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transverse stability of the ship. Calculation of the main meta-parameters is invariant to different vessels Vessel stability in accessible words

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 FROM. 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 FROM 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 AT- 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 - Z 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
, as well as
, where
, where - the weight of the transported cargo, kN; - the distance that the cargo CG has moved in the vertical direction, m.

Methods for determining the center of magnitude (c.v.) and the center of gravity (c.g.) of the vessel

To determine the position of any point on the ship, including c. t. and c. c., use a system of coordinate axes fixedly connected with the ship's hull.

The vertical axis OZ is taken as the line of intersection of DP with the plane of the midsection - frame, for the longitudinal - horizontal axis OX - the line of intersection of DP with the main plane and for the transverse - horizontal axis OY - the line of intersection of the midsection - frame with the main plane. In this case, the direction of the axis OX is taken as the positive direction of the axes - bringing in, OY - to the starboard side, OZ - up. The position of the points g and c of interest to us can be found from approximate and exact dependencies. Approximate methods for determining the coordinates c. in. C coordinate. in. along the width of the vessel, due to the symmetry of the vessel with respect to Dp, it should always be in the plane of the diameter, i.e. y c \u003d 0.

If this equality is not present, then the ship will be heeled.

The coordinate of the point c along the length of the vessel x c is always close to the middle of the vessel, if there is no trim on the bow or stern, and changes its position from the midship frame within small limits. Usually x c varies from +0.02L to -0.035L, where L is the length of the vessel.

C coordinate. in. the height of the ship can vary within the following limits: for ships with a rectangular cross section z c = 0.5T, where T is the draft of the ship; for vessels with a triangular cross section, z c will be equal to? T from the main plane, i.e. z c =0.66T, thus this coordinate depends on the shape of the cross section, and hence on the corresponding coefficients of completeness.

Determining the coordinates of the center of magnitude (c.v.) and the center of gravity (c.g.) The center of gravity (g) of a vessel without inclination, i. floating in an equilibrium position, must always be on the same vertical with the center of magnitude (c). This is achieved by the appropriate arrangement of cargoes on the ship, and in this case y c = 0.

Position of point g in height, i.e. its applicate z g , depends on the location of the cargo on the ship relative to its height and can be expressed in fractions of the height of the ship's side H by the dependence

where k is an experimental coefficient, the value of which is recommended for empty cargo ships 0.35 × 0.5, for towing propeller vessels 0.60 × 0.70.

For loaded cargo ships, as well as for passenger ships with high superstructures, the value of z g can be more than H, i.e. k>1.0 .

To accurately determine the values of the coordinates of the center of gravity - z g and x g, the ship is divided into weight articles, the distances of the centers of gravity of these weight articles from the main plane and the plane of the midship - frame are determined.

After all weight loads are determined, the shoulders of their center of gravity are found and the moments of forces are calculated, the coordinate of the center of gravity along the length of the vessel x g is determined by the formula

where UM n - the sum of the moments of all the forces of the weight articles in the bow of the vessel relative to the plane of the midship - frame;

UM k - the sum of the moments of all the forces of the weight articles in the stern of the vessel relative to the plane of the midship - frame.

The sign (+) will indicate that the abscissa of the center of gravity is located in the bow of the vessel, and the sign (-) that it is located in the stern of the vessel, since here the x-axis has a negative value.

The coordinate of the center of gravity in height z g is determined by the formula

where UM is the sum of the moments of all forces relative to the main plane.

The trapezoid rule, methods for determining the volumetric displacement of the vessel and drill

Volumetric displacement can be determined in various ways. Consider the simplest of them, providing a degree of accuracy sufficient for practice, a method based on the use of the trapezoid rule.

Initially, we apply the trapezoid rule to determine the areas of figures limited by curvilinear lines.

Let's divide the curvilinear figure (Figure 7) into n equal parts. The length of each such part will be, and the area u i of each part can be defined as the areas of trapezoids, the sides of which have ordinates for i, and heights for Dl.

Figure 7 - Scheme for calculating the area using the trapezoid method

Therefore, S \u003d u 1 + u 2 + ... u n-1 + u n or

Substituting in the formula the values for u in the form of areas of individual trapezoids, we obtain

This expression is called the formula of the trapezoid rule, in which y 0 + y 1 + y 2 + y 3 + .... + y n-1 + y n is the sum of the ordinates, denoted? 0;

It's called an amendment.

