Note: Descriptions are shown in the official language in which they were submitted.
11~6~3Zl
This invention relates to an improved tubular
projectile.
Projectiles are commonly launched at
supersonic velocities from a gun, a launcher rack, or the
like. They are intended to follow a desired trajectory from
a launch site or vehicle to a target or target area. The
trajectory wanted is often difficult to achieve. This may be
due to stringent trajectory requirements such as one with low
velocity decay to the target followed by a high velocity
decay and instability beyond the target to reduce the range.
In another situation there may be a need to maximize the
range. It may also be due to extraneous and disruptive
forces generated either during launching, or_in free flight.
A basic object of the present invention is to
provide an improved low cost tubular projectile particularly
suitable for training purposes. Another object is to
provide a tubular projectile having a configuration
and internal profile which is tailored to satisfy predetermined
trajectory requirements. A more specific objective is to pro
vide for a controIled variation in the drag forces applicable
~ .
to the tubular projectile in supersonic free flight, to cause
that projectile to follow a preselected flight pattern.
- The importance Df tailoring t~e trajectory of a
projectile to meet specified requirements is especially notable
in, although not limited to, the case of projectiles
designed as practice rounds. It is desirable that there
~;? be a rapid velocity decay after the projectile has trave~led
;1 the maximum useful distance thereby to reduce the size of
the danger zone. If the tubular projectile is to have value
as a training device it must be possible to match its
;
ballistic trajectory fairly closely to that of the actual
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weapon which it is designed to simulate e.g. an armour
piercing discarding sabot (otherwise known by the acronym
APDS). The break-up pattern upon impact, the range of the
projectile following ricochet and the cost of the projectile
are also factors to be considered.
The present invention, in part, involves the
discovery that, in order for a tubular projectile to perform
satisfactorily, the projectile must be arranged such that
supersonic flow conditions are established within the central
passageway of the projectile immediately after launch thereby
to achieve low drag conditions over the first portion of the
flight path. The invention further provides for the projectile
configuration to be arranged such that ~8 the velocity
decays down to a certain flight Mach number, choked flow
conditions are suddenly established within the central
passageway. This choked flow condition sets up a normal
shock wave ahead of the pro~ectile causing relatively rapid
velocity decay thus limiting the range of the projectile.
Although the prior art has provided a plurality
of types of tubular projectiles, none of the known prior art
designs are capable of providing the range limiting
transition from low drag conditions to high drag in accordance
with the invention. In fact, it can be shown that virtually
all of the prior art tubular designs have choked subsonic
flow in the central passageway at start up, and consequent
high drag through the entire flight path. This condition
is totally unacceptable especially for high kinetic energy
rounds which require high velocities at the target.
In one aspect the invention provides a projectile
adapted to be fired at supersonic velocity from a gun barrel
and comprising: a tubular body of substantially circular
cross-section having a leading inlet end and a trailing exit -~
end and a central passageway extending therethrough; the
leading end of the body being of a shape such that the internal
- . . : . .. ... . .
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diameter of the central passageway decreases from the leading
inlet end to a throat region, the ratio~-of the cross-sectional
area of said passageway in the throat region (At) to the cross-
sectional area of said passageway at the leading inlet end (Ai)
being sufficiently large and being so related to the projectile
velocity at launch as to enable a normal shock wave to pass
through the throat region to establish supersonic flow in said
passageway and thus provide a relatively low aerodynamic drag
after launching, with said ratio At/Ai also being a value less
than 1.0 so that as the velocity of the projectile decreases to
a predetermined flight Mach number, the shock wave is expelled
from the passageway to establish choked flow conditions in said
passageway and relatively high aerodynamic drag whereby to limit
the range of the projectile.
In a further aspect the invention provides a
projectile adapted to be fired at a supersonic velocity from
a gun barrel and comprising: a tubular body of substantially
circular cross-section having a leading inlet end and a trailing
exit end and a central passageway extending therethrough; the
leading end of the body being in the form of an annular wedge
with the internal diameter of the central passageway decreasing
from the leading inlet end to a throat region, the ratio of the
cross-sectional area of said passageway in the throat region
(At) to the cross-sectional area of said passageway at the
leading inlet end (Ai) being sufficiently greater than that
defined by the equation:
( t) (I ' 2 ~1) (y~2 Y2
Ai min (~ ) y~l/2(y-1) ~ -
where M = Mach number at launch (for M> 1.0) -
= Ratio of specific heats
. as to enable a normal shock wave to pass through the throat
~ :: : .. - -
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region to establish supersonic flow in said passageway and
thus provide a reIatively low aerodynamic drag after launching,
with said ratio At/Ai also being a value less than 1.0 so that
as the velocity of the projectile decreases to a predetermined
flight Mach number, the shock wave is expelled from the passage-
.way to establish choked flow conditions in said passageway and
relatively high aerodynamic drag whereby to limit the range of
the projectile.
Once the At/Ai ratio has been selected such asto ensure supersonic flow in the central passageway of the
projectile at the designated launch velocity, it is possible
: to predict the velocity (or Mach number) at which choked flow~ .
conditions are established in the passageway as the velocity of
the projectile decays during flight. The equation which
relates the theoretical "choking" flight Mach number to the
selected At/Ai ratio is as follows:
; ' ' ' . . ,
At
~ = ,
Ai ~ y~ 2~Y+1/2(y-1~
. ..._
~ Y21
(M = flight Mach number
~ = ratio of specific heats)
In a fu~ther feature the wall thickness ratio of
- the projectile t/R is from 0.15 to 0.45 where:
t = maximum wall thickness,
R = maximum radial distance from projectile axis
to outside surface of projectile.
- 4
:. . , , -
; ~ : . -- .
- . .. . . .
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In a .still further feature the annular wedge at
the leading end of the body is an internal wedge defining a
generally sharp leading edge arranged to e~able an oblique
shock wave to attach itself to the leading edge after launching
to assist in providing low aerodynamic drag on the projectile.
In a preferred form of the invention the annular
wedge at the leading end of the body is a composite wedge
defining a generally sharp leading edge of the projectile with
the included angle of such composite wedge being sufficiently
small as to enable an oblique shock wave to attach itself to
said leading edge after launching to assist in providing low
aerodynamic drag on the projectile.
The projectile may also include a pusher base and
a gas seal arrangement mounted to the trailing end of the
tubular body to transfer gaseous pressures in the gun barrel to
a driving force on said body and being separable from the latter
:
by virtue of stagnation pressures acting on the pusher base
after launch.
