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    "verified_text": "-general agreement is to be found in the treatment of exterior ballistics in various countries, but the subject of interior ballistics, which is concerned with the behaviour of the gun, the projectile and the charge while the projectile is in the bore, has led to widely different methods of treatment. i. interior ballistics the complexity of the physical laws involved makes the sub- ject a very difficult one to treat with completeness theoretically, and the aim of most methods is to produce a solution which 1s in general accord with the theoretical laws as at present known, but which contains undetermined coefficients whose values can be adjusted so as to make the calculated results agree with the observed results in particular cases. the maximum pressure inside the bore, and the muzzle velocity of the projectile on leaving the bore, are the quantities whose deter- mination 1s most important—any rational method with two unde- termined coefficients can be used to give agreement with observed results for maximum pressure and muzzle velocity in the case of one particular gun, charge and projectile. the soundness of the method employed will determine whether the results for other guns, charges and projectiles can be predicted with reasonable accuracy. in view of the divergencies between published systems, it is convenient here merely to indicate the systems of equations used, and the physical laws involved in any method, and then to give references to detailed accounts of each method. laws of burning of propellants——the actual problem of interior ballistics is, of course, that of the mode of expansion of gases under high pressure. the problem is slightly different from those ordinarily met with in the domain of physics in that the mass of the gas is constantly being added to as the propellant burns, so that the first problem to be solved is the rate at which the propellant burns, and the dependence of this rate on the existing conditions of pressure, temperature and density. modern propellants are for the most part colloids, and charges are composed of a number of grains of definite size and shape. for colloid propetlants piobert’s ‘ law of burning by parallel layers \"’ is well established. this law states that at any instant during the burning of the grain the thickness burnt through in the direction normal to the exposed surface is the same over the whole surface. the rate of burning of the thickness is a property of the propellant. most methods assume that this rate of burning is a function of the pressure only and, following sarrau (1876), use the relation cy _ bp a) where y is the fraction of the norma! thickness burnt off at time #, p is the mean pressure at that time, and b and a are constants de- pending upon the propellant. . the systems in use differ chiefly in the choice of a; some values in use are:— (1) a=1 (english and american practice; also used by char- bonnier and sugot—france). (2) a=% (gossot and liouville—france). function of form.—equation (1) yields information as to the rate at which the thickness is burnt away. obviously the rate at which the volume is being consumed, that is to say the rate at which gas is being produced, will further depend upon the area exposed to burning at any instant. thus, for a charge of given mass, one com- posed of small grains would be expected to be burnt more quickly than one composed of large grains. thus the form of the grain must be taken into account. the different forms employed may be divided into three main groups:— (1) those which burn with a continually decreasing surface. to this group belong all solid grains and short cylinders with an axial perforation. (2) those which burn with a practically constant surface, such as long thin tubes. ballistics (3) those which burn with an increasing surface to a certain stage, the grain then breaking up into other forms quite different from the original shape. an example of this type is a cylindrical grain pierced longitudinally by a number of holes. since it is the volume of charge burnt that determines the amount of gas present at any instant, it becomes necessary to determine a relation between z, the fraction of the charge burnt, and y, the frac- tion of the thickness of the grain burnt. charbonnier introduced the idea of a “ function of form.” let sp be the initial surface of the grain, and s the surface when a fraction z of the grain is burnt. then, depending upon the geometry of the grain, it is casy, using the law of burning by parallel layers to deduce a relation of the type > = so-b(z) (2) (z) is the “ function of form.” the combination of equations (1) and (2) leads to an equation adz ree eiahe _- — a of qm up ez) (3) where c is a constant. this is the required equation giving the rate of consumption of the charge at any instant. in england and elsewherea somewhat different practice is adopted. instead of considering the surface exposed at any instant it is usual to deduce, from the geometry of the grain, an equation of form of the type : z= fq) (4) and to use equations (1) and (4) in combination instead of the single equation (3), as in charbonnier's system. : characteristic equation of the gases-expansion equation.