OTIC FILE COPY ( )

TECHNICAL REPORT BRL-TR-3180

BRLLoUt)

SOLUTION OF THE LONG ROD PENETRATION EQUATIONSC14

.. ,,WILLIAM P. WALTERSTEVEN B.SOLETES DTIC

ELECTECE27inlMD

DECEMBER 1990 8 B

AMPiOVID FORM f PUC mASN; D r1SMJUTION UNLUMTID.

U.S. ARMY LABORATORY COMMAND

BALLISTIC RESEARCH LABORATORYABERDEEN PROVING GROUND, MARYLAND

, 12

NOTICFS

Destroy this report when it is no longer needed. DO NOT return it to the originator.

Additional copies of this report may be obtained from the National Teclr-ical Information Service,U.S. Dc;artment of Commerce, 5285 Port Royal Road, Springfield, VA 22161.

The firdl.ngs of this report are not to be construed as an official Department of the Army position,unless sc designated by other authorized documents.

The use of trade names or manufacturers' names in this report does not constitute indorsement ofany commercial product.

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11. SUPPLECNTARY NIED

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TABLE OF CONTENTS

1. INTRODUCTION ............................................... I

2. THE SOLUTION ................................................ I

3. SPECIAL CASE SOLUTIONS ...................................... 8

4. EXTENDING THE SOLUTION ..................................... . 11

5. CONCLUSIONS ................................................ 13

6. REFERENCES .................................................. 15

APPENDIX: FORTRAN SOURCE CODE LISTING ...................... 17

DISTRIBUTION ................................................ 33

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INTENTIONALLY LEFT BLANK.

iv

1. INTRODUCTION

The impact of a long, slender, eroding rod at high speed on a thick semi-infinite target was

initially fomrnuated by Alekseevski (1966) and Tate (1967, 1969). The governing equations, using the

notation of Wright and Frank (1988), are:

L = V- U, (1)

L - -Y /p, (2)

112 p, (U-V + Y =112 Pr Va + R, (3)

and

P = v. (4)

where U is the speed of the mear of the penetrator, L is the instantaneous penetrator length, V Is the

penetration velocity, P Is the depth of penetration, p, is the penetrator density, Y is the penetrator yield

stress, Pt is the target density, and R is the target resistance. In the equations, a dotted quantity

represents the dme derivative, d/dt.

Wright and Frank (1988) and Frank and Zook (1987) discuss these equations in detail, Including

the assumptions made in the derivation and approximate solutions. Our intent is to analyze the

mathematical, not the physical, aspects of the equations (1) through (4). Basically, L, U, V, and P are

the unknown, dependent variables, t (the time) is the independent variable, and p,, Y, pt, and R are

known constants.

2. THE SOLUTION

An exact analytical solution of equations (1) - (4) for L, U, V. and P is now obtained. From

equation (3)

u-_yU2 +E(l-y) (5)l -y

where, if U0 is the Initial (known) penetrator velocity, v=V/Uo, u=U/Uo, y)pt/p,, and

E-.2(R - Y)/(pr U'2). The minus sign is chosen for the radical in the solution of the quadratic

I

equation (3), to guarantee that V remans less than U, with both U and V real. Note that for the case

R > Y (Z > 0), the minimum admissable value of v is zero, corresponding to the moment that

penetration ccases, which occurs at u= T For the case of R < Y (E < 0), the minimal admissable

value of u is- 4 -, in order to keep the root real in (5). In this case v = u. corresponding to the

situation where rod erosion ceases and rigid body penetration commences.

Next. from equation (2), with K = Yl(p, U.2),

L = -K U,. (6)

Differentiation gives

L Ui L= O.

and the solution for dL/dt, eliminating L, Is

L = U oa

Alternately, from equations (1) and (5),

0" (-7)-• -FUjly

Combining these two expressions to eliminate dL/dt gives

a Y uld avy u + E(I -Y)T" K(I -y) K( -1y) (7)

Straightforward integration yields

ln(4(F +Y {V" +E¢ '.•( -y )

. U 2_ U yu2 + ÷(l-y) .G, (8)2K(l -y) 2K(I -y)

2

where 0 is a constant of integration which results from evaluation of the integral at the onset of

penetration, when u - I and a- a .. Note that 0o, which equals (I/UO)dU/dt1o, and has dimensions

of [IAI, can be evaluated from equation (6) as uo = -K UJLO. The constant G may be expressed as

G -N+InM +1nii,,

where- I..

}EI(2Jf 4,

No V-f + E(1- _ Y2K (s -y) 2K( 8 -y)

By substituting the constant A for the exponent. T.(2K•/y), equation (8) may be expressed in the

following form:

f,[ {uFY7+ yU,+ 0E( -I ~ A11n"

A* M

- yu_2 u 2 y u) +E(I -) +N,2K(I -y) 2K(U -y) (9)

By introducing the following transformation variable z,

v/_Uuyq VyU 2 +E(1-,Y) (10)

equation (9), under the transformation (10), yields

(z(A-1)/2 + E(1 -y)Z (A -3)/2) cxp(Bz - C/z) dz = F dt, (11)

3

where

B.

8KDo-Y)(FiT + 1)

8 K

and

FF -4sM/expE -

The use of the transformation (10) produces a differential equation (II), in which the variables, z

and t, are now separable. If Integrated from time 0 to some finite time t in the penetration process, the

limits on z will vary from its initial value, when u = 1, of

to some intermediate value z. For the case of R > Y, the terminal value of time at which the

governing equations are applicable occurs when penetration ceases at v = 0, in which case u =4'

and the terminal value of z, expressed as z., is given by

z. E,)'+ 1)2

For the case of R < Y, the long rod penetration equations are only valid (without modification) to the

time at which the penetrator begins rigid body penetration, in which case u = v = '-1, and the

terminal value of z becomes

Z.+ )2

Note that z is always positive, since Z is positive for R > Y, and (-I:) is also positive for R < Y.

- ~---~---.-- -- -4

Equation (11) may be further simplified by letting

*-(A+IYd2

in the first integral over z and

om z(A-l

in the second integral over z. Under these transformations, and letting E1=2/(A+I) and E2=2/(A-I), the

integration of equation (11) reduces to

E, fexp(B8' - -') dO

+ E(l y)E2 .exp(Be• -CO-4)dO a F f dt. (12)

The solution is now reduced to a straightforward integration, though it requires evaluation of an

exponential integral which, in theory, is a tabulated function of five input parameters. Defining the

function, W, as

W(BC,E.y,,y2 ) - E f exp(By'-Cy-L)dy, (13)YJ

the solution for t becomes simply

t - [w(B,C,E,,1 .,.) + E(I --y)w(Bc,E2.e,, )]/F.

In practice, our function is evaluated by expanding the exponential function in equation (13) in a

power series and integrating term by term, to the desired degree of precision. The result is z as an

implicit function of the time variable, t.

5

The number of power series terms required for convergence of our W function varies a great deal

with the input conditions to the problem. In particular, the evaluation of our W function by way of

power series can be exacerbated for problems where the penetrator velocity overwhelms the strengths

of the rod and target materials. Fortunately, problems in this velocity range ame generally beyond the

range of interest, for typical long rod penetrator impacts.