:
. The projectile may also include a driver band
!mounted to an outside surface of the tubular body and adapted
to engage with rifling grooves in a gun barrel to impart spin
to the projectile.
In accordance with a further feature said pusher .
base includes a leading end of ogive configuration which, prior
to separation, is enclosed within said tubular body.
Further features of the invention are set forth
in the following disclosure and in the claims appended hereto.
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In drawings which illustrate embodLments of
the invention:
Fig. 1 is a longitudinal section view of a tubular
projectile assembly taken along the axis of the projectile;
Fig. 2 is a view similar to Fig. 1 showing the
flight configuration of the projectile;
Fig. 3 is an elevation view of the trailing end
of the projeetile;
Fig. 4 shows the projectile in flight with oblique
shock waves attached to the leading and trailing ends thereof;
Fig. 5 is a view similar to Fig. 4 but showing a
normal shock wave ahead of the projectile; -
Figs.6(a) to ~(d) illustrate the shock swallowing
and shock expelling features of a projeetile made in aceordanee
with the present invention;
Fig. 7 is a graph bearing eurves illustrating
the effeet of throat to intake area ratio on the shock expelling
and swallowing processes at various flight Mach numbers;
Fig. 8 is a graph illustrating typical drag
coefficient variation in relation to Mach number at several
;; . .
wall thickness ratios for a typical tubular projectile;
Fig. 9 is a graph illustrating the variation of
ballistie eoeffieient with wall t~bkness ratio at various flight
Maeh numbers for a typical tubular projectilæ;
. Fig. 10 is a sehematie view of a portion of a
tubular projeetile with symbols applied thereto illustrating
the various dimensions of the projeetile;
Fig. 11 is a portion of a test report whieh shows
points of strike of projeetiles on a target;
Figs.12(a) to 12~e) illustrate various tubular -
projeetile eonfigurations used in field trials;
Fig. 13 is a plot of the trajectories of various
projectil~s;
, . .
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Fig. 14 is a plot of the velocity history of the
projectile shown in Fig. 12(b),
Fig. 15 illustrates the drag coefficient variation
with flight Mach number for a plurality of projectile configurations;
Fig. 16 is a longitudinal section view of a modified
form of projectile;
Before describing the theoretical considerations
governing the design of the tubular projectile, reference
will now be had to Figs. 1-3 which illustrate a typical
embodiment of the invention. Fig. 1 shows the assembly
complete including the projectile body 10, driving band 12, and
pusher base 14. A front protective co~er 16 of plastics
material breaks up and separates from body 10 immediately after
launch. Figures 2 and 3 illustrate the flight configuration of
the projectile i.e. the projectile body 10, per se.
The projectile body 10 is of circular cross- - -
section and has a central passageway 18 therein of circular
cross-section. The frontal portion of the body 10 is shaped
to define an annular composite wedge portion 20. This
composite wedge portion includes an internal wedge having an
annular wall22 which is at an angle to the longitudinal axis
of the projectile and thus tapers inwardly from the leading
edge 26 of the projectile to a throat portion 25 which
commences at region 24, and an external wedge having an
annular wall28 which is also at an angle to the axis of the
projectile and thus tapers outwardly and rearwardly from the
leading edge 26 to the region 30 where it meets the
cylindrical outer wall 32 of the projectile. The throat
portion 25 is of constant diameter from 24 to the trailing end
28 of the projectile. The apex of the annular internal and
external wedges actually lies a short distance forwardly of
~ 6 ~
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the leading edge 26 due to the fact that the latter is rounded
to a very small radius, as seen in cross section, for practical
reasons.
The trailing end section of the projectile is
stepped inwardly at 30 and the exterior wall of the stepped-in
portion is knurled to provide a good grip between the projectile
10 and the annular driving band 12 which is press fitted tightly
over the inwardly stepped portion. The driving band 12 includes
an annular recess 32 which retains pusher base 14 in position.
The trailing edge of driving band 12 includes an annular lip 13
which acts as a gas seal during launching.
The pusher base 14 abuts against the trailing
end 28 of projectile 10 and includes an annular projection 36
thereon having a groove therein containing an annular sealing
ring 38. Ring 38 helps to prevent blow-by of gases during
the launching of the projectile. As is well known in the art,
the pusher base functions to transform gaseous pressure
within the launch tube or gun barrel to a driving force which
accelerates the projectile assembly for launching. The driving
band 12, being of relatively soft material, (such as a suitable
plastics material) engages the riflings formed in the gun
barrel and imparts spin to the projectile thereby to help
stabilize same during flight. After launching, centrifugal
forces effect separation of the driving band 12 following
which the stagnation pressure which builds up in the interior
of the projectile act to force the pusher base 14 off the
projectile. ;
The tubular projectile 10 is intended to be
launched at supersonic velocities"lsually between Mach 4 and 4.5.
Thus, supersonic flow fields are associated with the tubular
projectile 10, and these can have two different structures.
At the higher velocity ranges, the flow field establishes an
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oblique shock ~aYe structure (providing the throat to intake
area ratio (At~Ai) is large enough as will be explained fully
hereafter in which`a compression shock wave is attached to the
leading edge 26, is followed by a region of expansion, and
then by a recompression shock wave attached to the trailing
edge 28. Such an oblique shock wave structure can be seen, for
instance, in Figure 4. A supersonic flow field in which an
oblique shock wave structure is formed associated with supersonic
flow inside the tubular projectile is associated with low drag
forces.
The velocity of the projectile decreases with
range and at a predetermined Mach number, which depends on the
At/Ai ratio,the flow field associated with the projectile 10
suddenly changes to one exhibiting a strong normal shock wave
(or bow wave) that is detached from the leading edge 26 of the
tubular projectile 10. This is best seen at Figure 5. The
presence of a strong normal shock wave detached from the leading
edge 26 indicates that choked flow conditions exist within that
projectile. Choked flow conditions tend to give the impression
;~ that the projectile is a solid cylinder, and in any case, impose
large drag forces on the projectile.
