Analysis of the Hydrostatic Parameters of a Floating Body with Cylindrical Stabilising Elements

1. Introduction

Historically, the extraction of mineral resources in the form of sands and gravels from below the water surface [1,2] was originally called water dredging [3] and included a complex of works involving disconnecting, transporting and storing the extracted mineral. In the present day, the operations specified above are carried out by surface mining. The desire to reduce the economic costs associated with transporting the dredged material to the shore of the mining lake has led to the total automation of the dredging process. Automation thus brings with it a range of technical equipment that enables the continuous routing of the mining material from the floating mining equipment to the stationary sorting plants installed on the shore of the mining area. Therefore, in inland navigation, in addition to basket, suction and dredge floating dredgers, technical vessels such as floating gravel treatment plants, bucket dredgers, floating cranes and last but not least floating conveyor belts are also used [4‒7], see Figure 1(a).

In the design of technical vessels [8], it is necessary to achieve the required design and dimensional parameters in order to ensure that these devices are sufficiently stable and provide seaworthiness throughout their area of operation for a particular navigation area. Of the wide range of technical navigation requirements imposed on a given floating device, it is essential to ensure buoyancy [9,10] and sufficient stability [11]. Buoyancy [12,13] is defined by the ability of a vessel to remain in a state of equilibrium when placed on a water surface. Buoyancy [14] is defined as the ability of a body to float on a still liquid surface by the action of hydrostatic buoyancy [11]. Buoyancy is one of the basic properties of a floating body, characterized by the ability of the body to remain in an equilibrium state when immersed in a liquid. The buoyancy reserve is defined as the mass amount of load (which the floating body is capable of carrying) that will cause the total immersion of the floating body in the liquid [15].

Stability [16,17] is generally defined as the ability of a vessel to return to its initial equilibrium position after a deflection, if the external forces of a static or dynamic nature cease to act on it [18]. The relative position of the centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$ of the floating body of gravity ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}\left[\mathrm{N}\right]$, called the metacentre $\mathrm{“}\mathrm{M}\mathrm{”}$, see Figure 1(b) (intersection of the axis $\mathrm{“}\mathrm{o}\mathrm{”}$ of the floating body and a vertical line $\mathrm{“}\mathrm{V}\mathrm{M}\mathrm{”}$ passing through the centre of gravity $\mathrm{“}\mathrm{V}\mathrm{”}$of the displaced fluid of gravity ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ in an inclined position by an angle $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$)has a fundamental influence on the stability of the floating body ). Stability defines the equilibrium positions of floating bodies: a) stable − metacentre $\mathrm{“}\mathrm{M}\mathrm{”}$ lying above the centre of gravity, a pair of forces is generated that try to return the floating body to a vertical position; b) unstable − metacentre $\mathrm{“}\mathrm{M}\mathrm{”}$ lies below the centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$, the deflection causes the body to overturn; and c) indifferent − centre of gravity and metacentre $\mathrm{“}\mathrm{M}\mathrm{”}$ merge (e.g. homogeneous cylinder), the body remains in any position).

Stability [19‒21] falls inherently into a broader group of navigation properties of floating bodies. Stability of a floating body [22,23] is defined as the ability of a floating body to return to its equilibrium position after deflection if the external forces causing the deflection cease to act on it.

According to the direction of deflection of the floating body, two variants of stability are recognized, “transverse” stability [3,24,25] and “longitudinal” stability.

Depending on the amount of deflection of the floating body, we distinguish between “initial” stability and “high angle of heel” stability.

According to the time effect of external forces and the influence of inertia of masses, it is possible to define “static” stability (which is defined by the magnitude of the return moment when the body is deflected from the equilibrium position) and “dynamic” stability (which is determined by the magnitude of the work that the floating body is able to absorb when deflected from the equilibrium position).

The behaviour of a body in a fluid is determined by the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ and the gravitational force ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}\left[\mathrm{N}\right]$, which is described by Archimedes’ law. According to the correlation of these forces (or according to a comparison of the density of the body ${\mathsf{\rho }}_{\mathrm{T}}\left[\mathrm{k}\mathrm{g}·{\mathrm{m}}^{-3}\right]$ and the density of the fluid $\mathsf{\rho }\left[\mathrm{k}\mathrm{g}·{\mathrm{m}}^{-3}\right]$), three basic situations can occur: a) the body sinks (force condition ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}>{\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$, density condition ${\mathsf{\rho }}_{\mathrm{T}}>\mathsf{\rho }\left[\mathrm{k}\mathrm{g}·{\mathrm{m}}^{-3}\right]$); b) the body floats (force condition ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}={\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$, density condition ${\mathsf{\rho }}_{\mathrm{T}}=\mathsf{\rho }\left[\mathrm{k}\mathrm{g}·{\mathrm{m}}^{-3}\right]$); c) the body floats on the surface (force condition ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}<{\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$, density condition ${\mathsf{\rho }}_{\mathrm{T}}<\mathsf{\rho }\left[\mathrm{k}\mathrm{g}·{\mathrm{m}}^{-3}\right]$).

The floating of a body on the surface is understood as a process when the body is partially immersed in the liquid and partially protrudes above the free surface of the liquid [26]. An important factor for bodies floating on the surface is that a small deflection (by an angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$) of the body from its equilibrium position produces a moment of force ${\mathrm{M}}_{\mathrm{F}}\left[\mathrm{N}·\mathrm{m}\right]$, which returns the deflected body to its equilibrium position.

A body floats on the surface when the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ exerted at the centre of gravity $\mathrm{“}\mathrm{V}\mathrm{”}$ on a fully immersed body is greater than its gravity ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}\left[\mathrm{N}\right]$, exerted at the centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$. In this case, the body emerges from the liquid sufficiently so that the buoyancy ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ acting on its immersed part just equals the gravity of the body ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}\left[\mathrm{N}\right]$, i.e. so that the resultant of the forces acting on the body is zero. For equilibrium surface floating to occur, it is necessary for the resulting moment of the forces ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}·{\mathrm{s}}_{\mathrm{a}}={\mathrm{F}}_{\mathrm{B}}·{\mathrm{s}}_{\mathrm{a}}=0\mathrm{N}$ acting on the body to be zero [10,27,28].

Buoyancy and gravity must act in one vertical line, which is called the “axis of buoyancy”. Thus, both the centre of mass (centre of gravity) $\mathrm{“}\mathrm{T}\mathrm{”}$ of the body and the centre of buoyancy, which is the centre of mass (centre of gravity) $\mathrm{“}\mathrm{V}\mathrm{”}$ of the displaced fluid, must lie on the axis of buoyancy. If the floating body is homogeneous, then every geometric axis of symmetry of the body can theoretically be an axis of floating. However, true surface floating occurs only for the axis with respect to which surface floating is “stable” or “indifferent” [27,28]. The surface floating of a body is stable [10,29], when at any deflection of the floating body from its equilibrium position the pair of forces consisting of buoyancy and gravity of the body tries to return the body to its equilibrium position. The surface floating of a body (e.g. a homogeneous sphere or a homogeneous cylinder for rotation about the axis of the cylinder) is indifferent [11,30] when no moment of force is produced when the body is deflected from its equilibrium position.

The stability of surface floating can be investigated if we find the so-called “surface of gravities” in the body, on which lies the effect of buoyancy $\mathrm{“}\mathrm{V}\mathrm{”}$ for various rotations $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ of the floating body [28]. The surface centre of gravity $\mathrm{“}\mathrm{V}\mathrm{”}$ is the centre of gravity of the fluid displaced by the body, see Figure 1(b). Figure 1(b) shows the cross-section of a cube, running through its geometric (and, due to the assumed homogeneity, its material) centre $\mathrm{“}\mathrm{T}\mathrm{”}$ perpendicular to its two edges. The intersections $\mathrm{“}\mathrm{p}\mathrm{”}$ with the surface centres of gravity are marked on the cross-sections (sections running thus are called the main sections). When the cube is deflected from the equilibrium position, the force pair of the body’s gravity ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}\left[\mathrm{N}\right]$ (acting in the centre of mass $\mathrm{“}\mathrm{T}\mathrm{”}$) and buoyancy ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ leads to the resumption of the surface floating. If floating on the narrower side, the same force pair overturns the block.

The term “metacentre” is introduced to capture the degree of stability [27,28]. Metacentres are usually defined as the centres of curvature of the main normal sections of the centre of gravity surface where the $\mathrm{“}\mathrm{o}\mathrm{”}$ axis of float intersects it. Figure 1(b) shows the metacentres $\mathrm{“}\mathrm{M}\mathrm{”}$ for the illustrated main sections of the block. The metacentre $\mathrm{“}\mathrm{M}\mathrm{”}$ in this case falls at the centre of mass (centre of gravity) of the body $\mathrm{“}\mathrm{T}\mathrm{”}$.

These specific designs of floating bodies can significantly affect the buoyancy distribution, the position of the centre of gravity and the overall stability behaviour of the system, but available literature only marginally addresses this issue. Insufficient knowledge of these influences poses a risk in the design and operation of the equipment, especially in terms of the possibility of loss of stability under variable operating conditions. Detailed research on this issue is therefore necessary to gain a deeper understanding of the behaviour of these systems, to refine design methods and to improve the safety and reliability of floating conveyors.

The issue of stability of floating belt conveyors has not yet been fully investigated systematically, especially in the case of floating body designs using floats with circular cross-sections [27]. These specific designs of floating bodies can significantly affect the buoyancy distribution, the position of the centre of gravity and the overall stability behaviour of the system, but the available literature only marginally addresses this area. Insufficient knowledge of these influences poses a risk in the design and operation of the equipment, especially in terms of the possibility of loss of stability under variable operating conditions. Detailed research on this issue is therefore necessary to gain a deeper understanding of the behaviour of these systems, to refine design methods and to improve the safety and reliability of floating conveyors.

The current state of knowledge in the field of stability of floating bodies has been analysed on the basis of available literature. It was found that the vast majority of the publications searched focus mainly on the issue of buoyancy and stability of ships [8,15], while the specific area of floating conveyor belts remains virtually uncovered in professional sources. This disparity points to an existing gap in research and highlights the need for a focused study of the stability of these devices, whose design and operating conditions differ from conventional vessels in many respects.

2. Materials and Methods

The detailed description of the procedures and methods used in this section is intended to ensure the transparency of the research, to enable reproducibility of the results and to support their use in subsequent studies or practical applications.

The analytical determination of the transverse stability of a floating body consisting of floats with a circular cross-section is, from a practical point of view, quite complex, time-consuming and requires knowledge of integral calculus. A more appropriate way to determine the coordinates of the centre of gravity of the buoyant force is to use a 3D CAD modeler, such as SolidWorks, Autodesk Inventor or Fusion 360.

Generative Artificial Intelligence (GenAI) was not used to generate text, data or graphics in the drafting of this paper.

