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Initial stability of the vessel
1. General concept about stability
Stability is the ability of a vessel to resist forces that deflect 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 ship is stable. The property, which in mechanics is called the stability of equilibrium, in the theory of the ship is usually called stability. Thus, buoyancy provides the conditions for the equilibrium position of the ship with a given landing, and stability - the preservation of this position.
The stability of the vessel changes with an increase in the inclination angle and at some of its value is completely lost. Therefore, it seems expedient to study the stability of the ship at small (theoretically infinitely small) 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 the vessel at small angles of inclination (initial stability) and stability at large angles of inclination.
When considering low inclinations, it is possible to make a number of assumptions that make it possible to study the initial stability of the vessel in the framework of the linear theory and obtain simple mathematical dependences of its characteristics. The stability of the ship at large angles of inclination is studied according to a refined nonlinear theory. Naturally, the property of stability of the vessel is uniform and the accepted division is purely methodological in nature.
When studying the stability of a vessel, consider its inclinations in two mutually perpendicular planes - transverse and longitudinal. When the ship is inclined in the transverse plane, determined by the heel angles, its lateral stability is studied; at inclinations in the longitudinal plane, determined by the trim angles, its longitudinal stability is studied.
If the inclination of the vessel occurs without significant angular accelerations (pumping of liquid cargo, slow inflow of water into the compartment), then the stability is called static.
In some cases, the forces tilting the vessel act suddenly, causing significant angular accelerations (a squall of wind, a wave roll, etc.). In such cases, dynamic stability is considered.
Stability is a very important seaworthy property of a vessel; together with buoyancy, it ensures the navigation of the vessel in a given position relative to the surface of the water, which is necessary to ensure progress and maneuver. A decrease in the stability of the vessel can cause an emergency list and trim, and a complete loss of stability can cause its overturning.
In order to prevent a dangerous decrease in the stability of the vessel, all crew members must:
Always have a clear idea of the stability of the vessel;
Know the reasons that reduce stability;
Know and be able to apply all means and measures to maintain and restore stability.
2. Equal volumetric inclination of the vessel. Euler's theorem
The stability of the vessel is studied at the so-called equal-volume inclinations, at which the value of the underwater volume remains unchanged, and only the shape of the underwater part of the vessel changes.
Let's introduce the basic definitions related to the inclinations of the vessel:
Inclination axis - the line of intersection of the planes of the two waterlines;
Inclination plane - 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;
Equal-volume waterlines - waterlines that cut off equal-size wedge-shaped volumes when the vessel tilts, one of which, when the vessel is inclined, enters the water, and the other exits the water.
Fig. 1. Consideration of Euler's theorem
With a known initial waterline, Euler's theorem is used to construct a waterline equal to it. According to this theorem, at an infinitely small inclination of the vessel, the planes of equal-volume waterlines intersect in a straight line passing through their common geometric center (center of gravity), or the axis of the infinitesimal equal-volume inclination passes through the geometric center of the initial waterline area.
Euler's theorem can be applied to finite small inclinations with the same small error, the smaller the angle of inclinations.
It is assumed that the accuracy sufficient for practice is provided at inclinations I 1012 0 and Ш 23 0. It is within these angles that the initial stability of the vessel is considered.
As you know, when a vessel is sailing without a heel and with a trim close to zero, the ordinate of the geometric center of the waterline area is yf = 0, and the abscissa is xf 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 equal-volume inclination is perpendicular to the DP and is offset from the pl. midship - frame at a distance x f (Fig. 1).
The quantity x f is a function of the ship's draft d. The dependence x f (d) is shown on the curves of the elements of the theoretical drawing.
When the vessel 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. Metacentres and metacentric radii
Suppose that the vessel from the initial position without heel and trim makes transverse or longitudinal equal-volume inclinations. In this case, the plane of longitudinal inclinations will be the vertical plane, which coincides with the DP, and the plane of transverse inclinations, the vertical plane, which coincides with the plane of the frame passing through the CV.
Transverse 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 rV also lies in the DP (Fig. 2). When the vessel is laterally inclined by an angle I, the shape of the immersed volume changes, the CV moves in the direction of inclination from point C to point C I, and the line of action of the buoyancy force will be inclined to the DP at an angle I.
The point of intersection of the lines of action of the buoyancy force at an infinitely small transverse equal-volume inclination of the vessel is called the transverse metacentre (point m in Fig. 2). The radius of curvature of the CV trajectory r (the elevation of the transverse metacentre above the CV) is called the transverse metacentric radius.
In the general case, the trajectory of a CV is a complex spatial curve and each angle of inclination corresponds to its own position of the metacentre (Fig. 3). However, for small equal-volume inclinations with a known approximation, it can be assumed that the trajectory
CV lies in the plane of inclination and is an arc of a circle centered at point m. Thus, we can assume that in the process of a small transverse equal-volume inclination of the vessel from a straight position, the transverse metacentre lies in the DP and does not change its position (r = const).
Fig. 2. Movement of the CV at low inclinations
Fig. 3. Moving CV at high inclinations
Fig. 4. To the derivation of the expression for the transverse metacentric radius
The expression for the transverse metacentric radius r will be 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 emerged from the water to the side that entered the water (Fig. 4).
According to the well-known theorem of mechanics, when moving a body belonging to a system of bodies, the center of gravity of the entire system is shifted in the same direction parallel to the movement of the body, and these displacements are inversely proportional to the gravity 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 CV (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: С С И =
For an element of vessel length dx, assuming that the wedge-shaped volume has a triangle shape in the plane of the frame, we obtain:
or at low angle
If by, then:
dv b = y 3 AND dx.
