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in dropping into a layer of water with a higher density, a submarine acquires positive buoyancy, and negative buoyancy when it passes into a layer of water of lower density.
In determining the specific density of sea water, which depends on its temperature and salinity y = y(t°,5%), the sea water density table can be used (See Chapter XI, Section 69).
12. Change in Buoyancy as a Function of
Hull Compression at Depth
When a submarine dives, her hull begins to experience pressure from all sides from a water column, which increases with an increase in submergence depth. The water pressure forces acting on the hull decrease volume Vp, resulting in a loss of buoyancy by the submarine. If the submarine possesses initial volume V,
pi then her volume decreases to H Vp in diving to depth H, and the loss of buoyancy is expressed as follows
where ß is the compression factor per meter of submergence of the submarine, assumed to be 0.000018. Example: A submarine with pressure hull volume Vp
= 1000 m3 dives to depth H = 100 m. The loss of buoyancy from hull compression is
Thus, in diving to great depth, a submarine acquires negative buoyancy, due to hull compression.
13. Submergence of a Submarine Under the Influence
of Residual Negative Buoyancy
With negative buoyancy, two forces will act on a diving submarine: force p, equal to the residual negative buoyancy acting vertically downward; and water resistance, acting in a direction opposite from movement of the submarine, i.e., vertically upward, and equal to kSu, where k is a coefficient which depends on the shape of the submarine hull, hull projections and volumes of the superstructure; S is the area of projection of the submarine onto a horizontal plane, and vis the submergence rate of the submarine.
Force kSu increases with an increasing submergence rate, and at a certain moment becomes equal to force p, i.e., p - kSu = 0. Hence the maximum submergence rate of the submarine will be
From this moment, movement of the submarine will be uniform.
As the submarine shifts from full buoyancy condition to periscope depth, we can assume a value of 0.06 for coefficient k, and 0.04 with further submergence.
SECTION 6. STABILITY OF A SUBMARINE
1. Stability Conditions
Stability is the ability of a submarine to return to a state of equilibrium once the forces causing deviation from this state are no longer acting.
In the absence of heeling forces, two forces act on the submarine: the force of gravity (weight) P, applied at the center of gravity Go, and the force of buoyancy D, applied at the center of buoyancy Co. Due to equilibrium of the submarine, both of these forces are equal and lie on the same vertical.
When an external pair of forces which do not change the weight and center of gravity of the submarine act on the submarine, the submarine heels at angle 0 (Fig. 3).
The center of gravity of the submarine remains at point Go, and the center of buoyancy Co shifts in the direction of the heel to point C1, due to the change in the shape of the submerged volume.
As a result of the influence of the two forces, which are not in the same plane, a pair of forces with righ ting moment DGOK is formed, which tends to return the submarine to its initial position. In this case the submarine acquires positive stability.
If under the influence of these forces a pair with moment DGK is formed, tending to rotate the submarine in the direction of heel, the submarine acquires negative stability (Fig. 4).
If after inclination of the submarine the center of buoyancy Co is on the same vertical with the center of gravity Go, then the righting moment of the pair of forces will be equal to zero and the submarine will remain in an inclined position, possessing neutral stability (Fig. 5).
With slight inclination of the submarine, the force of buoyancy D intersects the vertical and forms point M, which is called the transverse metacenter. The distance from this point to the center of gravity of the submarine MG, is called the transverse metacentric height h, or p - a, which characterizes the stability of the submarine.
Distance p from the transverse metacenter M to the center of buoyancy Co is called the transverse metacentric radius. The distance between the center of gravity Go and the center of buoyancy Co is designated by the letter a.
If point M lies above the center of gravity Co, the submarine possesses positive stability (Fig. 6). If point M lies below the center of gravity, it possesses negative stability (Fig. 4).
If the submarine is inclined, the righting moment M, is equal to
transverse metacentric radius;
This expression is called the metacentric transverse stability formula. The value of the transverse metacentric height p-a for various classes of submarines is not the same and varies within the limits 0.25-0.50 m.
3. Change in Transverse Stability Due to a
Vertical Load Displacement
With a load displacement P, along the vertical to height hy, the weight of the submarine and her load waterline remain unchanged. The metacenter also remains the same. The center of gravity of the submarine, on the basis of the theory of displacement of the center of gravity, shifts in the direction of load displacement:
Since stability is determined by the distance between the metacenter and the center of gravity, with load displacement along the vertical the new me tacentric height
MG1 = p-a GOG1
In displacing the load upward from the initial position, the change in stability GOG, is customarily designated with a “_” sign, and with a “+” sign when it is displaced downward.
From this expression it is evident that with upward load displacement the stability of the submarine decreases, and increases with downward displacement.
4. Change in Transverse Stability in Diving
Ballasting the tanks produces a change in weight of the submarine, its volume of displacement, and the actual waterline area, which in turn displaces three points: center of gravity Go, center of buoyancy Co, and metacenter Mo.
Since the stability of a submarine is determined by the relative position of points Go, Co and Mo, their position will change until the submarine is at full submergence.
When a submarine dives, the following changes occur in the position of points Go, Co and Mo:
1) with an increase in the submerged volume, the center of buoyancy Co will be displaced upward;
2) with the main ballast tanks full, the center of gravity Go will first drop, then rise;
3) metacentric radius p will decrease due to a decrease in the actual waterline area, but metacenter Mo will approach the center of buoyancy; with full submergence, metacenter M will coincide with the center of buoyancy Co (Fig.
The stability of a submerged submarine determines the relative position of points Go and Co
Let us examine two positions of the center of gravity Go and center of buoyancy Co.
a) The center of gravity is above the center of buoyancy (Fig. 8).
When the submarine is inclined, the buoyancy force and weight of the submarine create a pair of forces tending to capsize the submarine. In this case, the submarine possesses negative stability.
b) If the center of gravity is below the center of buoyancy, the buoyancy forces and weight of the submarine create a pair of forces tending to restore the submarine to its initial position. In this case, the submarine possesses positive stability (Fig. 9).