The entire value in square brackets is the corrected amount and is denoted? correct, then the expression for the area of a curvilinear figure can be abbreviated as follows

All calculations are most conveniently carried out in tabular form (Table 1).

When calculating the volumetric displacement of a vessel, it is necessary to calculate the volume of its underwater part, limited by the surface of the vessel and the plane of the effective waterline.

Knowing the size of the vessel and its shape when calculating the volumetric displacement, according to the rule of trapezoids, it is assumed that the volumetric displacement V is replaced by the sum of the volumes V 1 + V 2 + V 3 + .... + V n-1 + V n into which the underwater part is divided ships equally spaced from one another by planes parallel to the plane of the midship - frame, or the plane of the current waterline.

Table 1 - Calculating the area using the trapezoid method

Consider the case when the ship, having a length along the waterline L, draft T, is cut into n compartments by planes parallel to the plane of the midship - frame, as shown in Figure 8 with the distance between the compartments.

Figure 8 - Cross-section of the vessel by planes parallel to the plane of the midship frame

Denoting the volumes of the vessel compartments between the zero and the first section through V 1, between the first and second through V 2, etc., we write the expression for the volume of the underwater part of the vessel

V=V 1 +V 2 +V 3 +…+V n-1 +V n .(30)

The volumes of the selected compartments of the vessel can be determined as the product of half the sum of the areas of the frames and the distance between them DL, after which the equation takes the form

or, by analogy with the previous one, we will have

where F 0 +F 1 +….+F n is the sum of the areas of the frames;

Amendment;

the expression in square brackets is the corrected amount.

To determine the area of the frames F i (Figure 9), due to the symmetry of the vessel with respect to DП, only half of the area of the frame is determined, and then the result is doubled. In this case, the draft T is divided into m equal parts and the ordinates are drawn through the division points y 0, y 1 ...., y m, the areas limited by these ordinates will be f 1 , f 2 , ...., f m . Distances between ordinates

Figure 9 - Scheme for calculating the area of the frame

By analogy with the previous equation for determining the area of the frame F i will have the form

where is the double corrected sum obtained by first summing the ordinates along the frames, and then the frames along the length of the vessel.

Volumetric displacement can be obtained by dissecting the ship with equally spaced planes parallel to the main plane, and then summing up the compartments formed by these planes (Figure 10).

In this case, the draft T is divided into m equal parts, resulting in a series of waterline areas S spaced apart from each other.

Figure 10 - Section of the vessel by planes parallel to the main plane

Similarly to the previous expression for determining the volumetric displacement of the vessel will have the form

The area of each of the waterlines S 0 , S 1 , .... S m is determined by the dependence

where is the double corrected sum obtained by first summing up the ordinates over the waterlines, and then the waterlines over the ship's draft.

It is easy to see that the result of determining the volume displacement in the two cases will be the same.

Calculations of the volumetric displacement of the vessel are always carried out in tabular form (Table 2).

In this table, from the theoretical drawing of the vessel, the values \u200b\u200bof the ordinates y are entered for each waterline for each frame on one side. Do they sum up the ordinates horizontally and vertically, for each sum find corrections as the sums of extreme ordinates, find the corrected sums? correct In horizontal lines, calculate the area of each frame by multiplying the value? correct on DT (distance between waterlines), and in vertical columns calculate the area of each waterline by multiplying the corresponding values? corrected on DL (distance between calculated frames).

In the lower right corner of the table, the corrected sum of the column amounts and, at the same time, the corrected sum of the row sums of the CU are obtained. This value should be the same both vertically and horizontally, which is a kind of control over the correctness of calculating the volumetric displacement.

Table 2 - Calculation of the areas of frames, waterlines and displacement of the ship

No. of design frames	waterline number	Amendment	Corrected amount?	Frame area F=2DT?y




Amendment Corrected amount? Waterline area

By calculating the value of the double corrected sum?? , determine the volumetric displacement by the formula

Using the data on the values of the areas of the frames obtained in the table, they usually build a curve for changing these areas along the length of the vessel. Such a curve is called a drill along the frames. To do this, the length of the vessel L is plotted on a scale, on which the position of all equally spaced design frames from F 0 to F n is plotted. On the restored ordinates, on an appropriate scale, the values of the submerged area of the corresponding frames F are plotted. The curve connecting the ends of these ordinates is called the drill line along the frames (Figure 11).