` The choking phenomena of the tubular projectile has
been substantiated in wind tunnel tests using models and
flow visualization techniques and has been further demonstrated
in actual field trials. It is thus possible in accordance
with this invention to increase the drag forces on the tubular
~ projectile considerably, and suddenly. It is also possible to
; tailor the design configuration of the tubular projectile in such
~ - 8
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a way as to provide low drag over the first portion of
the xange that is to be followed by a natural transition
at a predetermined threshold or critical velocity (and Mach
number) to choked flow conditions which impose very high drag
forces on the projectile and so limit its range of flight. ,
The utility of this transition feature will become more
apparent hereafter.
The basic structural and functional feaitures
of a typical projectile made in accordance with the principles
of the invention have been briefly described above. The
following description will illustrate various important
theoretical and practical considerations involved in the design
of a typical spin stabilized tubular projectile ~given the
acronym STUP for convenience) to be used as a practice round,
which practice round is to be used to simulate a typical
APDS (armour piercing discarding sabot) projectile. The
description will make specific reference to the design of a
105 mm. projectile but it is bo be understood that the
invention is not to be limited to this size , but rather
extends to all practical sizes.
It should be understood that the flow properties of
a STUP are somewhat more critical than for conventional ogive
shapes. Specific design criteria has to be well understood and
used in the basic designs to fulfill the objec~ives. A STUP
designed as a practice round for 105mm APDS training, for
example, has to have several important features such as:
(a) trajectory match with the APDS in practice distances up
to 2500 meters; (b) small safety range; tc) minimum
richochet range; and (d) low cost.
g
. ~ . . . ~ .
. , . ,. , , :; ~ . - ,-
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The most important parameter to consider in the
initial design phase is the trajectory match with the APDS.
In theory, this is achieved by matching precisely the launch,
inertial properties, aerodynamic and dynamic stability
properties of the practice round with the APDS (i.e. APDS
is the weapon). In practice, this is not possible even with
a STUP. So, a compromise in the above design goals has to
be considered to achieve an acceptable trajectory match
with the APDS. This is a compromise between the muzzle
velocity, ballistic coefficient CDA/W, (CD-= drag coefficient,
A = area of pro~ectile based onmaximum outside diameter of flight
configuration, W = total weight of projectile _flight
configura~tion~, time of flight, inertial properties and
dynamic stability. To achieve low cost thé design calls for
a projectile with the minimum number of components;,which
dictates a full-calibre round (i.e. the APDS is sub-calibre
and it has a complicated and expensive sabot).
To achieve a trajectory match of the APDS with
a full-calibre STUP, the drag must be low and it is imperative
that supersonic flow be started in the central passageway of
the projectile as soon as thè round leaves the gun mu~zle;
otherwise, the drag will be too high. The flow starting
process, also termed the shock swallowing process, is described
below and can be used to establish a minimum throat to intake
area ratio based on the maximum muzzle Mach number. Re~erence
should now be had to the Figures 6(a) and 6(b) and the graphs
shown in Figures 6(d) and 7. Fig. 6 schematically illustrates
a tubular projectile in accordance with the invention having I -
a composite annular leading end wedge defining an inlet area
':
:
I
!
.,. ' ~
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Ai and a throa~ ~rea in the central passageway designated by At.
The internal flow process of a STUP is basically
that of a supersonic diffuser and the reverse De Laval nozzle.
Since the flow-is starting from rest at the muzzle of the gun,
a normal shock wave has to pass through the throat section to
establish supersonic flow in the central passageway and thus
low drag conditions.
The starting or shock swallowing process involves a
consideration of the governing equations for mass continuity,
momentum, and energy. Reference may be had to the following:
A. Hermann, "Aerodynamics of Supersonic Diffusers".
-- B. Donovan, A.F., Lawrence, H.R., "Aerodynamic Components of
- Aircraft at High Speed", Princeton University Press,1957.
C. Shapiro, H., "Compressible Fluid Flow", Vol. I, The
Ronald Press Company, New York.
As the flow (shown by the arrow) accelerates to
supersonic speeds, a normal shock appears in front of the inlet
as shown in Fig. 6~a). As the Mach number increases the
normal shock moves towards the inlet edge. The flow behind the
shock wave is subsonic and accelerates towards Mach number
one at the throat section. The amount the flow accelerates
depends on the geometry or the area ratioibetween the inlet and
throat. At some higher Mach number the shock wave will become
:, . , ............... _ ~ , . . . . . .
attached to the inlet edge. At this condition the Mach number at
~;, . . ., __
the throat is one or less. If the shock wave moves inside the
edge, even just a small distance, it will be 'swallowed' sinoe
it can be shown that this is an unstable region and thus
supersonic flow conditions are established. However, if -
~ the throat Mach number reaches one before the shock wave is
: attached to the edge it means that the Mach number is one at
.'' ~' ~ .
~0643Zl
the throat and the flow is choked. The additional mass flow
escapes around the edge as shown in Fig. 6(b). Even if the
Mach number is increased further the shock will never reach the
edge and the flow will not start. This is the condition for high
drag.
The equation which defines the minimum theoretical
_ _ _ , ., . . _ .......... ....
throat to intake area ratio for starting can be derived from the
governing equations which yield ;
- (At) (1 .,. Y21 M2) (y~12 y_l)l/y-
Ai i (~ M2) /?(Y
where: _
= ratio of specific heats
M = Mach number (M > 1.0) at launch or muzzle velocity
This equation défines curve A cn Figure 7.
The above equation and the curve shown in
Fig. 7 can readily be used to determine the minimum theoretical
throat to inlet area ratio (At/Ai)necessary to obtain
supersonic flow in the central passageway at the launch
velocity. However, in practice,the minimum (AtJAi) ratio is
always chosen so as to be somewhat greater than that
indicated by the equation since the boundary layer ha~ the
effect of making the throat area slightly smaller. To
illustrate this, we will consider the (At/Ai) ratio needed
for a tubular projectile which is being designed to match
~approximately) the flight characteristics of a 105mm APDS,
~ The muzzle velocity of the 105 mm APDS is 4850 ft/s (1478.3m/s)
!~ or Mach number 4.3. Prom the plot of At/Ai versus Mach nu~ber,
for the flow to start~the At/Ai ratio has to be at least 0.66.
A~ noted above, this design criteria does not account for the
boundary layer thickness which affects the effective minimum
':
. ,- . , . . - . , . , : : . , - :
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throat area. In the STUP model shown in Figs. 1-3 the
minimum throat area would be at the very rear face of the
model as the boundary layer thickness increases with length
and depends to some extent on ambient conditions as well,
particularily the temperature. Thus, the STUP design has
to have an At/Ai ratio which is larger than .66 for the
flow to start. For the STUP Model in question an At/Ai ratio
of 0.7 was selected. Generally speaking, an At/Ai ratio some
5~ or 6~ greater than the theoretical minimum given by the
equation is sufficient to account for the boundary layer
thickness.
