Damaging Effects of Weapons and Ammunition. Igor A. Balagansky

Damaging Effects of Weapons and Ammunition - Igor A. Balagansky


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of firing and the dispersion along the X‐axis characterizes the deviation along the firing line, the Y‐axis thus characterizes the deviation across the firing line.

Schematic illustration of setting up the coordinate system on the picture plane.

      Source: From Wentzel [2].

      (I.10)normal phi left-parenthesis x comma y right-parenthesis equals StartFraction 1 Over 2 pi normal sigma Subscript x Baseline normal sigma Subscript y Baseline EndFraction e Superscript minus one half left-bracket left-parenthesis x minus x overbar right-parenthesis squared slash normal sigma Super Subscript x Super Superscript 2 Superscript plus left-parenthesis y minus y overbar right-parenthesis squared slash normal sigma Super Subscript y Super Superscript 2 Superscript right-bracket Baseline comma

      where x overbar comma y overbar are the coordinates of the dispersion center. They characterize a systematic firing errors. If there is no systematic error, these values are equal to zero; σx, σy – standard deviations in the 0X, 0Y axes, respectively.

      In the practice, it is not the values of σx, σy that are usually used, but the so‐called probable (median) deviations of 0X, 0Y axes, which are denoted by Ex and Ey , respectively:

      Median deviations are convenient because they correspond to the principal half‐axis of the dispersion ellipse, within which exactly half of all hits lie. The law of dispersion, in this case, takes the following form:

normal phi left-parenthesis x comma y right-parenthesis equals StartFraction rho squared Over normal pi upper E Subscript x Baseline upper E Subscript y Baseline EndFraction e Superscript minus rho squared left-bracket left-parenthesis x minus x overbar right-parenthesis squared slash upper E Super Subscript x Super Superscript 2 Superscript plus left-parenthesis y minus y overbar right-parenthesis squared slash upper E Super Subscript y Super Superscript 2 Superscript right-bracket Baseline period

      It is usually assumed that errors along the firing line (Ex) do not depend on errors across the firing line (Ey). Therefore, these values can be considered independently of each other.

      I.3.4 Scheme of Two Groups of Errors

      Random shot error consists of several components of random errors: target coordinate error, correction for meteorological and ballistic factors, technical dispersion caused by differences in weight and shape of projectiles, etc. In a single shot, each component of the total shot error is repeated only once, in which case it is said that there is one group of errors. When several shots are fired at the same target, some components of the total shot error caused by common sources are repeated, while other components caused by different sources are not repeated. For example, when shooting at an unobserved target from the same gun with the same scope setting, all the shots will repeat the target coordinate error and will not repeat the error caused by variations in the shape and weight of the projectiles. Please note that in this case, the target coordinate error, although repeated from shot to shot, is an accidental rather than systematic error.

      If we return to the case of firing from one gun at an unobserved target (uncontrolled firing), we can consider two groups of random errors: data preparation errors (Ex0, Ey0) – group error, and technical dispersion (Bd, Bs) – individual error. Then the total median error of the shot along the firing line

      (I.12)upper E Subscript x Baseline equals StartRoot upper E Subscript x Baseline 0 Superscript 2 Baseline plus Ð’ Subscript d Superscript 2 Baseline EndRoot comma

      across the firing line

      (I.13)upper E Subscript y Baseline equals StartRoot upper E Subscript y Baseline 0 Superscript 2 Baseline plus Ð’ Subscript s Superscript 2 Baseline EndRoot period

      An estimate of the values of aggregated median errors when an artillery division is firing after complete preparation of the initial data can be obtained from Table A.1 in the Appendix.

      The degree of dependence between the shots can be characterized by the correlation ratio of the hit points coordinates. In the scheme of two error groups, the correlation ratio for each pair of shots is equal to the ratio of the square of the group error probable deviation to the square of the total error probable deviation:

      (I.14)normal mu Superscript left-parenthesis x right-parenthesis Baseline equals StartFraction upper E Subscript x Baseline 0 Superscript 2 Baseline Over upper E Subscript x Baseline 0 Superscript 2 Baseline plus upper B Subscript d Superscript 2 Baseline EndFraction comma

      (I.15)normal mu Superscript left-parenthesis y right-parenthesis Baseline equals StartFraction upper E Subscript y Baseline 0 Superscript 2 Baseline Over upper E Subscript y Baseline 0 Superscript 2 Baseline plus upper B Subscript s Superscript 2 Baseline EndFraction period

Schematic illustration of effect of the group and individual shooting errors.

      Source: From Wentzel [2].