In this section, subsection 2.1 outlines the methodology for determining the coordinates $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the centre of gravity of the $\mathrm{“}\mathrm{V}\mathrm{”}$ buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ displacement and the stability arm ${\mathrm{s}}_{\mathrm{a}}\left[\mathrm{m}\right]$ for three different positions ${\mathrm{h}}_{\mathrm{G}}\left[\mathrm{m}\right]$ of the centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$ of the floating body ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}\left[\mathrm{N}\right]$, based on a 3D model of the floating body created in the SolidWorks software environment.

The total gravity ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}$ [N] acting on the floating body is assumed. The gravity ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}$ [N] acts at the centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$, which is ${\mathrm{h}}_{\mathrm{G}}$ [m] away from the bottom plane of the floating body, see Figure 2. In the equilibrium position ($\mathsf{\varphi }=0\mathrm{d}\mathrm{e}\mathrm{g}$), the horizontal axes of the two floats with a circular cross-section are parallel to the water surface. The buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$, acting at the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$, must be in equilibrium with the gravity ${\mathrm{G}}_{\mathrm{F}\mathrm{B}}\left[\mathrm{N}\right]$ (1).

$${\mathrm{F}}_{\mathrm{B}}={\mathrm{G}}_{\mathrm{F}\mathrm{B}}\left[\mathrm{N}\right]\Rightarrow \mathsf{\rho }·\mathrm{g}·{\mathrm{V}}_{\mathrm{p}}=\mathsf{\rho }·\mathrm{a}·\mathrm{g}·\left({\mathrm{S}}_1+{\mathrm{S}}_2\right)={\mathrm{G}}_{\mathrm{F}\mathrm{B}}\left[\mathrm{N}\right]$$

(1)

where $\mathsf{\rho }\left[\mathrm{k}\mathrm{g}·{\mathrm{m}}^{-3}\right]$ is the density of water; ${\mathrm{V}}_{\mathrm{p}}\left[{\mathrm{m}}^3\right]$ is the total volume of the immersed parts of both floats; ${\mathrm{S}}_1\left[{\mathrm{m}}^2\right]$ is the footprint of the immersed part of the left float; ${\mathrm{S}}_2\left[{\mathrm{m}}^2\right]$ is the footprint of the immersed part of the right float; ${\mathrm{h}}_{\mathrm{p}}\left[\mathrm{m}\right]$ is the depth of immersion of the floats in the equilibrium position; $\mathrm{D}\left[\mathrm{m}\right]$ is the diameter of the float with a circular cross-section; $\mathrm{a}\left[\mathrm{m}\right]$ is the length of the float.

2.1. Coordinates of the Centre of Gravity of Displacement Determined from the 3D Model Created in SOLIDWORKS

In the following text, the coordinates $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the centre of gravity $\mathrm{“}\mathrm{V}\mathrm{”}$ of the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ and the stability arm ${\mathrm{s}}_{\mathrm{a}}\left[\mathrm{m}\right]$ are given for three possible variants of immersion of the floating body ${\mathrm{h}}_{\mathrm{p}}\left[\mathrm{m}\right]$ and for three different distances ${\mathrm{h}}_{\mathrm{G}}\left[\mathrm{m}\right]$ of the centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$, when the floating body is deflected from the equilibrium state by an angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$. The coordinates of the centre of gravity of the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ are read in SolidWorks software [31] (version Premium 2012×64, edition SP5.0) in the “ANALYSIS-PHYSICAL PROPERTIES menu.

2.1.1. Transverse Heel of the Floating Body When the Floats Are Immersed ${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$

Phase 1 starts at the equilibrium position of the floating body $\mathsf{\varphi }={\mathsf{\varphi }}_0=0\mathrm{d}\mathrm{e}\mathrm{g}$, see Figure 3(a) and ends with the right side of the cylindrical float immersed below the surface $\mathsf{\varphi }={\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ (2), see Figure 3(b).

$$\mathrm{t}\mathrm{a}\mathrm{n}\left({\mathsf{\varphi }}_1\right)={\displaystyle \frac{\mathrm{D}/2}{\mathrm{a}/2}}\Rightarrow {\mathsf{\varphi }}_1=\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{\mathrm{D}}{\mathrm{a}}}\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$$

(2)

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m};\mathrm{m}\right]$ of the centre of gravity $“\mathrm{V}\mathrm{”}$ of the buoyant force (displacement) ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ given in Table 1 are read from the 3D model of the floating body created in SolidWorks (version Premium 2012×64, edition SP5.0) [31] for the angle of deflection of the floating body from the equilibrium state $\mathsf{\varphi }\in \left(0÷{\mathsf{\varphi }}_1\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

Figure 4 presents the size of the stability arm ${\mathrm{s}}_{\mathrm{a}}\left[\mathrm{m}\right]$ [32], when the floating body is deflected from the equilibrium state by an angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, for an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=795\mathrm{m}\mathrm{m}$, for different distances ${\mathrm{h}}_{\mathrm{G}}$ [m] of the centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$ from the bottom plane of the floating body.

Phase 2 begins when the floating body is deflected from its equilibrium position $\mathsf{\varphi }={\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, see Figure 5(a), and ends with the loss of buoyancy $\mathsf{\varphi }={\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

${\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is the angle of heel of the floating body at which the stability arm size ${\mathrm{s}}_{\mathrm{a}\mathrm{n}\mathrm{d}}\left[\mathrm{m}\right]$ becomes zero $\left({\mathrm{s}}_{\mathrm{a}\mathrm{n}\mathrm{d}}=0\mathrm{m}\right)$, and the stability of the floating body is lost [9,28]. When the floating body deflects by an angle of ${\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ the reciprocating moment [27] that would return the floating body to its equilibrium position ceases to act ($\mathsf{\varphi }={\mathsf{\varphi }}_0=0\mathrm{d}\mathrm{e}\mathrm{g}$, see Figure 3a), and the floating body tips over [33].

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m};\mathrm{m}\right]$ of the centre of gravity $“\mathrm{V}\mathrm{”}$ of the buoyant force (displacement) ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ given in Table 2 are read from the 3D model of the floating body created in SolidWorks (version Premium 2012×64, edition SP5.0) for the angle of deflection of the floating body from the equilibrium state $\mathsf{\varphi }\in \left({\mathsf{\varphi }}_1÷{\mathsf{\varphi }}_{\mathrm{k}}\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

2.1.2. Transverse Heel of the Floating Body When the Floats Are Immersed ${\mathrm{h}}_{\mathrm{p}}<{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$

Phase 1 starts at the equilibrium position of the floating body $\mathsf{\varphi }={\mathsf{\varphi }}_0=0\mathrm{d}\mathrm{e}\mathrm{g}$, see Figure 6(a) and ends with the left side of the cylindrical float immersed below the surface $\mathsf{\varphi }={\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ (3), see Figure 6(b).

$$\mathrm{t}\mathrm{a}\mathrm{n}\left({\mathsf{\varphi }}_1\right)={\displaystyle \frac{{\mathrm{h}}_{\mathrm{p}}}{\frac{\mathrm{a}}{2}}}={\displaystyle \frac{\frac{\mathrm{D}}{2}-\mathrm{s}}{\frac{\mathrm{a}}{2}}}={\displaystyle \frac{\mathrm{D}-2·\mathrm{s}}{\mathrm{a}}}\Rightarrow {\mathsf{\varphi }}_1=\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{\mathrm{D}-2·\mathrm{s}}{\mathrm{a}}}\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$$

(3)

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m};\mathrm{m}\right]$ of the centre of gravity $“\mathrm{V}\mathrm{”}$ of the buoyant force (displacement) ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ given in Table 3 are read from the 3D model of the floating body created in SolidWorks (version Premium 2012×64, edition SP5.0) for the angle of deflection of the floating body from the equilibrium state $\mathsf{\varphi }\in \left(0÷{\mathsf{\varphi }}_1\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

Figure 7 presents the size of the stability arm ${\mathrm{s}}_{\mathrm{a}}\left[\mathrm{m}\right]$ (4) when the floating body is deflected from its equilibrium state by an angle of $\mathsf{\varphi }\mathrm{ }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, for various distances of the ${\mathrm{h}}_{\mathrm{G}}\left[\mathrm{m}\right]$ centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$, for the depth of immersion of the floating body of ${\mathrm{h}}_{\mathrm{p}}=0.5\mathrm{m}$ in the equilibrium state, see Figure 6(a).

$${\mathrm{s}}_{\mathrm{a}}={\mathrm{x}}_{\mathrm{T}}·\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathsf{\varphi }\right)+{\mathrm{y}}_{\mathrm{T}}·\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathsf{\varphi }\right)-{\mathrm{h}}_{\mathrm{G}}·\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$$

(4)

Phase 2 begins when the floating body is deflected from its equilibrium position $\mathsf{\varphi }={\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ and ends and ends with the right side of the cylindrical float submerging below the surface $\mathsf{\varphi }={\mathsf{\varphi }}_2\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ (5), see Figure 8(a).

$$\mathrm{t}\mathrm{a}\mathrm{n}\left({\mathsf{\varphi }}_2\right)={\displaystyle \frac{{\mathrm{D}-\mathrm{h}}_{\mathrm{p}}}{\frac{\mathrm{a}}{2}}}={\displaystyle \frac{\mathrm{D}-\left(\frac{\mathrm{D}}{2}-\mathrm{s}\right)}{\frac{\mathrm{a}}{2}}}\Rightarrow {\mathsf{\varphi }}_2=\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{\mathrm{D}+2·\mathrm{s}}{\mathrm{a}}}\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$$

(5)

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m};\mathrm{m}\right]$ of the centre of gravity $“\mathrm{V}\mathrm{”}$ of the buoyant force (displacement) ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ given in Table 4 are read from the 3D model of the floating body created in SolidWorks (version Premium 2012×64, edition SP5.0) [11] for the angle of deflection of the floating body from the equilibrium state $\mathsf{\varphi }\in \left({\mathsf{\varphi }}_1÷{\mathsf{\varphi }}_2\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

Phase 3 begins when the floating body is deflected from its equilibrium position $\mathsf{\varphi }={\mathsf{\varphi }}_2\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ and ends with the loss of buoyancy $\mathsf{\varphi }={\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m};\mathrm{m}\right]$ of the centre of gravity $“\mathrm{V}\mathrm{”}$ of the buoyant force (displacement) ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ given in Table 5 are read from the 3D model of the floating body created in SolidWorks (version Premium 2012×64, edition SP5.0) for the angle of deflection of the floating body from the equilibrium state $\mathsf{\varphi }\in \left({\mathsf{\varphi }}_2÷{\mathsf{\varphi }}_k\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

2.1.3. Cross Heel of the Floating Body when The Floats Are Immersed ${\mathrm{h}}_{\mathrm{p}}>{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$

