CONTRACT REPORT BRL-CR-605

1938 - Serving the Army for Fifty Years - 1988

DYNAMICS OF A BALLOTING PROJECTILELflIN A MOVING GUN TUBE

N

K. A. ANSARIJ. W. BAUGH, JR.

DECEMBER 1988 T I CMAR 2 31989

H

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BALLISTIC RESEARCH LABORATORY

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Dynamics of a Balloting Projectile in a Moving Gun Tube

12. PERSONAL AUTHOR(S)K.A. Ansari and J.W. Baugh, Jr.

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19JABSTRACT (Continue on reverse if necessary and Identify by block number)

In recent years, there have been significant strides made in modeling gun dynamics.Current capabilities include the use of flexible gun tube and projectile descriptions.However, modeling the interface conditions between the tube and projectile is still anarea of very active research. Highly detailed finite element models, which are currentlyunder development at the BRL and at other agencies will lead to a much better understandingof the important parameters involved in the balloting of projectiles. These models, however,require both substantial computer resources and time. Thus there is a need for thedevelopmeit of siinplified tube/projectile interface descriptions which both realisticallyrepresent the interface and are easily implemented. The bourrelet/gun-bore and obturatormodels incorporated in this study reasonably satisfy these requirements.

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19. ABSTRACT - cont'd

4'A nonlinear dynamic analysis of a balloting-projectile is presented in this report. Themathematical model used is a six-degree-of-freedom coupled model of the projectile and guntube system. The effects of obturator flexibility and projectile impact with the gun boreat the bourrelet are included in the analysis, and the nonlinear differential equations ofmotion of the system are derived using a Lagrangian formulation. Gaussian elimination and

-- ewmark's constant-average-acceleration method are employed to obtain a solution. Finally,numerical results for some specific test cases are obtained and discussed.

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DYNAMICS OF A BALLOTING PROJECTILE INA MOVING GUN TUBE

by

KA AnsariJW. Baugh, Jr.

Pacific Northwest LabRichland, Washington

Prepared for

U.S. Army Ballistic Research LaboratoryAberdeen Proving Grounds

Aberdeen, Maryland

January 1986

"TA"LE OF =TITs

PAGE

LIff OF LJZSTeATINSo...o.o. ............... ,,,.,,,°., ..... .o,..... 5

I * INmftCDIJ(I... , ..... ,,,,....,,,..... .... ,,, .. *.. ,,, ....... ,....

IIe LITERATUXJU WI. °° ....... ° ...... °°..... eg......

IIIo b'YSTM/ DE:ZCiIPTIWLg.g. gg.o,°.g.. ... .o. .. e.... ...... eses..e.o .010.

IV . THE SYSTEM KINEIIC fm& o.oso...a ° .... .°....gga......a....o°.10

V, a OURK&Ltr/GU--BOMA CUNTACr MWEL.... ....... ................. 13

Vl. ORTURATOe 140ULo, ......................... o......,.1

VII, TilE SYSTEI POTITL:.L smalY ........... , ..... ......... ,,...

VILL. MU TIONS OF H0ION... ........ *... ... 0. ......

IX. SOLUTION TECHNIQUE ............... ..... o .......... o........... 19

I. €CIUKUTiER CODE DESCRIPTIO( ...... .. *.,.... ....... 2

XI NUMERICAL RES9ULTS AND DISCUSSION ......... ..................... 21

Ill. CONCLUSIONS AND REICO( NDATIONS FOR FJURTER STUDY ............... 32

A(WOWLIW&4..so4•rr ..... ... e.e...... e......a........e.a... ....... 34

RItFE Di (28. .aa..........a...... a.......a.........a...3

APPEiDIX A ..... **a.......... a ................. g... sa....... . 37

APPWDIX B....... e.g ...... .......... ..a.......ea.

APPII Co* *a....*.*.*... ag.......... as ee..... a ag o.. a.. ......... 4

APPENDIX ........ a ....................... ............. 5l

LIST OF SYMJBOLS ....... a ................ a.....aa a a7

DISTRIfLJTI(N LIST ............ o .............. a....a...............i9

3

• , i' ' II II " T •

LIST OF ILLUSTRATIONS

Figure

1 Displaced Position of Gun Bore . ....... ........ .11

2 Projectile with Bourrelft and Obturator within Gun Bore . . . . . . 11

3 Body-fixed Axes and Projectile Displacements. . . . . . . . . . . . 12

"4 Obturator and Bourrelet/Gun-Bore Contact Models . . . . . . . . . . 13

5 Time History of Gas Pressure on Projectile. . . . . . . . . . . . . 23

6 Projectile Foundation Moment. . . . . . . . . . . . . . . . . . . . 23

7 Time History of Horizontal Gun Displacement (X ). . . . . . . . . . 24

8 Time History of Vertical Gun Displacement (Y ). . . . . . . . .... 24

9 Time History of X-Displacement of Projectile (x p) . . . . . . . . . 2510 Time History of Gun Angular Displacement () for Case la .....p. 26

11 Time History of Gun Angular Displacement (e) for Case lb. . . . . . 26

12 Time History of Gun Angular Displacement (e) for Case 2l. . . . . . 27

13 Time History of Gun Angular Displacement (e) for Case 2b. . . . . . 27

14 Time History of Y-Displacement of Projectile (ye) for Case I& . . . 28

14 Time History of Y-Displacement of Projectile (y ) for Case lb . . . 28

16 Time History of Y-Displacement of Projectile (yp) for Case 2a . . . 29

17 Time History of Y-Displacement of Projectile (yp) for Case 2b . . . 29

18 Time History of Yawing Motion of Projectile (a) for Case la . . . 30

19 Time History of Yawing Motion of Projectile (a) for Case lb. . . . 30

20 Time History of Yawing Motion of Projectile (a) for Case 2a . . . . 31

21 Time History of Yawing Motion of Projectile (a) fvr Case 2b . . . . 31

22 Bourrelet Impacting Bore . . . . . . . . . . . . . . . . . . . . . 44

- I, INTRODUCTION

The yawing or wobbling motion of a projectile within a gun tube is an

important consideration in internal ballistics and is known as balloting.