` It should be noted that the (At/Ai) ratio design
margin is relatively small for most existing gun systems. This
margin in the At/Ai~ratio for launch Mach numbers from 1 to~4.3
is 1 to 0.66, excluding boundary layer considerations. The
At/Ai ratio for launch Mach numbers of 1 to 5 is 1 to about .65,
also excluding boundary layers considerations. For At/Ai values
below these limits the flow will not start at launch and the drag
will be high. Since the At/Ai ratio is the square of the throat
to intake diameter ratio (dt/di)2, it can be observed from ex-
amination of the STUP model shown in Figs. 1 to 3, for example,
that the difference between the throat and intake diameter is
relatively small.
The (At/Ai) ratio is also of great significance
in limiting the range of the projectile. The process by
which the projectile of the invention "expels" the shock
wave and thuR establishes choked flow conditions within the
central passageway at a predetermined Mach number will now be
described.
With the condition of the flow started the !
Mach nu~ber throughout the central passageway is supersonic
as shown in Fig. 6(c). Contrary to subsonic flow, the
~ach number in the converging section decreases towards the
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throat and its value is also related to the throat to intake
area ratio. As Mach number decreases (i.e. as the projectile
loses speed in flight) towards one at the throat, the shock
; wave wlll appear in the throat, but it will be coming from
the rear or trailing edge of the projectile. At a slightly
lower Mach number, the shock wave will move into ~he
converging section and this being an unstable condition,
it will only stablize itself in front of theprojectile as shown
- in Fig. 6(a). This is the high drag condition which is
required to decrease the range. The equation which defines
the shock expelling process as a function of Mach number is
as follows:
At M
A~ ( y l 2)Y~ (Y-~)
;2 (M = flight Mach number
~ = ratio of specific heats)
This equation defines curve B on Figure 7.
The above described processes of swallowing
and expelling the shock at selected Mach numbers has
application in most direct fire gun systems where the overall
range of the weapon is large with respect to the maximum -
target range (e.g. a 105 mm Tank Gun). In general, these
processes provide for low drag (flow started) over the
first part of the flight path to the target followed by a sudden
~rah~ition to high drag (shock expelled) conditions to reduce
the range.
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Thus, by virtue of the shock swallowing and
expelling feature of the invention, a STUP(or tubular
projectile)can be designed with a much lower drag
coefficient than a conventional projectile at high Mach
numbers (at A in Fig. 6(d). At a predetermined Mach number the
coefficient of drag CD rises sharply to a high level (at B
in Fig. 6(d). These features help one to tailor the design
_ I
of the STUP to a seIected drag curve. (For some applications
this may be relatively easy while it is more difficult for
others which call for very low drag and a high choking Mach
number.)
Returning again to the practical example of the
105 mm Tank Gun STUP Practice Round, an (At/Ai) ratio of 0.7
was selected so as to provide for swallowing of the shock at
start-up or launch. From the plot of (At/Ai) vs. Mach number-
in-Fig.7 it can be seen that for this model, choking will
occur at about Mach 1.8 or somewhat greater (depending on
boundary layer effects). Reference to various tests carried
out, which verify that this choking effect does take place as
predicted,will be made later on in this disclosure.
With further reference to Fig. 7 (curve B) it
will be noted that as the ~At/Ai) ratio approaches one, the
`Mach number at which choking occurs also approaches one. If the
central passageway through the projectile is simply a uniform
diameter bore, i.e. if no "throat" is provided, choking will
not occur at all at supersonic velocities (assuming no boundary
layer effects); hence it is essential that the projectile have
an internal configuration arranged such that the choking
phenomena can take place e.g. by providing an annular internal
wedge or some other configuration capable of producing a
contraction of the flow. A co~posite leading end wedge is
desirable in most cases as it can readily be tailored to provide
the desired flow pattern but it should be realized that
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10643Z~
a fully operable projectile can be designated with an
external surface which is cylindrical or even slightly tapered
throughout its length toward the trailing end.
F~rther important considerations are the wall
thickness ratio t/R (where t= maximum wall thickness and R= one
half maximum outside diameter of projectile) t and the size of the
wedge angle(s) at the leading end of the projectile. The wedge
angles should be kept reasonably small to ensure that
attached oblique shock conditions at the leading edge are
achieved to minimize pressure drag. In the case of a composite
wedge design as shown in Figs. 1-3 the included angle 9 (see
Fig.2) between the inside and outside wedges should be less ,-
than about 15 ahd better still less than about 10. At the
same time the included angle will usually be greater than about
5. In the case of a design having an internal leading end
wedge only it is recommended that the wedge angle ~i.e. the
angle between the annular wall defined by the wedge and the
projectile axis) be less than about 10, and preferably from
about 3 to 5. An overly blunt leading edge may result in
detached shock wave conditions and high drag so therefore the ~ -
leading edge should be reasonably sharp; a "knife edge" leadingedge
howe~er is not necessary and for practical purposes the edge
may be rounded off to a small radius, e.g. .005 inches, to
reduce the possibility of edge damage during handling.
It is known in the art that the range of a
projectile is decreased by increasing the ballistic coefficient
CDA/W. In a tubular projectile, the coefficient of drag CD is
increased by increasing the wall thickness. Increasing the
wall thickness also increases the weight. There is, for any
particular tubular projectile, a wall thickness ratio t/R where
CDA/W is optimum and the retardation during flight is minimum.
The wall thickness ratio t/R is a parameter which is therefore
useful in expressing the various relationships which exist.
- 16
-: . . . - . . . , -
. . :
- - . .: , , . . .: ., - . . .
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~ The wall thickness ratio (t/R) is chosen so as to
achieve a minimal ballistic coefficient CDA/W. From tests of
various tubular projectiles, curves relating drag coefficient
CD, Mach number and (t/R) have been developed. Figure 8
shows generally the type of relationship existing between these
variables for tubular projectiles of the type belng discussed
herein. Experience has shown that (t/R) should be between about
0.15 and 0.45 in order that the drag coefficient may remain
within acceptable limits. A graph illustrating generally the
relationships between the ballistic coefficient CDA/W and
thickness ratio (t/R) for tubular projectiles of the type being
discussed herein is given in Fig. 9 which further illustrates
the importance of selecting the appropriate thickness ratio
so as to minimize the ballistic coefficient. (The curves shown
in Figs. 8 and 9 will vary depending on the exact configuration
of the projectile and are given here only for purposes of
illustration).
