ampersand c period circ semicolon left-parenthesis 3.33 right-parenthesis dot ModifyingAbove normal Ф With ampersand c period circ semicolon left-parenthesis 3.17 right-parenthesis equals 0.056 comma 4th Row upper F left-parenthesis 2 slash 3 u Subscript max Baseline right-parenthesis equals 1 minus ModifyingAbove normal Ф With ampersand c period circ semicolon left-parenthesis 2.67 right-parenthesis dot ModifyingAbove normal Ф With ampersand c period circ semicolon left-parenthesis 2.83 right-parenthesis equals 0.125 period EndLayout"/> 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 umax . Hence,
(I.35)
Knowing the function of F(u) distribution of U value for one shot, it is easy to find the probability Ru 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
This approximate method of finding the law of distribution of the damage fraction is suitable only for the case of one shot and cannot be extended to the case of several shots at the same area target.
I.3.7.2 Determining the Average Damage Fraction at Multiple Shots
There are methods to estimate the effectiveness of damaging area targets, suitable for both single and multiple shots. We won't be considering them here.
Let us consider here only an approximate method for determining the average fraction of the area damaged by several independent shots if the damage fraction by one shot is known. With an increasing number of shots n, the average fraction of the damaged area Md grows not in proportion to the number of shots n, but more slowly. There are two reasons for this:
with several shots fired, individual projectile areas may overlap;
the average Mn fraction of damaging cannot exceed one at any number of shots and at unlimited increasing n can only approach to one asymptotically.
The exact relationship between Mn and the number of shots n is quite complex. However, numerous calculations have shown that for practical needs, it can in most cases be replaced by function like
where M is the average damaged fraction of a single shot.
Formula (I.36) is not accurate enough if the target size is large compared to the dispersion area of the projectiles.
I.4 Self‐assessment Questions
1 What is the difference between the terms “effectiveness of the damaging effect” and “battle effectiveness”?
2 List the main types of ammunition and their damaging effects.
3 Describe the main types of degrees of damage.
4 Name and describe the generalized characteristics of damaging effects for contact ammunition.
5 What is the accumulation of damage?
6 Name and describe the generalized characteristics of the damaging effects of remote munitions.
7 How do the specified target sizes differ from their actual sizes for contact and remote ammunition?
8 List the kinds of targets and typical efficiency indicators of their damage.
9 List the main reasons for the dispersion of the ammunition when firing.
10 What is a picture plane and how is it built?
11 What's the dispersion law of ammunition when firing?
12 What are the similarities and the differences between systematic and group shooting errors?
13 What kind of shots are defined as dependent and why?
14 Why are probable deviations rather than standard deviations used in efficiency calculations?
15 How are generalized damage characteristics used in calculations of battle effectiveness?
References
1 1 Balagansky, I.A. and Merzhievsky, L.A.