This motion is a function of a number of small, difficult-to-measure parame-

ters such as manufacturing tolerances, lack of concentricity of the engraving

of the obturator, projectile and tube deformation, obturation of the propel-•lant gases and obturator wear, Certain combinations of these parameters can-potentially cause malfunctions to occur which include poor projectile launch,

damage to the fuse mechanism and explosion In the bore. The yawing motion of

the projectile leads to yaw at shot exit, and the resultant inaccuracy of the

-round. It can also lead to muzzle wear which can be significant for large

bore dipmeters and muzzle velocities. As a wobbling projectile moves down a

gun barrel, several forces act on it influencing its motion. These can be due

to imbalances, asymmetries, clearances and deformation in the projectile

itself, engraving and ?rictional forces, propellant gas blowby, tube vibra-

tion, recoil, and coupling between the projectile and the tube motion.

Depending upon the magnitudes Involved, these can lead to excessive local tube

wear, excessive projectile engraving, malfunctioning of projectlie components,

and undesirable in-bore motion. They can also cause improper exterior ballis-

tic free flight of the projectile. Differences occur between the point of aim

and the point of impact, which can be ascribed to the motion of the gun

itself, the eerodynamic forces acting and the yawing motion of the projectile

in the gun bore. Although these differences may be only as large as a few

mils'. they must, obviously, be minimized. In order for this to be effected, a

realts~fc prediction of the yawing motion of the projectile in the gun tube

becrines necessary.

7

II. LITERATURE SURVEY

In orders to predict the first maximum yaw exterior to the gun, Reno 1

carries out at rigorous Lagrangian treatment of the angular motion of the

projectile in the bore. He assumes that the projectile is centered at the

rotating Land, the plane of yaw rotates with the rifling, the impact of the

bourrelet is normal to the tube and rebound is described by a coefficient ofrestitution. Friction between the projectile and the bore is ignored. The

projectile is approximated by a single degree of freedom pendulum system, and

a closed-form solution is obtained. However, the initial yaw observed during

development of a 36-inch mortar in 1944 was greater than that predicted byReno's theory. Furthermore, scratches on the bore surface Indicated that the

projectile precessed in a direction opposite to the spin Imparted by the

rifling. In an attempt to provide an explanation, Thomas 2 generalizes Reno's

approach by removing the constraint on the orientation of yaw and deduces the

motion of the plane of yaw. His results show little difference in the yaw;

the orientation of the plane of yaw, however, is quite different. Darpas 3

studies a situation in which the bourrelet touches the bore throughout theprojectile travel within the gun tube and concludes that the gyroscopic effect

due to the projectile being,cucked In the bore, is a predominant one. The

eccentricity of the center of gravity and the droop of the tube may also not

be negligible. Perdreaville 4 presents an analysis of the projectile ballotingprohlemi using Lagrange's equations and Euler angles. The projectile is

assumed to have no center of mass offset and to be In dynamic balance about

the principal geometric axes. The equations are written to represent complete

1F. V. Reno, "The Motion of the Axis of a Spinning Sheli Inside the Bore of aGun," BRL Report No. BRL-R--320, 1943 (AD# 491839).

rL.E. Thoame, "The Motion of the Axis of a Spinning SheZZ Inside the Bore of aGun," BRL Report No. BRL-R-544, 1946 (PB# 22102).

3JG. Darpae, TransZated by H.P. Hitchcock, "Transverse Forces on ProjectilesWhich Rotate in the Barrel," BEL Report No. BRL-MR-1208, 1959 (Memorial del'artillerie francaise, 31:19, No. 1, 1957) (AD* 218873).

4F.J. Perdreaville, "Analysis of the Lateral Motion of a ProjectiZle in the GunTube," Sandia Laboratories, Albuquerqe, NM, SC-RR-7T0O71, 1971.

8

lateral dynrmlc freedom of the projectile bourrelet within the bore. Impact

is includeu and the equations are set-up so that the motion of the gun tube

can also be introduced. Chu and Soechtings present the same theory although

they resort to Euler's dynamical equations. Perdreaville 6 extends his analy-sis of Reference 4 to include the effects due to an unbalanced projectile in a

rigid gun tube. In a subsequent document7 he develops equations of motiondescribing the lateral motion of an artillery projectile as it moves down anelastic gun tube that vibrates laterally. However, he does not generate anynumerical results in either of these reports. Langhaar and Boresi 8 analyze

the problem of dynamics of a projectile in a concentric flexible moving tube.The motion of the tube is accounted for by using the Kirchhoff-Clebsch theory

of deformed rods. However, the effect of projectile balloting is ignored and

no numerical results are shown.

