Wilfrid Richmond.

The Americana: a universal reference library, comprising the arts ..., Volume 10 online

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and the value of K as the result of the most
reliable experiments will be shown. We dis-
tinguish

(a) The firings of the Gavre Commission,

(b) The English firings of 1904-6.

(c) The English and Russian firings of
1866-70.

(d) The Dutch firings of 1884.

(e) The Krupp firings of 1875-81.

and tabulate below the values of 10000 K in
order to avoid ciphers.
10000 K



v

300...

600...

900...
1200...
1500. . .
1800...
2100...
2400...
2700. . .
3000...
3300...
3600...
3900...



(a)

.461

.414

.528

1. 156

1.355

1.325

1.252

1. 178

1. 123

1.083

1.056

1. 041

1033



(b)
• 749
.568

.531
1. 129

1.365
1. 316
1.258
1. 177
1. 117
1.078
1045
1.015
.989




(e)
.170
.469
.607
1. 134
1. 314

5:28

t.219
1. 168



It thus appears that K is by no means con-
stant. It is to be noted that, at the low ve-
locities, the results are discordant, due to the
difficulty of accurately measuring the small
differences in velocity in finding the retarda-
tion. At the highest velocities the discrepancies
are probably due to the conclusions being
drawn from a relatively small number of shots.
Considering the wide intervals of time separat-
ing the firings, the results are remarkably con-
sistent

As already indicated, various laws have
been deduced from one or more sets of firings
but a comparison of them shows considerable
discrepancy and none of them is consistent with
the firings quoted when they are all considered.



In most cases attempts have been made to
produce simplicity rather than extreme ac-
curacy. This presents advantages in computa-
tion without necessarily sacrificing practical ac-
curacy. In other cases the formulas are pro-
hibitively complex. In the case of the Gavre
firings no attempt at formulation was made
and the integrations were performed by quadra-
tures.

A careful consideration of the data here
quoted in brief, leads to the conclusion that the
desired simplicity and all accuracy that is pos-
sible (in a case where the true law is not
exactly known but must be inferred from the
firings themselves) will be secured by employ-
ing expressions of the form

F(v)—Av n
using a number of values of A and n sufficient
to reproduce the mean results of firings with
accuracy. The values of A and n are tabulated
together with the limits of velocity between
which they hold.

Velocity limits log A n

o to 723 5.67368-10 2.0

723 to 904 3. 10040-10 2.9

904 to 1000 1.03105-10 3.6

1000 to 1198 7.20620-20 4'875

Iio8toi325 2.36265-10 3-2

1325 to 1497 5.06095-10 7/3

1497 to 1630 6.12840-10 2.0

1630 to 2000 6.84220-10 16/9

2000 to 2589 7.65989-10 1.53

2589 to 3276 7. I93I3-IO 5/3 7

3276 to 4000... 6. 80254^10 16/9

When the laws are in these forms the pro-
cesses of computation by Siacci's method for
direct fire are greatly simplified since we then
have

SM = (» — 2)Au«~*
(except when n^a, when it takes the form

log e u\

I



T(*0=



(n—i)Au n - 1



11 \ — — 2g —
nu) — nAu*



A(u)
When n = 2



«(» — l)^K 2n - 2



[ am ~!aw

and the principal and secondary tables may be
calculated by using these expressions with the
values of A and n belonging to the value of «
under consideration, a being here treated as if
if were v. (Sec article on Ballistics.)