Integrating, we get:
v b = AND y 3 dx, or:
where J x = ydx is the moment of inertia of the waterline area relative to the longitudinal central axis.
Then the expression for moving the CV will look like:
As can be seen from Fig. 5, at a small angle I
S S I r I
Comparing the expressions, we find that the transverse metacentric radius:
r =
Application of the transverse metacentre:
z m = z c + r = z c +
Longitudinal inclination
Fig. 6. To the derivation of the expression for the longitudinal metacentric radius
By analogy with the transverse inclinations, the point of intersection of the lines of action of the buoyancy force at an infinitely small longitudinal equal-volume inclination of the vessel is called the longitudinal metacentre (point M in Fig. 6). The rise of the longitudinal metacentric over the CV is called the longitudinal metacentric radius. The value 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.
Application of the longitudinal metacentre:
z m = z c + R = z c +
Since the area of the waterline is elongated in the longitudinal direction, then J yf is much larger than J x and, accordingly, R is much larger than r. The R-value is 1 2 the length of the ship.
The metacentric radii and the applicability of the metacentres are, as will be clear from the subsequent consideration, important characteristics of the stability of the ship. Their values are determined when calculating the elements of the immersed volume and for a ship sailing 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 curves of the elements of the theoretical drawing.
4. Condition of initial stability of the vessel
Metacentric heights
Let us find the condition under which the vessel, floating in a state of equilibrium without heel and trim, will have initial stability. We assume that the weights do not shift when the vessel is inclined and the CG of the vessel remains at the point corresponding to the initial position.
When the ship is tilted, 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 metacentre.
Fig. 6. First case of vessel stability
There are three typical cases of the state of the ship for which the impact of the moment of forces P and gV on it is qualitatively different. Let's consider them using the example of transverse inclinations.
1st case (Fig. 6) - the metacentre is located above the CG, ie. 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 metacentre (point m) is located above the center the gravity of the vessel. The moment of forces P and rV tends to return the ship to its initial 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 vessel is inclined, the resulting moment of forces P and gV straightens the vessel, and therefore it is stable. In this case, regardless of the size of the displacement of the center of magnitude during inclination, the pair of forces always tends to straighten the ship. This is because the G point lies below the C 0 point. Such a low position of the center of gravity, which ensures unconditional stability on ships, is difficult to implement constructively. This center of gravity can be found in particular on sailing yachts.
Fig. 7. Second and third cases of vessel stability
2nd case (Fig. 7, a) - the metacentre is located below the CG, ie. 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 metacentre coincides with the CG, i.e. z m = z g. In this case, when the ship is inclined, the forces P and gV 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 a case of indifferent equilibrium.
From the point of view of the theory of the vessel, in accordance with the definition of the stability of the vessel, the vessel 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 metacentre above the CG. The vessel is laterally stable if
and longitudinal stability if
From here the physical meaning of the metacentre 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 metacentre and the ship's CG at W = I = 0 is called the initial metacentric height or simply the metacentric height. The transverse and longitudinal planes of inclination of the vessel correspond to the transverse h and longitudinal H metacentric heights, respectively. 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 over the CV.
As you can see, h and H differ only in metacentric radii, since b is the same value.
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 it was considered, when the vessel is inclined, a pair of forces acts, the moment of which characterizes the degree of stability.
At small equal-volume 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 shoulder of the moment = l And is called the shoulder of the lateral stability.
From the right-angled triangle mGK we find that
l И = h sinИ, then:
m И = P h sinИ = гV h sinИ
Or, taking into account small values of I and taking sinII 0 / 57.3, we obtain the metacentric formula for lateral stability:
m И = гV h И 0 / 57.3
Considering by analogy the inclination of the ship in the longitudinal plane (Fig. 8), it is easy to obtain the metacentric formula for the longitudinal stability:
M W = P l W = dV N sin W = dV N W 0 / 57.3,
where М Ш - longitudinal restoring moment, and l Ш - shoulder of longitudinal stability.
Fig. 8. Lateral inclination of the vessel
In practice, the stability coefficient is used, which is the product of the displacement by the metacentric height.
Lateral stability coefficient
K I = gV h = P h
Longitudinal stability coefficient
K W = dV N = P N
Taking into account the stability coefficients, the metacentric formulas take the form
m И = К И И 0 / 57.3,
M W = K W W 0 / 57.3
Meta-centric stability formulas, which give a simple dependence of the restoring 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.
Fig. 9. Longitudinal inclination of the vessel
In particular, these formulas can be used to determine the angle of roll or the angle of trim, which the vessel will receive from the effect of a given heeling or trimming 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 reverse in sign of the restoring moment m I (M W), increasing in magnitude with an increase in the roll angle (trim). The increase in the roll (trim) angle will continue until the restoring moment becomes equal in magnitude to the heeling moment (trimming moment), i.e. until the condition is met:
m I = m cr and M W = M diff.
After that, the vessel will sail with the roll (trim) angles:
And 0 = 57.3 m cr / gV h,
W 0 = 57.3 M diff / gV H
Assuming in these formulas I = 1 0 and W = 1 0, we find the values of the moment of heeling the vessel by one degree, and the moment of trimming the vessel 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 by one centimeter m is also used. At a small value of the angle Ш, when tg Ш Ш, Ш = (d n - d k) / L = D f / L.
Taking this expression into account, the metacentric formula for the longitudinal restoring moment will be written as:
M W = M diff = gV H D f / L.