—the next step in the solution of the problem is to consider the behaviour of the gases when produced. most methods take as characteristic equation a modified form of van der waal’s equation, namely, p(v—y) = rt (5) with the usual notation, due allowance being made, in calculating v, for the space occupied by the unburnt propellant. v and 7, of course, depend upon the amount of gas produced, that is, upon z, and v also depends upon x, the distance the projectile has moved at time ¢, and upon the initial air-space. once burning is complete the relation between v and x is dv = a-dx (6) where a is the area of cross-section of the bore. some methods assume that the expansion during burning is isothermal and equation (5) 1s written | p(v—n) => (7) where a is a constant. b eceule vk others assume that the expansion is adiabatic, or partly adiabatic and partly isothermal, and write p(v~n)* =x (8) where k is some constant intermediate in value between unity and the ratio of the specific heats of the gases. whatever assumptions are made, the object is to produce a relation between p, z and x which shall hold during burning. this relation is the ‘‘ expansion equation.’’ the motion of the projectile can now be considered in two stages, namely, during burning and after burning. motion while the charge 1s being consumed.—lit is most convenient to take either z, or y, as the independent variable during this stage, as each ranges from 0 to i during the stage, and supplies definite limits for any necessary integrations. | the variables concerned are p, x, fand z, if charbonnier's func- tion of form is used. thus three equations will be necessary in order to complete the system. | . equation (3) between z and ¢, and the expansion equation between p, z and x, are two, and the third is supplied by an equation of mo- tion or energy, which, in its simplest form, can be written dx where m is the mass of the projectile. if equation (4) is used as the equation of form, equations (1) and (4) replace equation (3), and the four equations now provide the complete system for the five variables p, z, vy, x and # there is no difficulty in the integration of the equations. | motion after the charge is consumed.—most methods take this phase as being adiabatic expansion and write (10) pc(v—n)* = const. where ¥ is the ratio of the specific heats of the products of combus- tion. this, together with the equation of motion, forms the com- plete system for this phase. the most convenient independent vari- able to use is x, the shot travel, or, which is much the same thing, to use v, the volume. methods differ in the choice of y, some taking it to be 1-2, others leaving it undetermined, and fixing its value in par- ticular cases by the comparison of calculated values with observed values of the velocity. most methods agrce in taking y as constant over the range of temperatures occurring in a gun. secondary considerations.—so far nothing has been said con- cerning the energy expended in 319 (1) engraving the driving band of the projectile; (2) friction be- tween the band and the rifling; (3) rotation of the projectile; (4) recoil of the gun; (5) kinetic energy of the gases; (6) heat losses by conduction through the material of the gun. it is in allowing for these losses that the difficulty of reconciling calculated results with observed results arises. in general, the heat losses by conduction are considered negligible, and the others are usually compensated for by writing, instead of m, the mass of the projectile, a fictitious mass #’, given by the equation m’ = itm+yuc) (11) where 7 is a correction coefficient and is greater than unity, and ye ts the mass of the charge. o<. po 1. another point to be considered is the pressure at which the pro- jectile commences to move, known as the “ shot-start ’’ pressure. an obvious simplification is to take this as zero; this assumption is made in many methods in use. hadcock (proc. roy. soe., a, vol. 94) in his system allows for a definite shot-start pressure. thermodynamical method.—a method differing in principle from most others is that of henderson and hasse (prec. roy. soc., a, vol. 100). the authors give first of all an account of the chemistry of the explosion, and finally evolve an expression for the heat- content of the gases as a function of the absolute temperature only. by the use of an equation of form, a rate of burning law, and equating the energy lost by the gases to the kinetic energy of the projectile (allowing for losses by a fictitious mass m’), the system of equations is complete. full details of the solution are given in the paper cited. this method further differs from others in that no assumption is made as to the constancy of the ratio of the specific heats. monomial formulae.