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14.7. the task of processing the results of experimental firing (bombing).

Lecture



One of the important practical problems arising in the study of issues of shooting and bombing is the task of processing the results of experimental shooting (bombing).

Here we will distinguish between two cases: the firing of shock projectiles and the firing of remote projectiles.

When firing assault projectiles, dispersion is characterized by the law of distribution of a system of two random variables: the abscissas and the ordinates of the hit point on a certain plane (real or imaginary). When firing remote projectiles dispersion is spatial in nature and is described by the law of distribution of the system of three coordinates of the point of rupture of the projectile.

We first consider the problem of processing the firing of shock shells. Let produced   14.7.  the task of processing the results of experimental firing (bombing). independent shots on some flat target and coordinates are recorded   14.7.  the task of processing the results of experimental firing (bombing). hit points (Fig. 14.7.1):

  14.7.  the task of processing the results of experimental firing (bombing). .

  14.7.  the task of processing the results of experimental firing (bombing).

Fig. 14.7.1.

Assuming that the distribution law of the system   14.7.  the task of processing the results of experimental firing (bombing). normal, you want to find estimates for its parameters: the coordinates of the center of dispersion   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). corner   14.7.  the task of processing the results of experimental firing (bombing). determining the direction of the main dispersion axes   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). and main shots   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). .

We begin by considering the simplest case when the direction of the main axes of dispersion is known in advance. This case is often found in practice, since usually the direction of the main dispersion axes is determined by the shooting conditions themselves (for example, during bombing, the direction of flight and perpendicular to it; in aerial shooting, the direction of transverse velocity of the target and perpendicular to it, etc.). In this case, the task of handling firing is greatly simplified. Knowing in advance at least tentatively the direction of the main axes, one can choose coordinate axes parallel to them; in such a coordinate system, the abscissa and ordinate of the point of entry are independent random variables, and their distribution law is determined by only four parameters: the coordinates of the center of dispersion and the main standard quadratic deviations   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). . Estimates for these parameters are determined by the formulas

  14.7.  the task of processing the results of experimental firing (bombing). (14.7.1)

Let us consider a more complicated case when the direction of the main axes of dispersion is not known in advance and must also be determined from experience. In this case, estimates of all five parameters should be determined: the coordinates of the center of dispersion   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). corner   14.7.  the task of processing the results of experimental firing (bombing). and principal mean square deviations   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). (fig. 14.7.2).

  14.7.  the task of processing the results of experimental firing (bombing).

Fig. 14.7.2.

Estimates for the coordinates of the center of dispersion in this case are determined in the same way as in the previous case, using the formulas

  14.7.  the task of processing the results of experimental firing (bombing). ;   14.7.  the task of processing the results of experimental firing (bombing). . (14.7.2)

Let us turn to the evaluation of the angle   14.7.  the task of processing the results of experimental firing (bombing). . Suppose that the directions of the main axes of dispersion are known, and draw through the point   14.7.  the task of processing the results of experimental firing (bombing). main axles   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). (fig. 14.7.2). In system   14.7.  the task of processing the results of experimental firing (bombing). coordinates of a random point   14.7.  the task of processing the results of experimental firing (bombing). will be:

  14.7.  the task of processing the results of experimental firing (bombing).

or

  14.7.  the task of processing the results of experimental firing (bombing). (14.7.3)

Obviously the magnitude   14.7.  the task of processing the results of experimental firing (bombing). will have a mathematical expectation of zero:

  14.7.  the task of processing the results of experimental firing (bombing). .

Since the axes   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). - main scattering axes, magnitudes   14.7.  the task of processing the results of experimental firing (bombing). are independent. But for quantities subject to normal law, independence is equivalent to uncorrelatedness; therefore, we only need to find such an angle   14.7.  the task of processing the results of experimental firing (bombing). at which values   14.7.  the task of processing the results of experimental firing (bombing). not correlated. This value determines the direction of the main axes of dispersion.

Calculate the correlation moment values   14.7.  the task of processing the results of experimental firing (bombing). . Multiplying equalities (14.7.3) and applying to the product of their mathematical expectation operation, we get:

  14.7.  the task of processing the results of experimental firing (bombing).

  14.7.  the task of processing the results of experimental firing (bombing). .

Equating this expression to zero and dividing both parts by   14.7.  the task of processing the results of experimental firing (bombing). , we have:

  14.7.  the task of processing the results of experimental firing (bombing). . (14.7.4)

Equation (14.7.4) gives two angle values   14.7.  the task of processing the results of experimental firing (bombing). :   14.7.  the task of processing the results of experimental firing (bombing). and   14.7.  the task of processing the results of experimental firing (bombing). differing by   14.7.  the task of processing the results of experimental firing (bombing). . These two angles determine the directions of the main axes of dispersion.

Replacing in equality (14.7.4)   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). their estimates, we obtain the estimate for the angle   14.7.  the task of processing the results of experimental firing (bombing). :

  14.7.  the task of processing the results of experimental firing (bombing). .

Find the estimates for the main standard quadratic deviations.   14.7.  the task of processing the results of experimental firing (bombing). ,   14.7.  the task of processing the results of experimental firing (bombing). . For this we find the variances   14.7.  the task of processing the results of experimental firing (bombing). given by formulas (14.7.3), according to the theorem on the dispersion of a linear function:

  14.7.  the task of processing the results of experimental firing (bombing). ;

  14.7.  the task of processing the results of experimental firing (bombing). ,

where do we find the estimates for the main variances:

  14.7.  the task of processing the results of experimental firing (bombing). (14.7.5)

Estimates for the main standard deviations are expressed by the formulas:

  14.7.  the task of processing the results of experimental firing (bombing). (14.7.6)

Let us write separately a full summary of the formulas for the treatment of firing at a flat target in the case when the direction of the main axes of dispersion is not known in advance. Estimates of the desired parameters are determined by the formulas:

  14.7.  the task of processing the results of experimental firing (bombing). (14.7.7)

Where

  14.7.  the task of processing the results of experimental firing (bombing). (14.7.8)

In conclusion, it should be noted that it makes sense to take up firing on the full formulas (14.7.7) only when the number of experiments is large enough (of the order of many dozen; only in this case is the angle   14.7.  the task of processing the results of experimental firing (bombing). can be evaluated with reasonable accuracy. With a small number of observations value   14.7.  the task of processing the results of experimental firing (bombing). , obtained by processing, is largely random.

The task of firing remote projectiles, we consider here only in the simplest case, when the direction of the main axes of dispersion (at least tentatively) is known in advance. As a rule, the tasks of processing experiments occurring in the practice of firing with remote projectiles are of this type. Then you can choose the coordinate axes parallel to the main axes of dispersion and consider the three coordinates of the point of rupture of the projectile as independent random variables.

Let the result   14.7.  the task of processing the results of experimental firing (bombing). independent shots recorded coordinates   14.7.  the task of processing the results of experimental firing (bombing). distance points of remote projectiles

  14.7.  the task of processing the results of experimental firing (bombing).

in coordinate system with axes parallel to the main scatter axes. Estimates for the parameters of the normal law are determined by the formulas:

  14.7.  the task of processing the results of experimental firing (bombing). (14.7.9)

We will not stop on solving the problem of shooting with remote projectiles in the case when the directions of the main scattering axes are not known beforehand, since in practice this problem is relatively rare.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis