The stability of the vessel is the main forces and conditions of equilibrium. Vessel static stability diagram
- Depending on the plane of inclination, there are lateral stability when heeling and longitudinal stability at trim. With regard to surface ships (vessels), due to the elongation of the shape of the ship's hull, its longitudinal stability is much higher than the transverse one, therefore, for the safety of navigation, it is most important to ensure proper transverse stability.
- Depending on the magnitude of the inclination, stability is distinguished at small angles of inclination ( initial stability) and stability at large angles of inclination.
- Depending on the nature of the acting forces, static and dynamic stability are distinguished.
Initial lateral stability
With a roll, stability is considered as initial at angles up to 10-15 °. Within these limits, the restoring force is proportional to the angle of heel 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 magnitude, but changes in shape. Equal-volume inclinations correspond to equal-volume waterlines, cutting off equal immersed hull volumes. The line of intersection of the planes of the waterlines is called the axis of inclination, which, with equal volume inclinations, passes through the center of gravity of the waterline area. With transverse inclinations, it lies in the diametrical plane.
Free surfaces
All the cases discussed above assume that the center of gravity of the ship is stationary, that is, there are no loads that move when tilted. But when such weights are present, their influence on stability is much greater than the others.
A typical case is liquid cargoes (fuel, oil, ballast and boiler water) in partially filled tanks, that is, with free surfaces. Such loads are capable of overflowing when tilted. If the liquid cargo fills the tank completely, it is equivalent to a solid fixed cargo.
Influence of free surface on stability
If the liquid does not fill the tank completely, that is, it has a free surface that always occupies a horizontal position, then when the vessel is tilted at an angle θ the liquid overflows in the direction of inclination. The free surface will take the same angle relative to the design line.
Levels of liquid cargo cut off equal volumes of tanks, that is, they are similar to waterlines of equal volume. Therefore, the moment caused by the transfusion of liquid cargo when heeling 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 x i x θ,Where i x- the moment of inertia of the area of the free surface of the liquid cargo relative to the longitudinal axis passing through the center of gravity of this area, γ- specific gravity of the 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 x i x θ = P(h − γ x i x /γV)θ = Ph 1 θ,Where h- transverse metacentric height in the absence of transfusion, h 1 = h − γ g i x /γV- actual transverse metacentric height.
The influence of the overflowing load gives a correction to the transverse metacentric height δ h = − γ x i x /γV
The densities of water and liquid cargo are relatively stable, that is, the main influence on the correction is 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 value of the correction is that the presence of free surfaces is always reduces stability. Therefore, organizational and constructive measures are being taken to reduce them:
Dynamic stability of the vessel
Unlike static, the dynamic effect of forces and moments imparts significant angular velocities and accelerations to the ship. 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 ship'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 the inclination increases, the restoring moment increases, but at the beginning, up to the angle θ st, at which m cr = m θ, it will be less heeling. Upon reaching the angle of static equilibrium θ st, the kinetic energy of rotational motion will be maximum. Therefore, the ship will not remain in the equilibrium position, but due to the kinetic energy it will roll further, but slower, since the restoring moment is greater than the heeling one. The previously accumulated kinetic energy is repaid by the excess work of the restoring moment. As soon as the magnitude of this work is sufficient to completely extinguish the kinetic energy, the angular velocity will become equal to zero and the ship will stop heeling.
The largest angle of inclination that the ship receives from the dynamic moment is called the dynamic angle of heel. θ dyn. In contrast to it, the angle of heel with which the ship will sail under the action of the same moment (according to the condition m cr = m θ), is called the static bank angle θ st.
Referring to the static stability diagram, work is expressed as the area under the restoring moment curve m in. Accordingly, the dynamic bank angle θ dyn can be determined from the equality of areas OAB And BCD corresponding to the excess work of the restoring moment. Analytically, the same work is calculated as:
,on the interval from 0 to θ dyn.
Reaching dynamic bank angle θ dyn, the ship does not come into equilibrium, but under the influence of an excess restoring moment, it begins to straighten rapidly. In the absence of water resistance, the ship would enter into undamped oscillations around the equilibrium position when heeling θ st / ed. Physical Encyclopedia
Vessel, the ability of the vessel to resist external forces that cause it to heel or trim, and return to its original equilibrium position after the termination of their action; one of the most important seaworthiness of a ship. O. when heeling ... ... Great Soviet Encyclopedia
The quality of the ship is to be in balance in a straight position and, being taken out of it by the action of some kind of force, return to it again after the termination of its action. This quality is one of the most important for the safety of navigation; there were many… … Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
G. The ability of the vessel to float upright and to straighten up after tilting. Explanatory Dictionary of Ephraim. T. F. Efremova. 2000... Modern Dictionary Russian language Efremova
Stability, stability, stability, stability, stability, stability, stability, stability, stability, stability, stability, stability (
Vessel stability at low inclination angles (θ less than 120) is called initial, in this case the restoring moment depends linearly on the angle of heel.
Consider the equal volume inclinations of the ship in the transverse plane.
In doing so, we will assume that:
the angle of inclination θ is small (up to 12°);
the section of the curve CC1 of the CV trajectory is an arc of a circle lying in the plane of inclination;
the line of action of the buoyancy force in the inclined position of the vessel passes through the initial metacenter m.
Under such assumptions, the total moment of a pair of forces (forces of weight and buoyancy) acts in the plane of inclination on the arm GK, which is called the static stability arm, and the moment itself - restoring moment and is designated Mv.
Мv = Рhθ.
This formula is called metacentric formula for transverse stability.
With transverse inclinations of the vessel at an angle exceeding 12 °, it is not possible to use the above expression, since the center of gravity of the inclined waterline area is shifted from the diametrical plane, and the center of magnitude does not move along an arc of a circle, but along a curve of variable curvature, i.e. metacentric the radius changes its value.
To solve stability issues at large angles of heel, static stability diagram (DSO), which is a graph expressing the dependence of the shoulders of static stability on the angle of heel.
The diagram of static stability is constructed using pantocarrens - graphs of the dependence of the stability shoulders of the form lf on the volumetric displacement of the vessel and the angle of heel. Pantocarrens of a particular ship are built in the design bureau for heel angles from 0 to 900 for displacements from an empty ship to a ship's displacement in full load (there are tables of curved elements of the theoretical drawing on the ship).
Rice - a - pantocarenes; b - graphs for determining the shoulders of static stability l
To build a DSO, you need:
on the abscissa axis of the pantocaren, set aside a point corresponding to the volumetric displacement of the vessel at the time of completion of loading;
restore the perpendicular from the obtained point and read off the values of lf from the curves for roll angles of 10, 200, etc.;
calculate the shoulders of static stability according to the formula:
l = lf – a*sinθ = lf – (Zg – Zc) *sinθ,
where a \u003d Zg - Zc (at the same time, the applicate of the ship's CG Zg is found from the calculation of the load corresponding to a given displacement - they fill in a special table, and the applicate of the CV Zc - from the tables of curved elements of the theoretical drawing);
construct a curve lf and a sinusoid a*sinθ, the differences in the ordinates of which are the shoulders of static stability l.
To plot the static stability diagram, on the abscissa axis lay the roll angles θ in degrees, and along the ordinate axis - the shoulders of static stability in meters. The diagram is built for a certain displacement.
On fig. certain states of the vessel are shown at various inclinations:
position I (θ = 00) corresponds to the position of static equilibrium (l= 0);
position II (θ = 200) - a shoulder of static stability appeared (1 = 0.2 m);
position III (θ = 370) - the static stability arm has reached its maximum (I = 0.35 m);
position IV (θ = 600) - the static stability arm decreases (I = 0.22 m);
position V (θ = 830) - the static stability arm is equal to zero. The ship is in a position of static unstable equilibrium, since even a slight increase in heel will cause the ship to capsize;
position VI (θ = 1000) − the static stability arm becomes negative and the vessel capsizes.
Starting from positions large, than position III, the vessel will not be able to independently return to the equilibrium position without applying an external force to it.
Thus, the vessel is stable within the angle of heel from zero to 83°. The point of intersection of the curve with the abscissa axis, corresponding to the angle of capsizing of the vessel (θ = 830) is called chart sunset point, and this angle chart sunset angle.
Maximum heeling moment Мcr max , which the vessel can support without capsizing corresponds to the maximum static stability arm.
Using the static stability diagram, it is possible to determine the heeling angle from the known heeling moment M1, which has arisen under the action of wind, waves, cargo displacement, etc. To determine it, a horizontal line is drawn from point M1 until it intersects with the curve of the diagram, and from the resulting point, a perpendicular is lowered to the abscissa axis (θ = 260). The reverse problem is solved in the same way.