Figure 11 - Drill on the frames

This drill has the following properties:

1. The area of the figure, limited by the line L, the extreme ordinates and the drill along the frames, calculated according to the trapezoid rule, is numerically equal to the volumetric displacement of the vessel;

2. Abscissa c.t. this area expresses the abscissa of the c.v. ship, i.e. X s

3. The coefficient of completeness of the combat area along the frames is nothing more than the coefficient of the longitudinal completeness of the volumetric displacement of the vessel

4. The drill on the frames gives a visual representation of the nature of the distribution of volumetric displacement along the length of the vessel, which must be known when calculating the strength of the vessel.

Similarly, a curve is constructed for changing the areas of waterlines depending on the ship's draft (Figure 12). Such a curve is called a drill line along the waterlines. To do this, on any scale, the ship's draft T is laid down, on which the positions of all equally spaced waterlines from S 0 to S m are plotted. On a different scale, on each abscissa restored from the corresponding waterline, the value of its area is plotted. The curve connecting the ends of these abscissas is called the drill line along the waterlines. It has the following properties:

1. The area of \u200b\u200bthe figure, limited by the line T, extreme abscissas and drill along the waterlines, calculated according to the trapezoid rule, is numerically equal to the volumetric displacement of the ship;

Figure 12 - Drilling along the waterlines

2. The ordinate of the center of gravity of the area is equal to the ordinate of the center of magnitude of the vessel Z s.

3. The coefficient of completeness of the drill area along the waterlines is the coefficient of the vertical completeness of the ship's displacement

4. The curve gives a visual representation of the nature of the distribution of volumetric displacement along the height of the vessel, which is important to know to characterize the smoothness of the ship's contours.

1. Stability of a surface floating body

2. Stability of the surface - floating body

Surface - floating body under the influence of any external forces can tilt in one direction or another. The ability of a body to return to its original position is called its stability.

A floating body or ship has three characteristic points: the center of gravity g, the center of magnitude c and the metacenter m. The center of gravity g of a dry cargo vessel does not change its position during rolling. The center of magnitude, when the vessel is tilted, moves in the direction of inclination, while the line of action of the Archimedean force intersects the navigation axis "0 - 0" at a point called the metacenter. The position of the metacenter does not remain constant when the vessel is tilted. However, at angles not exceeding u = 15 o, the position of the metacenter almost does not change and is taken unchanged. In this case, the center of magnitude c moves approximately along the arc of a circle described from the point m with radius r and is called the metacentric radius. The stability of the ship depends on the relative position of the centers c,g,m.

Suppose we have a ship that has received a roll at an angle and< 15 о (рисунок 13). Для надводно - плавающих тел Архимедова сила D всегда равна силе веса G. Эти две силы образуют пару сил, стремящуюся вернуть судно в первоначальное (нормальное) положение. Таким образом, рассматриваемый случай является случаем остойчивого положения судна.

Let us depict the second case (Figure 14), when the center of gravity g will be located on the navigation axis above the center of magnitude c. In this case, the resulting moment when the vessel is tilted at an angle and tends to return the vessel to its normal position, i.e. and in this case we have a stable position of the ship.

Figure 13 - Vessel stability when the center of gravity is below the center of magnitude.

Figure 14 - Stability of the vessel with the center of gravity below the metacenter, but above the center of magnitude

However, it is easy to see that, under equal conditions, the stability in the second case is less than the stability in the first case, since the arm of the pair of forces, and hence the restoring moment in the first case, will be greater.

And, finally, consider the third case, when the center of gravity will be located above the metacenter m (Figure 15). The resulting pair of forces tends to tilt the ship even more. In this case, there are no forces capable of returning the ship to its normal position. We have a case of unstable position of the vessel. Having considered three cases with a vessel having a different position of the center of gravity, we can say that the higher the center of gravity of the vessel, the less its stability. Therefore, in order to increase the stability of bodies, one should always strive to lower their center of gravity.

