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Initial stability of the vessel
1. General concept about stability
Stability is the ability of a ship to resist forces that deviate it from its equilibrium position, and to return to its original equilibrium position after the action of these forces ceases.
The equilibrium conditions of the vessel are not sufficient for it to constantly float in a given position relative to the surface of the water. It is also necessary that the balance of the vessel be stable. The property, which in mechanics is called stability of equilibrium, in ship theory is usually called stability. Thus, buoyancy provides the conditions for the equilibrium position of the vessel with a given landing, and stability ensures the preservation of this position.
The stability of the vessel changes with increasing angle of inclination and at a certain value it is completely lost. Therefore, it seems appropriate to study the stability of the vessel at small (theoretically infinitesimal) deviations from the equilibrium position with I = 0, W = 0, and then determine the characteristics of its stability, their permissible limits at large inclinations.
It is customary to distinguish between the stability of a vessel at small angles of inclination (initial stability) and stability at large angles of inclination.
When considering small inclinations, it is possible to make a number of assumptions that make it possible to study the initial stability of the vessel within the framework of linear theory and obtain simple mathematical dependencies of its characteristics. The stability of the vessel at large angles of inclination is studied using a refined nonlinear theory. Naturally, the stability property of a vessel is uniform and the accepted division is purely methodological in nature.
When studying the stability of a vessel, its inclinations in two mutually perpendicular planes - transverse and longitudinal - are considered. When the ship tilts in the transverse plane, determined by the roll angles, its lateral stability is studied; when inclined in the longitudinal plane, determined by the trim angles, its longitudinal stability is studied.
If the ship tilts without significant angular accelerations (pumping liquid cargo, slow flow of water into the compartment), then stability is called static.
In some cases, the forces tilting the ship act suddenly, causing significant angular accelerations (wind squall, wave roll, etc.). In such cases, dynamic stability is considered.
Stability is a very important seaworthiness property of a vessel; together with buoyancy, it ensures the vessel floats in a given position relative to the surface of the water, necessary to ensure movement and maneuver. A decrease in the stability of the vessel can cause an emergency roll and trim, and a complete loss of stability can cause it to capsize.
To prevent a dangerous decrease in the stability of the vessel, all crew members are obliged to:
Always have a clear understanding of the vessel's stability;
Know the reasons that reduce stability;
Know and be able to apply all means and measures to maintain and restore stability.
2. Equal volumetric inclinations of the vessel. Euler's theorem
The stability of a vessel is studied under so-called equal-volume inclinations, in which the value of the underwater volume remains unchanged, and only the shape of the underwater part of the vessel changes.
Let us introduce the basic definitions related to the inclination of the vessel:
The inclination axis is the line of intersection of the planes of two waterlines;
The inclination plane is a plane perpendicular to the inclination axis, passing through the CV corresponding to the initial equilibrium position of the vessel.;
Inclination angle - the angle of rotation of the vessel about the inclination axis (the angle between the waterline planes), measured in the inclination plane;
Equivolume waterlines are waterlines that cut off wedge-shaped volumes of equal size when the ship tilts, one of which enters the water when the ship tilts, and the other comes out of the water.
Rice. 1. Consideration of Euler’s theorem
Given a known initial waterline, Euler's theorem is used to construct a waterline of equal volume to it. According to this theorem, with an infinitesimal inclination of the vessel, the planes of equal-volume waterlines intersect along a straight line passing through their common geometric center (center of gravity), or the axis of infinitesimal equal-volume inclination passes through the geometric center of the area of the original waterline.
Euler's theorem can also be applied for finite small inclinations with the smaller the error, the smaller the inclination angle.
It is assumed that accuracy sufficient for practice is ensured at inclinations I 1012 0 and Ш 23 0. Within these angles, the initial stability of the vessel is considered.
As is known, when a vessel is sailing without a list and with a trim close to zero, the ordinate of the geometric center of the waterline area y f = 0, and the abscis x f 0. Therefore, in this case we can assume that the axis of the transverse small equal-volume inclination lies in the DP, and the axis of the longitudinal small equivolume inclination is perpendicular to the DP and offset from the square. midship - frame at a distance x f (Fig. 1).
The value x f is a function of the vessel's draft d. The dependence x f (d) is presented on the curves of the elements of the theoretical drawing.
When the ship is inclined in an arbitrary plane, the axis of equal volume inclinations will also pass through the geometric center (center of gravity) of the waterline area.
3. Metacenters and metacentric radii
Let us assume that the ship, from its initial position without roll or trim, makes transverse or longitudinal equal volume inclinations. In this case, the plane of longitudinal inclinations will be a vertical plane that coincides with the DP, and the plane of transverse inclinations will be a vertical plane that coincides with the plane of the frame passing through the CV.
Lateral inclinations
In the upright position of the vessel, the CV is in the DP (point C) and the line of action of the buoyancy force gV also lies in the DP (Fig. 2). When the vessel is tilted transversely at an angle I, the shape of the immersed volume changes, the CV moves in the direction of inclination from point C to point C I, and the line of action of the buoyancy force will be inclined to the DP at an angle I.
The point of intersection of the lines of action of the buoyancy force at an infinitesimal transverse equal-volume inclination of the vessel is called the transverse metacenter (point m in Fig. 2). The radius of curvature of the trajectory of the CV r (the elevation of the transverse metacenter above the CV) is called the transverse metacentric radius.
In the general case, the CV trajectory is a complex spatial curve and each inclination angle corresponds to its own position of the metacenter (Fig. 3). However, for small equal-volume inclinations, with a known approximation, it can be assumed that the trajectory
The CV lies in the plane of inclination and is an arc of a circle with its center at point m. Thus, we can assume that during a small transverse equal-volume inclination of the vessel from a straight position, the transverse metacenter lies in the DP and does not change its position (r = const).