—sarrau and others have attempted to ex- press the muzzle velocity and maximum pressure by means of mo- nomial formulae obtained empirically. such formulae have no theoretical foundation, and are of very restricted application, inas- much as they are the equivalent of the differential coefficients of muzzle velocity and the maximum pressure with respect to the vari- ables involved for one particular set of values of those variables. they can be relied upon to give accurate results only for cases vary- ing very little from the particular one upon which they were based. bibliography.—bianchi, nozioni fondimentali di balistica in- terna (1914, 2nd ed., revised by g. madaschi); p. charbonnier, balistique interieure (1908); desmazicres, ‘‘ note sur l'etat actuel de la balistique intericure,\"”’ revue d'artillerie, vol. 85 (1920); ‘‘ examen des principales etudes theoriques de balistique interieure publiees a letranger de 1913 4 1922,\" memortal del'’artillerie francaise, tome il]. (1924); gossot and liouville, les effets des explosifs (1919); gutierrez, baltstica interior (spain, 1924); hadcock, internal ballistics,” prec. roy. soc., a, vol. 94 (1918); haesen, balistique interieure (1904); henderson and ilasse, “‘ the thermodynamical theory of explosions,” proc. roy. soc., a, vol. 100. hunt and wright, “ the internal ballistic problem for long elliptic cords,” journal of the royal artillery (feb. 1922); love and pidduck, “ lagrange’s ballistic problem,” phil. trans. roy. soc., a, vol. 222 (1922); proudman, “ the principles of internal ballistics,’’ proc. roy. soc., a, vol. 100; schweikert, innere ballistik (leipzig, 1923); sugot, “les formules de charbonnier,” afemortal de l’artillerie navale (1913); cours de baltstique (1918); tschappat, text book of ordnance and gunnery (1917). il. exterior ballistics modern developments in artillery material, and more espe- cially the requirements of anti-aircraft gunnery, have rendered essential a knowledge of the elements at all points of a trajec- tory, and not merely those at the point of fall, as heretofore. law of air reststance-—the air resistance r, depends upon 1. the velocity, v, of the projectile. 2. thecalibre, d, of the projectile. 3. the shape of the projectile. 4. the steadiness in flight. 5. the physical characteristics of the air tn which the projectile is moving, namely, the density, viscosity and clasticity. for a projectile of given shape and steadiness, dimensional con- siderations lead to the form | saeco 6 pdeep(- . “ where p=the density of the air, a =the velocity of sound in air (a measure of the elasticity), y=the kinematic viscosity, (1) f is a function of the two arguments z and id both of no dimensions, a v fis called the drag coefficient. at velocities less than the critical velocity ¥ | foasd and the expression (1) reduces to the form r = a-p-du-p, the ordinary expression for the resistance of a viscous fluid. 320 at velocities greater than the critical velocity, ia f =a vd --b, where b is some constant. as v increases the value of the second term becomes of more and more importance, till at velocities of the order of 600 ft./sec. f is practically constant, and the expression (1) leads to a quadratic law of resistance. at still higher velocities, the argument se has little effect, and in most ballistic work it is assumed that r « p-d-v-f(=): (2) experiments of bashforth, and later those of the gavre commis- sion, 1888, were all directed to the determination of *f(\") as a func- tion of v only, whereas it appears from the foregoing that the value of a, in other words an elasticity effect, also should be taken into account. thus the effect of a change of temperature will be twofold —firstly, it will affect the density, p, and secondly it will affect the elasticity, and hence a. density of the air.—since the resistance depends upon the air density, any small arc method will demand a knowledge of the density of the air at each point of the trajectory. as is well known, this density diminishes with height, and it is usual to assume a stand- ard density lapse, given by a formula of the type = poe hv (3) where p=density at a height y, ; po=density at ground level, and # is a constant. this may be expressed more generally in the form = po hy), (4) where h(y) is some function of the height: h(y) may be given as a formula or in tabular form. trajectories are calculated for this assumed density structure, and any variations from this assumed law are allowed for in differ- ential variations. ballistic coefficient.—from (2) the resistance to a projectile of given shape and steadiness is given by rec pd?-v*-f(=). as in 3-271 it is assumed that the shape and steadiness can be allowed for by coefficients « and o respectively, and the retardation, r, experienced by a projectile of mass m, and coefficients « and o is given by vv nop dvs (-) mt (5) r the quantity aah is denoted by c, and is called the ballistic coefficient, c, as thus defined, depends upon the density, and there- fore upon the height. with this notation, (5) becomes i() i) choice of the resistance law.