BALLISTICS. Ballistics is a branch of Applied Physics which deals with the motion of projectiles. Its chief application is to artillery projectiles and rifle bullets, though latterly a new application has been found in aerial bombs. There are two branches of the subject; one deals with the motion of the pro jectile while it is in the gun and is called Internal Ballistics; the other, called external ballistics, deals with the motion after the projectile has left the muzzle.
There are two distinct stages in the motion of the projectile in the gun; the first stage begins at the moment the charge is ignited and ends at the moment the charge is completely con sumed; at this moment the projectile enters upon the second stage of its motion, which ends when the projectile leaves the gun. As soon as the charge is ignited gas is given off by the burn ing propellant and the pressure in the chamber rises; the rate of burning of the propellant increases as the pressure increases, so that gas is more and more rapidly evolved. The rise in pressure is consequently rapid at first. The chamber is sealed by means of a copper (or other soft metal) band, known as the driving band, which surrounds the projectile near its base. As soon as the pressure in the chamber is high enough to overcome the shearing resistance of this band the projectile begins to move and the driving band, though still effectively sealing the bore, is engraved by the rifling. The effect of the movement of the projectile is to increase the volume behind it and thus to check the rapid rise in pressure; the result is that when the projectile has moved a short distance the pressure has reached its maximum value and thereafter decreases.
The rate of decrease in pressure is slow at first, then more rapid, and finally slower again. The end of the first stage, that is to say, the completion of burning, may occur at any point in this sequence of pressure changes, depending mainly on the size and shape of the powder grains. The time, as well as the position of the projectile in the bore, when combustion is com plete, varies widely according as different guns and different am munition are used.
In the second stage of the motion the gas expands under its pressure, which is now falling off more rapidly; the projectile continues to accelerate but the acceleration continually de creases. Finally, the projectile leaves the bore, still slightly accelerated by the escaping gas, some two or three hundredths of a second after the charge is ignited. In addition to the forward motion a spin is imparted to the projectile by the action of the spiral grooves of the bore (the rifling) on the driving band.
During the part of the motion just described the effects of the weight of the projectile and the air resistance are small com pared with the enormous force of propulsion exerted by the gas.
When the projectile has left the muzzle, however, these effects become all-important.
The chief effect of the weight of the projectile, i.e., the force exerted on it by gravity, is to curve the trajectory downwards; it also tends to retard the projectile during its upward motion and to accelerate it during its descent.
The effect of the reaction of the air is much more complicated. Not only does it tend to retard the projectile in its motion along the trajectory but it also has the effect of diverting the axis from the trajectory so that the nose tends to move upwards relative to the centre of gravity. This upward tendency is counteracted by the spin of the projectile and the combined effect (as may be seen by experiment with a gyroscope) is that the nose moves laterally. In fact, as the projectile moves along, the axis pre cesses round the trajectory, the nose being now above, now to one side, now below, now to the other side of the trajectory, and so on. This oscillatory motion of the axis gives rise to air reactions which tend to reduce or damp the motion; but owing to the unsymmetrical nature of the latter the mean position of the f is a constant called the force of the explosive. Let be the cubical capacity of the chamber of the gun Sl the cross-sectional area of the grooved bore x the distance the projectile has moved along the bore at time t 8 the specific gravity of the propellant.
Then the volume occupied by the gases at time t is Most authors make the approximation a= i/o thus making rt constant; the error thus involved is generally small.
T and can now be eliminated from equation (I), which then takes the form = (4) where L is replaced by the kinetic energy of the projectile, mass m, velocity u, and y = This equation, which is called the fundamental equation, was first given by Resal in 1864 and is still in general use. The most serious discrepancy between this theory and actual results arises from neglecting the kinetic energy of the charge. The usual method of correcting this is to replace m in equation (4) by m+qw where q is a fraction which is determined empirically. Other discrepancies arise from neglecting: (a) Rotational energy of the projectile.
(b) Kinetic energy of the recoiling gun.
(c) Resistance of the air to the projectile.
(d) Work done by friction between the driving band and the bore.
(e) Loss of heat by conduction through the gun.
(f) Dissociation of the gases.
(g) Variation of the specific heats.
Allowance for (a), (b), (c) and (d) is generally made by using a further modification of the mass of the projectile, replacing m by where ΅ = i(m+qw) and i, called factor of effect, is some what greater than unity.
Allowance for (e), (f) and (g) is generally made by modifying the value of f and by taking a mean value of y.
The Dynamical Equation.For the solution of the problem two more relations are required between the variables of equation (4). One of these is obtained from the Newtonian relation between force and acceleration, namely, du (5) dx The other is derived from the laws of combustion of the .pro pellant.
Combustion of the Propellant.Here again simplifying assumptions are made; unlike the foregoing, however, they have considerable experimental verification.
The assumptions are: (I) That the whole of the charge is ignited simultaneously; That the combustion of each piece of propellant takes place in parallel layers (Piobert's Law, 1840); (3) That the rate of burning along the normal to the surface of each piece is proportional to some power of the pressure. (Vieille's Law, 1893.) Let 1 be the least distance, measured along the normal to the surface of 1, piece, for which the piece is completely consumed and let yl be the distance burnt at time t; then if the charge consists of pieces of equal size and similar shape, z=o when y=o z = I when y= 1.
Modern propellants are made in simple geometrical shape; for such f(y) is generally a linear function of y.
The general form of z in these circumstances is z=ay{ I (6) where a, X and a are constants which satisfy the condition a(I X--a) = I.
The following table gives a, X and a for typical shapes of propellant: a X a Tube, annulus + I I o length 2 k k diameter or side + 2 I thickness 2l k k(2+k) k(1+k) In most practical cases I/k may be neglected compared with unity; in these circumstances z is at most a quadratic function of y. From the last assumption, Where w and it are constants depending solely on the nature of the propellant. The values of n determined by various authorities range between o-5 and 1.
Charbonnier expresses the relation between z and p in slightly different form.
Then In deducing these values i/k was neglected compared with I in the case of cord and in the case of disc.
When n = I a complete analytical solution can be obtained; for this reason many authors make the approximation n= I and adjust the constant w (or A) to give true rate of burning at mean pressure; the error involved is thus reduced to a minimum, but for some propellants, notably nitro-cellulose, it is never theless considerable.
In measuring rapid changes of pressure the inertia of the mov ing parts of the recording system may be the source of serious errors. To overcome this Petavel (19o5) replaced the crusher by a strong tube of steel which was compressed longitudinally by the action of the gases on a piston fixed at one end; the actual compression was of the order of o.oi inch, and was recorded optically.
where pl is the final pressure (i.e., when z= 1). Equation (8) then becomes ap = Ap1p" 4(p/p1) dt By comparing the relation obtained experimentally with this relation between p and t the values of A (whence w) and n can be determined.
This method is also used to determine Ack(z) for pieces of irregular shape of propellant for which it is already known. The value of y at the high temperatures encountered in explo sions is not easily obtained by experiment. A mean value, deduced from thermo-chemical theory, is generally adopted. Thus, for cordite M.D., Henderson and Hasse give the fol lowing values: Absolute Temperature y 3,000 1.186 2,500 1.210 2,000 1.242For this propellant the mean value 1.2 is usually adopted. The most reliable experimental verification of the theory is obtained by comparing the calculated with the observed muzzle velocity in a variety of cases.
He suggests a propellant which is made up in layers, the core having high explosive force and high rate of burning, the layer enclosing this having these prop erties in somewhat lower degree and so on, the outmost layer having comparatively low explosive force and low rate of burning.
Such variations in these properties are obtained by modifying the chemical constitution of the explosive; to produce such a propellant which is sufficiently stable for magazine storage is a problem of no little difficulty in explosive chemistry.
Hutton's method consisted in measuring the velocity of the cannon-ball at the muzzle and at a known distance from the muzzle. The former was determined from the recoil of the gun, which was hung from suitable supports; the latter was deduced from the swing of a ballistic pendulum which received the ball.
If m= mass of projectile = muzzle velocity = velocity at distance x from muzzle, the mean drag, R=-- Provided x is reasonably small this value may be taken as the actual drag for mean velocity, v = z (v1+v2).
From the time of Hutton to the present day experiments in America and on the Continent to determine the drag have been based on this principle, namely, to determine the velocity at two points a known distance apart on an approximately horizontal trajectory. A large number of instruments for measuring the velocity of a projectile between two given points have been in vented during this time; for an exhaustive account of these see Negrotto, Balistica Experimental y Aplicada.
Since 1865 experiments conducted in England have been based on a different principle. In that year Bashforth invented a chron ograph which enabled him to measure the times at which a projectile passed a number of equidistant points along an ap proximately horizontal trajectory. The velocity and retardation were deduced from these times by the method of finite differ ences.
The most exhaustive experiments yet undertaken are those of 0. von Eberhard, at Krupp's in 1912; projectiles of many shapes and sizes were used and the drag as a tabulated function of velocity was deduced for each type. At the present time (1928) experiments are in progress in America, with a new type of chronograph. The projectile is magnetized and fired through solenoids placed a few feet apart, thus inducing a minjite elec tric pulse in each. These solenoids are connected through an amplifier to an oscillograph which records the time interval of the projectile between the solenoids; from this the velocity can be deduced. A similar method, but based on the Bashforth prin ciple, is used in England.
In France, in 1917, experiments were undertaken in a new direction. Instead of using a moving projectile the thrust on a stationary projectile in a current of air moving at high velocity was directly measured. Similar experiments have been conducted in England and America. The projectile (or scale-model thereof) is supported by means of a steel spindle fixed to the centre of the base in prolongation of its axis; this spindle is attached at its other end to a mechanism designed to measure the thrust on the projectile. Compressed air issues from a reservoir through an orifice, which is so designed that a steady stream flows past the projectile at high velocity. The temperature and velocity of the air in the stream are calculated from the pressure and tempera ture of the compressed air. The result of all such experiments is that numerical values of the drag are determined at a series of velocities for various types of projectile.
Dimensional considerations lead to the form R= (v/a) for the drag, in air of density p, on a projectile of diameter d moving with velocity v, a being the velocity of sound in air. The function Ay! called the drag coefficient, is plotted for various shapes of projectiles in fig. 6, which is based on the results of Eberhard's 1912 experiments.
It is evident that the drag coefficient is a different function of v for each shape of projectile. For pointed projectiles, however, as a first approximation, the drag can be expressed in the form R= (v/a) where f(v/a) is the same function for all shapes, k is a constant which is determined by the shape and o- is another constant (introduced by Bashforth) to allow for unsteadiness in flight.
Writing F(v) = (v/a) M= mass of projectile C= kapce (C is called the ballistic coefficient) the retardation of the projectile due to the drag can be expressed in the form The Differential Equations of Motion.To obtain a first approximation, the projectile is treated as a heavy particle mov ing under the influence of the drag and gravity.
Let x= the horizontal distance and y = the vertical distance of the projectile from the muzzle at time 1, v= the velocity at that time and inclination of v (fig. 7).
Then the equations of motion of the projectile are The functions 1(u), T(u), S(u) and A (u) can be calculated by any suitable process of numerical integration and tabulated. The values of X and ,u chosen by Siacci are such that 9p Solution by Small Arcs.--Modern gunnery demands a fairly accurate knowledge of the co-ordinates of all points on a series of trajectories for which O ranges from o° to 9o°. In order to furnish this knowledge it is necessary to solve the differential equations by small arcs, due al lowance being made for variation in air density (and possibly elas ticity, by varying a) with height. The problem may be stated as follows : Given the elements x, y, v, 0, t, and C at the beginning of an arc of the trajectory, and one element (the independent variable) at the end of the arc, to determine the other elements at the end of the arc. In French methods 0 is chosen as independent vari able; in England and America t is used.
The density at height y is generally assumed to be of the form p= Poe where is the density at mean sea level and h is a constant. The retardation is then where is the value of C at mean sea level. In existing methods variation in a is not allowed for; there would however be no difficulty in including this variation if it could be expressed either numerically or analytically as a function of y.
Then P1= - - etc., -- T and with the aid of these the effect of any known wind structure can be determined.
The effect of a wind at right-angles to the plane of the tra jectory is determined as follows: Let the wind Ow be blowing between heights y and Y cor responding to the time interval t to T on the trajectory. Relative to the air the projectile is deflected horizontally through an angle bw/q at time t and therefore through a lateral distance (X x)8w/q at time T. During this time the displacement of the air is (T 1)8w; hence the total lateral displacement at time Tis in the direction of the wind.
It follows that the weighting factors for the periods /1-12, etc., are of the form (X T t (X x)/q Range Tables.The chief object of all such calculations is to produce a range table which gives, in convenient form, all the information concerning the ballistic properties of an equipment that the practical artillerist is likely to require.
A limited number of experimental trials with the equipment are first undertaken; these are analysed to determine the value of C at a number of angles of departure. Owing to the approxi mation made concerning k and to the variation of a the value of C generally varies from trajectory to trajectory; it is therefore necessary to carry out trials at a number of angles of departure. The calculation of the range table is then based on the values of C thus determined.
A secondary reaction, known as the Magnus effect, can be represented by a force at right-angles to the plane of yaw. This force, which does not generally act through the centre of gravity, is the result of the unsymmetrical distribution of pressure arising from the circulation of air round the projectile. There are, in addition, two damping couples which tend to reduce the spin and the angular motion of the axis respectively.
Experiments to determine these forces and couples as functions of velocity and yaw are described by Fowler, Gallop, Lock and Richmond (see Bibliography). Quantitative knowledge is at present far from complete and the principal effect, the lateral drift, is determined for each equipment by experimental trials. It is a matter of experience that the drift is represented fairly accurately- for the formulae D= tan Oo where K, and are constants determined by experiment, X is the range, T the time of flight and Oo is the angle of departure.