Rice. 2. Movement of the central wheel at low inclinations
Rice. 3. Movement of the central point at large inclinations
Rice. 4. To derive the expression for the transverse metacentric radius
The expression for the transverse metacentric radius r is obtained from the condition that the axis of the small transverse equal-volume inclination of the vessel lies in the DP and that with such an inclination the wedge-shaped volume v is, as it were, transferred from the side that left the water to the side that entered the water (Fig. 4).
According to the well-known theorem of mechanics, when a body belonging to a system of bodies moves, the center of gravity of the entire system will move in the same direction parallel to the movement of the body, and these movements are inversely proportional to the gravitational forces of the body and the system, respectively. This theorem can be extended to the volumes of homogeneous bodies. Let's denote:
С С И - movement of the central point (geometric center of volume V),
b - displacement of the geometric center of the wedge-shaped volume v. Then, in accordance with the theorem
from where: S S I =
For the vessel length element dx, assuming that the wedge-shaped volume has the shape of a triangle in the plane of the frame, we obtain:
or at low angle
If by, then:
dv b = y 3 And dx.
Integrating, we get:
v b = And y 3 dx, or:
where J x = ydx is the moment of inertia of the waterline area relative to the longitudinal central axis.
Then the expression for moving the CV will look like:
As can be seen from Fig. 5, at small angle I
S S I r I
Comparing the expressions, we find that the transverse metacentric radius is:
r =
Applicate of the transverse metacentre:
z m = z c + r = z c +
Longitudinal inclinations
Rice. 6. To derive the expression for the longitudinal metacentric radius
By analogy with transverse inclinations, the point of intersection of the lines of action of the buoyancy force at an infinitesimal longitudinal equal-volume inclination of the vessel is called the longitudinal metacenter (point M in Fig. 6). The elevation of the longitudinal metacenter above the CV is called the longitudinal metacentric radius. The magnitude of the longitudinal radius is determined by the expression:
R = ,
where J yf is the moment of inertia of the waterline area relative to the transverse central axis.
Applicate of the longitudinal metacentre:
z m = z c + R = z c +
Since the area of the waterline is elongated in the longitudinal direction, J yf is much greater than J x and, accordingly, R is much greater than r. The value of R is 1 2 ship lengths.
Metacentric radii and applicates of metacenters are, as will be clear from subsequent consideration, important characteristics of the stability of the vessel. Their values are determined when calculating the elements of the immersed volume and for a ship floating without heel and trim, they are represented by the curves J x (d), J yf (d), r(d), R(d) in the drawing of the curved elements of the theoretical drawing.
4. Condition initial stability vessel
Metacentric heights
Let us find a condition under which a ship floating in a state of equilibrium without roll or trim will have initial stability. We assume that the loads do not shift when the ship tilts and the center of gravity of the ship remains at the point corresponding to the initial position.
When the ship tilts, the force of gravity P and the buoyancy force rV form a pair, the moment of which acts on the ship in a certain way. The nature of this effect depends on the relative position of the CG and the metacenter.
Rice. 6. First case of ship stability
There are three possible characteristic cases of the state of the vessel for which the influence on it of the moment of forces P and rV is qualitatively different. Let's consider them using the example of transverse inclinations.
1st case (Fig. 6) - the metacenter is located above the CG, i.e. z m > z g . In this case, a different location of the center of magnitude relative to the center of gravity is possible.
I. In the initial position, the center of magnitude (point C 0) is located below the center of gravity (point G) (Fig. 6, a), but when inclined, the center of magnitude shifts towards the inclination so much that the metacenter (point m) is located above the center the gravity of the vessel. The moment of forces P and rV tends to return the vessel to its original equilibrium position, and therefore it is stable. A similar arrangement of points m, G and C 0 is found on most ships.
II. In the initial position, the center of magnitude (point C 0) is located above the center of gravity (point G) (Fig. 6, b). When the ship tilts, the resulting moment of forces P and rV straightens the ship, and therefore it is stable. In this case, regardless of the size of the displacement of the center of magnitude during tilting, the pair of forces always tends to straighten the ship. This is explained by the fact that point G lies below point C 0. Such a low position of the center of gravity, ensuring unconditional stability on ships, is difficult to implement structurally. This arrangement of the center of gravity can be found in particular on sailing yachts.
Rice. 7. Second and third cases of ship stability
2nd case (Fig. 7, a) - the metacenter is located below the CG, i.e. z m< z g . В этом случае при наклонении судна момент сил Р и гV стремится еще больше отклонить судно от исходного положения равновесия, которое, следовательно, является неустойчивым. В этом случае наклонения судно имеет отрицательный восстанавливающий момент, т.е. оно не остойчиво.
3rd case (Fig. 7, b) - the metacenter coincides with the CG, i.e. z m = z g . In this case, when the ship tilts, the forces P and rV continue to act along the same vertical, their moment is equal to zero - the ship will be in a state of equilibrium in the new position. In mechanics, this is the case of indifferent equilibrium.
From the point of view of the theory of the ship, in accordance with the definition of ship stability, the ship in the 1st case is stable, and in the 2nd and 3rd cases it is not stable.
So, the condition for the initial stability of the vessel is the location of the metacenter above the CG. A ship has lateral stability if
and longitudinal stability, if
From here the physical meaning of the metacenter becomes clear. This point is the limit to which the center of gravity can be raised without depriving the vessel of positive initial stability.
The distance between the metacenter and the center of gravity of the vessel at W = I = 0 is called the initial metacentric height or simply metacentric height. The transverse and longitudinal planes of inclination of the vessel correspond, respectively, to the transverse h and longitudinal H metacentric heights. It's obvious that
h = z m - z g and H = z m - z g, or
h = z c + r - z g and H = z c + R - z g,
h = r - b and H = R - b,
where b = z g - z c is the elevation of the CG above the CV.
As you can see, h and H differ only in metacentric radii, because b is the same quantity.
Therefore H is much larger than h.
b = (1%) R, therefore in practice it is considered that H = R.