Assuming in the formula D f = 1 cm = 0.01 m, we get:
m D = 0.01 g V N / L.
With known values of m 1 0, M 1 0 and m D, the roll angle, the trim angle and the trim from the impact on the vessel of a given heeling or trim moment can be determined by simple dependencies:
And 0 = m cr. / m 1 0; W 0 = M diff / M 1 0; D f = M diff / 100 m D
In the above reasoning, it was assumed that the ship in the initial position (before the impact of m cr or M diff) was sailing straight and on an even keel. If, in the initial position of the vessel, the roll and trim were different from zero, then the found values of I 0, W 0 and D f should be considered as additional (dI 0, dW 0, dD f).
Using the metacentric stability formulas, it is also possible to determine what necessary heeling or trimming moment must be applied to the vessel in order to create a given heel angle or trim angle (for the purpose of sealing a hole in the side skin, painting or inspecting propellers). For a vessel floating in the initial position without heel or trim:
m cr = gV h And 0 / 57.3 = m 1 0 And 0;
M diff = gV N W 0 / 57.3 = M 1 0 W 0
or M diff = 100 D f m D
In practice, metacentric stability formulas can be used at small angles of inclination (AND< 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 ... Shape stability and load stability
Consideration of this issue makes it possible to establish the nature of stability, to find out the physical reasons for the appearance of the restoring moment when the ship is tilted. In accordance with the metacentric stability formulas (angles И and Ш are expressed in radians):
m И = гV h И = гV (r - b) И = гV r И - гV b И;
M W = dV H W = dV (R - b) W = dV R W - dV b W
Thus, the restoring moments m И, М Ш and the shoulders of static stability l И, l Ш are the algebraic sum of their components:
m And = m f + m n; M W = M f + M n;
l И = l f И + l n И; l W = l f W + l n W,
where are the moments
m f = gV r And;
M f = gV R W,
it is customary to call the moments of stability of the form, moments
m n = - gV b I;
M n = - gV b W,
moments of stability of the load, and the shoulders
l f I = m f / gV;
l f W = M f / gV,
transverse and longitudinal arms 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 moments of form and load can be represented as:
m f = g J x I,
M f = g J yf W;
m n = - гV (z g - z c) И,
М n = - гV (z g - z c) Ш
By its physical nature, the moment of stability of the form always acts in the direction opposite to the inclination of the vessel, and, therefore, always ensures stability. It is calculated in terms of the moment of inertia of the waterline area relative to the inclination axis. It is the stability of the shape that predetermines a significantly greater longitudinal stability in comparison with the transverse one. J yf "J x.
The moment of stability of the load due to the position of the CG above TsV b = (z g - z c)> 0 always reduces the stability of the vessel and, in essence, it is provided only by the stability of the form.
It can be assumed that in the absence of a waterline, for example, for a submarine in a submerged position, there is no form moment (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 measuresship
Landing the ship straight and on an even keel
In cases where the vessel is sailing with slight angles of heel and trim, the measures of initial stability can be determined using metacentric diagrams.
With a given mass of the vessel, the determination of the initial stability measures is reduced to the determination of the metacentric applicates (or metacentric radii and CV applicates) and the CG applicates.
Fig. 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 the metacentric diagram included in the curves of the elements of the theoretical drawing. According to the metacentric diagram (Fig. 10), it is possible not only to determine z c and r, but with a known CG applicability, to find the transverse metacentric height of the vessel.
In fig. 10 shows the sequence of calculating the transverse metacentric height of the vessel when receiving cargo. Knowing the mass of the received cargo m and the application of its center of gravity z, it is possible to determine the new application of the ship's CG z g 1 by the formula:
z g 1 = z g + (z- z g),
where z g is the application of the ship's CG before receiving the cargo.
Landing the boat with trim
When the ship is trimmed, 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 lines are fuller than the bow lines, therefore, an increase in the lateral stability of the vessel should be expected with a differential at the stern, and with a differential at the bow, a decrease in the lateral stability of the vessel.
Fig. 11. Diagram of Firsov - Gundobin
To calculate the transverse metacentric height of the vessel, taking into account the trim, the Firsov - Gundobin diagrams, the initial stability of the KTIRPiH and interpolation curves are used.
The Firsov - Gundobin diagram (Fig. 11) differs from the Firsov diagram in that it contains the curves z m and z c, the values of which are determined from the known draft of the ship by the bow and stern.
The diagram of the initial stability KTIRPiH (Fig. 12) allows to determine the applicate of the ship's metacentre z m by the known mass D and the abscissa of its center of gravity x g.
Using the diagram of interpolation curves (Fig. 13), it is possible to find the transverse metacentric radius r and the applicate of the center of magnitude of the vessel z c at known ship draft forward and stern.
The diagrams shown in Fig. 11-13, allow you to find z m for any ship landing, including on an even keel. Therefore, they make it possible to analyze the effect of trim on the initial lateral stability of the vessel.
Fig. 12. Diagram of the initial stability of a trawler of the "Karelia" type
stability ship metacentre cargo
Fig. 13. Diagram for determining z c and r
8 ... Impact of movement of goods on boarding andstability of the vessel
To determine the landing and stability of the vessel with arbitrary movement of goods, it is necessary to consider separately the vertical, lateral horizontal and longitudinal horizontal movement.
