Why do I need a waterline on ships. Waterline area elements. New explanatory and derivational dictionary of the Russian language, T.F. Efremova

Waterline

Waterline applied to the ship's hull (black)

Waterline (Dutch. waterlinie) - the line of contact of a calm water surface with the hull of a floating vessel. Also - in the theory of the ship, an element of the theoretical drawing: a section of the hull with a horizontal plane.

The following waterlines are distinguished:

• constructive waterline (CWL) - that is, calculated, determined for the full load of the vessel;
• cargo waterline - calculated for a predetermined load and sailing conditions;
• operating waterline - current, at a given load and conditions;
• theoretical waterlines - a set of sections at equal distances, forming one of the types of theoretical drawing: plan.

The effective waterline is determined by the shape of the vessel, its average density, as well as the degree of water disturbance in the given basin. The waterline area is used to calculate the hull fullness factor. The shape of the waterline area, or rather its moment of inertia, is a factor that determines the stability of the shape. Obviously, depending on the conditions of load, roll and trim, the shape of the waterline area, and with it the stability, can change.

The waterline length serves as a characteristic linear dimension in determining the Froude number for displacement vessels, and, accordingly, their theoretical speed.

Load line

Load line (Plimsoll line)

All commercial vessels must carry a mark on board called load line (eng. load line, Plimsoll line). This mark defines the level to which the vessel can be safely loaded, i.e. cargo waterline... When the boat is loaded, it sinks deeper into the water and the mark drops closer to the surface of the water.

Before this mark became mandatory, many ships were lost due to congestion. Sometimes the reason for overloading is the desire to receive additional profit from transportation, and sometimes the difference in water density - depending on its temperature and salinity of the vessel's sediment, can significantly change.

The British politician Samuel Plimsol proposed a system of universal marking of ships, which made it possible to determine the maximum load of a ship depending on the season and region.

Load line letters mean:

Storms are frequent in winter. A high wave can rock the boat or flood the deck, so additional buoyancy is required. The North Atlantic is a particularly stormy area, plus the danger of icing - there should be even more buoyancy. Tropical waters, on the contrary, are calm, there you can safely load the vessel.

The other two grades - F and TF - correspond to S and T, converted to the density of fresh water.

Literature

• // Encyclopedic Dictionary of Brockhaus and Efron: In 86 volumes (82 volumes and 4 additional). - SPb. , 1890-1907.

Wikimedia Foundation. 2010.

Synonyms:

See what "Waterline" is in other dictionaries:

Waterline ... Spelling dictionary-reference

- (Dutch and English water water, and Latin linea feature). The line along which a vessel with luggage can submerge in water. Dictionary of foreign words included in the Russian language. Chudinov AN, 1910. WATERLINE from English. water, water, and lat. linea, hell. Damn ... Dictionary of foreign words of the Russian language

- (Water line) curve obtained when the surface of the ship's hull is crossed by a horizontal plane parallel to the water level. See the theoretical drawing of the ship. Samoilov K.I. Marine dictionary. M. L .: State Naval Publishing House ... ... Marine Dictionary

- (from the netherl. water water and lijn line) the line of contact of a calm water surface with the hull of a floating vessel. The load waterline marked by the load line coincides with the water surface when the vessel is fully loaded and corresponds to ... ... Big Encyclopedic Dictionary

A line along the side of the vessel that defines the maximum draft of the vessel when fully loaded. Dictionary of business terms. Academic.ru. 2001 ... Business Glossary

WATERLINE, waterline, wives (Dutch. waterlinie) (n.). The line along the side to which the ship is submerged. Ushakov's explanatory dictionary. D.N. Ushakov. 1935 1940 ... Ushakov's explanatory dictionary

- [te], and, wives. (specialist.). The line along the side, until the swarm of the vessel is immersed in the water at normal draft. Freight in. (coinciding with the water surface when the vessel is fully loaded). Ozhegov's Explanatory Dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Ozhegov's Explanatory Dictionary

Female, mor. the line on the hull of the ship along which it sits in the water; loading, loading, draft. This feature is calculated by the builder in advance and is indicated on the drawing of the ship. Spirit level man., Dutch. a projectile showing a plane in level, how it stands ... ... Dahl's Explanatory Dictionary

Modern approaches to shipbuilding require a continuous search for original technical solutions to gain superiority over potential enemies in the oceans. And more and more designers are turning to projects of multihull watercraft - catamarans and trimaran. Suffice it to recall the littoral ships of the US Navy of the "Independence" type or the latest Russian development "Rusich-1". Doctor of Technical Sciences Viktor Dubrovsky tells how else you can improve the technical characteristics of multihulls by means of an original solution - reducing the waterline area.