Because B is always positive, the first part of the exponential term grows with z. As B is made

parabolically larger by increasing the penetrator striking velocity Uo, more terms are required to make

the power series converge. Because it is a binomial that needs to be exponentiated, use of n terms in

the exponential expansion requires that n(n-l)12 monomials be evaluated. Similarly, the coefficient for

each of the n highest order monomi-ds requires (2n) operations to evaluate. Thus, the computational

effort required to evaluate our W function varies greatly with initial conditions to the problem.

Typical penetration problems involving significant, but not total, penetrator erosion require that

10 to 20 exponential terms be evaluated in order to keep the relative error of the time variable in the

fifth decimal place. Such calculations require mere seconds of computation on a PC. As

hypervelocity conditions ame approached, the number of exponential terms required for the same

convergence epsilon may exceed 200 (recall that 200 exponential terms implies 200 x 199/2 = 19,900

monomials), requiring several minutes on a PC. Fortunately, this solution technique need not be

pursued for problems in hypervelocity since, under these conditions, the long rod penetration equations

approach the standard Bernoulli flow conditions, which may be readily solved by hand.

Having t as a function of z, the normalized rod speed, u, follows from

equation (10) as

u W z-E(1 (14)

Equation (5) may then be employed to obtain the normalized penetration velocity, v. The rate of rod

erosion comes from equation (1), in the form

L L 0(v-u) .

6ý- 7.

The kene, ator length, L, may be obtained in the following fashion. From equations (1) and (2), one

obtains

L (u-v)14

Substituting for v and u give the following:

L, yuk ku2 + _(I _-)

L K(I--y) K(l-y)

Note the identical form of this nelatior and equation (7). As a result, the solution loo,.. nearly

identical to equation (9), including the definition of constants A, M, and N:

-In(LL) - U2 E(i -MY)

+ - U Tu2uE(TT-) +N.2K(I-y) 2K(1-y)

Finally, the penetration, P. is obtained as follows:

I I

P-fVdr. Uo v (dt/dz)dz. (15)0 0

The quantity dt/dz has been previously obtained in equation (11) as

dt . (Z(A'.)I2 + E(i -Y)z(A-3)/2)exp(Bz - C/z)dz F

We may cxpress v in terms of z, using equations (5) and (14). as follows:

7

These substitutions into (15) produce the following expression for P:

PM F I[ 1 - zA12Z(A4) E(1 '2)(l-4)-4Y2]exp(Bz - C/z) dz.o 2(-y)

This equation is similar to Equation 11, in that It may be solved directly for penetration P. in this case,

in terms of serveral W functions.

The results of the present solution have been compared with the original results presented by Tate

(1967), in which he numerically integrated the penetrtion.time history of a duralumin rod strinking a

polythene target, for two different target resistance values. The curves resulting from the present

analytical solution achieve a direct overlay to Tate's numerically integrated solution.

Finally, the source code listing for the penetration equations considered in this report is given in

the Appendix.

3. SPECIAL CASE SOLUTIONS

Two special cases are considered:

A) p, = p, = p. and Y - R = a

B) P, = Pt = P.

SPECIAL CASE A: p, = p, = p, and Y = R = a.

Equation (3) reduces to

(u-v)2 a v,

with the non-trivial solution u 2v. Differentiating equation (2). and combining the result with

equation (I), as before, yields:

S

where

4aH.-•

Integratng this equation and evaluating the constant of integration at time equal 0, where 0 6 ,.

which has the value d, = (.H U.)/(4L.), rsults In

Integrating again for u gives the result in terms of our W function as

t- xp- IH)W(I/1H.,O2,. 1,u),

which gives u implicitly as a function of time, t. As mentioned above, u 2v applies, and thus

determines penetration rate v. Also,

-U.u

La-H V.L *

follow directly. To determine penetration P. employ the tactic of equation (15), namely

P a f Vd Ujfv (dtldu) du.0 1

Penetration rate, v, is known direcy In terms of u, equal to (u/2), and dt/du is simply I/, given by

d ru2-l1

Thus, making use of the term d, - (-H U.)/(4L,), the penetration may be computed as

9

SPECIAL CASE B: p, = p, = p.

Equation (3), which is no longer quadratic, as In the general case, yields

V 1( -E/M).2

Differentiating equation (2), and combining the result with equation (1), and the expression for v

above, yields

S2 2u-

Direct integration results in

i M 1 4. exp I K] ,

where the exponent A has the value, analogous to the general case derivation, of 11(2K). The

variables are separable, and

"fUA exp U[ - du -4. fdt.

This integral is evaluated using the same procedure used for equation (11). By letting•} Us (A*I)

the integv, l reduces to

1 f exp tI (AAi )- I- 1)t

(A+l- ) L 4K

the evaluation of which may be expressed in terms of our W [unction as

I l exp[-IIK]W((4K)--.,O, 2/(A+ 1)' 1, u(-,.,,).

The evaluation of the rcmaining variables v, dL/dt, L, P follows an approach analogous to the

general case.

109

4. EXTENDING THE SOLUTION

The long rod equations (I)-(4) do not hold for the complete penetration process. In particular, if

R > Y, the equations are only valid until penetration ceases (V = 0), even though rod erosion is still

occuring (U > 0). On the other hand, for R < Y, the equations (1)-(4) are only valid until rod erosion

ceases (V = U). even though rigid body penetration proceeds (V = U > 0).

When either of these limiting conditions occurs, the equations (1)-(4) can no longer remain valid

without some sort of alteration, in order to prevent the situations V < 0 or U < V. To illustrate this

point, consider adding the constraint of setting V to zero, when R > Y, to obtain

L =-U,

L U =-Y/p,.

1/2 p, U2 + Y = R. and

or setting the term V = U, if R < Y, to obtain

L =0,

LU = Y/p,,

Y = 112 p, U2 4 R. and

PU.

Since V has been eliminated as a dependent variable, leaving just U, L, and P as unknowns, the

stipulation of four governing equations overconstrains the problem. If no othcr conditions are

changed, then the third equation of each set (modified Bernoulli equation) amounts to setting the rod

and/or penetration velocity to a constant, which is clearly incorrect. One possibility involves the

elimination of the modified Bernoulli equation directly, on the assumption that the U, V relationship it

defines is replaced by the rigidity constraints: V = 0 if R > Y, or V = U if R < Y.

Another possible remedy to this problem, and the alternative being proposed, involves introducing

an additional unknown, in the form of a material resistance R or Y, to vary from its initial value of Ro

or Y., respectively.

The rationale for this step Is given now. The term R and Y represent dcviatoric stress resistances

built up in the target and penetrator, respectively, which act as part of a force balance in the modified

Bernoulli in equation (3). For the case of Ro > YO, the target penetration ceases at some point while

rod erosion continues. After this point of target rigidity, the target is elastic. Thus, the resistive stress

11

offered by the target (R) will decrease from its full plastic value (R), to an elastic value exactly equal

to the penetrator yield strength (Y,). just as the penetrator velocity becomes identically zero. For the

alternative case of k < YO, the rod erosion ceases at some point while rigid body penetration

continues. After this point of penetrator rigidity, the penetrator is elastic. Thus, the resistive stress

offered by the penetrator (Y) will decrease from its full plastic value (Y,) to an elastic value exactly

equal to the target yield strength (R,). just as the penetration velocity becomes identically zero.

These equation sets may then be solved to deternine the residual rod erosion (if R > Y) or

residual penetration (if R < Y), and the associated event duration, using the appropriate initial

conditions, which resulted from the terminal conditions associated with the previous solution of

Equations (1)-(4). The boundary conditions are given below, with subscript r referring to conditions at

the onset of rigidity (target rigidity if R > Y or rod rigidity if R < Y), and subscript x referring to

conditions of U = V =0.