The flight weight W of the pxojectile is dictated
by the interior ballistic limits of the gun system. In the
example we are considering (the 105 mm STUP Practice Round, for
the 105 mm Tank Gun L7Al, the maximum shot weight of the APDS
(projectile plus sabot) allowa~le to achieve a muzzle velocity
of 1478.3 m/sec is about 13 lbs. In this case then, the
shot weight of the STUP practice round (projectile plus pusher
base and rider band etc.) cannot exceed 13 lbs if the-initial
velocity of the tubular projectlle is to match that of the
weapon which it is designea to simulate.
- 17 -
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Another important phase of a STUP design procedure
is estimating the dynamic stability. This involves the ratio
between the gyroscopic and aerodynamic moments. In summary, the
gyroscopic stability factor must be greater than one for the
projectile to be dynamically stable. The gyroscopic stability
factor Sg is defined as follows:
I 2 p2
Sg= ~ :
where~ = 8 pd3 v2 Cm ~,
: and , :
x ~ axial mo~ent of inertia
Iy - transverse moment of inertia
p - angular velocity
p - air density
d - max. body diameter
. . V - velocity a~ the mu~le
~ :
., Cm ~ static moment coefficient
In the 105 mm Tank Gun L7Al for example, the
parameters p, p, V would be the same for both STUP and the APDS. ~`
But the value f Ix is much greater for a full-calibre STUP than
a sub-calibre APDS while Iy would be of the same order. Then
from the equation above the ratio (IX2/I ) would be much larger
i for STUP than the APDS. The value of Cm is very difficult to
?i - estimate but it is of the same order for both projectiles. Then -
it can be shown that the 105 mm STUP Practice Round has a
much larger Sg(i.e. Sg >> 1) than the APDS and this has to
be considered in~the trajectory match. The STUP tends to fly
a flatter trajectory so it does not fall vertically as much
as the APDS with range. The fineness ratio ~/D (where ~=
.
- 18
- 106432~
projectile length~and D - maximum outside diameter is
theoretically limited by the maximum allowable stability
factor Sg. In practice the,(/D ratio may vary from about 2
to about 5. Finally, in the process of matching the
trajectory of the APDS with STUP, a compromise between
muzzle velocity, time of flight, ballistic coefficient and
dynamic stability is made and this is achieved through
theoretical estimation and full scale experimental iteration
techniques.
The basic equations for determining the drag co-
efficient and the ballistic coefficient are given below. These
equations may be used in a simple computer code e.g. (APL
language) to arrive at a design which fulfills the trajectory
requirements. It should be emphasized that the invention is
not to be limited to any particular method of computation.
Conventional mathematical techniques based on known aerodynamic
principles may be employed to optimize the design; however,
the summary of the governing e~uations involved and the
simple computer code used will be of assistance to those
skilled in the art in designing a STUP to meet a particular
set of requirements. Reference is had to Fig. 10.
The basic aerodynamic principles may be had
from the following references: -
A. Amcs Research Staff, "Equations, Tables and Charts for
Comprcssible ilow", NACA Report 1135, 1953.
B. ~oerner, S., "Fluid-Dynamic Drag".
. C. ~ACA R~l L53C02.
.
-- 19 --,
1(~643Zl
The nomenclature to be used in the governing
equations is as follows:
L c Projectile length
Di = Intake Dlameter
Dt = Inside or ~hroa~ diameter
Do = Outside Diameter .
t = Wall thickness
- R = One half outside diameter Do
0O = Outside wedge angle
0i = Inside wedge angle
A ~ Reference Area (~Do2/4)
M ~ Mach Nunlber
V = Velocity ` . -:
., .
, ~ ~ Ratio of Specific Heats
, . .
' Cp = Coefficient of Pressure
CD = Coefficient of Pressure Drag - Outside Wedge
ÇD = Coefficient of Pressure brag - Inside Wedge ~ :
Pi , : ,
Cf = Coefficient of Friction
Re ~ Reynolds Number .
. Cp - Coefficient of Base Pressure
I p - A,ir,Density. - - ---~~~~
, ~ ~ Air viscosity
- 20
.
10643Zl
The following assumptions are ma~e in the
mathematical analysis:-
(a) Two-dimensional fluid flow (only valid for t/R< .5).
(b) Oblique shock wave attached to leading edge of
projectile.
(c) No choking i.e. supersonic flow in central
passage of the projectile.
.
(d) Zero angle of attack.
(e) Geometry as shown in Fig. 10 (single or ~-
composite wedges).
The coefficient of drag (CD) is expressed as follows:
CD = CD (prcssure) + CD (friction) ~ CD (base)
e coefficients are based on total projccted area (~Do2/4)
(a) Pressure Drag Coefficient (CD ) is expressed as follows:
Prom Ref. A for a two-dimensional wedge
: C 2 ~ + (y+l)M4 ~ 2~1) ~2 + 1 r( ~ ~8
P (M2 l,l,2 . 2(~2 l,2 (M2-l)7~2 L 16
7+12y-3y2 M6 + 3 (y+l~M4~2M2 ~ 3~ 93
FOT the outside wedge (~ = ~0)
P E ( )
. .
,,
- 21 -
~, -
: - . .
.
1064321
For the i~side we~ge (~
D = Cp
and D CD CD
p po Pi
(b) Skin Friction Drag Coefficient (~D )is given by the
following:
From Ref. B for tubulent flow
` Cf = K Cf'
where K = (1 ~ .15 M2) 432
.. ...
R = pVL~ !
The skin friction is based on wetted area Sw ~ ~L(D ~ Dt) ~-
and CD = Cf4L
; (c) Base Drag Coefficient (CD )is determined as follows:
` The base pressure coefficient for a two-dimensional
body is gi~en in Ref.C. Base drag data was also obtained in wind
tunnel tests using 105 mm models. The wind tunnel results corre-
lated well with the data of Ref.C. The following function was
derived from the data:
PB Ao f AlM + A2M + A3M 3 1 . 5 < M ~ 4 . 5
where Ao = .6331
Al . 33257
., I .
A2 = .06619 . .
3 - 004
- 22 -
,'' ~ 1~
I .