Introduction

Objects with a small waterline area include semi-submersible (usually drilling) platforms and ships (ships) with a small water-plane area ships.

In fig. 1 shows a diagram of the appearance of a semi-submersible platform. In the working position, the waterline is located approximately in the middle of the height of the pillars (columns) connecting the pontoons with the upper structure, in the stowed position - slightly below the upper decks of the pontoons.

Semi-submersible platforms in the world have been used since the 50s, since that time more than 300 such objects of a fairly large displacement have been built. Practice has shown that they can constantly be in the harshest seas of the planet and work most of the time, including during very intense waves. In fig. 2 shows a double hull with a small waterline area (LWA).

Research, design and construction of SMPVs began in the 60s, since then more than 70 such ships have been built in the world, mainly of small displacement, often used as experimental ones.

Already these illustrations reveal the main difference between objects with a small waterline area: a decrease in displacement volumes near the waterline with compensation for these volumes due to more submerged parts of the ship.

Currently, displacement volumes crossing the free surface are commonly referred to as "struts" (for ships) or "columns" (for platforms). Underwater volumes today do not have an established name: they speak of "pontoons" for platforms and vessels, "underwater hulls", "underwater volumes", etc. for ships.

In the author's publications since 1978, the following terminology has been used for ships: each hull consists of a surface platform - a rack (s) - a gondola (the latter term was borrowed from aviation). The same terminology is used below.

In addition, to characterize the location of the hulls relative to each other and relative to the water surface, the following terms are used: lateral clearance (usually the distance between the diametrical planes of the hulls); vertical clearance (distance of the platform bottom from the calculated waterline); longitudinal clearance (the distance between the midships of the hulls if they are shifted longitudinally).

The noted feature of the contours affects all the technical and operational qualities of the vessels. In addition, like all multihull objects, SMPVs are distinguished by an increased deck area relative to the volumetric displacement. Therefore, like all multihulls, SMPVs are effective for transporting a low weight payload that requires large deck areas or large volumes for its placement, i.e. "light" cargo. These include passengers in cabins, rolling equipment, light containers, research laboratories, weapon systems, primarily aviation. Therefore, in particular, it is most rational to design SMPV on the basis of a given initially required deck area.

Dimensional ratios and types of PWM

The specific distribution of the displacement volumes also determines the specifics of the proportions of the dimensions of the GMS.

For the convenience of using the internal volumes of the nacelles and improving the manufacturability of their assembly, it is advisable to provide uninterrupted flow around the ends: choose a semi-elliptical nose shape and a tapered one - aft. The rest of the length is a cylinder. As a result, the coefficient of completeness of the nacelle and the hull as a whole becomes dependent on the elongation of the nacelle L / D, where L is the length, D is the diameter of the nacelle.

The reduced waterline area requires increased hull spacing to provide the required initial lateral stability. These and other, described below, features of the architectural and structural type determine the ratio of the main dimensions that are not typical for single-hull ships and for multi-hull ships with traditional contours. The most likely values \u200b\u200bof these ratios are given below when considering the features of the deck area and the initial stability of various SMPVs.

Until now, to one degree or another, several types of SMPWs have been studied, although only double-hull ones are used in practice (most of the more than 70 SMPWs built in recent years are dupluses, in the stated terminology). In fig. 3 shows the studied types of SMPV.

It should be noted that the shown terminology proposed by the author in 1978 is not generally accepted. For example, in Japan, all double-hull ships are called catamarans, regardless of the shape of the hulls. It seems that the selection of two types of double-hull SMPV makes the classification more accurate. SMPV with one long strut in each hull was first built in Holland, the name of this first ship was proposed by the author as a common one for ships of this architecture. The term "trisek" was proposed by the authors of the first two-hull SMPV with two short struts as part of each hull, built in the USA: "THREE SECTIONS", i.e. platform and two underwater volumes.

