to integrate the upscaling assessment expertise in the design process? Eco-design approach ! 5 archetypes of upscaling have been identified in the literature. What should be the scope of such a prospective analysis ? SubRQ1.2: What are the prevalent phenomena to focus on? Which metrics/indicators to describe them? Riondet L, Rio M, Perrot-Bernardet V, Zwolinski P. 2022. For an upscaling assessment integration in product. 32th CIRP Design Conference. Upscaling is represented with an arrow from subject to its result Types Type of entity on which focus is made A group of technologies with varying levels of performance A complex system : Large scale system with high number of interconnected subsystems/technologies A technology or a group of the same technology Case study: LCA applied to the upscaling of photovoltaic systems VDMA -International Technology Roadmap for Photovoltaic (ITRPV) -2019 Results -Figure 21 Ritzer et al. (2021) -Upscaling of perovskite solar modules: The synergy of fully evaporated layer fabrication and all-laser-scribed interconnections Perspectives 1. Application of different LCA modes (dynamic, prospective, absolute, regionalized) to assess the upscaling of PV systems, Activity Browser Steffen et al. 2015 updated with Persson et al. 2022 https://www.stockholmresilience.org/research/planetary-boundaries.html Paper Riondet et al. (2023) in preparation. Upscaling of emerging technologies and environmental assessment: review and perspectives for designers and LCA practitioners. of Solar Photovoltaic: deployment, investment, technology, grid integration and socio-economic aspects. A Global Energy (2019) -Figure 6 photovoltaics using absolute LCA. 30th CIRP Life Cycle Engineering Conference. 2. Validation of the LCA methodology (scope, metrics), 3. Development of a designer-oriented tool on and Preliminary results


Jacques Mégel

Janis Kliava

On the buoyancy force and the metacentre On the buoyancy force and the metacentre On the buoyancy force and the metacentre On the buoyancy force and the metacentre

We address the point of application A of the buoyancy force (also known as the Archimedes force) by using two different definitions of the point of application of a force, derived one from the work-energy relation and another one from the equation of motion. We present a quantitative approach to this issue based on the concept of the hydrostatic energy, considered for a general shape of the immersed cross-section of the floating body. We show that the location of A depends on the type of motion experienced by the body. In particular, in vertical translation, from the work-energy viewpoint, this point is fixed with respect to the centre of gravity G of the body. In contrast, in rolling/pitching motion there is duality in the location of A ; indeed, the work-energy relation implies A to be fixed with respect to the centre of buoyancy C, while from considerations involving the rotational moment it follows that A is located at the metacentre M. We obtain analytical expressions of the location of M for a general shape of the immersed cross-section of the floating body and for an arbitrary angle of heel. We show that three different definitions of M viz., the "geometrical" one, as the centre of curvature of the buoyancy curve, the Bouguer's one, involving the moment of inertia of the plane of flotation, and the "dynamical" one, involving the second derivative of the hydrostatic energy, refer to one and the same special point, and we demonstrate a close relation between the height of M above the line of flotation and the stability of the floating body. Finally, we provide analytical expressions and graphs of the buoyancy, flotation and metacentric curves as functions of the angle of heel, for some particular shapes of the floating bodies, viz., a circular cylinder, a rectangular box, a parabolic and an elliptic cylinder.

Clearly, h

F is a resultant of all elementary hydrostatic forces applied to the surface of the immersed body. A force is fully described by (i) its norm, (ii) its direction and (iii) its point of application. Note that the point (iii) is not included in the above definition of the Archimedes' Law. On the other hand, it would be surprising if such a fundamental epistemological question as that of the point of application of the buoyancy force were not at all discussed in the literature.

By definition of the resultant force, the point of application of h F should be defined by requiring that if we apply such a force, the (macroscopic) behaviour of the body will be the same as that caused by the ensemble of elementary hydrostatic forces. Note, however, that this definition does not guarantee the uniqueness of this point; indeed, one may quite well conceive it to be different in different physical situations.

An analysis of the abundant bibliography concerned with the stability of immersed bodies

shows that many authors prefer to elude the question of the exact location of the point of application of the Archimedes' force, e.g., speaking of "the line of action" of this force. 2, p. 73 What is really shown is only that the line of application of the Archimedes' force passes through the centre of buoyancy, so that one may guess that A can be located anywhere on this line. If this statement is sufficient in statics, it is certainly not sufficient in dynamics. Indeed, one might argue that a force applied to a rigid body, as a sliding vector, obeys to the principle of transmissibility stating that "the condition of motion of a rigid body remains unchanged if a force F of a given magnitude, direction and sense acts anywhere along the same line of action…". 3 However, in this relation, an important amendment had been made already more than a century ago, namely, that "we may imagine a force to be applied at any point in the line of its direction, provided this point be rigidly

I. Introduction I. Introduction I. Introduction I. Introduction

"If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the "If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the "If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the "If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards by a force equal to the difference be difference be difference be difference between its weight and the weight of the fluid displaced tween its weight and the weight of the fluid displaced tween its weight and the weight of the fluid displaced tween its weight and the weight of the fluid displaced." ." ."

." 1, p. 257 One of the most long-standing -for 22 centuries -and, perhaps, one of the best-known laws of physics is the Archimedes' Law stating that a body immersed in a fluid is subjected to a force equal to the weight of the displaced fluid and acting in opposite direction to the force of gravity. This force is called "buoyancy force" or "Archimedes' force"; we denote it by h F , the subscript "h" standing for "hydrostatic", and its point of application will be referred to as A.

connected with the first point of application". [START_REF] Wormell | Wormell's mechanic[END_REF] Below, we will show that, while a floating body can be considered as rigid, the point of application of h F is not necessarily rigidly connected with it because the displaced fluid is evidently not rigid. Therefore the question of the point of application of this force is quite legitimate and meaningful.

In a quite natural way one is tempted to relate the point of application of h F to one of the three following remarkable points of a floating body.

(i) The centre of gravity of the body, denoted by G (the distinction between the centre of gravity and that of mass/inertia is of no relevance for the present analysis). To our knowledge, such assignment has never been suggested before.

(ii) The centre of buoyancy (centre of gravity of the displaced fluid) denoted by C (from careen) or by B (from buoyancy); we prefer using the former notation. Note that C sometimes is defined as the geometrical centre of the displaced fluid; meanwhile, for a homogeneous fluid (the case considered below) both definitions coincide. Most frequently, the point of application of h F is assigned to the centre of buoyancy, e.g., see Refs. [5 (p. 58), 6, 7].