Having the principal and secondary tables
it is at once practicable to predict what will be
the behavior of a given projectile under
assumed conditions. It is usual in direct fire
to have what Is called a range table, calculated
for average air density and presuming a certain
projectile and muzzle velocity. The table gives,
for different ranges or distances at equal in-
tervals, the angles with the horizontal at which
the projectile is launched, and has a number of



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other columns giving the time of flight, the
angle of fall, the maximum height to which
the projectile ascends, the striking velocity, the
perforation of armor, in inches; the effect on
the range, of an accelerating or retarding wind,
of a density of the air different from that
presumed in the table, and of a change in
muzzle velocity. The deviating effects of cross-
wind and drift are also given, usually in degrees.
The character of the range-table and of the
information which it contains will be influenced
by the mount and surroundings. Naval guns
will need a different form of range table from
that required for coast defense guns, because
of the stable platform of the latter as com-
pared with the rolling and pitching platform
afforded by the warship. This makes it neces-
sary to discuss the several methods of fire;
but before doing so it will be necessary to con-
sider equations already deduced in the article
on Ballistics. It was there found that



tane



tamp sxn2<p

and if * is the angular elevation of a point
distant x feet from the gun, and y feet above
its level, and if 9& is the value of 9 which
would place the projectile at the foot of the
ordinate y then

sin 2 <p x = aC
and



tan s



sin 2<px



tan <p sin 2<p

or, if all the angles are small

— = — —
9 1 9

or 9 x = 9 — e

This amounts to saying that in order to shoot
at a point above or below the level of the gun
the line of sight (which is the line from the
eye through the sights and the target) must
make with the axis of the bore of the piece
the angle given in the range table for the dis-
tance at which the target is. This idea is
familiar to every one who uses small arms,
the angle being set off by a leaf graduated in
ranges and constituting the rear sight of the
gun. The same condition obtains in the case of
naval and field artillery guns, which use a sight
graduated in ranges, aim being taken in a
similar manner. Where, however, a gun is
mounted on land on a permanent carriage, at a
fixed height above the water, the range-scale is
usually much larger and is graduated to cor-
respond to values of <p rather than <p — e, and
the scale is set by a man who attends to this
alone while the gun pointer is concerned only
with following the target horizontally. This
independence of function is a great advantage.
On movable platforms, such as ships, inde-
pendence is sought by means of two telescopic
sights, one for the pointer who follows the
target horizontally and the other for a second
pointer who follows it vertically. It is easily
seen that where a single gun pointer has to do
both, good shooting is exceedingly difficult.

The error committed by assuming the
simple relation

9 x = 9 ~ «



may be ascertained by substituting this value
for <p&> obtaining the equation

tane sin 2(<p — e )

tan <p "~" sin 2<p
or

2 cos*<p (tan<p — tans) = sin 2 (<p — e)
and it is at once seen that this equation is true
when

(a) <p = e
or (o) e = o

These conditions are approximately realized in
direct fire with high velocities. They are not
generally realized in curved or high angle fire,
the former condition being never realized on
account of the low muzzle velocities employed.
The actual error in <p in any case may be found
by assuming <p and e and calculating sin 2<p
from the relation

sin 2(<p — e)
*' n 2 ^= ; tone



tan <p
the assumed values of e and <p being used in
the second member to calculate sin 2<p, and
thence <p to be compared with the assumed 9.

The actual calculation of e, the angle of
position, is needed only in the case of sea coast
guns mounted on a stable platform on land.
The value of e in that case is found from the
height of the gun above the water increased
by the curvature of the earth in feet for each
range. The curvature in feet may be readiJy
calculated and is given by the formula

\iooo/

R being the range in yards, and K the curva-
ture in feet. If h be the height above the water,

y = —(h + K)
and

(h + K)



= 0.2154



tan e =



3R



The curvature for various ranges is given in
the following table:

R(yds tf(fcet) #(yds) A;(feet)

1,000 0.22 9,000 17.45

2,000 0.86 ro,ooo 21.54

3,000 1.04 ri,ooo 26.06

4,000 3.45 12,000 31.02

5»ooo 5.39 13,000 36.40

6,000 7.75 14,000 42.22

7,000 10.55 * I5f0oo 48.47

8,000 13.79 16,000 55.14

Using these values of e for the different
ranges, the elevation tp for the range scales is
found and applied. The range scales having
been graduated for the conditions presumed in
the range-table, it is next necessary to find and
provide for the application of corrections to be
made in order to lay the gun properly for the
actual atmospheric and other conditions that
obtain at the time of firing. In the case of
naval and field artillery guns differential for-
mulas suffice, as they must rely largely on ob-



