Damaging Effects of Weapons and Ammunition. Igor A. Balagansky
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Название: Damaging Effects of Weapons and Ammunition
Автор: Igor A. Balagansky
Издательство: John Wiley & Sons Limited
Жанр: Химия
isbn: 9781119779551
isbn:
For intermediate values 0 < u < umax, 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 pm (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 Lx × Ly ).
A similar geometrical formation can be used to find the distribution function F(u) for any intermediate value 0 < u < umax . 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 < umax.
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 αu, βu, γu, δu – 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 = umax/3 and u = 2umax/3.
Example
One shot is fired at a target with dimensions Tx = 2; Ty = 1. Damage zone has dimensions Lx = Ly = 3. There is no offset of the aim point. We need to draw on four points the function F(u) of distribution of damage fraction U.
Solution
The creation of zones is shown in Figure I.19. The epicenter hitting in the zone (pm) – the inner part of ABCD rectangle – corresponds to covering the whole target (u = 1). Hitting in the zone (p0) (outer part of ABCD rectangle) means no target coverage (u = 0). We find the probability of hitting in the zones (p0) and (pm), as well as the outer parts of the dotted rectangles corresponding to u = umax/3 and u = 2umax/3