How to find the center of size of a ship. Metacentric height - criterion of vessel stability: formula

The longitudinal stability of a vessel is much higher than its transverse stability, so for safe navigation it is most important to ensure proper transverse stability.

Depending on the magnitude of the inclination, stability at small angles of inclination is distinguished ( initial stability) and stability at large inclination angles.

Depending on the character active forces distinguish between static and dynamic stability.

Static stability- is considered under the action of static forces, that is, the applied force does not change in magnitude. Dynamic stability- is considered under the action of changing (i.e. dynamic) forces, for example wind, sea waves, cargo movement, etc.

Initial lateral stability

Initial lateral stability. System of forces acting on the ship

During roll, stability is considered as initial at angles up to 10-15°. Within these limits, the righting force is proportional to the roll angle and can be determined using simple linear relationships.

In this case, the assumption is made that deviations from the equilibrium position are caused by external forces that do not change either the weight of the vessel or the position of its center of gravity (CG). Then the immersed volume does not change in size, but changes in shape. Equal-volume inclinations correspond to equal-volume waterlines, cutting off immersed volumes of the hull of equal magnitude. The line of intersection of the waterline planes is called the inclination axis, which, with equal volume inclinations, passes through the center of gravity of the waterline area. With transverse inclinations, it lies in the center plane.

Free surfaces

All the cases discussed above assume that the center of gravity of the vessel is stationary, that is, there are no loads that move when tilted. But when such loads exist, their influence on stability is much greater than others.

A typical case is liquid cargo (fuel, oil, ballast and boiler water) in tanks that are partially filled, that is, with free surfaces. Such loads can overflow when tilted. If the liquid cargo fills the tank completely, it is equivalent to a solid fixed cargo.

Effect of free surface on stability

If the liquid does not completely fill the tank, i.e. has a free surface that always occupies a horizontal position, then when the ship tilts at an angle θ the liquid flows towards the inclination. The free surface will take the same angle relative to the KVL.

Levels of liquid cargo cut off equal volumes of tanks, i.e. they are similar to equal volume waterlines. Therefore, the moment caused by the overflow of liquid cargo during a roll δm θ, can be represented similarly to the moment of shape stability m f, only δm θ opposite m f by sign:

δm θ = - γ f i x θ,

Where i x- moment of inertia of the free surface area of the liquid load relative to the longitudinal axis passing through the center of gravity of this area, γ f- specific gravity of liquid cargo

Then the restoring moment in the presence of a liquid load with a free surface:

m θ1 = m θ + δm θ = Phθ − γ f i x θ = P(h − γ f i x /γV)θ = Ph 1 θ,

Where h- transverse metacentric height in the absence of transfusion, h 1 = h − γ f i x /γV- actual transverse metacentric height.

The effect of the iridescent weight gives a correction to the transverse metacentric height δ h = - γ f i x /γV

The densities of water and liquid cargo are relatively stable, that is, the main influence on the correction is exerted by the shape of the free surface, or rather its moment of inertia. This means that the lateral stability is mainly affected by the width, and the longitudinal length of the free surface.

The physical meaning of the negative correction value is that the presence of free surfaces is always reduces

In contrast to the static effect, the dynamic effect of forces and moments imparts significant angular velocities and accelerations to the vessel. Therefore, their influence is considered in energies, more precisely in the form of the work of forces and moments, and not in the efforts themselves. In this case, the kinetic energy theorem is used, according to which the increment in the kinetic energy of the vessel's inclination is equal to the work of the forces acting on it.

When a heeling moment is applied to the ship m cr, constant in magnitude, it receives a positive acceleration with which it begins to roll. As you tilt, the restoring moment increases, but at first, up to the angle θ st, at which m cr = m θ, it will be less heeling. Upon reaching the static equilibrium angle θ st, the kinetic energy of rotational motion will be maximum. Therefore, the ship will not remain in the equilibrium position, but due to kinetic energy it will roll further, but slowly, since the righting moment is greater than the heeling moment. The previously accumulated kinetic energy is extinguished by the excess work of the restoring torque. As soon as the magnitude of this work is sufficient to completely extinguish the kinetic energy, the angular velocity will become zero and the ship will stop heeling.

