Methods for determining the center of magnitude (c.v.) and center of gravity (c.g.) of a vessel. Metacentric height - criterion of vessel stability: formula How does the position of the center of magnitude affect stability
The theory of lateral stability considers the inclination of the ship occurring in the midship plane, and an external moment, called the heeling moment, also acts in the midship plane.
Without limiting ourselves to small inclinations of the vessel for now (they will be considered as a special case in the section “Initial Stability”), let us consider the general case of heeling of the vessel under the action of an external heeling moment 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 midsection plane. When exposed to this heeling moment, the ship has a constant roll to the opposite side, the magnitude of which is determined by the wind force and the righting moment on the part of the ship.
In the literature on ship theory, it is customary to combine in the figure two positions of the ship at once - straight and with a list. The heeled position corresponds to a new position of the waterline relative to the ship, which corresponds to a constant submerged volume, however, the shape of the underwater part of the heeled ship no longer has symmetry: the starboard side is submerged more than the left (Fig. 1).
All waterlines corresponding to one value of the vessel’s displacement (at constant weight of the vessel) are usually called equal volume.
The accurate representation in the figure of all equal-volume waterlines is associated with great calculation difficulties. In ship theory, there are several techniques for graphically depicting equal-volume waterlines. At very small angles of heel (at infinitesimal equal-volume inclinations), one can use a corollary from L. Euler’s theorem, according to which two equal-volume waterlines, differing 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 its validity, since each waterline has its own center of gravity of the area).
Scheme of formation of the restoring moment
If we abstract from the real distribution of forces of the ship's weight and hydrostatic pressure, replacing their action with concentrated resultants, we arrive at the diagram (Fig. 1). At the center of gravity of the vessel, a weight force is applied, directed in all cases perpendicular to the waterline. In parallel to it, there is a buoyancy force 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 are independent of each other, they no longer act along one line, but form a pair of forces parallel and perpendicular to the acting waterline B 1 L 1. Regarding weight force R we can say that it remains vertical and perpendicular to the surface of the water, and the tilted ship deviates from the vertical, and only the convention of the drawing requires that the vector of the weight force be deviated from the center plane. The specifics of this approach are easy to understand if you imagine a situation with a video camera mounted on a ship, showing on the screen the surface of the sea inclined at an angle equal to the angle of roll of the ship.
The resulting pair of forces creates a moment, which is usually called restoring moment. This moment counteracts the external heeling moment and is the main object of attention in the theory of stability.
The magnitude of the restoring moment can be calculated using the formula (as for any pair of forces) as the product of one (either of two) forces and the distance between them, called static stability shoulder:
Formula (1) indicates that both the shoulder and the moment itself depend on the angle of roll of the vessel, i.e. represent 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 cargo along the height of the vessel, for example, when there is excess cargo on the deck) turns out to be quite high, then a situation may arise when the weight force is to the right of the line of action of the supporting force. Then their moment will act in the opposite direction and will contribute to the ship's heeling. Together with the external heeling moment, they will capsize the ship, since there are no other counteracting moments.
It is clear that in this case this situation should be assessed as unacceptable, since the vessel does not have stability. Consequently, with a high center of gravity, the ship may lose this important seaworthiness quality - stability.
On sea-going displacement vessels, the ability to influence the stability of the vessel, to “control” it, is provided to the navigator only through the rational placement of cargo and reserves along the height of the vessel, which determine the position of the vessel’s center of gravity. Be that as it may, the influence of the 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 the draft. Human influence on the shape of the hull ends at the design stage of the vessel.
Thus, the vertical position of the center of gravity, which is very important for the safety of the ship, is in the “sphere of influence” of the crew and requires constant monitoring through special calculations.
To calculate the presence of “positive” stability of a vessel, the concept of metacenter and initial metacentric height is used.
Transverse metacenter- this is the point that is the center of curvature of the trajectory along which the center of the value moves when the ship heels.
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 vessel without a roll is usually called initial transverse metacenter.
The distance between the center of gravity of the vessel and the initial metacenter in a particular loading option, measured in the center plane (DP), is called initial transverse metacentric height.
The figure shows that the lower the center of gravity is located in relation to the constant (for a given draft) initial metacenter, the greater will be the metacentric height of the vessel, i.e. the greater is the leverage of the restoring moment and this moment itself.
Dependence of the righting moment arm on the position of the vessel's center of gravity.
Thus, the metacentric height is an important characteristic that serves to control the stability of the vessel. And the greater its value, the greater at the same roll angles will be the value of the righting moment, i.e. resistance of the ship to heeling.
For small heels of the vessel, the metacenter is approximately located at the site of the initial metacenter, since the trajectory of the center of magnitude (point WITH) is close to a circle and its radius is constant. From a triangle with a vertex at the metacenter, a useful formula follows that is valid at small roll 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 metacentric height do not depend on the weight of the vessel and its displacement, but represent universal stability characteristics with which the stability of vessels of different types and sizes can be compared.
Static stability arm
So for ships with a high center of gravity (timber carriers), the initial metacentric height takes 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 righting moment is determined mainly by the magnitude of the vessel’s displacement R, then the static stability arm is a “control variable” that affects the range of torque changes M in at a given displacement. And from the slightest changes l(θ) Due to inaccuracies in its calculation or errors in the initial information (data taken from ship drawings, or measured parameters on the ship), the magnitude of the moment significantly depends M in, which determines the vessel’s ability to resist inclinations, i.e. determining its stability.
Thus, the initial metacentric height plays the role of a universal stability characteristic, allowing one to judge its presence and size regardless of the size of the vessel.
If we follow the stability mechanism at large roll angles, new features of the righting moment will appear.
For 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 point. As a rule, this radius increases with the roll of the vessel and the transverse metacenter (as the beginning of this radius) leaves the center plane and moves along its trajectory, tracking the movements 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 righting moment (and its shoulder l(θ)) remain the only characteristics of ship stability at high inclinations.
However, in this case, the initial metacentric height does not lose its role as a fundamental initial characteristic of the stability of the vessel as a whole, since the order of magnitude of the righting moment depends on its value, as on a certain “scale factor,” i.e. its indirect effect on the stability of the vessel at large angles of roll remains.