Phase 1 starts in the equilibrium position $\mathsf{\varphi }={\mathsf{\varphi }}_0=0$ deg, see Figure 8(b), and ends with the upper right part of the cylindrical float immersed below the water surface $\mathsf{\varphi }={\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ (6), see Figure 9(a).

$$\mathrm{t}\mathrm{a}\mathrm{n}\left({\mathsf{\varphi }}_1\right)={\displaystyle \frac{\mathrm{D}-\left(\frac{\mathrm{D}}{2}+\mathrm{s}\right)}{\frac{\mathrm{a}}{2}}}={\displaystyle \frac{\frac{\mathrm{D}}{2}-\mathrm{s}}{\frac{\mathrm{a}}{2}}}\Rightarrow {\mathsf{\varphi }}_1=\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{\mathrm{D}-2·\mathrm{s}}{\mathrm{a}}}\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$$

(6)

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m};\mathrm{m}\right]$ of the centre of gravity $“\mathrm{V}\mathrm{”}$ of the buoyant force (displacement) ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ given in Table 6 are read from the 3D model of the floating body created in SolidWorks (version Premium 2012×64, edition SP5.0) [11] for the angle of deflection of the floating body from the equilibrium state $\mathsf{\varphi }\in \left(0÷{\mathsf{\varphi }}_1\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

Figure 10 presents the size of the stability arm ${\mathrm{s}}_{\mathrm{a}}\left[\mathrm{m}\right]$ when the floating body is deflected from its equilibrium state by an angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, for distances of the ${\mathrm{h}}_{\mathrm{G}}=1600\mathrm{m}\mathrm{m}$ centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$, for the depth of immersion of the floating body of ${\mathrm{h}}_{\mathrm{p}}=1.09\mathrm{m}$ in the equilibrium state, see Figure 9(a).

Phase 2 begins with the rising of the left cylindrical float above the surface $\mathsf{\varphi }={\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ (6), see Figure 9(a), and ends with the immersion of the right cylindrical float below the water surface $\mathsf{\varphi }={\mathsf{\varphi }}_2\mathrm{ }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ (7) (${\mathsf{\varphi }}_2=28.59\mathrm{ }\mathrm{d}\mathrm{e}\mathrm{g}$), see Figure 9(b).

$$\mathrm{t}\mathrm{a}\mathrm{n}\left({\mathsf{\varphi }}_2\right)={\displaystyle \frac{\frac{\mathrm{D}}{2}+\mathrm{s}}{\frac{\mathrm{a}}{2}}}={\displaystyle \frac{{\mathrm{h}}_{\mathrm{p}}}{\frac{\mathrm{a}}{2}}}\Rightarrow {\mathsf{\varphi }}_2=\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{2·{\mathrm{h}}_{\mathrm{p}}}{\mathrm{a}}}\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$$

(7)

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m};\mathrm{m}\right]$ of the centre of gravity $“\mathrm{V}\mathrm{”}$ of the buoyant force (displacement) ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ given in Table 7 are read from the 3D model of the floating body created in SolidWorks (version Premium 2012×64, edition SP5.0) [11] for the angle of deflection of the floating body from the equilibrium state $\mathsf{\varphi }\in \left({\mathsf{\varphi }}_1÷{\mathsf{\varphi }}_2\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

Phase 3 begins when the floating body is deflected from its equilibrium position $\mathsf{\varphi }={\mathsf{\varphi }}_2\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, see Figure 9(b), and ends with the loss of buoyancy $\mathsf{\varphi }={\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m};\mathrm{m}\right]$ of the centre of gravity $“\mathrm{V}\mathrm{”}$ of the buoyant force (displacement) ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ given in Table 8 are read from the 3D model of the floating body created in SolidWorks (version Premium 2012×64, edition SP5.0) for the angle of deflection of the floating body from the equilibrium state $\mathsf{\varphi }\in \left({\mathsf{\varphi }}_2÷{\mathsf{\varphi }}_k\right)\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

Figure 11 presents the size of the stability arm ${\mathrm{s}}_{\mathrm{a}}\left[\mathrm{m}\right]$ [32], when the floating body is deflected from the equilibrium state by an angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, for an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=500;795;1090\mathrm{m}\mathrm{m}$, for distances ${\mathrm{h}}_{\mathrm{G}}$ = 1600 mm of the centre of gravity $\mathrm{“}\mathrm{T}\mathrm{”}$ from the bottom plane of the floating body.

In the following text, subsection 2.2 gives the analytically calculated coordinates of the centre of gravity of the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ for three possible variants of the buoyancy of the floating body ${\mathrm{h}}_{\mathrm{p}}\left[\mathrm{m}\right]$, when the floating body is deflected from the equilibrium state by an angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

2.2. Analytical Determination of the Coordinates of the Centre of Gravity of a Floating Body

The floating body, a floating conveyor belt, consists of two floats on which the actual technology for transporting the mining material is placed. The float consists of a cylindrical body whose geometry is determined, from a lateral view, see Figure 12, by the radius of the floats with a circular cross-section $\mathrm{R}={\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$and the length $\mathrm{a}\left[\mathrm{m}\right]$ of the cylindrical float. The depth of immersion (called draft) of a floating body in the equilibrium state ($\mathsf{\varphi }=0\mathrm{d}\mathrm{e}\mathrm{g}$) is defined by the expression ${\mathrm{h}}_{\mathrm{p}}\left[\mathrm{m}\right]$ in the coordinate system of Figure 2(a), or by the expression $\mathrm{s}\left[\mathrm{m}\right]$ in Figure 12. $\mathrm{s}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ is the vertical distance of the left front surface of the cylindrical float of the floating body of the water surface plane from the longitudinal axis. The origin of the $“0”$ coordinate system is chosen at the centre of the height of the left front wall (circular surface in plan view) of the floating body and is oriented according to Figure 12.

The following text defines the procedure for calculating the position of the centre of gravity of displacement $“\mathrm{V}\mathrm{”}$ of a cylindrical float, when it is deflected from the equilibrium position by an angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, and the values of the coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{V}};{\mathrm{y}}_{\mathrm{V}}\right\}[\mathrm{m};\mathrm{m}]$ of displacement $“\mathrm{V}\mathrm{”}$ for the defined values of the heel angle $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ of the floating body.

The solution to calculating the position of the centre of gravity of the displacement $“\mathrm{V}\mathrm{”}$ of a floating body consisting of cylindrical floats can be divided into three basic phases. These three phases result from the existence of a limiting angle ${\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, see Figure 12.

The angle ${\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is subject to a relation (3) expressed from the geometric dimensions of the float Figure 3 and Figure 12. The limiting angle ${\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is defined as the angle subtended by a line segment in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ drawn from the upper right corner of the cylindrical float to its lower left corner.

From a general point of view, the heel angle $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ of a floating body can take three limiting values. For Phase 1, the angle $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is defined as $\mathsf{\varphi }\le {\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, see Phase 1 below for more details; for Phase 2, the angle $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is defined as ${\mathsf{\varphi }}_1<\mathsf{\varphi }\le {\mathsf{\varphi }}_2\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, see Phase 2 below for more details; and for Phase 3, the angle $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is defined as $\mathsf{\varphi }\le {\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, see Phase 3 below for more details.

Phase 1 occurs when the heel angle of the floating body $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is less than, or equal to, the limiting angle ${\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ and greater than $0\mathrm{d}\mathrm{e}\mathrm{g}$, see relation (8).

$${0<\mathsf{\varphi }\le \mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$$

(8)

If the floats with a circular cross-section of the floating body are immersed below the water surface at exactly half of their diameter, i.e. ${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$, see Figure 3(b), the angle ${\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ can be determined according to (2).

If the floats with a circular cross-section of the floating body are immersed below the water surface below half of their diameter, i.e. ${\mathrm{h}}_{\mathrm{p}}<{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$, see Figure 6(b), the angle ${\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ can be determined according to (4).

If the floats with a circular cross-section of the floating body are immersed below the water surface above half of their diameter, i.e. ${\mathrm{h}}_{\mathrm{p}}>{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$, see Figure 9(a), the angle ${\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ can be determined according to (6).

Phase 2 occurs when the angle $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ of heel of the floating body is greater than the limiting angle ${\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ and less than or equal to the angle ${\mathsf{\varphi }}_2\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, see (9).

$${\mathsf{\varphi }}_1<\mathsf{\varphi }\le {\mathsf{\varphi }}_2\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$$

(9)

If the floats with a circular cross-section of the floating body are immersed below the water surface below half of their diameter, i.e. ${\mathrm{h}}_{\mathrm{p}}<{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$, see Figure 8(a), the angle ${\mathsf{\varphi }}_2\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ can be determined according to (5).

If the floats of a circular cross-section of the floating body are immersed below the water surface above half of their diameter, i.e. ${\mathrm{h}}_{\mathrm{p}}>{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$ (see Figure 9(b)), the angle ${\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ can be determined according to (7).

Phase 3 occurs in the opposite case, if the angle of heel $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ of the floating body takes on a value greater than the limiting angle ${\mathsf{\varphi }}_2\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ and is less than or equal to the limiting (critical) angle ${\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, see (10). The critical angle ${\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is the angle at which loss of buoyancy of the floating body occurs.

$${\mathsf{\varphi }}_2<\mathsf{\varphi }\le {\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$$

(10)

In the case where the floats with a circular cross-section of the floating body are immersed below the water surface at exactly half of their diameter, i.e. ${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$, the angle ${\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is defined by Table 2. In this state, the angle is ${\mathsf{\varphi }}_2={\mathsf{\varphi }}_1\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

If the floats with a circular cross-section of the floating body are immersed below the water surface below half of their diameter, i.e. ${\mathrm{h}}_{\mathrm{p}}<{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$, the angle ${\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is defined by Table 5.

If the floats with a circular cross-section of the floating body are immersed below the water surface above half of their diameter, i.e. ${\mathrm{h}}_{\mathrm{p}}>{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$, the angle is ${\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ defined by Table 8.

Now the three basic phases will be subjected to a closer analysis. For this purpose the solution of the position of the centre of gravity of the $“\mathrm{V}\mathrm{”}$ floating body must be further broken down for:

Phase 1, when the angle of heel of the floating body $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ lies in the interval defined by equation (8). This situation requires breaking down the further solution procedure into three states: 1a), 1b) and 1c), see below. The three different directions of the solution to determine the position of the centre of gravity of displacement of the floating body arise from the position of point $\mathrm{S}\left(\mathsf{\varphi }\right)$ and the position of point ${\mathrm{S}}_1\left(\mathsf{\varphi }\right)$, defined as the point of intersection of the plane of the water surface with the imaginary vertical plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ (see Figure 13) drawn at a distance of [m] from the origin of the coordinate system (the plane tangent to the right frontal circular wall of the cylindrical floats).