It must be remembered that first you should perform calculations related to the change in stability (vertical movement, lifting of the load)
Verticalmovement of cargo
From point 1 to point 2 does not create a moment capable of tilting the vessel, and therefore, its landing does not change (if only the stability of the vessel remains positive). This movement only leads to a change in height in the position of the center of gravity of the vessel. It can be concluded that this movement leads to a change in the stability of the load while the stability of the form remains unchanged. The displacement of the center of gravity is determined by the well-known theorem of theoretical mechanics:
дz g = (z 2 - z 1),
where m is the mass of the cargo being moved,
D is the mass of the vessel,
z 1 and z 2 - CG applications of the load before and after the movement.
The increment in metacentric heights will be:
dh = dH = - dz g = - (z 2 - z 1)
After moving the cargo, the vessel will have a transverse metacentric height:
The vertical movement of the load does not lead to a significant change in the longitudinal metacentric height, due to the smallness of dH in comparison with the value of H.
Fig. 14. Vertical movement of cargo
Fig. 15. Lateral horizontal movement of the load
Suspended loads
They appear on the ship as a result of lifting cargo from the hold onto the deck, taking catch, picking nets using cargo arrows, etc. A suspended load (Fig. 16) has an effect on the stability of a vessel similar to that of a vertically displaced one, only the stability change occurs instantly at the moment of its separation from the support. When lifting a 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. You can estimate the change in metacentric height using the formula
where l = (z 2 - z 1) is the initial length of the suspension of the load.
On small boats, in low stability conditions, lifting of cargo with ship's arrows can be a significant hazard.
Lateral horizontal movement of the load
The lateral horizontal movement of the cargo mass m (Fig. 17) leads to a change in the ship's heel as a result of the resulting moment m cr with the 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 position of the CG of the load before and after movement.
Taking into account the equality of the heeling m cr and the restoring moments m AND, using the metacentric stability formula, we obtain:
Дh sinИ = m l y cosИ, whence
tgИ = m l y / Дh.
Considering that the roll angles are small, we can assume that tgI = I = I 0 / 57.3, and the formula will take the form
And 0 = 57.3 m l y / Dh.
If, before the cargo was moved, the ship had a list, then in this formula the angle should be considered as an increment dI 0
Fig. 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 W = M dif, we get:
tg W = m l x / DN, or
W 0 = 57.3 m l x / DN.
In practice, longitudinal inclinations are more often estimated by the size of the 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 cargo scale and KETCH)
m D = 0.01 g V H / L (kN m / cm);
m D = 0.01 DN / L = 0.01 DR / L (t m / cm),
since H R we get
D f = m l x / m D (cm).
Change in draft 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 vessel 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 ship's metacentric height is discussed in detail in 7.2.
9 ... Influence of accepting small cargo on boarding and stability of the vessel
Changing the landing of the vessel upon receiving the cargo was considered 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 line with the CG of the waterline area at the point with the applicate z.
As a result of an increase in draft, the volumetric displacement of the vessel will increase by dV = m / s and an additional buoyancy force rdV will appear, applied in the CG layer between the waterlines WL and W 1 L 1.
Fig. 19. Acceptance of small cargo on board
Considering the vessel straight-sided, the applicate of the CG of the additional buoyancy volume will be equal to d + dd / 2, where the draft increment is determined by the known formulas dd = m / sS or dd = m / q cm.
When the vessel is inclined at an angle I, the force of the weight of the load p and the equal buoyancy force r dV make up a pair of forces with the arm (d + dd / 2 -z) sinI. The moment of this pair dm И = р (d + dd / 2 - z) sin И increases the initial restoring moment of the vessel m И = гV h sin И, therefore, the restoring moment after receiving the load becomes equal to
m AND 1 = m AND + dm AND, or
(rV + r dV) (h + dh) sin И = rV h sin И + r dV (d + dd / 2 - z) sin И,
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 the 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) cargo.
It can be seen from the formula that
dh< 0 при z >(d dd / 2 - h) and
dh> 0 for 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 (limiting) plane.
The neutral plane is the plane, the reception of the cargo on which does not change the stability of the vessel. Reception of cargo above the neutral plane reduces the stability of the vessel, below the neutral plane increases it.
10 ... Influence of liquid cargo on the stability of the vessel
The ship carries a significant amount of liquid cargo in the form of fuel, water and oil supplies. If the liquid cargo fills the tank as a whole, its effect on the stability of the ship is similar to that of an equivalent solid cargo of mass
m w = with w v w.
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 the ship can also appear as a result of extinguishing fires and damage to the hull. Free surfaces have a strong negative effect on both the initial stability and the stability of the vessel at high inclinations. When the vessel is tilting, the liquid cargo, which has a free surface, flows towards the inclination, thus creating an additional moment that heels the vessel. The emerging moment can be considered as a negative correction to the ship's restoring moment.
Fig. 20. Influence on the initial stability of the free surface of a liquid cargo
Free surface effect
The influence of the free surface (Fig. 20) will be considered when the ship is landing straight ahead and on an even keel. Suppose that in one of the tanks of the ship there is a liquid cargo with a volume of v w, which has a free surface. When the vessel is inclined 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 And, we can assume that this movement occurs along an arc of a circle of radius r 0 centered at the point m 0, at which the lines of action of the weight of the liquid intersect before and after the inclination of the vessel. Analogous to 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 a suspended one, where l = r 0, and m = c w v w.
Fig. 21. Curves of dimensionless coefficient k
Using the formula for the suspended load, we obtain the formula for the effect on the stability of the free surface of the liquid:
As can be seen from the formula, it is i x that affects 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 taking 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, affects i x and, consequently, dh. Thus, it is necessary to be especially wary of free surfaces in wide compartments.
