The most important objectives of ballistic investigation at the present time are those concerned with uniformity. It is of greater practical interest, for example, to learn how to cause the gun, as a heat engine, to execute its cycle uniformly from round to round than it is to increase, or even to maintain, a particular thermodynamic efficiency. Statistical evidence indicates that such development is a primary requisite, in many cases, for marked improvement in the precision of gunnery, however good projectile design and nominal exterior ballistic performance may be. This refers to normally accurate conditioning with respect to fire control and does not mean, of course, that moderate or large-range deviations always originate in variations other than those of flight characteristics. It is rather that deviations from exterior uniformity of the latter nature are usually the direct result of some abnormality developing initially within the gun.
One of the difficulties in establishing satisfactory ballistic engineering practice occurs through the frequent necessity for acceptance of inadequate sampling in the essential experimental work. The number of specific trials which can be made per condition is sometimes too small to secure very reliable results, particularly in the case of large-scale ordnance. This is not entirely a matter of prohibitive expense. For example, a practical limitation may be introduced by a rapidly changing status of the bore as the number of cycles is increased. The excessive rate of change of bore contour and the extreme nature of the erosion phenomena associated with the gun, comparing other heat engines, are a direct result of the tremendous rates at which energy is dissipated. During the working cycle of a major-caliber gun the average “horsepower” of the system may have a value of the order of that for the combined power plants of all of the ships of the United States Navy. For modern, high-power guns it is perhaps thirty or forty horsepower per pound of motor. Automatically, and essentially beyond control, there occur in connection with the various phenomena differences of state which are important and which are not necessarily subject to reliable compensation in the interpretation or prediction of results. The numbers of rounds which may be taken in small groups as identically conditioned are therefore somewhat limited.
As an example of one statistical dilemma, it may be noted that the contribution to interior ballistic dispersion of factors not inherently associated with any one powder as distinguished from another is often sufficient to mask such variations as might be attributed to differences in powders. Tests for uniformity can thus be significant, under those circumstances, only if very large samples are fired; that is, tests to distinguish powders as of good or of poor regularity,* etc. But tests of quite satisfactory magnitude may not only introduce appreciable probability of varied conditioning during their development, but will also sometimes seriously deplete the supply of material in the lot.
The preceding notes with respect to the sampling problems of large-scale ballistics are possibly sufficient to emphasize the practical importance of having a substantial basis for the association of experiments conducted at reduced scale or at different scales. It is unavoidable that the engineering structure of ballistics shall depend to a considerable extent on empirical summaries. Except for the solution of certain problems of exterior flight under a limited range of conditions, for which the dynamics of the spinning top provides a close analogue, there is little in the way of normally complete analytic reduction of the principal ballistic problems. Conditions usually employed are sufficiently extreme, in fact, to render the ordinary machinery of dynamics inadequate as a vehicle for rigorous solution and, in most cases, difficult of statement. Treatment of ideal or simplified special cases is not often of great practical value because of the extent of departure from actual experience which is necessary in order to accomplish reduction.
As is well known, it is occasionally possible to facilitate the arrangement of empirical summaries through the identification of certain ratios as independent variables, the latter being dimensionless combinations of the significant specific variables of condition which, together, determine the characteristics of interest. A representative coefficient is accordingly defined as a function of the ratios and, if actually descriptive of the general range of observed data within the order of experimental error, the conditions for reduced-scale or model tests are prescribed for a limited field. This procedure is typical of much engineering analysis, particularly in applications of hydrodynamics. If properly verified, it affords a valuable condensation of information and a convenient graphical basis for continued modification in accordance with latest experimental averages.
Penetration of Armor Plate
The curves of Fig. 1 define an armor penetration coefficient, a function of the
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ratio of plate thickness and projectile diameter and of the angle 0; it represents limit velocities (Fj) required for complete penetration of plate of thickness e, with obliquity of impact 6 (the angle between tangent to trajectory and normal to plate) when attacked by a projectile of diameter d and weight M. This coefficient
- f(e/d,q)=KM1/2Vt cosq/e1/2d
was developed through the assumption (of the essential variables as)
Vt cos q=Vn(M,q,e,d,To)
in which T0 is a quantity having the dimensions of energy, K=T0/ed2 representing the specific quality of the plate; similar projectiles are implied. The derivation on this assumption has been outlined in an earlier discussion.3 It is more general than the first coefficient defined therein, since recent experiments have shown that it is capable of predicting the order of results with dissimilar projectiles, although the latter may require moderate adjustment of the coefficient curves for precise representation.