Condition 1a) Both cylindrical floats at equilibrium ($\mathsf{\varphi }=\mathrm{d}\mathrm{e}\mathrm{g}$) of the floating body are immersed below the water surface by a degree of $\mathrm{s}=0\mathrm{m}\mathrm{m}$ (${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{-\mathrm{D}}{2}}=-\mathrm{R}\left[\mathrm{m}\right]$), see Figure 13.

The position of point $\mathrm{S}\left(\mathsf{\varphi }\right)$ located on the axis $\mathrm{“}\mathrm{y}\mathrm{”}$ of the coordinate system in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be calculated according to (11), see Figure 13.

The position of point ${\mathrm{S}}_1\left(\mathsf{\varphi }\right)$ located on the plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ (plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ is parallel to the axis $\mathrm{“}\mathrm{y}\mathrm{”}$ at a distance of $\mathrm{a}\mathrm{n}\mathrm{d}\left[\mathrm{m}\right]$ from the origin of the coordinate system) in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$, can be determined according to (12).

$$\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)={\displaystyle \frac{-\mathrm{s}\left(\mathsf{\varphi }\right)}{\frac{\mathrm{a}}{2}}}\Rightarrow \mathrm{s}\left(\mathsf{\varphi }\right)={\displaystyle \frac{-\mathrm{a}}{2}}·\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$$

(11)

$${\mathrm{s}}_1\left(\mathsf{\varphi }\right)=\mathrm{s}\left(\mathsf{\varphi }\right)+\mathrm{a}·\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)={\displaystyle \frac{\mathrm{a}}{2}}·\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$$

(12)

According to Figure 13, point $\mathrm{S}\left(\mathsf{\varphi }\right)$ in state 1a) is located in the interval$\mathrm{S}\left(\mathsf{\varphi }\right)\in \left[-\mathrm{s}\left(\mathsf{\varphi }\right);0\right]$ and point ${\mathrm{S}}_1\left(\mathsf{\varphi }\right)$ is located in the interval ${\mathrm{S}}_1\left(\mathsf{\varphi }\right)\in \left[0;{\mathrm{s}}_1\left(\mathsf{\varphi }\right)\right]$.

It follows from the above (for the state defined as 1a) that the volume of the immersed parts of the floats ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ of the floating body, determined by the sum of the two triple integrals, can be determined according to the equation (13).

$$\mathrm{V}\left(\mathsf{\varphi }\right)=2·{\int }_{-\mathrm{R}}^{\mathrm{s}\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+2·{\int }_{\mathrm{s}\left(\mathsf{\varphi }\right)}^{{\mathrm{s}}_1\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-{\displaystyle \frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}\left[{\mathrm{m}}^3\right]($$

(13)

The coordinate $\left\{{\mathrm{x}}_{\mathrm{T}}\right\}\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to relation (14), see Supplementary Materials to article No. 1 (Chapter I), see Table 9.

$${\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)={\displaystyle \frac{{\int }_{-\mathrm{R}}^{\mathrm{s}\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{x}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+{\int }_{\mathrm{s}\left(\mathsf{\varphi }\right)}^{{\mathrm{s}}_1\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{x}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}{{\int }_{-\mathrm{R}}^{\mathrm{s}\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+{\int }_{\mathrm{s}\left(\mathsf{\varphi }\right)}^{{\mathrm{s}}_1\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}}\left[\mathrm{m}\right]$$

(14)

The coordinate $\left\{{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to relation (14), see Supplementary Materials to article No. 1 (Chapter I), see Table 9.

$${\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)={\displaystyle \frac{{\int }_{-\mathrm{R}}^{\mathrm{s}\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{y}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+{\int }_{\mathrm{s}\left(\mathsf{\varphi }\right)}^{{\mathrm{s}}_1\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{y}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}{{\int }_{-\mathrm{R}}^{\mathrm{s}\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+{\int }_{\mathrm{s}\left(\mathsf{\varphi }\right)}^{{\mathrm{s}}_1\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}}\left[\mathrm{m}\right]$$

(15)

Condition 1b) The floats with a circular cross-section of the floating body are in the equilibrium state of the floating body immersed under the water surface by a degree of $\mathrm{s}=-295\mathrm{m}\mathrm{m}$ (${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{D}}{2}}-\mathrm{s}\left[\mathrm{m}\right]$), see Figure 14.

The position of point ${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)$ in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be expressed according to (16), see Figure 14. The position of point ${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)$ in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be determined according to (17).

$${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)=\mathrm{s}\left(\mathsf{\varphi }\right)-\mathrm{s}=-\left[{\displaystyle \frac{\mathrm{a}}{2}}·\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)+\mathrm{s}\right]\left[\mathrm{m}\right]$$

(16)

$${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)={\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)+\mathrm{a}·\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)={\displaystyle \frac{\mathrm{a}}{2}}·\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)-\mathrm{s}\left[\mathrm{m}\right]$$

(17)

According to Figure 14, point ${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)$ in state 1b) is located in the interval${\mathrm{S}}^{*}\left(\mathsf{\varphi }\right)\in \left[-\mathrm{R};\mathrm{s}\right]$ and point ${\mathrm{S}}_1^{\mathrm{*}}\left(\mathsf{\varphi }\right)$ is located in the interval ${\mathrm{S}}_1^{\mathrm{*}}\left(\mathsf{\varphi }\right)\in \left[\mathrm{s};\mathrm{R}-\mathrm{s}\right]$.

For state 1b), the volume of the immersed parts of the ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ floating body can be determined according to equation (13) by replacing the expressions $\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (11), ${\mathrm{s}}_1\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (12) with the expressions ${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (16), ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (17), see Supplementary Materials to article No. 1 (Chapter II).

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\},\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to the relation (14) and (15) by replacing the expressions $\mathrm{s}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (11), ${\mathrm{s}}_1\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (12) with the expressions ${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (16), ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (17), see Supplementary Materials to article No. 1 (Chapter II), see Table 10.

State 1c) The floats of the floating body are immersed below the water surface by a degree of $\mathrm{s}=295\mathrm{m}\mathrm{m}$ (${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{D}}{2}}+\mathrm{s}\left[\mathrm{m}\right]$) in the equilibrium state of the floating body.

The position of point ${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)$ in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be expressed according to (18), see Figure 15.

The position of point ${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)$ located on the plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ (plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ is parallel to the axis $\mathrm{“}\mathrm{y}\mathrm{”}$ at a distance of $\mathrm{a}\mathrm{n}\mathrm{d}\left[\mathrm{m}\right]$ from the origin of the coordinate system) in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be determined according to (19).

$${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)=\mathrm{s}\left(\mathsf{\varphi }\right)+\mathrm{s}=\mathrm{s}-{\displaystyle \frac{\mathrm{a}}{2}}·\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$$

(18)

$${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)={\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)+\mathrm{a}·\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)={\displaystyle \frac{\mathrm{a}}{2}}·\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)+\mathrm{s}\left[\mathrm{m}\right]$$

(19)

According to Figure 15, point ${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)$ is located in the interval${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)\in \left[\mathrm{R}-\mathrm{s};\mathrm{s}\right]$ and point ${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)$ is located in the interval ${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)\in \left[\mathrm{s};\mathrm{R}\right]$.

For state 1c), the volume of the immersed parts of the ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ floating body can be determined according to equation (13) by replacing the expressions $\mathrm{s}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (11), ${\mathrm{s}}_1\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (12) with the expressions ${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (18), ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$(19), see Supplementary Materials to article No. 1 (Chapter III), see Table 11.

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\},\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to the relation (14) and(15) by replacing the expressions $\mathrm{s}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (11), ${\mathrm{s}}_1\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (12) with the expressions ${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (18), ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (19), see Supplementary Materials to article No. 1 (Chapter III), see Table 11.

Phase 2, when the angle of heel of the floating body $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ lies in the interval defined by equation (9). This situation requires breaking down the further solution procedure into two directions 2a) and 2b), see below. The two different directions of the solution to determine the position of the centre of gravity of displacement of the floating turntable arise from the position of point $\mathrm{S}\left(\mathsf{\varphi }\right)$, and point ${\mathrm{S}}_1\left(\mathsf{\varphi }\right)$ as a point defined as the point of intersection of the water surface plane with the imaginary vertical plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ drawn through the right wall of the float at a distance a [m] from the origin of the coordinate system.

2a) The floats of the floating body are immersed below the water surface by a degree of$\mathrm{s}=-295\mathrm{m}\mathrm{m}$ (${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{D}}{2}}-\mathrm{s}\left[\mathrm{m}\right]$) in the equilibrium state of the floating body.

The position of point ${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)$ located in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be determined by (16) and Figure 16.

The position of point ${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)$ located on the plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ (plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ is parallel to the axis $\mathrm{“}\mathrm{y}\mathrm{”}$ at a distance of $\mathrm{a}\mathrm{n}\mathrm{d}\left[\mathrm{m}\right]$ from the origin of the coordinate system), in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be determined by (17) and Figure 16.

For condition 2a), the volume of the immersed parts of the floats ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ of the floating body can be determined according to the equation (13) by replacing the expressions $\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (11), $\mathrm{ }{\mathrm{s}}_1\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (12) with the expressions ${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (16) ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (17), see Supplementary Materials to article No. 2 (Chapter I).

It follows from the above (for the state defined as 2a) that the volume of the immersed parts of the floats ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ of the floating body, determined by the sum of the two triple integrals, can be determined according to the equation (20).

$$\mathrm{V}\left(\mathsf{\varphi }\right)=2·{\int }_{-\mathrm{R}}^{{\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-{\displaystyle \frac{\left[\mathrm{y}-{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}\left[{\mathrm{m}}^3\right]$$

(20)

The coordinate $\left\{{\mathrm{x}}_{\mathrm{T}}\right\}\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to relation (21), see Supplementary Materials to article No. 2 (Chapter I), see Table 12.

$${\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)={\displaystyle \frac{{\int }_{-\mathrm{R}}^{{\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{x}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}{{\int }_{-\mathrm{R}}^{{\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}}\left[\mathrm{m}\right]$$

(21)

The coordinate $\left\{{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to relation (22), see Supplementary Materials to article No. 2 (Chapter I), see Table 12.

$${\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)={\displaystyle \frac{{\int }_{-\mathrm{R}}^{{\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{y}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}{{\int }_{-\mathrm{R}}^{{\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}}\left[\mathrm{m}\right]$$

(22)

2b) The floats of the floating body are immersed below the water surface by a degree of$\mathrm{s}=295\mathrm{m}\mathrm{m}$ (${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{D}}{2}}+\mathrm{s}\left[\mathrm{m}\right]$) in the equilibrium state of the floating body.

The position of point ${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)$ located in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be determined by (18) and Figure 17.

The position of point ${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)$ located on the plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ (plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ is parallel to the axis $\mathrm{“}\mathrm{y}\mathrm{”}$ at a distance of $\mathrm{a}\left[\mathrm{m}\right]$ from the origin of the coordinate system), in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be determined by (19) and Figure 17.