The greatest angle of inclination that a ship receives from a dynamic moment is called the dynamic angle of heel θ din. In contrast, the angle of roll with which the ship will float under the influence of the same moment (according to the condition m cr = m θ), is called the static roll angle θ st.

If we refer to the static stability diagram, the work is expressed by the area under the righting moment curve m in. Accordingly, the dynamic roll angle θ din can be determined from the equality of areas OAB And BCD, corresponding to the excess work of the restoring torque. Analytically the same work is calculated as:

in the range from 0 to θ din.

Having reached the dynamic bank angle θ din, the ship does not come into equilibrium, but under the influence of an excess righting moment begins to accelerate to straighten. In the absence of water resistance, the ship would enter into undamped oscillations around the equilibrium position when heeling θ st Marine Dictionary - Refrigerated vessel Ivory Tirupati initial stability is negative Stability is the ability of a floating craft to resist external forces causing it to roll or trim and return to a state of equilibrium after the end of the disturbance... ... Wikipedia

A vessel whose hull rises above the water when moving under the influence of a lifting force created by wings submerged in the water. The patent for the vessel was issued in Russia in 1891, but these vessels began to be used in the 2nd half of the 20th century... ... Great Soviet Encyclopedia

An all-terrain vehicle capable of moving both on land and on water. An amphibious vehicle has an increased volume of a sealed body, which is sometimes supplemented with mounted floats for better buoyancy. Moving on water... ... Encyclopedia of technology

- (Malay) type of sailing vessel, lateral stability to the horn is provided by an outrigger float, attached. to the main body with transverse beams. The vessel is similar to a sailing catamaran. In ancient times, P. served as a means of communicating about the Pacific Ocean... ... Big Encyclopedic Polytechnic Dictionary

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amphibian- (from the Greek amphíbios leading a dual lifestyle) seaplane equipped with a land landing gear and capable of being based both on the water surface and at land airfields. The most common are A. boats. Taking off from the water... ... Encyclopedia "Aviation"

META CENTER

(Metacenter) - the point of intersection of the normals to the planes of the ship’s waterlines when it is tilted, drawn through the centers of gravity of the underwater volumes (centers of magnitude). Distinguish transverse M. - when the ship tilts about its longitudinal axis and longitudinal M. - when tilted about the transverse axis. For any vessel (both surface and underwater), the M. must be located above the center of gravity.

Samoilov K. I. Marine Dictionary. - M.-L.: State Naval Publishing House of the NKVMF of the USSR, 1941

Metacenter

the center of curvature of the trajectory of movement of the center of magnitude when the ship (floating body) tilts. When tilting from side to side, the position of the metacenter is different from the position of the metacenter when tilting from bow to stern. Accordingly, a distinction is made between transverse and longitudinal metacenters. If the ship’s center of gravity lies below the metacenter (primarily the transverse one), then when the ship tilts, a pair of forces will act on it, returning it to its original position. Therefore, the metacenter should be considered as the limit to which the vessel’s center of gravity can be raised (for example, when consuming supplies, unloading), without depriving the latter of its positive stability.

EdwART. Explanatory Naval Dictionary, 2010

Synonyms:

See what "META CENTER" is in other dictionaries:

Metacenter... Spelling dictionary-reference book

- (Greek, from meta, and kentron center). The center of gravity in stable equilibrium, usually located outside the real center. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. METACENTER Greek, from meta, and kentron, center ... Dictionary of foreign words of the Russian language

Point, the stability of the equilibrium (stability) of a floating body depends on the position of the swarm. In equilibrium, on a floating body, in addition to the force of gravity P applied at the center of gravity (CG) of the body (Fig.), there is also a buoyant force A, the line of action to... ... Physical encyclopedia