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CrtJNKERY



servation of the fall of the shots, m applying
corrections. In the case of field artillery
operating in mountainous country the height
above sea-level, in the absence of readings of
barometer and thermometer, should be con-
sidered, as it makes a great deal of difference
amounting in a 3-inch gun with 1700 f. s. muz-
zle velocity to a range increase given by
Rh
&0000
in which h is the height above sea level in feet
and R the range in yards. Thus, at a range of
5000 yards, the effect is 500 yards for a height
of 8000 feet above sea level.

The effect of the barometer and ther-
mometer readings on the ballistic coefficient is

shown in the table of values of -r given

o

below. (See article Ballistics.)
Values of —■ .

V



Ther-




Bafpmeter




mometer








Fahr.




Inches






28


29


30


31


— 20


.89


.86


83


.81


— IO°


• 91


.88


• 8S


.83





.93


.90


.87


ft


xo°


95


.92


.89


20°

3 °:


.97
99


:S


• 91
93


.88 .
.90


40 p


I.02


• 98


.95


.93


50°


X.04


1. 00


97





6o°


1.06


1. 01


99


70°


J. 08


1.04


1. 01


.98


So°


I. II


1.07


1.03


1. 00





I. 13


1.09


1.06


x.02


TOO°


I. 16


1. 12


1.08


1.04



and for n = 4/3

As these equations each contain only V 9 <p, X,
and C, it is plain that differential formulas for
range changes may be readily found as fol-
lows:

for * = 4

2dV * s*n 2f dX i—M ( dV

+ — thtti— ~=- — rr-|(* — *)-y



£SJf 2 if



and for n = 4/3

2dV d sin 2<p

V



M



dX dC\
' AT — C J



stn 2<p



dX
"X s



><i



-A/)j(H_



2)



dV



■ X c \



Placing the coefficient of the second member
equal to Q we may write generally

2dV dsin2 9 M I dV ^X_JC>

V T sin 2 <p IT V V } V^IC Cf

and by interpolating in the denominator of Q,
we find

, Q=Z 8(1 — J#)_



In order to obtain in a general way a con-
ception of the effect produced by atmospheric
conditions it will be desirable to note that for
velocities between 4000 and 1500 feet per second
average values of A and n may be taken as a
basis for differential formulas, thus essentially
covering the case of powerful seacoast and
naval guns.

Between 1700 and 500 feet per second an
average will be used to cover the case of field
guns and between 800 and zero the values of
A and n are constant. The value of n averages
about 1.7 for high power guns; and about 3.0
for direct fire field guns.

Placing

AV"-*^r =m

we find from the secondary functions for the
point of fall,

2 ft — 2

— 2(n — i)m>

and this expression assumes very simple forms
for n = 4 and n = 4/3 giving
for 11 = 4

M gx c

Vol. 10 — 20



in which



Placing



so that



3MC4 — n) +3* — 4



, f - 9X
M <~~ V**n2<p



i + Q



-=L



- 3A/U — ») + 3* — 4
X -~M(4-3w) +3»+4
the differential formula assumes the form



dC T dsin2<p



dV



+ ia — (.-L)j-



X K J C sin 2 9 m *- " x " ""> V

Knowing V, <p , and X, and using the mean n
applicable in the case, the change in range
to a change in V, <p, or C may be
C may change due to several causes



incident

found.

since






so that



dC

C ''



it



, dW U. s^»m



2dd
d



In case the weight of the projectile changes,
and the powder charge remains the same, an
increase in w will increase C, but diminish V*
If the charge is changed to meet the change in
w and keep V the same, it is only necessary to
consider the change in C. Changes in the pow-
der charge and projectile affecting V will be
considered later.