—as has been pointed out, all pre- vious experiments have been directed towards the expression of the (6) r : : v ; drag coefficient as a function of v alone, and not of aa further, in the application of these experimental results to the compilation of tables for use in the calculation of trajectories, attempts have been made to give the law of resistance a functional form. thus, the results of shoeburyness experiments of 1906 have been expressed in the form r=az\", where values of a and n were given for different regions of velocity. a and # were so chosen that the aggregate was a continuous function, but the differential coefficients were discontinuous at the points where a and » changed—the ‘'n- junctions \" as they are called. this being so, any attempt at check- ing step-by-step integration by the method of differences will break down at these points. a in france the method has been to utilise the gavre commission results: these results have been smoothed, and attempts have been made to express them by a continuous function which also has con- tinuous differentia! coefficients. one such formula, based on the gavre results, is that of le chef d'escadron demogue, and is as follows:— vi-+0-0392(~) log f(z) = log] 0-2550+ -tan ty’ (7) 27226+494 07 ai brno 900 7 9—339 (4 where fan v’ is expressed in minutes of arc and v= s50 (where v is in metres per sec.). ballistics all small arc methods used at present depend on a resistance law (or tables of resistance) given as a function of v only. when experi- ; oe, 8 mental methods yield a law based on the argument 7” it will be essential to consider the value of @ at each point of the trajectory, thereby taking into account the clasticity effect. with this remark, the following work will refer to the resistance law as p.d?.v*.f(@), since existing tables afford this information only. equation (6) will thus become 12-f(v) the differential equations of motion of the projectile-—treating the projectile as a heavy article, the equations of motion are of the form x\"=—r cos (9) g ay\" =—g—rsinod (10) x’ =v cos @. (11) y’ =v sin @ (12) xy, xe, vy, have the usual significance, and @ is the angle that the tangent to the trajec- tory makes with the horizontal. from the above equations of motion, the following equation follows immediately d(v cos @) _ d@ 0 equation (13) contains only the two variables v and @ (if, for the rv, (13) moment, the dependence of c on y is neglected). it is the differ- ential equation of the hodograph and serves as the starting point for most of the small-arc methods, and for this reason is sometimes referred to as the principal equation. integration by small arcs.—briefly stated, the problem is: given the elements x, y, v, 6, and the ballistic coefficient at the beginning of an arc of the trajectory and one element at the end of the arc (this element thus fixing the size of the arc), to determine the other ciements at the end of the arc. (see fig. 1.) the methods in use differ chiefly in the clement used to fix the arc; this element is usually referred to as the independent variable. the french method ts to use 6 as the independent variable, choos- ing the size of arc so as to have a given total curvature, usually a whole number of degrees. the method used is the afethode gui. m.! in great britain and the u.s.a. time is used as the independent variable. italian methods make use of a generalisation of siacci's method (bianchi), in which the chord and its perpendicular are taken as the co-ordinate axes for each arc. according to cranz, the ger- mans make use of a purely graphical method of integration. integration in small arcs—time as independent variable,-—the suffix o will refer to values at the beginning of the arc, and the suffix i to those at the end. , g will be used to denote v cos 6, the horizontal component of velocity. equations (8) and (13) lead to d(v cos @) 2 vif(v) de cg (14) also from the equations of motion ge (15) d@—s gcos 8 5 tlence il 8) eee q _ adlvcosd) _ _vflojcos o dt db c me) q 1 a f i 1 4g _ =i sec 0f0)0 (17) go @ bo leading to ate it _ xe5f@, _ yy ‘is a @ where 6, », and c are appropriate mean values estimated from the values for preceding arcs; f(v) is at once obtained from tables. it follows from (15) that a; 1 { sec? @de@ = — £-dt (19) bo lo e . = _§g integrating fan @yo—tan@, = eo (20} further m1 —%0 = f(h—b) (21) vi —mo = tan (x1 — xo) (22) ee ee pee ee ee 1 garnier-haag-marcus. a full account of it is to be found in garnier: calcul des trajectoires par arcs successifs—gauthier- villars (1919), balloon—balmont equations (18), (20), (21) and (22) are the working equations. (t; —to) is chosen so that the above equations are true to some prede- termined numerical accuracy. as a first approximation, the