The point on whose position the stability of the equilibrium of a floating body depends. For a body with a longitudinal plane of symmetry, the metacenter is the point of intersection with this plane of the resultant forces of fluid pressure on the body... Big Encyclopedic Dictionary

METACENTER, metacenter, male (from Greek meta outside, behind and Latin centrum center) (physical). The point of intersection of a vertical line passing through the center of gravity of a floating body (ship) with the plane of the immersion line (with the waterline). Ushakov's Explanatory Dictionary... Ushakov's Explanatory Dictionary

Male, fur. center of gravity, outside the center of volume, magnitude; | mor. the point of mutual intersection of the plumb line passing through the center of gravity of the ship and the direction of lateral water pressure when the ship is tilted; The vessel must always be loaded so that the center... ... Dahl's Explanatory Dictionary

Noun, number of synonyms: 1 point (100) ASIS Dictionary of Synonyms. V.N. Trishin. 2013… Synonym dictionary

Lateral inclination of a floating vessel. The metacenter is designated M. The center of magnitude is designated B Metacenter (from Greek μετα through and Latin centrum center) center of curvature trajectory ... Wikipedia

LECTURE No. 4

General provisions of stability. Stability at low inclinations. Metacenter, metacentric radius, metacentric height. Metacentric formulas of stability. Determination of landing parameters and stability when moving cargo on a ship. Influence on the stability of loose and liquid cargo.

Inclining experience.

Stability is the ability of a ship, removed from a position of normal equilibrium by any external forces, to return to its original position after the cessation of the action of these forces. External forces that can displace a ship from a position of normal equilibrium include: wind, waves, movement of cargo and people, as well as centrifugal forces and moments that arise when the ship turns. The navigator is obliged to know the characteristics of his vessel and correctly assess the factors affecting its stability.

There are transverse and longitudinal stability. The lateral stability of a vessel is characterized by the relative position of the center of gravity G and center of magnitude WITH. Let's consider lateral stability.

If the ship is tilted on one side at a small angle (5-10°) (Fig. 1), the central point will move from point C to point . Accordingly, the supporting force acting perpendicular to the surface will intersect the center plane (DP) at the point M.

The point of intersection of the vessel's DP with the continuation of the direction of the supporting force during a roll is called initial metacenter M. Distance from the point of application of the supporting force WITH to the initial metacenter is called metacentric radius .

Fig.1 – C static forces acting on the ship at low heels

Distance from initial metacenter M to the center of gravity G called initial metacentric height .

The initial metacentric height characterizes the stability at small inclinations of the vessel, is measured in meters and is a criterion for the initial stability of the vessel. As a rule, the initial metacentric height of motor boats and speedboats is considered good if it is greater than 0.5 m, for some ships it is permissible less, but not less than 0.35 m.

A sharp tilt causes the ship to roll and the period of free rolling is measured with a stopwatch, that is, the time of full swing from one extreme position to the other and back. The transverse metacentric height of the vessel is determined by the formula:

, m

Where IN- width of the vessel, m; T- rolling period, sec.

To evaluate the results obtained, use the curve in Fig. 2, built according to data dacha designed boats.

Ri.2 – W Dependence of the initial metacentric height on the length of the vessel

If the initial metacentric height , determined by the above formula, will be below the shaded line, which means that the ship will have a smooth roll, but insufficient initial stability, and sailing on it can be dangerous. If the metacenter is located above the shaded strip, the vessel will be characterized by rapid (sharp) rolling, but increased stability, and therefore, such a vessel is more seaworthy, but its habitability is unsatisfactory. The optimal values will be those that fall within the shaded band area.

The roll of the ship on one side is measured by the angle between the new inclined position of the center plane with the vertical line.

The heeled side will displace more water than the opposite side, and the center of gravity will shift towards the heel. Then the resultant forces of support and weight will be unbalanced, forming a pair of forces with a shoulder equal to

The repeated action of weight and support forces is measured by the righting moment:

Where D- buoyancy force equal to the weight of the vessel; l- stability arm.