Target Rizd: R > Y (V = 0)

Initial Conditions Final Conditions

U = Ut U = 0L = Lr L = LtP = P, P P,t = t = tR = Re R =Yo

Y = Y0 Y = Yo

Penetrator Rigid: R < Y (V = U)

Initial Conditions Final Conditions

U= U, U= 0Lm L, LrP= P, P=P,t = t4 t =txR Ro R= RoY= YO Y= Ro

For the case of R > Y, the solution is expressable in terms of the W function described in the

original section of this report. Non-dimensional terms (e.g., K, y, u) defined earlier are also

employed, to give the solution as:

12

0. L W-[(2K.)-102,

The term K, refers to the value of K at its orginal constant value, when R=R,, and Y =Y. For the

case of R < Y, the solution is determined a$:

P. - P, - (L,/y) In (Y.R.)

17tan 2KoRo

The results are again compared to Tate (1967), In which the penetration of a duralumin rod into

polythene was modeled for two values of target reslstance-14 and 27 tons per square inch. Tate

pointed out that these two values produced curves which straddled the actual penetration-time data. In

Tate's formulation, the equations are only valid to the point where rigid-body penetration commences

which, for the case in question, accounts for only 80% of the total penetration. Using the current

formulation, for computing the residual, rigid-body penetration of the duralunin rod, the experimental

value of penetration may be used to compute the exact value of target resistance required. For this

case, using the units of Tate, a target resistance of 20 tons per square inch was computed as necessary

to produce the experimentally observed residual penetration, measured from Tate's graph as 6.15

inches. The penetration process is predicted to stop at 242 psec after impact, thus implying that rigid

body penetration takes 42% of the total time of penetration.

5. CONCLUSIONS

An analytical solution to the long rod penetration equations for long rod penetration is offered.

The general case is solved as well as two special cases in which some of the target and penetrator

parameters (e.g., density and/or strength) are equal. This analytical solution allows a faster and easier

solution of the penetration equations, since stability considerations associated with any numerically

integrated solution are avoided.

Additionally, the proposed modification to the original model permits the computation of residual

rod erosion and residual penetration, which offer additional information about the penetration process,

not available in the original penetration equations formulated by Tate.

13

INTEMFONALLY LEFr BLANK.

14

6. REFERENCES

Alekseevski, V. P. "Penetration of a Rod into a Target at High Velocity." Combustion. Explosion.An Shock Waves, vol. 2, pp. 99-106, 1966.

Frank, K., and J. Zook. "Energy Efficient Penetration and Perforation of Targets in the HypervelocityRegime." Internaionl Journal of Impact EnaineerinM. vol. 5, pp. 277.284, 1987.

Tate, A. "A Theory for the Deceleration of Long Rods After Impact." Journal of the Mechanics andPhysics of Solids, vol. 15, pp. 387-399, 1967.

Tate, A. "Further Results in the Theory of Long Rod Penetration." Journal of the Mechanics andPhysics of Solids, vol 17, pp. 141-150, 1969.

Wright, T.W., and K. Frank. "Approaches to Penetration Problems." BRL-TR-2957, U.S. ArmyBallistic Research Laboratory, Aberdeen Proving Ground, MD, December 1988.

15

INTENTONALLY LEFT BLANK.

16

APPENDIX:FORTRAN SOURCE CODE LISTING

17

ImNTmToNALLY LEFT BLANK.

18

ptrogra tateCC CODED BY Steven B. Seqletee, AUGUST-SEPTEMBER, 1990.CC THIS PROGRAM IMPLEMENTS WALTERS' AND SEGLETES' ANALYTICALC SOLUTION OF THE TATE EQUATIONS:CC dL/dt - V - UCC L dU/dt - -Y/@rC 2 2C 1/2 Or (U-V) + Y -12 ft V + RCC dP/dt -VCC WHERE:C V - ROD VELOCITYC U - PENETRATION VELOCITYC Y - TATE ROD STRENGTH PARAMETERC @r - ROD DENSITYC R - TATE TARGET STRENGTH PARAMETERC ft - TARGET DENSITYC L - ROD LENGTHC P - PENETRATIONC t - TIMECC --- VARIABLES--ICC **w**DIMENSIONAL TATE CONSTANTS*****CC R - TARGET RESISTANCEC Y - ROD RESISTANCEC RHOT - TARGET DENSITYC RHOR - ROD DENSITYC Lo - INITIAL ROD LENGTHC Uo - INITIAL ROD VELOCITYCC *****NON-DIMENSIONAL CONSTANTS*****CC GAMMAC SIGMAC KC AC BIGBC CC MC NC F - A CONGLOMERATION OF CONSTANTS, APPEARING OCCASIONALLYC ZO - INITIAL VALUE OF Z (WHEN U - Uo)C ZX - TERMINAL VALUE OF Z (WHEN V - 0)CC *****OUTPUT VARIABLES*****CC Z() - ARRAY OF TRANSFORMATION VARIABLES, AT WHICH SOLUTIONC IS TO BE EVALUATEDC To - ARRAY OF TIMES, CORRESPONDING TO THE TRANSFORMATIONC Z VARIABLESC U - ROD VELOCITYC V - PENETRATION VELOCITYC Ldot - RATE OF ROD EROSIONC L - LENGTH OF UNERODED RODC P() - ARRAY OF PENETRATION VALUES CORRESPONDING TO THEC TRANSFORMATION Z VARIABLES

19

C XTAIL - COORDINATE OF THE REAR END OF THE RODCC *****BOOKKEEPING VARIABLESCC MXPS - MAXIMUM NUMBER OF TIMES, LESS 1, AT WHICH SOLUTION MAYC BE EVALUATEDC EPSILON - RELATIVE ERROR TO BE USED IN TESTING FOR SERIESC CONVERGENCE OF THE W INTEGRALC NTERMS - NUMBER OF TERMS REQUIRED TO CONVERGE THE WC INTEGRALC NTAVG - AVERAGE NUMBER OF TERMS REQUIRED TO CONVERGE THE WC INTEGRAL, OVER THE 5 INTEGRAL EVALUATIONS REQUIRED.C NPASS - NUMBER OF TIMES AT WHICH THE SOLUTION IS EVALUATEDC (MAY NOT EXCEED MXPS+l)C I - A LOOP INDEXC ANSWER - A ONE CHARACTER RESPONSE TO THE QUERY FOR SCREEN OR FILEC OUTPUTCC *****OTHER VARIABLES*****CC ZZ - SCALAR VERSION OF Z, CORRESPONDING TO THE SINGLE Z ELEMENTC OF CURRENT INTERESTC DT - USER SPECIFIED DELTA TIME, AT WHICH HE HOPES TO HAVE SOLUTIONSC EVALUATED (NOTE THAT ACTUAL SOLUTION INTERVAL IS ONLYC APPROXIMATELY THIS VALUE)C DZ - INCREMENT IN Z CORRESPONDING ROUGHLY TO THE TIME INCREMENT DTC Ur - NON-DIMENSIONAL ROD VELOCITY (U/Uo)C Vr - NON-DIMENSIONAL PENETRATION VELOCITY (V/Vo)CC D - AN EXPONENT ON Z, APPEARING IN THE W INTEGRALC TERMx() - WHERE x IS 1, 2 OR 3... ARRAYS CONTAINING EVALUATIONS OFC THE W INTEGRAL, EACH ARRAY ELEMENT WITH DIFFERENTC LIMITS OF INTEGRATION, AS GIVEN BY THE Z ARRAYC ROOTGAMMA - SQUARE ROOT OF GAMMA, WHICH OCCURS FREQUENTLYC ROOTTERM - A SQUARE ROOT TERM, APPEARING OCCASIONALLYC ROOTZ - SQUARE ROOT OF Z, APPEARING OCCASIONALLYC