According to the static stability diagram, it is possible to determine the value of the initial metacentric height, to find which it is necessary:
from a point on the x-axis corresponding to a bank angle of 57.3° (one radian), restore the perpendicular;
from the origin, draw a tangent to the initial section of the curve;
measure the segment of the perpendicular enclosed between the abscissa axis and the tangent, which is equal to the ship's metacentric height on the scale of the stability arms.
LECTURE №4
General provisions of stability. Stability at low inclinations. Metacenter, metacentric radius, metacentric height. Metacentric stability formulas. Determination of landing parameters and stability when moving cargo on a ship. Influence on the stability of loose and liquid cargoes.
Rolling experience.
Stability called the ability of a ship, brought out of a position of normal equilibrium by any external forces, to return to its original position after the termination of these forces. External forces that can take the ship out of normal equilibrium include: wind, waves, movement of goods and people, as well as centrifugal forces and moments that occur when the ship turns. The navigator is obliged to know the features of his vessel and correctly assess the factors affecting its stability.
Distinguish between transverse and longitudinal stability. The transverse stability of the vessel is characterized by the relative position of the center of gravity G and center of magnitude WITH. Consider lateral stability.
If the vessel is heeled to one side at a small angle (5-10°) (Fig. 1), the CV will move from point C to point . Accordingly, the support force acting perpendicular to the surface will cross the diametrical plane (DP) at the point M.
The point of intersection of the ship's DP with the continuation of the direction of the support force during roll is called initial metacenter M. Distance from the point of application of the support force WITH to the initial metacenter is called metacentric radius .
Fig.1 - C static forces acting on a ship at low heels
Distance from the initial metacenter M to the center of gravity G called initial metacentric height
.
The initial metacentric height characterizes the stability at low 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 boats is considered good if it is more than 0.5 m, for some ships it is permissible less, but not less than 0.35 m.
A sharp inclination causes the ship to roll and the stopwatch measures the period of free roll, that is, the time of full swing from one extreme position to another and back. The transverse metacentric height of the vessel is determined by the formula:
,
m
Where IN- ship's width, m; T- pitching period, sec.
The curve in Fig. 1 serves to evaluate the obtained results. 2, built according to the data 
country-designed boats.
Ri.2 - H 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 bar, which means that the vessel will have a smooth roll, but insufficient initial stability, and navigation 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 habitability on it is unsatisfactory. Optimal values will fall within the zone of the shaded band.
The list of the vessel on one of the sides 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 CV will shift in the direction of the roll. 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 the weight and support forces is measured by the restoring moment:
.
Where D- buoyancy force equal to the ship's weight force; l- Stability shoulder.
This formula is called the metacentric stability formula and is valid only for small heeling angles, at which the metacenter can be considered constant. At large angles of heel, the metacenter is not constant, as a result of which the linear relationship between the restoring moment and the angles of heel is violated.
Small ( 
) and big ( 
) metacentric radii can be calculated using the formulas of Professor A.P. Van der Fliet:
;
.
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 rolling.
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 weighing 75 kg, sitting on the 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 symmetrical about the diametrical plane. The loading of the ship should be checked by drafts measured from both sides. The inclinometer is installed strictly perpendicular to the DP so that it shows 0 °.
After that, it is necessary to move loads (for example, people) at pre-marked distances until the inclinometer shows 10 °. An experiment for verification should be carried out as follows: heel the ship on one side, and then on the other side. Knowing the fixing moments of the heeling ship at various (up to the largest possible) angles, it is possible to construct a static stability diagram (Fig. 3), which will allow assessing the stability of the ship.
Fig.3 - Diagram of static stability
Stability can be increased by increasing the width of the vessel, lowering the CG, and installing stern boules.
If the ship's CG is located below the CG, then the ship is considered to be very stable, since the support force during roll does not change in magnitude and direction, but the point of its application shifts towards the ship's inclination (Fig. 4, a). Therefore, when heeling, a pair of forces is formed with a positive restoring moment, tending to return the ship to a 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 heavy keel yachts and not typical for larger conventional hull boats.
If the CG is located above the CG, 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 with the inclined position of the vessel, the line of action of the support force crosses the diametrical plane above the center of gravity (Fig. 4, b).