This formula is called the metacentric stability formula and is valid only for small roll angles, at which the metacenter can be considered constant. At large roll angles, the metacenter is not constant, as a result of which the linear relationship between the righting moment and roll angles is violated.

Small ( ) and big ( ) metacentric radii can be calculated using the formulas of Professor A.P. Fan der Fleet:

;
.

By the relative position of the cargo on the ship, the navigator can always find the most favorable value of the metacentric height, at which the ship will be sufficiently stable and less subject to pitching.

The heeling moment is the product of the weight of the cargo moved across the vessel by a shoulder equal to the distance of movement. If a person weighs 75 kg, sitting on a bank will move across the ship by 0.5 m, then the heeling moment will be equal to 75 * 0.5 = 37.5 kg/m.

To change the moment that heels the ship by 10°, it is necessary to load the ship to full displacement completely symmetrically relative to the center plane. The vessel's loading should be checked by drafts measured on both sides. The inclinometer is installed strictly perpendicular to the DP so that it shows 0°.

After this, you need to move loads (for example, people) at pre-marked distances until the inclinometer shows 10°. The test experiment should be carried out as follows: tilt the ship on one side and then on the other side. Knowing the fastening moments of a ship heeling at various (up to the greatest possible) angles, it is possible to construct a static stability diagram (Fig. 3), which will allow assessing the stability of the ship.

Fig.3 – Static stability diagram

Stability can be increased by increasing the width of the vessel, lowering the center of gravity, and installing stern buoys.

If the CG of the vessel is located below the CV, then the vessel is considered to be very stable, since the supporting force during a roll does not change in magnitude and direction, but the point of its application shifts towards the tilt of the vessel (Fig. 4, a). Therefore, when heeling, a pair of forces is formed with a positive restoring moment, tending to return the ship to its normal vertical position on a straight keel. It is easy to verify that h>0, with a metacentric height of 0. This is typical for yachts with a heavy keel and is not typical for larger vessels with a conventional hull design.

If the CG is located above the CV, then three cases of stability are possible, which the navigator should be well aware of.

1st case of stability

Metacentric height h>0. If the center of gravity is located above the center of magnitude, then when the vessel is in an inclined position, the line of action of the supporting force intersects the center plane above the center of gravity (Fig. 4, b).

Fig. 4 – Case of a stable vessel

In this case, a couple of forces with a positive restoring moment is also formed. This is typical for most conventionally shaped boats. Stability in this case depends on the hull and the position of the center of gravity in height. When heeling, the heeling side enters the water and creates additional buoyancy, tending to level the ship. However, when a ship rolls with liquid and bulk cargo that can move towards the roll, the center of gravity will also shift towards the roll. If the center of gravity during a roll moves beyond the plumb line connecting the center of magnitude with the metacenter, then the ship will capsize.

2nd case of an unstable vessel in indifferent equilibrium

Metacentric height h= 0. If the CG lies above the CG, then during a roll the line of action of the support force passes through the CG MG = 0 (Fig. 5).

Fig. 5 – Case of an unstable vessel in indifferent equilibrium

In this case, the CV is always located on the same vertical with the CG, so there is no recovering pair of forces. Without the influence of external forces, the ship cannot return to an upright position. In this case, it is especially dangerous and completely unacceptable to transport liquid and bulk cargo on a ship: with the slightest rocking, the ship will capsize. This is typical for boats with a round frame.

3rd case of an unstable ship with unstable equilibrium

Metacentric height h<0. ЦТ расположен выше ЦВ, а в наклонном положении судна линия действия силы поддержания пересекает след диаметральной плоскости ниже ЦТ (рис. 6). Сила тяжести и сила поддержания при малейшем крене образуют пару сил с отрицательным восстанавливающим моментом и судно опрокидывается.