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Where the gun is fixed in position incre-
mental changes in range as a result of changes
in V and C are calculated by working out sepa-
rate range tables presuming a changed V or C;
and the results are applied in practice to the
range scales by suitable mechanical computa-
tions. The corrections must also include the
effects of a change in the height of tide, of an
accelerating or retarding wind, and of travel of
a moving target during the time of operation.
The accuracy obtainable with a system of this
character is illustrated by recent firing at Fort
Monroe with two 12-inch guns fired at a mov-
ing target 5 miles distant, the battery actually
firing seven out of eight shots through a net
screen 20 yards long and 10 yards high, in 4
minutes 31 seconds, the projectiles weighing
half a ton each.

The effect of wind on range and across the
range is readily determined by considering the
relative motion of the air particles and the pro-
jectile, the entire problem being one of vectors ;
the effect on the range of a wind blowing W
feet per second is that due to a change in
velocity and one in elevation given by the ex-
pressions

dV = — Wcos<p

dtp — -r =— {a <p in radians)

the wind W being an accelerating, component,
the travel of the wind in the time of flight is
WT feet and the total effect is therefore
given by



AX

X '



WT _W cos<p



j 2 _fi(i-L) \



+ ^?XL



The effect of a cross wind W is simpler in
form and is given by



D = w ( T -vh V )



Cross effects are best expressed as angles as
they are so observed at the battery. The value
of the deflection in radians is, therefore,

D W tVTcostp \
X sa Vcos<p[— l )-

and in degrees

^ - . W iVTcos*, \i8o°
Deflection = 77—- ( —T — 1 )

There is one other deflection, called drift,
which is caused by the combined action of
gravity, air resistance, and the rotation of the
projectile about its longer axis. The results of
the forces acting are essentially

(a) A retarding force.

(b) A deviating force.

(c) A disturbing couple.

We are, in practice, principally concerned
with the deviating force, which may be shown
to cause a lateral acceleration



dH
d(*



^0-/O



cos O
v



Upon noting that (see Ballistics)
vdO



dt



-= — g cose



we find

dfi <*



„. 2V * dd

■A./. • —

n w dt
or upon integration between <p and
ds 2V d*

77-U-*). —•-(,-*)

In the above expressions (1 — K) is a con-
stant dependent upon the shape of the head of
the projectile, its length in calibers and its
radius of gyration; and n is the number of
calibers over which the projectile travels in
making one complete turn on its axis at the
time of leaving the gun. Their values are
ordinarily

1 — K = 04S to 0.20
n=20 to 30
(usually 25)

ds
Taking the mean value of ^-between <p and 0,

we find

z Vd*

7 = (ir-K)— (<p-6)

ore From this,

x "" (I " _/C) tcm 1

Upon considering that in the act of obeying
the deviating force the projectile encounters a
lateral air resistance, and if the angular
drift be designated D', and r^ be the retarda-
tion in the path of the projectile it is seen that
r* sin D' is the mean lateral retardation due
to this cause. Since D' is small we may substi-
tute its radian measure for its sine and the
cross-retardation becomes Ur^ Since the time
of flight, t, from <p to is the same for both
retardations, the distance effects should be pro-
portional to the mean retardations; that is, in
the ratio D' :i. The loss of range due to
r f is Vt cos (p — x and hence that due to the
cross-retardation is



■>■{



Vt cosy



)



Hence finally



™ * *, /Vt cos a \



x Vt cos<p

d* secw

For the point of fall this becomes

D' = (1 — K) — (f + w) stc<p

If <p and w are in radians D' is in radians; if
they are in degrees D' is given in degrees.



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by



The correction for height of tide is given For curved and high angle fire the form



AR=— cot(a> — e)



AR being the range connection in yards and h
the height in feet of the tide above the reference
mark assumed for the range tables. The same
formula is used to compute the danger space
which is the space over which a target of
height h is in danger of being hit It is that
part of the range beyond the target when the
projectile just grazes the top of the target.