In addition, in the English-language literature, all three-hull ships are called trimaran, regardless of the shape and size ratios. On the contrary, in Russian practice since the 70s (studies of the propulsion of high-speed river vessels by A.G. Lyakhovitsky) the name "trimaran" has been applied to three-hull vessels with the same hulls of conventional hulls. Therefore, a separate name for three-body SMPVs with identical bodies seems appropriate.

SMPV have both common features that distinguish them from single-hull ships and from multihull ships with conventional contours, and specific to each type. These features are discussed in more detail below. It should be noted that almost every feature of a particular type of vessel can be favorable, unfavorable or neutral for a particular purpose of use. All of these issues are briefly discussed below.

Here, a single-hull object of equal displacement is conventionally used as a base for comparison, although in practice, when choosing ship options at the very beginning of its design, it is also necessary to consider comparable types of multi-hull ships with traditional contours.

It should be especially noted that each SMPV can be designed so that at full displacement the vessel will have a draft along the top of the gondolas, which expands the possibilities of using shallow water areas and ports. At the same time, to increase seaworthiness in waves, it is necessary to provide for the reception of ballast water. It is clear that the volume of this ballast corresponds to the volume of the submerged part of the racks, i.e. relatively small in relation to the total displacement of the vessel.

However, the strong influence of relatively small amounts of ballast on the landing of SMPV is a significant inconvenience of its operation. If not foreseen, the mere use of fuel while sailing will lead to unacceptable changes in landing, primarily roll and trim. Therefore, for example, one of the world's first SMPVs, the Japanese passenger ferry, had an automatic ballasting system to maintain the required landing variation limits during operation.

How it works

1. Deck area

While volume redeployments have the greatest impact on hydrostatics and hydrodynamics, it is more convenient from a design point of view to start by looking at relative deck areas. This consideration is based on the above-mentioned system of the most probable ratios of dimensions, which determine the specifics of this type of ships.

The main results of such assessments are shown in Table 1.

Ship type	Relative length of one body	Probable dimension ratios	Deck area
Monohull		L / B \u003d 8; A D ~ 0.8
Trisek or duplus		L SW \u003d 0.64 * L; B OA \u003d (0.3 ÷ 0.5) * L SW;	(0.19 ÷ 0.32) * L 2
Small waterline hull and two outriggers		L M \u003d 0.8 * L; L M / B M \u003d 8; L A \u003d (0.3 ÷ 0.4) * L M; B OA \u003d (0.3 ÷ 0.4) * L M;	(0.13 ÷ 0.16) * L 2
		L 1 \u003d 0.35 * L; A D ~ 0.75; L OA \u003d 1.6 * L 1; B OA \u003d (0.6 ÷ 0.8) * L 1;	(0.25 ÷ 0.35) * L 2

Table 1.

Here: L, V, B - length, displacement, width of a comparable single-hull vessel, AD - coefficient of completeness of the upper deck; B1, BOA - width of one body and overall width; LSW - length at the waterline; LO-length of the outrigger; LM is the length of the main body; lMON, l1 is the relative length of a single-hull vessel and one hull of multi-hull vessels.

Obviously, with an equal number of decks, the SMPV will have, in one way or another, increased, in comparison with a single-hull high-speed vessel, the area of \u200b\u200bthe decks and the internal volume of the surface. That is why a large payload is always located in the surface platform connecting the hulls.

2. Initial stability and emergency landing

Longitudinal stability of SMPV is noticeably lower than that of a comparable conventional vessel. Therefore, in contrast to the current situation, when the longitudinal stability is not standardized for any types of ships, when designing the SMPV, it is necessary to take some approximate limits of the longitudinal metacentric height. Taking into account the ratio of the overall dimensions in the plan, it seems convenient to choose the longitudinal height of the double-hull SMPV 2 times greater than the transverse one, and 3 times greater - for the three-hull SMPV.