(iii) The metacentre denoted by M. This point is of a special interest, since, in order to assert the stability of the body against overturn, M, defined in the vicinity of equilibrium, should be located above G. We have found only one textbook suggesting, without demonstration, M as a candidate for the point of application of h F , see Ref. [8, footnote p. 26]. On the other hand, in a very interesting while, unfortunately, not easily accessible paper, Herder and Schwab [START_REF] Herder | On dynamically equivalent force systems and their application to the balancing of a broom or the stability of a shoe box[END_REF] suggest a distinction between "statically equivalent" and "dynamically equivalent" resultant forces. The former are defined for a fixed position of the body (statics) in which case only a line and not a point of application of the resultant can be defined. The latter are related to the stability of a nominal state with respect to small variations about this state. In this case the point of application of the resultant force is essential, and these authors define it as the "dynamically equivalent" point of application. For the particular case of floating parallepiped ("shoe-box") Herder and Schwab deduce from considerations based on the hydrostatic energy that the "dynamically equivalent" point of application of the buoyancy force is the metacentre.

It has seemed quite tempting to generalize this approach to the general case of the floating body of arbitrary shape, inclined through an arbitrary angle.

The concept of metacentre dates back to Bouguer's Traité du Navire, de sa construction et de ses mouvemens (1746) [START_REF] Bouguer | Historical roots of the theory of hydrostatic stability of ships (Max-Planck-Institut für Wissenschaftsgeschichte Preprint[END_REF] . Using the prefix µετά = "beyond", he had designated a specific point M of a floating body, defined as the intersection of two vertical axes passing through the centre of buoyancy C (the centre of gravity of the displaced fluid) at two slightly different angles of heel. Besides, Bouguer formulated the well-known theorem relating the distance between C and M to the ratio of the moment of inertia of the plane of flotation and the volume of the displaced fluid.

Three years later, Euler in Scientia navalis (1749) gave a general criterion of the ship stability, based on the restoring moment: [START_REF] Euler | Scientia Navalis seu Tractatus de Construendis ac Dirigendis Navibus[END_REF] the ship remains stable as far as the couple weight (applied at G) and the buoyancy force (whose line of application passes through C) creates a restoring moment. A change of sign of the latter results in capsizing, and its vanishing in inclined position (at equilibrium it vanishes by definition) corresponds to the overturn angle.

The problem of stability of the floating bodies, which can be traced back to Archimedes himself, see [1, On floating bodies, Book I, pp. 253-262; Book II, pp. 263-300], has never ceased to interest scientists and engineers [START_REF] Rawson | Basic ship theory[END_REF][START_REF] Derrett | Ship stability for masters and mates[END_REF][START_REF] Taylor | The principles of ship stability[END_REF][START_REF] Bertram | Practical ship hydrodynamics[END_REF][START_REF] Blagoveshchensky | Theory ship motions (Transl. from the 1 st Russian[END_REF] , in particular, in the relation to the metacentre, e.g., see 7, [START_REF] Ch | Applications de géométrie et de mécanique, à la marine, aux ponts et chaussées, etc… pour faire suite aux développements de géométrie[END_REF][START_REF] Gilbert | How things float[END_REF][START_REF] Erdös | Floating equilibrium of symmetrical objects and the breaking of symmetry, Part 1: Prisms[END_REF][START_REF] Angremond | Breakwaters and Closure Dams[END_REF] and has become an important part of academic studies [START_REF] White | Fluid mechanics[END_REF][START_REF] Giles | Theory and problems of fluids mechanics and hydraulics (Schaum's outline series[END_REF][START_REF] Benson | University physics[END_REF][START_REF] Comolet | Mécanique expérimentale des fluides, T. I: Statique et dynamique des fluides non visqueux[END_REF][START_REF] Besant | A treatise on hydromechanics Part I: Hydrostatics[END_REF] . Meanwhile, there has been no significant progress in this field since the original Bouguer's Treatise 10 and its reformulation in terms of a "novel geometry" in the Dupin's textbook [START_REF] Ch | Applications de géométrie et de mécanique, à la marine, aux ponts et chaussées, etc… pour faire suite aux développements de géométrie[END_REF] , relating the metacentric curve, i.e., the loci of M, with the buoyancy curve formed by the loci of C, the former being the evolute of the latter.

The aim of the present work is to elucidate the question of the point of application of the buoyancy force in relation with that of ship stability. We consider two different approaches to the definition of the point of application of a resultant force based, first, on the work-energy relation and, second, on the equation of motion. We obtain appropriate expressions of the hydrostatic energy and the location of the characteristic points for a floating body of rather general shape. On this basis we show that the location of the point of application of the buoyancy force depends on the type of motion experienced by the body. In particular, in pure translation this point is fixed with respect to the centre of gravity G while in rolling or pitching motion it is fixed with respect to the centre of buoyancy C (from the viewpoint of the work-energy relation) or located at the metacentre M (from the viewpoint of the rotational moment).

In the framework of the formalism developed in this work, we present a quantitative analysis of the concept of the metacentre. We calculate the location of the "geometrical", Bouguer's and "dynamical" metacentres for an arbitrary angle of heel and show that all three definitions, in fact, result in one and the same metacentric curve. Finally, in the Appendix we apply the general relationships to determine the location of C and M for several particularly simple shapes of the floating body.

II. Buoyancy force and hydrostatic energy II. Buoyancy force and hydrostatic energy II. Buoyancy force and hydrostatic energy II. Buoyancy force and hydrostatic energy Consider a floating, i.e., partially immersed in a fluid, rigid body, e.g., a vessel. In such situation, strictly speaking, one should also take into account the atmospheric pressure experienced by the part of the body situated above the waterline. However, as far as the specific density of the air remains much lower than that of the fluid, as we have assumed in the analysis given below, the atmospheric pressure can be neglected. In the formulae given below the surface S and the volume V concern only the submerged part of the body, V being equal to the volume of the displaced fluid.

The hydrostatic force exerted on an element dS of the immersed surface of the body is

h d d p = - F
S where p is the hydrostatic pressure. The minus sign in this expression is due to the fact that, by convention, dS is directed outwards a closed surface while the hydrostatic force is directed inwards the body. Integrating h dF over the immersed surface and applying the gradient theorem to pass from a surface integral to an integral over the immersed volume V yields the buoyancy force as: One defines the plane of flotation as "the plane in which the body is intersected by the surface of the liquid" 22, p. 67 . The line of flotation is defined as the intersection of the plane of flotation with a vertical cross-section of the body; the centre of the former is referred to as the centre of flotation and denoted by O.

