In the Naval Institute issue for January, 1919, whole No. 191, is published an article by Captain E. F. Eggert, Construction Corps, U. S. Navy, descriptive of a method of computing trajectories by mechanical integration. The method is essentially similar to that already in use by the Ordnance Department of the army, and now being adopted at the naval proving grounds for long-range work. The present system was developed by Professor F. R. Moulton, Professor of Astronomy at Chicago University, while serving as major in the ordnance reserve corps of the army during the war. And it is gratifying to find that from within the regular navy should come a scheme whose general principles are identical. Major Moulton's method, however, has several refinements of detail so that it will be of interest to reproduce it. Likewise Professor G. A. Bliss, at the Aberdeen Proving Ground, has developed a system of differential corrections to the undisturbed trajectory from which so-called weighting factor curves are drawn. From these differential corrections and weighting factor curves the calculated trajectories can be corrected accurately for all variations from standard conditions—muzzle velocity, angle of departure, wind, density of the air, and particularly wind and density which vary at different altitudes.
The reasons why the old methods of Alger and Ingalls must be discarded for long-range work have been clearly shown. The quantities ? and fa are not known, and any assumptions of their values are bound to be in error. The only way of finding them is by calculating backwards from observed ranges, and even this cannot be done accurately unless the observed ranges are corrected for all variations from standard conditions. By mechanical integration these errors are entirely eliminated and the method is equally accurate for all angles of elevation from 0° to 90°.
The Three Basic Assumptions
There are three assumptions that have to be made in all work in exterior ballistics.
1. It must be assumed that the axis of the shell stays close enough to the tangent to the trajectory to insure that the resistance of the air acts along this tangent. We know that this is not the case in the horizontal plane for we get the phenomenon of drift, but by the present method of calculation, trajectories have been plotted and found to agree accurately with the trajectories plotted from observations of the bursts of explosive shell timed to burst at various points in their flight. In the vertical plane, therefore, this assumption appears to be generally true.
2. It must be assumed that the experiments from which the resistance of the air has been calculated are accurate, and that results hold for all types of shell provided the proper ballistic coefficient is used. No experimental data of this nature can be considered absolute and slight changes will come as new experiments are carried out. The old resistance formulae of Mayevski have been superseded by new tables in which the function is continuous. This is called the G-function and is actually the retardation caused by air of unit density on a projectile of unit ballistic coefficient divided by the velocity at which the shell is traveling. The whole table is in metric units.
3. The last assumption is that the tables for the density of the air at all altitudes are accurate. This assumption is less open to criticism than either of the above, for in the first place a great deal of experimentation has been carried out along this line, and in the second place the differential corrections take care of differences in the density of the upper air from those assumed in the original computation. This function is called the H-function. In the old method the constant f, or fa, is a mean value of this H-function.
The Computation of the Undisturbed Trajectory
These assumptions are fundamentally those of Captain Eggert with the difference that the retardation values are derived from more recent experiments. Since the G-function is the retardation divided by the velocity a new symbol has to be introduced to replace R, the retardation. This symbol has been denoted as F and is equal to For any shell, whose ballistic coefficient is C, at any height, and traveling at any velocity, also FV= R. Using the usual system of coordinates with the X-axis horizontal and positive down the range and the Y-axis vertical and positive upwards, let us denote the accelerations and velocities parallel to the X-axis as X" and X' respectively and those parallel to the Y-axis as Y" and Y'. Now the only forces acting on the shell in space are the resistance of the air and gravity. Resolving the retardation or negative acceleration caused by the air into its vertical and horizontal components, Rxand Ry, we get
These are Captain Eggert's equations in a slightly different form, and the only method of solving them is by mechanical integration.
Before going on with the integration process let us first examine the function F. F is a product of three things, the reciprocal of the ballistic coefficient, the G-function, and the H-function. C is known for the shell and is constant throughout the trajectory. G is determined by the velocity. Except at the muzzle of the gun the velocity itself is not known, but its two components X' and Y’ are. however, so that squaring and adding these two components will give us the square of the velocity. To simplify the looking up of G, the G-table is made with as argument and the table itself gives the logarithm of G. H is determined completely by the altitude Y' and the logarithm of H is tabulated with Y as argument. The logarithm of F is the sum of the logarithms of G and H and the colog. of C. A small form, Fig. I, has been arranged so that this work can be done very simply with four-place tables. It will be found quicker to use the four-place tables than a slide rule and table of squares, for a four-place table can be worked entirely with one hand.
The process of integrating mechanically these equations with accuracy is slightly more complicated than Captain Eggert assumes. If values of FX' at the beginning and end of a small time interval, ?t, are known he states that the integral of FX’, or in other words
The first two terms only of this parenthesis are used by Captain Eggert and at least near the beginning of the trajectory, when values are changing rapidly, considerable errors will creep in thereby. To evaluate ?Y' at any time interval, the same formula is used by substituting for FX' and its difference, —(FY' + g) and its differences. And again for getting ?X and ?Y, from X' and Y’ and their differences.
