2. Relative Speed and Relative Course

Suppose two ships are maneuvering with respect to each other (Fig. 93). We shall consider the motion of ship M in its relative motion with respect to ship K, i.e., conditionally assuming the latter to be fixed.

In order to determine relative speed and direction of relative movement from the direction of movement of ship M, we plot speed vector m and obtain point a, to which we add speed vector of the reference ship V. Connecting the origin of vector m (point Mo) with the origin of vector V (point b), we obtain the magnitude and direction of vector V. The direction of vector V will be called the relative course and designated as K..

3. Basic Maneuvering Triangles

ρ

Position triangle M。KM (Fig. 93) has as its apexes the following points: 1) Ko - reference ship; from this point we read the movable systems of coordinates in which the relative motion of the maneuvering ship is examined; 2) M1 final relative position of the maneuvering ship;

3) Mo - initial relative position of the maneuvering ship.

The sides of the position triangle are as follows:

1) KoMo - initial distance Do, plotted along the line of initial bearing Bo; 2) KoM-final distance D1, plotted along the line of final bearing B1; 3) MOM - relative motion vector 5.

Movement triangle MoM,M1 has the following sides:

1) Sk 2) Sm 3) S

m

t absolute movement vector of the reference ship;

t

t

absolute movement vector of the maneuvering ship;

- relative movement vector.

4. Direct Construction of a Speed Triangle

Speed vectors for both ships-the maneuvering ship Vm and the reference ship VK- -are plotted from the same origin (point A) in the direction of actual motion of the ships (Fig. 94). Relative speed vector V, connects the end of vector with the end of vector m2

Direct construction of a speed triangle is used in solving problems on a maneuvering board. Direct construction of a movement triangle is accomplished in a similar manner.

5. Inverse Construction of a Speed Triangle

Speed vectors for both ships-the maneuvering ship m and reference ship -are constructed according to the direction of actual motion of the ships, but in such a way that their ends converge at point C (Fig. 95). Relative speed vector V connects the origins of vectors V and Vm, whereby its direction is from the origin of the vector of the maneuvering ship (point A) toward the end of the vector of the reference ship (point D).

The method of inverse construction of a movement triangle is similar to the preceding. Inverse construction of a speed triangle is used in solving problems on a chart. Overall control over the correctness of solution of speed and movement triangles, irrespective of the method used to construct them, is always achieved by observing the equality

When both ships are moving in a straight line, the basic maneuvering properties, reciprocal bearing, and bearing rate can be expressed through the relative motion properties (Fig. 96), whereby:

Range rate (ORR) is a projection of the relative speed vector in the initial bearing:

Reciprocal bearing (ORB) is a projection of the relative speed vector in a

direction perpendicular to the initial bearing:

When a ship is maneuvering on a straight course with respect to a fixed point, its angle on the bow and distance are constantly changing. In the process, the change in angle on the bow is equal to the change in bearing

When two ships maneuver on straight courses with respect to each other, their angles on the bow qk and 9m, as well as relative angle on the bow change continuously, whereby

will

In problems involving two ships moving in a straight line, we must work with the following parameters:

D1;

courses and speeds of the guide and maneuvering ship Kk, Km, Vk, Vm; 2) the initial and final bearings and distances between ships Bo, B1, Do,

3) change in bearing (or angle on the bow);

4) duration of the maneuver t.

Various methods can be used to solve maneuvering problems. The simplest and most accessible of them is the method of graphical solution on a maneuvering board or chart.

8. Maneuvering Board

The maneuvering board is a grid, consisting of a series of concentric circles equidistant from one another, and radial straight lines drawn from a common

center every 5 or 10° and used to read courses and bearings.

On the outer circle of the maneuvering board, there are 1o divisions, plotted clockwise from 0 to 360°. On the inner side of the outermost circle there are also 1° divisions, but the reciprocal of the former. They are used to take reciprocal bearing readings. Numbers indicating the ordinals of the concentric circles, beginning at the center, are plotted on the main radii of the maneuvering board, with readings 0°, 90°, 180° and 270°.

In order to avoid errors in using the maneuvering board to solve maneuvering problems, the following rules must be kept in mind:

1) in constructing a position triangle, the reference ship K is assumed to be fixed at the center of the maneuvering board; in order to plot the relative position of the maneuvering ship, the bearing and distance are plotted from the center of the maneuvering board. The beginning of the relative movement vector So always coincides with the initial relative position of the maneuvering ship M., and the end coincides with the final relative position of the maneuvering ship M1;

P

2) the speed triangle is constructed in the center of the maneuvering board; vectors m and 1⁄2 are drawn from the center of the maneuvering board in the direction of the courses of the ships Km and K, and the ends of vectors Vm and are connected by vector V, the direction of which is from the end of the speed vector of the reference ship to the end of vector m

9. Solving Problems Involving Both Ships Moving

in a Straight Line on a Maneuvering Board

The sequence in the solution of problems is as follows.

Let us assume that the reference ship is at the center of the maneuvering board (point K). From Bo and Do, after plotting them from the center of the plotting board, we plot the initial relative position of the maneuvering ship Mo. Similarly we plot the designated relative position of the maneuvering ship M1.

In the center of the maneuvering board, using the direct construction method, we construct speed triangle Kab (Fig. 97). The vector enclosing the ends of vectors and m is the relative speed vector V; it must be parallel to the relative movement vector 5, in the position triangle.

ρ

We determine the time required for the maneuver from the equation:

Two situations are possible, depending upon the ratio of speeds between the maneuvering ship and the reference ship: