Page:EB1922 - Volume 30.djvu/432

392 Integration of the Differential Equations of the Variations. By using equation (21) with (22) and (26) we make a step-by-step numerical integration for any assumed variation. The system re- quires a separate integration corresponding to each assumed varia- tion of conditions in each trajectory.' It is always assumed that the original trajectory has been constructed and that we know its ele- ments. The integration may proceed forward or backward from any point, as at the muzzle or the end of the trajectory, where we know or may estimate the values of disturbing variations and the effects produced by them.

Bliss's Method. As may be inferred, the method for computing differential corrections, just described, involves a large amount of work. A method discovered by Professor G. A. Bliss and improved by Dr. T. H. Gronwall, in which use is made of a system of linear differential equations adjoint to the linear differential equations of the variations, as given by equations (13) and (14), reduces the work required to the extent that after the original trajectory has been computed, one numerical integration of the system will suffice for the computation of the corrections for all the variations. The method is, therefore, invaluable when a large number of differential variations are to be worked out.

Tabulated Differential Corrections in Ballistic Tables. Certain differential corrections are conveniently tabulated, in separate columns of ballistic or range tables as follows:

(a) Range and deflection corrections for the rotation of the earth as functions of the geographical latitude of the gun, azimuth of the plane of fire, and the three standard parameters of the trajectory, muzzle velocity, angle of departure and ballistic coefficient.

(6) Corrections in range for variations in assumed air density throughout layers at convenient altitude intervals.

(c) Corrections in range for a component of wind in the plane of fire, throughout layers at convenient altitude intervals.

(d) Corrections in deflection for a component of wind at right angles to the plane of fire throughout layers at convenient altitude intervals.

In addition, corrections for variations in initial components of muzzle velocity, and variations in ballistic coefficient, may be obtained by interpolation in the main columns of the tables. A variation in air density at the gun may be corrected for in this way by determining its effect on the ballistic coefficient and making the corresponding interpolation in the table. This assumes that any change in air density at the gun is accompanied by a corresponding change aloft according to the law given by the H function. Of the first list of variations referred to, more will be said below.

Effect of Wind. Any wind acting on the projectile in flight may be resolved into two components: one along and the other perpendicular to the plane of fire. It is convenient to dp this in considering the effects of winds, and we thus have range winds and cross winds.

Uniform Range Wind. Corrections due to rear or head winds may be handled by equations (21) and (22) and (26). In using these equations we must merely remember to increase or decrease the velocity with respect to the ground by the wind velocity when it is desired to get from the Tables, I. or III., the corresponding functions. Aside from this, a correction due to wind may be handled in the same manner as a correction due to variations in any of the initial conditions, air density, etc.

Variable Wind. The direction and velocity of the wind will seldom be uniform throughout the trajectory. The velocity of the wind and also its direction near the surface of the earth is frequently influenced by local causes, such as the presence of hills, trees, houses, etc., to such an extent as to give no indication of the true average values during the flight of the projectile. Under normal conditions the wind may change both in direction and velocity as we go upward. Cases in which there is complete reversal of the wind well within the maximum ordinate of the trajectory are not unusual. The change in direction may also be accompanied by a change in velocity. In the preparation of range tables it is necessary to correct in some way for the effect of this sort of wind. The method usually followed is to divide the air above the earth's surface into zones of height, say 250 metres. By observation, the direction and velocity of the wind in each zone are determined.

For any assumed trajectory let AR be the range correction of a uniform range wind of I metre per second, acting throughout. Now, dividing the trajectory into zones of height (as shown in fig. 4) let ARi be the total range effect produced by a wind of I metre per second blowing in the first zone and no wind in the other zones. The ARi correction can be computed by numerical integration of equations (21) and (22) to the limits of the zone, using the I metre wind. With the corrections found for that point, the integration is continued in still air until the projectile again enters the first zone. With these last corrections and the wind again acting, the integration is continued and the final correction AR, determined.

In the same manner the correction AR 2 for a wind of I metre oer second, acting in the second zone, and no wind in the other zones, is determined. We then have

(27) AR = AR, + AR 2 + +AR n

If ui, us, ua, etc., represent the wind velocities in the various zones,

and AR represents the total range effect due to them, we may put approximately

(28) AR = u,AR,+w 2 AR 2 +

+ ......... fn Un) AR

istic wind. It is the wind which, if

FIG. 4.

In this equation some of the winds may be rear and others head, so the terms should be taken with their proper signs. Weighting Factors and Ballistic Wind. If we place

I \ t ARl t AR2 *

(2 9) /' = AR? /2= AR- etC -

the ratios fi,/a, etc., are called weighting factors since they show the fractional part of the total wind effect that is produced in each zone. Using these factors we may write

(30) R = (.fi i+/2 2 The factor in brackets is the ballistic

blowing uniformly throughout the trajectory, would produce the same range effect that is produced by the variable winds actually blowing.

In the preceding discussion we have considered only rear or head, that is range, winds.

Cross Wind. In the discussion of the trajectories so far given, no account has been taken of forces which tend to move the projectile from the plane of fire. Aside from drift, the principal cause of deflec- tion from the plane of fire is the existence of a cross wind component. While the deflection due to drift is constant for any one trajectory for a given gun and projectile and is determined once for all by experiment, that due to cross wind varies with the velocity of the wind aS well as with the elevation and azimuth of the gun. If we let z represent the distance in metres the projectile is blown from the plane of fire at any instant by a cross wind TV, distances and winds to the right being taken as positive, we will have z', the velocity from the plane of fire, and z", the acceleration produced by the component of air resistance normal to the plane of fire. The velocity of the pro- jectile with respect to the air will be z' w.

Now it will be sufficiently exact to consider the motion per- pendicular to the plane of fire in the same manner in which we considered the horizontal motion in the plane of fire in equation (9)., remembering that the velocity with respect to the air is z' w. We may then write,

(31) z"=-E(z'-w).

Combining this equation with the relation x" = -E x' from (9) we obtain after reduction and division by x' 2 ,

(32)

Upon integration from o to t this becomes,

z' _ w^_ w_

(33) 2 - F *'

or

(34)

Integrating again we obtain,

(35) *=

which makes the total deflection at the end of the trajectory,

(36)

-7 T W V

z _ wT ___x.

In this expression w T is the total motion of the air with respect to the ground in the time of flight T. The deflection of the projectile

is less than the motion of the air by the amount ,pX, which is the

deflection at the total range X that would be caused by a change of

w azimuth by the angle whose tangent is ,.

Cross wind weighting factprs and the ballistic cross wind are determined in the manner described for range winds.

Effect of Curvature of the Earth. While in the example of the numerical integration of a trajectory given above, and also in the construction of ballistic tables, the effect of curvature of the earth is not taken into account, it would be quite possible, still retaining