Below we will consider two types of motion of a partially immersed body: a vertical translation and an oscillation about one of the horizontal axes of a floating body. In most cases a floating vessel is much longer in one of the principal horizontal directions than in the perpendicular one, and the oscillations about the longitudinal and transversal axes are respectively called rolling and pitching. We will not explicitly consider the pitching; indeed, its characteristics can be readily obtained from those of the rolling motion. The latter will be systematically described in two interrelated coordinate systems, see

( ) ( ) ( ) ( ) O O O O cos sin sin cos X x x z z Z x x z z ϑ ϑ ϑ ϑ = - + - = -- + - . (II.3)
We suppose that, as is most often the case, the fluid can be considered as incompressible and homogeneous, so that the hydrostatic pressure depends only on the immersion depth:

f p gZ µ = -
where f µ is the specific density of the fluid and g is the acceleration of gravity. Then eqs.

(II.1), (II.2) become:

f h f d Z Z V g V gV µ µ = = ∫∫∫ F u u (II.4) and f h d V E g Z V µ = -∫∫∫ . (II.5)
III. III. III. III. Two definitions of the point of application of a force Two definitions of the point of application of a force Two definitions of the point of application of a force Two definitions of the point of application of a force By definition of the centre of buoyancy, its coordinates are

C C C 1 1 1 d ; d ; d V V V X X V Y Y V Z Z V V V V = = = ∫∫∫ ∫∫∫ ∫∫∫ , (III.1)
therefore, from eqs. (II.5) and (III.1),

h f C h C E gVZ F Z µ = - = - . (III.2)
The concept of the hydrostatic energy allows defining the point of application of the buoyancy force from the usual work-energy relation, e.g., see ref. [22]. The elementary work is defined as the scalar product of a force with an elementary displacement of its point of application; on the other hand, for a force deriving from a potential energy it equals the diminution of this energy. In the present case,

h h h A A h d d d W F Z E δ = ⋅ = = - r F . (III.3)
If the displacement of the point A matches that of a definite point of the body, the latter point can be identified as the point of application of h F . As far as the above identification is based on the workenergy relation, the corresponding point will be referred to as the "energetical" point of application of the buoyancy force. This definition applies for any type of motion.

Another possibility of defining the point of application of h F is based on the principle that the rotational moment of a force vanishes in its point of application. This definition is applicable to types of motion where a rotational component is present, in particular, to rolling/pitching. A floating body experiences both the buoyancy force h F and the force of gravity g F , and its total potential energy is the sum of the hydrostatic and gravitational energies, total h g

E E E = + . The couple h g + F F produces a rotational moment (torque) total h G g A = ∧ + ∧ M
r F r F where r r r r A and r r r r G are position vectors of A and G with respect to an arbitrary coordinate system; obviously, total M does not depend on the choice of the origin of coordinates.

The strategy usually adopted in mechanics is to separate a general motion of the body into translation of an arbitrary point, chosen inside or outside the body, and rotation about this point. Most often the centre of rotation is chosen at G, in which case G 0 = r , but this choice is not compulsory.

For a floating body the location of G depends not only of its shape but also of the distribution of weights inside the hull, which changes with the ship loading. Therefore, we have chosen to focus on the buoyancy force and the related hydrostatic energy without systematically calling to mind the force of gravity and its potential energy.

However, the rotational moment of the single force h F does depend on the origin of r r r r A , therefore, in considering only the buoyancy force and the hydrostatic energy, the choice of the origin of coordinates is physically meaningful. The famous Euler's theorem 11 states the following.

"T T T

The oscillatory movement of a floating body (roll he oscillatory movement of a floating body (roll he oscillatory movement of a floating body (roll he oscillatory movement of a floating body (rolling ing ing ing or pitch or pitch or pitch or pitching ing ing ing) can be described as a rotation about ) can be described as a rotation about ) can be described as a rotation about ) can be described as a rotation about the ce the ce the ce the centre ntre ntre ntre

" " " "

Choosing O as the origin of A r , the rotational moment of the buoyancy force, h M is calculated from a variation of E h:

( ) ( ) h A h A h A h h d d d d d E = - ⋅ = - ∧ ⋅ = - ⋅ ∧ = - ⋅ r F r F r F M Θ Θ Θ (III.4)
where the elements of the angle vector ( )

, , 0 ϕ ϑ Θ =
are angles of rotation about the X and Y axes. In deriving eq. (III.4) we have used the expression of variation of a position vector r r r r in an

infinitesimal rotation, d d = ∧ r r Θ
. From the latter equation one gets

h h E Θ = -∇ M
where elements of the gradient Θ ∇ are derivatives with respect to the angles of rotation. In the particular case of rolling, the only non-vanishing element

of h M is h h Y E ϑ ∂ = -∂ M u . (III.5)
On the other hand, h M is the sum of moments of the elementary hydrostatic forces, so, for a homogeneous fluid it can be calculated as

h h f d d S S g Z µ = ∧ = ∧ ∫∫ ∫∫ M r F r S . (III.6)
Applying the curl theorem and taking into account that the curl of the radius vector is identically null,

0 0 0 0 ∇ ∧ ≡ r , we get ( ) ( ) f h f d d V Y X V g Z X V g Y V µ µ = - ∇ ∧ = - + ∫∫ ∫ ∫ ∫∫ u u M r . (III.7)
By definition of the moment of the buoyancy force, ( )

h A h A A f h A Z Y X gV F X Y µ = ∧ = - + ∧ = M r r u u u F . (III.8)
From eqs. (III.7) and (III.8) we obtain the coordinates of A in the horizontal plane as the first moments of the immersed volume about the X and Y axes. These coordinates coincide with the corresponding coordinates of C, cf. eqs. (III.1):

A A 1 1 d ; d V V X X V Y Y V V V = = ∫∫∫ ∫∫∫ . (III.9)
X A and Y A give the lengths of the lever arms of h F for rolling and pitching, respectively. Therefore, h F is applied along the vertical line passing through C. Meanwhile, as one might expect, the vertical coordinate

Z A of the point of application of h F remains undefined.
Interestingly, if we attempt to determine the centre of gravity of a body from vanishing of the corresponding rotational moment in a given position, its vertical coordinate Z G will also remain undefined.

Meanwhile, this difficulty is readily overcome by rotating the body through an arbitrary angle about any non-vertical axis. From the viewpoint of such experience, the point of application of the weight can be defined as the "intersection of all vertical lines passing through the centre of gravity for different angular positions of the body". The same procedure can be used to specify Z A , and as will be discussed later, the "intersection of all vertical lines passing through the centre of gravity of the displaced fluid (i.e.

the centre of buoyancy) for different angular positions of the body", by definition, indicates the metacentre.

