Damaging Effects of Weapons and Ammunition. Igor A. Balagansky

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.

      Example

      One shot is fired at a target with dimensions T_x = 2; T_y = 1. Damage zone has dimensions L_x = L_y = 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 (p_m) – 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 (p_m)_, as well as the outer parts of the dotted rectangles corresponding to u = u_max/3 and u = 2u_max/3

The graph of the function F(u), drawn on four points, is shown in Figure I.20.

      Knowing the distribution function F(u) of value U for one shot, it is easy to find its average value (expected value) M = M[U]. For random variables of mixed type, the expected value consists of two parts: sum and integral. The sum is applied to those values that have nonzero probabilities (i.e. where the distribution function makes a jump) and the integral to the area where the distribution function is continuous.

Figure I.19 Illustration to the example: the creation of the law of distribution of the fraction of damaged area.

      Source: From Wentzel [2].

Figure I.20 Illustration to the example: function of distribution of the damaged area.

      Source: From Wentzel [2].

In our case, there are two jumps in the F(u) distribution function: at 0 and u_max . Hence,

      (I.35)

      Knowing the function of F(u) distribution of U value for one shot, it is easy to find the probability R_u that for one shot, the damaged fraction U will be not less than the set value u:

      Example

      For the previous example, we need to find the probability that at least 80% of the target area will be damaged.

      Solution

      According to the graph in Figure I.20 at u = 0.8, we have