The effect of projectiles for guns at the sea
coast and on warships is measured partly by
their ability to remain entire while passing
through armor and the materials of which
fortifications are composed, and partly by their
capacity to carry explosive. The shell is
relatively weak for armor perforation, but has
a larger cavity and can carry more explosive,
while the shot is better adapted to the work
of perforation but has a smaller cavity. The
formulas used in coast artillery range tables
are, for armor piercing shot:
For Krupp Cemented Armor
Below 1800 feet per second:



d ~-«^" 1000
Above 1800 feet per second:

* 7 \ & 1 1000 3 f

in which

/ = thickness of armor in inches.
d = caliber of projectile in inches.
w= weight of projectile in pounds.
v = striking velocity in feet per second.

These formulas are for normal impact; that
is, for a projectile striking armor in a direction
perpendicular to its surface. Where the im-
pact is oblique, the path of the projectile being
inclined at an angle a to the normal to the
plate, the following percentages should be sub-
tracted :



a Subtract

o° 0%

5° 0%

io° 1%

15° 2%

20° 4%



a

2 K

K
K

45



Subtract
6%
8%
11%
15%
19%



For the perforation of deck steel by mortar
projectiles,



7"= [54175-ioh- 8 / 1 ypjj^



for normal impact, that is for a oo° angle of
fall, the deck being horizontal. For any other
angle of fall subtract percentages as follows:

Angle



Angle

of fail Subtract



of fall Subtract



45°


20%


50°


16%


SS*


12%


6o°


Q%


6s°


6%



70 d
75°
8o°
8 S °
oo°



4%
2%
0%
0%
0%



gx*



y = xtan <p- -yi—i



{ lllC ?}



has given most satisfactory results in practice.
As an illustration, it was found practicable to
compute zone range tables from ojo feet per
second initial velocity to 1500 feet per second
initial velocity for the 12-inch mortar, using
projectiles weighing from 700 to 1046 pounds
and having heads of widely different shapes.
It was found that the angle of fall, the striking
velocity and the maximum ordinate were in
very close accord with calculations by quad-
ratures for the same V, tp, and X, using rigidly
accurate methods. The simplicity of the equa-
tion above recommends it because of the ready
calculation of the elements at any point, x, y,
of the trajectory, and because differential
formulas for range changes are easily deduced.
The value of K is a function of both V and f ,
depending to a great extent upon the steadiness
in flight of the projectile. Its value .must be
found from proving ground firings, and
formulated in terms of V and <p.

Inspection of the equation at once shows
that this trajectory is an hyperbola.

The formulas for range changes are
dX , „*C t % dsin2 9 \ , ^dV



+ (,_„)«■}



and as n is here 2.0



dX
X



dC



= (1— M) 7r+Af



d sin 2ip



dV



H-d-M)-^



dK
K



in which
Af=



gX



V*sin 29

Representing 1 — Kx by m, we find upon
solving the ballistic problem (see Ballistics)
x(i— Af)=X(i — m)

tana _ m 1— m»

tan <p 1 — M m*



zflcos*



V*cos* <p
VT cos<p



=m»



x Vikd+ Vm)

I U

y=xtony> 1 1 — YZT

For the point of fall

tanm 1

ranp M

y = o
Vo> *cos 2 w



1— in



M



V*cos* <p
VTcostp __



= Af»



Va/(i + V M)



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For the highest point or summit,
=o .
x* i

X = i + y/JT



I



Also



X tantp (i.-h V3f)i

3

vf = MiV*cosW

A = J



Mortars and howitzers, to which the equa-



power guns only a single velocity is used.

The penetrative effects of projectiles en
various materials are indicated below:

Wrought iron :

(x) = "Jr ' \"§$r) 3 • • .Tresidder
Simple steel :



/O.T =



rfO.76



V
1022



.De Marre



Sand:



t w ( / v \ 2)

r =8i -^ log] 1 +4-6(— ) {..(Parodi)

For masonry, brick, concrete, clay, oak, pine,
the penetration for sand should, according to
Ronca and Bassani, be multiplied, respectively,



3 4 •
2 to 3. ". — ; for ordinary



(A)
(B)



r— 3A/ + S

L ~ 13 — SM



(C) L =

In all cases



6M + 2
10 — 2M

ciX



M = -



V 2 sin2y

The use of a constant value of n in each of
these cases permits the definite formulation of
expressions for the remaining elements. Thus



tan 6 —tantp —



gx*



2V t C0S t <p

9*



f=



V % cos t <p

0.