This report presents a dynamic analysis of a balloting projectile in a

moving gun tube. A six degree of freedom mathematical model for the

projectile/gun-tube system is resorted to and the nonlinear differential

equations of motion derived using a Lagrangian approach. The effects ofobturator flexibility, and projectile impact with the bore at the bourrelet orat the obturator along with subsequent rebound are included. The nonlinear

equations of motion are solved using Gaussian elimination and the constant-average-acceleration integration scheme. Finally, numerical results for some

sample test cases are obtained and discussed.

5S.R. Chu and F.K. Soechting, "Traneveree Motion of an Acoelerating Shell,"

Picatinny Arsenal, Dover, NJ, PA-TR-4314, 1972 (AD# 894572).

6F.J. PerdreavilZe, "Analysis of the Lateral Motion of an Unbalanced

Projectile in a Rigid Gun Tube," Sandia Laboratories, Albuquerque, NM, 8711$,SAND 74-0361, 2974.

7F.J. Perdreavi~le, "Analysis of the Lateral Motion of an UnbalancedProjectile in an Elastic Gun Tube," Sandia Laboratories, Albuquerque, NM,87215, SAND 74-0362, 1974.

8R.L. Langhaar and A.P. Boresi, "Dynamics of a Projectile in a Concentric

Flexible Tube," BLM Applied Mechanics Consultants, Contract Report No.ARBRL-CR-00501, February 2983.

9

III. SYSTEM DESCRIPTION

The mathematical model employed is a six-degree-of-freedom coupled modelof the gun-tube and projectile system. An inertial reference frame XYZ isselected with the gun-tube center of mass at (Xg, Y ). The bore may bedisplaced from its original position due to recoil and vibration with thedisplaced position described by (Xg, Y9g e) as shown in Figure 1. A typicalprojectile is represented in Figure 2. The origin of the body-fixed axes(x,y,z) is fixed on the bore centerline as shown in Figure 3. The motion ofthe projectile is described by the coordinates (xp, yp) of its center of massrelative to the gun and the yawing angle a about the orthogonal axis z. Theprojectile is assumed to have no center of mass offset and to be in dynamicbalance about its principal axes. Because of recoil and vibration, the gunbore can have axial as well as lateral and rotational motion. As the pro-jectile moves down the gun bore, the bourrelet as well as the obturator canimpact with the gun bore and then rebound from it. This effect can be includ-ed by using simple restoring springs to represent the interaction between thebore and the projectile. Both the gun tube and projectile are assumed tobehave as rigid bodies and rifling of the projectile is not considered.

IV. THE SYSTEM KINETIC ENERGY

The total kinetic energy of the projectile/gun-tube system is

T T Tgun + Tprojectile

= M mg (og2 + 9g2) + Ig ;2

+ i mp (tp2 + p2) + p a2 (1)

where mg. mp represent the masses of the gun tube and projectile and Ig, Iprepresent the mass moments of inertia about axes orthogonal to the plane ofmotion at the respective centers of gravity. Expressions for the quantitiesp and tp are derived in Appendix A.

10

Y

Inertial Frame of Reference X

Zz

Figure 1. Displaced Position of Gun Bore

Gun BoreBourrelet

Obturator

Figure 2. Projectile with Bourrelet and Obturator within Gun Bore

11

CL V

0I

V. BOURRELET/GUN-BORE CONTACT MODEL

When the projectile contacts the gun bore, the rebound force at the

. .bourrelet can be included with the help of a simple spring deflection model,as shown in Figure 4, representing the stiffness characteristics of themetallic part of the bourrelet which comes into contact with the gun tube uponi--.mpact. In this situation, the contribution to the potential energy of the

.system would be

2"Vbc- kbc 6bc (2)

where kbc is the appropriate spring constant and abc represents the displace-

ment of the projectile into the gun bore, that is, the distance from the borewall to the undistorted location of the bourrelet as determined by the orien-tation of the projectile axis. This displacement is (see Appendix B)

6bc "yp + sip s - R cl,bc (3)

where xp represents the distance between the projectile CG and the plane ofthe bourrelet and R clbc is the radial clearance between the bourrelet and thebore. Contact between the bourrelet and the bore occurs when

6bc >1 0 (4)

Gun Bore

1/2 k 1/2 ki,! ~~1/2 k ý: k

i11/2 k \1//2 k.;

1/ k.- Projectile

Figure 4. Cbturator and Bourrelet/Gun-Bore Contact Models

13

-This contact also contributes a friction force applied to the projectile in

the negative x direction, its magnitude being

Ff 1cbc kbc 6bc

where "bc is the coefficient of friction for the bourrelet/gun-bore contact.