It will be noted that the condition for extension in scale (with one class of plate and projectiles) is that the ratio of plate thickness to projectile diameter be maintained at the same value and, also, the obliquity of impact. It is required to determine one set of experimental F curves for each class of projectiles and of plate characteristics. Within a reasonable range of conditions, model tests with thinner plates and smaller projectiles may be accepted as reliable criteria of the results to be expected for the same ratios at full scale. The application of the empirical framework employed is accordingly not merely as a basis for general representation of penetration data but also in prescribing what section of data (from previous experiments) is most legitimately used (i.e., similarly conditioned) for direct comparison in order to improve the statistical significance of a particular test. Its reliability is gaged entirely by the practical results of repeated trial. The form indicates, however, the nature of an appropriate distribution of qualifying conditionings for such trial. The dispersion of plotted points for the present collection of experimental data is measured by a probable deviation from the curves of 2-4 per cent; this is fairly small in view of the relatively large probable error of any individual limit determination, particularly at the higher obliquities. The dispersion is, for example, very much less in the case of the points for normal or nearly normal impact.
In addition to the data which determine the limit velocity functions, there has been obtained a series of values of velocity remaining after the projectile has passed through armor plate. These can now be measured with satisfactory accuracy by oscillographic registration of the times at which sets of double contact screens behind the plate are bridged by the projectile, and they define the absorption functions. Energy loss increases and momentum absorption decreases as the striking velocity is increased beyond that of the limit for the plate. From the limit and the absorption functions it has been possible to establish a laminated or divided-armor coefficient. The latter expresses the relative effectiveness of various armor distributions with respect to solid plate of equal weight.
The following penetration formulas have been more or less generally used in this country or abroad for the representation of plate data and are reviewed at this point in order to facilitate comparison with the preceding description.
- T=K1ed2(e/d)1/3/cos2q ;
or V = K1 t e1/2d(e/d)1/6/M1/2cosq
(III) V1=Ce*7d*75f1(q)/M1/2
(VI)Vt= k(de2/M)1/3 (for normal impact).
The notation is that previously given, k, K, and C indicating constants and T the kinetic energy at impact. It will be observed that the first expression, (II), which originated with the firm of Krupp, is stated in a form somewhat similar to that defining the coefficient (I), but restricting the latter by analogy to f= (e/d)1/6 and no variation of the coefficient with q (the obliquity) is assumed. The formula (III) is essentially that of J. de Marre, but with the introduction of the term f1(q). The quantity C is taken to be a characteristic constant representative of specific plate quality. This develops from the assumption that the striking energy required for complete penetration is proportional to some power of the projectile diameter and of the plate thickness. It is now identified empirically by the exponents noted in the above form. The use of such a set of constants, if of inexact value, or if more fundamentally incorrect because of a requirement for series representation, will result in C values which vary among test combinations of d, M, and e. This would occur not because the plate quality actually varies to that extent but because, in effect, C is not constant over a variety of test conditions. As has been observed heretofore4 the formula (IV), which was stated by Lieutenant Colonel Moisson, may be restated (for q = 0) in accordance with the coefficient (I), in which case
f(e/d) = ( e/d)1/6*(M/d3)1/6
Inasmuch as the second factor in this expression is a constant for similar projectiles, the Moisson formula is a special case of that of Krupp.