So, to control the stability of the vessel before loading, it is necessary at the first stage to estimate the value of the initial transverse metacentric height h 0, using the expression:
where z G and z M0 are applicates of the center of gravity and the initial transverse metacenter, respectively, measured from the main plane in which the beginning of the OXYZ coordinate system associated with the vessel is located (Fig. 3).
Expression (4) simultaneously reflects the degree of participation of the navigator in ensuring stability. By choosing 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 of settlement d, called curves of theoretical drawing elements.
Further control of the vessel's stability is carried out according to the methods of the Maritime Register of Shipping (RS) or according to the methods of the International Maritime Organization (IMO).
Initial transverse metacentric height
Static stability diagram
Righting moment arm l and the moment itself M in have a geometric interpretation in the form of a Static Stability Diagram (SSD) (Fig. 4). DSO is graphical dependence of the restoring moment arm l(θ) or the moment itselfM in (θ) from roll angle θ .
This graph, as a rule, is depicted for a ship’s roll only to the starboard side, since the whole picture when a ship rolls to the left side for a symmetrical ship differs only in the sign of the moment M in (θ).
The importance of DSO in the theory of stability is very great: it is not only a graphical dependence M in(θ); The DSO contains comprehensive information about the state of the vessel's loading from the point of view of stability. The ship's DSO allows you to solve many practical problems on a given voyage and is a reporting document for the ability to begin loading the ship and sending it on a voyage.
The following properties can be noted as DSO:
· The DSO of a particular vessel depends only on the relative position of the vessel’s center of gravity G and the initial transverse metacenter m(or metacentric height value h 0) and displacement R(or draft d avg) and takes into account the availability of liquid cargo and supplies using special adjustments,
· the shape of the hull of a particular vessel is manifested in the DSO over the shoulder l(θ), rigidly connected to the shape of the body contours , which reflects the displacement of the center of the quantity WITH towards the side entering the water when the vessel is heeling.
metacentric height h 0, calculated taking into account the influence of liquid cargo and reserves (see below), appears on the DSO as the tangent 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 = 1 rad (57.3 0) and construct a triangle with a hypotenuse tangent to the DSO at θ = 0, and horizontal leg θ = 57.3 0. The vertical (opposite) leg should be equal to the metacentric height h 0 on 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 the DSO reflects, in a sense, the unchanged shape of the ship’s hull through the value l(θ);
metacentric height h 0 actually determines the type and extent of the DSO.
Roll angle θ = θ 3, at which the DSO graph intersects the x-axis is called the sunset angle of the DSO. Sunset angle θ 3 determines only the value of the roll angle 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 during a roll
θ = θ 3 will not be correct, since the capsizing of the vessel begins much earlier - soon after overcoming the maximum point of the DSO. Maximum point of DSO ( l = l m (θ m)) indicates only the maximum distance between the weight force and the supporting force. However, the maximum leverage l m and maximum angle θm are important quantities in stability control and are subject to verification for compliance with relevant standards.
DSO allows you to solve many problems of ship statics, for example, determining the static angle of roll of a ship under the influence of a constant (independent of the ship’s roll) heeling moment M cr= const. This heel angle can be determined from the condition that the heeling and righting moments are equal M in (θ) = M cr. In practice, this problem is solved as the task of finding the abscissa of the intersection point of the graphs of both moments.
Interaction of heeling and righting moments
The static stability diagram reflects the ship's ability to generate a righting moment when the ship is tilted. Its appearance has a strictly specific character, corresponding to the loading parameters of the vessel only on a given voyage ( R = Р i ,h 0 =h 0i). The navigator, who is involved in planning the loading voyage and stability calculations on the ship, is obliged to build a specific DSO for two states of the ship on the upcoming voyage: with the original location of the cargo unchanged and at 100% and 10% of the ship's stores.
In order to be able to construct static stability diagrams for various combinations of displacement and metacentric height, he uses auxiliary graphic materials available in the ship's documentation for the design of this vessel, for example, pantokarens, or a universal static stability diagram.
Pantocarena
Pantocares are supplied to the ship by the designer as part of information on stability and strength for the captain. Pantocarena are universal graphs for a given vessel, reflecting the shape of its hull in terms of stability.
Pantokarens (Fig. 6) are depicted in the form of 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 ).
Pantocarena
The shape arm is the distance by which the buoyancy force will move relative to the original center of magnitude C o when the ship rolls (Fig. 7). It is clear that this displacement of the center of magnitude is associated only with the shape of the body and does not depend on the position of the center of gravity in height. A set of shape arm values at different heel angles (for a specific vessel weight P=P i) are removed from the pantocaren graphs (Fig. 6).
To determine the stability arms l(θ) and construct a static stability diagram for 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:
Shape and weight stability arms
Having performed calculations for two load states ( R zap.= 100% and 10%), two DSOs are constructed on a blank form, characterizing the stability of the vessel on this voyage. It remains to check the stability parameters for their compliance with national or international standards for the stability of sea vessels.
There is a second way to construct a DSO, using the universal DSO of a given vessel (depending on the availability of specific auxiliary materials on the ship).
Methods for determining the center of magnitude (c.v.) and center of gravity (c.g.) of a vessel
To determine the position of any point on the ship, including the c. t. and c. c., use a system of coordinate axes fixedly connected to the ship’s hull.
The vertical axis OZ is taken to be the line of intersection of DP with the midship-frame plane, the longitudinal - horizontal axis OX is the line of intersection of DP with the main plane, and the transverse - horizontal axis OY is the line of intersection of the midsection - frame with the main plane. In this case, the positive direction of the axes is taken to be the direction of the axis OX - bringing in, OY - to the starboard side, OZ - up. The position of the points g and c of interest to us can be found using approximate and exact dependencies. Approximate methods for determining the coordinate c. V. Coordinate c. V. along the width of the vessel, due to the symmetry of the vessel relative to DP, it should always be in the diametrical plane, i.e. y c =0.
If this equality does not exist, then the ship will be tilted.
The coordinate of point c along the length of the vessel x c is always close to the middle of the vessel, if there is no trim at the bow or stern, and changes its position from the midship frame within small limits. Typically xc varies from +0.02L to -0.035L, where L is the length of the vessel.