integer MXPSPARAMETER (MXPS-200)double precision gamma, sigma, K, A, BigB, c, F, zO, zx, M, Ncommon /wconstants/ gamma, sigma, K, A, BigB, c, F, zO, zx, M, Ndouble precision R, Y, rhot, rhor, Lo, Uocomnmon /tconstants/ R, Y, rhot, thor, Lo, Uodouble precision rootgomma, z(O:MXPS), zz, D, terml(O:MXPS),

term2(0:MXPS), term3(0:MXPS), EPSILON, dt,& dz, t(O:MXPS), U, V, Ldot, L, P(O:MXPS), xtail,& rootterm, rootz, Ur, Vr, Cl, Einteger nterms, ntavg, npass, icharacter*l answerPARAMETER (EPSILON-l.OD-3)

CI format V Enter approximate dt: '1)2 format (lx,' z t U V

& 'dL/dt L P Xtail')3 format (f6.2,ip, e9.2,2elO.3,elO.2,2elO.3,ell.3)4 format ('xxxxx',' 8',/,

& 'R,Y,@t,@r - ',ip4elO.3,/,

Iu'lI 'U',/,,

& 'dL/dt',/,S'L',/,

20

i ' Xtail')5 format (lp,8O10. 3 )6 format (' Screen Only or File Output too (S/F)? ')7 format (e Illegal Choice...TTry again...')8 format (/,' Average number of series terms used in solution - '1,13

& ,/,' Number of times at which solution was evaluated - ',i3)9 format (' Requested timestep requires more than ',i5,

& ' solutions...',/,' Please increase MXPS PARAMETER')10 format (al)11 format (' Solutions have been requested at more than',i4,' times.'

& ,/,' MXPS parameter must be increased in order to proceed.')12 format (' Penetrator goes rigid, since U - V.')13 format (' Rod continues to erode in rigid target, since V - 0.')

CC INPUT TATE CONSTANTS AND INITIALIZE CONSTANTS FOR CURRENT MODEL

call getconstantsrootgamma - dsqrt (gauma)

CC DETERMINE DESIRED DT STEP SIZE (NOT NEEDED TO SOLVE THE EQUATIONS;C RATHER, IT PROVI7ES CONVENIENCE OF OUTPUT SPECIFICATION TO THEC USER)

write (*,1)read (*,*) dt

CC INITIALIZE VARIABLES

npaas - 1CC DETERMINE OUTPUT MODE (S FOR SCREEN ONLY; F FOR FILE OUTPUT TOO)

70 write (*,6)read (*,10) answer

C CONVERT ANSWER TO UPPER CASEif (answer .eq. 'a') then

answer - 'S'else if (answer .eq. 'f') then

answer - 'F'end ifif (answer .eq. 'F') then

CC SET UP OUTPUT FILE 'TATE.OUT'

open (2,file-'tate.out',access-'append')write (2,4) R, Y, rhot, rhor

else if (answer .ne. 'S') thenwrite (*,7)goto 70

end ifCC FOR FIRST ITERATION, START AT ZO

z(0) - zOCC AS LONG AS Z IS ABOVE MINIMUM LIMIT, PROCEED WITH THE FOLLOWING:

100 if (z(npass-.1) .gt. zx) thenCC COMPUTE DZ REQUIRED TO PRODUCE DESIRED T (APPROXIMATE ONLY)C BY EVALUATING DERIVATIVE AT MIDDLE OF Z INCREMENT

zz - z(npass-l)do 105 i - 1, 2

dz - dt6 * F / dexp(BigB*zz - c/zz)

I / ( zz**((A-l.)/2.)6 + sigma*(l.-garma) * zz**((A-3.)/2.))

if (-dz .gt. zz) thenzz - zz/2.

21

elsez2 - zz + dz/2.

end if105 continue

CC DETERMINE NEW Z.C IF Z FALLS BELOW MINIMUM LIMIT, RESET Z TO MINIMUM LIMIT

z(npass) - z(npass-1) + dzif (z(npass) .1t. zx) z(npass) - zK

CC INCREMENT COUNTER ON NUMBER OF TIMES AT WHICH SOLUTION ISC COMPUTED

npass - npase + 1CC CHECK TO SEE THAT ARRAY LIMITS HAVE NOT BEEN EXCEEDED

if (npass .qt. MXPS) thenwrite (*,11) MXPSstop

end ifgoto 100

end ifCC EVALUATE T, FOR THE GIVEN Z, REQUIRING SOLUTION OF THEC W INTEGRAL

D - (A-1.)/2.call wintegral (terml,npass,BigB,c,D,z0,z,nterms,EPSILON)ntavg - nterms

CD - (A-3.)/2.call wintegral (term2,npass,BigB,c,D,zO,z,nterms,EPSILON)ntavg - ntavg + nterm3

Cdo 110 i - 0, (npass-1)

t(i) - (terml(i) + uigma*(1.-gamma)*term2(i)) / F110 continue

CC EVALUATE PENETRATION, REQUIRING SOLUTION OF W INTEGRAL

D - (A )/2.call wintegral (terml, npass,BigB,c,D,zO,z,nterms,EPSILON)ntavg - ntavg + nterms

CD - (A-2.)/2.call wintegral (term2,npass,BigB,C,D,zO,z,nterma,EPSILON)ntavg - ntavg + nterms

CD - (A-4.)/2.call wintegral (term3,npass,BigB,c,D,zO,z,nterms,EPSILON)ntavg - (ntavg + nterms) / 5

Cdo 115 i - 0, (npass-l)

P(i) - UO / F * ((1-rootgamma)/(2.*rootgafrmra*(l.-gamma)) * termil(i) -

sigma * term2(i) -& sigma**2*(i.+rootgamruaa)*(1.-gamma)/(2.*rootgammaa)& *term3(i))

115 continueCC SET UP HEADER COLUMNS

write (*,2)CC DETERMINE REST OF PARAMETERS

do 120 1 - 0, (ndass-1)zz - z(i)

22

CC EVALUATE U, FROM THE GIVEN 2

Ur * (zz - sigma*(1.-ganma)) / (2.*rootganma*dsqrt(zz))U " Ur * Uo

CC EVALUATE V FROM THE COMPUTED U

Vr - (Ur - daqrt(qfamm&*Ur**2 + sigma* (1.-namma))) / (I.-gamma)V Vr * Uo

CC EVALUATE Ldot FROM COMPUTED U AND V

Ldot - V - UC -C EVALUATE ROD LENGTH FROM COMPUTED U

rootterm - dsqrt(qamha*Ur**2 + sigima*(1.-qamma))rootz Ur*rootgamma + roottermL - Lo * dexp(Ur*(rootterm - Ur*qamma)/(2.*K*(l.-gamma)))

• (rootz)**A / (dexp(N) * M)CCC COMPUTE TAIL LOCATION OF ROD

xtail - PWi) - LCC WRITE TO SCREEN

write (*,3) z(i), t(i), U, V, Ldot, L, P(i), xtailC OUTPUT RESULTS

if (answer .eq. 'F') thenCC WRITE TO FILE

write (2,5) z(i), t(i), U, V, Ldot, L, P(i), xtailend if

120 continueCC COMPLETE THE PROBLEM TO ALLOW...

i - npassz(i) - 0.U 0.V- 0.Ldot - 0.