VI. OBTURATOR MODEL

An obturator model can be built Into the balloting analysis by including

transverse springs as shown in Figure 4, in the plane of the obturator,

simulating the properties inherent to it. The spring k0 represents the

stiffness of the plastic band while k0 represents the stiffness of the metal-

lic part of the obturator which should be included in the obturator stiffness

when the metal impacts the gun bore. The restoring forces in these springs

will tend to bring back the projectile CG towards the centerline of the gun

bore in its motion down the barrel. The spring stiffness values are to be

obtained either experimentally or analytically. The potential energy contri-

bution due to these transverse springs will be

obt J k 62 (no impact)Vobt 0 ½k 0

(6)k4 2 + ( k (6 )2 (upon impact)

Vobt 0 k 0 o 0 - 2clo

where 6o is the projectile displacement at the obturator, k and k representthe stiffness of the plastic band and metallic part respectively and RCl'o is

the radial clearance between the obturator and the bore. The displacement 60

is given by

6o "lyp- sin (7)

where 1 is the distance between the projectile CG and the plane of the

obturator. Contact between the metallic part of the obturator and the gun

bore occurs when

6o0 - R clo 4 0 (8)

14

VII. THE SYSTEM POTENTIAL ENERGY

The total potential energy of the system is given by

V -V +V +V +V +VSystem Gun Proj Bourrelet Contact Obt Found Moment

*mg g Yg +mp g Yp + 62 +* ko 62 + ik ( - Rclo). 2

g g p b bc0 0 0 0 c

+ n (9)1 n

where the third term is to be included only when there is contact between thebourrelet and the bore, the fifth term is to be included only when the metal-

lic part of the obturator impacts the gun bore, and the last term is a power

series due to the foundation moment which is a large nonlinear effect generat-ed by the projectile as it wobbles down the gun bore and which can be measured

experimentally. The quantities mg and mp represent the gun mass and -theprojectile mass respectively and n denotes the number of terms in the power

series representation of the foundation moment, which is the resistance to

wobbling provided by the stiffness of the obturator drive band.

VIII. EQUATIONS OF MOTION

Lagrange's equations of motion are9

dT (IL)-1L - (r 1,i 2# *..6) (0Tt r "q qr " rnc

where L is the Lagrangian function, Qrnc is the non-conservative generalized

force acting in each coordinate direction qr, and 4r denotes the generalizedvelocity associated with the generalized coordinate qr" The Lagrangian L Is

the difference between the system kinetic and potential energies, and is, in

this case

9L. Meirovitoh, "AnaZytioal Methods in Vibrations," MaamiZZan, New York, 1969,pp. 30-50.

15

L T-V (Um ( + 9 2)+ I + 2

+ iM (2 . 11 + 2&2j ;p p p

(mig g Y + mp g Y + i kbc 62bc + i ko 2

2k (a " R 2 + E 1 a n(11)0 0 l to n n

where the quantities f p and tp are derived in Appendix A, 6bc is given inAppendix B and so and R¢1,o are as defined in the section on Obturator Model.

Application of Lagrange's equations yields the following nonlinear,coupled equations of motion for the projectile/gun-bore system

ql = X9 : (m9 + mp) (x p sine + yp Cos e + mp Xp cos e

m p yp sin e e pXp ;2 cos e + 2m p p sine a mpyp sine

+ 2 mppg cos 0 (12)

2 a Y 9: (m +m + ) M g + mp (xp cos e-yp sin e) e+mp xp sin e

+m cos e 9mpXp •2 sine - p+ mpy cos e

+ 2 mpyp; sine - (m + m )g (13)

q3 • : mp (xp sin e + yp cos e) Xg + mp (xp cos e - Yp sin e) Yg

+ (1Ig + M p X p + Mp yp2 • m p y p Xp +mp PX p yp

+mp (2 i (xp p + yp S p) - 2g 9(ip sin e + yp cos e)

+ (xp p Cosg [(ýp Cos e -p sin 0)

- p sin e + y cos e)1- m { [A Yp - 9g xp)

g p + Yp)] sin e - [URgXp + Ygp)

+ ( p - 1gp )] cos e0 : 0 (14)

16

q4 Xp: mp (X +Y +e [Xp(cos e - sine) -yp(sin e cos e)]

+ (sin e + cos e) Xp + (cose - sin e) yp - 2 4p6 (sine + cos e)

+ 2 Ap• (cos e -sin e) - (xp+ yp) j2 cos e + (y -x p) 42 sine

SXp ;2 p - p 0(gCOS - 1g sin e) i + g sin e)

Sfx(t) - ubc kbc( Yp + Lp aJ - Rcl.bc sgn p (15)

q5 p: mp (-Xsin e + g cos e + xp e + yp + 2 ; -p 4e sin e

-g 1 Cos e - y p 2 + 4 (2g cos e + t sin e)

+ g cos e} + kbc (y p L+ a - pclbc sgn (yp + Lp a))

+ (k0 + k;) (y - L a) - Rcl'o k; sgn (yp - to a) fy(t) (16)

q6 • a: lp + kbc (yp + Lpa - Rclbc sgn (yp + Lp a)) Lp

- (k0+ k;) to(Yp "-L0

+ Rclo k' Lo sgn L - £ 0a) + Z a 2an'z + c1; + c2 j+ m (t) (17)

Equations (12-17) can be written down in the form

(M]{q) + CC] {4) + [K] (q) {F} (18)

where [MI, [C), and [K] are 6 x 6 matrices, (F) is a force column matrix and{q} is a vector of generalized coordinates. The elements of these matrices

are as follows:

M11 - (mg + m) M13 a mp (xp sine + yp Cos e)

M14 -mp cos ,M15 U - -p sine ,

M2 2 ' (mg + m) M2 3 0 mp (xp cos e - yp sin e)

M?4 a 1p sine M2 5 - mp cos e

M31 - - mp (xp sin e + yp cos e) ,

M32 mnp (x p cos e - yp sin e) ,

17

433a I+ K(2 + Y 2 M34 lh.yp (19)M3 -9 + Vp5X p ,M4 - y

35-m , *4 x mp , M*42 mp

M43 a mp [xp (cosO - sin e) - p (sine + cos e)]