Therefore, we need an equation of motion involving

I

where I is the tensor of inertia about O and the elements of the angle vector Θ , the second derivative of Θ , are angular accelerations around the X and Y axes. In order to account for a small rotation about the state of the body described by this equation, we take the differentials of its both sides:

( ) ( ) h A h h A d d d d Θ = = ∧ ∧ = ∧ ∧ M r F F r I Θ Θ . (III.10)
Developing the double vector product and making use of the fact that h F is vertical while dΘ is horizontal, yields

( ) ( ) ( ) h A h A h A h d d d d = ⋅ - ⋅ = - ⋅ M r F r F r F Θ Θ Θ . (III.11)
In the case of the rolling motion, from eqs. (III.5) and (III.11) we get for the sought-after scalar product:

h h A h A h d ² ² d Y M E Z F ϑ ϑ ∂ = - = ⋅ = ∂ r F . (III.12)
Thus, defining the vertical coordinate of the point of application of h F , we are led to the following theorem:

Th Th Th
The scalar product of the buoyancy force and the position vector of its point of application from the origin at the e scalar product of the buoyancy force and the position vector of its point of application from the origin at the e scalar product of the buoyancy force and the position vector of its point of application from the origin at the e scalar product of the buoyancy force and the position vector of its point of application from the origin at the centre of flotation is given by minus the derivative of the rotational moment with respect to the centre of flotation is given by minus the derivative of the rotational moment with respect to the centre of flotation is given by minus the derivative of the rotational moment with respect to the centre of flotation is given by minus the derivative of the rotational moment with respect to the angle of angle of angle of angle of roll or pitch or by roll or pitch or by roll or pitch or by roll or pitch or by the second deri the second deri the second deri the second derivative of the hydrostatic energy with respect to this angle vative of the hydrostatic energy with respect to this angle vative of the hydrostatic energy with respect to this angle vative of the hydrostatic energy with respect to this angle.

This result can be considered as a particular case of the general analysis of the "dynamically equivalent" point of application given by Herder and Schwab [START_REF] Herder | On dynamically equivalent force systems and their application to the balancing of a broom or the stability of a shoe box[END_REF] .

Eqs. (III.9) and (III.12) are necessary and sufficient to determine all the three coordinates of the point of application of h F . As this determination is based on dynamical equations involving the rotational moment, this point will be referred to as the "dynamical" point of application of the buoyancy force.

It is easy to show that the second derivative of the total potential energy can be expressed as

( ) ( ) total total A G h A G h d ² ² d Y M E Z Z F ϑ ϑ ∂ = - = - ⋅ = - ∂ r r F , (III.13)
directly yielding the vertical distance between G and A.

In what follows we present a model calculation of the location of the different points of application of the buoyancy force. The calculations refer to the immersed part of the transversal crosssection of the floating body for brevity denoted by "immersed cross-section", so, the results given below refer to a portion of the body of unit length along its longitudinal axis. The characteristics of the whole body can be readily obtained by weighted integration along the y/Y axis. In the general case the shape of the immersed cross-section can be quite complicated. It is not necessarily supposed to be symmetric with respect to the z axis, e.g., the case of a Venetian gondola; moreover, it can be delimited by a multi-valued function, e.g., see Figure 2. However, it can always be represented as a combination of pieces of two different types:

(i) those fully immersed, delimited from the top and from the bottom by functions

( ) z f x + = and ( ) z f x - =
, respectively, and along the horizontal axis by the abscissas x inf and x sup ;

(ii) those partially immersed, delimited from the top by the line of flotation and from the bottom by a one-value function

( ) z f x =
and along the horizontal axes by the abscissas x P and x Q .

The expressions of areas and coordinates of the centres of buoyancy for such pieces are given in Appendix I.

IV IV IV IV. The point of application of the buoyancy force in translation . The point of application of the buoyancy force in translation . The point of application of the buoyancy force in translation . The point of application of the buoyancy force in translation

For definiteness, we consider a vertical ascending motion of the body (a "dry-docking"). This problem is most conveniently treated in the body frame at 0 ϑ = . We assume that the body is in equilibrium; therefore, the force of gravity is supposed to be applied along the vertical line passing through C.

First we consider a fully immersed piece of type (i) elevated through a small vertical distance d .

b Its area and hence h F remain constant, cf. eqs. (A.1), and from eq. (III.2) we get therefore, the point of application of h F is rigidly connected with C. Meanwhile, in this case C is rigidly connected with the immersed piece itself, so, A can be assigned to any point located on the vertical line through C. One can see that for a fully immersed piece only the line and not the point of application of h F can be defined.

h h C C C h h d d d d E F z F z F z = - - = - . (IV.1)
Next, we treat a partially immersed piece of type (ii), see Figure 3. In the starting position the line of flotation is located at O 0 z = , so, the buoyancy force is, cf. eq. (II.4),

Q Q P P 0 h f f d d d x x x f x F z x f x g g µ µ = = - ∫ ∫ ∫ . (IV.3)
and the hydrostatic energy is, cf. eq. (II.5), (  ) (IV.8)

Q Q P P 0 2 1 h f f 2 d d d x x x f x g E z z x f g x µ µ = - = ∫ ∫ ∫ . ( IV
Q Q P P d 0 h f f d d d d d d x x b x f f x E z b z z g x x b g f µ µ -       = - + - ≈         ∫ ∫ ∫ ∫ . ( IV
As in the case of a fully immersed piece, A is rigidly connected with the partially immersed piece in translational motion.

For an immersed cross-section of a general shape the coordinates of A are weighted sums of the corresponding coordinates of separate pieces, therefore A remains rigidly connected with such a piece. As G is rigidly connected with the immersed body and, in translation, G is located on the axis of application of h F , one can consider that h F is applied at G. This assignment being based on the workenergy relation, the corresponding point of application of h F will be referred to as an "energetical" one.

On the other hand, C is not rigidly connected with a partially immersed piece; therefore, in this case it would be an error to consider that h F is applied at C. Indeed, from eqs. (III.2) and (IV.7),

h h C C h h d d d d E F z F z F b = - - = - , (IV.9)
and substituting F h , E h , dF h and dE h from the respective expressions, one gets It can be easily shown that if the contour of the immersed cross-section ( ) f x is a singlevalued function, C dz is always smaller than or equal to 1 2 db . Indeed, from eq. (IV.10) one has ( ) ( )

Q Q P P 2 1 C 2 2 d d 1 d d x x x x z f x f x b L                           -  = ∫ ∫ . ( IV
Q Q P P 2 2 d d x x x x f x x L f x x     ≤       ∫ ∫ . (IV.12)
In the integral form this inequality was first published on p. 4 of the paper by Bouniakowsky 23 , but in can be directly deduced from the Cauchy-Schwarz inequality valid for any square-integrable functions. One can see that

1 2 C d d z b ≤
, the limiting case

1 2 C d d z b =
occurring for the rectangle, as discussed before.