C ... RIGID BODY ROD TO COME TO STOP, IF R < Yif (sigma .1t. 0.) then

write (*,12)L- LPMi) - PUi-1) + L/gamma * dlog(Y/R)C1 - dsqrt( 2.*K*R/(qamma1Y) )t(i) - t(i-1) + 2.*L/(gamma-Uo*Cl) *

& datan (Ur/Cl)CC ... OR NON-PENETRATING ERODING ROD TO COME TO STOP, IF R > Y

elsewrite (*,13)L - L * dexp ((Y-R)/Y)P(i) - PUi-1)z(0) - 0.BigB - l./(2.*K)c- 0.E- 2.call wintegrate (terml,l,BigB,c,E,Ur,z,nterms,EPSILON)t(i) = t(i-1) - L/(2.*K*Uo) * terml(0)

end ifxtail - P(i) - L

C WRITE TO SCREENwrite (*,3) z(i), t(i), U, V, Ldot, L, P(i), xtail

23

C OUTPUT RESULTSif (answer .eq. 'F') then

CC WRITE TO FILE

write (2,5) z(i), t(i), U, V, Ldot, L, P(M), xtailend if

CC FINISHED ITERATING... CLOSE UP SHOP.

write (*,8) ntavg, npassif (answer .eq. 'F') close (2)

.stopend

subroutine getconstantsCC ACQUIRE THE TATE CONSTANTS NEEDED BY THE CODE, AND INITIALIZEC CORRESPONDING CONSTANTS FOR PRESENT SOLUTIONCC ----- VARIABLES---CC *****DIMENSIONAL TATE CONSTANTS*****CC R - TARGET RESISTANCEC Y - ROD RESISTANCEC RHOT - TARGET DENSITYC RHOR - ROD DENSITYC Lo - INITIAL ROD LENGTHC Uo - INITIAL ROD VELOCITYCC *"***NON-DIMENSIONAL CONSTANTS*****CC GAMMAC SIGMAC KC AC SIGBC CC MC NC ZO - INITIAL VALUE OF Z (WHEN U - Uo)C ZX - TERMINAL VALUE OF Z (WHEN V - 0)C

" "'OTHER VARIABLES*****

- MINIMUM VALUE OF U, WHEN V - 0

tible precision gains, sigma, K, A, BigB, C, F, z0, zx, M, Nc..mmon /wconstants/ gqmma, sigma, K, A, Bi9B, _, F, z0, zx, M, Ndouble precision R, Y, rhot, rhor, Lo, Uocommon /tconstants/ R, Y, rhot, rhor, Lo, Uodouble precision Utain

C1 format (' Enter the target & rod resistances (R & Y): ')2 format (' Enter the target & rod densities: ')3 format (' Enter the initial rod length and velocity: ')4 format (Ip,' The target resistance,',el0.3,

& ', must not equal the rod resistance.')5 format (lp,' The initial rod velocity,',elO.3,£ ', must exceed the minimum value,',elO.3,'.')

CC READ PARAMETERS REQUIRED IN TATE EQUATIONS

write (*,l)read (,') R, Y

24

write (*,2)read (*,*) rhot, rhorwrite (*,3)read (*,*) Lo, Uo

CC CHECK FOR DATA INCONSISTENCIESCC TARGET RESISTANCE MUST NOT EQUAL ROD RESISTANCE, FORC SOLUTION TO TATE EQUATIONS TO BE MEANINGFUL

if (R .eq. Y) thenwrite (*,4) Rstop

"end ifCC Umin OCCURS WHEN PENETRATION VELOCITY V - 0 FOR R > Y, ANM U - VC FOR R < Y. IT ROUGHLY CORRESPONDS TO THAT VELOCITY WHERE THE RODC BOUNCES OFF THE TARGET, WITHOUT MAKING PENETRATION. MAKE SUREC THAT Uo EXCEEDS THE VALUE OF Umin.

sigma - 2. * (R-Y) / (rhor*UO**2)gamma - rhot / rhorif (sigma .t. 0.) then

Umin - Uo * deqrt(sigma)else

Umin - Uo * dsqrt(-sig=a/ganu)end ifif (Uo .lt. Umin) then

write (*,5) Uo, Uminstop

end ifCC INITIALIZE REST OF CONSTANTS (DERIVED FROM TATE CONSTANTS)C

call initializeC

returnend

subroutine initializeCC INITIALIZE CONSTANTS USED IN SOLUTION OF TATE PROBLEM,C GIVEN THE TATE CONSTANTSCC .-.-- VARIABLESmCC *****DIMENSIONAL TATE CONSTANTS*****CC R - TARGET RESISTANCEC Y - ROD RESISTANCEC RHOT - TARGET DENSITYC RHOR - ROD DENSITYC Lo - INITIAL ROD LENGTHC UO - INITIAL ROD VELOCITYCC *****NON-DIMENSIONAL CONSTANTS*****CC GAMMAC SIGMAC KC AC BIGBC CC MC N

25

C Zo - INITIAL VALUE OF Z (WHEN U - Uo)C zx - TERMINAL VALUE OF Z (WHEN V - 0)CC *****OTHER VARIABLES*****CC ROOTGAMMA - SQUARE ROOT OF GAMMA, WHICH OCCURS FREQUENTLYC ROOTTERM - A SQUARE ROOT TERM, APPEARING OCCASIONALLYC ROOTZO - SQUARE ROOT OF ZO, APPEARING OCCASIONALLYC Udotor - ORIGINAL ACCELERATION OF ROD / UoC

double preclsion gans", sigma, K, A, BigB, c, F, z0, ix, M, Ncommon /vconstants/ gam, sigma, K, A, BigB, c, F, a(, zo,, N, Ndouble precision R, Y, rhot, rhor, Lo, Uocommon /tconstants/ R, Y, rhot, rhor, Lo, Uodouble precision rootgamma, rootterm, rootz0, Udotor

CC GAMMA DEFINED IN ROUTINE GETCONSTANTS.. .NO NEED TO REINITIALIZEC gannra - rhot / rhorCC TEMPORARY VARIABLE...

rootgamra - dsq-t (ganna)C SIGMA DEFIN9D IN ROUTINE GETCONSTANTS.. .NO NEED TO REINITIALIZEC sigma - 2. *. (R-Y) / (rhor*Uo**2)

K - Y / (rhor*Uo**2)A - sigma / (2. * K * zootgamma)BigB - 1. / (8. * K * rootgamlaa * (rootgalmma + 1.))c - sigma'*2 * (1. - gamma) * (rootgamma + 1.) /

& (8. * K * rootgasmua)CC ZO IS THE INITIAL VALUE OF Z, CORRESPONDING TO THE SITUATIONC OF U - Uo.

rootterm - daqrt(gamma + sigma*(l.-gawnna))rootz0 - rootgamma + roottermzO - rootzO**2

CC M AND N ARE CONSTANTS APPEARING IN THE EQUATION FOR dU/dt (AND THUSC THE EQUATION FOR 11(t).