5. Metacentricstability formulas and their practical application
As was discussed, when the ship is tilted, a pair of forces acts, the moment of which characterizes the degree of stability.
For small equivolume inclinations of the vessel in the transverse plane (Fig. 8) (CV moves in the plane of inclination), the transverse restoring moment can be represented by the expression
m И = P = rV,
where the moment arm = l And is called the lateral stability arm.
From the right triangle mGK we find that
l И = h sinИ, then:
m И = P h sinИ = gV h sinИ
Or taking into account the small values of I and taking sinII 0 /57.3, we obtain the metacentric formula for lateral stability:
m И = gV h И 0 /57.3
Considering by analogy the inclination of the vessel in the longitudinal plane (Fig. 8), it is not difficult to obtain a metacentric formula for longitudinal stability:
M Ш = P l Ш = gV Н sin Ш = gV Н Ш 0 /57.3,
where M Ш is the longitudinal restoring moment, and l Ш is the longitudinal stability arm.
Rice. 8. Lateral inclination of the vessel
In practice, the stability coefficient is used, which is the product of displacement and metacentric height.
Lateral stability coefficient
K I = gV h = P h
Longitudinal stability coefficient
K Ш = gV Н = Р Н
Taking into account the stability coefficients, the metacentric formulas will take the form
m I = K I I 0 /57.3,
M W = K W W 0 /57.3
Metacentric stability formulas, which give a simple dependence of the righting moment on the force of gravity and the angle of inclination of the vessel, make it possible to solve a number of practical problems arising in ship conditions.
Rice. 9. Longitudinal inclination of the ship
In particular, using these formulas it is possible to determine the angle of heel or trim angle that the ship will receive from the influence of a given heeling or trim moment, with a known mass and metacentric height. The inclination of the vessel under the influence of m cr (M diff) leads to the appearance of a restoring moment m I (M W) of the opposite sign, increasing in magnitude with increasing angle of roll (trim). The roll (trim) angle will increase until the righting moment becomes equal in magnitude to the heeling moment (trim moment), i.e. until the condition is met:
m I = m cr and M Ш = M diff.
After this, the ship will float with angles of roll (trim):
And 0 = 57.3 m cr / gV h,
W 0 = 57.3 M diff / gV N
Assuming I = 1 0 and W = 1 0 in these formulas, we find the values of the moment of heeling the ship by one degree, and the moment trimming the ship by one degree:
m 1 0 = gV h = 0.0175 gV h,
M 1 0 = gV H = 0.0175 gV H
In some cases, the value of the moment of trimming the vessel per centimeter m D is also used. At a small value of the angle Ш, when tg Ш Ш, Ш = (dн - dк)/L = Df/L.
Taking this expression into account, the metacentric formula for the longitudinal restoring moment will be written as:
M W = M diff = gV N D f / L.
Assuming in the formula D f = 1 cm = 0.01 m, we obtain:
m D = 0.01 gV N/ L.
With known values of m 1 0, M 1 0 and m D, the heel angle, trim angle and trim from the influence of a given heeling or trim moment on the vessel can be determined by simple dependencies:
And 0 = m cr. / m 1 0 ; W 0 = M diff / M 1 0 ; D f = M differential / 100 m D
In the above reasoning, it was assumed that the ship in its initial position (before the influence of m cr or M diff) floated straight and on an even keel. If in the initial position of the vessel the roll and trim differed from zero, then the found values of I 0, Ш 0 and D f should be considered as additional (dI 0, dSh 0, dD f).
Using metacentric stability formulas, you can also determine what necessary heeling or trim moment must be applied to the ship in order to create a given heel angle or trim angle (for the purpose of repairing a hole in the side plating, painting or inspecting propellers). For a vessel floating in its initial position without roll or trim:
m cr = gV h I 0 /57.3 = m 1 0 I 0;
M diff = gV N W 0 /57.3 = M 1 0 W 0
or M diff = 100 D f m D
It is practically permissible to use metacentric stability formulas at small angles of inclination (I< 10 0 12 0 и Ш < 5 0) но при условии, что при этих углах не входит в воду верхняя палуба или не выходит из воды скула судна. Они справедливы также при условии, что восстанавливающие моменты m И и М Ш противоположны по знаку моментам m кр и М диф, т.е., что судно обладает положительной начальной остойчивостью.
6 . Form stability and load stability
Consideration of this issue makes it possible to establish the nature of stability and to clarify the physical reasons for the occurrence of a righting moment when the ship tilts. In accordance with the metacentric formulas of stability (angles I and W are expressed in radians):
m I = gV h I = gV (r - b) I = gV r I - gV b I;
M Ш = gV Н Ш = gV (R - b) Ш = gV R Ш - gV b Ш
Thus, the restoring moments m И, М Ш and static stability arms l И, l Ш represent the algebraic sum of their components:
m I = m f + m n; M Sh = M f + M n;
l I = l f I + l n I; l Ш = l f Ш + l n Ш,
where are the moments
m f = gV r I;
M f = gV R Ш,
It is customary to call the moments of shape stability, moments
m n = - gV b I;
M n = - gV b Ш,
moments of load stability, and the shoulders
l f I = m f / gV;
l f Ш = M f / gV,
transverse and longitudinal shoulders of shape stability, shoulders
l n I = - m n / gV;
l n W = - M n / gV,
transverse and longitudinal load stability arms.
b = z g - z c,
where J x and J yf are the moment of inertia of the waterline area relative to the transverse and longitudinal central axis, respectively, then the shape and load moments can be represented as:
m f = g J x I,
M f = g J yf Ш;
m n = - gV (z g - z c) And,
M n = - gV(z g - z c) Ш
By its physical nature, the moment of shape stability always acts in the direction opposite to the inclination of the vessel, and, therefore, always ensures stability. It is calculated through the moment of inertia of the waterline area relative to the inclination axis. It is the stability of the shape that predetermines significantly greater longitudinal stability compared to transverse stability because J yf » J x .