Erosion Rates and Life of Gun
Although the modern gun employs what appears to be a highly effective gas seal around its piston, in the form of the forced and engraved rotating band, there is, nevertheless, a very small annular orifice during certain phases of the cycle. The point at which maximum pressure is attained corresponds to a projectile displacement of several calibers, ordinarily. Immediately subsequent to the full engraving of the band, a displacement of a few inches at most, the gun tube begins to expand appreciably from the position of close contact with the engraved surface. Thus from a projectile orientation near the origin of bore until it is well beyond the position of maximum pressure, the seal will not be tight. The same thing is true as the projectile reaches the vicinity of the muzzle, but for a different reason. The very great quantity of gas which expands from the muzzle orifice at extremely high velocity as the rear of the band breaks contact with the bore, causes rapid erosive wear of the bore surface at the muzzle. As this develops farther in the bore with each firing, the point at which the high-velocity expansion becomes seriously effective progresses inward from the muzzle. It is fairly evident, then, that the region of maximum rates of wear will include both the muzzle and the origin of bore if, as herein assumed, the principal agency of erosion is the mechanical action of the high-velocity flow of hot gases over the surface, the magnitude of wear varying with the quantity of gas, the temperature of the adjacent gas layers, the velocity of flow, and the contour of the surface. The rate at the origin has been found to be a generally significant criterion of the wear phenomena throughout the bore. That the origin is the position of maximum rate, is the result of several factors of conditioning which tend to enhance the intensity of erosive action in the vicinity. The surface of the bore is here in contact with gases at higher temperatures than in any other phase of the cycle; that is, at the point where the absorption of energy from the gas in external work on the moving system is still small and thus where the temperature most nearly approaches that of combustion for the powder. Further, the contour of the bore at this point has a discontinuous slope; the presence of a sharp corner enhances the mechanical action of the gases in flow and increases the temperature reached by the bore surface, the latter because of the greater ratio of heating area to that of the section available for conducting the heat away from the “point.”
The discussion of the preceding paragraph emphasizes the primary function of increase in the temperature of combustion of the powder and of increased quantity of gases expanding and circulating at high velocity, as important agencies in producing higher erosion rates. The decrease in rate which occurs as the gun wears may be due to the fact that the bore contour near the origin gradually develops a less severe slope, not reaching the minimum diameter for a considerable distance. Not only is the mechanical action of the impinging gases less, accordingly, but the seal of the rotating band should be relatively better over the first few calibers than in a new gun. This follows from the result that the initial engraving does not cut to full depth, and the gun expansion is compensated as the projectile is displaced along the bore to points of smaller diameter, the greater residual section of band being more nearly sufficient to fill the bore. General Charbonnier has suggested a point of view regarding erosive action6 which attributes the results primarily to the secondary turbulent motion of the gases behind the projectile, and does not accept that the gas actually expands through the “moving orifice.” The latter consideration is contrary to present photographic evidence, however.
Not only is the rate of enlargement of the bore at the origin found to be a valuable index of certain characteristics of the gun system, but the total enlargement at this point is a satisfactory measure of probable loss of muzzle velocity (with a fixed charge) at various stages in life. Thus a generalization regarding erosion rates can be made to establish not only a basis for life-of-gun estimation but also for the prediction of some of the characteristics of velocity-loss erosion curves.6 It identifies a scale of “equivalent round” ratios by which the relative effect of charges other than standard (full) value may be incorporated in the gun record.
In recent summaries of enlargement data it has been found advantageous to define a wear or enlargement coefficient
(V) F(C/C0, l/d)=E/k(w/d3) rdp
E enlargement per round at origin of bore
d diameter of bore
w weight of projectile
C weight of powder charge
C0 powder charge for datum velocity
l/d length of bore in calibers
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k, p, r constants
Contours of the logarithm of this function are shown in Fig. 2. The rate E is very rapidly increasing in C/C0 but decreases with increasing l/d. The latter statement is contrary to the general impression that longer bores are associated inherently with greater erosion rates. On the basis of direct comparison of data for varied length of gun and by reference to results at reduced charge, it is justified to conclude that the moderate increment in time interval in the bore (either for the reduced velocity or because of the longer bore) is insignificant in effect comparing that of even a slight change in charge. In other words, the important thing about a long-bore gun is that, because of the additional travel and therefore greater energy extraction, it is possible to reduce the charge (the quantity of gas) and still have the same muzzle velocity.