It is usual to begin the computation by using values of ?t of ½ second, or ¼ second. The smaller the interval, the smoother will run the differences. As soon as the second or third differences, depending upon the number of significant figures employed, begin to run uniformly the interval may be extended. When ½-second values of ?t have been used at the beginning, the increase is made to 1 -second intervals. Again, these intervals are extended to 2 seconds when indicated by the smoothness of the results. Mistakes are detected by irregularities in the differences, and each integration can be checked from the computation (Fig. 1). The important point is to remember the integration follows the equation (b), and that the functions themselves have the algebraic sign from the original equations (a). Also, that the difference columns keep the proper algebraic sign, noting that first differences, for example, are the result of substracting the value of the function at the previous time interval from its value at this time interval. Similarly for second and other differences. Most of the numerical work can be done mentally and results written down directly. As a matter of practice the algebraic signs are rarely written into the form; they must be carefully watched, however. With experience
This is the fundamental equation of short-arc integration and is accurate for this type of function. As written in the text, the equation is simpler and perhaps more easily understood.
and X and Y are zero; likewise t = o. X gradually increases throughout the trajectory. Y increases to the maximum ordinate, Y0, and then decreases to zero. —FY' — g must be watched for algebraic sign as the shell approaches the maximum ordinate, and so must the sign of the integral of —FY' — g. Y rarely becomes zero the second time, when the shell lands, on an even time interval: double interpolation is here necessary. Likewise, for Y', X', X, T and F. By double interpolation approximate values only are obtained. Exact values are found by getting the mean value of the vertical velocity, Y', during the last time interval. Dividing the last positive value of Y by this mean vertical velocity, gives the time taken by the projectile to reach the ground from that point. This time interval added to the time of the last computed point gives the total time of flight. Exact values of X, X', and Y' are obtained by multiplying the mean rate of change of these quantities, between the last computed point and the total time, by the fractional time interval obtained above. Applying these
After the trajectory has been computed, the values of F are carried to five significant figures and should run very smoothly. They form an excellent check on the accuracy of the computation.
In working long-range trajectories the values of Y must be corrected for the curvature of the earth. Likewise the effect of gravity changes in a long trajectory, and has both a vertical and horizontal component which enter into the values of X" and Y" For these corrections, tables have been prepared which the computer keeps at hand and uses as the occasion demands.
As a matter of interest, the same trajectory used by Captain Eggert has been computed by Major Moulton's method and is shown as Fig. 2.
Corrections to the Trajectory
The differential corrections to the trajectory have been designed to take care of
(a) differences in angle of departure.
(b) differences in muzzle velocity.
(c) change in range of wind component in the plane of fire.
(d) variations in density, or ballistic coefficient.
(e) cross-wind component.
(f) effect of rotation of the earth, on both range and deflection.
The equations themselves and the adjoint system of equations used in solving them are too long and complicated for the purpose of this article. The final equations which are ordinarily used, are two
Weighting-Factor Curves
The method of handling the results of the computations of the differential corrections is to plot the values of the functions against the ratio of the various ordinates to the maximum ordinate. When it is understood that the numerical values of the different terms of equation (1) at the time t represent the effect of the unit disturbance upon that portion of the trajectory lying above the particular value of Y, it is evident that by employing percentage values of the maximum ordinate, results for different elevations may be plotted on the same sheet.
Fig. 7 is the weighting-factor curve for the density or a change in the ballistic coefficient.
The dotted line curve in each of these figures is the limiting curve approached by each particular curve as the elevation approaches zero. Such limiting curves are readily constructed and, as may be seen from the figures, are fairly suitable for low elevations, say up to 10°.
The Ballistic Wind
In order to apply the weighting-factor curves thus constructed, it is necessary to know what the wind is doing in the upper atmosphere. This information was obtained daily on the Western Front and its source and distribution became a highly organized service. A special name was given to this data, and it was called "Sondage." The method was simple, and its application to proving ground conditions presents no difficulties. For the measurement of these wind-currents a little balloon filled with hydrogen is released, and simultaneous observations of its position at given time intervals are made from two stations. With these observations the position of the balloon in space can be plotted at the end of known intervals of time. Thus the horizontal distance between any two such plotted points may be taken as the horizontal movement of the air at the average altitude of the two points, during the known time interval. Suitable scales and constant time intervals between observations enable such wind currents to be converted into velocities per unit of time, usually meters per second. At the same time the azimuths of the balloon's position give, by their difference, the direction of the wind.