The moment of stability of the load due to the position of the CG above the CV b = (z g - z c) > 0, always reduces the stability of the vessel and, in essence, it is ensured only by the stability of the shape.
It can be assumed that in the absence of a waterline, for example, in a submarine in a submerged position, there is no moment of shape (J x = 0). In the submerged position, the submarine, due to the ballasting of special tanks, has a CG position below the CV, as a result of which its stability is ensured by the stability of the load.
7 . Determination of initial stability measuresvessel
Landing the vessel straight and on an even keel
In cases where the ship sails with small angles of heel and trim, measures of initial stability can be determined using metacentric diagrams.
For a given mass of the vessel, determining the measures of initial stability comes down to determining the applicate of metacenters (or metacentric radii and applicates of CV) and the applicate of CG.
Rice. 10. Metacentric diagram
Applicate CV z c and metacentric radii r, R are characteristics of the immersed volume of the vessel and depend on the draft. These dependencies are presented on a metacentric diagram included in the curved elements of the theoretical drawing. Using the metacentric diagram (Fig. 10), you can not only determine z c and r, but with a known CG applicate, find the transverse metacentric height of the vessel.
In Fig. Figure 10 shows the sequence of calculating the transverse metacentric height of the vessel when receiving cargo. Knowing the mass of the accepted cargo m and the applicate of its center of gravity z, we can determine the new applicate of the ship's center of gravity z g 1 using the formula:
z g 1 = z g + (z- z g),
where z g is the vessel’s center of gravity before receiving the cargo.
Landing the vessel with trim
When a vessel is sailing with trim, fuller sections of the hull enter the water, which leads to an increase in the waterline area (shape stability) and, accordingly, the transverse metacentric height. In fishing vessels, the stern contours are fuller than the bow ones, so one should expect an increase in the lateral stability of the vessel when trimmed to the stern, and a decrease in the lateral stability of the vessel when trimmed to the bow.
Rice. 11. Firsov-Gundobin diagram
To calculate the transverse metacentric height of the vessel, taking into account the trim, Firsov-Gundobin diagrams, initial stability KTIRPiKh and interpolation curves are used.
The Firsov-Gundobin diagram (Fig. 11) differs from the Firsov diagram in that it contains curves z m and z c, the values of which are determined from the known drafts of the vessel bow and stern.
The KTIRPiKh initial stability diagram (Fig. 12) allows you to determine the applicate of the ship’s metacenter z m from the known mass D and the abscissa of its center of gravity x g.
Using the diagram of interpolation curves (Fig. 13), with known drafts of the vessel bow and stern, it is possible to find the transverse metacentric radius r and the applicate of the center of the vessel's size z c.
The diagrams shown in Fig. 11-13, allow you to find z m for any landing of the vessel, including on an even keel. Consequently, they make it possible to analyze the effect of trim on the initial lateral stability of the vessel.
Rice. 12. Diagram of initial stability of the Karelia type trawler
stability vessel metacenter cargo
Rice. 13. Diagram for determining z c and r
8 . Impact of cargo movement on landing andship stability
To determine the landing and stability of the vessel during arbitrary movement of cargo, it is necessary to consider separately vertical, transverse horizontal and longitudinal horizontal movement.
It must be remembered that first you should perform calculations related to changes in stability (vertical movement, lifting a load)
Verticalcargo movement
From point 1 to point 2 does not create a moment capable of tilting the ship, and therefore, its landing does not change (unless the stability of the ship remains positive). Such a movement only leads to a change in the height of the position of the vessel’s center of gravity. It can be concluded that this movement leads to a change in load stability while maintaining unchanged shape stability. The displacement of the center of gravity is determined by the well-known theorem of theoretical mechanics:
dz g = (z 2 - z 1),
where m is the mass of the transported load,
D is the mass of the vessel,
z 1 and z 2 - CG load applicates before and after movement.
The increment of metacentric heights will be:
dh = dN = - dz g = - (z 2 - z 1)
The vessel after moving the cargo will have a transverse metacentric height:
Vertical movement of the load does not lead to a significant change in the longitudinal metacentric height, due to the smallness of dN compared to the value of H.
Rice. 14. Vertical movement of cargo
Rice. 15. Transverse horizontal movement of cargo
Suspended loads
They appear on the ship as a result of lifting cargo from the hold onto the deck, receiving the catch, retrieving nets using cargo booms, etc. A suspended load (Fig. 16) has the same effect on the stability of a vessel as a vertically moved one, only the change in stability occurs instantly at the moment it is separated from the support. When lifting the load, when the tension in the pendant becomes equal to the weight of the load, the center of gravity of the load moves from point 1 to the suspension point (point 2) and further lifting will not affect the stability of the vessel. The change in metacentric height can be estimated using the formula
where l = (z 2 - z 1) is the initial length of the load suspension.
On small boats, in conditions of reduced stability, lifting cargo with ship's booms can pose a significant danger.
Transverse horizontal movement of cargo
Transverse horizontal movement of a cargo mass m (Fig. 17) leads to a change in the roll of the vessel as a result of the resulting moment m cr with a shoulder (y 2 - y 1) cosI.
m cr = m (y 2 - y 1) cosI = m l y cosI,
where y 1 and y 2 are the ordinates of the CG position of the load before and after movement.
Taking into account the equality of the heeling moment m cr and the righting moment m And, using the metacentric stability formula, we obtain:
Дh sinИ = m l y cosИ, from where
tgI = m l y /Дh.
Considering that the roll angles are small, we can assume that tgИ = И = И 0 /57.3, and the formula takes the form
And 0 = 57.3 m l y / Дh.
If before moving the cargo the ship had a list, then in this formula the angle should be considered as an increment dI 0
Rice. 17.