Fig.4 - The case of a stable vessel
In this case, a pair of forces with a positive restoring moment is also formed. This is typical of most conventionally shaped ships. Stability in this case depends on the body 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 vessel rolls with liquid and bulk cargoes capable of moving in the roll direction, the center of gravity will also shift in the roll direction. 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 unstable sudok with indifferent equilibrium
metacentric height h= 0. If the CG lies above the CG, then during the roll the line of action of the support force passes through the CG MG=0 (Fig. 5).
Fig.5 - The case of an unstable ship with indifferent equilibrium
In this case, the CV is always located on the same vertical with the CG, so there is no restoring pair of forces. Without the influence of external forces, the ship cannot return to a straight position. In this case, it is especially dangerous and completely unacceptable to transport liquid and bulk cargoes 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 in 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 ray 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 falls, the more stable the ship. 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 fastening of the cargo, it is possible to move it on the ship. Let us consider the influence of this factor on the change in the landing parameters of the vessel and its stability.
Vertical movement of cargo.
Fig.1 - The effect 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 
move to a point 
, and the vertical itself will pass, as before, through the center of magnitude 
.
The second equilibrium condition will be met: 
.
Since in our case both conditions of equilibrium are met, we can conclude: when the cargo is moved vertically, the ship does not change its equilibrium position.
Consider the change in the initial transverse stability. Since the shapes of the volume of the ship's hull submerged in water and the area of the waterline have not changed, the position of the center of magnitude 
and the transverse metacenter remains unchanged when moving the load vertically. Only the ship's CG moves, which will result in a decrease in the metacentric height 
, and 
, where 
, Where 
- the weight of the transported cargo, kN;
- the distance that the cargo CG has moved in the vertical direction, m.
So the new value 
, where the sign (+) is used when moving the load up, and the sign (-) is used down.
It can be seen from the formula that the vertical movement of the load upwards causes a decrease in the lateral stability of the vessel, and when moving down, the lateral stability increases.
The change in stability is equal to the product 
. The change in transverse stability will be relatively smaller for a ship with a large displacement than for a ship with a small one, therefore, on ships with a large displacement, the movement of goods is safer than on small ships.
Transverse horizontal movement of the load.
Cargo movement 
from a point 
exactly 
(Fig. 2) at a distance 
will cause the ship to roll at an angle 
and the displacement of its CG in the direction parallel to the line of movement of the load.
Fig. 2 - The occurrence of a heeling moment during the transverse movement of the load
Leaning into a corner 
, the ship comes to a new equilibrium position, the ship's gravity 
, now applied at the point 
and sustaining power 
, applied at the point 
, act along one vertical perpendicular to the new waterline 
.
The movement of the load leads to the formation of a heeling moment:
,
Where 
- shoulder of cargo movement, m.
Restoring moment according to the metacentric formula of stability
.
Since the ship is in equilibrium, then 
and , whence the angle of heel during the transverse movement of the load 
. Since the roll angle is small, 
.
If the vessel already has an initial angle of heel, then after the horizontal movement of the cargo, the angle of heel will be 
.
Stability called the ability of the ship to resist the forces that deviate it from the equilibrium position, and return to its original equilibrium position after the termination of these forces.
The equilibrium conditions of the vessel obtained in Chapter 4 "Buoyancy" are not sufficient for it to constantly float in a given position relative to the water surface. It is also necessary that the balance of the vessel is stable. The property, which in mechanics is called the stability of equilibrium, in the theory of the ship is usually called stability. Thus, buoyancy provides the conditions for the equilibrium position of the vessel with a given landing, and stability ensures the preservation of this position.
The stability of the vessel changes with an increase in the angle of inclination and at a certain value it is completely lost. Therefore, it seems appropriate to study the stability of the vessel at small (theoretically infinitesimal) deviations from the equilibrium position with Θ = 0, Ψ = 0, and then determine the characteristics of its stability, their permissible limits at large inclinations.
It is customary to distinguish vessel stability at low inclination angles (initial stability) and stability at high inclination angles.
When considering small inclinations, it is possible to make a number of assumptions that make it possible to study the initial stability of the vessel within the framework of the linear theory and obtain simple mathematical dependences of its characteristics. Vessel stability at large angles of inclination is studied using a refined non-linear theory. Naturally, the stability property of the ship is unified and the accepted division is purely methodological.
