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REPO REPORT RT TYPE TYPE
12
RIFL RIFLIN IN
WIST WIST
C OV OV E E D
FmaI
Febr Februa uary ry 1995 1995 ;UBTITLE
N D D AT AT E
5.
ESIG ESIG
FUND FUNDIN IN
NUMB NUMBER ER
AMCMS: AMCMS: 6111.02.H6 6111.02.H611.10 11.10 6. AUTH AUTHOR OR(S (S
oyce oyce W. oane oane
REPORT REPORT NUMBER NUMBER
U.S. U.S. Army Army ARDE ARDE Bene Bene Labo Labora rato tori ries es AMST AMSTAA-AR AR-C -CCB CB-O -O Wate Waterv rvli liet et NY 1218 121899-40 4050 50
9. SPON SPONSO SORI RING NG
O NI NI TO TO RI RI N
ARCCB-1R-9S011
G E C Y N A E (S (S ) A N
10. SPONSORIN SPONSORING/MO G/MONITOR NITORING ING AGEN AGENCY CY REPO REPORT RT NUMB NUMBER ER
A DD DD R S S( S( E
U.S. U.S. Army Army ARDE ARDE Case Comb Combat at Arma Armame ment nt Cent Center er Pica Picati tinn nn Arse Arsena nal, l, NJ 0780 078066-50 5000 00
11 SUPP SUPPLE LEME MENT NTAR AR
12a, 12a, DIST DISTRI RIBU BUTI TION ON
NOTE NOTE
AVAI AVAILA LABI BILI LITY TY
1 2b 2b . D IS IS T I BU BU TI TI O
STAT STATEM EMEN EN
C OD OD E
Appr Approv oved ed fo publ public ic rele releas ase; e; dist distri ribu buti tion on unli unlimi mite te
13 ABST ABSTRA RACT CT ( Ma Ma XI XI mu mu m 2 0
w or or d
This This re or co si er th effi effici cien en solu soluti tion on will comp comput ut th rifl riflin in cUIV cUIVew ewhi hich ch ro uc
14Rl 14RlHB HBJE JECT CT
th foll foll in ro lem: lem: give give pres pres ur trav travel el hape hape at or virtually proj projec ecti tile le torq torq cUIV cUIV of an desi desire re shap shape: e:
TERM TERM
ing, ing, Twis Twist, t, Prog Progre ress ssiv ive, e, Gain Gain Torq Torque ue
pa ti ular ular ro nd
32 1 6 P RI RI C C OD OD E
17.
SECURITY SECURITY CLASSIFICAT CLASSIFICATION ION
UN ClASSIFIED N S ~ 7 54 54 00- 0, 0, -2 -2 B O - 55 55 0
SECURITY SECURITY CLASSIFICAT CLASSIFICATION ION
UNCIASSIFIED
19.
SECURITY SECURITY CLASSIFICAT CLASSIFICATION ION OF ABST ABSTRA RACT CT
UNCLASSIFIED
2 0 L IM IM IT IT AT AT IO IO N O F A BS BS T A C
_ill ann P~,?,~:rlO€"O
2gB· 2gB· "2
ro tl\
AI\j)L
St
i~~·'t
ACKNOWLEDGEMENT...................................................
..... ... ... ... ...... ... ... ... ... ............ ... ... ...
, .... ...... .... .. ...... ...... .. .......... .. .. .. .. .. ..
1
ACKNOWLEDGEMENT
Fi
INTRODUCTION
considered.
ca
ic
ed re en ra ed
ro ec les, les, co tant tant
is rifl rifl ng
as ma
at ry
he
vari variou ou reas reason ons. s.
fort fortun unat atel el estr estr ct ve base base re
es re
mo
effo effort rt at aria ariabl bl tw
te ni e. Fo in ta ce
lt ug
rifl riflin in
esig esig
av
ed anal anal ic
tr itio itio al prog progre ress ss
tw
form form
or
rifl rifl ng oe
ere. ere.
As
\ [_
Unannounced Just Justit itic ic Lo
[J
._=~~:.~:~~.~~~_~
Distribution
. . -
Availabi~t~_~~~~.:~_ ... ... Dist
Avail
ai·"r~/cr
Special
. .
