Читать книгу Damaging Effects of Weapons and Ammunition - Igor A. Balagansky - Страница 35

Consider the area target T, on which a single shot is fired, with damage area L (Figure I.11). Let us condition all linear dimensions to express not in meters, but in probable deviations on corresponding axes. When it is necessary to express them in meters, we will mark it with a special symbol (m):

(I.32)

So, one shot is considered, in which the damage zone L is reset on target T so that the random point O₁ (the epicenter of the explosion) is dispersed around the origin of coordinates O according to normal law. The probable deviations, chosen as units, are equal to one. It is necessary to find the average damaged fraction U.

Consider a random overlapping area S_d of the damage zone L with the target T (see the shaded area in Figure I.12) and the damaged fraction U:

(I.33)

Figure I.11 Mutual position and sizes of the target and the damage zone.

Source: From Wentzel [2].

Figure I.12 Random area overlapping the damage zone with the target.

Source: From Wentzel [2].

The value of U depends on the random position of the epicenter O₁ . If point O₁ lies far away from the target (Figure I.12a), the damaged fraction will be equal to zero. If point O₁ lies as shown in Figure I.12b, then the damage zone and the target will overlap partially. Finally, if the damage zone lies as shown in Figure I.12c, the maximum possible area will be covered. In this case (when the damage zone is smaller than the target), this maximum fraction is equal to the ratio of the damage zone area to the target area:

Figure I.13 Complete coverage of the target area by a damage zone.

Source: From Wentzel [2].

There is another case when the damage zone is large and the target is small and all can be covered by the damage zone (Figure I.13). In this case,

One or another, at any ratio of damage zone and target size, there is some maximal value of u_max of damaged fraction U.

A random value U is a so‐called mixed type value that has separate values with finite probabilities other than zero and intervals where the distribution function is continuous and only a certain probability density corresponds to each individual value. The distribution function (integral distribution law) for such random variables has breaks (jumps) in several points, and in the intervals between them grow continuously [4].

Remember that the integral law of distribution of the portion of the damaged area at one shot determines the probability of the occurrence that the portion of the damaged area U will be less than that specified by argument u.

Figure I.14 shows the form of the value distribution function U – a fraction of the damaged area at one shot. The extremes 0 and u_max values of U have probabilities other than zero. At these points, the distribution function has breaks (jumps). The jumps are equal to the probabilities of these values:

For intermediate values 0 < u < u_max, the distribution function is continuous and the probability of each individual value is zero.

At a single shot, it is not difficult to build an exact law of U value distribution.

For example, let's look at the size ratio of the target T and the damage zone L (dashed rectangle), as shown in Figure I.15. It is clear that the probability of p₀ is nothing more than the probability of the epicenter O₁ hitting the zone shaded by Figure I.15 – the outer part of rectangle A′B′C′D′.

Figure I.14 The function of the damaged fraction distribution with one shot.

Source: From Wentzel [2].

Figure I.15 Illustration for creating a distribution law of a portion of the damaged area.

Source: From Wentzel [2].

Similarly, the probability p_m (Figure I.16) can be found as the probability of O₁ hitting the shaded area of ABCD corresponding to the maximum overlapping area (in this case L_x × L_y ).

A similar geometrical formation can be used to find the distribution function F(u) for any intermediate value 0 < u < u_max . To do this, you need to draw a series of curves (geometric places of the center of the damage zone) corresponding to the same damage fraction u. The probability of obtaining the damaged fraction less than u is the probability of the point hitting O₁ to the outer part of the corresponding curve.

Figure I.16 Illustration for creating a distribution law of portion of the damaged area.

Source: From Wentzel [2].

Figure I.17 Creation of the law of distribution of the fraction of damaged area for the intermediate values 0 < u < u_max.

Source: From Wentzel [2].

However, establishing the exact shapes of these curves is a rather difficult task, and, taking into account that the rectangular configuration of the target and the damage zone is chosen enough freely, it is possible to replace all such curves with rectangles (see the dotted line in Figure I.17). The probability of the point O₁ hitting the outer part of each of such rectangles corresponding to the given value of the damaged fraction u is calculated by the formula

(I.34)

where α_u, β_u, γ_u, δ_u – are the coordinates limiting the rectangle corresponding to this value of u in the 0x and 0y axes (Figure I.18).

Figure I.18 Determination of probability obtaining a given fraction of damaged F(u) using the normalized Laplace function.

Source: From Wentzel [2].

For a rough approximation of the function F(u), four points are almost enough: two boundary and two middle points, for example for u = u_max/3 and u = 2u_max/3.