For condition 2b), the volume of the immersed parts of the floats ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ of the floating body can be determined according to the equation (13) by replacing the expressions s$\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (11), $\mathrm{ }{\mathrm{s}}_1\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (12) with the expressions ${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$, (18) ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (19), see Supplementary Materials to article No. 2 (Chapter II).

The coordinates $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\},\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to the relation (14) and(15) by replacing the expressions $\mathrm{s}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (11), ${\mathrm{s}}_1\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (12) with the expressions ${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (18), ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (19), see Supplementary Materials to article No. 1 (Chapter III), see Table 11.

It follows from the above (for the state defined as 2b) that the volume of the immersed parts of the floats ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ of the floating body, determined by the sum of the two triple integrals, can be determined according to the equation (23).

$$\mathrm{V}\left(\mathsf{\varphi }\right)=2·{\int }_{-\mathrm{R}}^{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+2·{\int }_{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}^{\mathrm{R}}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-{\displaystyle \frac{\left[\mathrm{y}-{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}\left[{\mathrm{m}}^3\right]$$

(23)

The coordinate $\left\{{\mathrm{x}}_{\mathrm{T}}\right\}\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to relation (24), see Supplementary Materials to article No. 2 (Chapter II), see Table 13.

$${\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)={\displaystyle \frac{{\int }_{-\mathrm{R}}^{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{x}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+{\int }_{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}^{\mathrm{R}}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{x}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}{{\int }_{-\mathrm{R}}^{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+{\int }_{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}^{\mathrm{R}}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}}\left[\mathrm{m}\right]$$

(24)

The coordinate $\left\{{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to relation (24), see Supplementary Materials to article No. 2 (Chapter II), see Table 13.

$${\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)={\displaystyle \frac{{\int }_{-\mathrm{R}}^{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{y}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+{\int }_{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}^{\mathrm{R}}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{y}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}{{\int }_{-\mathrm{R}}^{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}+{\int }_{{\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)}^{\mathrm{R}}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}}\left[\mathrm{m}\right]$$

(25)

Phase 3, when the angle of heel of the floating body $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ lies in the interval defined by equation (10). This situation requires breaking down the further solution procedure into three directions 3a, 3b) and 3c), see below. The three different directions of the solution to determine the position of the centre of gravity of displacement of the floating turntable arise from the position of point $\mathrm{S}\left(\mathsf{\varphi }\right)$, and point ${\mathrm{S}}_1\left(\mathsf{\varphi }\right)$ as a point defined as the point of intersection of the water surface plane with the imaginary vertical plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ drawn through the right wall of the float at a distance a [m] from the origin of the coordinate system.

3a) The floats of the floating body are immersed below the water surface by a degree of$\mathrm{s}=0\mathrm{m}\mathrm{m}$ (${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$) in the equilibrium state of the floating body.

The position of point S$\left(\mathsf{\varphi }\right)$ located in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be determined by (11) and Figure 18.

The position of point ${\mathrm{S}}_1\left(\mathsf{\varphi }\right)$ located on the plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ (plane $\mathrm{“}\mathsf{\tau }\mathrm{”}$ is parallel to the axis $\mathrm{“}\mathrm{y}\mathrm{”}$ at a distance of $\mathrm{a}\left[\mathrm{m}\right]$ from the origin of the coordinate system), in the plane $\mathrm{“}\mathrm{x}-\mathrm{y}\mathrm{”}$ can be determined by (12) and Figure 18.

According to Figure 18, point $\mathrm{S}\left(\mathsf{\varphi }\right)$ in state 3a) is located in the interval$\mathrm{S}\left(\mathsf{\varphi }\right)\in \left[-\mathrm{s}\left(\mathsf{\varphi }\right);-\mathrm{R}\right]$ and point ${\mathrm{S}}_1\left(\mathsf{\varphi }\right)$ is located in the interval ${\mathrm{S}}_1\left(\mathsf{\varphi }\right)\in \left[\mathrm{R};{\mathrm{s}}_1\left(\mathsf{\varphi }\right)\right]$.

It follows from the above (for the state defined as 1a) that the volume of the immersed parts of the floats ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ of the floating body, determined by the sum of the two triple integrals, can be determined according to the equation (26).

$$\mathrm{V}\left(\mathsf{\varphi }\right)=2·{\int }_{-\mathrm{R}}^{\mathrm{R}}\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-{\displaystyle \frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}\left[{\mathrm{m}}^3\right]$$

(26)

The coordinate $\left\{{\mathrm{x}}_{\mathrm{T}}\right\}\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to relation (27), see Supplementary Materials to article No. 3 (Chapter I), see Table 14.

$${\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)={\displaystyle \frac{{\int }_{-R}^R\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{x}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}{{\int }_{-\mathrm{R}}^R\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}}\left[\mathrm{m}\right]$$

(27)

The coordinate $\left\{{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m}\right]$ of the centre of gravity of displacement $\mathrm{“}\mathrm{V}\mathrm{”}$ can be expressed according to relation (28), see Supplementary Materials to article No. 3 (Chapter I), see Table 14.

$${\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)={\displaystyle \frac{{\int }_{-\mathrm{R}}^R\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{y}·\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}{{\int }_{-\mathrm{R}}^R\left[{\int }_{-\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}^{\sqrt{{\mathrm{R}}^2-{\mathrm{y}}^2}}\left({\int }_0^{\mathrm{a}-\frac{\left[\mathrm{y}-\mathrm{s}\left(\mathsf{\varphi }\right)\right]}{\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathsf{\varphi }\right)}}\mathrm{d}\mathrm{x}\right)·\mathrm{d}\mathrm{z}\right]·\mathrm{d}\mathrm{y}}}\left[\mathrm{m}\right]$$

(28)

For state 3b), the volume of the immersed parts of the floats of the ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ floating body can be determined according to equation (20) by replacing the expressions $\mathrm{s}\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (11), $\mathrm{ }{\mathrm{s}}_1\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (12) with the expressions ${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (16), ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$(17), see Supplementary Materials to article No. 3 (Chapter II). Submersion of the floats in equilibrium state $\mathrm{s}<0\mathrm{ }\left[\mathrm{m}\right]$ (${\mathrm{h}}_{\mathrm{p}}<{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$), see Figure 6(a) and Figure 19.

Point ${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)$ is located according to Figure 19 in the interval${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)\in \left({\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right);-\mathrm{R}\right)$, point ${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)$ is located according to Figure 19 in the interval${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)\in \left(-\mathrm{R};\mathrm{R}\right)$.

Table 15. Coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ of the floating body during phase 3 of the deflection ${\mathsf{\varphi }}_2\le \mathsf{\varphi }\le {\mathsf{\varphi }}_{\mathrm{k}}$ at an immersion depth of $\mathrm{s}=-295\mathrm{m}\mathrm{m}$, ${\mathrm{h}}_{\mathrm{p}}=500\mathrm{m}\mathrm{m}$.

$\mathrm{a}=4\mathrm{m}$, $\mathrm{D}=1.59\mathrm{m}$, ${\mathrm{h}}_{\mathrm{p}}=0.5\mathrm{m}$, $\mathrm{s}=-0.295\mathrm{m}$
$\mathsf{\varphi }$	$\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$	30	35	40	45	50	51.17 ${=\mathsf{\varphi }}_{\mathrm{k}}$1
${\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$ (27)	${10}^{-3}·\mathrm{m}$	903.69	891.42	892.29	898.84	907.97	910.32
${\mathrm{x}}_{\mathrm{T}}={\displaystyle \frac{\mathrm{a}}{2}}-{\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$1		109.31	1,108.58	1,107.71	1,101.16	1,092.03	1,089.68
${\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$ (28)		‒183.79	‒142.94	‒114.23	‒92.67	‒75.66	‒72.15
${\mathrm{y}}_{\mathrm{T}}=\left|{\displaystyle \frac{-\mathrm{D}}{2}}-{\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)\right|$1		611.21	652.06	680.77	702.33	719.34	722.85
$\mathrm{V}\left(\mathsf{\varphi }\right)$ (26)	${10}^{-6}·{\mathrm{m}}^3$	5,913,195.68	6,269,213.43	6,546,147.04	6,770,780.82	6,959,273.25	6,999,357.43

¹ see Table 5.

Table 15. Coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ of the floating body during phase 3 of the deflection ${\mathsf{\varphi }}_2\le \mathsf{\varphi }\le {\mathsf{\varphi }}_{\mathrm{k}}$ at an immersion depth of $\mathrm{s}=-295\mathrm{m}\mathrm{m}$, ${\mathrm{h}}_{\mathrm{p}}=500\mathrm{m}\mathrm{m}$.

$\mathrm{a}=4\mathrm{m}$, $\mathrm{D}=1.59\mathrm{m}$, ${\mathrm{h}}_{\mathrm{p}}=0.5\mathrm{m}$, $\mathrm{s}=-0.295\mathrm{m}$
$\mathsf{\varphi }$	$\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$	30	35	40	45	50	51.17 ${=\mathsf{\varphi }}_{\mathrm{k}}$1
${\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$ (27)	${10}^{-3}·\mathrm{m}$	903.69	891.42	892.29	898.84	907.97	910.32
${\mathrm{x}}_{\mathrm{T}}={\displaystyle \frac{\mathrm{a}}{2}}-{\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$1		109.31	1,108.58	1,107.71	1,101.16	1,092.03	1,089.68
${\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$ (28)		‒183.79	‒142.94	‒114.23	‒92.67	‒75.66	‒72.15
${\mathrm{y}}_{\mathrm{T}}=\left|{\displaystyle \frac{-\mathrm{D}}{2}}-{\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)\right|$1		611.21	652.06	680.77	702.33	719.34	722.85
$\mathrm{V}\left(\mathsf{\varphi }\right)$ (26)	${10}^{-6}·{\mathrm{m}}^3$	5,913,195.68	6,269,213.43	6,546,147.04	6,770,780.82	6,959,273.25	6,999,357.43

¹ see Table 5.

For state 3c), the volume of the immersed parts of the floats of the ${\mathrm{V}}_{\left(\mathsf{\varphi }\right)}\left[{\mathrm{m}}^3\right]$ floating body can be determined according to equation (20) by replacing the expressions $\left(\mathsf{\varphi }\right)\left[\mathrm{m}\right]$ (11), $\mathrm{ }{\mathrm{s}}_1\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (12) with the expressions ${\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$ (18), ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\mathrm{ }\left[\mathrm{m}\right]$(19), see Supplementary Materials to article No. 3 (Chapter III). Submersion of the floats in equilibrium state $\mathrm{s}>0\mathrm{ }\left[\mathrm{m}\right]$ (${\mathrm{h}}_{\mathrm{p}}>{\displaystyle \frac{\mathrm{D}}{2}}\left[\mathrm{m}\right]$), see Figure 8(b) and Figure 20.