In the general case the relation between C dz and db can be very different. For instance, consider the immersed cross-section shown in Figure 5, looking like the bulb-shape keel of certain yachts participating in the America's cup regatta [START_REF] Van Oossanen | Recent developments in the design of America's cup yachts[END_REF] . Surprisingly, in this case one obtains C d d z b ≈ -, so, when the body is elevated the centre of buoyancy is lowered over the same distance ! V. Duality of the poi V. Duality of the poi V. Duality of the poi V. Duality of the point of application of the buoyancy force in nt of application of the buoyancy force in nt of application of the buoyancy force in nt of application of the buoyancy force in rolling/pitching motion rolling/pitching motion rolling/pitching motion rolling/pitching motion Now we examine the location of the point of application in rolling/pitching. In the general case, in this type of motion a rotation through an angle ϑ is combined with a vertical translation of the centre of buoyancy, see Figure 6. At an arbitrary angle of heel the line of flotation is described by the equations

Q P O O Q P tan z z z z x x x x ϑ - -= = - - (V.1)
where the coordinates of O are ( ) ( )

1 O P Q 2 1 O P Q 2 x x x z z z = + = + . (V.2)
The curve described by O in the course of the rolling/pitching motion is called flotation curve.

The area of the immersed cross-section and the coordinates of the centre of buoyancy are given by eqs. (A.1), (A.3) and (A.4).

V.

( ) ( ) flotation h h C h C O C O sin cos E F d F x x z z ϑ ϑ   = = - --   (V.3)
and in the Earth frame (reminding that by definition of the Earth frame, O 0 Z = ) cf. eq. (II.3),

h h C E F Z = - . (V.4)
In order to calculate the derivative of E h with respect to ϑ , we consider a small variation of the angle of roll and make use of the fact that the area of the immersed cross-section remains constant in the course of the rolling/pitching motion:

d 0 dϑ ′ ≡ = A A (V.5)
(here and below the primes denote derivatives with respect to ϑ ). Substituting i ii + A = A A from eqs. (A.1) and applying the Leibniz rule to calculate the derivatives of the parameter-depending integrals, we get ( )

P Q P Q tan 0 x x z z ϑ ′ ′ ′ ′ + -- = . (V.6)
Taking the derivative of eq. (V.1) yields ( )

P Q P Q tan cos L x x z z ϑ ϑ ′ ′ ′ ′ - -+ = . (V.7)
The angular derivatives of the couples of variables (x P , z P ) and (x Q , z Q ) are related through the equation of the function

( ) z f x =
. Using the chain rule for the derivatives of composite functions,

P P Q Q P Q d d ; d d f f z x z x x x ′ ′ ′ ′ = = , (V.8)
from eqs. (V.6) and (V.7) we get a system of equations including only two unknown derivatives, P x ′ and Q x ′ . Resolving this system yields

1 1 P Q 2 2 P Q ; d d sin cos sin cos d d L L x x f f x x ϑ ϑ ϑ ϑ ′ ′ = = - - - . (V.9)
From these expressions the derivatives of O x and O z , respectively, O x ′ and O z ′ are straightforward.

By the way, the centre of flotation has a very particular role in this motion. Under a small variation of the angle of roll, d ϑ ϑ ϑ → + , its coordinates change as follows:

O O O O O O d ; d x x x z z z ϑ ϑ ′ ′ → + → + . (V.10)
The initial and the new lines of flotation are described by, cf. eq. (V.1), x ′ and C z ′ in the derivative of eq. (V.3) with respect to ϑ , we get

( ) ( ) h h C O C O cos sin E F x x z z ϑ ϑ ′   = - + -   . (V.14)
Surprisingly, this equation looks as if in eq. (V.3) only the trigonometric functions were ϑ -dependent, which is generally not the case.

Transforming to the Earth frame, cf. eq. (II.3), yields

h h C E F X ′ = . (V.15)
Obviously, this result is consistent with the relation

C C d d Z X ϑ = - (V.16)
arising from the fact that the rotation occurs about O and O 0 X = .

The rotational moment and the lever arm of the buoyancy force with respect to O are, cf.

eq. (III.5),

h h C h C ; M F X a X = - = -. (V.17)
In the position of stable equilibrium at 0 ϑ = the total potential energy total h g

E E E = +
of a floating body is minimal and the rotational moment of the couple h g + F F vanishes. However, the hydrostatic energy at equilibrium does not necessarily take an extreme (minimal or maximal) value, so that in the general case the moment of the buoyancy force with respect to O, h M does not vanish. If at 0 ϑ = E h has a minimum, a deviation from equilibrium produces a restoring moment (of opposite sign to that of ϑ ). On the contrary, if at 0 ϑ = E h has a maximum, an overturning moment (of the same sign as that of ϑ ) is produced.

The elementary work of the buoyancy force in a small rotation about O is expressed as, cf.

eq. (III.3),

h h A h d d W F Z E δ ϑ ′ = = - . (V.18)
Comparing with eqs. (V.15), (V.16) yields

A C d d Z Z = , (V.19)
i.e., the vertical displacement of the point of application of h F coincides with that of C. Therefore, A is rigidly connected with C, and, as far as A and C are located on the same vertical line, one can consider that C is the "energetical" point of application of h F in rolling/pitching motion. This result is different from that for the translational motion, in which case one can consider that h F is applied at G, cf.

eq. (IV.8).

V.

( ) ( ) h h C O C O 3 O 1 2 O 1 sin cos cos sin L E F x x z z x z ϑ ϑ ϑ ϑ   ′′ ′ ′   = - - + -    -  - A . (V.20)
In the latter formula the terms in O x ′ and O z ′ describe the displacement of the point O with respect to the body frame. Meanwhile, eqs. (III.4) and (III.12), describing the angular derivatives of E h , refer to a coordinate system with origin at the centre of rotation. Therefore, eq. (V.20) should be considered in a coordinate system centered at O, in which case O x ′ and O z ′ vanish, and we get the "truncated" second derivative of E h :

( ) ( ) h h C O O 3 C 1 12 sin cos L E F x x z z ϑ ϑ -   ′′   = - - +     A . (V.21)
In the Earth frame h E ′′ is further simplified to

3 1 1 h C 2 h L E F Z ′′ = +           A , (V.22)
yielding the vertical distance from the line of flotation to A, cf. eq. (III.12),:

A flotation 3 C h A h 1 12 E L Z F d Z = ′′ = = + A . (V.23)
On the other hand, the distance of C from the line of flotation is -Z C (Z C is negative in the system of coordinates chosen), therefore,

3 A 1 C A C 12 L d Z Z = -= A . (V.24)
In accordance with the Bouguer's theorem [START_REF] Bouguer | Historical roots of the theory of hydrostatic stability of ships (Max-Planck-Institut für Wissenschaftsgeschichte Preprint[END_REF] , this expression corresponds to the metacentric distance, vide infra. Therefore, one can conclude that the "dynamical" point of application of the buoyancy force coincides with the metacentre.