The lateral stability of SMPV determines the ratio of their overall dimensions in the plan, see Table 2, where examples of SMPV of various types with the same displacement are considered. To clarify the place of the SMPV in the general row of multihull vessels, the table also includes vessels with a traditional shape of contours: a catamaran (double-hull), a trimaran (three identical hulls) and a vessel with outriggers (a large central and two small side hulls). For simplicity, the requirement to ensure the initial lateral stability of the SMPV is the same as that of the comparable single-hull vessel.

Main dimensions and initial lateral stability of 1000-ton vessels of various types (outrigger dimensions in brackets):

Ship type	Single-body (high-speed)	Catamaran			Trimaran		Traditional center building + 2 outriggers	Center. Hull with MPV + 2 outriggers
Length of one body, m		65, 80					95 (30)	65 (35)
Overall length, m		65, 80
Width of one body, m		6, 4					7 (1)	7 (1.5)
Overall width, m		18, 16
Waterline area, kV m		2 x310, 2x 250
Design draft, m
Center height, m
Board height, m
Center of mass height, m
Transverse metacentre. .radius, m
Transverse metacentre. height, m
Longitudinal metacentre. radius, m
Longitudinal metacentre. height, m

* - to the bulkhead deck.
Table 2.
Analysis of the given data shows that the transverse size of the SMPV is chosen according to a completely different principle than the same dimensions of multihull vessels with traditional contours. The overall width of the SMPV is determined by the requirement for a certain initial stability. On the contrary, the distance between housings of traditional shape is chosen as minimum acceptable in order to reduce their hydrodynamic interaction, which is usually unfavorable, i.e. according to the requirements of speed. At the same time, the lateral stability of all vessels with traditional hulls, except for outrigger ones, is much higher than that of the comparable single-hull. Moreover, the initial lateral stability of the catamaran, if necessary, can be equal to the longitudinal stability, and even slightly exceed it. The stability of an outrigger vessel is comparable to that of a monohull or slightly more, if necessary.

The longitudinal stability of the SMPV is significantly less than that of all other types of ships, both single-hull and multi-hull. This circumstance strongly affects many characteristics of the PMS.

First of all, we note that a decrease in stability leads to difficulties in limiting the angle of emergency roll (trim): flooding of the same volume leads to a significantly greater roll or trim of the NMP than in a single-hull vessel of comparable displacement. However, it is usually not difficult to maintain a minimum freeboard if the bulkhead deck is the upper deck connecting the superstructure hulls.

The lack of lateral stability of the SMPV can be partially compensated by the collapse of the struts near the surface platform, which provides an increase in the area of \u200b\u200bthe stability diagram. But the main thing is that all multihulls have an impenetrable platform connecting the hulls. This volume sharply reduces the angles of roll and trim, as soon as its sides or ends begin to enter the water. The likelihood of flooding in the event of an accident is also significantly reduced, since usually the cutouts in the platform are far enough from the sides and ends.

Ensuring the emergency stability of the SMPV usually also does not cause problems, as soon as the watertight surface platform begins to enter the water.

Filling the compartments (usually at the ends) with non-combustible floating blocks (or large granules in nets to simplify movements during repairs) can be recommended as a significant constructive measure to ensure an emergency landing of SMPVs.

Usually the size of outriggers is small and comparable to the size of statistically possible holes in accidents. This means that in the event of an accident, complete flooding of the outrigger is likely, that is, a significant loss of the waterline area and stability. In turn, this means that usually the lateral stability must be provided by a single outrigger. However, filling the outriggers with floating materials reduces the size, self-drag and weight of the outriggers.

Thus, the emergency landing and stability of the SMPV, like most multihull ships, hardly corresponds to the concepts underlying the rules that were previously created for single-hull ships. As a result of the absence of specific stability rules, any SMPV turns out to be an experimental object, that is, all its characteristics are determined by calculations and are consistent with the corresponding Register for each project separately.

3. Seaworthiness

The high seaworthiness of the SMPV is their main difference and greatest advantage. The differences in the geometry and stability of the SMPV described above also determine the features of seaworthiness.

It is known that own rolling periods strongly affect seaworthiness. These periods are determined by the ratio of the restoring and inertial forces and moments. For pitching, it is the ratio of the longitudinal stability and the moment of inertia of the masses (including the added mass of water) about the transverse axis.