According to Figure 20, it applies that ${\mathrm{s}}_1^{\ast }\left(\mathsf{\varphi }\right)\in \left(-\mathrm{R};\mathrm{R}\right)$.

Point ${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)$ is located according to Figure 20 in the interval${\mathrm{S}}^{\ast }\left(\mathsf{\varphi }\right)\in \left({\mathrm{s}}^{\ast }\left(\mathsf{\varphi }\right);-\mathrm{R}\right)$, point ${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)$ is located according to Figure 20 in the interval${\mathrm{S}}_1^{\ast }\left(\mathsf{\varphi }\right)\in \left(-\mathrm{R};\mathrm{R}\right)$.

Table 16. Coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ of the floating body during phase 3 of the deflection ${\mathsf{\varphi }}_2\le \mathsf{\varphi }\le {\mathsf{\varphi }}_{\mathrm{k}}$ at an immersion depth of $\mathrm{s}=295\mathrm{m}\mathrm{m}$, ${\mathrm{h}}_{\mathrm{p}}=1090\mathrm{m}\mathrm{m}$.

$\mathrm{a}=4\mathrm{m}$, $\mathrm{D}=1.59\mathrm{m}$, ${\mathrm{h}}_{\mathrm{p}}=1.09\mathrm{m}$, $\mathrm{s}=0.295\mathrm{m}$
$\mathsf{\varphi }$	$\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$	30	35	40	42.09 ${=\mathsf{\varphi }}_{\mathrm{k}}$1
${\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$ (27)	${10}^{-3}·\mathrm{m}$	1,349.98	1,277.20	1,223.50	1,204.92
${\mathrm{x}}_{\mathrm{T}}={\displaystyle \frac{\mathrm{a}}{2}}-{\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$1		650.13	722.80	776.50	795.08
${\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$ (28)		‒108.99	‒93.20	‒80.08	‒75.19
${\mathrm{y}}_{\mathrm{T}}=\left|{\displaystyle \frac{-\mathrm{D}}{2}}-{\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)\right|$1		686.01	701.80	714.92	719.81
$\mathrm{V}\left(\mathsf{\varphi }\right)$ (26)	${10}^{-6}·{\mathrm{m}}^3$	9,971,338.97	9,615,317.52	9,338,385.45	9,239,227.27

¹ see Table 8.

Table 16. Coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ of the floating body during phase 3 of the deflection ${\mathsf{\varphi }}_2\le \mathsf{\varphi }\le {\mathsf{\varphi }}_{\mathrm{k}}$ at an immersion depth of $\mathrm{s}=295\mathrm{m}\mathrm{m}$, ${\mathrm{h}}_{\mathrm{p}}=1090\mathrm{m}\mathrm{m}$.

$\mathrm{a}=4\mathrm{m}$, $\mathrm{D}=1.59\mathrm{m}$, ${\mathrm{h}}_{\mathrm{p}}=1.09\mathrm{m}$, $\mathrm{s}=0.295\mathrm{m}$
$\mathsf{\varphi }$	$\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$	30	35	40	42.09 ${=\mathsf{\varphi }}_{\mathrm{k}}$1
${\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$ (27)	${10}^{-3}·\mathrm{m}$	1,349.98	1,277.20	1,223.50	1,204.92
${\mathrm{x}}_{\mathrm{T}}={\displaystyle \frac{\mathrm{a}}{2}}-{\mathrm{x}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$1		650.13	722.80	776.50	795.08
${\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)$ (28)		‒108.99	‒93.20	‒80.08	‒75.19
${\mathrm{y}}_{\mathrm{T}}=\left|{\displaystyle \frac{-\mathrm{D}}{2}}-{\mathrm{y}}_{\mathrm{T}}\left(\mathsf{\varphi }\right)\right|$1		686.01	701.80	714.92	719.81
$\mathrm{V}\left(\mathsf{\varphi }\right)$ (26)	${10}^{-6}·{\mathrm{m}}^3$	9,971,338.97	9,615,317.52	9,338,385.45	9,239,227.27

¹ see Table 8.

The coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the buoyant force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$ of the floating body during deflection $0\le \mathsf{\varphi }\le {\mathsf{\varphi }}_{\mathrm{k}}\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ at various immersion depths with $\left[\mathrm{m}\right]$ are presented in Figure 21.

3. Results

The transverse stability of the floating body was investigated in laboratory conditions on a test rig, whose 3D model created in SolidWorks software (version Premium 2012×64, edition SP5.0) [11] is presented in Figure 22. The test rig consists of a watertight tank 1 and a floating body 2 (which consists of two floats made of PLEXI pipe of 80/72 mm outer/internal diameter [34], length 200 mm and two connecting parts), which is mechanically attached to two force transducers 3, type MCF-100 N [35].

Both floats with a circular cross-section of the floating body 2 are fitted with 75-80 mm twin-screw sleeves 4 [36] with a hexagonal nut length of 18 mm with internal threading M8/M10 length 8 mm/10 mm, see Figure 22.

According to Archimedes’ law, the floating body of the test device with a mass of ${\mathrm{m}}_{\mathrm{F}\mathrm{B}}={\displaystyle \frac{{\mathrm{G}}_{\mathrm{F}\mathrm{B}}}{\mathrm{g}}}\mathrm{}\left[\mathrm{k}\mathrm{g}\right]$ will immerse below the water surface up to the volume $\mathrm{V}\left[{\mathrm{m}}^3\right]$, at which it is buoyed by a buoyant force corresponding to the gravity of fluid of the same volume as the immersed part of the body.

In the equilibrium state of the floating body of the test device (angle $\mathsf{\varphi }=0\mathrm{d}\mathrm{e}\mathrm{g}$) with a mass of ${\mathrm{m}}_{\mathrm{F}\mathrm{B}}=0.931\mathrm{k}\mathrm{g}$ , the front walls of both floats (diameter $\mathrm{D}\left[\mathrm{m}\right]$ and length of one float $\mathrm{a}\left[\mathrm{m}\right]$, see Figure 2) of the floating body are immersed below the water surface by a degree of $\mathrm{S}\left[{\mathrm{m}}^2\right]$. If the floating body is designed so that its centre of gravity lies on the vertical axis of the axis of symmetry of the floating body, see Figure 23, then the longitudinal axes of both floats are parallel to the water surface.

A schematic diagram of the measurement chain for laboratory measurement of tensile forces ${\mathrm{F}}_{\mathrm{B}\mathrm{i},\mathrm{j}\left(\mathrm{s}\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$, when a floating body is laterally deflected out of equilibrium by an angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, is shown in Figure 24. The measurements were carried out while the cylindrical floats were immersed in equilibrium state to a depth of $\mathrm{s}=20\mathrm{m}\mathrm{m};0\mathrm{m}\mathrm{m};-20\mathrm{m}\mathrm{m}$ below the water level of the laboratory device tank, see Figure 22.

A load sensor cable equipped with a D-Sub plug 4 was plugged into the socket of the measuring module BR4-D 4 [37] of the strain gauge apparatus DS NET during the laboratory measurements, see Figure 24. A PC 8 (ASUS K72JR-TY131 laptop) was connected to the DS NET strain gauge 6 using a network cable with RJ-45 connectors 7 at both ends.

The total immersion volume of a floating body $\mathrm{V}\left[{\mathrm{m}}^3\right]$ (1) is given by the sum of the immersion volumes of the two floats $\mathrm{V}={\mathrm{V}}_1+{\mathrm{V}}_2\mathrm{ }\left[{\mathrm{m}}^3\right]$. Assuming that ${\mathrm{V}}_1={\mathrm{V}}_2\left[{\mathrm{m}}^3\right]$ and ${\mathrm{S}}_1={\mathrm{S}}_2={\displaystyle \frac{\mathrm{S}}{2}}\left[{\mathrm{m}}^2\right]$ (see Figure 2), and knowing the value of the intrinsic mass of the floating body of the test device ${\mathrm{m}}_{\mathrm{F}\mathrm{B}}=0.931\mathrm{k}\mathrm{g}$, it is possible to calculate in the Mathcad software environment (version 14.0.0.163) [39] using the command “Given−Find” the central angle$\mathsf{\alpha }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ of the immersed area (circular segment) of one float from relation (30), and the depth ${\mathrm{h}}_{\mathrm{p}\left(\mathrm{m}\right)}\left[\mathrm{m}\right]$ of the immersed float (height of the circular segment) from relation (31), or${\mathrm{s}}_{\left(\mathrm{m}\right)}\left[\mathrm{m}\right]$ according to (32).

$$\mathsf{\rho }·\mathrm{V}=\mathsf{\rho }·\mathrm{S}·\mathrm{a}={\mathrm{m}}_{\mathrm{F}\mathrm{B}}\Rightarrow \mathrm{S}={\displaystyle \frac{{\mathrm{m}}_{\mathrm{F}\mathrm{B}}}{\mathsf{\rho }·\mathrm{a}}}={\displaystyle \frac{0.931}{1000·0.2}}=46.6·{10}^{-4}{\mathrm{m}}^3$$

(29)

$${\mathrm{S}}_1\left(={\displaystyle \frac{\mathrm{S}}{2}}\right)={{\displaystyle \frac{1}{2}}·\left({\displaystyle \frac{\mathrm{D}}{2}}\right)}^2·\left[\mathsf{\alpha }-\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathsf{\alpha }\right)\right]\left[{\mathrm{m}}^2\right]\Rightarrow \mathsf{\alpha }=173.3\mathrm{d}\mathrm{e}\mathrm{g}$$

(30)

where $\mathsf{\alpha }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ is the central angle of the circular segment [32]; $\mathrm{D}=80\mathrm{m}\mathrm{m}$.

$${\mathrm{S}}_1={\left({\displaystyle \frac{\mathrm{D}}{2}}\right)}^2·\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{\frac{\mathrm{D}}{2}-{\mathrm{h}}_{\mathrm{p}\left(\mathrm{m}\right)}}{\frac{\mathrm{D}}{2}}}\right)-\left({\displaystyle \frac{\mathrm{D}}{2}}-{\mathrm{h}}_{\mathrm{p}\left(\mathrm{m}\right)}\right)·\sqrt{2·{\mathrm{h}}_{\mathrm{p}\left(\mathrm{m}\right)}·{\displaystyle \frac{\mathrm{D}}{2}}-{\mathrm{h}}_{\mathrm{p}\left(\mathrm{m}\right)}^2}\left[{\mathrm{m}}^2\right]\Rightarrow {\mathrm{h}}_{\mathrm{p}\left(\mathrm{m}\right)}=37.7\mathrm{m}\mathrm{m}$$

(31)

$${\mathrm{s}}_{\left(\mathrm{m}\right)}={\displaystyle \frac{-\mathrm{D}}{2}}+{\mathrm{h}}_{\mathrm{p}\left(\mathrm{m}\right)}=-40+37.7=-2.3\mathrm{m}\mathrm{m}$$

(32)

A buoyant force of ${\mathrm{F}}_{\mathrm{B}\left(\mathrm{m}\right)}={\mathrm{m}}_{\mathrm{F}\mathrm{B}}·\mathrm{g}=0.931·\mathrm{g}=9.13\mathrm{N}$, Figure 25(b) is applied to the floating body of the test device, consisting of cylindrical floats with a diameter of $\mathrm{D}=80\mathrm{m}\mathrm{m}$ and length of $\mathrm{a}=200\mathrm{m}\mathrm{m}$, at an immersion depth of ${\mathrm{s}}_{\left(\mathrm{m}\right)}=-2.3\mathrm{m}\mathrm{m}$ $\left({\mathrm{h}}_{\mathrm{p}}=37.7\mathrm{m}\mathrm{m}\right)$, see Figure 25(a).