144 . mp (sine + cos e) , M45 - mp (cos e- sin e)

M51 " -mp sine M5 2 p Cos e,

1453 ,, mp Xp , M5 5 mp M66 a 'p

C14 1 -2 mp sine C15 , -2 mp 4 cos e

C23 - 2 m (p cos 0 -p sin e)

C3 1 , - {((;p sin e + yp cos e) + (xp cos e - yp sin e)} mp

C3 2 * {(; p cos e - 'p sin e) - i (xp sin e +yp Cos e)} mp,

C33 - - mp (( g yp - 9 gXp) sine - (i9xp + tgyp) cos el

C34 a mp (0g sin e -t 9 cos e),

C35 - m (0' sin e + R sin e) ,35 g g

C4 1 -mp 6 sin e ,C 4 2 mp e cos e, (20)

C43 -mp (xp4 + p) , C44 - 2m p (cos e - sin e),

C = -2i m (sin e + cos e)45 p

C51 - -r e cos e c52 , -p sine ,

C53 - mp (g cos e + ',g sin e) , C54 - 2mp

C66 - c1 +c2 2

18

K14 = -mpJ 2 cose K1 5 m;2 stin

K24 a -nm;2 sn 0 K2 5 -m 2 cos 0 ,

K3 4 a 2mp ; Xp I K3 5 -m2m p ,

K44 a 'mj 2 (cos 0 + sin 6), K 4 * 2 (sine cos 0)

K5 5 -mp8 2 + k bc+ko+ , (21)

p 0 0'

K6 kbc p 0 (k + ' -0+ n 0 -K 65 k kbc Lp .(k° 0÷ k ý) 10

K k 2 + (ko+ k;) Z2 +• I n-n266 bc p 0 o 0 "

"" 2 a -(mp + mg) g,

F, 4 -mp g sin 6+ fx(t) .bc kbc (Iyp + RRcl,bc) sgn ,(22)

F5 . -nap g Cos 0 + kbc Rcl,bc sgn (yp + Ipa) + fy(t)' (yy

+ Rclo k; sgn . 0 a )

F6 a m(t) + kbc Rcl,bc Ip sgn (yp + LOU) - Rcl'o k' Lo sgn (y p 0

All other elements are zero.

IX. SOLUTION TECHNIQUE

The equations of motion (Equations 18) for the projectile/gun tube system

are coupled, nonlinear differential equations which can be solved only numer-

ically. If the values of q's and q's are known at any instant, their substi..

tution in (M], [C],.[K and (F) will lead to a set of 6 simultaneous algebraic

equations with the q's as the unknowns. Elements of these matrices are first

computed from given initial values of q's and q's at the first time step and

then the simultaneous equhtions generated are solved by Gaussian elimination

19

t. o find the q's which lead to the solution for the subsequent time step.

Essentially, we have a set of six coupled nonlinear differential equations to

be solved at ehch time step. The solution to these equations may be obtained

by Newmark's constant average-acceleration method of integration10 which is a

self-starting technique that yields a step-by-step solution up to a specified

time limit, once the initial positions and velocities are supplied. The time

step can be monitored during the process of computation if needed. At impact

--and rebound between the projectile and the gun bore, either at the bourrelet

or at the obturator, the solution of the equations of motion need not be

interrupted although additional forces will come into play from the activation

of springs kbc and k; and from the frictional effect between the bourrelet and

the bore when the bourrelet slides on the walls of the gun tube. The projec-

tile/gun tube dynamic system response can then be studied at any desired time

and time histories of all coordinates describing the system behavior can

easily be obtained.

The first consideration in selecting a numerical integration scheme for

multi degree of freedom systems, as Craig 1 1 points out, should be its stabil-

ity. In most cases, it is desirable to use a method that is unconditionally

stable such as the constant-average-acceleration method. This method also

produces no amplitude error, that is, there is no numerical dissipation,

regardless of the time step size. The period error, too, is small. However,

for some multi degree of freedom systems, it is desirable to have numerical

dissipation to filter out the response of less accurate higher modes, which is

similar to truncating higher modes in a mode-superposition solution. Accord-

ing to Craig, Newmark's linear acceleration method has an advantage over his

constant-average-acceleration method in that it provides this damping for

higher modes. One difficulty, however, is that it is not unconditionally

stable. In order to ensure an accurate solution to the balloting projectile

problem, both of these methods were resorted to and, for small time steps, no

10 X.J. Bathe and E.L. WiZon, "NumerioaZ Methode in Finite EZement AnaZylyi,"Prentice Hall, Englewood Cliffe, N.J., 1976, pp. 322-325.

1 1R.R. Craig, Jr., "StruoturaZl Dynamios - An Introduction to ComputerMethods," John Wiley, New York, 1981, pp. 461-463.

20

difference in the results was noted. It was therefore decided to use the

constant-average-acceleration method for this work because of its uncondition-

al stability.

X. COMPUTER CODE DESCRIPTION

All computer code used in this study was written in Pascal on the

Hewlett-Packard 9a16 Series 200 Desk Top Computer. The main program, NEWMIARK,

performs numerical integration based on either the linear acceleration or

constant-average-acceleration method. MOPBOX (Matrix OPerations tool BOX)provides various low-level routines to manipulate matrix and vector entities,

including addition, subtraction, multiplication, etc. There is also a routine

that solves linear algebraic equations by Gauss-Jordan Elimination. Graphics

support is provided by the PLOTPAC Graphic Plotting Package. For a complete

listing of the code, see Appendix D - Computer Program Listing.