Let us determine how much the loss of stability in a rectangular tank decreases after the installation of 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 the corrections to the metacentric height before and after the installation of bulkheads will be
dh / dh n = i x / i x n = (n +1) 2.
As can be seen from the formulas, the installation of one bulkhead reduces the effect 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, where the upper curve corresponds to an asymmetrical trapezoid, the lower one is symmetrical. For practical calculations, the coefficient k, regardless of the shape of the surface area, it is advisable to take as for rectangular surfaces k = 1/12.
In ship conditions, the effect of liquid cargo is taken into account using the tables given in the “Ship Stability Information”.
Table 1
Correction for the effect of free surfaces of liquid cargo on the stability of a BMTR type vessel "Mayakovsky"
|
Correction, m, dh |
Displacement of the vessel, m |
||||||
The tables give corrections to the metacentric height of the vessel dh for a set of tanks, which, under operating conditions, may turn out to 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 a metacentric height correction of less than 1 cm are not taken into account in the calculations.
Depending on the type of amendments, the metacentric height of the ship, taking into account the influence of liquid cargo in partially filled tanks, is found by the formulas
h = z m - z g - dh;
h = z m - z g - dm h /
As can be seen, free surfaces seem to raise the ship's center of gravity or decrease its transverse metacentre by the value
dz g = dz m = dh = dm h /
The manifestation 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
dH = - with w 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 OY). 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 result in 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 |
Applicat CG, m |
Moment mx, tm |
Moment mz, tm |
Free surface corrections, 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 |
Fig. 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 is inclined (Fig. 22). Accordingly, the moment of inertia of the free surface will also undergo a sharp decrease, and, therefore, the correction to the metacentric height. Those. an ineffective loss of stability is observed, which practically can be ignored.
To reduce the negative impact on the stability of the vessel of overflowing liquid cargo on it, the following constructive and organizational measures can be provided:
Installation of longitudinal or transverse bulkheads in tanks, which allows to sharply reduce the intrinsic moments of inertia i x and i y;
Installation in tanks of longitudinal or transverse diaphragms-bulkheads with small openings in the lower and upper parts. In case of sharp inclinations of the vessel (for example, when rolling), the diaphragm acts as a bulkhead, since the liquid flows through the holes rather slowly. From a constructive point of view, diaphragms are more convenient than impermeable bulkheads, since when installing the latter, the systems for filling, draining and ventilation of tanks are significantly complicated. However, with prolonged inclinations of the vessel, the diaphragms, being permeable, cannot reduce the effect of the overflowing liquid on stability;
When accepting liquid cargo, ensure complete filling of tanks without the formation of free surfaces of the liquid;
When consuming liquid cargo, ensure complete drainage of tanks; “Dead stocks” of liquid cargo should be kept to a minimum;
Ensure dry holds in ship compartments where liquid with a large free surface area can accumulate;
Strictly follow the instructions for the reception and consumption of liquid cargo on board.
Failure to perform the listed organizational measures by the crew of the vessel can lead to a significant loss of stability of the vessel and cause an accident.
11 ... Experienced definition of metacentricthe height and position of the center of gravity of the vessel
When designing a ship, its initial stability is calculated for typical load cases. The actual stability of the constructed vessel differs from the calculated one due to the calculation errors and deviations from the design, made during the construction. Therefore, on the ships, an experimental determination of the initial stability is carried out - inclining, with the subsequent calculation of the position of the ship's CG.
The following should be subjected to inclining:
Serial-built ships (the first and then every fifth ship in the series);
Each new ship of non-serial construction;
Every ship after refurbishment;
Vessels after major repairs, re-equipment or modernization with a change in displacement by more than 2%;
Vessels after stowage of permanent solid ballast, if the change in the center of gravity cannot be accurately determined by calculation;
Vessels whose stability is unknown or needs to be checked.
Inclining is carried out in the presence of the Register inspector in accordance with the special “Instruction for Inclining of the Register vessels”.
The essence of inclining is as follows. Inclining is carried out on the basis of the equality m cr = m И, which determines the equilibrium position of the vessel with a roll И 0. The heeling moment is created by the movement of loads (inclination) along the width of the vessel at a distance l y; within small inclinations of the vessel:
m cr = m l y.
Then from the equality m l y = сV h И 0 / 57.3
find that h = 57.3 m l y / sVI 0.
The elevation of the ship'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 absence or smallness of trim are determined using the curves of the elements of the theoretical drawing according to the value of the displacement V. In the presence of a trim, these values must be determined by a special calculation. Displacement V is found on the Bonjean scale based on the measurement of the bow and stern draft of the vessel by the indentation marks. The seawater density is determined using a hydrometer.
The mass of the roll ballast m and the transfer arm l y are set, the value of the roll angle And 0 is measured.
Before heeling, the load of the vessel should be as close as possible to its unladen displacement (98 104%). Metacentric height the vessel must be at least 0.2 m. To achieve this, ballast is allowed.
Supply items and spare parts must be in their proper places, cargo must be secured, and tanks for water, fuel, oil must be drained. Ballast tanks, if filled, must be pressed in.
Crenballast is stacked on the open deck of the vessel on both sides on special racks in several rows relative to the DP. The mass of the inclined ballast carried across the vessel should provide a bank angle of about 3 0.
To measure the roll angles, prepare special weights (at least 3 meters long) or inclinographs. The use of ship's inclinometers for measuring angles is unacceptable, since they give a significant error.