Coordinate c. V. the height of the vessel can vary within the following limits: for vessels with a rectangular cross-section z c = 0.5T, where T is the draft of the vessel; for ships with a triangular cross-section z c will be equal to? T from the main plane, i.e. z с =0.66Т, thus this coordinate depends on the shape of the cross section, and therefore on the corresponding completeness coefficients.
Determination of the coordinates of the center of magnitude (c.v.) and the center of gravity (c.t.) The center of gravity (g) of a vessel located without inclination, i.e. floating in an equilibrium position, must always be on the same vertical with the center of magnitude (c). This is achieved by appropriate arrangement of cargo on the ship, and in this case y c = 0.
The position of point g in height, i.e. its application 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 the experimental coefficient, the value of which is recommended for empty cargo ships 0.35?0.5, for towing screw ships 0.60?0.70.
For loaded cargo ships, as well as for passenger ships with high above-deck superstructures, the value of z g can be more than N, 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 items, the distances of the centers of gravity of these weight items from the main plane and the midship plane - frame are determined.
After all the weight loads have been determined, the shoulders of their center of gravity have been found and the moments of force have been calculated, the coordinate of the center of gravity along the length of the vessel x g will be determined by the formula
where UM n is the sum of the moments of all the forces of the weight elements in the bow of the vessel relative to the midship - frame plane;
UM k - the sum of the moments of all the forces of the weight elements in the stern of the ship relative to the midship plane - the frame.
The (+) sign will indicate that the abscissa of the center of gravity is located in the bow of the ship, and the (-) sign that it is located in the stern of the ship, since here the x-axis has a negative value.
The coordinate of the center of gravity along the 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 trapezoidal rule, methods for determining the volumetric displacement of a vessel and drill
Volumetric displacement can be determined in various ways. Let's consider the simplest of them, which provides a degree of accuracy sufficient for practice, a method based on the use of the trapezoidal rule.
Initially, we apply the trapezoidal rule to determine the areas of figures bounded by curved 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 are ordinates y i, and heights Dl.
Figure 7 - Scheme for calculating area using the trapezoidal method
Therefore, S = š 1 + š 2 + ... š n-1 + š n or
Substituting the values for u into the formula in the form of areas of individual trapezoids, we obtain
This expression is called the trapezoidal rule formula, in which y 0 +y 1 +y 2 +y 3 +….+y n-1 +y n is the sum of ordinates, denoted by? 0 ;
It's called an amendment.
The entire value in square brackets is the corrected amount and is indicated by? corrected, then the expression for the area of a curvilinear figure can be written abbreviated as follows:
It is most convenient to carry out all calculations 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 current waterline.
Knowing the dimensions of the vessel and its outline when calculating the volumetric displacement, according to the trapezoidal rule, we proceed from the fact that the volumetric displacement V is replaced by the sum of volumes V 1 +V 2 +V 3 +….+V n-1 +V n into which the underwater part is divided of the vessel with planes equidistant from one another parallel to the midship plane - the frame, or the plane of the existing waterline.
Table 1 - Calculation of area using the trapezoidal method
Let's consider the case when the ship, having a waterline length L, draft T, is cut into n compartments by planes parallel to the midship frame plane, as indicated in Figure 8 with the distance between the compartments.
Figure 8 - Section of the vessel with planes parallel to the plane of the midship frame
Having denoted the volumes of the ship's compartments between the zero and first sections through V 1, between the first and second through V 2, etc., we write an expression for the volume of the underwater part of the ship
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 - the sum of the areas of the frames;
Amendment;
the expression in square brackets is the corrected amount.
To determine the frame areas F i (Figure 9), due to the symmetry of the vessel relative to DP, only half the frame area is determined, and then the result is doubled. In this case, the draft T is divided into m equal parts and the ordinates y 0, y 1 ...., y m are drawn through the division points, the areas limited by these ordinates will be f 1, f 2, ...., f m. Distances between ordinates
Figure 9 - Scheme for calculating the frame area
By analogy with the previous one, the equation for determining the frame area 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 cutting the ship with equidistant 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 a vessel with planes parallel to the main plane
Similar to the previous one, the 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 the ordinates along the waterlines, and then the waterlines along the vessel's draft.
It is easy to see that the result of determining the volumetric displacement in 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 ordinate values y are entered for each waterline for each frame on one side. Sum the ordinates horizontally and vertically, find the corrections for each sum as the sum of the extreme ordinates, find the corrected sums? corr. In horizontal lines, calculate the area of each frame by multiplying the value? isp on DT (distance between waterlines), and in vertical columns calculate the area of each waterline by multiplying the corresponding values? isp on DL (distance between design 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 CU line amounts are obtained. This value should be the same both vertically and horizontally, which is a kind of control for the correctness of calculation of volumetric displacement.
Table 2 - Calculation of the areas of frames, waterlines and displacement of the vessel
|
No. of design frames |
Waterline no. |
Amendment |
Corrected amount?y |
Frame area F=2ДT?y |
|||||
|
Amendment Corrected amount?y Waterline area |
By calculating the value of the double corrected amount?? , determine the volumetric displacement using the formula
Using the values of frame areas obtained in the table, a curve of changes in these areas along the length of the vessel is usually constructed. Such a curve is called a line along the frames. To do this, the length of the vessel L is plotted on some scale, on which the position of all equally spaced design frames from F 0 to F n is plotted. On the reconstructed ordinates, the values of the immersed area of the corresponding frames F are plotted on the appropriate scale. The curve connecting the ends of these ordinates is called a line along the frames (Figure 11).
Figure 11 - Formation along frames
This drill has the following properties:
1. The area of the figure, limited by the line L, the outer ordinates and the line along the frames, calculated according to the trapezoidal rule, is numerically equal to the volumetric displacement of the vessel;
2. Abscissa c.t. this area expresses the abscissa of the c.v. vessel, i.e. X with
3. The coefficient of completeness of the combat area by frames is nothing more than the coefficient of longitudinal completeness of the volumetric displacement of the vessel
4. Construction by frames gives a clear idea of the nature of the distribution of volumetric displacement along the length of the vessel, which is necessary to know when calculating the strength of the vessel.