Fig.6 - C beam of an unstable ship in unstable equilibrium

The analyzed cases show that the ship is stable if the metacenter is located above the ship's CG. The lower the CG goes, the more stable the ship is. In practice, this is achieved by placing cargo not on the deck, but in the lower rooms and holds.

Due to the influence of external forces on the ship, as well as as a result of insufficiently strong securing of the cargo, it is possible for it to move on the ship. Let us consider the influence of this factor on changes in the landing parameters of the vessel and its stability.

Vertical movement of cargo.

Fig. 1 – The influence of vertical movement of the load on the change in metacentric height

Let us determine the change in the landing and stability of the vessel caused by the movement of a small load in the vertical direction (Fig. 1) from the point exactly . Since the mass of the cargo does not change, the displacement of the vessel remains unchanged. Therefore, the first equilibrium condition is satisfied:
. It is known from theoretical mechanics that when one of the bodies moves, the CG of the entire system moves in the same direction. Therefore, the ship's CG will move to a point , and the vertical itself will pass, as before, through the center of the quantity .

The second equilibrium condition will be met:
.

Since in our case both equilibrium conditions are met, we can conclude: When the load moves vertically, the ship does not change its equilibrium position.

Let's consider the change in initial lateral stability. Since the shape of the volume of the ship’s hull immersed in water and the area of the waterline have not changed, the position of the center of the value and the transverse metacenter remains unchanged when the load moves vertically. Only the ship's CG moves, which will result in a decrease in metacentric height
, and
, where
, Where - weight of the transported cargo, kN; - the distance by which the CG of the load has moved in the vertical direction, m.

Stability is one of the most important seaworthiness of a vessel, which is associated with extremely important issues regarding navigation safety. Loss of stability almost always means the death of the ship and, very often, the crew. Unlike changes in other seaworthiness, the decrease in stability is not visible, and the crew of the ship, as a rule, is unaware of the impending danger until the last seconds before capsizing. Therefore, the greatest attention must be paid to the study of this section of the theory of the ship.

In order for a ship to float in a given equilibrium position relative to the surface of the water, it must not only satisfy the conditions of equilibrium, but also be able to resist external forces tending to take it out of the equilibrium position, and after the cessation of the action of these forces, return to its original position. Therefore, the balance of the ship must be stable or, in other words, the ship must have positive stability.

Thus, stability is the ability of a vessel, brought out of a state of equilibrium by external forces, to return to its original equilibrium position again after the action of these forces ceases.

The stability of the vessel is associated with its balance, which serves as a characteristic of the latter. If the ship's balance is stable, then the ship has positive stability; if its equilibrium is indifferent, then the ship has zero stability, and, finally, if the ship's equilibrium is unstable, then it has negative stability.

Tanker Captain Shiryaev

This chapter will examine the lateral inclinations of the ship in the midship frame plane.

Stability during transverse inclinations, i.e. when a roll occurs, is called transverse. Depending on the angle of inclination of the vessel, lateral stability is divided into stability at small angles of inclination (up to 10-15 degrees), or the so-called initial stability, and stability at large angles of inclination.

The tilting of the ship occurs under the influence of a pair of forces; the moment of this pair of forces, causing the vessel to rotate around the longitudinal axis, will be called heeling Mkr.

If Mcr applied to the ship increases gradually from zero to the final value and does not cause angular accelerations, and therefore inertia forces, then stability with such an inclination is called static.

The heeling moment acting on the ship instantly leads to the occurrence of angular acceleration and inertial forces. The stability manifested by such an inclination is called dynamic.

Static stability is characterized by the occurrence of a restoring moment, which tends to return the vessel to its original equilibrium position. Dynamic stability is characterized by the work of this moment from the beginning to the end of its action.

Let us consider the uniform transverse inclination of the vessel. We will assume that in the initial position the ship has a straight landing. In this case, the supporting force D' acts in the DP and is applied at point C - the center of the vessel’s size (Center of buoyancy-B).