When studying the stability of a vessel, its inclinations are considered in two mutually perpendicular planes - transverse and longitudinal. When the vessel is tilted in the transverse plane, determined by the angles of heel, it is studied lateral stability; with inclinations in the longitudinal plane, determined by the trim angles, study it longitudinal stability.
If the inclination of the ship occurs without significant angular accelerations (pumping liquid cargo, slow water flow into the compartment), then stability is called static.
In some cases, the forces tilting the vessel act suddenly, causing significant angular accelerations (wind squall, wave surge, etc.). In such cases, consider dynamic stability.
Stability is a very important nautical property of a vessel; together with buoyancy, it ensures the navigation of the vessel in a given position relative to the surface of the water, which is necessary to ensure propulsion and maneuver. A decrease in the ship's stability can cause an emergency roll and trim, and a complete loss of stability can cause it to capsize.
In order to prevent a dangerous decrease in the ship's stability, all crew members must:
always have a clear idea of the ship's stability;
know the reasons that reduce stability;
know and be able to apply all means and measures to maintain and restore stability.
In the theory of lateral stability, ship inclinations are considered that occur in the midship plane, and an external moment, called the heeling moment, also acts in the midship plane.
Without limiting ourselves to small ship inclinations for the time being (they will be considered as a special case in the section “Initial stability”), let us consider the general case of ship heeling due to the action of an external heeling moment that is constant in time. In practice, such a heeling moment can arise, for example, from the action of a constant wind force, the direction of which coincides with the transverse plane of the vessel - the midship plane. Under the influence of this heeling moment, the ship has a constant roll to the opposite side, the value of which is determined by the wind force and the restoring moment from the side of the ship.
In the literature on the theory of the ship, it is customary to combine two positions of the ship in the figure at once - straight and rolled. The banked position corresponds to a new position of the waterline relative to the vessel, which corresponds to a constant submerged volume, however, the shape of the underwater part of the banked vessel no longer has symmetry: the starboard side is submerged more than the port side (Fig.1).
All waterlines corresponding to one value of the ship's displacement (at a constant weight of the ship) are called equal volume.
The exact image in the figure of all equal-volume waterlines is associated with great computational difficulties. In ship theory, there are several methods for graphical representation of equal volume waterlines. At very small angles of heel (at infinitely small equal-volume inclinations), one can use the corollary from L. Euler's theorem, according to which two equal-volume waterlines that differ by an infinitely small angle of heel intersect along a straight line passing through their common center of gravity of the area (for finite inclinations, this the statement loses force, since each waterline has its own center of gravity of the area).
If we ignore the actual distribution of the forces of the ship's weight and hydrostatic pressure, replacing their action with concentrated resultant forces, then we come to the scheme (Fig. 1). At the ship's center of gravity, a weight force is applied, directed in all cases perpendicular to the waterline. Parallel to it, the buoyancy force acts, applied in the center of the underwater volume of the vessel - in the so-called center of magnitude(dot WITH).
Due to the fact that the behavior (and origin) of these forces do not depend on each other, they no longer act along the same line, but form a pair of forces parallel and perpendicular to the acting waterline V 1 L 1. With regard to the strength of the weight R we can say that it remains vertical and perpendicular to the surface of the water, and the heeled vessel deviates from the vertical, and only the convention of the figure requires that the vector of the weight force be deflected from the diametrical plane. It is easy to understand the specifics of this approach if we imagine a situation with a video camera mounted on a ship, showing on the screen the sea surface tilted at an angle equal to the ship's roll angle.
The resulting pair of forces creates a moment, which is commonly called restoring moment. This moment counteracts the external heeling moment and is the main object of attention in the theory of stability.
The value of the restoring moment can be calculated by the formula (as for any pair of forces) as the product of one (any of the two) forces and the distance between them, called shoulder of static stability:
Formula (1) indicates that both the shoulder and the moment itself depend on the ship's roll angle, i.e. are variable (in the sense of roll) quantities.
However, not in all cases the direction of the restoring moment will correspond to the image in Fig.1.
If the center of gravity (as a result of the peculiarities of the placement of goods along the height of the vessel, for example, with an excess of cargo on the deck) is quite high, then a situation may arise when the weight force is to the right of the line of action of the support force. Then their moment will act in the opposite direction and will contribute to the heeling of the vessel. Together with the external heeling moment, they will capsize the vessel, since there are no other opposing moments anymore.