In the transition from a single-hull traditional object to a double-hull SMPV, stability decreases more than the moment of inertia of the masses. As a result, the pitching period of the double-hull SMPV increases by about 2 times.

With regard to rolling, the picture is the opposite: with approximately the same initial stability, the moment of inertia of the masses (including the attached one) about the longitudinal axis sharply increases. As a result, the proper rolling period of the SMPV is also about 2 times longer than that of a comparable single-hull object. These relationships are shown in Fig. 4.

It is clear that such significant differences greatly change the behavior of the PWM in waves. So, if single-hull vessels usually fall into resonance due to longitudinal rolling on oncoming waves, then SMPV - at the same and close to it heading angles. Sufficiently large PMSWs rarely fall into resonance when moving lagged to the wave. The heaving amplitudes of the PWM without dampers in resonance modes are greater than those of comparable vessels of other types, but the accelerations in this mode are very small.

In fig. 5 shows the pitching amplitudes of two 100-ton boats in oncoming waves. These data were obtained from the results of testing the models of the duplus and the catamaran, however, the amplitudes of the second can be quite accurately considered equal to the amplitudes of a single-hull vessel of the same length and displacement.

The dependence of the pitching on the speed of the duplus in the oncoming waves, which is completely unusual for objects with traditional contours, is obvious: the amplitudes decrease with increasing speed.

Unfortunately, the amplitudes of the vertical acceleration of pitching depend on the speed differently, see fig. 6.

Obviously, with the usual speed limitation on oncoming waves by acceleration values, the duplus has a significant advantage in terms of the attainable travel speed.

Already the first full-scale tests of SMPV showed that, in terms of seaworthiness, such a vessel is comparable to a traditional one-hull 5-15 times larger displacement (depending on the ratio of the relative areas of the waterline). In fig. 7 shows the amplitude of the pitching of the semi-scale model of the PWM on natural roughness with working and non-working stabilizers.

In 1978, the author published and in 2000 detailed the method of "folding" all information about seaworthiness, allowing it to be characterized by one number. This "seaworthiness factor" is the average annual probability of meeting the given seaworthiness standards by the considered vessel in the given water area.

These calculations show that the SMPV becomes practically "all-weather" with a displacement of about 5-6 thousand tons.

4. Running on calm water

A separate SMPV body usually differs from the same traditional one by an increased wetted surface and a reduced coefficient of residual resistance. It should be remembered that these values \u200b\u200bare interdependent in the usual system for predicting the towing resistance of a full-scale object: if the wetted surface is artificially increased, then the residual resistance coefficient, as a relative value, decreases - while the absolute value of this resistance component remains unchanged.

Fig. 8 contains a comparison of the relative values \u200b\u200bof the wetted surface of the hulls of two types: traditional and with a small waterline area.

In fig. 9 shows the residual drag coefficients of conventional hulls and hulls with a small waterline area.

In essence, it is possible to compare the running performance of hulls of different types only at the level of designed ships of the same purpose. At the same time, the other side of the flow around two or three hulls that make up a multihull vessel will be noticeable, including the SMPV: the hydrodynamic interaction of the hulls, first of all, the wave systems generated by them. The interaction features are varied and depend on the number, relative position, dimensions and shape of the cases.

It can be assumed that the maximum of the upper curve corresponds to a Froude number of about 0.5 along the length of the strut, of which there are two on the body of this type of SMPV.

An interesting example of "longitudinal interaction is the option of replacing each duplus body with two shorter bodies of the same type. In this case, the Froude number along the length of one part of such a tandem will be 1.5 - 1.7 times greater than the original body. And if the original body moved at a relative speed of about 0.5, that is, on the “hump” of the wave drag, then shorter hulls in tandem will move already in the overhill zone. Together with a decrease in the wetted surface with a decrease in elongation, such a transition can be effective for reducing towing drag.

In addition to the "longitudinal" interaction, there is also the interaction of two bodies, placed at a certain (stability) distance from each other.

In this case, favorable interaction is observed in rather narrow ranges of relative velocity (from 0.33 to 0.43 and 0.2 to 0.25); the rest of the studied range of relative velocities is characterized by unfavorable - to one degree or another - interaction of wave systems. At high speeds, the interaction tends to zero.