Similarly, a curve of changes in waterline areas depending on the draft of the vessel is constructed (Figure 12). This curve is called the waterline line. To do this, the draft of the vessel T is plotted on some scale, on which the positions of all equidistant waterlines from S 0 to S m are plotted. On another 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 waterline line. It has the following properties:
1. The area of the figure, limited by the T line, the extreme abscissas and the line along the waterlines, calculated according to the trapezoidal rule, is numerically equal to the volumetric displacement of the vessel;
Figure 12 - Combat line along the waterline
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 c.
3. The coefficient of completeness of the formation area along the waterlines is the coefficient of vertical completeness of the vessel’s displacement
4. The curve gives a visual representation of the nature of the distribution of volumetric displacement over the height of the vessel, which is important to know to characterize the smooth contours of the vessel.
1. Stability of a surface floating body
2. Stability of a surface floating body
A 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 ship does not change its position when rolling. When the ship tilts, the center of magnitude moves in the direction of tilt, while the line of action of the Archimedean force intersects the “0 - 0” axis of navigation at a point called the metacenter. The position of the metacenter does not remain constant when the ship tilts. However, at angles not exceeding u = 15°, the position of the metacenter remains almost unchanged and is accepted as unchanged. In this case, the center of the quantity c moves approximately along a circular arc described from point m with radius r and is called the metacentric radius. The stability of the vessel depends on the relative position of the centers c, g, m.
Suppose we have a ship that has received a list 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 axis of navigation above the center of value c. In this case, the resulting moment when the ship tilts at an angle tends to return the ship to its normal position, i.e. and in this case we have a stable position of the ship.
Figure 13 - Stability of the vessel when the center of gravity is below the center of magnitude.
Figure 14 - Stability of the vessel when the center of gravity is below the metacenter, but above the center of magnitude
However, it is easy to notice that, under equal conditions, the stability in the second case is less than the stability in the first case, since the leverage of the pair of forces, and therefore 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 an unstable vessel position. Having considered three cases with a ship that had a different position of the center of gravity, we can say that the higher the center of gravity of the ship, the less its stability. Consequently, to increase the stability of bodies, one must always strive to lower their center of gravity.
Figure 15 - Stability of the vessel 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 relationship between 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 the relative position of centers c and g, when center c lies on the axis of swimming below center g.
Thus
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 swimming axis is considered to be the metacentric height h. There is the following relationship between h,r and a If we now turn our attention again to the cases of ship position considered above, 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 and, i.e. when h=0 or r= a, we will have the case of an unstable position of the vessel, since in this case the lines of action of the Archimedean force D and the force of gravity of the vessel G will coincide and, therefore, no restoring moment can be formed. This case in the theory of swimming is called an indifferent state. During the operation of ships, it is sometimes necessary to switch from linear movement to movement along a curve and vice versa. This is possible provided that external forces are applied to the ship, the moments of which will force 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 ship's course 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 only be used on self-propelled ships with two propulsors. With the help of propulsion devices, the ship changes course if the stops from the propulsion T are unequal 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 are the stops of the left and right movers; l is the distance between the axes of the propellers. This moment forces the ship to change its course. If T 1 = T 2, the ship will rotate in place without receiving forward motion. If T 1 >T 2, the vessel, in addition to rotation under the influence of torque, will also have forward motion, and if T 1<Т 2 судна, кроме вращения, будет иметь и поступательное движение назад. Usually, to turn the vessel, a steering device is used, which is, in the most general case, a vertical plate (rudder blade) located in the flow 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 turning the rudder If the rudder is deflected from the diametral by an angle b, then at a vessel speed V, according to the laws of hydromechanics, a hydrodynamic pressure force acts on the rudder, the magnitude of which can be determined using the Jossel formula where R a is the water pressure on the rudder blade; F is the area of the underwater part of the rudder blade; V—vessel speed; b - rudder shift angle (angle of deviation from the diametral); k b - experimental coefficient depending on the angle b, it represents the pressure per 1 m 2 of the rudder blade area at a ship speed of 1 m/sec. The value of k b is determined by the empirical formula The value of k is recommended to be 400 n/m 3 for single-screw ships, and 225 n/m 3 for twin-screw ships. When the rudder is shifted to an angle b on the ship, in addition to the resistance force R and the stop T, which are mutually balanced (with uniform movement), 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, b is the length of the rudder blade, and l is the length of the vessel, due to which the value is neglected. After substituting the value of P a into equation (48), it is clear that if the ship is moving at a constant speed, the magnitude of the moment depends on the product cosб sinb. This product reaches its maximum at b = 36 o. It follows that there is no point in deflecting the rudder blade by more than 35-36 degrees, since the moment of rotation of the vessel does not increase. 2., drifting the ship in the opposite direction of the rudder turn. In order to verify this, let us apply forces Ra at point g, directed in opposite directions. This will not disturb the balance of the vessel. 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 its 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 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 drag force. is the force that causes a ship to change its original course. The considered complex scheme of interaction of the forces arising in connection with the shifting of the rudder blade to angle b also determines a very complex path of movement of the vessel. It is customary to consider three periods of vessel motion. The first is maneuverable, when the rudder is shifted and when, under the influence of force, the ship suffers a lateral drift. The second is evolutionary, which continues until the ship begins to rotate uniformly around a fixed axis. The third 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 during its full turn is called the ship's circulation (Figure 21), and its diameter is the circulation diameter. The time during which the ship makes a complete revolution is called the circulation period. The smaller the circulation diameter, the better the vessel’s agility; therefore, agility is one of the most important qualities of rafting vessels that have to work in timber rafting roads 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, m2; l,T - length and draft of the vessel, m; OB - maneuvering period when lateral drift occurs, numerically equal to k; BC is an evolutionary period.
The longitudinal stability of a vessel is much higher than its transverse stability, so for safe navigation it is most important to ensure proper transverse stability.
- Depending on the magnitude of the inclination, stability at small angles of inclination is distinguished ( initial stability) and stability at large inclination angles.
- Depending on the nature of the acting forces, static and dynamic stability are distinguished.
Initial lateral stability
Initial lateral stability. System of forces acting on the ship
During roll, stability is considered as initial at angles up to 10-15°. Within these limits, the righting force is proportional to the roll angle and can be determined using simple linear relationships.