NEW•ARK, as well as its supporting modules, MOPBOX and PLOTPAC, were

tested using the example of a linear, two degree of freedom system given in

Bathe & Wilson10 on pp. 324-325. The input used is shown on page 76, Appendix

D. Results from NEWMARK were in exact agreement with those given in this

reference.

XI. NUMERICAL RESULTS AND DISCUSSION

The balloting projectile model and analysis developed have been utilized

to simulate the dynamical behavior of an APFSDS-T (Armor Piercing Fin Stabil-ized Discarding Sabot with a Tracer) projectile in a 120 mm MiA1 Tank Main GunTube in the following situations:

Case 1 Projectile CG at the Obturator (to "O)

(a) Initial condition: yobt * 0 , 0 - 0.054 deg , correspond-

ing to a maximum tilt of the projectile with no displacement at

the obturator.

21

-(b) Initial condition: 'yobt ` -o0026 in , • * 0.054 deg ,

corresponding to same projectile tilt as in (a) in conjunction

with a displacement at the obturator equal to one-half of the

-radial clearance between the projectile and the gun bore at the

obturator.

1C6:e 2 Projectile CG Between the Obturator and the Bourrelet, 2 inchesfrom Obturator (1- M 2 Inchesi

(a) Same as in 1(a).

(b) Same as in 1(b).

Other data used is as follows:

Gun weight = 4000 lbs

Projectile weight = 16 lbs

Projectile cross-sectional area a 17.53 in2

I p - 0.583 lb-in-sec2

1 9 - 150.0 lb-in-sec2

c 1 a 71 lb-in-sec, approximately 5% of

critical

c 2 -0

kbc = 5.0 x 107 lbs/in

ko =5.0 x 106 lbs/in

ko- -1.0 x 108 lbs/in

L p =5.5 inches

Rcl, bc = 0.0052 inches

Rcl, o a 0.0052 inches

"bc X 0

Time history of gas pressure p(t) on the projectile as given in Figure 5.

The curve for p(t) shown is a modification that results from inclusion of

the effect due to friction at the obturator/bore interface.

Projectile foundation moment as given in Figure 6.

The time step used for all solutions is 0.005 msec.

22

50,000

40,000

30,000

J 20,000

10,000

0 0 1 2 3 4 6 6-

Time (msec)

Figure 5. Time History of Gas Pressure on Projectile

80,000

45,000

30,000 -

'015,000

00

Projectile Angular Displacement a (degrees)

Figure 6. Projectile Foundation Moment

23

Gun Horizontal and Vertical Displacements and Projectile x-DisplacementResponse

The time ýistories of the horizontal gun displacement X (t), the vertical

gun displacement Y (t) and the x-displacement x p(t) of the projectile generat-ed by the analysis are shown in Figures 7-9. These are similar for all the

cases analyzed and no significant changes are noted from one case to the next.

00.00

-0.250 -

.0 -0.500

-0.750

.1.000

-1.2500 1 2 3 4 5 6 7

Time (meec)

Figure 7. Time History of Horizontal Gun Displacement (X )

0.000

.0.002

C". -0.004

ES-0.006-

.0-0008-

-0.0100 1 2 3 4 5 8 7

Time (msec)

Figure 8. Time History of Vertical Gun Displacement (Y )

24

350

300 -

j200

150 -

1100 -

50 -

0 0 1 2 3 4 5 6 7Time (msec)

Figure 9. Time History of X-Displacement of Projectile (x p)

Gun Angular Displacement Response

The time histories of the angular displacement e(t) of the gun for the

various cases are represented in Figures 10-13. For a projectile with its CG

located at the obturator, and initially cocked in the bore with no displace-

ment at the obturator, the angular gun displacement 9(t) gradually increases

with time as shown In Figure 10 until the projectile exits the bore with no

impact occurring between the bourrelet and the bore at any time at all. When

an initial vertical displacement is imposed at the obturator, oscillations of

the tube result from continual impact between the bourrelet and the bore as

shown in Figure 11. For the case in which the projectile CG is located away

from the obturator and the projectile is initially cocked in the bore with no

displacement at the obturator, the gun angle increases until the first impact

between the bourrelet and the bore and then begins to oscillate as seen in

Figure 12. With an additional initial vertical displacement imposed at the

obturator, the oscillations are seen to be small until the first moment of

impact between the bourrelet and the bore, following which oscillation ampli-

tudes increase with further impacts.

25

0.0000125

I 0.0000100-

S0.0000075

0.0000050-

S0.0000025

0.00000000 1 2 3 4 5 6 7Time (msec)

Figure 10. Time History of Gun Angular Displacement (e) for Case la

0.000025

0.000020-

I 0.000005-

0.000010

. -0.000005-a

-0.000010

0 1 2 3 4 5 6 7

Time (msec)

Figure 11. Time History of Gun Angular Displacement (e) for Case lb

Projectile y-Displacement Response

For case 1(a), Figure 14 essentially reveals a low-frequency type oscil-

latory behavior that results from the projectile not impacting the g~in bore at

all. For case 1(b), owing to continual impact between the bourrelet and the

bore, the oscillations are seen to be of higher frequency as shown in Figure

15. For case 2(a), Figure 16 shows a gradual fall of displacement until the

occurrence of the first impact between the bourrelet and the bore leading to

26

,, _ •1

0.0000100.0000080.000008-

-0.000002 --90.000004-

-0.0008 -

0 1 2 3 4 5 a 7

Time (mlec)

Figure 12. Time History of Gun Angular Displacement (e) for Case 2a

0.0000300.000025-

-- 0.000020.00015-

S0.000010 -0-- 000005-

E 0.000000

• 0,00005 -

-0.00010 -

9

-0.000025 .