In practice there are several details that are of interest. First the base line must be of known length and azimuth. While two stations are sufficient for any one flight, yet their location at the corners of an equilateral triangle, if possible, will be convenient and will eliminate the case of a surface wind blowing directly towards one observer. Telephone communication allows simultaneous readings at constant time intervals. One station is the "master" and shall be the point of release of the balloon, or the "origin." The balloons are of thin rubber and colored either red or white. When empty they are of either 6-inch or 9-inch radius. They are inflated by hydrogen until the proper "free lift" is obtained. Some very interesting data has been collected during the war on the rate of ascent of the balloons. While of course two observers make results independent of such rate, yet a working formula has been derived which makes it possible to get fairly accurate results, with one theodolite. The instruments used are special theodolites with continuous tangent screws for both azimuth and elevation. An open sight is provided for use by a second observer who helps to keep the balloon in the field of the instrument. This is important after the balloon has reached a considerable altitude, for then once lost it is seldom recovered again.
The plotting may be done in several ways: by universal drafting machine, by protractors, or by specially designed curves for solving the problem mathematically. If the base line be laid out to scale on a plotting board and graduated circles with moveable arms placed around the two stations, then by setting the arms for the azimuth from each station the horizontal trace of the balloon's position is gotten. Then with a properly graduated scale and the vertical angle from either station, the balloon's height above such horizontal trace may be found. Knowing each position of the balloon and its time interval after release, the average velocity and direction of the wind in the different zones can be quickly found. Such winds are then resolved parallel to and across the line of fire, and are at once available for correcting the fall of shot by means of the weighting-factor curves, Figs. 5 and 6.
The Ballistic Density
The ballistic density is obtained by means of observations of temperatures at different altitudes. An airplane fitted with a special thermometer is sent up and readings made at each 1000 feet up to the maximum ordinate, or as high as the machine can go. With this data and suitable meteorological tables, the density of the air at the various altitudes can be calculated. The ratio of the change of the density with altitude, to the standard density is desired in order to apply the third term of equation (1); or ?H/H=?Y. By plotting both the logarithms of the actual density and of the standard density against the altitude, it can be shown that the difference between the two curves multiplied by a constant will give the value of ?Y. Then by plotting an auxiliary curve with values of ?Y as ordinates and Y as abscissae, the ratio of the change of density to the standard density for all altitudes can be gotten. This curve is applicable to all guns firing during the time of day for which the curve holds true. For any zone the value of ?Y can be obtained, and is then multiplied by its proper weighting factor for that zone, as shown in Fig. 7. And the sum of the products gives the percentage variation by which to multiply
Before proceeding with the application of the weighting-factor curves, it should be said that the ballistic wind and density maybe determined several times during the day. At proving grounds guns are not fired on elevations which will give a maximum ordinate greater than the height to which balloons have been observed or airplanes have ascended. In practice in the field it has been found possible to extend the plot of either the balloons or the airplane observations, by extrapolation and formulae, to a considerable distance above the highest observation, with acceptable results.
The Application
The proving ground use of the method above described is mainly to provide suitable data from which to construct range tables. Suppose that it is desired to extend the range table for a 16”/45 gun from 15° to 40° elevation. With the best possible value of the coefficient of form from previous rangings, the given muzzle velocity, and an elevation of 20°, compute the trajectory by short-arc integration. Compute the differential corrections and draw the weighting-factor curves. Send up pilot balloons and airplanes for ballistic wind and density, just before firing. Then fire at least 5 rounds on 20° elevation, and make ranging observations. If the computed range is grossly at variance with the observed range, then a serious error has been made in the assumed value of i. A new trajectory must then be computed using a more appropriate value of i; and the differential corrections and weighting-factor curves must be recomputed. With these weighting-factor curves and the differential corrections, the observed ranges are corrected to standard conditions; that is, to the range with an undisturbed trajectory. Since the third term of equation (1) gives the value of ?R due to a variation of one per cent in either the density of the air or the ballistic coefficient, the difference between the corrected range and the calculated range divided by the value of this
It will thus be appreciated that for range-table computation the method herein described permits the relatively quicker Siacci method to be used, and that by the addition of the curve of ?i against ? the results are then freed of the errors analyzed so ably by Captain Eggert.
In conclusion, it may be said that several refinements have been omitted entirely, and that no discussion of the actual derivation of the G and H tables has been attempted. Further, if the method given for handling the altitude factor, fa, is not agreed in, it may be stated that a new equation for evaluating it has recently been developed, which allows a computation of ?i independently of the altitude-factor.
It would also be unfair to the originators of the method not to say that its flexibility permits of considerable modification of the basic tables and the introduction of new ones, as experimental work in exterior ballistics continues.
In general, it would seem unwise to attempt any long-range firings for range-table data without using this, or a similar method to compensate for the variations of both wind and density with increasing altitudes. It is not deemed probable that the method has any direct application to shipboard conditions.