Inclining is carried out in calm weather when the ship is heeled no more than 0.5 0. The depth of the water area should exclude touching the ground or finding a part of the hull in muddy soil. The vessel should be able to roll freely, for which the slack of the mooring lines should be provided and the vessel should not touch the wall or the hull of another vessel.
Experience consists of side-to-side rolls of rolls performed on command and measurements of the roll angle before and after the transfer.
Determination of the initial stability over the rolling period is carried out on the basis of the well-known "captain's" formula:
where f And - the period of the ship's own onboard vibrations;
C And - inertial coefficient;
B is the breadth of the vessel.
It is recommended to determine the period of the ship's side roll at each inclining test, and for ships with a displacement of less than 300 tons, its determination is mandatory. The means for determining f And is an inclinograph or stopwatches (at least three observers).
The rocking of the vessel is carried out by coordinated jumps of the crew from side to side in time with the vibrations of the vessel until the inclination of the vessel by 5 8 0. The captain's formula allows, at any state of the vessel's load, to approximately determine the metacentric height when it is on sea waves. It should be remembered that for the same ship, the value of the inertial coefficient C And is not the same, it depends on its loading and the placement of cargo. As a rule, the inertia coefficient of an empty vessel is greater than that of a loaded one.
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The main characteristic of stability is restoring moment, which must be sufficient for the vessel to withstand the static or dynamic (sudden) action of heeling and trimming moments arising from the displacement of loads, under the influence of wind, waves and other reasons.
The heeling (trimming) and restoring moments act in opposite directions and are equal at the equilibrium position of the vessel.
Distinguish lateral stability corresponding to the inclination of the vessel in the transverse plane (roll of the vessel), and longitudinal stability(trim of the vessel).
The longitudinal stability of sea-going vessels is deliberately ensured and its violation is practically impossible, while the placement and movement of cargo leads to changes in the lateral stability.
When the vessel is inclined, its center of magnitude (CV) will move along a certain curve called the CV trajectory. At a low inclination of the vessel (no more than 12 °), it is assumed that the trajectory of the CV coincides with a flat curve, which can be considered an arc of radius r centered at point m.
The radius r is called transverse metacentric radius of the vessel, and its center m - initial metacentre of the vessel.
Metacenter - the center of curvature of the trajectory along which the center of the C value moves during the inclination of the vessel. If the inclination occurs in the transverse plane (roll), the metacentre is called transverse, or small, with inclination in the longitudinal plane (trim) - longitudinal, or large.
Accordingly, there are transverse (small) r and longitudinal (large) R metacentric radii, representing the radii of curvature of the trajectory C with roll and differential.
The distance between the initial metacentre m and the ship's center of gravity G is called initial metacentric height(or simply metacentric height) and denoted 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 metacentre and the ship's center of gravity. M.V. is a measure of the initial stability of the ship, which determines the restoring moments at low heel or trim angles.
With increasing m.v. the stability of the vessel is increased. For a positive stability of the vessel, it is necessary that the metacentre is above the CG of the vessel. If m.v. negative, i.e. the metacentre is located below the ship's CG, the forces acting on the ship form not a restoring, but a heeling moment, and the ship floats with an initial heel (negative stability), which is not allowed.
OG - elevation of the center of gravity above the keel; OM — elevation of the metacentre above the keel;
GM is the metacentric height; CM - metacentric radius;
m - metacentre; G - center of gravity; С - center of magnitude
There are three possible cases of the location of the metacentre m relative to the center of gravity of the vessel G:
the metacentre m is located above the CG of the vessel G (h> 0). At low inclination, the forces of gravity and buoyancy create a pair of forces, the moment of which tends to return the ship to its original equilibrium position;
The CG of the vessel G is located above the metacentre 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 CG and metacentre m coincide (h = 0). The vessel will behave unstable, as there is no shoulder of a pair of forces.
The physical meaning of the metacentre is that this point serves as the limit to which the center of gravity of the vessel can be raised without depriving the vessel of positive initial stability.
By the relative position of the cargo on the ship, the boatmaster can always find the most favorable value of the metacentric height, at which the ship will be sufficiently stable and less subject to rolling.
The heeling moment is the product of the weight of the cargo moved across the vessel by the shoulder equal to the distance of movement. If a person weighing 75 kg, sitting on the bank will move across the vessel by 0.5 m, then the heeling moment will be equal to 75 * 0.5 = 37.5 kg / m
Fig 91. Static stability diagram
To change the moment that heels the vessel by 10 °, it is necessary to load the vessel to full displacement, completely symmetrically relative to the centreline plane.
The loading of the vessel should be checked by the draft measured from both sides. The inclinometer is installed strictly perpendicular to the center line so that it shows 0 °.
After that, it is necessary to move loads (for example, people) at pre-marked distances until the inclinometer shows 10 °. The experiment for verification should be carried out as follows: bank the vessel on one side and then on the other side.
Knowing the fastening moments of the vessel heeling at different (up to the maximum possible) angles, it is possible to construct a diagram of static stability (Fig. 91), which will assess the stability of the vessel.
Stability can be increased by increasing the width of the vessel, lowering the CG, and arranging stern boules.
If the center of gravity of the vessel is located below the center of magnitude, then the vessel is considered very stable, since the support force during roll does not change in magnitude and direction, but the point of its application shifts towards the inclination of the vessel (Fig. 92, a).
Therefore, when heeling, a pair of forces is formed with a positive restoring moment, striving to return the ship to a normal vertical position on a straight keel. It is easy to verify that h> 0, with the metacentric height being 0. This is typical for yachts with a heavy keel and not typical for larger vessels with a conventional hull arrangement.