following mean values are taken q = 2(n +40) tan 6 1(tan 0, +ian 6) ¥ = 20\" +40) v= gsecd the procedure may be summarised as follows:— (1) from previous arcs estimate fan6, c, and v and so, from equa- tion (18) and the known value go, calculate q. (2) with this value of m, calculate q sec 6, which should now be checked against the estimated value of 2. (3) if the two values do not agree, use the calculated value of v as a fresh estimate of z, and repeat the process till no difference (to the degree of accuracy employed) is obtained. (4) the values of x1, 3: and 6 are at once obtained from the work- ing equations, and the next arc can now be calculated. it will be found necessary to use small values of (4—f)), say 4 second, when » is near the velocity of sound in air, as f(z) will be changing rapidly in this region. the above procedure makes use of estimates from the preceding arcs. in the case of the first few arcs, therefore, the procedure is somewhat different. from equations (14) and (15) ff 5 . eis from the initial conditions, namely, the muzzle velocity, angle of departure, and value of c at the ground level, an estimate of 5g and therefore of g for the first arc can be made. ilence, from equa- tions (20) and (22) (tan 6,;—tan 00) and (j1— 30) are obtained. these values iead to values of c, sec@, and v sufficiently accurate for a first working of the arc. from the results, better estimates are made and the arc is reworked, the process being repeated till the required degree of agreement between estimated and calculated values is obtained. the second and third arcs are worked on similar lines, except that the results for the first arc facilitate the estimation of 6¢. the fourth and succeeding arcs can now be worked by the general process. differential variations.—in the calculation of trajectories a cer- tain set of standard conditions has been assumed, namely: (1) earth motionless. (2) no wind. (3) standard density structure. (4) muzzle velocity, v, and the ballistic coefhcient, c, are constant and (23) re ae ; known. (5) further, if a admitted as the argument of the drag coefficient, a standard temperature (or elasticity) structure will have to be specified in addition to the density structure. the effect of small variations in the initial conditions, v, c and @ can be found by interpolation between calculated trajectories. for other variations, it has been found desirable to consider variations in the elements of the trajectory as functions of the variations from normal conditions to which they are due, and to solve the differ- ential equations of the variations by an extension of “ small-are ” methods. such a method is that discovered by professor g. a. bliss and improved by dr. t. h. gronwall. effect of wind.—the effect of a uniform wind throughout the trajectory can be calculated by a simple consideration of the velocity of the shell relative to the air, it being borne in mind that this is the velocity for which ballistic tables are calculated. wind velocities being small compared with projectile velocities, it will be seen that the uniform wind may be resolved into two com- ponents, one in the plane of fire, and affecting the range, but not the line, and the other perpendicular to this plane, and affecting the line only. in general, of course, the direction and speed of the wind will seldom be uniform throughout the trajectory. it being assumed, however, that the wind structure can be determined by observation, the method of correction is exhibited in the following paragraph:— weighting factors: application to conditions varying throughout the trajectory.—let there be some condition e (maybe wind, den- sity, elasticity, etc.) which is not that for which the normal trajec- tories were calculated, and which does not vary by the same amount from norma! throughout the trajectory, but whose variation from normal at any particular point is known. the trajectory is divided into a number of zones, generally 10, of equal height. the mean variation from normal of e in each zone is known. let arg be the range correction for a variation ae of e from normal throughout the whole trajectory. let ar, be the total range effect produced by a variation ae in the p zone alone, e being considered normal in all the other zones. (this can easily be done by small arc methods.) hence ar, = =z 321 . ar, ar, the ratios ar. ar, | are called weighting factors, since they show the weight of the varia- tion ae in each zone. now let the variation of e from normal be different in each zone, ...de, in the 1st zone, 6e2 in the 2nd zone, and so on. then, the variations possessing the proportional property of ist order variations, are denoted by the quantities #1, w2, and be: ar range effect in ist zone = aon = wou be a aro range effect in 2nd zone = ap or: = wikis ae and, by summation, ar, 19 total range cffect, ar = ae ou wp-bep, i 10 * . a ¥ that is to say =w,-se, is the variation in e which, if uniform ae throughout the