0 1 2 3 4 5 aTime (msec)

Figure 13. Time History of Gun Angular Displacement (e) for Case 2b

the subsequent oscillatory behavior pattern depicted. For case 2(b) (seeFigure 17) the behavior is seen to be a sinusoidal motion superimposed upon an

almost linear decay of amplitude until the first impact, after which time, an

increase in frequency occurs.

27

0.000076

0.000050

-0.0

-0.000100c-0.00007 5 -

.0.000126-

-0.00010-0 1 2 3 4 5 6 7Time (msec)

Figure 14. Time History of Y-Displacement of Projectile (y ) for Case la

0.00250.0020-0.0015

. 0.00100.0005

-0.0000~ 0.0005.0.0010

-0.0015

-0.0020-0.0025-0.00301

0 2 3 4 5 6 7Time (mgec)

Figure 15. Time History of Y-Displacement of Projectile (y ) for Case lb

Projectile Yaw Response

The time histories of the yawing motion of the projectile a(t) generated

for the various cases are represented in Figures 18-21. For case 1(a), theresponse is a low frequency, sinusoidal type of motion as shown in Figure 18,which results from the projectile not impacting the gun tube at all. For case1(b) (see Figure 19), because of continual impact between the bourrelet andthe bore, oscillations of larger frequency occur. For case 2(a) (see Figure

28

.0.020

0.0016

S0.0010

0.0006

0.0000

-0.0010

. .0.0015

-.000,0

0.0026 1 2 3 4 6

Time (mse)

Figure 16. Time History of Y-Displacement of Projectile (yp) for Case 2a

0.003

3 0.001.0.001

.-0.03

-0,0034-

0 1 2 3 4 5 6 7

Time (muec)

Figure 17. Time History of Y-Displacement of Projectile (y ) for Case 2b

20), the bourrelet is initially in contact at the top of the bore. As theprojectile travels down the gun barrel, the bourrelet drops making impact

twice before exiting. For case 2(b) (see Figure 21), pre-impact oscillations

are relatively small, but become large following bourrelet contact.

29

o• 0.0010

0.0008

•'- 0.0006.! 0.0004-

* 1 0.0002-

0.0000

-0.0002 -

-0.0005-"10•.0000

.0.0005 -

-0.00100 1 2 3 4 6 6 7

Time (maec)

Figure 18. Time History of Yawing Motion of Projectile (a) for Case la

0.0016

V 0O0005

.-0.0010-

0 1 2 3 4 6 6 7Time (maec)

S~Figure 19. Time History of Yawing Motion of Projectile (a) for Case lb

Effect of Using a Smaller Time Step for Integration

In order to determine the effect of a smaller time step on the dynamic

behavior of the system, case 2a is also analyzed using At - .0005 msec. There

is no appreciable change in results except that the oscillation period becomes1.1slightly smaller, which is in agreement with Craig's observation

30

0.00100.000e80.00060.0004-

10.00020.0000

-0.0002 -

. -0.0004.

is -0.0006 -

-0.0008-

-0.00100 1 2 3 4 5 6 7

Time (msec)

Figure 20. Time History of Yawing Motion of Projectile (a) for Case 2a

0.001500.00125

. 0.00100

0.00076

0.000500.00025

0.00000-0.00025 -

-0.000W0-0.00075 .

-0.00125 0 - _ t I I

0 1 2 3 4 5 6 7

Time (msec)

Figure 21. Time History of Yawing Motion of Projectile (a) for Case 2b

Other Observations

The results shown indicate that the projectile motion oscillates about

the limits set by the clearance between the projectile bourrelet and the gun

bore. The maximum longitudinal deceleration of the gun is 220 g's and occurs

at about 2 msec for all the cases analyzed. The maximum lateral acceleration

or deceleration of the gun is lowest (1 g) for Case la, In which the projectile

CG is at the obturator and the projectile is initially cocked in the bore with

31

no vertical displacement at the obturator. It is largest (14 g) for Case 2b,in which the projectile CG is 2 inches away Irom the obturator and the pro-Jectile is Initially cocked in the bore with a vertical displacement at theobturator. The maximum projectile longitudinal acceleration is seen to be

.... 61s7,000 g, the same for all cases. The maximum projectile lateral accelerationor deceleration is the lowest (45 g) for Case la and highest (48,000 g) forCase 2b. The maximum projectile yaw rate is the lowest (0.8 rads/sec) forCase 2a and highest (9.5 rads/sec) for Case 2b, with the maximum yaw being ofthe order of .001 radians for all cases. Details are shown in Table 1. Ingeneral, the projectile whose CG is situated away from the obturator and which

is initially cocked in the bore with a prescribed vertical displacement at theobturator (Case 2b) seems to exhibit the most vigorous dynamic behavior, as

expected.