In this case, the assumption is made that deviations from the equilibrium position are caused by external forces that do not change either the weight of the vessel or the position of its center of gravity (CG). Then the immersed volume does not change in size, but changes in shape. Equal-volume inclinations correspond to equal-volume waterlines, cutting off immersed volumes of the hull of equal magnitude. The line of intersection of the waterline planes is called the inclination axis, which, with equal volume inclinations, passes through the center of gravity of the waterline area. With transverse inclinations, it lies in the center plane.
Free surfaces
All the cases discussed above assume that the center of gravity of the vessel is stationary, that is, there are no loads that move when tilted. But when such loads exist, their influence on stability is much greater than others.
A typical case is liquid cargo (fuel, oil, ballast and boiler water) in tanks that are partially filled, that is, with free surfaces. Such loads can overflow when tilted. If the liquid cargo fills the tank completely, it is equivalent to a solid fixed cargo.
Effect of free surface on stability
If the liquid does not completely fill the tank, i.e. has a free surface that always occupies a horizontal position, then when the ship tilts at an angle θ the liquid flows towards the inclination. The free surface will take the same angle relative to the KVL.
Levels of liquid cargo cut off equal volumes of tanks, i.e. they are similar to equal volume waterlines. Therefore, the moment caused by the overflow of liquid cargo during a roll δm θ, can be represented similarly to the moment of shape stability m f, only δm θ opposite m f by sign:
δm θ = - γ f i x θ,Where i x- moment of inertia of the free surface area of the liquid load relative to the longitudinal axis passing through the center of gravity of this area, γ f- specific gravity of liquid cargo
Then the restoring moment in the presence of a liquid load with a free surface:
m θ1 = m θ + δm θ = Phθ − γ f i x θ = P(h − γ f i x /γV)θ = Ph 1 θ,Where h- transverse metacentric height in the absence of transfusion, h 1 = h − γ f i x /γV- actual transverse metacentric height.
The effect of the iridescent weight gives a correction to the transverse metacentric height δ h = - γ f i x /γV
The densities of water and liquid cargo are relatively stable, that is, the main influence on the correction is exerted by the shape of the free surface, or rather its moment of inertia. This means that the lateral stability is mainly affected by the width, and the longitudinal length of the free surface.
The physical meaning of the negative correction value is that the presence of free surfaces is always reduces
In contrast to the static effect, the dynamic effect of forces and moments imparts significant angular velocities and accelerations to the vessel. Therefore, their influence is considered in energies, more precisely in the form of the work of forces and moments, and not in the efforts themselves. In this case, the kinetic energy theorem is used, according to which the increment in the kinetic energy of the vessel's inclination is equal to the work of the forces acting on it.
When a heeling moment is applied to the ship m cr, constant in magnitude, it receives a positive acceleration with which it begins to roll. As you tilt, the restoring moment increases, but at first, up to the angle θ st, at which m cr = m θ, it will be less heeling. Upon reaching the static equilibrium angle θ st, the kinetic energy of rotational motion will be maximum. Therefore, the ship will not remain in the equilibrium position, but due to kinetic energy it will roll further, but slowly, since the righting moment is greater than the heeling moment. The previously accumulated kinetic energy is extinguished by the excess work of the restoring torque. As soon as the magnitude of this work is sufficient to completely extinguish the kinetic energy, the angular velocity will become zero and the ship will stop heeling.
The greatest angle of inclination that a ship receives from a dynamic moment is called the dynamic angle of heel θ din. In contrast, the angle of roll with which the ship will float under the influence of the same moment (according to the condition m cr = m θ), is called the static roll angle θ st.
If we refer to the static stability diagram, the work is expressed by the area under the righting moment curve m in. Accordingly, the dynamic roll angle θ din can be determined from the equality of areas OAB And BCD, corresponding to the excess work of the restoring torque. Analytically the same work is calculated as:
,
in the range from 0 to θ din.
Having reached the dynamic bank angle θ din, the ship does not come into equilibrium, but under the influence of an excess righting moment begins to accelerate to straighten. In the absence of water resistance, the ship would enter into undamped oscillations around the equilibrium position when heeling θ st Marine Dictionary - Refrigerated vessel Ivory Tirupati initial stability is negative Stability is the ability of a floating craft to resist external forces causing it to roll or trim and return to a state of equilibrium after the end of the disturbance... ... Wikipedia
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The main characteristic of stability is righting moment, which must be sufficient for the vessel to withstand the static or dynamic (sudden) action of heeling and trim moments arising from the displacement of cargo, under the influence of wind, waves and other reasons.
The heeling (trimming) and righting moments act in opposite directions and are equal in the equilibrium position of the vessel.
Distinguish lateral stability, corresponding to the inclination of the vessel in the transverse plane (vessel roll), and longitudinal stability(ship trim).
The longitudinal stability of sea vessels is obviously ensured and its violation is practically impossible, while the placement and movement of cargo leads to changes in lateral stability.
When the ship tilts, its center of magnitude (CM) will move along a certain curve called the CM trajectory. With a small inclination of the vessel (no more than 12°), it is assumed that the trajectory of the central point coincides with a flat curve, which can be considered an arc of radius r with a center at point m.
The radius r is called transverse metacentric radius of the vessel, and its center m - initial metacenter of the ship.
Metacenter - the center of curvature of the trajectory along which the center of magnitude C moves during the process of tilting the ship. If the inclination occurs in the transverse plane (roll), the metacenter is called transverse, or small, while the inclination in the longitudinal plane (trim) is called longitudinal, or large.
Accordingly, transverse (small) r and longitudinal (large) R metacentric radii are distinguished, representing the radii of curvature of the trajectory C during roll and trim.
The distance between the initial metacenter t and the center of gravity of the vessel G is called initial metacentric height(or simply metacentric height) and are designated by the letter h. The initial metacentric height is a measure of the ship's stability.
h = zc + r - zg; h = zm ~ zc; h = r - a,
where a is the elevation of the center of gravity (CG) above the CV.
Metacentric height (m.h.) - the distance between the metacenter and the center of gravity of the vessel. M.v. is a measure of the initial stability of the vessel, determining the righting moments at small angles of roll or trim.