Note that Herder and Schwab 9 have carried out a similar calculation in the particular case of a rectangular immersed cross section in equilibrium position. However, in their equations (53, 55), corresponding to the above eqs. (V.22), (V.23), incorrectly appears the distance between G and M, in contradiction with the Euler's theorem. In fact, the latter distance should be calculated from the second derivative of the total potential energy, cf. eq. (III.13).

The existence of two different points of application of h F (C and M) deduced from two different definitions ("energetical" and "dynamical") for the same type of motion may seem paradoxical. While these two points result in identical rotational moments, because the corresponding lever arm is the same, their locations on the vertical axis are different.

In order to understand the cause of this duality, the reader can consider the apparently trivial case of the point of application of the force of gravity g F exerted on the floating body in rolling/pitching motion. Applying to g F the same formalism, as before for h F , cf. eqs. (V.3), (V.4), (V.14), (V.20), (V.22), (III.12), (V.23), shows that both definitions of the point of application of g F converge to the centre of gravity of the body.

In fact, the different behaviour of In fact, the different behaviour of In fact, the different behaviour of In fact, the different behaviour of g F and and and and h F is due to the fact that the shape of the immersed volume does not is due to the fact that the shape of the immersed volume does not is due to the fact that the shape of the immersed volume does not is due to the fact that the shape of the immersed volume does not remain constant in the course of rolling/pitching motion. The corresponding variation gives rise to the additional term in remain constant in the course of rolling/pitching motion. The corresponding variation gives rise to the additional term in remain constant in the course of rolling/pitching motion. The corresponding variation gives rise to the additional term in remain constant in the course of rolling/pitching motion. The corresponding variation gives rise to the additional term in eq. eq. eq. eq. ( ( ( (V V V V. . . . 22 22 22 22) ) )

) and ff., depending on the length of the line of flotation L and responsible for the duality of the point of and ff., depending on the length of the line of flotation L and responsible for the duality of the point of and ff., depending on the length of the line of flotation L and responsible for the duality of the point of and ff., depending on the length of the line of flotation L and responsible for the duality of the point of application of application of application of application of h F . . . .

For a fully immersed body 0 L = , so that M is merged with C, cf. eq. (V. 24) Actually, in the literature one can find three different definitions of the metacentre and the metacentric curve. We have chosen to designate them as the "geometrical" one (the evolute of loci of the centre of buoyancy), the Bouguer's one (relating the metacentric distance to the moment of inertia of the plane of flotation) and the "dynamical" one (suggested by the finding by Herder and Schwab that the "dynamically equivalent" point of application of the buoyancy force is the metacentre). [START_REF] Herder | On dynamically equivalent force systems and their application to the balancing of a broom or the stability of a shoe box[END_REF] Previously, the "geometrical" metacentre has been defined for any angle of heel whereas the Bouguer's metacentre was considered only in the vicinity of equilibrium and the "dynamical" metacentre only in the particular case of a "shoe-box" in equilibrium [START_REF] Herder | On dynamically equivalent force systems and their application to the balancing of a broom or the stability of a shoe box[END_REF] . Besides, the existing demonstrations of the Bouguer's theorem are employing ad hoc geometrical constructions and are far from being rigorous, see Refs. In this context, we would like to clarify the following two points:

(i) Do the three definitions of the metacentre address one and the same point of the floating body and if yes, is this true for any angle of heel? (ii) What is the physical meaning of the metacentric curve for an arbitrary angle of heel?

To begin with, we express the relation between the metacentric curve ( )

M M z x
and the buoyancy curve ( )

C C
z x by means of the usual definition of the evolute of a curve as the locus of its centers of curvature: 25

M C C M C C ; N N x x z z z x D D ′ ′ = - = + . (VI.1)
where

2 2 C C C C C C ; N x z D x z x z ′′ ′′ ′ ′ ′ ′ = + = - . (VI.2)
The expressions of C

x ′′ and C z ′′ are calculated by taking the derivatives of eqs. (V.13) with respect to ϑ and inserting P x ′ and Q x ′ from eqs. (V.9), see Appendix I. Substituting these expressions in eq.

(VI.2) yields

( ) ( ) 3 2 2 M M C 1 C geometrical M C M C 12 cos z z L d z z x x ϑ - = - + - = = A . (VI.5)
This distance is exactly the same as the distance from C to the "dynamical" point of application of the buoyancy force, cf. eq. (V.24), therefore the "dynamical" metacentre coincides with the "geometrical" one. Now let us demonstrate the Bouguer's theorem for an arbitrary angle of heel. The statement of this theorem is as follows:

"The metacentric The metacentric The metacentric The metacentric distance is equal to the ratio of the moment of inertia I distance is equal to the ratio of the moment of inertia I distance is equal to the ratio of the moment of inertia I distance is equal to the ratio of the moment of inertia I of the plane of flotation with respect to a of the plane of flotation with respect to a of the plane of flotation with respect to a of the plane of flotation with respect to a horizontal axis and the immersed volume V of the displaced fluid" horizontal axis and the immersed volume V of the displaced fluid" horizontal axis and the immersed volume V of the displaced fluid" horizontal axis and the immersed volume V of the displaced fluid": : : :

M C Bouguer I d V = .
(VI.6)

As before, we consider the immersed cross-section of the floating body. In this case eq. (VI. which proves the theorem. As the right-hand side of this equation coincides with that of eq. (VI.5), the "Bouguer's" metacentre coincides with the "geometrical" one.

We conclude that for any angle of heel, all three above-mentioned definitions of the metacentre are equivalent: where max L is the total length of the body. The pitching metacentric distance can be determined in a similar way.

From eqs. (V.15), (V.17

VII. Conclusion VII. Conclusion VII. Conclusion VII. Conclusion

We have shown that the location of the point of application of the buoyancy force depends not only on the type of motion of the floating body (translation or rolling/pitching) but, in the latter case, also on the definition of this point. In translation this point remains fixed with respect to the centre of gravity of the body while in rolling/pitching it is subject to a duality. Namely, from the viewpoint of the work-energy relation it is fixed with respect to the centre of buoyancy while from the viewpoint of the rotational moment it is located at the metacentre. This peculiarity of the buoyancy force is due to the fact that, whereas the immersed body can still be considered as a rigid one, the shape of the displaced fluid does not remain constant in the course of the motion. Indeed, in the case of a completely immersed body, the shape of displaced fluid remains fixed, so that metacentre and the centre of buoyancy coincide.