106~321
Other Formulas which are well known to those
skilled in the art are as follows:
Ba11istlc Coefficient
CDA 2
W (ft /lb)
Retardation
dV _ ~16.1 p V CDAj/W (ft/sec/ft)
The above equations have been used in a computer
program (APL Language) and for.convenience the program listing
is given below
APL LISTING
.
INPUT nATA(lines 1! 2 and 4)j
.~. . .
L - Model Length (in)
DO - Model Outside Diameter (in)
T0R - Wall Thickness Ratio ~t/Rl
AOR -Intake area Ratio (At/Ai) .
(Area of throat/Area of'in~ake)
TETAO - Outside Wedge Angle (deg)
TETAI -.Inside Wedge Angle (deg)
D~NS - Material Densi*y ~lb~in3)
PI.TCH - Gun Rifling Pitch (~t3
~iO - Atmospheric Density ~.002378 slugs/ft3)
MU - Viscosity (3.719 x 10-7 slugs/ft sec)
.S - Speed of Sound tlll7 ft~sec~
VEL - Velocity tft/sec)
Ao,A1,A2,A3 - Constants in the polynomial that defines the base drag
coefficient
. .
OUTPUT DAl`A tLines 57 to 62
.
T - Wall Tl~ickness (in~
DT - Inside or Throat Diameter (in)
Dl - Intake Diameter (in)
TO - ~lickness Outside Intake Dia~neter (in)
Tl - Thickness Inside Intake Diameter (in) : -
LOD - Model Fineness Ratio
Longth of Outside Parallel Section (in)
L2 - Length o Outside Wedge (in) l:
L1 -- - Length o~ I~side W~dge lin)
-- 23 --
10643Zl
` ~ . .
. 2
L4 _ iength of ~nsidc Parallel Section (in)
V - Material Volume (in3
- W - Model ~eight (lb?
CPO - Coefficient of Pressure - Outsi.de Wed~e
C.DPO - Coefficient of Pressure Drag - Outside Wedge
, CPI - Coefficicnt of Pressure - Inside Wedge
CDPI - Coefficient.of Pressure Drag - Inside Wedge
CDP - Coefficient of Pressure Drag
RE - Reynolds Number - -
CF - Coefficient of Friction ~.
CDF - Coefficient of Friction Drag
~; PPB - Base Pressurc Coeffici.ent
~ CDB . - Coefficient of Base Drag
M - Mach Nu~ber .
~, C~ - Coefficient of Drag
- 13allistic Coefficient (ft2/lb) ,
VELD. - Velocity Retardation (ft/sec/l00 ft)
'!'.' , SPIN- Angular Velocity (reYlsec)
VT.- Tan~entiai Velocity (ft/sec)
STRESS - Trangential lhoop) Stress (lb/in2)
.
Noto: Reference area used in the aerodynamics is based on outside diameter.
.
.
; , . . .
: .
.~ , . ': ~' ' .
.,', ' ' ' "~, . ' '.
., . '
., ~ , .
, ~ 24 ~ ~
;,
,; ' ~:.
, .
.. . .
.'.~
' '
10643Z~ -
.
APL LISTIhlG
.
i VSTUP2 [ O 3 V Model 2 6
VS~UP2
[13 L,DO,TO~,AOR,TETAO,TETAI,DEOS,PITCH
L2~ RHO,M~,S,VEL
, [3] .~VEL.S
[4] A0,A1,A2,A3
[53 DELO+TETAOxol.l8o
~63 DELI+TETAIxo1 180
[7] T+TORXDO.2
C8] DT~DO~2xT
[9~ DI+DTX(1.AO~)*.5
[ 103 TO+(DO-DI)-2
TI+(DI-DT).2
[123 L2+TO 3oDELO
[13] L3+TI-30DELI
[14] L1+L-l2 `
[15] L4(L2-L3
[16] Vl+(olXLl 4)X(DO*2)-DT*2 j
[17] V2+.?618XL2x(DO*2)+(DoxDI)+DI*2
[18] V3+.26l8xL3x(DI*2)+(DIxDl7)+DT*2
[193 V4+olxL4x(DT*2).4 '~20] V+V1+V2+(-V3)-V4 -
[21] W+DE~SxV
[26] M2l(M*2)-1
[27] CP1+2.M2*.5 (
[28] CP2+((2.4X~*4)-4XM2).2x~2*2
[293 CP3+(1-(~2*(7-2)))X(.36xM*8)-(l~493x~*6)+(3~6x~l*4)-(2x~*2)+4-3
30] CPO+(CP1xDELO)+(CP2xDELO*2)+CP3xDELO*3
[31] CDPO+CFOX1~(DI~DO)*2
[323 CPI+(CP1xDELI)+(CP2xDELI*2)+CP3xDELI*3
t33~ CDPI+CPIX((DI*2)-DT*2) DO*2 '
[3 4 3 CDP+CDPO~CDPI ,
[36J RE+RHoxvELx(L~l2).M
C373 CFP+((3~46X(10ORE))-5~6)*-2
[38] K+(1+.15X~*2)*-.432 ` '
C39J CF+~xCFP
[ 403 CUF+CFX4XLx(DO+DT)~DO*2
[41~ PPB~A0+(A1x~)+(A2x~*2)+A3xh~*3
[42] CDB+PPBX1-(DT.DO)*2 i:
[ 4 3 ] CD~CDP+CDF+CDB
[44~ CDA+(CDxolx(Do*2)t4).w i
[45~ CDA+CDA~144
C46] VELD+16~1xRHOxVELxCDAx100
C473 SPI~+VEL~PITCH
[~8] SPI~R+SPINx2
[4gJ VT+SPINPXDO~24
[50J STRESS~(DEOSX(VT*2)~ 32~2)x12
[ 513 LOD+L~DO -
L573 T,DT.DI,TO,TI,LOD -
C583 L1~L2~L3~L4~V~W~VCs~cs~wcT
[ S 9 ~ CPO . CDPO . CPI, CDPI s CDP
[60] RE,CF,CDP ~ -
C613 PPL~CDB
L62] ~.CD.CDA.YELD - , I
-V . .
- . . .
. . ~ .