RIFL RIFLIN IN
SYNT SYNTHE HESI SI
Let P(x)
be by the
followin followin equation equations: s: P A n S in in 4J 4J =
dt
where
I>
he roje roject ct
an
is it velo veloci city ty
Let f(x) f(x);;;:r8(x)
First, /,
-;;;:v-;;;:-f(x)
dx
dt
/,
-(vf(x»
dt
dx
and dv
dvld
dt
dx
-;;;:v-;;;:--v
2d
nSi
_1 m!!._v
dx
L;;;:nrCost/J;;;:iv !!'_(vfl(X»
rdx
Solv Solvin in fo iv
Cos¢dx
(vfl(X»
AP(x)-
IV
Tan~_(vf'(x» dx
!.m-v dx
but Tan~=t'(x)
or -P(x) =-
In egra egra in
+-'-(vf'(x)l
this this we av
where
v(x)
Let
P(x) x.
shap shap
func functi tion on L(x)
Reca Recall llin in
el(x)
that that
fl(x)
l(u)
lex)
;:"P(u)du
and
Hence, v(x)
an
ther theref efor or fl(x)
em
2A
U(x) (;r; l(u)U(u)
Letting J(x)
/? A(u)U(u)
fl(x) 2AR2
J(x)
(I) 2A R'1I(l)J(l)
-------""-~~-''-''rU(l)J(l)
Therefore jl(X)
fl(l)J(I)U(x)J(x) U(I)J(I)I(x)
where g(x)
(x), getting
J(l)l(x) J(1)I(x)
mp by
x),
J(x)
=;:%
J(x) (esp (espec ecia iall ll
Recall ll that that J. Reca
A(u)lJ(u)dU
A(O)=O
x.
Since
Hence, J(x)
=;:%
A ( U ) lJ lJ ( U ) J 1 ( u )d )d U : ; P K ( u )d )d I (u (u )
where K(x)
21(x)lJ(x) p(x)
by I'Ho I'Hosp spit ital al's 's rule rule J, J(x)
where
x)
(O
fo is give give
diff differ eren enti tiab able le
and
Inde Indeed ed g(O),
,t,
x:
A (X (X )
AI(O)
P(x) P(x)
Xpl(O)
l(x)2:
!..X2pl(0)
J(x)-
p'(O)
211(0) 211(0)U(0 U(0 :K(O) :K(O) pl(O)
I(x)
and (0): (0): K( )I(I )I(I
J(I)
P(O) =0 I(x)
U(x) K(x)
1(0)=0 J;xp(u)du.
~f'(X) 2.l(x)U(x) P(x)
K(O) K(O) 211( 211(0) 0)U( U(0) 0) pl(O) Six)»foxJ(u)dK(u)
J(x)
K(x) K(x)J( J(x) x) Sex) Sex)
g(x) (0)
r(x)
i:~~=~ K(O)J(l) J(/)
JI(l) U ( X ) g(X) U(l)
f1x)
f1Qg(x) {~f'(l)r(l-g(xY~)
fix) fojl(u)du
Th eq atio ations ns through U. Fu
efin efinin in
implic icit itly ly beca becaus us (x do so impl (x vergen ence ce (x). Co verg
ex reme remely ly rapi rapid, d, taki taki
(x),
is roug roughl hl
cons consta tant nt near near unit unit fo al prac prac ical ical rifl riflin in co figu figura rati tion ons. s.
mi
Jt(x)
kP(x)
fl(x)=r(l)
Hence, U(x)
c on on s t = = U
K(x)=2kU=const
S(x) ;;"ldK =0 J(x)
x)
( x) x ) = 1 (1 ( 1 ) 2k 2k UJ U J (x (x ) = 1 2kUl(I)J(x)
Therefore JI(X)
(imm (immed edia ia
conv conver erge ge ce
co stan stan
JI(l)
wist wist rifl riflin ing) g)
Now, Now, reca recall ll ha
where 2A
JI(1)/(1)
" ' ''" ! . .
rU(l)J(l)
/PUA· defi define ne
(AocP),
~rf'(I)J(I)_(~rf'(l)
2klP 2klP Pc
(a)
cons consta tant nt twis twis torq torque ue q:_E_
/J
I). IJ(l)U(l) IPIJ(l)
is
a=l
rt ally ally inde indepe pend nden en
of
andf(l). O"s.