A variant of the "longitudinal" interaction is the influence of the longitudinal shift of the central body of a three-body object on the total value of its residual resistance coefficient.

The available test results of a large domestic series of SMPV models make it possible to evaluate all possible options for the dimensions and relative position of the hulls at the early stages of design.

The longitudinal position of the outriggers has the greatest influence on the residual resistance of the outrigger vessel.

As for the propellers, the same types can be used for SMPVs as for traditional ships and ships, most often placed one on each of the two hulls or one at the aft hulls of three-hull objects, or one or two at the stern of the central hull. ships with outriggers. Since SMPVs can have an increased design draft, at least when moving at sufficient depths, the propellers of these objects usually have increased diameters, which has a positive effect on the propulsion coefficient. Another feature of the MPS is a higher viscous associated flow and a reduced suction coefficient, which also means an increase in the propulsion coefficient.

A unique series of SMPV models tested at the A.N. Krylov Central Research Institute in the 70s makes it possible to predict the towing resistance of vessels of various types at the early stages of design (without additional tests before the technical design stage).

5. Durability

The complete scheme of forces and moments acting on multihull ships, including SMPV, is rather complicated. However, in the early design stages, the main external load is the lateral horizontal force and the transverse bending moment determined by it, Fig. ten.

The greatest transverse loads act when parked logged to waves, which is the design case for transverse strength.

The most effective counteraction to the general lateral loads are transverse bulkheads located along the entire height of the side of the SMPV, Fig. 11, and their associated attached skin straps.

The placement of bulkheads providing transverse strength, each of which should be from side to side and from the bottom to the upper deck, should be started at the first stages of the design of the general arrangement. If such a bulkhead is to be permeable, the loss of its strength due to cutouts must be compensated for by reinforcements.

For double-hull SMPVs, longitudinal strength is less important than for traditional ships, mainly because the length of the hulls is shorter with the same displacement. The longitudinal strength of triple-hull and outrigger SMPVs plays a significant role and should be checked, as with traditional hulls. A common difference is the drop in the longitudinal bending moment of the SMPW with an increase in speed - in traditional ships, the longitudinal bending moment increases with an increase in speed on headwind. The most loaded section of the SMPV is usually the horizontal section of each strut at the point where its vertical collapse begins. The design of the rack must be smooth - to prevent stress concentration in the most loaded section.

If we estimate the required thickness of the strut plating in the most loaded section and take this thickness as the average, and then determine the overall dimensions of all parts of the structure, we can estimate the mass of the hull structures of the SMPV, see Fig. 12.

Usually, the mass of the hull structures of the SMPV in relation to the displacement is greater than that of comparable conventional ships, but less in relation to the deck area.

The SMPV with outriggers have the smallest relative mass.

7. Design

To take into account the peculiarities of the PWM, the author proposed a special algorithm for their design. One of the main input data in this algorithm is the area of \u200b\u200bdecks required to perform the tasks of the vessel.

As a rule, the projected SMPV does not have prototypes, or access to the relevant information is impossible. Therefore, the dimensions are chosen by the variant method when calculating the main technical and operational qualities by direct calculations. The diagram of the corresponding algorithm is shown in Fig. 13.

The result of domestic studies of the characteristics of SMPVs since the end of the 60s has become the possibility of developing early stages of projects for ships of any purpose. During the indicated time, the author proposed many options for SMPV and other multihull ships, see Fig. fourteen.

1. The main advantage of ships with a small waterline area is high seaworthiness, comparable to the seaworthiness of traditional ships 5-15 times larger displacement.

2. The available domestic materials of tests, calculations and methodological developments make it possible to carry out the early stages of projects of such ships without additional tests and calculations.

The widespread use of vessels with a small waterline area is recommended in all cases where high seaworthiness increases the efficiency of the fleet. To demonstrate the effectiveness of the use of such vessels, it is recommended to use the seaworthiness comparison methodology, "folding" all information into one figure, "seaworthiness factor".

Victor Dubrovsky

Literature

1. "Multihull ships", collection, comp. and ed. Dubrovsky V.A. ed. "Shipbuilding", 1978, 297 pp.