~0643Z~
Example Model 26
Input(l)~o 4~127 0.2125 0.7 3 3.5 ~.282 6.25
(2)0.002378 3.719~ 7 1117 485~
(4)~.63~31 0.33257 0 06619 0.004
Output(57)~.43~5 3-.25 3.8845 0.12125 ~.~1725 2.4231
(58)7.6863 2.3137 5.1869 2.8733 40.082 11.303
0.42706 0.12043 0 12043
(59)0.02835 0.0032339 0.033815 0.0089873 0.012221
(60)2,s843E7 0.0013934 0.02414
(61)o~10973 0.041681
(62)4,342 0.078042 0.000~414 11.91
' ~ . ,
By following an iterative procedure on those input
- parameters which can be varied e.g. TOR, TETAO, TETAI, the
following data was produced:
(See Input Data)
L ~ 10.0 or L = 10.0 ins.
DO ~-- 4.127 or Do o 4.127ins = 105 mm
; TOR C 0.213 or t/R= 0.213
.. ~
.
AOR ~-- 0.7 ) or At/Ai=O7
TETAO~-3 or 0O = 3
TETAI~- 3.5 or ei = 3.5
DENS ~- .282 = (mat.dens.)=.282 lb/in.3
PITCH~-6.25 = ~rifling pitch)=6.25 ft.
RHO ~ .002378 = (atoms density in slugs/ft.3)
MU ~ 3.719E 7 = (air viscosity)3.719xl0 7
slugs/ft.sec~
S ~- 1117 = (speed of sound)1117 ft/sec.
VEL ~- 4850 = (muzzle velocity)~4850 ft/sec.
.
.A, .
Ao,Al,A2,A3 ~- .6331, -.33257,.06619,-.004
and (See Output Data) (Pariial Listing Only)
.
.. . .
;~ . -
.5
- 26 -
. ,
,5
,, .
'. ` '' .''~'i.' ' ' . '' ' "'` :
': ,. ' . ': , , ,
-'' . . ,
` . .
- ~0643Zl
DT ~ 3.250 or throat diameter = 3.250 ins.
DI < 3.88 or intake diameter = 3.88 ins.
LOD<- 2.42 or fineness ratio (L/D) = 2.42
L~ 7.81 or length of outside parallel section = 7.81 ins.
L2 ~-- 2.29 or length of outside wedge , = 2.29 ins.
~ 5.19 or - length of inside wedge = 5.19 ins.
L3
W ~- - 9.88 or weight of projectile = 9.88 lbs.
The remaining output data is not given here. The
above partial listing gives the basic parameters for the 105 mm
STUP Practice Round design being considered here by way of example.
It will be noted here that the optimum wall thickness ratio
(t/R) is calculated to be 0.213; the optimum wedge angles were
found to be 3 deg. for the outside wedge and 3.5 deg. for the
ins~de wedge. Other pertinent dimensions are given above.
These dimensions were applied to the particular embodiment shown
in Figs. 1 to 3 to provide a successful 105 mm practice round.
The material used for the projectile was AISI 1018 hard drawn
steel. The leading edge 26 shown in Fig. 2 was rounded off
to a small radius (i.e. .005 in. radius).
The accuracy of the STUP as compared with
.
conventional projectiles i.e. APDS and TPDS (Target practice
' discarding sabot) has been demonstrated in various trials. The
~ following is a portion of a test record, relating to the
development testing of the 105 mm ST W described above
- 27 _
:
: ,,., . ~ ~ . .. .
i06432~
Weapon used: 105 mm Tank Gun
Target: 20' x 20' at 1000 metres
Ammunition: 1) 105 mm STUP - B ~ounds
2) 105 mm APDS/T C35Al - R Rounds
3) 105 mm TPDS~T C-36 - W Rounds
The test was carried out with the following sight
settings: ~ine - 0 mils; Elevation - 2.0 mils. The test is
outlined in the following table and the points of strike of
- the test rounds are plotted in Fig. 11~ It will be seen that
the STUP practice rounds made in accordance with the invention
are at least as accurate as the conventional APDS and TPDS
rounds. , ;
TABLE ~
' ' ' ',',
.. . .
~` Rds. No. MuzzleTerminal Point of StrikeSpin
Velocity Velocity ~
meter/sec meter/sec Horizontal Vertical rev/sec.
- : .
, W-l 1500.01287.9 10.9 11.5
' W-2 1501.41292.4 10.9 10.0
R-l 1468.91360.5 7.7 11.3 8I5
, , ! .
B7L1 1459.01316.1 10.2 13.0 774
B7F1 1482.91342.2 12.2 16.7 770
B7F2 1478.31341.9 11.4 9.0 815
, B8L1 1459.21327.6 10.5 11.2 789
.; .
R-2 1474.61378.7 11.0 10.9 811
t
~ B8F1 1474.21340.7 12.0 13.0 854
.~. , .
B8F2 1466.61334.0 10.0 8.7 812
Remarks: l-Point of Aim: ~or:10.0 ft. Vert: 10.0 ft.
,:
~, '
2-Trunnion height 4'4-1/2" Height of aiming mark from
ground 13' 2".
' 3-Strike co-ordinates measured from left bottom corner
of target. I
- 28 -
.
~0643Zl
The criticality of the (At/Ai) ratio including
the choking phenomena has been demonstrated in various trials.
Reference will be had to a set of field trials using both
conventional APDS projectiles and several varieties of STUP
projectiles. Fig. 12 illustrates the several models tested.
The models shown in Figs. 12(a) and 12(b) were provided with ~ ~
composite leading end wedges, Fig.12(a) with a composite I -
wedge CW(S) and Fig. 12(b) with a slightly modified wedge
CW(M) while the model of Fig. 12(c) was provided only with
an internal leading end wedge t~W). The three models of
Figs. 12(a),-12(b) and 12(c) were provided also with
external trailing end wedges; this helps to reduce the base
drag coefficient somewhat but does not have a substantial
effect on the overall performance in flight. TXe basic
dimensions of the sevqral models shown in Fig. 12 are as
follows: (See Fig. 10 which illustrates how these
dimencionc ar- applie~.3
'''
: .
. . .
~ . .
;
- 29 -
~ .
~ ~ .
~064321
Fig. 1 2(a) CW(S) ~ig.12(b)_CW(M)- F g.12(c) IW_
L 10.0 in. 10.0 in. 10.0 in.
L2 2.12 in. 1.414 in.
L3 1.878 in. 2.134 in. 2.64 in.
Di 3.578 in. 3.651 in.' 3.80 in.
Dt 3 050 in. 3.050 in. 3.05 in.