M - (rootz0)**AN - (rootterm - gamma) / (2.*K*(1.-ga.ima))

CUdotor -Y / (rhor * Lo * Uo)F - 4. * Udotor * M * rootgamma * dexp(N - sigma/(4. * K))

Cif (R .gt. Y) then

CC FOR R > Y:C ZX IS THE TERMINAL VALUE OF Z, WHEN V - 0. FROM -.A•£ EQUATION,C WE SEE THAT V - 0 WHEN U/Uo - ROOT(SIGMA). WHEN U IS THIS VALUE,C Z TAKES ON THE FOLLOWING VALUE:

zx sigma * (rootgamma + 1.)**2else

CC FOR R < Y:C ZX IS THE TERMINAL VALUE OF Z, WHEN V - U. FROM TATE EQUATION,C WE SEE THAT V - U WHEN U/Uo - ROOT(-SIGMA/GAMMA). WHEN U ISC THIS VALUE, Z TAKES ON THE FOLLOWING VALUE:

zx - -sigma * (rootgamma + 1.)**2end if

Creturnend

subroutine wintegral (w, npass, B, C, D, z1, z2, n, eps)

26

C SUBROUTINE TO EVALUATE THE W INTEGRAL:

CC z2C ( D -1C I z exp (Dz- Cz )dzCC zlCC TO DO SO, TRANSFORM THE INTEGRAL WITH THE FOLLOWING TRANSFORMATIONC AND INTEGRATE THAT FUNCTION, AS FOLLOWS:CC DlI 1/(D+I) DC LET y - z THEN z - y AND dy - (D+l) a ds.CC LET E - 1/(D+1). THE INTEGRAL THEN BECOMESCC y 2

C ( E -EC EI exp (Byy- Cy ) dyCC ylCC ---" VARIARLES----CC *****BOOKKEEPING VARIABLESCC MXPS - MAXIMUM NUMBER OF TIMES, LESS 1, AT WHICH SOLUTION CANC BE EVALUATED17 EPS - RELATIVE ERROR TO BE USED IN TESTING FOR SERIESC CONVERGENCE OF THE W INTEGRALC N - NUMBER OF TERMS REQUIRED TO CONVERGE THE NC INTEGRALC I - A LOOP INDEXC NPASS - NUMBER OF Z VALUES AT WHICH TO EVALUATE SOLUTIONCC *****OTHER VARIABLES*****CC W0 - ARRAYS CONTAINING EVALUATIONS OF THE W INTEGRAL, EACHC ARRAY ELEMENT WITH DIFFERENT LIMITS Or INTEGRATION, ASC GIVEN BY THE Z2 ARRAYC Zi - LOWER LIMIT OF INTEGRATIONC Z2() - ARRAY CONTAINING UPPER LIMITS OF INTEGRATIONC B - CONSTANT MULTIPLIER OF Z IN EXPONENTIAL TERM OF WC INTEGRALC C - CONSTANT MULTIPLIER OF 1/Z IN EXPONENTIAL TERM OF WC INTEGRALC D - EXPONENT OF Z IN W INTEGRALC E - TRANSFORMED VARIABLE EQUAL TO 1/(D+l)C Y1 - TRANSFORMED ZI LIMIT OF INTEGRATIONC Y2() - TRANSFORMED Z2() ARRAY LIMITS OF INTEGRATIONC DPLUS1 - INTERMEDIATE VARIABLEC

integer MXPSPARAMETER (MXPS-200)integer n, i, npassdouble precision w(0:MXPS), B, C, D, zl, z2(0:MXPS), epsdouble precision E, yl, y2(0:MXPS), DpluslDplusl - D÷I.yl - zl**Dplusldo 100 i - 0, (npa&s-1)

100 y2(i) z2(i)**DpluslE - l./Dplusl

27

call wintegrete (w, npags, B, C, 9, yl, y2, n, eps)returnend

subroutine wintegrate (w, npass, B, C, E, yl, y2 , n, eps)CC FUNCTION TO EVALUATE THE W INTEGRAL (WITHC TRANSFORMATION), BY SERIES EXPANSION:CC y2C (E -aC ZI exp(By-Cy )dyCC ylCC THE SERIES EXPANSION OF THE EXPONENTIAL 13:CC 2 3C exp (x) - 1 + x + x /2! + x /3! ...C

c I I I IC TERM # 0 1 2 3CC WHEN THE ABSOLUTE VALUE OF A GIVEN TERM Of THIS EXPANSION ISC LESS THAN THE ERROR "epC" TIMES THE CURRENT FUNCTION VALUE,C THE SERIES IS TRUNCATED. THE NUMBER OF TERMS REQUIRED TOC SUM WITHIN THIS ERROR VALUE IS RETUR.4ED IN VARIABLE "n".CC NOTE THAT INTEGRAL Of TERM 0 OF THE EXPANSION IS (y2-yl) WHENC EVALUATED BETWEEN yl AND y2.CC REMEMBER ALSO, THAT TERM "x" IN THIS EXPANSION IS REALLY A TERMC LIKE (A + B). THUS, THE POLYNOMIAL EXPANSION OF TERMS IN THEC EXPONENTIAL ARE:CC n n n k (n-k)C (A + B) - SUM ( )ABC k-O kCC WHERE:CC n n!C ( . .------------C k (n-k)! k!C

integer MXPSPARAMETER (MXPS-200)integer n, k, npass, idouble precision w(0:MXPt.), B, C, E 1,1 2(0MXPS), epadouble precision f, fi, exponent, exp2,

value(0 MXPS), tvalue(O:MXPS), minlimitCC W INTEGRAL ONLY INTENDED FOR POSITIVE LIMIT OF INTEGRATION

minlimit - yldo 80 i - 0, (npass-l)

80 if (y2 (i) .1t. minlimit) minlimit - y2(i)C CHECK TO SEE IF INTEGRATION LIMITS ARE OK

if (minlimit .1t. 0.) thenwrite (*,'(" Negative integration limits not allowed.'')')stop

end ifC

28.

C INITIALIZE COUNTER FOR # OF TERMS OF EXPONENTIAL EXPANSION USEDn 0

CC EVALUATE THE INTEGRAL OF THE FIRST TERM DIRECTLY:

do 90 1 - 0, (npass-1)90 w(i) - (y2(i) - yl)

CC FOR EVERY ADDITIONAL TERM OF THE EXPONENTIAL EXPANSION, DO THEC FOLLOWING:C INCREMENT TERM COUNTER

100 n - n +.1- -C

C INITIALIZE INTEGRAL ASSOCIATED WITH THIS EXPONENTIAL TERM TO ZEROdo 110 £ - 0, (npass-l)