0'
x=O
Ais
having Ab
(0 cons consid ider erab ably ly larg larger er than than
Ali (l).
0'
proj projec ec il
RE
wh le
UR
vi
R AV AV E
rq eles eles
F UN UN C
obviou ousl sl impr imprac acti tica ca (l), an obvi
xit. xit.
ON
If
10
P(x)=axe
ii
a, b,
P(X1)=Pl pl(X (X )=P2 (Xl) an
(PI)
pressure ( P 2 )
peci pecify fy th
(x
This This repr repres esen enta tati tion on seem seem insu insuff ffic icie ient ntly ly flex flexib ible le howe howeve ver, r, so th foll follow owin in piec piecew ewis is expo expone nent ntia ia repr repres esen enta tati tion on is sugg sugges este ted: d:
where
(x)
-b
PlX1)=Pl Plx1)=O lX )=P2 lX )=P2 lx )=Plx PlxJ=P3
11
Ir~2) c2
p}(x):
and
PIe
C}'
Therefore
12
Letting
Clearly,
an
ince ince
1,/( 1,/(00 00)= )=00 00
Diff Differ eren enti tiat atin ing, g, we have have
j'(x) jll(X}
rrlna+p rr(lna)2
Hence,
Theref Therefore ore
since/ since/(O) (O)=O, =O,f(O f(O
O,f(x)
13
ha
m, fix)
(m
t:::
¥I (m)( (m)( -m
Hence,
-2f(m) fll(m)
(or exactly. Now, k= In
In m= In
f(m)
'"
fl/(m)
/JIn -1=k-klnk-I Ina
-Bm -1= e m m a a'"(1n
)2=emlna(ln )2=emlna(ln
)2=k(1n
ln
111(m)
(In
al
redu reduce ce to zz
In
(I k+J2 k+J2(I (I k~
is z.... z--=zfl(z)
-1 j;
filln
14
)2
P2(x): (x)=a
222
-b
and
Now,
Define/by
Clearly,
15
:)
-
x,
-_
Therefore,
Now,
an
ince ince
li
x, -_ :;
Furthermore,
Therefore, conc conclu lude de that that provided
~=O
[(z)
Taking z=
init initia iall lly, y, we have have
z, Computing z/I(1.)-f(Z);;;;P3-
Summ Summar ariz iz ng th resu result lt of th
ecti ection on
Pix) P(x)
8p;e
Pix)
if if
g(x):;;; 1-(-)
-
/3=ljX '1XV'1
Ina
(c lna-l)-rJ C1
lna+/3
...
[1-(1
18
RIFL RIFLIN IN
ANAL ANALYS YSIS IS
iv ~(v!,( ~(v!,(x»= x»=
dx
2rj'(x)
2r!'(x) dx
L(x)=
dx
(vj'(X»2
2Af'(X)2](X)2) 2rj'(x) dx U(X)2
/,(xi
AR2
Fo co stan stan
2vj'(x)~(vj'(x»
(U)du.
wist wist rifl rifl ng wher wher f(x)=constant=k,
av
l+(k:r If
L, :!!!' :!!!'_'! _'!!' !'_(U2v _(U2v
2Adx
an sinc sinc dx
'f'< x»:-. rL rv
2v x)
vf x» ;;:;'!!__(V ' !!__(V fl(X»
hich hich ives ives
20
u.x)f/(X)
REFERENCES
Da id Fi la so Pr va Wa ervl ervlie ie Y, 1993 1993
Comm Commun unic icat atio ion, n, U.S. U.S.
21
rm
RD C, Bene Bene
abor aborat ator or es
APPENDIX
9JnSS9Jd
~
m
~
~
~
~
anbJo.l
N
~
~--------------------~~
aA. nO 5u! Jl
adolS
It
anbJo..L 26
al\Jn~
5u!
adolS
.
.
.
anbJol.
.
.
5u!
adOI 29
....
COPIES IE
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