Rice. 1

Let us assume that the vessel, under the influence of a heeling moment, has received a transverse inclination at a small angle θ. Then the center of the magnitude will move from point C to point C 1 and the supporting force, perpendicular to the new existing waterline B 1 L 1, will be directed at an angle θ to the center plane. The action lines of the original and new direction of the support force will intersect at point m. This point of intersection of the line of action of the supporting force at an infinitesimal equal-volume inclination of a floating vessel is called the transverse metacentre.

We can give another definition to the metacenter: the center of curvature of the curve of displacement of the center of magnitude in the transverse plane is called the transverse metacenter.

The radius of curvature of the curve of displacement of the center of a quantity in the transverse plane is called the transverse metacentric radius (or small metacentric radius). It is determined by the distance from the transverse metacenter m to the center of magnitude C and is denoted by the letter r.

The transverse metacentric radius can be calculated using the formula:

i.e., the transverse metacentric radius is equal to the moment of inertia Ix of the area of the waterline relative to the longitudinal axis passing through the center of gravity of this area, divided by the volumetric displacement V corresponding to this waterline.

Stability conditions

Let us assume that the ship, which is in a direct equilibrium position and floating along the waterline of the overhead line, as a result of the action of the external heeling moment Mkr, has heeled so that the original waterline of the overhead line with the new existing waterline B 1 L 1 forms a small angle θ. Due to the change in the shape of the hull part submerged in water, the distribution of hydrostatic pressure forces acting on this part of the hull will also change. The center of the vessel's size will move towards the roll and move from point C to point C 1.

The supporting force D', remaining unchanged, will be directed vertically upward perpendicular to the new effective waterline, and its line of action will intersect the DP at the original transverse metacenter m.

The position of the ship's center of gravity remains unchanged, and the weight force P will be perpendicular to the new waterline B 1 L 1. Thus, the forces P and D', parallel to each other, do not lie on the same vertical and, therefore, form a pair of forces with the arm GK, where point K is the base of the perpendicular lowered from point G to the direction of action of the supporting force.

The pair of forces formed by the weight of the vessel and the supporting force, tending to return the vessel to its original equilibrium position, is called a restoring pair, and the moment of this pair is called the restoring moment Mθ.

The issue of stability of a heeled ship is decided by the direction of action of the righting moment. If the righting moment tends to return the ship to its original equilibrium position, then the righting moment is positive, the stability of the ship is also positive - the ship is stable. In Fig. Figure 2 shows the location of the forces acting on the ship, which corresponds to a positive righting moment. It is easy to verify that such a moment occurs if the CG lies below the metacenter.

Rice. 2

Rice. 3

In Fig. Figure 3 shows the opposite case, when the restoring moment is negative (the center of gravity lies above the metacenter). It tends to further deflect the ship from its equilibrium position, since the direction of its action coincides with the direction of action of the external heeling moment Mkr. In this case, the ship is not stable.

Theoretically, it can be assumed that the restoring moment when the vessel tilts is equal to zero, i.e. the force of the weight of the vessel and the supporting force are located on the same vertical, as shown in Fig. 4.

Rice. 4

The absence of a righting moment leads to the fact that after the heeling moment ceases, the ship remains in an inclined position, i.e., the ship is in indifferent equilibrium.

Thus, according to the relative position of the transverse metacenter m and C.T. G can be judged on the sign of the righting moment or, in other words, on the stability of the vessel. So, if the transverse metacenter is above the center of gravity (Fig. 2), then the ship is stable.

If the transverse metacenter is located below the center of gravity or coincides with it (Fig. 3, 4), the ship is not stable.

This gives rise to the concept of metacentric height: transverse metacentric height is the elevation of the transverse metacenter above the center of gravity of the vessel in the initial equilibrium position.

The transverse metacentric height (Fig. 2) is determined by the distance from the center of gravity (i.e. G) to the transverse metacenter (i.e. m), i.e., the segment mG. This segment is a constant value, since and C.T. , and the transverse metacenter do not change their position at small inclinations. In this regard, it is convenient to accept it as a criterion for the initial stability of a vessel.