Longitudinal horizontal movement of cargo
Longitudinal horizontal movement of the load (Fig. 18) leads to a change in the trim of the vessel and the transverse metacentric height. By analogy with the previous case with M Ш = M diff, we obtain:
tg Ш = m l x /DN, or
W 0 = 57.3 m l x / DN.
In practice, longitudinal inclinations are often assessed by the amount of trim
D f = Ш 0 L /57.3, then
D f = m l x L /DN,
where L is the length of the vessel.
Using the moment differentiating the vessel by 1 cm (included in the load scale and KETCH)
m D = 0.01 gV N/ L (kN m/cm);
m D = 0.01 DN/ L = 0.01 DR/L (t m/cm),
since Н R we get
D f = m l x / m D (cm).
Change in settlement during longitudinal movement of cargo:
dd n = (0.5L - x f) Df/ L,
dd k = - (0.5L + x f) Df/ L.
Then the new draft of the ship will be:
d n = d + dd n = d + (0.5L - x f) Df/ L,
d k = d + dd k = d - (0.5L + x f) Df/ L;
where x f is the abscissa of the longitudinal inclination axis.
The effect of trim on the metacentric height of the ship is discussed in detail in 7.2.
9 . The effect of accepting a small load on the landing and stability of the vessel
Changing the ship's position when accepting cargo was discussed in 4.4. Let us determine the change in the transverse metacentric height dh when receiving a small load of mass m (Fig. 19), the center of gravity of which is located on the same vertical with the CG of the waterline area at the point with applicate z.
As a result of the increase in draft, the volumetric displacement of the vessel will increase by dV = m / s and an additional buoyancy force g dV will appear, applied in the CG of the layer between the waterlines WL and W 1 L 1 .
Rice. 19. Acceptance of small cargo onto the ship
Considering the vessel to be straight-sided, the applicative CG of the additional volume of buoyancy will be equal to d + dd /2, where the increment in draft is determined by the well-known formulas dd = m/ cS or dd = m / q cm.
When the ship is tilted at an angle I, the force of the weight of the load p and the equal buoyancy force g dV form a pair of forces with a shoulder (d + dd /2 -z) sinI. The moment of this pair dm I = p (d + dd /2 - z) sin And increases the initial righting moment of the vessel m I = gV h sin And, therefore the righting moment after receiving the load becomes equal
m And 1 = m And + dm And, or
(gV + g dV)(h + dh) sin I = gV h sin I + g dV(d + dd /2 - z) sin And,
passing to mass values, we get
(D + m)(h + dh) sin I = D h sin I + m (d + dd /2 - z) sin I.
From the equation we find the increment of metacentric height dh:
For the general case of accepting or removing a small load, the formula will take the form:
where + (-) is substituted when accepting (removing) the load.
From the formula it is clear that
dh< 0 при z >(d dd /2 - h) and
dh > 0 at z< (d дd /2 - h), а
dh = 0 at z = (d dd /2 - h).
The equation z = (d dd /2 - h) is the equation of the neutral (limit) plane.
The neutral plane is a plane on which the acceptance of a load does not change the stability of the vessel. Receiving a load above the neutral plane reduces the stability of the vessel, below the neutral plane increases it.
10 . Effect of liquid cargo on ship stability
The vessel carries a significant amount of liquid cargo in the form of fuel, water and oil reserves. If a liquid cargo fills a tank entirely, its effect on the ship's stability is similar to that of an equivalent solid cargo of
m f = c f v f.
On a ship, there are almost always tanks that are not completely filled, i.e. the liquid has a free surface in them. Free surfaces on a ship can also appear as a result of firefighting and hull damage. Free surfaces have a strong negative impact on both the initial stability and the stability of the vessel at high inclinations. When the ship tilts, the liquid cargo, which has a free surface, flows towards the tilt, creating an additional moment that heels the ship. The resulting moment can be considered as a negative amendment to the righting moment of the vessel.
Rice. 20. Influence on the initial stability of the free surface of a liquid load
Free surface influence
We will consider the influence of the free surface (Fig. 20) when landing the ship straight and on an even keel. Let us assume that in one of the tanks of the ship there is a liquid cargo with a volume v l and having a free surface. When the ship tilts at a small angle I, the free surface of the liquid will also tilt, and the center of gravity of the liquid q will move to a new position q 1. Due to the smallness of the angle I, we can assume that this movement occurs along a circular arc of radius r 0 with a center at point m 0 at which the lines of action of the weight of the liquid intersect before and after the inclination of the vessel. By analogy with the metacentric radius
r 0 = i x /v w,
where i x is the intrinsic moment of inertia of the free surface of the liquid relative to the longitudinal axis (parallel to the coordinate axis OX). It is easy to see that the case under consideration has the same effect on stability as the suspended case, where l = r 0, and m = c w v f.
Rice. 21. Curves of the dimensionless coefficient k
Using the formula for a suspended load, we obtain the formula for the influence on the stability of the free surface of the liquid:
As can be seen from the formula, it is i x that influences stability.
The moment of inertia of the free surface is calculated by the formula
where l and b are the length and width of the surface, and k is a dimensionless coefficient that takes into account the shape of the free surface.
In this formula, you should pay attention to the last factor - b 3, that the width of the surface, to a greater extent than the length, influences i x and therefore dh. Thus, you need to be especially wary of free surfaces in wide compartments.
Let us determine how much the loss of stability in a rectangular tank will decrease after installing n longitudinal bulkheads at equal distances from each other
i x n = (n +1) k l 3 = k l b 3 /(n +1) 2 .
The ratio of amendments to the metacentric height before installation and after installation of bulkheads will be
dh / dh n = i x / i x n = (n +1) 2 .
As can be seen from the formulas, installing one bulkhead reduces the influence of the free surface on stability by 4 times, two - by 9 times, etc.
The coefficient k can be determined from the curve in Fig. 21, in which the upper curve corresponds to an asymmetrical trapezoid, the lower one to a symmetrical one. For practical calculations, the coefficient k, regardless of the shape of the surface area, should be taken as k = 1/12 for rectangular surfaces.