110 tvalue(i) - 0.CC LOOP OVER EACH POLYNOMIAL EXPANSION TERM, FOR THIS EXPONENTIALC TERM

do 200 k - 0, nCC GET THE CONSTANT ASSOCIATED W/ THIS EXPONENTIAL/POLYNOMIALC PRODUCT TERM... IT IS GIVEN BY:C f - B~k * (-C)^(n-k) / (n-k)! k!C THE FOLLOWING TECHNIQUE REDUCES PROBABILITY OF NUMERICALC OVERFLOW, BY INTERSPERSING DIVISION AND MULTIPLICATION

f - 1.do 120 i - 1, max0(k, (n-k))

fi - dfloat(1)if (i .1e. k ) f - f * b /fiif (U .1e. n-k) f - f * (-c)/fi

120 continueif (f .ne. 0.) then

CC GET THE EXPONENT ON Y ASSOCIATED W/ THIS POLYNOMIAL EXPANSIONC TERM

exponent - E * (2*k - n)CC BEGIN THE INTEGRATION OF THIS POLYNOMIAL EXPANSION TERM:CC y2C ( n k (n-k) E(2k-n) y2C I (-1) B C y ( exponentC I-------------------------- dy - I f y dyC ) (n-k)! k!C yl ylC

if (exponent .ne. -1.) thenC IF EXPONENT NOT EQUAL TO -1, THENCC y2C ( exponent f (exponent+1) 1y2C I f y dy---------------- y IC ) (exponent + 1) lylC ylCC FOR NORMAL POLYNOMIAL INTEGRATION...C GET THE EXPONENT AFTER INTEGRATION

exp2 - exponent + 1.C GET THE TOTAL COMBINED CONSTANT AFTER INTEGRATION

f - f / exp2do 150 i - 0, (npass-1)

C EVALUATE THE I;4TEGRAL BETWEEN Y1 AND Y2value(i) - f * (y2(i)**exp2 - yl**exp2)

29

C LUMP THIS INTWMTIOI VALUE FOR THIS POLYNOMIAL TERM INTOC THE SUN FOR THE EXPONENTIAL TERM

tvaluo(i) - tvalue(i) 4 value(i)150 continue

elseC OTHERWISE, IF EXPONENT - -1, THENCC y2C ( exponentC I f y dy - f ln(y2/yl)

C ylCC FOR LOGARITHMIC INTEGRATION...C THE TOTAL COMBINED CONSTANT AFTER INTEGRATION IS JUST F

do 160 1 - 0, (npaee-1)C EVALUATE THE INTEGRAL BETWEEN Y1 AND Y2

value(i) - f * dloq(y2(i)/yl)C LUMP THIS INTEGRATION VALUE FOR THIS POLYNOMIAL TERM INTOC THE SUM FOR THE EXPONENTIAL TERM

tvalue(i) - tvalue(i) + value(i)160 continue

end ifend if

C200 continue

CC LUMP THIS INTEGRATION VALUE FOR THE EXPONENTIAL TERM INTOC THE SUM FOR THE W INTEGRAL

do 250 1 - 0, (npass-1)250 w(i) - w(i) + tvalue(i)

CC CHECK FOR SERIES CONVERGENCE AGAINST USER SPECIFIED EPSILON;C IF TERM DOESN'T CONVERGE FOR ANY OF THE LIMITS OF INTEGRATION,C THEN COMPUTE ANOTHER EXPONENTIAL SERIES TERM FOR *ALL* THEC LIMITS OF INTEGRATION

do 300 i - 0, (npass-1)300 if (dabs(tvalue(i)) .gt. dabs(eps*w(i))) goto 100

CC MODIFY INTEGRAL VALUE TO ACCOUNT FOR E CONSTANT IN FRONTC OF INTEGRAL

do 350 i - 0, (npass-1)350 w(i) - E * W(i)

returnend

30

No of NoofQoW QWKzm 9o .midn

2 Adwinlsator 1 DkeucorDefewe Technical Info Center US Army Avidaon ResearhAThN: DTIC-DDA ad Technology AcivityCameron Station ATTN: SAVRT-R (Lilrary)Alexandria, VA 22304.6145 MS 219-3

Ames Roamch CanoHQDA (SARD-TR) MoMfeIS MWeld CA 940354000WASH DC 20310-0001 1 CommandeCommander US Army Missile CommandUS Army Materiel Command ATTN: AMSMI.RD-CS.R (Du")ATMN: AMCDRA-ST Redstone Arsenal AL 35898-50105001 Eisenhower AvenueAlexandkia, VA 22333-0001 1 Command

US Army Tank-Autmotive CommandCommander AMlN: AMSTA-TSL (Technical Library)US Army Laboratory Command Wmma , MI 48397.500ATTN: AMSLC-PLAdelphi, MD 20783-1145 1 Dire r

US Army TRADOC Analysis Command2 Commander ATIN: ATAA-SL

US Anmy. ARDEC White San&s Mik Range, NM 88002-5502ATIN: SMCAR-IMI.IPicatinny Arsenal, NJ 07806-5000 (hft OW Commadlt

US Army Infantry School2 Commander AMTT: ATSH.-CD (Security Mgr.)

US Army, ARDEC Fort BRmnin, GA 31905-5660AT7N: SMCAR-TDCPicatinny Arsenal, NJ 07806-5000 GWY, .,l Commandant

US Amy Inftntry SchoolDirector ATMN: ATSH-CD-CSO-ORBenet Weapons Laboraory Fort Banning, GA 31905-5660US Army, ARDECATTN: SMCAR-CCB-TL I Air Fore Ann ent LaboratoryWatervliet, NY 12189-4050 AMrN: AFATI.DLODL

Eoni AFB, FL 32542-5000Commander

US Army Armament, Munition Aberdeen Provin% Groundand Chemical Command

ATTN: SMCAR-ESP-L 2 Dir, USAMSAARock Island, IL 61299-5000 ATTN: AMXSY-D

AMXSY.MP, H. Cohen

Commander I Cdr, USATECOM

US Amy Aviation Systems Command ATTN: AMSTE.TDAITN: AMSAV-DACL 3 Cdr, CRDEC, AMCCOM4300 Ooodfellow Blvd. ATIN: SMCCR-RSP-ASt. Louis, MO 63120-1798 SMCCR-MU

SMCCR-MSII Dir, VLAMO

AMTN: AMSLC-VL.D

31

No."of No.Of

Conies Qr~niuz2ii 99imsQuA~

2 Director I DirectorDARPA US Army Missile and Space IntelligenceATTN: J. Richardson Center

MAI R. Lundberg ATTN: AIAMS-YDL1400 Wilson Blvd. Redstone Arsenal, AL 35898-5000Arlington, VA 22209-2308

I Commander2 US Amy MICOM USACECOM

Library R&D Technical LibraryATTN: AMSMI-RD-TE-F, Matt H. Triplett ATTN: ASQNC-ELC-I-T, Myer CenterRedstone Arsenal, AL 35898-5250 Fort Monmouth, NJ 07703-5301

2 TACOM RD&E Center 5 Naval Weapons CenterATTN: AMCPM-ABMS-SA, John Rowe ATTN: Code 3263, Tucker T. Yee

AMSTA-RSS, K. D. Bishnoi Code 3268, Don ThompsonWarren, Ml 48397-5000 Code 6214, W. J. McCarer

Code 40572 Commander Code 45, Tech Library

Armament RD&E Center China Lake, CA 93555-6001US Army AMCCOMATTN: SMCAR-CCH-V, M. D. Nicolich 3 Naval Surface Warfare Center

SMCAR-FSA-E, W. P. Dunn ATIN: Code 0-22, Charles R. GamenPicatinny Arsenal, NJ 078065000 Code 0-33, Linda F. Williams

Code H-il, Mary Jane Sill4 US Army Belvoir RD&E Center Dahlgren, VA 22448.5000

ATTN: STRBE-NAE, Bryan WestlichSTRBE-JMC, Terilee Hanshaw 18 Naval Surface Warfare CenterSTRBE-NAN, ATTN: 0-402, Pao C. Huang