If the transverse metacenter is located above the center of gravity of the vessel, then the transverse metacentric height is considered positive. Then the condition for the stability of the vessel can be given in the following formulation: the vessel is stable if its transverse metacentric height is positive. This definition is convenient in that it allows one to judge the stability of the vessel without considering its inclination, i.e., at a roll angle of zero, when there is no righting moment at all. To establish what data is necessary to obtain the value of the transverse metacentric height, let us turn to Fig. 5, which shows the relative location of the center of magnitude C, the center of gravity G and the transverse metacenter m of a vessel having positive initial lateral stability.

Rice. 5

The figure shows that the transverse metacentric height h can be determined by one of the following formulas:

h = Z C ± r – Z G ;

h = Z m – Z G .

The transverse metacentric height is often determined using the last equality. The applicate of the transverse metacenter Zm can be found from the metacentric diagram. The main difficulties in determining the transverse metacentric height of a vessel arise when determining the applicate of the center of gravity ZG, which is determined using a summary table of the vessel's mass load (the issue was discussed in the lecture -).

In foreign literature, the designation of the corresponding points and stability parameters may look as shown below in Fig. 6.

Rice. 6

where K is the keel point;
B – center of buoyancy;
G—center of gravity;
M – transverse metacentre;
CV – applicate of the center of magnitude;
KG – applicate of the center of gravity;
KM - applicate of the transverse metacenter;
VM – transverse metacentric radius (Radius of metacentre);
BG – elevation of the center of gravity above the center of magnitude;
GM – transverse metacentric height.

The static stability arm, denoted in our literature as GK, is denoted in foreign literature as GZ.

Lateral inclinations

In the upright position of the vessel, the CV is in the DP (point C) and the line of action of the buoyancy force gV also lies in the DP (Fig. 2). When the vessel is tilted transversely at an angle I, the shape of the immersed volume changes, the CV moves in the direction of inclination from point C to point C I, and the line of action of the buoyancy force will be inclined to the DP at an angle I.

The point of intersection of the lines of action of the buoyancy force at an infinitesimal transverse equal-volume inclination of the vessel is called the transverse metacenter (point m in Fig. 2). The radius of curvature of the trajectory of the CV r (the elevation of the transverse metacenter above the CV) is called the transverse metacentric radius.

In the general case, the CV trajectory is a complex spatial curve and each inclination angle corresponds to its own position of the metacenter (Fig. 3). However, for small equal-volume inclinations, with a known approximation, it can be assumed that the trajectory

The CV lies in the plane of inclination and is an arc of a circle with its center at point m. Thus, we can assume that during a small transverse equal-volume inclination of the vessel from a straight position, the transverse metacenter lies in the DP and does not change its position (r = const).

Rice. 2.

Rice. 3. CV movement at high inclinations

Rice. 4.

The expression for the transverse metacentric radius r is obtained from the condition that the axis of the small transverse equal-volume inclination of the vessel lies in the DP and that with such an inclination the wedge-shaped volume v is, as it were, transferred from the side that left the water to the side that entered the water (Fig. 4).

According to the well-known theorem of mechanics, when a body belonging to a system of bodies moves, the center of gravity of the entire system will move in the same direction parallel to the movement of the body, and these movements are inversely proportional to the gravitational forces of the body and the system, respectively. This theorem can be extended to the volumes of homogeneous bodies. Let's denote:

С С И - movement of the central point (geometric center of volume V),

b - displacement of the geometric center of the wedge-shaped volume v. Then, in accordance with the theorem

from where: S S I =

For the vessel length element dx, assuming that the wedge-shaped volume has the shape of a triangle in the plane of the frame, we obtain:

or at low angle

If by, then:

dv b = y 3 And dx.

Integrating, we get:

v b = And y 3 dx, or:

where J x = ydx is the moment of inertia of the waterline area relative to the longitudinal central axis.

Then the expression for moving the CV will look like:

As can be seen from Fig. 5, at small angle I