It is clear that in this case this situation should be assessed as unacceptable, since the ship does not have stability. Consequently, with a high position of the center of gravity, the ship may lose this important seaworthiness - stability.
On offshore displacement ships, the ability to influence the stability of the vessel, “control” it, is provided to the navigator only by rational placement of cargo and reserves along the height of the vessel, which determine the position of the center of gravity of the vessel. Be that as it may, the influence of crew members on the position of the center of magnitude is excluded, since it is associated with the shape of the underwater part of the hull, which (with a constant displacement and draft of the vessel) is unchanged, and in the presence of a roll of the vessel it changes without human intervention and depends only on draft. Human influence on the shape of the hull ends at the design stage of the vessel.
Thus, the position of the center of gravity in height, which is very important for the safety of the vessel, is in the “sphere of influence” of the crew and requires constant monitoring through special calculations.
For the calculation control of the vessel's "positive" stability, the concept of the metacenter and the initial metacentric height is used.
Transverse metacenter is a point that is the center of curvature of the trajectory along which the center of magnitude moves when the vessel rolls.
Consequently, the metacenter (as well as the center of magnitude) is a specific point, the behavior of which is exclusively determined only by the geometry of the shape of the vessel in the underwater part and its draft.
The position of the metacenter, corresponding to the landing of the ship without a roll, is commonly called initial transverse metacenter.
The distance between the ship's center of gravity and the initial metacenter in a specific loading option, measured in the center line (DP), is called initial transverse metacentric height.
The figure shows that the lower the center of gravity is in relation to the constant (for a given draft) initial metacenter, the greater the metacentric height of the vessel, i.e. the larger is the shoulder of the restoring moment and this moment itself.
Thus, the metacentric height is an important characteristic that serves to control the ship's stability. And the greater its value, the greater at the same roll angles will be the value of the restoring moment, i.e. resistance of the vessel to heeling.
With small ship heels, the metacenter is approximately located at the site of the initial metacenter, since the trajectory of the center of magnitude (points WITH) is close to a circle, and its radius is constant. A useful formula follows from a triangle with a vertex at the metacenter, which is valid for small bank angles ( θ <10 0 ÷12 0):
where is the roll angle θ should be used in radians.
From expressions (1) and (2) it is easy to obtain the expression:
which shows that the static stability arm and the metacentric height do not depend on the weight of the ship and its displacement, but are universal stability characteristics that can be used to compare the stability of ships of different types and sizes.
So for ships with a high center of gravity (timber carriers), the initial metacentric height takes on the values h 0≈ 0 - 0.30 m, for dry cargo ships h 0≈ 0 - 1.20 m, for bulk carriers, icebreakers, tugs h 0> 1.5 ÷ 4.0 m.
However, the metacentric height should not take negative values. Formula (1) allows us to draw other important conclusions: since the order of magnitude of the restoring moment is determined mainly by the displacement of the vessel R, then the static stability arm is a “control variable” that affects the range of torque change M in for this displacement. And from the slightest change l(θ) due to inaccuracies in its calculation or errors in the initial information (data taken from the ship's drawings, or measured parameters on the ship), the magnitude of the moment significantly depends M in, which determines the ability of the vessel to resist inclinations, i.e. determining its stability.
Thus, the initial metacentric height plays the role of a universal stability characteristic, which makes it possible to judge its presence and magnitude, regardless of the size of the vessel.
If we follow the mechanism of stability at large angles of heel, then new features of the restoring moment will appear.
With arbitrary transverse inclinations of the vessel, the curvature of the trajectory of the center of magnitude WITH changes. This trajectory is no longer a circle with a constant radius of curvature, but is a kind of flat curve that has different values of curvature and radius of curvature at each of its points. As a rule, this radius increases with the roll of the vessel and the transverse metacenter (as the beginning of this radius) leaves the diametrical plane and moves along its trajectory, tracking the movement of the center of magnitude in the underwater part of the vessel. In this case, of course, the very concept of metacentric height becomes inapplicable, and only the restoring moment (and its shoulder l(θ)) remain the only characteristics of the ship's stability at high inclinations.