With increasing m.v. The stability of the vessel increases. For positive stability of the ship, it is necessary that the metacenter be above the center of gravity of the ship. If m.v. negative, i.e. the metacenter is located below the center of gravity of the ship, the forces acting on the ship form not a restoring moment, but a heeling moment, and the ship floats with an initial roll (negative stability), which is not allowed.
OG – elevation of the center of gravity above the keel; OM – elevation of the metacenter above the carina;
GM - metacentric height; CM – metacentric radius;
m – metacenter; G – center of gravity; C – center of magnitude
There are three possible cases of the location of the metacenter m relative to the center of gravity of the vessel G:
the metacenter m is located above the center of gravity of the vessel G (h > 0). With a small inclination, the forces of gravity and buoyancy forces create a pair of forces, the moment of which tends to return the ship to its original equilibrium position;
The ship's CG G is located above the metacenter m (h< 0). В этом случае момент пары сил веса и плавучести будет стремиться увеличить крен судна, что ведет к его опрокидыванию;
The ship's center of gravity G and the metacenter m coincide (h = 0). The ship will behave unstable, since the shoulder of the couple of forces is missing.
The physical meaning of the metacenter is that this point serves as the limit to which the ship’s center of gravity can be raised without depriving the ship of positive initial stability.
§ 12. Seaworthiness of ships. Part 1
Both civilian vessels and military ships must have seaworthiness.A special scientific discipline deals with the study of these qualities using mathematical analysis - ship theory.
If a mathematical solution to the problem is impossible, then they resort to experiment to find the necessary dependence and test the conclusions of the theory in practice. Only after a comprehensive study and experience testing of all the seaworthiness of the vessel do they begin to create it.
Seaworthiness in the subject “Ship Theory” is studied in two sections: statics and dynamics of the vessel. Statics studies the laws of equilibrium of a floating vessel and the associated qualities: buoyancy, stability and unsinkability. Dynamics studies a ship in motion and considers its qualities such as controllability, pitching and propulsion.
Let's get acquainted with the seaworthiness of the vessel.
Buoyancy of the vessel is called its ability to float on water at a certain draft, carrying intended loads in accordance with the purpose of the vessel.
A floating ship is always acted upon by two forces: a) on the one hand, weight force, equal to the sum of the weight of the vessel itself and all cargo on it (calculated in tons); the resultant of the weight forces is applied to ship's center of gravity(CG) at point G and is always directed vertically downwards; b) on the other hand, maintaining forces, or buoyancy forces(expressed in tons), i.e., the water pressure on the submerged part of the hull, determined by the product of the volume of the submerged part of the hull by the volumetric weight of the water in which the ship floats. If these forces are expressed by the resultant applied at the center of gravity of the underwater volume of the vessel at point C, called center of magnitude(CV), then this resultant will always be directed vertically upward in all positions of the floating vessel (Fig. 10).
Volumetric displacement is the volume of the immersed part of the hull, expressed in cubic meters. Volumetric displacement serves as a measure of buoyancy, and the weight of water displaced by it is called weight displacement D) and is expressed in tons.
According to Archimedes' law, the weight of a floating body is equal to the weight of the volume of liquid displaced by this body,
Where y is the volumetric weight of sea water, t/m 3, taken in calculations to be equal to 1.000 for fresh water and 1.025 for sea water.
Rice. 10. Forces acting on a floating ship and the points of application of the resultant forces.
Since the weight of a floating vessel P is always equal to its weight displacement D, and their resultants are directed opposite to each other along the same vertical, and if we designate the coordinates of points G and C along the length of the vessel, respectively x g and x c, along the width y g and y c and along height z g and z c , then the equilibrium conditions of a floating vessel can be formulated by the following equations:
P = D; x g = x c .
Due to the symmetry of the ship relative to the DP, it is obvious that points G and C must lie in this plane, then
Y g = y c = 0.
Typically, the center of gravity of surface vessels G lies above the center of magnitude C, in which case
Sometimes it is more convenient to express the volume of the underwater part of the hull through the main dimensions of the vessel and the coefficient of overall completeness, i.e.
Then the weight displacement can be represented as
If we denote by V n the total volume of the hull up to the upper deck, provided that all side openings are closed watertight, we obtain
The difference V n - V, representing a certain volume of the waterproof hull above the load waterline, is called reserve buoyancy. In the event of an emergency ingress of water into the ship's hull, its draft will increase, but the ship will remain afloat, thanks to its reserve of buoyancy. Thus, the greater the height of the freeboard impermeable side, the greater the reserve of buoyancy. Consequently, buoyancy reserve is an important characteristic of a vessel, ensuring its unsinkability. It is expressed as a percentage of normal displacement and has the following minimum values: for river vessels 10-15%, for tankers 10-25%, for dry cargo ships 30-50%, for icebreakers 80-90%, and for passenger ships 80-100 %.
Rice. 11. Construction along frames
The weight of the vessel P (weight load) And the coordinates of the center of gravity are determined by a calculation that takes into account the weight of each part of the hull, mechanisms, pieces of equipment, supplies, supplies, cargo, people, their luggage and everything on the ship. To simplify calculations, it is planned to combine individual specialty titles into articles, subgroups, groups and workload sections. For each of them, the weight and static moment are calculated.
Considering that the moment of the resultant force is equal to the sum of the moments of the component forces relative to the same plane, after summing up the weights and static moments over the entire vessel, the coordinates of the vessel’s center of gravity G are determined. Volumetric displacement, as well as the coordinates of the center of the value C along the length from the midsection x c and along height from the main line z c is determined from a theoretical drawing using the trapezoidal method in tabular form.
For the same purpose, they use auxiliary curves, the so-called construction curves, also drawn according to the data of the theoretical drawing.
There are two curves: formation along the frames and formation along the waterlines.
Construction on frames(Fig. 11) characterizes the distribution of the volume of the underwater part of the hull along the length of the vessel. It is built in the following way. Using the method of approximate calculations, the area of the immersed part of each frame (w) is determined from a theoretical drawing. The length of the vessel is plotted along the abscissa axis on the selected scale and the position of the frames of the theoretical drawing is plotted on it. On the ordinates reconstructed from these points, the corresponding areas of the calculated frames are plotted on a certain scale.
The ends of the ordinates are connected by a smooth curve, which is the line along the frames.