Figure 15 - Vessel stability when the center of gravity is above the metacenter

The different influence of a pair of forces on the stability of floating bodies depends on the relative position of the center of gravity g and the metacenter m. When the metacenter is located above the center of gravity, the body is stable and when the metacenter is located below the center of gravity, it is not stable. This can also be characterized by the ratio of r and a, where a is the distance between the center of gravity and the center of magnitude. It is generally accepted that a positive value of a corresponds to such a mutual position of the centers c and g, when the center c lies on the axis of navigation below the center g.

In this way

when r>a - the ship is stable (cases 1 and 2),

at r

The distance between the center of gravity and the metacenter on the navigation axis is considered to be the metacentric height h. Between h, r and a there is the following relationship

If we now again turn our attention to the above cases of the position of the vessel, we will notice that for the first and second cases h>0, and for the third, the metacentric height h< 0. Следовательно, знак при h характеризует остойчивость судна. Положительное значение метацентрической высоты характеризует остойчивое положение судна, а отрицательное значение метацентрической высоты - неостойчивое.

And, finally, when the metacenter m coincides with the center of gravity of the ship when it is tilted at an angle u, i.e. when h=0 or r= a, we will have a case of an unstable position of the ship, since in this case the lines of action of the Archimedean force D and the ship's gravity G will coincide and, therefore, no restoring moment can be formed. This case in the theory of swimming is called the indifferent state.

During the operation of ships, it may be necessary to switch from straight-line motion to curved motion and vice versa. This is possible provided that external forces are applied to the ship, the moments of which will cause the ship to deviate from the original direction of movement.

The ability of a ship to change direction and move along a curved path is called agility.

Changing the course of the vessel can be achieved in two ways - either with the help of propulsion devices, or with the help of special steering devices. The first method can be applied only on self-propelled ships with two propellers. With the help of propulsion devices, the ship changes course if the stops from the propeller T are not the same in size or if they are directed in opposite directions (Figure 16)

Figure 16 - Vessel agility

In this case, a moment is created from a pair of forces, the numerical value of which can be determined by the formula:

where T 1 and T 2 - stops of the left and right movers;

l is the distance between the axes of the propellers.

This moment causes the ship to change its course.

If T 1 =T 2 , the ship will rotate in place without receiving translational motion. If T 1 >T 2, the ship, in addition to rotation under the action of the moment, will also have translational forward movement, and if T 1<Т 2 судна, кроме вращения, будет иметь и поступательное движение назад.

Usually, a steering device is used to turn the vessel, which is, in the most general case, a vertical plate (rudder blade) located in the stream behind the stern of the vessel (Figure 17). The rudder blade can rotate around the o-axis. The plate, together with other devices for attaching and turning it, is called a rudder.

Figure 17 - Forces acting on the ship when the rudder is turned

If the rudder is deflected from the diameter by an angle b, then at the speed of the vessel V, according to the laws of hydromechanics, the hydrodynamic pressure force acts on the rudder, the value of which can be determined by the Jossel formula

where R a - water pressure on the rudder blade;

F is the area of the underwater part of the rudder blade;

V- ship speed;

b - the angle of the rudder blade (the angle of deviation from the diameter);

k b is an experimental coefficient depending on the angle b, it represents the pressure per 1 m 2 of the rudder blade area at a vessel speed of 1 m / s.

The value of k b is determined by the empirical formula

The value of k is recommended to be taken for single-screw vessels 400 N/m 3 , and for twin-screw vessels 225 N/m 3 . When the rudder is shifted to an angle b on the ship, in addition to the resistance force R, stop T, which are mutually balanced (with uniform motion), the following forces also act:

1. A pair of forces forming a moment M. The numerical value of this moment is determined by the dependence

In this formula, the value is much smaller, in - the length of the rudder blade, and l - the length of the vessel, due to which the value is neglected. After substituting the value of P a into equation (48), it can be seen that if the ship moves at a constant speed, the magnitude of the moment depends on the product of cosb sinb. This product reaches its maximum at b = 36 o. It follows from this that there is no point in deflecting the rudder blade by more than 35-36 °, since the moment of rotation of the vessel does not increase in this case.

2., demolishing the vessel in the opposite direction of the rudder. In order to verify this, let us apply at the point g the forces Pa, directed in opposite directions. The balance of the ship will not be disturbed by this. One force Ra applied at point g, together with the force Ra acting on the rudder blade, forms a pair of forces. Let's break it down into components.