In shipboard conditions, the influence of liquid cargo is taken into account using the tables given in the “Vessel Stability Information”.
Table 1
Correction for the influence of free surfaces of liquid cargo on the stability of a vessel of the BMTR “Mayakovsky” type
|
Amendment, m, dh |
Vessel displacement, m |
||||||
The tables give corrections to the metacentric height of the vessel dh for a set of tanks, which, due to operating conditions, may be partially filled (Table 1) to the lateral stability coefficient dm h = dh = c w i x for each tank separately (Table 2). Tanks with adjustments to the metacentric height of less than 1 cm are not taken into account in the calculations.
Depending on the type of corrections, the metacentric height of the vessel, taking into account the influence of liquid cargo in partially filled tanks, is found using the formulas
h = z m - z g - dh;
h = z m - z g - dm h /
As you can see, free surfaces seem to increase the center of gravity of the vessel or reduce its transverse metacenter by an amount
dz g = dz m = dh = dm h /
The appearance of the free surface of the liquid cargo also affects the longitudinal stability of the vessel. The correction to the longitudinal metacentric height will be determined by the formula
dN = - s f i y /,
where i y is the intrinsic moment of inertia of the free surface of the liquid relative to the transverse axis (parallel to the coordinate axis of the OU). However, due to the significant value of the longitudinal metacentric height H, the dH correction is usually neglected.
The considered change in stability from the free surface of the liquid occurs when its volume is from 5 to 95% of the volume of the tank. In such cases, the free surface is said to lead to an effective loss of stability.
table 2
Correction for the influence of free surfaces of liquid cargo on the stability of the vessel m/v "Alexander Safontsev"
|
Name |
Abscissa CG, m |
Application CG, m |
Torque mx, tm |
Moment mz, tm |
Corrections for free surfaces, tm |
||||
|
Tank car DT No. 3 |
|||||||||
|
Tank car DT No. 4 |
|||||||||
|
Tank car DT No. 5 |
|||||||||
|
Tank car DT No. 6 |
|||||||||
|
Tank car DT No. 35 |
Rice. 22. Case of ineffective loss of stability
If there is only a very thin layer of liquid in the tank, or the tank is filled almost to the top, then the width of the free surface begins to decrease sharply when the vessel tilts (Fig. 22). Accordingly, the moment of inertia of the free surface will also undergo a sharp decrease, and, consequently, the correction to the metacentric height. Those. There is an ineffective loss of stability, which can practically be ignored.
To reduce the negative impact of overflowing liquid cargo on the ship’s stability, the following design and organizational measures may be provided:
Installation of longitudinal or transverse bulkheads in tanks, which makes it possible to sharply reduce their own moments of inertia i x and i y;
Installation of longitudinal or transverse diaphragm bulkheads in tanks, having small holes in the lower and upper parts. When the ship tilts sharply (for example, when pitching), the diaphragm acts as a bulkhead, since liquid flows through the holes quite slowly. From a structural point of view, diaphragms are more convenient than impermeable bulkheads, since when installing the latter, the filling, draining and ventilation systems of tanks become significantly more complicated. However, during prolonged inclinations of the vessel, the diaphragms, being permeable, cannot reduce the effect of the overflowing liquid on stability;
When accepting liquid cargo, ensure that the tanks are completely filled without the formation of free liquid surfaces;
When consuming liquid cargo, ensure that the tanks are completely drained; “dead stocks” of liquid cargo should be minimal;
Ensure dry holds in ship compartments where liquid can accumulate with a large free surface area;
Strictly follow the instructions for receiving and disposing of liquid cargo on board the ship.
Failure by the ship's crew to carry out the listed organizational measures can lead to a significant loss of ship stability and cause an accident.
11 . Experimental determination of metacentricheight and position of the vessel's center of gravity
When designing a vessel, its initial stability is calculated for typical load cases. The actual stability of the constructed vessel differs from the calculated one due to calculation errors and deviations from the design made during construction. Therefore, on ships, an experimental determination of initial stability is carried out - inclination, followed by calculation of the position of the ship's CG.
The following should be heaped:
Serial-built vessels (the first and then every fifth vessel in the series);
Each new vessel of non-serial construction;
Each vessel after restoration;
Vessels after major repairs, re-equipment or modernization with a change in displacement of more than 2%;
Vessels after laying permanent solid ballast, if the change in the center of gravity cannot be determined sufficiently accurately by calculation;
Vessels whose stability is unknown or needs to be tested.
Inclining is carried out in the presence of a Register inspector in accordance with the special “Instructions for inclining Register vessels”.
The essence of heeling is as follows. The inclination is carried out on the basis of the equality m cr = m I, which determines the equilibrium position of the vessel with a roll I 0. The heeling moment is created by moving loads (incline ballast) across the width of the vessel at a distance l y ; within small inclinations of the ship:
m cr = m l y.
Then from the equality m l y = сV h И 0 /57.3
they find that h = 57.3 m l y /сVI 0.
The elevation of the vessel's CG above the main plane z g and the abscissa of the CG x g are determined from the expressions:
z g = z c + r - h; and x g = x c .
The values of z c , r and x c in the case of absence or small trim are determined using the curved elements of the theoretical drawing according to the displacement value V. In the presence of trim, these values must be determined by a special calculation. Displacement V is found on the Bonjean scale based on measuring the draft of the vessel bow and stern along the marks of the recess. The density of sea water is determined using a hydrometer.
The roller ballast mass m and the transfer arm l y are specified, and the roll angle I 0 is measured.
Before heeling, the vessel's load should be as close as possible to its light-ship displacement (98-104%). The metacentric height of the vessel must be at least 0.2 m. To achieve this, ballast is allowed.
Supplies and spare parts must be in their normal places, loads must be secured, and tanks for water, fuel, and oil must be drained. When filled, ballast tanks must be pressed in.
