1. The Le Duc ballistic formula was discussed in the PROCEEDINGS in whole No. 138, by the late Professor Alger, and in whole No. 143, by Mr. G. W. Patterson. It calculates velocities and pressures in guns. Taking the notation of Mr. Patterson's article, we have for our naval guns:
Letter | Unit | Meaning |
? | Lbs. | Weight of smokeless powder in charge. |
? | Lbs. | Weight of projectile. |
? | … | Density of loading= 27.68 ? / S |
S | Cu. in. | Chamber capacity. |
? | … | A powder constant (largest for slow powders) |
v | F.-s | Muzzle velocity. |
u | Ft. | Travel of projectile in bore. |
A | Sq. in. | Cross-section of bore. |
g | F.-s.s. | Acceleration of gravity (32.155 at N. P. G.). |
? | … | Specific gravity of powder. |
P | Tons, sq. in. | Effective pressure on base of projectile. |
v=au/b+u (I)
a=6823 ?1/12 (?/p)½ (II)
b= ?(1- ?/ ?) (S/p)2/3 (III)
P=a2 x b x p x u/2240 x g x A x (b+u)3 (IV)
The values of the constant in (II) and the exponent in (III) are due to Mr. Patterson. He also gives the, relation that in our powder ? is equal to
12.3728—.132 (per cent total volatiles) /7.4
and states that pressure gauges usually read 1.12P.
2. The commonest use of such formulae is to see how a new powder being proved suits the gun. To do this, we take the charge that gives service velocity and compute a pressure curve (using the factor 1.12), and plotting this together with the strength curve we can see how the factor of safety varies along the bore, and whether it is everywhere above its prescribed minimum. The maximum pressure computed from weight of charge and the velocity will rarely agree perfectly with the gauges. How to get the most probable curve is then the question. Multiplying all ordinates of the curve by the factor which would make the computed maximum pressure agree with the gauge pressure is not permissible, for this would change the area of the curve, and the pressure curve's area is fixed by the muzzle energy.
3. The purpose of this note is to describe a simple way of forcing a curve that will check up with the observed gauge pressures without violating the formula. We work from charge and observed pressure instead of from charge and velocity.
In Professor Alger's article we see that the maximum pressure comes where u=b/2, so that (IV) becomes
P= 4/27 x a2 x p/2240 x g x A x b (V)
Introducing the factor 1.12, which represents ail average value, and transposing, we have
b= 4/27 x p x 1.12a2/2240 x g x A x Pmax (gauge) (V)
We can then substitute the value of a from (II), and replace ? by its equivalent 27.68?/S at the same time.
b= 4/27 x 1.12 x 6823 2 x 27.681/6 x ?1/6 ? / 2240 x g x A x S1/6 x Pmax (gauge) (VII)
so that
b= Q?7/6 / P max (gauge) (VIII)
where Q = [ 4/27 x 1.12/2240 x 38232/ 32.155 x 27.681/6 / S1/6 ],which is different for each gun; and, from (IV),
P max (gauge) = R x u / (b+u)2 (IX)
where R is a constant for a whole pressure curve, and of such value that it brings a pressure which is in agreement with the gauge pressure when u=b/2.
As a result we can tabulate log Q for all service guns, and use a printed form like the following:
log ? | ?= | Gun Powder Date |
1/6 log ? | P max (gauge) |
|
Colog |
| |
Log Q |
| |
log b | b= b/2= |
|
| 3 b/2 |
|
R= P max (gauge) x (3b/2)3 / b/2 = PuN = R x uN / (b+uN)
u1 b | u2 b | u3 b | u4 b |
b+u1 P1 | b+u2 P2 | b+u3 P3 | b+u4 P4 |
It is seen that b is obtained in three references to the tables and the other steps are done most conveniently on the slide rule or omnimeter. The first point can be obtained with practice in a little over two minutes from starting to write down the pressure and weight of charge; the other points follow much more rapidly, once b and R have been obtained for the first. Nothing has been changed in the formula in this application; it is merely one way of arranging the work to get a special result with rapidity.