Do 3.800 in.~ 3.800 in. 3.800 in.
t 0.375 in. 0.375 in. 0.375 in.
R 1.900 in. l.goo in. 1.900 in.
~o 3O 3O OO
ei 8 8 8
At/Ai 0.727 0.700 0.640
t/R 0.197 0.197 0.197
L/D 2.63 2.63 2.63
.-. .~ . ., ---
. . . ,~ .
The models shown in Fig. 12 were all fired at equal
;~ velocities under similar conditions and the flight paths tracked , -with doppler and tracking radar. The flight paths were compared
with the flight path of a conventional weapon(A.P.D.S.)fired under
the same conditions and with the same shot weight. The flight
; paths of the several projectiles are shown in Fig. 13. The
.; .
internal wedge model, having an At/Ai ratio of only 0.640 was
choked at all times and had an overall range of less than 20,000
ft. The two composite wedge models, having At/Ai ratios above
the critical values,were unchoked immediately after launch but
,became choked at about 7.0 to 7.2 seconds after launch and had
ranges of 45,000 ft. approx. for CW(M) and 50,000 ft. approx.
for CW(S). The conventional APDS had a range of about
72,000 ft. The velocity history curves of the composite wedge
models CW(M) are shown in Fig. 14. The inflection point on
the curve demonstrates that choking did occur as predicted.
The drag coefficients of the several models shown in Fig. 12
'
~, , ................ .- .......... - :
.
~06432~
were calculated against flight Mach number and the results are
shown in Fig. 15. The two composIie wedge models having
At/Ai ratios of .727 and .700 respectively show a sharp
transition from low to high drag at slightly less than Mach 2
which is in accordance with the theoretical predictions. The
internal wedge model (IW) was choked and exhibited high drag
at all Mach numbers. The drag coefficient of the APDS
showed a gradual increase with decreasing Mach number as
anticipated.
As noted previously, the tubular projectile of j
the invention can be varied in design considerably so long as
the basic design criteria established above are observed.
For example, the trailing end can be blunt, as shown in
Figs. 1-3 or it can have a relatively sharp edge (see '
Figs. 12(a), 12(b), f~r exampie). The internal passageway
of the projectile need not be precisely cylindrical, e.g. -
a small gradual increase in diameter toward the trailing end
may be beneficial in certain designs and in like manner the
diameter of the exterior surface may be gradually decreased
(tapered) toward the trailing end. U~ually the degree of
taper for the inside and outside surfaces does not exceed
2 or 3 degrees. The leading end of the projectile is preferably
provided with a composite wedge as shown in Figs. 1-3 for
,. - . .. , ............ .. .. . . . ..... .. ... ... .. ,, . .
example. However, the external leading end wedge is not an
essential feature of the invention. In either case the
generatrix of the leading wedge portion may be straight, as
shown in Figs. 1 and 2, or curved in some suitable fashion.
In all cases however, the At/~i ratio must be large enough
to ensure that supersonic flow conditions in the central
, ~ ~ ., ,. ' :
3i
_ . . . ~ . . .................................. . , _
~ -
10643Zl
passageway can be achieved at the launch velocities in
question. The provision of a trailing edge may be helpful
in many cases where it is desired to reduce the base drag.
The modification shown in Fig. 16 may be useful
in certain instances. This projectile 50 is basically the
same as that shown in Figures 12~a), 12(b). The embodiment
of Figure 16, however, has a trailing end wedge section adapted
to support a pusher base 52 in the form of a conventional non-
tubular projectile the~leading end portion of same having a
conventional ogive shape. Pusher base 52 is configured to
have a radially extending shoulder 54 adapted to abut against
a bearing seat 56. Although not shown in Figure 16, it is
common for the trailing end wedge on the projectile to
releasably carry at least a driver band (not shown) and prefer-
. ; ~
ably a sealing ring as well. Such a driver band and sealing ringfunction in the same manner as described with reference to
Figures 1-3. Once the tubular projectile 50 has been launched
it establishes a normal shock wave just ahead of the leading
I .
edge thereof. As a result, high stagnation pressures are
establ~shed within the central opening of the projectile which
pressures cause separation of the pusher base 52. Because the
drag forces on the conventional projectile, i.e., the pusher
base 52, exceed those on the tubular projectile 50, each
pro~ectile will follow its own trajectory. This technique can
be used for launching a tubular projectile from an aircraft
where the pusher base has to be designed to follow a stable
trajectory; otherwise it might be ingested by the engines.
Tubular projectiles as envisaged herein take
full advantage of the spin inherent in a spin-stabilized
projectile. That spin may be used to inhibit ric~chet of
the present tubular projectile beyond the desired target area.
By providing ~uitable riflings in the launch barrel or tube,
- 3 ~ -
~ : - ,. - .
1064321
the tubular projectile is launched with a spin rate in the
range of about 500 to 1,000 rps (30,000 to 60,000 rpm), and
more preferably, in the order of about 750 rps. Rates of spin
in this magnitude generate stresses in the shell-like body
section which are in the order of 60,000 to 65,000 psi.
- Naturally, a projectile according to this invention, will be
made of a material seIected so that the material can withstand
the stresses generated by these high rates of spin. Successfully
tested prototypes of the tubular projectile have been made of
AISI 4340 steel, an annealed alloy steel. Another acceptable
type of steel and which is preferred on account of its lower
cost, is AISI 1018 steel. The latter is a plain carbon, hard
drawn steel having a yield strength in the range of 65,000 to
70,000 psi.
¦ It is highly desirable that the material of which
the tubular projectile is made be selected so that the yield
strength of same will exceed by only a small margin, the
calculatable stresses imposed on the surface of the tubular
projectil~ when launched with the desired rate of spin. When
such a spinning tubular projectile encounters a target or other
object in the target area, additional loads and stresses due
to the impact will be imposed on same and the additional stress
generated by the impact will cause failure of the projectile.
It has been found that the body section of the tubular projectile
will crac~, with these cracks being propagated to cause
breakup of the projectile in a manner broadly similar to the
peeling of a banana. As the projectile body section breaks up
in that manner, a greatly increased aerodynamic drag is imposed
on the resulting fragments. Consequently, the fragments slow
down rapidly, and any tendency for excessive ricbchet beyond
the target area is severely cut back, and perhaps even
eliminated. Most importantly, consistent reliability in
controlling unwanted ricochet beyond the target area is
obtainable.
_ 33_