The inline ballast is laid on the open deck of the vessel on both sides on special racks in several rows relative to the DP. The mass of the incline ballast transferred across the vessel should ensure a heel angle of about 3 0 .
To measure roll angles, special scales (at least 3 meters long) or inclinographs are prepared. The use of ship inclinometers to measure angles is unacceptable, since they give a significant error.
Inclining is carried out in calm weather when the ship roll is no more than 0.5 0. The depth of the water area should prevent touching the ground or finding part of the hull in muddy soil. The vessel must be able to heel freely, for which slack should be provided in the mooring lines and the vessel should not touch the wall or hull of another vessel.
The experience consists of transferring the roller ballast from side to side on command and measuring the roll angle before and after the transfer.
Determination of initial stability based on the rolling period is made on the basis of the well-known “captain’s” formula:
where f I is the period of the vessel’s own onboard oscillations;
C I - inertial coefficient;
B is the width of the vessel.
It is recommended to determine the rolling period of a ship during each inclining experiment, and for ships with a displacement of less than 300 tons, its determination is mandatory. The means for determining fI is an inclinograph or stopwatches (at least three observers).
The rocking of the vessel is carried out by coordinated runs of the crew from side to side in time with the vibrations of the vessel until the vessel tilts by 5 8 0. The captain's formula allows, under any state of the ship's load, to approximately determine the metacentric height when it is in waves. It must be remembered that for the same vessel the value of the inertial coefficient C I is not the same; it depends on its loading and placement of cargo. As a rule, the inertial coefficient of an empty ship is greater than that of a loaded one.
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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).
Longitudinal stability 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 low inclination, 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.
By the relative position of the cargo on the ship, the navigator can always find the most favorable value of the metacentric height, at which the ship will be sufficiently stable and less subject to pitching.
The heeling moment is the product of the weight of the cargo moved across the vessel by a shoulder equal to the distance of movement. If a person weighs 75 kg, sitting on a bank will move across the ship by 0.5 m, then the heeling moment will be equal to 75 * 0.5 = 37.5 kg/m.
Figure 91. Static stability diagram
To change the moment that heels the ship by 10°, it is necessary to load the ship to full displacement completely symmetrically relative to the center plane.
The vessel's loading should be checked by drafts measured on both sides. The inclinometer is installed strictly perpendicular to the center plane so that it shows 0°.
After this, you need to move loads (for example, people) at pre-marked distances until the inclinometer shows 10°. The test experiment should be carried out as follows: tilt the ship on one side and then on the other side.
Knowing the fastening moments of a ship heeling at various (up to the greatest possible) angles, it is possible to construct a static stability diagram (Fig. 91), which will evaluate the stability of the ship.
Stability can be increased by increasing the width of the vessel, lowering the center of gravity, and installing stern buoys.
If the center of gravity of the vessel is located below the center of magnitude, then the vessel is considered very stable, since the supporting force during a roll does not change in magnitude and direction, but the point of its application shifts towards the tilt of the vessel (Fig. 92, a).
Therefore, when heeling, a pair of forces is formed with a positive restoring moment, tending to return the ship to its normal vertical position on a straight keel. It is easy to verify that h>0, with the metacentric height equal to 0. This is typical for yachts with a heavy keel and is not typical for larger vessels with a conventional hull structure.
If the center of gravity is located above the center of magnitude, then three cases of stability are possible, which the navigator should be well aware of.
The first case of stability.
Metacentric height h>0. If the center of gravity is located above the center of magnitude, then when the vessel is in an inclined position, the line of action of the supporting force intersects the center plane above the center of gravity (Fig. 92, b).
Rice. 92. The case of a stable ship
In this case, a couple of forces with a positive restoring moment is also formed. This is typical for most conventionally shaped boats. Stability in this case depends on the hull and the position of the center of gravity in height.
When heeling, the heeling side enters the water and creates additional buoyancy, tending to level the ship. However, when a ship rolls with liquid and bulk cargo that can move towards the roll, the center of gravity will also shift towards the roll. If the center of gravity during a roll moves beyond the plumb line connecting the center of magnitude with the metacenter, then the ship will capsize.
The second case of an unstable vessel in indifferent equilibrium.
Metacentric height h = 0. If the center of gravity lies above the center of magnitude, then during a roll the line of action of the supporting force passes through the center of gravity MG = 0 (Fig. 93).
In this case, the center of magnitude is always located on the same vertical as the center of gravity, so there is no recovering pair of forces. Without the influence of external forces, the ship cannot return to an upright position.
In this case, it is especially dangerous and completely unacceptable to transport liquid and bulk cargo on a ship: with the slightest rocking, the ship will capsize. This is typical for boats with a round frame.
The third case of an unstable vessel in unstable equilibrium.
Metacentric height h<0. Центр тяжести расположен выше центра величины, а в наклонном положении судна линия действия силы поддержания пересекает след диаметральной плоскости ниже центра тяжести (рис. 94).
The stability of a vessel is a property due to which the vessel does not capsize when exposed to external factors (wind, waves, etc.) and internal processes (displacement of cargo, movement of liquid reserves, the presence of free surfaces of liquid in compartments, etc.). The most comprehensive definition of ship stability may be the following: the ability of a ship not to capsize when exposed to natural sea factors (wind, waves, icing) in its assigned navigation area, as well as in combination with “internal” reasons caused by the actions of the crew
This feature is based on the natural property of an object floating on the surface of water - it tends to return to its original position after the cessation of this influence. Thus, stability, on the one hand, is natural, and, on the other, requires regulated control on the part of the person taking part in its design and operation.
Stability depends on the shape of the hull and the position of the vessel's center of gravity, therefore, by correctly choosing the shape of the hull during design and correctly placing cargo on the vessel during operation, it is possible to ensure sufficient stability to guarantee the prevention of capsizing of the vessel under any sailing conditions.