The concept of non-uniqueness of the point of application of a resultant force seems quite unusual; nevertheless, as we have shown, this non-uniqueness is an inherent feature of the buoyancy ( )

sup sup inf inf Q Q flotation P P i ii O d d d d d cos d x x f x f x x x z x f x z x f f x z x z L f x ϑ + - + - = = - = = - ∫ ∫ ∫ ∫ ∫ ∫ A A (A.1)
where x inf and x sup are the extreme abscissas of the fully immersed pieces, and ( )

P P P ,
x z , ( )

Q Q Q , x z
and L are, respectively, endpoints and the length of the line of flotation for a body inclined through an angle ϑ :

Q P cos x x L ϑ - = . (A.2),
The coordinates of the centre of buoyancy of the immersed cross-section are calculated by adapting to the two-dimensional case the general formulae eqs. (III.1), (III.9). The horizontal coordinates of C for the pieces of types (i) and (ii) are ( )

( ) sup sup inf inf Q Q flotation i P i P i 3 1 O O 1 C 2 2 C 2 1 1 1 1 1 1 d d d d d sin cos ² d x x f x f x x x z x f x x z x f f x x x z x x L x x f x x x ϑ ϑ + - + - = = -     = = + + -       ∫ ∫ ∫ ∫ ∫ ∫ A A A A . (A.3)
The corresponding vertical coordinates of C are

( ) ( ) ( ) sup sup inf inf Q Q flotation i P i P i 1 2 2 1 O 2 2 2 1 O O 2 1 2 C 2 2 2 1 C 1 1 1 1 1 1 d d ² ² d d d s sin co d x x f x f x x x z x f x z z x f f x z z x z L L z z f f x z z ϑ ϑ + - + - = = -         = -           = + - + ∫ ∫ ∫ ∫ ∫ ∫ A A A A .(A.4)
The expressions of the second derivatives of the coordinates of the centre of buoyancy used in Section VI are as follows: In accordance with eq. (V.3), the depth of C below the line of flotation is

L x f f x x L z f f x x ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ
4 3 3 flotation C sin 2 co 2 s sin d R α α α α     = -     -   . (A.12)
Both distances and the hydrostatic energy

flotation h f C E m g d = A
are independent of ϑ . As a consequence, the lever arm of the buoyancy force vanishes.

The floating equilibrium is stable if G is located below the centre of the circle (the axis of the cylinder). x l = . The buoyancy and the metacentric curves as functions of the angle of heel are, cf. eqs.

(A.3), (A.4), (VI.4), ( ) ( )

1 1 1 C C 12 2 24 ² ² tan ; tan ² l l x z b b b ϑ ϑ ϑ ϑ = = -+ (A.13) and ( ) ( ) 3 1 1 1 M M 12 2 24 ² ² 3 tan ; 1 cos ² l l x z b b b ϑ ϑ ϑ ϑ     = - = -+ -      . (A.14)
Both curves are illustrated in Figure 10. The metacentric curve is "V"-shaped, and for the metacentric distance we get The lever arm of the buoyancy force, in accordance with eq. (V.17), is

2 1 1 2 1 2 2 2 h cos 1 1 1 sin l b a b ϑ ϑ         - +          =  . (A.17)
The angular dependence of a h is shown in Figure 11, left. For larger bodies E h is minimal at equilibrium, 0 ϑ = and increases with the angle of heel, cf. eq. (A.16); the sign of a h is opposite to that of ϑ , corresponding to a restoring moment. For narrower body an opposite behaviour is observed: at equilibrium E h is maximal, so that a h and ϑ have the same sign, corresponding to an overturning moment. In the vicinity of equilibrium the change in the sign of a h takes place at a critical value As a difference from the rectangular box, for the parabolic cylinder the lever arm of the buoyancy force, cf. eq. (V.17), for 0 ϑ > is always positive, resulting in an overturning moment Consider an elliptic cylinder, see Figure 15, of cross-section described by where a and b are, respectively, the semi-major and semi-minor axes. As in the case of the circular cylinder, the origin of coordinates is chosen at the centre of the transversal cross-section. To simplify the formulae, we limit ourselves to the case of half-submerged cylinder, so that the line of flotation always passes through its centre and the area of its immersed half is ).

In a position inclined through an angle ϑ the line of flotation meets the ellipse at the points The buoyancy and metacentric curves are shown in Figure 16 for rolling about two equilibrium states, respectively, with the major axis parallel and perpendicular to the line of flotation. In the first case the metacentric curve is "Λ"-shaped while in the second case it is "V"-shaped. From eqs. (A.34), (A.35) the metacentric distance is, cf. eq. (VI.5),

2 )Figure 1 .

 21 Figure 1. Two different representations of the coordinate systems defined in the text. Left: The floating body turned anticlockwise with respect to the horizontal line of flotation. Right: The line of flotation turned clockwise with respect with the vertical floating body. The profile of the immersed cross-section is described by the function

Figure 1 .

 1 Figure 1. The main axes of the first one, for brevity referred to as the Earth frame, are denoted by majuscules and chosen as follows: the X and Y axes are respectively directed along the width and the length of the body and the Z axis is vertical ascending; the corresponding unit vectors are denoted by , , X Y Z u u u . The origin of coordinates is chosen at the instantaneous position of the centre of flotation O, and the horizontal XOY plane coincides with the plane of flotation. The second coordinate system, referred to as the body frame, is rigidly related to the body. Its axes are denoted by minuscules, with the y axis along the longitudinal axis of the body and the x and z axes turned through an angle ϑ (angle of heel) with respect to the X and Z axes of the Earth frame; the corresponding unit vectors are denoted by , , x y z u u u . The origin of coordinates is chosen at the location of O at equilibrium, therefore, at equilibrium both frames coincide and for an arbitrary angle of heel the relation between the respective coordinates is

hM

  and containing Z A . In eq. (III.8) Z A is eliminated because of the properties of the vector product A h ∧ r F , so, in a somewhat intuitive way, one may suggest an equation satisfied by the corresponding scalar product A h ⋅ r F . It can be obtained considering the equation of rotation of a solid: h = M Θ

Figure 2 .

 2 Figure 2. Example of an immersed cross-section, (i) and (ii) representing fully and partially immersed pieces.

Figure 3 .

 3 Figure 3. Immersed cross-section delimited by the single-valued function

. 4 )

 4 When the piece is elevated through d 0 b > , the line of flotation moves down to d is the length of the line of flotation for 0 ϑ = .The corresponding change in E h in linear approximation is, cf. eq. (II.5),

. 6 )

 6 From eqs. (IV.3) and (IV.6) one gets:

. 10 )

 10 Obviously, only for a fully immersed body, when 0 L = , one gets C d d z b = . In other cases the relation between the displacements of the body and of its centre of buoyancy can be very different. displacement of C is only half of that of G (and of the point of application of the buoyancy force).

Figure 4 .