However, at the same time, the initial metacentric height does not lose its role of being the fundamental initial characteristic of the stability of the vessel as a whole, since the order of magnitude of the restoring moment depends on its value, as on a certain “scale factor”, i.e. its indirect influence on the stability of the vessel at large angles of heel remains.
So, to control the stability of the vessel, carried out before loading, it is necessary at the first stage to evaluate the value of the initial transverse metacentric height h 0, using the expression:
where z G and z M 0 are the applicates of the center of gravity and the initial transverse metacenter, respectively, counted from the main plane in which the origin of the OXYZ coordinate system associated with the ship is located (Fig. 3).
Expression (4) simultaneously reflects the degree of participation of the navigator in ensuring stability. By selecting and controlling the position of the vessel's center of gravity in height, the crew ensures the stability of the vessel, and all geometric characteristics, in particular, Z M 0, must be provided by the designer in the form of graphs from settlement d, called curved elements of a theoretical drawing.
Further control of the vessel's stability is carried out according to the methodology of the Maritime Register of Shipping (RS) or the methodology of the International Maritime Organization (IMO).
Restoring moment arm l and the moment M in have a geometric interpretation in the form of a static stability diagram (DSD) (Fig.4). DSO is graphic dependence of the restoring moment arm l(θ) or the very momentM in (θ) from the angle of heel θ .
This graph, as a rule, is depicted for the ship's roll to starboard only, since the whole picture for a list to port for a symmetrical ship differs only in the sign of the moment M in (θ).
The value of DSO in the theory of stability is very large: it is not only a graphic dependence M in(θ); The DSO contains comprehensive information on the status of the ship's loading in terms of stability. The DSO of the vessel allows solving many practical problems in this voyage and is a reporting document for the ability to start loading the vessel and sending it on a voyage.
The properties of DSO are as follows:
- DSO of a particular ship depends only on the relative position of the ship's center of gravity G and initial transverse metacenter m(or the value of the metacentric height h 0) and displacement R(or draft d cf) and takes into account the presence of liquid cargoes and stocks with the help of special amendments,
- the shape of the hull of a particular ship is shown in the DSO over the shoulder l (θ), rigidly connected with the shape of the hull contours , which reflects the displacement of the center of magnitude WITH towards the side entering the water when the ship is heeling.
- metacentric height h 0, calculated taking into account the influence of liquid cargoes and reserves (see below), appears on the DSO as the tangent of the slope of the tangent to the DSO at the point θ = 0, i.e.:
To confirm the correctness of the construction of the DSO, a construction is made on it: the angle is set aside θ \u003d 1 rad (57.3 0) and build a triangle with a hypotenuse tangent to the DSO at θ = 0, and a horizontal leg θ = 57.3 0. The vertical (opposite) leg should be equal to the metacentric height h 0 axis scale l(m).
- no actions can change the type of DSO, except for changing the values of the initial parameters h 0 And R, since DSO reflects in a sense the invariable shape of the ship's hull through the value l (θ);
- metacentric height h 0 actually determines the type and extent of the DSO.
Bank angle θ = θ 3, at which the DSO graph crosses the abscissa axis, is called the sunset angle of the DSO. sunset angle θ 3 determines only the value of the angle of heel at which the weight force and the buoyancy force will act along one straight line and l(θ 3) = 0. Judge the capsizing of the vessel when heeling
θ = θ 3 will not be true, since the capsizing of the vessel begins much earlier - shortly after the maximum point of the DSO is overcome. DSO maximum point ( l = l m (θ m)) indicates only the maximum removal of the weight force from the support force. However, the maximum leverage l m and maximum angle θm are important values in the control of stability and are subject to verification for compliance with the relevant standards.
DSO allows you to solve many problems of ship statics, for example, to determine the static angle of the ship's heel under the action of a constant (independent of the ship's roll) heeling moment M cr= const. This angle of heel can be determined from the condition of equality of the heeling and righting moments M in (θ) = M cr. In practice, this problem is solved as a problem of finding the abscissa of the intersection point of the graphs of both moments.
The static stability diagram reflects the vessel's ability to create a righting moment when the vessel is tilted. Its appearance has a strictly specific character, corresponding to the loading parameters of the vessel only in this voyage ( R = Ri , h 0 = h 0 i). The navigator involved in planning the loading voyage and stability calculations on the ship is obliged to build a specific DSS for the two states of the ship on the upcoming voyage: with the initial position of the cargo unchanged and at 100% and at 10% of ship stores.