V cos<p
v=V cosy secO. Qt

which (placing w = /£F»- 2 ^)



_ X

Gl ~~ n(»— i)m*



In — %



^ I+(n _ 2)m |_ r

— I— 2(» — i)m
0-= (n-i)m ■]{x+(»-a)i»U-«-i}



0*= { 1+ (n — 2)»»}2-n

For the three cases under consideration
these expressions assume the following definite
forms :

n = 1.7 ; /o0 /4 = 708693 — 10
n — 3.0; /o£.<4 =2.91280— 10
*=2.o; /o^M = 5.67368— 10



by ~» r - * T to — .
J 3 9 3 2

4
earth by — .
J 3
For closely approximating to the correct
calculation of the elements of a trajectory the
following average laws are given:

(A) For high power guns of the Coast Ar-
tillery and Navy:

F(v) = [7.08693 — 10] v 1 - 7

(B) For direct fire Field Artillery guns:

F{v) — [2.91280 — 10] v*

(C) For all mortars and howitzers:

F(v) = [5.67368— 10] v 2
In the three cases the value of L in the dif-
ferential formulas becomes
y __ 6.9A/ + 1.1



n=i.7; m= [7.08693—10] c ^
n = 3.o; m z



Vx
'■ [2.91280 — 10] —



<*=T



n = 2.o; m= [5.67368— 10]— ■£
For 11 = 1.7, for any point x, y

" "J (I— 0.3m) 3 —I 1.4ml

10 j -IZ 1

Ql= vf^\ (l ~~ a3w) 3 "" l )

10 J, T L

0»= "7^" j(i— a3m)3



-\



Q*=(i— 0.3m) 3
For the point of fall, whatever the value
of n

V 2 sin 2<p



<2x =



9X



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2Q9 , tan op
~%- =1 +

\2* tan<p

- VT coso

Qi— v



__ v w cosw



V costp



For n = 3.o

Qi=i + im + im 2
0*=i -f m + Jm 2

1

°*— i+m



For the muzzle velocity, V, x = X, and

*» *» 8l ** £"

Hence

28* wk« # jc ;



Place



L - A



WF»



For n = 2, as already shown a hyperbolic
trajectory should be used, and the values there
given are applicable.

The problems in gunnery which depend on * nen
interior ballistics are for the most part those
related to changing the charge, weight of
projectile, muzzle velocity and maximum
pressure. The formulas of interior ballistics and finally
do not afford as convenient differential
formulas as are desirable, and semi-empirical
methods are of considerable value in classify-
ing proving ground results.

Such a method will now be shown:

From the record of firings with gun barrels
of different lengths as well as from the form-
ulas of interior ballistics, certain relations be-
tween travel and velocity are noted. They take
the general form



2430 P(C— r')
= £8.03,37-10]=^ ^



x _ T ML



L =



V — -L / B V



in which U is the velocity corresponding to
infinite travel; v, the velocity corresponding to

travel x feet; B the travel for which s> — —

2
(See Ballistics.)

Upon differentiating the above equation, it is
found that

_____ ftt/ 2 fl"*2n-l

and, for a maximum pressure,



It is a simple matter to compute a table of
values of L for equicrescent values of -y~ an< ^

IP

y x is readily found also. With such a table it

is practicable to reduce to simple form the in-
verse problem from firings affording data as to
V, P, w, C and c', and also a). The value of L

~ U

is calculated from the data and



and -p-are



£

X

taken from the table. The computations in
using the table may be simplified by placing



Online LibraryWilfrid RichmondThe Americana: a universal reference library, comprising the arts ..., Volume 10 → online text (page 81 of 185)