 4 Figure 4. Square immersed to a depth b. b ∆ and

Figure 5 .

 5 Figure 5. Cross-section of a bulb-shape keel. The profile of the keel and the bulb are described, respectively, by ( )1 4 19 20 2 1 x   -    and

Figure 6 .

 6 Figure 6. Two immersed cross-sections turned through an angle ϑ . Left: that of Figure3; right: the one delimited by the functions

  made use of the fact that, according to eq. (V.6centre of rotation (pivot point) of the line of flotation is the intersection of the initial and the new lines of flotation, therefore it verifies the system of equations (V.V.11), (V.12) provide a simple but quite general proof of the Euler's theorem, see Section III. Note that Euler himself had proved this theorem only in the vicinity of equilibrium.[START_REF] Euler | Scientia Navalis seu Tractatus de Construendis ac Dirigendis Navibus[END_REF] Taking the derivatives of eqs. (A.3), (A.4) with respect to ϑ and transforming the results by means of eqs. (V.

[ 8 (

 8 pp. 45-46)], [21 (pp. 81-82)].

  where I is the moment of inertia of the line of flotation with respect to O,

10 )

 10 Generalizing eqs. (VI.10) to the three-dimensional case, one should perform the integration along the longitudinal axis of the body to get the expression of the global rolling metacentric distance as

  ) and (V.23) it is seen that the metacentric height (vertical distance from the line of flotation to the metacentre) is directly proportional to the angular derivative of the lever arm of the buoyancy force: character (restoring or overturning) of the rotational moment of the buoyancy force depends on the sign (resp. positive or negative) of M flotation d, and the absolute value of the latter determines the rate of angular dependence of the former.

Figure 8 .

 8 Figure 8. Buoyancy and flotation curves for a floating circle with R = 100 and 60 α = °. The diamonds and circles indicate, respectively, the locations of C and O for 0 , 30 , 60 ϑ = °°° and 90°. The metacentric curve is reduced to a point (triangle).

Figure 9 .

 9 Figure 9. Floating square of side l. b is the immersion depth at equilibrium. Consider a floating body of rectangular cross-section (a box, see Figure 9) and denote l its width and b the immersion depth at equilibrium. The origin of coordinates is chosen at O. For definiteness we consider the case where two angles of the rectangle are immersed. Obviously, bl = A , 1 P 2x l = -

Figure 10 .

 10 Figure 10. Buoyancy and metacentric curves for a floating square with l = 200 and b = 100. The diamonds and triangles indicate, respectively, the locations of C and M for 0 ,15 , 30 ϑ = °°° and 45°. The flotation curve is reduced to a point (circle).

Figure 11 .Figure 11 ,Figure 12 .Figure 14 .

 11111214 Figure 11. Left: lever arm of the buoyancy force for a floating rectangle of width l and immersion depth b vs. the angle of heel. Right: metacentric height. The ϑ -values are limited by the requirement that of two angles of the rectangle be immersed. For 1 l b = one always has an overturning moment, for 2.25 l b = the moment is an overturning one for heel angles less than 31 ϑ ≈ ° and becomes a restoring one at larger ϑ . For 3 l b = one always gets a restoring moment.

12 ,Figure 15 .

 1215 Figure 15. Floating ellipse of semi-axes a and b.

  . (VI.4) one calculates the location of the metacentre:

l → . In this case, M coincides with C:

force. While this finding is not expected to bring about changes in practical applications, it has a certain fundamental and educational interest for the mechanics of floating bodies. It would be interesting to find out whether some other physical forces do possess a similar non-uniqueness of the point of application.

Using the general approach based on the hydrostatic energy formalism, we have shown that the various definitions of the metacentre ("geometrical", "Bouguer's" and "dynamical"), in fact, concern one and the same distinct point of the immersed body. This finding holds (i) for any shape of the immersed body and (ii) not only in the vicinity of equilibrium but also for any angle of heel.

Besides, from the viewpoint of the rotational moment the metacentre proves to be the point of application of the buoyancy force in the rolling/pitching motion. These findings shed new light on the long-standing concept of the metacentre.

Another, more practical, implication of this study concerns the criterion of ship stability in relation to the location of the metacentre. Indeed, the metacentric height is proportional to the angular derivative of the lever arm of the buoyancy force. Thus, the character (restoring or overturning) of the rotational moment of the buoyancy force depends on the sign (resp., positive or negative) of the height of the metacentre above the line of flotation, and the absolute value of this height determines the rate of its angular dependence.

The model developed in the present study allows one to get analytical expressions for the location of the metacentre for the floating body of an arbitrary shape. Thus, it presents a certain interest for teaching and practice of naval mechanics and engineering, as well.

Appendix I Appendix I Appendix I Appendix I: : : : Some Some Some Some expressions used in the main t expressions used in the main t expressions used in the main t expressions used in the main text ext ext ext

For the areas of fully and partially immersed pieces of the cross section of a floating body, respectively, type (i) and (ii), introduced in Section III, we get:

Appendix Appendix Appendix

Appendix II II II II: : : : Some special cases of immersed bodies Some special cases of immersed bodies Some special cases of immersed bodies Some special cases of immersed bodies

Here we consider the buoyancy, flotation and metacentric curves for several simple shapes of floating bodies. All subsequent results are readily obtained as particular cases of the general formulae derived above. The immersed area is ( )

and the abscissas of the endpoints of the line of flotation are

In the body frame both O and C describe circular arcs of equations Consider a floating cylinder of cross section described by the parabola (note that x C and x O coincide). The metacentric curve is given by, cf. eqs. (VI.4),

as in the case of floating box, it is "V"-shaped. The metacentric distance is, cf. eqs. (VI.5),

(A.36) The lever arm of the buoyancy force, in accordance with eq. (V.17), is ( ) Figure 17 shows the angular dependence of the locations of C and M with respect to the line of flotation as well as the lever arm of the buoyancy force with respect to O. At 0, 180 ϑ = ° the major axis is horizontal; the depth of C and the hydrostatic energy are minimal; the sign of a h is opposite to that of the deviation from equilibrium, corresponding to a restoring moment. At 90 ϑ = ° the major axis is vertical; the depth of C and the hydrostatic energy are maximal; the signs of a h and of the deviation from equilibrium, 90 ϑ -° are the same, corresponding to an overturning moment. However, stable equilibrium corresponds to the minimum of the total potential energy and not to that of the hydrostatic energy only. For 0 ϑ = the metacentric height is maximal and in the case shown in Figure 17 equilibrium remains stable for G located up to ca. 127 units above the line of flotation. On the other hand, for 90 ϑ = ± ° the metacentric height is minimal and in order to preserve stable equilibrium, G should be located deeper than ca. 63.7 units below the line of flotation.