By introducing an appropriate assumption with respect to permissible total enlargement (as a function of bore diameter), which is consistent with life-of-gun—enlargement data, a relative life formula is obtained. It has the form
(VI) L1/L2=P(x/y)K(l1/l2) U(d1/d2)
x and y being C/C0 values for the two guns. Fig. 3 shows the equivalent graphs. The function Z1/Z2 of the figure is identical with the product P(x/y) * U(d1/d2). The scale to be used on the various vertical lines through the d1/d2 values is calibrated by the intersections of the sloping lines radiating from a point at the left of the figure. Each of the latter corresponds to a specific value of the relative Z scale.
The above enlargement rate and life formulas do not include the pressure explicitly, a fact consistent with recent experimental results.
The following life formulas are here noted for comparison and reference:
(VII) L=K1/p1*7V2d(d—K)
(VIII) L=AxyCV2/d2l
x=x(p); y=y(d); C projectile weight/d3; A determined by properties of gun steel and rotating band; K1 constant. The first of these, (VII), was stated by Lieutenant Colonel Jones of the British Army. The second was proposed by Captain Justrow.7 Both forms define life distributions differing considerably from those of Fig. 3 and the present data for U. S. naval guns. It will be noted that the term CV2/l in the Justrow formula appears in a sense which is inherently reciprocal, and as a result the pressure function x(p) is probably distorted in representing a set of actual data. Application of this expression to conditions differing appreciably from those used in the derivation of its constants would therefore be questioned.
It has been proposed by Commander G. L. Schuyler, U. S. Navy, to substitute for C/C0 of the enlargement rate and life functions a term in velocity in order to gain the convenience of a formula referred to gun data most easily obtained (V, l, d, w). Having determined, by a method of successive approximations, a consistent set of slightly varying exponents, he has obtained an equivalent tabular representation of observed service-charge erosion rates which is in excellent agreement. The summary can probably be generalized to provide satisfactory representation of rates for reduced (and excess) velocity values, in which case it will be essentially equivalent to the F function of Fig. 2. A similar rate formula may be had by substituting a linear expression in V/V0 for C/C0 and writing the empirical equivalent of the curves of Fig. 2. The C/C0, V/V0 line of Fig. 3 is a close approximation for most guns, being a mean value. The result is
(IX) E = K(w/d3)q(d/l)s(V-k)r
K, k, r, q, s constants.
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In order to complete the collection of erosion and life formulas8 of possible interest to the U. S. naval service, the following additional empirical expressions are listed:
Erosion curves for service use have been obtained in these forms at the Proving Ground, the probable values of the constants determined by systematic adjustment with respect to Proving Ground and fleet data.
1 An outline suggestive of the fluctuation in dispersion measure to be expected among small samples selected at random from the same lot of material may be had by collecting certain data regarding differences. Although the least reliable of all dispersion criteria when applied to a specific small group of observations, the maximum difference in the sample, or “range,” is a fair index with respect to distribution characteristics among many samples. The probable difference (e.g., in muzzle velocity) for two rounds is 
2 r, r being the probable deviation of a single value from a large aggregate mean
r=.6745
, = 
2/n
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(the standard deviation). The probable deviation may also be obtained from (see Note 2 following)
d(r) being a deviation from the large group mean, and Sp-1=0P—0P-1, etc., successive differences (of observations). In the case of small samples, the observed standard deviation is less than the probable actual value of s, in the Pearson ratio (n — 2)/n a result of a tendency to deviate from a normal frequency distribution as the number of individuals per group is decreased (the large aggregate from which the samples are selected being normal). The observed dispersion measures will accordingly be somewhat less in the average than the true value representative of the system. The probable maximum difference among n rounds may be estimated by means of the value of p from pn(n-1)/2=.5 there being n(n-1)/2 differences, and the probable maximum is that magnitude as likely to be exceeded (at least once) as not. If the distribution of differences were normal, the probable ratio of maximum separation to probable (observed) deviation, P/r0, might be obtained from a normal probability table, entering with the computed values of p. For the purpose of this discussion (with n