If the center of gravity is located above the center of magnitude, there are three possible stability cases, which the skipper should be familiar with.
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 diametrical plane above the center of gravity (Fig. 92, b).
Fig. 92. Stable ship case
In this case, a pair of forces with a positive restoring moment is also formed. This is typical of most ships of the normal shape. Stability in this case depends on the body and the position of the center of gravity in height.
When heeling, the heeling side enters the water and creates additional buoyancy that tends to level the boat. However, when a ship heels with liquid and bulk cargo capable of moving towards the bank, the center of gravity will also shift towards the bank. If the center of gravity while heeling moves beyond the plumb line connecting the center of magnitude with the metacentre, the vessel will capsize.
The second case of an unstable vessel with an 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 line with the center of gravity, so there is no recovering pair of forces. Without the influence of external forces, the ship cannot return to a straight position.
In this case, it is especially dangerous and completely unacceptable to transport liquid and bulk cargo on the ship: with the slightest roll, the ship will capsize. This is typical of boats with a round frame.
The third case of an unstable vessel with 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 the vessel is called its property, due to which the vessel does not turn over when exposed to external factors (wind, waves, etc.) and internal processes (displacement of goods, movement of liquid reserves, the presence of free surfaces of liquid in compartments, etc.). The most capacious definition of a ship's stability can be the following: the ability of a ship not to roll over when exposed to natural sea factors (wind, waves, icing) in the navigation area assigned to it, 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 the water - it tends to return to its original position after the cessation of this impact. Thus, stability, on the one hand, is natural, and, on the other hand, it requires regulated control by a person taking part in its design and operation.
Stability depends on the shape of the hull and the position of the ship's CG, therefore, by choosing the correct shape of the hull in the design and the correct placement of cargo on the ship during operation, sufficient stability can be ensured to ensure that the ship is not capsized under all sailing conditions.
The inclination of the vessel is possible for various reasons: from the action of the incident waves, due to asymmetric flooding of the compartments during a breach, from the movement of goods, wind pressure, due to the reception or consumption of goods, etc. There are two types of stability: transverse and longitudinal. From the point of view of safe navigation (especially in stormy weather), the most dangerous are lateral inclinations. Lateral stability is manifested when the ship is heeled, i.e. when leaning on board. If the forces causing the inclination of the vessel act slowly, then the stability is called static, and if fast, then dynamic. The inclination of the vessel in the transverse plane is called the roll, and in the longitudinal plane - the trim; the angles formed in this case denote, respectively, O and y. Stability at small angles of inclination (10 - 12 °) is called initial stability.
(fig. 2)
Let us imagine that under the action of external forces the vessel has received a roll 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 has moved from the initial position C to the side of the ship's roll, to the center of gravity of the new volume - point C1. When the ship is tilted, gravity P applied at point G and support force D applied at point C, while remaining perpendicular to the new waterline В1Л1, form a pair of forces with the arm GK, which is a perpendicular dropped from point G to the direction of the supporting forces.
If we continue the direction of the support force from point C1 to the intersection with its original direction from point C, then at small heel angles corresponding to the conditions of initial stability, these two directions will intersect at point M, called the transverse metacentre.
The relative position of points M and G allows you to establish the following feature characterizing lateral stability: (Fig. 3)
- A) If the metacentre 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 the points M and G coincide, the forces P and D act along one vertical 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 seek to return to its original equilibrium position (Fig. 3 ).
Fig. 3
External signs of negative initial stability of the ship are:
- - sailing of the ship with a heel in the absence of heeling moments;
- - the desire of the ship to roll over to the opposite side when straightening;
- - transshipment from side to side during circulation, while the roll remains even when the ship enters a straight course;
- - a large amount of water in holds, platforms and decks.
Stability, which manifests itself in the longitudinal inclination of the vessel, i.e. with a differential, it is called longitudinal.
With the longitudinal inclination of the vessel at an angle w around the transverse axis Ts.V. will move from point C to point C1 and the support force, the direction of which is normal to the current waterline, will act at an angle w to the original direction. The lines of action of the original and new directions 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 metacentre M.
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 much larger than the transverse r. Roughly, the longitudinal metacentric radius R is considered to be approximately equal to the length of the vessel. Since the value of the longitudinal metacentric radius R is many times greater than the transverse r, the longitudinal metacentric height H of any vessel is many times greater than the transverse h. therefore, if the vessel has lateral stability, then longitudinal stability is guaranteed.
Factors affecting the stability of a ship that have a strong influence on the stability of a ship.
Such factors that must be taken into account when operating a small vessel include:
- 1. The stability of a vessel is most noticeably influenced by its width: the greater it is in relation to its length, side depth and draft, the higher the stability. The wider vessel has more restorative torque.
- 2. The stability of a small vessel is increased if the shape of the submerged part of the hull is changed at large heel angles. This statement, for example, is the basis for the action of side boules and foam fenders, which, when immersed in water, create an additional restoring moment.
- 3. Stability is impaired by the presence of side-to-side mirror fuel tanks on the ship, so these tanks should have baffles parallel to the ship's centreline or be tapered at their top.
- 4. Stability is most strongly influenced by the placement of passengers and cargo on the vessel, they should be placed as low as possible. It is not allowed to sit on board and move randomly on a small vessel during its movement. Weights must be securely secured to prevent their unexpected displacement from their regular places.