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. There are two types of stability: transverse and longitudinal. From the point of view of navigation safety (especially in stormy weather), the most dangerous are transverse inclinations. Transverse stability manifests itself when the ship rolls, i.e. when tilting it on board. If the forces causing the ship to tilt act slowly, then stability is called static, and if quickly, then dynamic. The inclination of the ship in the transverse plane is called roll, and in the longitudinal plane - trim; the angles formed in this case are designated O and y, respectively. Stability at small angles of inclination (10 - 12°) is called initial stability.
(Fig.2)
Let's imagine that, under the influence of external forces, the ship tilted at an angle of 9 (Fig. 2). 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 C1. When the vessel is in an inclined position, the gravity force P applied at point G and the supporting force D applied at point C, remaining perpendicular to the new waterline V1L1, form a pair of forces with the arm GK, which is a perpendicular lowered from point G to the direction of the supporting forces.
If we continue the direction of the support force from point C1 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 the transverse metacenter.
The relative position of points M and G allows us to establish the following feature characterizing lateral stability: (Fig. 3)
- A) If the metacenter is located above the center of gravity, then the restoring moment is positive and tends to return the ship to its original position, i.e., when heeling, the ship will be stable.
- B) If point M is below point G, then with a negative value of h0 the moment is negative and will tend to increase the roll, i.e. in this case the vessel is unstable.
- C) When points M and G coincide, forces P and D act along one vertical straight line, a pair of forces does not arise, and the restoring moment is zero: then the ship must be considered unstable, since it does not strive to return to its original equilibrium position (Fig. 3 ).
Fig.3
External signs of negative initial stability of a ship are:
- -- navigation of a ship with a roll in the absence of heeling moments;
- - the desire of the ship to roll over to the opposite side when straightening;
- - transferring from side to side during circulation, while the roll remains even when the ship enters a direct course;
- -- a large amount of water in holds, on platforms and decks.
Stability, which manifests itself during longitudinal inclinations of the vessel, i.e. when trimming, it is called longitudinal.
When the vessel is longitudinally tilted at an angle w around the transverse axis Ts.V. will move from point C to point C1 and the supporting force, the direction of which is normal to the existing waterline, will act at an angle w to the original direction. The lines of action of the original and new direction of the support forces intersect at a point. The point of intersection, the line of action of the supporting forces at an infinitesimal inclination in the longitudinal plane is called the longitudinal metacenter of M. seaworthiness, stability, and propulsion of the ship.
The longitudinal moment of inertia of the waterline area IF is significantly greater than the transverse moment of inertia IX. Therefore, the longitudinal metacentric radius R is always significantly larger than the transverse radius r. It is roughly assumed that the longitudinal metacentric radius R is approximately equal to the length of the vessel. Since the longitudinal metacentric radius R is many times greater than the transverse r, the longitudinal metacentric height H of any ship is many times greater than the transverse h. therefore, if the vessel has lateral stability, then longitudinal stability is certainly ensured.
Factors affecting ship stability that have a strong influence on the stability of the ship.
Factors that must be taken into account when operating a small vessel include:
- 1. The stability of a vessel is most significantly affected by its width: the larger it is in relation to its length, side height and draft, the higher the stability. A wider boat has a greater righting moment.
- 2. The stability of a small vessel increases if the shape of the submerged part of the hull is changed at large angles of heel. This statement, for example, is the basis for the action of side boules and foam fenders, which, when immersed in water, create an additional righting moment.
- 3. Stability deteriorates if the ship has fuel tanks with a surface mirror from side to side, so these tanks must have partitions installed parallel to the centerline of the ship, or be narrowed in their upper part.
- 4. Stability is most strongly influenced by the placement of passengers and cargo on the ship; they should be located as low as possible. On a small vessel, people should not be allowed to sit on board or move around arbitrarily while it is moving. Loads must be securely fastened to prevent them from unexpectedly moving from their normal places.
- 5. In strong winds and waves, the effect of heeling moment (especially dynamic) is very dangerous for the ship, therefore, as weather conditions worsen, it is necessary to take the ship to shelter and wait out the bad weather. If this is impossible to do due to the considerable distance to the shore, then in stormy conditions you should try to keep the ship “head to the wind”, throwing out the sea anchor and running the engine at low speed.
Excessive stability causes rapid rolling and increases the risk of resonance. Therefore, the register has established restrictions not only on the lower, but also on the upper limit of stability.
To increase the stability of the ship (increase the righting moment per unit of roll angle), it is necessary to increase the metacentric height h by appropriately placing cargo and supplies on the ship (heavier cargo at the bottom, and lighter cargo at the top). For the same purpose (especially when sailing in ballast - without cargo) they resort to filling ballast tanks with water.
- Depending on the plane of inclination there are lateral stability when heeling and longitudinal stability at trim. In relation to surface ships (vessels), due to the elongated shape of the ship's hull, its longitudinal stability is much higher than transverse stability, therefore, for navigation safety, it is most important to ensure proper lateral 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
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 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 flows towards the inclination. The free surface will take the same angle relative to the KVL.
Liquid cargo levels cut off equal volumes of tanks, that is, 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 stability. Therefore, organizational and constructive measures are being taken to reduce them:
Dynamic stability of the vessel
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 / ed. Physical encyclopedia
Vessel, the ability of a vessel to resist external forces causing it to roll or trim, and to return to its original equilibrium position after their action ceases; one of the most important seaworthiness qualities of a vessel. O. when heeling... ... Great Soviet Encyclopedia
The quality of a ship being in equilibrium in an upright position and, having been removed from it by the action of some force, returning to it again after its action ceases. This quality is one of the most important for navigation safety; there were many… … Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
G. The ability of a vessel to float in an upright position and to straighten itself after tilting. Ephraim's explanatory dictionary. T. F. Efremova. 2000... Modern explanatory dictionary of the Russian language by Efremova
Stability, stability, stability, stability, stability, stability, stability, stability, stability, stability, stability, stability (