8 —10) it is assumed that the approximation is adequate; if n = 8, P/r0 = 4.72, (P/s0 = 3.99), etc. The corresponding estimate of P/r is then 4.1. In about half of all 8-round samples examined, it is thus expected that the maximum separation in the group will be more than 4.7r0 (or than 4.1r). (Values of this ratio may be compared with the patterns or “rejection criteria” of Chauvenet, Vallier, Mazzuoli, and others for various n; the latter are defined by the probable number of rounds excluded rather than by a constant probability of including all.) If the probability of occurrence of a maximum separation equal to or less than k(N)r0 is changed from 1/2 to 9/10, the value of k(N) is about 6.1. Similarly, another tenth will have extreme separations less than about 3.7r0. Yet these differences in uniformity measure may not be interpreted as showing distinguishing characteristics of the material tested by firing eight rounds. The point may be further illustrated by reference to a definition of the standard error of the standard deviation s This is obtained as
If n = 10, Js = s/3.7. A 25-30 percent increase in dispersion measure is thus so frequent that no rejection limit within 1.25sa, (sa for average material) could serve any very useful purpose in an attempt to identify good materials based on a 10-round test. In fact, a fraction of the order of one-fourth of all such samples would involve fluctuations sufficient to show' at least +15-18 percent increase. Yet there is no legitimate criterion for distinguishing the assumed inferior powders from the results of such a sample even though the average dispersion (sa) should be adopted as the high limit. The fluctuation range in which 99.7 per cent of all 10-round samples may be expected to occur has an upper limit of the order of sa +3Js or about 1.8s for n = 10. The (a posteriori) probability is clearly not high that the “cause” of a particular small sample dispersion of large value, for example, a mean deviation equal to the limit prescribed for rejection, ±(JV+K), is actually a deviation in inherent quality represented by a full history performance JV³(JV+K). (The average full history dispersion of the class is JV.) The following possible alternatives may be noted as potential explanation of the observed result
there being a possible sampling fluctuation from the respective full history means in each case sufficient to give the result (JV+K) for the specific small sample. If the probability of each status C(6) is p(6), and the probability is P(6) that, if the status existed, the observation for one sample would be ±(JV+K+1/2), then the probability that the powder actually has a representative dispersion J ³±(JV+K), is of the order
Substituting estimated values of the individual probabilities, corresponding to a sample of size 5-8 rounds, and assuming values for probable sampling (fluctuation) deviation and probable actual deviation in powder uniformity which are fairly descriptive, it is obtained that the probability of the significant cause noted would lie in the range 1/4 to 1/10. The magnitude of K arbitrarily adopted for this illustration is such that a fraction a little less than 1/10 of samples fired would prescribe rejection if taken rigidly. A corresponding uncertainty exists regarding the significance of the samples showing acceptable dispersion measure. The probability of the sufficiency-of-quality-deviation cause increases, of course, as the observed dispersion assumes values of larger deviation from the acceptance limit, but is inadequate to provide a substantial basis on which the interests of economy or precision are really served by sorting powders for uniformity by means of very small sample firing tests. If circumstances render a uniformity criterion imperative, then it is essential to recognize the statistical requisites and to employ great care in conditioning the long series of rounds.
2It is of possible interest to note the significance of a considerable departure from unity in the value of the ratio (r0)2/r0. Such a deviation, the ratio being computed from data for a large number of observations, indicates the presence of a systematic factor in the experiment, such as the erosive wear of a gun and loss of velocity, relative motion of target, etc. The order of magnitude of the effect may be estimated as that value of a systematic correction which, applied to the series of observations, reduces the two measures of probable error to approximate equality.
3L. Thompson and E. B. Scott, “Penetration of Projectile.... Memorial de L’Artillerie Fran- caise, T. VI, 1927, (No. 4).
4Note (N.D.L.R.) p. 1258, reference (3).
5C. Cranz, (Interior) Ballistics V. 2, 1926, pp. 157, 433.
6See formula XIII.
7C. Cranz, pp. 158-9.
8Calibration scales for the graphical summaries, and constants of the formulas, may be obtained for service use from the Bureau of Ordnance, Navy Department.