trajectory, would produce the same effect as the given variations in the different zones. hence, given the weighting factors, and the variations in each zone, the problem is reduced to finding the effect of a uniform variation throughout the trajectory, and this follows from the principal variations already calculated. weighting factors are calculated and classified according to the height of the trajectory concerned. they can be calculated for range wind, cross wind, density,! and elasticity.? vhe following are typical values of cross wind weighting factors:— vertex no. of zone counting upwards from ground height io *25 -06 1,500 ft. 4,000 ft. 12,000 ft. o05 ‘o7 708 -06 ‘io “10 ‘io \"o7 | ‘07 | -09 | 09 08 | -08 08 | -08 | -09 | -09 ‘10 | +22 05 | 06 ‘07 | -07 | -08 | -08 -13, | °30 the above table brings out the greater importance of variations in the top zones compared with those near the ground. rotation of projectiles—yaw—drift—stability—thus far the projectile has been considered as a heavy particle, and no account has been taken of the rotation necessary to keep it point foremost, nor of the forces that may be called into play by any oblique presen- tation of the elongated projectiles. jump card? experiments have shown that the axis of the projectile does not always he along the tangent to the trajectory. the angle between them is called the angle of yaw. when yaw is developed the line of the resultant air pressure intersects the axis at a point in front of the centre of gravity, and this air pressure exerts a moment about the centre of gravity in such a direction as to increase the yaw. due to the rotation of the projectile about its axis, precession and nutation of the axis ensues. if the spin is insufficient, this moment may cause the projectile to become unstable. bibliography.—bairstow, fowlerand hartree, ‘‘ the pressure distribution on the head of a shell moving at high vetocities,”’ proc. roy. soc., a, vol. 97 (1920); bertagna, manuale del tiro (florence, 1923); charbonnier,”’ note sur l'etat actuel de la balistique extcrieure appliquee,”’ afemorial de lartilterte frangatse, tome ini. (1924); balistique exterieure (1921), cranz and becker, lehrbuch der ballistik, 5th ed.; dufrenois, risser and rousier, balistegue exterieure (1921); fowler, gallop, lock and richmond, “ the aerodynamics of a spinning shell,” phil. trans. roy. soc., a, vol. 221; fowler and lock, ‘‘ the aerodynamics of a spinning shell,” part lk., phil. trans. roy. soc., a, vol. 222; garnier, calcul des trajectoires par arcs successifs (1919); hunt, “ drift and stability of projectiles,” jour. of the royal artillery (1919); chapter on reac- tion of air to artillery projectiles in the afechanical properties of fluids (1923); negrotto, balistica experimenta ly a plicada (1920); ottenheimer, balistigue exterieure (1924); sugot, cours de balis- ligue (paris, 1918); vahlen, ballistik (1922). (a. v. k.) balloon: see airship. balmont, constantine (1867- }, was born june 3 1867 at gumishche, in the province of vladimir, on his father’s estate. a scholar of elizabethan drama and of shelley, he first became known as the translator of the latter and the apostle of 1in practice, temperature weighting factors (for density effect only) are calculated instead of density weighting factors, as it is easier to measure tempcrature variations from a standard tempera- ture structure than actually to measure densities. 2the application of elasticity weighting factors is in process of development. 3a very complete account of jump-card experiments and their analysis, and the conditions of stability, will be found in the ‘* aero- dynamics of a spinning shell,” by r. h. fowler, e. g. gallop, c. n. h. lock and h. w. richmond, f.r.s. (phil. trans. roy. soc., a, vol. 221), and part if. of a paper on the same subject by r. if. “ls fowler and c. n. h. lock (phil, trans. roy. soc., a, vol. 222). 22> his ideas. his extensive travels in south africa, mexico, new zealand and spain account for the exotic vein in his poetry. he produced his best work during the ’nineties and the early years of the present century. the very titles of the volumes pub- lished during this period—under the northern sky (1894), silence (1898), in boundless s pace (1895), the burning buildings (1900), only love (1903), let us be like the sun (1903)—indicate the stages of his development from pure aestheticism to an aggressive and partly anarchical nietzschean poetry, which brought him immense popularity and made him the acknowl- edged head of a younger generation of russian symbolists. this vogue has long since passed, but he remained one of the finest of modern russian lyric poets. the liturgy of beauty (1905), evil charms (1906), the bird of flame (1907) are representative of his later work. he has also written several volumes of prose. in 1918, shortly after the outbreak of the bolshevik revolution, he went to live in paris.",
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