Rice. 12. Drilling along the waterline.
Drilling along the waterline(Fig. 12) characterizes the distribution of the volume of the underwater part of the hull along the height of the vessel. To construct it, using a theoretical drawing, calculate the areas of all waterlines (5). These areas on a selected scale are laid out along the corresponding horizontal lines located along the draft of the vessel, in accordance with the position of a given waterline. The resulting points are connected by a smooth curve, which is the line along the waterlines.
Rice. 13. Cargo size curve.
These curves serve as the following characteristics:
1) the areas of each of the combat units express the volumetric displacement of the vessel on the appropriate scale;
2) the abscissa of the center of gravity of the combat area along the frames, measured on the scale of the length of the vessel, is equal to the abscissa of the center of magnitude of the vessel x c;
3) the ordinate of the center of gravity of the building area along the waterlines, measured on the draft scale, is equal to the ordinate of the center of the vessel's size z c. Cargo size is a curve (Fig. 13) characterizing the volumetric displacement of the vessel V depending on its draft T. Using this curve, you can determine the displacement of the vessel depending on its draft or solve the inverse problem.
This curve is constructed in a system of rectangular coordinates based on pre-calculated volumetric displacements along each waterline of the theoretical drawing. On the ordinate axis, on a selected scale, the draft of the vessel is plotted along each of the waterlines and horizontal lines are drawn through them, on which, also on a certain scale, the displacement value obtained for the corresponding waterlines is plotted. The ends of the resulting segments are connected by a smooth curve, which is called the load size.
Using the cargo size, it is possible to determine the change in the average draft due to the receipt or discharge of cargo or, based on a given displacement, to determine the draft of the vessel, etc.
Stability called the ability of a ship to resist the forces that caused it to tilt, and after the cessation of these forces, return to its original position.
The tilting of the vessel is possible for various reasons: from the action of oncoming waves, due to asymmetrical flooding of compartments during a hole, from the movement of cargo, wind pressure, due to the receipt or consumption of cargo, etc.
The inclination of the ship in the transverse plane is called roll, and in the longitudinal plane - d ifferent; the angles formed in this case are denoted by O and y, respectively,
There are initial stability, i.e. stability at small angles of heel at which the edge of the upper deck begins to enter the water (but not more than 15° for high-sided surface vessels), and stability at high inclinations .
Let's imagine that, under the influence of external forces, the ship tilted at an angle of 9 (Fig. 14). As a result, the volume of the underwater part of the vessel retained its size, but changed its shape; On the starboard side, an additional volume entered the water, and on the left side, an equal volume came out of the water. The center of magnitude moved from the original position C towards the ship's roll, to the center of gravity of the new volume - point C 1. When the vessel is in an inclined position, the gravity force P applied at point G and the support force D applied at point C, remaining perpendicular to the new waterline B 1 L 1 form a pair of forces with the arm GK, which is a perpendicular lowered from point G to the direction of the support forces .
If we continue the direction of the support force from point C 1 until it intersects with its original direction from point C, then at small roll angles corresponding to the conditions of initial stability, these two directions will intersect at point M, called transverse metacenter .
The distance between the metacenter and the center of magnitude MC is called transverse metacentric radius, denoted by p, and the distance between point M and the center of gravity of the vessel G is transverse metacentric height h 0. Based on the data in Fig. 14 we can form an identity
H 0 = p + z c - z g .
In a right triangle GMR, the angle at the vertex M will be equal to angle 0. From its hypotenuse and the opposite angle, one can determine the leg GK, which is shoulder m of a couple restoring a vessel GK=h 0 sin 8, and the restoring moment will be equal to Mvost = DGK. Substituting the leverage values, we get the expression
Mvost = Dh 0 * sin 0,
Rice. 14. Forces acting when the ship rolls.
The relative position of points M and G allows us to establish the following feature characterizing lateral stability: if the metacenter is located above the center of gravity, then the restoring moment is positive and tends to return the vessel to its original position, i.e., when heeling, the vessel will be stable, vice versa, if point M is located below point G, then with a negative value of h 0 the moment is negative and will tend to increase the roll, i.e. in this case the ship is unstable. A case is possible when points M and G coincide, forces P and D act along the same vertical line, a pair of forces does not arise, and the restoring moment is zero: then the ship should be considered unstable, since it does not strive to return to its original equilibrium position (Fig. 15).
The metacentric height for representative load cases is calculated during the design process of the vessel and serves as a measure of stability. The value of the transverse metacentric height for the main types of ships lies in the range of 0.5-1.2 m and only for icebreakers it reaches 4.0 m.
To increase the lateral stability of a vessel, it is necessary to reduce its center of gravity. This is an extremely important factor that must always be remembered, especially when operating a vessel, and strict records must be kept of the consumption of fuel and water stored in double-bottom tanks.
Longitudinal metacentric height H 0 is calculated similarly to the transverse one, but since its value, expressed in tens or even hundreds of meters, is always very large - from one to one and a half lengths of the vessel, then after the verification calculation the longitudinal stability of the vessel is practically not calculated; its value is interesting only in the case of determining the draft of the vessel bow or stern during longitudinal movements of cargo or when compartments are flooded along the length of the vessel.
Rice. 15. Transverse stability of the vessel depending on the location of the cargo: a - positive stability; b - equilibrium position - the ship is unstable; c - negative stability.
The issues of vessel stability are given exceptional importance, and therefore usually, in addition to all theoretical calculations, after the construction of the vessel, the true position of its center of gravity is checked by experimental inclination, i.e., lateral inclination of the vessel by moving a load of a certain weight, called incline ballast .
All previously obtained conclusions, as already mentioned, are practically valid at initial stability, i.e., at small angles of roll.
When calculating lateral stability at large angles of roll (longitudinal inclinations in practice are not large), the variable positions of the center of magnitude, metacenter, transverse metacentric radius and the arm of the righting moment GK are determined for various angles of roll of the vessel. This calculation is made starting from the straight position through 5-10° to the angle of roll when the righting arm turns to zero and the ship acquires negative stability.