To study the navigational qualities of a ship, it is necessary to know the values \u200b\u200bon which they depend. These values \u200b\u200binclude a group of indicators characterizing the geometry of the ship's hull and called - elements of the theoretical drawing; the latter are also called - hydrostatic indicators of the vessel.

The elements of the theoretical drawing include:

V	–	volumetric displacement, m 3;
z from	–	the applicate of the center of gravity of the immersed volume of the body (applicate of the center of magnitude - CV), m;
x from	–	abscissa CV, m;
x f	–	abscissa of the center of gravity of the waterline area, m;
S	–	waterline area, m 2;
w	–	submerged area of \u200b\u200bthe frame, m 2;
d, a, b	–	completeness coefficients: displacement, waterline area and submerged area of \u200b\u200bthe frame, respectively;
I x	–	moment of inertia of the waterline area relative to the longitudinal axis 0X, m 4;
I f	–	moment of inertia of the waterline area relative to the transverse axis passing through its center of gravity, m 4;
r	–	small (transverse) metacentric radius, m;
R	–	large (longitudinal) metacentric radius, m

The elements of the theoretical drawing are usually divided into two groups: buoyancy elements ( V, S, w, z from , x c , x f, a, d, b)and elements of initial stability ( I x, I f, r, R)... The use of buoyancy elements is shown in the Buoyancy section of this manual.

The main parameter characterizing the landing of the vessel (position of the vessel relative to the water) is its depth ( z). In the absence of roll and trim (landing straight ahead and on an even keel), penetration is the only landing parameter, and in case of an arbitrary landing, it is the main parameter. Taking into account the above, the values \u200b\u200bof the elements of the theoretical drawing are usually presented in the form of dependencies (curves) on immersion (Fig. 1.10).

In fig. 1.10 does not show the dependence of the change in the immersed area of \u200b\u200bthe frames ( w). As a base (argument) to represent the change wthe length of the waterline is taken ( L) for some value of immersion ( z). The graph of such a dependence (Fig. 1.11) is called the marching on the frames.

General expressions for buoyancy elements.To calculate the volumetric displacement, the coordinates of the center of magnitude and other buoyancy elements, a theoretical drawing is used.

We select from the underwater volume of the hull by two planes of frames spaced by an infinitely small amount dx element of this volume (Fig. 1.12, and). The volume of such an element will be w · dx, and the immersed volume of the vessel is determined by integrating this expression over the length of the vessel

Fig. 1.11. Combat on frames

To determine the abscissa of the center of magnitude (x c) we use the theorem that the static moment of volume ( V) relative to the midsection is equal to the total moment of its elements, i.e.

The applicate of the center of quantity is determined, similarly to (1.6), through the static moment of the volume relative to the main plane

The static moment of an elementary area (see Fig. 1.14) relative to the 0 axis Have is equal; and for the entire area of \u200b\u200bthe waterline we will have

Similarly, if in formula (1.7) the area of \u200b\u200bthe waterline is replaced by expression (1.10), we will have

(1.15)

General expressions for determining the completeness coefficients a, b, d, related to the elements of buoyancy are represented by formulas (1.1) (1.2) and (1.3); the use of the latter is possible with known values \u200b\u200b( S, V and w).

The general expressions presented above for determining the buoyancy elements contain a definite integral, which can have an exact solution if the function is given analytically.

The dependencies describing the theoretical surface of the ship's hull are set in the form of a drawing, i.e. graphically. In this case, the definite integral is calculated by approximate formulas (quadrature formulas). In ship theory calculations, quadrature formulas are called rules. In the practice of shipbuilding calculations, three rules have become widespread: the trapezium rule, the Simpson rule and the Chebyshev rule . The advantage of the trapezoid rule is simplicity and clarity; it is widely used in practice.

Trapezium rule. The essence of this rule and its application for calculating buoyancy elements are presented below.

If it is necessary to calculate a definite integral of the form, and the integrand function y \u003d f (x) is given in the form of a curve (Fig. 1.15), then the geometric expression of the integral will be the area ( AND), limited by a given curve, abscissa axis and end ordinates. For an approximate calculation of the area, it is divided into a series of trapezoids with the same height; in this case, calculating the integral is reduced to determining the area bounded by a broken line, i.e. to calculate the sum of the areas of trapeziums whose bases are ordinates at 0 , y 1 , … y n:

where is the height of the trapezoid; n - number of intervals.