With a known degree of $\mathrm{s}\left[\mathrm{m}\right]$ $\left({\mathrm{h}}_{\mathrm{p}}\left[\mathrm{m}\right]\right)$ immersion of the floats of the floating body at equilibrium ($\mathsf{\varphi }=0\mathrm{d}\mathrm{e}\mathrm{g}$) below the water surface, the floating body was deflected in the transverse direction by a known angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$. When the floating body was deflected out of its equilibrium state, the force sensors 3, see Figure 23, detected the magnitudes of buoyant forces ${\mathrm{F}}_{\mathrm{B}\mathrm{i},\mathrm{j}\left(\mathrm{s}\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ (where $\mathrm{“}\mathrm{i}\mathrm{”}$ is the force sensor number,$\mathrm{“}\mathrm{j}\mathrm{”}$ is the measurement number).

The results of the experimental measurements of buoyant force ${\mathrm{F}}_{\mathrm{B}\mathrm{i},\mathrm{j}\left(\mathrm{s}\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ at the depth of immersion of the floating body of the test device $\mathrm{s}=0\mathrm{m}\mathrm{m}$ below the water surface and the known value of the longitudinal deflection of the floating body from the equilibrium state $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ are shown in Table 17.

Figure 26(a) shows the values of buoyant forces ${\mathrm{F}}_{\mathrm{B}\mathrm{j}\left(0\right)0}\left[\mathrm{N}\right]$ measured by the MCF30-100 N force sensors, when the floats with a circular cross-section were not immersed below the water surface and at a float immersion depth of $\mathrm{s}=0\mathrm{m}\mathrm{m}$ , at a deflection angle of $\mathsf{\varphi }=0\mathrm{d}\mathrm{e}\mathrm{g}$.

Figure 26(b) shows the values of buoyant forces ${\mathrm{F}}_{\mathrm{B}\mathrm{i},3\left(0\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ measured by the MCF30-100 N force sensors at an immersion depth of $\mathrm{s}=0\mathrm{m}\mathrm{m}$ of the floats below the water surface, with gradual deflection $=0÷50\mathrm{d}\mathrm{e}\mathrm{g}$ of the floating body of the test device.

The results of the experimental measurements of buoyant force ${\mathrm{F}}_{\mathrm{B}\mathrm{i},\mathrm{j}\left(\mathrm{s}\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ at an immersion depth of the floating body of the test device $\mathrm{s}=0\mathrm{m}\mathrm{m}$ below the water surface and the known value of the longitudinal deflection of the floating body from the equilibrium state $\mathsf{\varphi }=0÷50\mathrm{d}\mathrm{e}\mathrm{g}$ are shown in Table 18.

A buoyant force${\mathrm{F}}_{\mathrm{B}\left(25\right)}=18.27\mathrm{N}$, see Figure 27(a), is exerted on the floating body of the test device, consisting of cylindrical floats with a diameter of $\mathrm{D}=80\mathrm{m}\mathrm{m}$ and length of $\mathrm{a}=200\mathrm{m}\mathrm{m}$, at an immersion depth of $\mathrm{s}=25\mathrm{m}\mathrm{m}$ $\left({\mathrm{h}}_{\mathrm{p}}=65\mathrm{m}\mathrm{m}\right)$ and own mass of the floating body of the test device.

Figure 28(a) presents the floating body of the laboratory immersion device $\mathrm{s}=25\mathrm{m}\mathrm{m}$.

Figure 28(b) shows the values of buoyant forces ${\mathrm{F}}_{\mathrm{B}\mathrm{i},1\left(25\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ measured by the MCF30-100 N force sensors at an immersion depth of $\mathrm{s}=25\mathrm{m}\mathrm{m}$ of the floats below the water surface, with gradual deflection $=0÷50\mathrm{d}\mathrm{e}\mathrm{g}$ of the floating body of the test device.

From three times ($\mathrm{n}=3$) repeated measurements under the same technical conditions, the arithmetic mean ${\mathrm{F}}_{\mathrm{B}\mathrm{i}\left(\mathrm{s}\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ and the marginal error ${\mathsf{\kappa }}_{\mathrm{i}\left(\mathsf{\beta },\mathrm{n}\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ (33) were calculated according to Student’s distribution [40].

$${\mathsf{\kappa }}_{\mathrm{i}\left(\mathsf{\beta },\mathrm{n}\right)\mathsf{\varphi }}={\mathrm{t}}_{\mathsf{\beta },\mathrm{n}}·\overline{\mathrm{s}}\left[\mathrm{N}\right]$$

(33)

where ${\mathrm{t}}_{\mathsf{\beta },\mathrm{n}}\left[-\right]$ is the Student’s coefficient (for the chosen risk $=5\mathrm{\%}$ and the number of measured values $\mathrm{n}=3$ can be determined according to [40] ${\mathrm{t}}_{\mathsf{\beta },\mathrm{n}}={\mathrm{t}}_{5\mathrm{\%},3}=4.3$); $\overline{\mathrm{s}}\left[\mathrm{N}\right]$ is the standard deviation of the arithmetic mean (34).

$$\overline{\mathrm{s}}\approx {\displaystyle \frac{5}{4}}·{\displaystyle \frac{{\displaystyle \sum _{\mathrm{j}=1}^3}\left|{\mathrm{F}}_{\mathrm{B}\mathrm{i},\mathrm{j}\left(\mathrm{s}\right)\mathsf{\varphi }}-{\mathrm{F}}_{\mathrm{B}\mathrm{i}\left(\mathrm{s}\right)\mathsf{\varphi }}\right|}{\mathrm{n}·\sqrt{\mathrm{n}-1}}}\left[\mathrm{N}\right]$$

(34)

From the measured mean value of the buoyant force ${\mathrm{F}}_{\mathrm{B}\left(\mathrm{s}\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$, it is possible to express the total immersed volume of the floats ${\mathrm{V}}_{\left(\mathrm{s}\right)\mathsf{\varphi }}\left[{\mathrm{m}}^3\right]$ of the floating body of the test device at an immersion depth of $\mathrm{s}\left[\mathrm{m}\right]$ and deflection angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ of the floating body from the equilibrium state according to the relation (35).

$${\mathrm{V}}_{\left(\mathrm{s}\right)\mathsf{\varphi }}={\displaystyle \frac{{\mathrm{F}}_{\mathrm{B}\left(\mathrm{s}\right)\mathsf{\varphi }}}{\mathsf{\rho }·\mathrm{g}}}\left[{\mathrm{m}}^3\right]$$

(35)

Table 19 shows the total immersed volume of the floats ${\mathrm{V}}_{\left(25\right)\mathsf{\varphi }}\left[{\mathrm{m}}^3\right]$ of the floating body of the test device at an immersion depth of $\mathrm{s}=25\mathrm{m}\mathrm{m}$ and deflection angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ of the floating body from equilibrium.

The results of the experimental measurements of buoyant force ${\mathrm{F}}_{\mathrm{B}\mathrm{i},\mathrm{j}\left(\mathrm{s}\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ at an immersion depth of the floating body of the test device $\mathrm{s}=-25\mathrm{m}\mathrm{m}$ below the water surface and the known value of the longitudinal deflection of the floating body from the equilibrium state $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ are shown in Table 20.

A buoyant force${\mathrm{F}}_{\mathrm{B}(-25)}=2.01\mathrm{N}$, see Figure 29(a), is exerted on the floating body of the test device, consisting of cylindrical floats with a diameter of $\mathrm{D}=80\mathrm{m}\mathrm{m}$ and length of $\mathrm{a}=200\mathrm{m}\mathrm{m}$, at an immersion depth of $\mathrm{s}=-25\mathrm{m}\mathrm{m}$ $\left({\mathrm{h}}_{\mathrm{p}}=15\mathrm{m}\mathrm{m}\right)$ and own mass of the floating body of the test device.

Figure 30(a) shows a floating body immersed $\mathrm{s}=-25\mathrm{m}\mathrm{m}$ beneath the water’s surface of the laboratory device in equilibrium position.

Figure 30(b) shows the values of buoyant forces ${\mathrm{F}}_{\mathrm{B}\mathrm{i},1\left(-25\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ measured by the MCF30-100 N force sensors at an immersion depth of $\mathrm{s}=0\mathrm{m}\mathrm{m}$ of the floats below the water surface, with gradual deflection $=0÷50\mathrm{d}\mathrm{e}\mathrm{g}$ of the floating body of the test device.

Table 21. Immersed volume of the floats ${\mathrm{V}}_{\left(-25\right)\mathsf{\varphi }}\left[{\mathrm{m}}^3\right]$ of the floating body of the test device at an immersion depth of $\mathrm{s}=-25\mathrm{m}\mathrm{m}$ and angle of deflection of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

$\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$	0	10	20	30	40	50
${\mathrm{F}}_{\mathrm{B}\left(-25\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$1	2.08	2.90	4.04	5.10	6.15	6.63
${\mathrm{V}}_{\left(-25\right)\mathsf{\varphi }}\left[{{10}^{-6}·\mathrm{m}}^3\right]$2	212.10	295.72	411.97	520.06	627.13	676.07
${\mathrm{V}}_{\mathrm{p}}\left[{10}^{-6}·{\mathrm{m}}^3\right]$3	260.98	298.50	420.20	572.29	705.79	794.42
${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$4	2.56	2.93	4.12	5.61	6.92	7.79

¹ see Table 20 and Table 2 according to (35), ³ according to SolidWorks, ⁴ according to (1).

Table 21. Immersed volume of the floats ${\mathrm{V}}_{\left(-25\right)\mathsf{\varphi }}\left[{\mathrm{m}}^3\right]$ of the floating body of the test device at an immersion depth of $\mathrm{s}=-25\mathrm{m}\mathrm{m}$ and angle of deflection of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$.