- 5. In case of strong wind and waves, the heeling moment (especially dynamic) is very dangerous for the vessel, therefore, with the deterioration of weather conditions, it is necessary to take the vessel to the shelter and wait out the bad weather. If this cannot be done due to the considerable distance to the coast, then in stormy conditions one should try to keep the ship "with its bow to the wind", throwing out the floating anchor and operating the engine at low speed.
Excessive stability will cause rapid rolling and increase the risk of resonance. Therefore, the register has established limitations not only for the lower, but also for the upper limit of stability.
To increase the stability of the vessel (increase the restoring moment per unit of the roll angle), it is necessary to increase the metacentric height h by appropriately placing cargo and stores on the vessel (heavier weights at the bottom, and lighter ones 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, a distinction is made between lateral stability when heeling and longitudinal stability at differential. As applied to surface ships (ships), due to the elongation of the shape of the ship's hull, its longitudinal stability is much higher than the transverse one, therefore, for the safety of navigation, it is most important to ensure proper lateral stability.
- Depending on the value of the inclination, stability is distinguished at small angles of inclination ( initial stability) and stability at large angles of inclination.
- Depending on the nature of the acting forces, static and dynamic stability are distinguished.
Initial transverse stability
When heeling, stability is considered as initial at angles up to 10-15 °. Within these limits, the restoring force is proportional to the roll angle and can be determined using simple linear relationships.
It is assumed that deviations from the equilibrium position are caused by external forces that do not change either the weight of the vessel or the position of its center of gravity (CG). Then the immersed volume does not change in magnitude, but changes in shape. Equal-volume inclinations correspond to equal-volume waterlines, cutting off the submerged volumes of the hull of equal size. The line of intersection of the waterline planes is called the inclination axis, which at equal-volume inclination passes through the center of gravity of the waterline area. At transverse inclinations, it lies in the diametrical plane.
Free surfaces
All the cases discussed above assume that the center of gravity of the vessel is stationary, that is, there are no weights that move when inclined. But when there are such loads, their effect on stability is much greater than the rest.
Typical cases are liquid cargo (fuel, oil, ballast and boiler water) in tanks that are partially filled, that is, with free surfaces. Such weights are capable of overflowing when tilted. If the liquid cargo fills the tank completely, it is equivalent to a solid fixed cargo.
Influence of the 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 pours in the direction of inclination. The free surface will take the same angle relative to the waterline.
Levels of liquid cargo cut off equal volumes of tanks, that is, they are similar to equal-volume water lines. Therefore, the moment caused by the overflow of liquid cargo when heeling δm θ, can be represented similarly to the moment of form stability m f, only δm θ the opposite m f by sign:
δm θ = - γ and i x θ,Where i x- the moment of inertia of the free surface area of the liquid cargo relative to the longitudinal axis passing through the center of gravity of this area, γ w- specific gravity of liquid cargo
Then the restoring moment in the presence of a liquid cargo with a free surface:
m θ1 = m θ + δm θ = Phθ - γ w i x θ = P (h - γ w i x / γV) θ = Ph 1 θ,Where h- transverse metacentric height in the absence of transfusion, h 1 = h - γ and i x / γV is the actual transverse metacentric height.
Influence of overflowing weight gives correction for lateral metacentric height δ h = - γ w i x / γV
The densities of water and liquid cargo are relatively stable, that is, the main influence on the correction is provided by the shape of the free surface, more precisely, its moment of inertia. This means that the lateral stability is mainly influenced by the width, and the longitudinal length of the free surface.
The physical meaning of the negative value of the correction is that the presence of free surfaces is always reduces stability. Therefore, organizational and constructive measures are taken to reduce them:
Dynamic stability of the vessel
In contrast to the static one, the dynamic action 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 of the kinetic energy of the inclination of the vessel is equal to the work of the forces acting on it.
When a heeling moment is applied to the ship m cr constant in magnitude, it receives a positive acceleration, with which it begins to roll. As the inclination increases, the restoring moment increases, but first, up to the angle θ st at which m cr = m θ, it will be less heeling. Upon reaching the angle of static equilibrium θ st, the kinetic energy of rotational motion will be maximum. Therefore, the vessel will not remain in the equilibrium position, but due to the kinetic energy will heel further, but more slowly, since the restoring 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 amount of this work is sufficient to completely dissipate the kinetic energy, the angular velocity will become equal to zero and the ship will stop heeling.
The largest angle of inclination that the boat receives from the dynamic moment is called the dynamic bank angle. θ din... In contrast, the roll angle with which the ship will sail under the influence of the same moment (by the condition m cr = m θ) is called the static roll angle θ st.
Referring to the static stability diagram, the work is expressed by the area under the restoring 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.
Reaching the dynamic roll angle θ din, the ship does not come to equilibrium, and under the influence of the excess restoring moment begins to accelerate to straighten. In the absence of water resistance, the vessel would enter into continuous oscillations around the equilibrium position when heeling θ art / ed. Physical encyclopedia
The ship, the ability of the ship to withstand external forces causing it to roll or trim, and return to the original equilibrium position after they cease to operate; one of the most important seaworthiness of a ship. O. when heeling ... ... Great Soviet Encyclopedia
The quality of the ship is to be in equilibrium in an upright position and, being driven out of it by the action of any force, to return to it again upon termination of its action. This quality is one of the most important for the safety of sailing; there were many… … Encyclopedic Dictionary of F.A. Brockhaus and I.A. Efron
G. Ability of the boat to float in an upright position and to straighten after inclination. Efremova's Explanatory Dictionary. T.F. Efremova. 2000 ... Modern explanatory dictionary of the Russian language by Efremova
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