Since half of each ordinate, except for the extreme ones, appears in the resulting expression twice, the formula can be converted to the form

and the half-sum of extreme ordinates, called the correction to the sum, as

The trapezoidal rule can be applied to calculate any definite integrals, with the integrand y \u003d f (x) can have any geometric or physical meaning.

Calculation of the area of \u200b\u200bthe frame. The frame is set by its outline on the "body" projection of the theoretical drawing (see Fig. 1.13). according to the trapezoidal rule, the area of \u200b\u200bthe frame is determined as the sum of the areas of trapeziums with the same height , i.e.

.

(1.20)

After transformations and notation (1.16) - (1.18) adopted according to the trapezoidal rule, expression (1.20) can be represented in the form

MODULE 3. ELEMENTS OF THEORETICAL DRAWING

Displacement curve and cargo dimension. Weight scale

To determine the draft by displacement, or, conversely, the displacement by draft, use displacement curve V (z).To construct it, it is necessary to calculate the integral with a variable upper limit:

where x n and x k -abscissas of the points of intersection of the waterlines with the lines of the stem and sternpost, respectively, during draft z.

Curve type V (z)shown in Fig. 6, which also shows the curves V in (z)and M (z)=ρV in (z)... Curve V in (z)characterizes the volumetric displacement, taking into account the protruding parts (skin, keels, etc.), and M (z) -displacement taking into account the density of water (mass).

Curve M (z)called cargo size.The density of the water depends on the area of \u200b\u200bnavigation, as well as on the temperature of the water (i.e., on the season), therefore, sometimes a series of curves are plotted M (z)for various ρ .

Fig. 6. Displacement curve and cargo dimension for a conventional vessel.

To determine V,x with,z with, you need to know the area of \u200b\u200bthe waterlines Sand abscissas x fthe centers of gravity of these areas. To calculate stability, calculate the moments of inertia of the waterline areas relative to the coordinate axes Oh ohand axes ff,passing through the center of gravity of the waterline area.

First, we find the elements of the waterline area for a ship sitting upright and on an even keel. Let's select an elementary area (Fig. 1) with a length dxand width 2y: dS \u003d 2ydx,then

. (1)

Fig. 1. To the definition of the elements of the area of \u200b\u200ba symmetrical waterline.

The abscissa of the center of gravity of the waterline area is

x f \u003d M y / S,(2)

where M y -static moment of the waterline area about the axis OU.For determining M ylet us first write down the expression for the static moment of the elementary area dS: dM y \u003d xdS \u003d x2ydx,from where

. (3)

Now we will obtain formulas for determining the axial moments of inertia of the area of \u200b\u200bthe waterline relative to the main central axes

Find the moment of inertia dI xelementary area dS, for which we use the formula known from theoretical mechanics for the moment of inertia of the area of \u200b\u200ba rectangle relative to the main central axis: where b \u003d dx, h \u003d2y, i.e.

.

. (4)

Moment of inertia of the waterline area S about the axis ff is equal

, (5)
Where I y -moment of inertia of the waterline area about the axis OUdefined by the formula

, (6) since the elementary moment of inertia of the area dSis equal ;Sx 2 f -portable moment of inertia.

During operation, the vessel can sail with an initial heel when the waterline is asymmetric relative to the DP. To calculate the area, static moments, moments of inertia and other elements for this case, we introduce the right at nand left y lordinates (Fig. 2).

Fig. 2. To the determination of the elements of the area of \u200b\u200ban asymmetrical waterline

According to fig. 2 expression for the area of \u200b\u200ban element, taking into account that at nis negative, can be written as dS= y n dx- y l dx=(y p - y l) dx , and the area of \u200b\u200bthe waterline as

... (7) Similarly for the static moment of the area S about the axis OUget

(8)

(9)

For an asymmetrical waterline, the static moment of the area about the axis Ohis not zero. The static moment for the right elementary area is

,

for the left -

,

total -

Then the formula for the total static moment will be written as

.(10)

The center of gravity Fthe area of \u200b\u200bthe waterline will be located from the DP at a distance

 

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