$\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$	0	10	20	30	40	50
${\mathrm{F}}_{\mathrm{B}\left(-25\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$1	2.08	2.90	4.04	5.10	6.15	6.63
${\mathrm{V}}_{\left(-25\right)\mathsf{\varphi }}\left[{{10}^{-6}·\mathrm{m}}^3\right]$2	212.10	295.72	411.97	520.06	627.13	676.07
${\mathrm{V}}_{\mathrm{p}}\left[{10}^{-6}·{\mathrm{m}}^3\right]$3	260.98	298.50	420.20	572.29	705.79	794.42
${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$4	2.56	2.93	4.12	5.61	6.92	7.79

¹ see Table 20 and Table 2 according to (35), ³ according to SolidWorks, ⁴ according to (1).

Figure 31(a) shows the mean values calculated according to Student’s distribution [39] of the values of buoyant forces measured by the ${\mathrm{F}}_{\mathrm{B}\left(\mathrm{s}\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ MCF30-100 N force sensors, when the floating body is deflected out of equilibrium.

Figure 31(b) shows the laboratory device for the detection of buoyant forces ${\mathrm{F}}_{\mathrm{B}\mathrm{i},\mathrm{j}\left(0\right)\mathsf{\varphi }}\left[\mathrm{N}\right]$ during deflection $\varphi \left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$of a floating body from its equilibrium position.

4. Discussion

The issue of longitudinal and transverse stability of floating belt conveyors using floating bodies with floats of circular cross-section [16,29] has not been comprehensively and systematically analysed so far, which is evidenced by the lack of relevant scientific articles and publications in this area.

The presented paper describes two methodological approaches to determine the position of the centre of gravity of the displacement of a floating body during its lateral deflection from equilibrium state.

The first method (see section 2.1) is based on the graphical and numerical determination of the centre of gravity of the displacement of the floating body, which is complemented by the use of 3D modelling tools (e.g. SolidWorks [33] or Inventor). The computationally demanding and time-consuming analytical procedure for determining the position of the centre of gravity of the displacement when the body is deflected from its equilibrium position can be effectively replaced by determining the centre of gravity based on a volume model created in a CAD system environment.

The results obtained for the coordinates of the centre of gravity of the displacement $\left\{{\mathrm{x}}_{\mathrm{T}};{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m};\mathrm{m}\right]$ (i.e. the centre of buoyancy force ${\mathrm{F}}_{\mathrm{B}}\left[\mathrm{N}\right]$) at each stage of the deflection of the floating body from the equilibrium state by a known magnitude of angle $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$, at three immersion depths ${\mathrm{h}}_{\mathrm{p}}\left[\mathrm{m}\right]\left(0.795;0.5;1.09\right)$ are given in Table 1 to Table 8.

Maximum stability arm value ${\mathrm{s}}_{\mathrm{a}}=329.68·{10}^{-3}\mathrm{m}$ is achieved for cylindrical floats with a diameter of $\mathrm{D}=1.59\mathrm{m}$, whose immersion below the water surface reaches exactly half the float diameter, i.e. ${\mathrm{h}}_{\mathrm{p}}={\displaystyle \frac{\mathrm{D}}{2}}=0.795\mathrm{m}$, at an angle of heel of $\mathsf{\varphi }=25\mathrm{d}\mathrm{e}\mathrm{g}$ (see Table 2), for the position of the centre of mass ${\mathrm{h}}_{\mathrm{G}}=1.6\mathrm{m}$.

Maximum stability arm value ${\mathrm{s}}_{\mathrm{a}}=540.25\mathrm{m}\mathrm{m}$ is achieved for cylindrical floats of diameter $\mathrm{D}=1.59\mathrm{m}$ whose submergence below the water surface does not exceed half the diameter, e.g. ${\mathrm{h}}_{\mathrm{G}}<{\displaystyle \frac{\mathrm{D}}{2}}=0.5\mathrm{m}$, at an angle of heel $\mathsf{\varphi }\approx 20\mathrm{d}\mathrm{e}\mathrm{g}$ (see Table 4), for the position of the centre of mass ${\mathrm{h}}_{\mathrm{G}}=1.6\mathrm{m}$.

Maximum stability arm size ${\mathrm{s}}_{\mathrm{a}\mathrm{n}\mathrm{d}}=701.01-{10}^{-3}\mathrm{m}$ is achieved for cylindrical floats of diameter $\mathrm{D}=1.59\mathrm{m}$ whose submergence below the water surface exceeds half the diameter, e.g. ${\mathrm{h}}_{\mathrm{G}}>{\displaystyle \frac{\mathrm{D}}{2}}=1.09\mathrm{m}$, at an angle of heel $\mathsf{\varphi }\approx 40\mathrm{d}\mathrm{e}\mathrm{g}$ (see Table 7), for the position of the centre of mass ${\mathrm{h}}_{\mathrm{G}}=1.6\mathrm{m}$.

The second method (see section 2.2) is based on the analytical determination of the coordinates of the centre of gravity of the displacement of the floating body. The coordinates of the centre of gravity of the displacement are identical to the values of $\left\{{\mathrm{x}}_{\mathrm{T}},{\mathrm{y}}_{\mathrm{T}}\right\}\left[\mathrm{m},\mathrm{m}\right]$determined by the “first method”.

Table 9 presents the coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{T}\left(\mathsf{\varphi }\right)},{\mathrm{y}}_{\mathrm{T}\left(\mathsf{\varphi }\right)}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the buoyant force of the floating body in phase 1 of deflection, at an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=795\mathrm{m}\mathrm{m}$.

Table 10 presents the coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{T}\left(\mathsf{\varphi }\right)},{\mathrm{y}}_{\mathrm{T}\left(\mathsf{\varphi }\right)}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the buoyant force of the floating body in phase 1 of deflection, at an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=500\mathrm{m}\mathrm{m}$, and Table 11 at an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=1090\mathrm{m}\mathrm{m}$.

Table 12 presents the coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{T}\left(\mathsf{\varphi }\right)},{\mathrm{y}}_{\mathrm{T}\left(\mathsf{\varphi }\right)}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the buoyant force of the floating body in phase 2 of deflection, at an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=500\mathrm{m}\mathrm{m}$, and Table 13 at an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=1090\mathrm{m}\mathrm{m}$.

Table 14 presents the coordinates of the centre of gravity $\left\{{\mathrm{x}}_{\mathrm{T}\left(\mathsf{\varphi }\right)},{\mathrm{y}}_{\mathrm{T}\left(\mathsf{\varphi }\right)}\right\}\left[\mathrm{m},\mathrm{m}\right]$ of the buoyant force of the floating body in phase 3 of deflection, at an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=795\mathrm{m}\mathrm{m}$, Table 15 at an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=500\mathrm{m}\mathrm{m}$, and Table 16 at an immersion depth of ${\mathrm{h}}_{\mathrm{p}}=1090\mathrm{m}\mathrm{m}$.

The aim of the experimental tests carried out on laboratory equipment (see Figure 22) was to determine the actual buoyancy force values based on direct measurements using force transducers [35], depending on the angle of $\mathsf{\varphi }\left[\mathrm{d}\mathrm{e}\mathrm{g}\right]$ inclination and at different levels of immersion ${\mathrm{h}}_{\mathrm{p}}\left[\mathrm{m}\right]$ of the floating body below the liquid surface [1].

Experimental measurements verified the correctness of the analytical determination of the position of the centre of gravity of the displacement of the floating body during its lateral deflection from the equilibrium position.

Based on Figure 26(b) and Figure 31 the conclusion can be formulated that when the floating body is submerged below the surface by a value exceeding half of the float diameter (circular cross-section), the magnitude of the buoyancy force decreases with increasing angle of inclination.

Based on Figure 28(b), it can be concluded that when a floating body is immersed below the surface by $\mathrm{s}>0\left(=25\mathrm{m}\mathrm{m}\right)$more than half the float diameter (circular cross-section), the magnitude of the buoyant force decreases with increasing angle of heel, see Figure (31).

Based on Figure 30(b), it can be concluded that when a floating body is immersed below the surface by a value $\mathrm{s}<0=-25\mathrm{m}\mathrm{m}$ not exceeding half the float diameter (circular cross-section), the magnitude of the buoyant force increases with increasing angle of heel, see Figure (31).

Future research directions in the area of floating conveyor belt buoyancy should also focus on a detailed analysis of the stability loss mechanisms in relation to the transverse stability of the floating body. The key topic is to deepen the understanding of critical states, in which the transition from steady state to unsteady behaviour occurs due to changes in buoyancy distribution and metacentre shift during transverse deflection of the floating body.

Furthermore, it is appropriate to extend the research by experimental and numerical modelling of the dynamic response of floating conveyors to external loads, including the effects of waves, uneven loading and changes in embedment. Attention should also be paid to the influence of the geometrical parameters of the floats and their arrangement on the so-called “metacentric height” [9,28] and the overall lateral stability of the system.

Optimization of the structural design to increase the stability margin is also a promising area, especially through modifications of the shape of the floats, their mutual configuration and the weight distribution of the conveyor [19,29]. At the same time, the development of advanced computational methods [18,19,22,30] can be recommended, which allow reliable prediction of stability limit states in real operating conditions [41].

5. Conclusions

In this paper, the stability analysis of floating bodies consisting of two parallel cylindrical floats with a circular cross-section is considered. The floating body acts as the supporting element of serially arranged belt conveyors for the continuous transport of the extracted granular substrate from the water environment.

A systematic mathematical apparatus for predicting the trajectory of the centre of gravity (centre of buoyancy) as a function of the angle of transverse heel of the body has been developed and verified. This procedure defines the methodology for calculating the stability arm as a critical criterion for assessing the hydrostatic stability of a structure.

The analytical modelling of the three variants of floating body immersion showed that the maximum value of the stability arm was achieved at the initial static immersion of the floating body, which takes exactly the value of half the float diameter.

Through strain gauge measurements on a physical laboratory model for three discrete levels of initial immersion, the correlation between the analytical model and the real behaviour of the system was verified.

The research quantified the nonlinear changes in buoyant force during heeling as a function of the initial dive footprint. The system exhibits a constant buoyant force during heeling only when immersed to the level of half the float diameter. At lower initial drafts, increasing roll generates an increase in total buoyancy, while at deeper initial drafts, the trend in buoyant force tends to decrease with increasing roll angle.

The obtained knowledge and the proposed computational procedures provide an exact engineering basis for the dimensioning and optimisation of a safe operating envelope for floating conveyor systems in terms of their hydrostatic stability.

Subsequent phases of research should focus on the transition from a static stability description to the nonlinear dynamic stability of the floating system. This dynamic approach allows the definition of complex system transfer functions and the setting of safe operating limits under realistic non-stationary conditions. A key area for future development is quantification of the total deformation work that the floating body is able to absorb when angularly deflected from the equilibrium position before reaching the capsize limit state. Another unexplored area is the modelling of the stochastic effects of the external environment, in particular the hydrodynamic forces of waves, wind shocks and variable force loads from the irregular flow of the conveyed bulk material on the conveyor belt.

This dynamic approach allows the definition of complex system transfer functions and the setting of safe operating limits under realistic non-stationary conditions.