The force increases the resistance to the movement of the vessel due to the braking effect of the rudder blade, which is at a certain angle b to the direction of movement. The force causes a lateral drift of the vessel (drift), the presence of which causes the occurrence of a lateral resistance force. is the force that causes the ship to change its original course. The considered complex scheme of interaction of the emerging forces in connection with the shifting of the rudder blade to the angle b also determines a very complex path of the vessel's movement. It is customary to consider three periods of the ship's movement.

The first is maneuverable, when the rudder is shifted and when, under the action of force, the ship receives lateral drift.

The second is evolutionary, which continues until the ship begins to rotate uniformly around a fixed axis.

The third one is steady, when all the forces acting on the ship and their moments are mutually balanced and the ship begins to move in a circle.

The curve described by the ship's center of gravity at full turn is called the ship's circulation (Figure 21), and its diameter is the circulation diameter. The time it takes for a ship to make a complete revolution is called its circulation period. The smaller the circulation diameter, the better the agility of the vessel, therefore, agility is one of the most important qualities of rafting vessels that have to work on timber rafting raids in water areas constrained by floating structures.

The circulation diameter can be determined by the formula

where S is the area of the rudder blade, m 2;

l,T - length and draft of the vessel, m;

OB - maneuvering period, when there is a lateral drift, numerically equal to k;

VS - evolutionary period.

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).

Scheme of the formation of a restoring moment

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 FROM).

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 vessel 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.

Dependence of the shoulder of the restoring moment on the position of the ship's center of gravity.

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 FROM) 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.

Shoulder of static stability

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.

In this way, 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 FROM 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 M0 are the applicates of the center of gravity and the initial transverse metacenter, respectively, measured 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 M0, 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).

Initial transverse metacentric height

Static stability diagram

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 shoulder 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 vessel is manifested 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 FROM 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 lm 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.

Interaction of heeling and restoring 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 = Р i ,h 0 =h 0i). 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

Pantocarenes are supplied to the ship by the designer as part of the stability and strength information for the captain. pantocarenes 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 - l f (P, θ ).

pantocarenes

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=P i) 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:

Shoulders stability of shape and weight

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).

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.

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 (that is, dynamic) forces, for example, wind, sea waves, cargo movement, etc.

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-sized 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, having 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.

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:

full pressing of tanks to avoid free surfaces
if this is not possible, filling under the neck, or vice versa, only at the bottom. In this case, any inclination sharply reduces the free surface area.
control of the number of tanks with free surfaces
breakdown of tanks by internal impenetrable bulkheads in order to reduce the moment of inertia of the free surface i x

Dynamic stability

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:

A θ = ∫ 0 θ m θ ∂ θ (\displaystyle A_(\theta )=\int _(0)^(\theta )m_(\theta )\partial \theta ) ,

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 with amplitude from 0 to θ dyn. But in practice, due to the resistance of the water, the oscillations quickly die out and it remains to float with a static heel angle. θ st.

The dynamic effect of the heeling moment is always more dangerous than the static one, as it leads to more significant inclinations. Within the rectilinear portion of the static stability diagram, the dynamic bank angle is approximately twice the static angle: θ dyn ≈ 2 θ st.

Transverse inclinations

In the upright position of the ship, CV is in the DP (point C) and the line of action of the buoyancy force rV also lies in the DP (Fig. 2). With the transverse inclination of the vessel 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 infinitely small transverse equal-volume inclination of the vessel is called the transverse metacenter (point m in Fig. 2). The radius of curvature of the CV trajectory 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, we can assume that the trajectory

The CV lies in the plane of inclination and is an arc of a circle centered at the point m. Thus, we can assume that in the process of a small transverse equal-volume inclination of the ship 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. four.

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 has left the water to the side that has entered the water (Fig. 4).

According to the well-known theorem of mechanics, when moving a body belonging to a system of bodies, 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 gravity forces of the body and the system, respectively. This theorem can also be extended to the volumes of homogeneous bodies. Denote:

C C I - displacement of CV (geometric center of volume V),

b - displacement of the geometric center of the wedge-shaped volume v. Then, according to the theorem

from: C C 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 a small angle And