According to the data of this calculation, for a visual representation of the stability of the vessel at large angles of heel, a static stability diagram(it is also called the Reed diagram), showing the dependence of the static stability arm (GK) or the righting moment Mvost on the roll angle 8 (Fig. 16). In this diagram, the heel angles are plotted along the abscissa axis, and the value of the righting moments or the arms of the righting pair are plotted along the ordinate axis, since in equal-volume inclinations, at which the displacement of the vessel D remains constant, the righting moments are proportional to the stability arms.
Rice. 16. Diagram of static stability.
A static stability diagram is constructed for each characteristic case of ship loading, and it characterizes the stability of the ship as follows:
1) at all angles at which the curve is located above the x-axis, the restoring arms and moments have a positive value, and the ship has positive stability. At those heel angles when the curve is located below the abscissa axis, the ship will be unstable;
2) the maximum of the diagram determines the maximum heel angle of 0 max and the maximum heeling moment when the vessel is statically tilted;
3) the angle 8 at which the descending branch of the curve intersects the abscissa axis is called sunset angle diagram. At this roll angle, the righting arm becomes zero;
4) if on the abscissa axis we plot an angle equal to 1 radian (57.3°), and from this point we construct a perpendicular to the intersection with the tangent drawn to the curve from the origin, then this perpendicular on the scale of the diagram will be equal to the initial metacentric height h 0 .
Stability is greatly influenced by moving, i.e., unsecured, as well as liquid and bulk cargoes that have a free (open) surface. When the vessel tilts, these loads begin to move in the direction of the roll and, as a result, the center of gravity of the entire vessel will no longer be at a fixed point G, but will also begin to move in the same direction, causing a decrease in the lateral stability arm, which is equivalent to a decrease in the metacentric height with all consequences arising from this. To prevent such cases, all cargo on ships must be secured, and liquid or bulk cargo must be loaded into containers that prevent any transfer or spilling of cargo.
With the slow action of forces that create a heeling moment, the ship, tilting, will stop when the heeling and righting moments are equal. Under the sudden action of external forces, such as a gust of wind, the pull of a tug on board, pitching, a broadside salvo from guns, etc., the ship, tilting, acquires angular speed and even with the cessation of the action of these forces will continue to roll by inertia for an additional angle until all its kinetic energy (living force) of the rotational motion of the vessel is used up and its angular velocity becomes zero. This tilting of the ship under the influence of suddenly applied forces is called dynamic tilt. If during a static heeling moment the ship floats, having only a certain roll of 0 ST, then in the case of dynamic action of the same heeling moment it can capsize.
When analyzing dynamic stability, for each displacement of the vessel, a dynamic stability diagrams, the ordinates of which represent, on a certain scale, the areas formed by the curve of the moments of static stability for the corresponding roll angles, i.e., they express the work of the righting pair when the vessel is tilted at an angle of 0, expressed in radians. In rotational motion, as is known, the work is equal to the product of the moment and the angle of rotation, expressed in radians,
T 1 = M kp 0.
Using this diagram, all issues related to the determination of dynamic stability can be resolved as follows (Fig. 17).
The roll angle with a dynamically applied heeling moment can be found by plotting the operation of the heeling pair on a diagram on the same scale; The abscissa of the intersection point of these two graphs gives the desired angle 0 DIN.
If in a particular case the fastening moment has a constant value, i.e. M cr = const, then the work will be expressed
T 2 = M kp 0.
And the graph will look like a straight line passing through the origin.
In order to construct this straight line on the dynamic stability diagram, it is necessary to plot an angle equal to a radian along the abscissa axis and draw an ordinate from the resulting point. Having plotted the value M cr on it on an ordinate scale in the form of a segment Nn (Fig. 17), it is necessary to draw a straight line ON, which is the desired graph of the operation of the heeling pair.
Rice. 17. Determination of the roll angle and maximum dynamic inclination using the dynamic stability diagram.
The same diagram shows the dynamic inclination angle 0 DIN, defined as the abscissa of the point of intersection of both graphs.
With an increase in the moment M cr, the secant ON can take a limiting position, turning into an external tangent OT drawn from the origin to the dynamic stability diagram. Thus, the abscissa of the tangent point will be the maximum limiting angle of the dynamic inclinations 0. The ordinate of this tangent, corresponding to the radian, expresses the maximum heeling moment at the dynamic inclinations M crmax.
When sailing, a ship is often exposed to dynamic external forces. Therefore, the ability to determine the dynamic heeling moment when deciding on the stability of a vessel is of great practical importance.
A study of the causes of ship deaths leads to the conclusion that ships mainly die due to loss of stability. To limit the loss of stability in accordance with various navigation conditions, the Register of the USSR has developed Stability Standards for transport and fishing fleet vessels. In these standards, the main indicator is the ship’s ability to maintain positive stability under the combined action of roll and wind. The vessel meets the basic requirement of the Stability Standards if, under the worst loading scenario, its M CR remains less than M OPR.
In this case, the minimum capsizing moment of the vessel is determined from static or dynamic stability diagrams, taking into account the influence of the free surface of liquid cargo, roll and elements of the calculation of the vessel's windage for various cases of vessel loading.
The standards provide for a number of requirements for stability, for example: M KR
the metacentric height must have a positive value, the sunset angle of the static stability diagram must be at least 60°, and taking into account icing - at least 55°, etc. Mandatory compliance with these requirements in all cases of loading gives the right to consider the vessel stable.
Unsinkability of the ship is called its ability to maintain buoyancy and stability after flooding of part of the interior with water coming from overboard.
The unsinkability of the vessel is ensured by the reserve of buoyancy and the preservation of positive stability in partially flooded rooms.
If the ship has a hole in the outer hull, then the amount of water Q flowing through it is characterized by the expression
where S is the area of the hole, m²;
G - 9.81 m/s²
N - distance of the center of the hole from the waterline, m.
Even with a minor hole, the amount of water entering the body will be so large that the sump pumps will not be able to cope with it. Therefore, drainage equipment is installed on the ship based on the calculation of only removing water entering after the hole has been repaired or through leaks in the joints.
To prevent the spread of water flowing into the hole throughout the ship, constructive measures are provided: the hull is divided into separate compartments watertight bulkheads and decks. With this division, in the event of a hole, one or more limited compartments will flood, which will increase the vessel's draft and, accordingly, reduce the freeboard and the vessel's buoyancy reserve.
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