Steven G. Bishop R-12,Josh Williams Bryan A. Baudler

Fort Belvoir, VA 22060-5166 Robert H. MoffeuRobert Garrett

Commander R-32, Thomas L. JunglingUS Army Research Office U-43, Richard CaminityPO Box 12211 R-lOA, W. ReedResearch Triangle Park, NC 27709-2211 John P. Maua

Paula WalterCommander Lisa MensiUS Army Foreign Science and Kenneth Kiddy

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E. JohnsonCommander W. BullockDet S, USAOG C. DickersonUSAINSCOM 10901 New Hampshire Ave.ATTN: IAGPC-S Silver Spring, MD 20903.5000Fort Meade, MD 20755

32

No. of No. of

2 Naval Weapons Support Center 14 Sandia National LaboratoesATTIN: John D. Barher ATTN: Division 9122,

Sung Y. Kim Robert 0. NellumsCode 2024 Jim HickersonCrane, IN 47522-5020 Vince Luk

Division 1533,C Civil Bngtneering Laboratory Martin KippNaval Construction Battalion Center Allen RobinsonATTM: Code L-56, Joel Young Division 2512,Port Hueneme, CA 93041 Willianl J. Andrzejewski

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ATTN: AFATI4DLJW, Tony RolletW. Cook Mike BurkettM. Nixon MS P940,

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Advanced Technology, Inc. I Explosive TechnologyATTN: John Adams ATTN: Michael L. KnacbelPO Box 125 PO Box KKDahlgren, VA 22448-0125 Fairfield, CA 94533

33

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14 Lawrence Livermore National Laboratories I Rockwell Missile Systems Division

ATTN: L-122, ATTN: Terry NeuhartBarry R. Bowman 1800 Satellite Blvd.Ward Dixon Duluth, GA 30136Raymond PierceRussell Rosisky 1 Rockwell IntljRockeodyne DivisionOwen J. Alford ATTN: James MoldenhauerDiana Stewart 6633 Canoga Ave (HB 23)Tony Vidlak Canoga Park, CA 91303

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A77N: D. Barette

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505 King Ave.Columbus, OH 43201 2 Physics International

ATTN: Ron FunstonDefense Technology International, Inc. Lamont GamettATMN: D. E. Ayer 2700 Merced St.The Stark House PO Box 501022 Concord Saweet San Leandro, CA 94577Nashua, NH 03060

2 Lockheed Missile & Space Co., Inc.

4 California Research and Tech. Corp. ATTN: S. Kusumi (0-81-11, Bldg 157)ATTN: Roland Franzen Jack Philips (0-54-50)

Dennis Orphal PO Box 3504Ron E. Brown Sunnyvale, CA 94088Mark Majerus

5117 Johnson DrivePleasanton, CA 94566

34

' , - . " '. :

No. of No. of

I Lockheed Missile & Space Co., Inc. 1 Martin Marietta AwoopeATTN: Richard A. Hoffman ATTN: Donald R. BraggSanta Cruz Facility PO Box 5837, MP 109Empire Orade Road Orlando, FL 32855Santa Cruz, CA 95060

1 SRI Inteniational2 Mason & Hanger - Silas Mason Co. ATTN: Dr. L. Seaman

ATTN: Thomas J. Rowan 333 Ravenaswod Ave.Christopher Vogt Menlo Park. CA 94025

Iowa Army Ammunition PlantMiddletown, IA 52638-9701 1 Northrop Corporation

Eiectro-Mechanical DivisionLockheed Engineering & Space Sciences ATIN: Donald L. HallATTN: Ed Cykowski, MS B-22 500 East Orangethorpe Ave.2400 NASA Road 1 Anaheim, CA 92801Houston, TX

3 Boeing Aerospace Co.2 Dyna East Corporation Shock Physics & Applied Math

ATTN: P. C. Chou Engineering TechnologyR. Ciccarelli ATMN: R. Helzer

3201 Arch St. J. ShraderPhiladelphia, PA 19104 T. Murray

PO Box 39992 E. I. DuPont De Nemoura & Company Seattle, WA 98124

ATThN: B. ScottL. Minor I McDonnell Douglas Astronautics Company

Security Director, Legal Department AMFN: Bruce L. CooperP0 Box 1635 5301 Bolsa AveWilmington, DE 19899 Huntington Beach, CA 92647

Aerojet Electro Systems Company 1 D.R. Kennedy and Associates Inc.ATTN: Warhead Systems, ATTN: Donald Kennedy

Dr. J. Carleone PO Box 4003PO Box 296 Mountain View, CA 94040.Azusa, CA 91702

I University of ColoradoPhysics International Company ATIN: Timothy MaclayTactical Systems Group Campus Box 431. NNT 3-41Eastern Division Boulder, CO 80309PO Box 1004Wadsworth, OH 44281-0904 1 New Mexico Institute Mining & Tech.

ATTN: David J. Chavez3 Alliant Techsystems, Inc. Campus Station, TERA Group

Defense Systems Division Socorro, NM 87801ATTN: G. Johnson

J. HoultonN. Berkholtz

7225 Northland DriveBrooklyn Park, MN 55428

35

No. of

Univerity of DaytonRoumch utnDocuwent ControL, W•h Brothm BranchATIN: Dr. S. J. BlemPO Box 283Dayton, OH 43409

4 Univardty of DWAw'wEDept. of Mechanical EngneuffnATTN: Prof. J. Vinoa

Prof. D. WilkinsProf. J. OuleupDew R. B. PipM

Newmk. DE 19716

36

No.o NO. of

2 D•fesm Ram ch Establihment Sufld 2 AB Bofiro/Ammunition DivisionATTN: Chrs Weickcrt ATTN: Jan hWulid

Davitd Msckay Anders NonleURalston, Afheta, TOJ 2NO Ralston BOX 900CANADA S-691 80 Bofors

SWEDENDefam Resech Establishmnt ValcatderATTN: Noebm Gas 1 Mesauabmiti-BOlkow-BlohmPO Box 8800 Dymmics DMiionCowmueW PQ, GOA IRO ATTN: Monfme HeldCANADA PO Box 1340

D8898 SchrabenhausenI Candian Arsenals, LTD GERMANYATTN: Pierre Pelletir5 Montee des Aemmx I TNO Prins Mum LaoratoryVill/e de OGdeur, PQ, J5Z2 ATTN: 1H. J. PamanCANADA PO Box 45

2280 AA R4j k2 Ernst Mach Instuce Lange Kleiwag 137

A7TN: A. J. Stlp E NEhTIERLANDSV. HohIfr

Eckerutaasm 4 4 ISD-7800 Prelburg i. Dr. Al 4.. WIlUhmicGERMANY H. J. Emnst

P. Y. Chanteret3 IABG V. bhim.,

ATTN: H. J. Rasiachen P68301 Saint-LouisW. Schittke C~dex. 12. nre de i'IndustrieF. Schawp" B.P. 301

Einsteinsmae 20 FRANCED-8012 Ouobrun B. MuenchenGERMANY I IFAM

ATiN: Leow MeyerRoyal Amament R&D Establishment Lesumer Heersrasse 36AT•'N: Ian Cullis Brena 77Fort Halstead GERMANYSevenooks, Kent TNI4 7BJENGLAND

Centre d'Etudes de GramatATIN: Gerald Solve46500 OmmatFRANCE

PRB S.A.ATrN: M. VansnickAvenue de Tervueren 168, Bte. 7Brussels, B-1150BELOIUM

37

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