In order to be able to build static stability diagrams for various combinations of displacement and metacentric height, he uses auxiliary graphic materials available in the ship's documentation for the project of this vessel, for example, pantokarens, or a universal static stability diagram.
Pantocarenes are supplied to the ship by the designer as part of the stability and strength information for the captain. are universal graphs for a given vessel, reflecting the shape of its hull in terms of stability.
Pantocarenes (Fig. 6) are shown as a series of graphs (at different heel angles (θ = 10,20,30,….70˚)) depending on the weight of the vessel (or its draft) of some part of the static stability arm, called the stability arm forms - lf(R, θ ).
The shoulder of the form is the distance that the buoyancy force will move relative to the original center of magnitude C ο when the vessel rolls (Fig. 7). It is clear that this displacement of the center of magnitude is associated only with the shape of the hull and does not depend on the position of the center of gravity in height. A set of shape shoulder values at different heel angles (for a specific vessel weight P=Pi) are removed from the pantocaren charts (Fig. 6).
To determine the shoulders of stability l(θ) and build a diagram of static stability in the upcoming voyage, it is necessary to supplement the form arms with weight arms l in which are easy to calculate:
Then the ordinates of the future DSO are obtained by the expression:
Having performed calculations for two load states ( R app.\u003d 100% and 10%), two DSOs are built on a blank form, characterizing the ship's stability in this voyage. It remains to check the stability parameters for their compliance with national or international standards for the stability of marine vessels.
There is a second way to build a DSS, using the universal DSS of a given ship (depending on the availability of specific auxiliary materials on the ship).
Universal DSO(Fig. 6a) combines the transformed pantocarenes to determine lf and graphs of weight shoulders lV(θ). To simplify the view of graphical dependencies lV(θ) (see formula (6)) it was necessary to make a change of variable q = sin θ , resulting in sinusoidal curves lV(θ) transformed into straight lines lV (q(θ)). But in order to do this, it was necessary to adopt an uneven (sinusoidal) scale along the x-axis θ .
On the universal DSO, presented by the ship designer, there are both types of graphic dependencies - l f (Р,θ) And l in (z G ,θ). Due to the change in the x-axis, the graphs of the shoulder shape l f no longer look like pantocarenes, although they contain the same amount of information about the shape of the body as pantocarenes.
To use the universal DSO, it is necessary to use a meter to remove from the diagram the vertical distances between the curve l f (θ, P *) and curve l in (θ, z G *) for several values of the ship's heel angle θ = 10, 20, 30, 40 ... 70 0 , which will correspond to the application of formula (6a). And then, on a clean DSO form, build these values as the ordinates of the future DSO and connect the points with a smooth line (the axis of roll angles on the DSO is now taken with a uniform scale).
In both cases, both when using pantocaren and when using a universal DSO, the final DSO should be the same.
On the universal DSO, sometimes there is an auxiliary axis of the metacentric height (on the right), which facilitates the construction of a specific straight line with the value z G * : corresponding to some value of the metacentric height h 0 * , because the
Let us now turn to the method of determining the coordinates of the ship's center of gravity XG And Z G. In the information about the stability of the ship, you can always find the coordinates of the center of gravity of an empty ship abscissa xG0 and ordinate z G 0.
The products of the ship's weight and the corresponding coordinates of the center of gravity are called the static moments of the ship's displacement. relative to the midship plane ( M x) and main plane ( Mz); for an empty ship we have:
For a loaded ship, these quantities can be calculated by summing up the corresponding static moments for all cargo, tank stores, ballast in ballast tanks, and empty ship:
For static moment MZ it is necessary to add a special positive correction, taking into account the dangerous effect of the free surfaces of liquid cargoes, stores and ballast, available in the ship's tank tables, ∆MZh:
This correction artificially increases the value of the static moment in order to obtain the worst values of the metacentric height, thereby the calculation is carried out with a safety margin.
Sharing now static moments M X And M Z correct for the total weight of the vessel in this voyage, we obtain the coordinates of the center of gravity of the vessel along the length ( XG) and corrected ( Z G correct), which is then used to calculate the corrected metacentric height h 0 correct:
and then to build a DSO. The value of Z mo (d) is taken from the curved elements of the theoretical drawing for a specific average draft.