For the projectile whose CG is at the obturator and which is initially

cocked in the bore but has no prescribed initial vertical displacement at theobturator, the results indicate that there is no impact at all between thebourrelet and the bore throughout the in-bore period. Consequently, Ecuation(17) which is the equation of motion corresponding to the sixth generalized

coordinate, namely, the projectile yawing motion, reduces to an uncoupled

second order differential equation in the coordinate a. If the foundationmoment curve is linear over the range of the in-bore period, as is the casehere, this equation of motion further reduces to a linear uncoupled secondorder differential equation in a. Thus, so long as the projectile CG issituated at the obturator and the obturator has no initial vertical displace-ment, the projectile could be assumed to be a single degree of freedom systemfor obtaining a good initial estimate of yawing response. However, for othersituations, a more rigorous multi degree of freedom system analysis would seemnecessary, as the results of this work indicate.

XII. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STUDY

In order to ensure accuracy in internal ballistics, a realistic predic-tion of the yawing motion of a projectile is necessary. A dynamic analysisincorporating a six degree of freedom mathematical model of a projectile/gun-

32

L~L.

1-4 '4 1% 0 @

V)t

00 4 4-A

4J L .45iu

In §a §

1.1- 4. oP-C - IrVE -i 4.' I M

w~~~ CT 0I

4-0

4J.C

V) w4 Ia A

-) E uLA 0L'J

K-J

4LO r-33

+- i--ube system is presented in this report. The nonlinear differential equations

of motion of the system are derived using a Lagrangian formulation with the

.-effects of obturator flexibility and projectile impact and rebound at the

-bourrelet and the obturator Included. Using Gaussian elimination and

- .. •:Iewnark's constant-average-acceleration integration scheme, the nonlinear

equations are solved to yield usable dynamical response information for some

specific test cases.

The analysis and results indicate that as long as the CG of the projec-

tile is located at the obturator and no initial vertical displacement is

prescribed at the obturator, the projectile can be treated as a simple single

degree of freedom system for estimating its yawing response. For more complex

situations, however, a more rigorous analysis employing a larger number of

degrees of freedom such as the one suggested in this report seems to be in

order.

The following recommendations for further work are bound to constitute

additional contributions to the general problem of the dynamics of a balloting

projectile in a moving gun tube.

1. Account for the three-dimensional motion of the projectile and the

gun employing Euler's angles.

2. Treat the projectile as a rigid body moving within an elastic gun

tube.

3. Investigate the effect of considering the projectile as a flexible

body.

4. Include the effect of rifling of the projectile.

ACKNOWLEDGMENT

The authors are grateful to E.M. Patton for providing insight into the

physical aspects of gun dynamics.

34

REFERENCES

1. F.V. Reno, "The Motion of the Axis of a Spinning Shell Inside the Bore ofa Gun," SRL Report No. BRL-R-320, 1943 (AD# 491839).

2. L.H. Thomas, "The Motion of the Axis of a Spinning Shell Inside the Boreof a Gun," BRL Report No. BRL-R-544, 1945 (PBS 22102).

3. J.G. Darpas, Translated by H.P. Hitchcock, "Transverse Forces on Projec-tiles Which Rotate in the Barrel," SRL Report No. BRL-MR-1208, 1959 (Memorialde l'artillerie francaise, 31:19, No. 1, 1957) (AD# 218873).

4. F.J. Perdreaville, "Analysis of the Lateral Motion of a Projectile in the

Gun Tube," Sandia Laboratories, Albuquerque, NM, SC-RR-710071, 1971.

5. S.H. Chu and F.K. Soechting, "Transverse Motion of an AcceleratingShell," Picatinny Arsenal, Dover, NJ, PA-TR-4314, 1972 (AD# 894572).

6. F.J. Perdreaville, "Analysis of the Lateral Motion of an UnbalancedProjectile in a Rigid Gun Tube," Sandia Laboratories, Albuquerque, NM, 87115,SAND 74-0361, 1974,

7. F.J. Perdreaville, "Analysis of the Lateral Motion of an UnbalancedProjectile in an Elastic Gun Tube," Sandia Laboratories, Albuquerque, NM,87115, SAND 74-0362, 1974.

8. H.L. Langhaar and A.P. Boresi, "Dynamics of a Projectile in a ConcentricFlexible Tube," BLM Applied Mechanics Consultants, Contract Report No. ARBRL-CR-00501, February 1983.

9. L. Meirovitch, "Analytical Methods in Vibrations," Macmillan, New York,1969, pp. 30-50.

10. K.J. Bathe and E.L. Wilson, "Numerical Methods in Finite Element Analy-

sis," Prentice Hall, Englewood Cliffs, N.J., 1976, pp. 322-325.

11. R.R. Craig, Jr., "Structural Dynamics - An Introduction to ComputerMethods", John Wiley, New York, 1981, pp. 461-463

35

36

APPENDIX A

PROJECTILE CENTER OF GRAVITY OiSPLACEMENTS AND VELOCITIES

37

38

APPENDIX A

PROJECTILE CENTER OF GRAVITY DISPLACEMENTS AND VELOCITIES

The position vector Ip of the projectile center of mass in the inertial

frame of reference can be written as

S ,p rP/G + rG (A-1)

when rp/G represents the position vector of the projectile center of mass with

respect to the gun tube center of mass, and *G represents the position vector

of the gun tube center of mass in the inertial frame of reference. Using iand 3 as unit vectors along X and Y axes and t and m as those along thebody-fixed x and y axes, we can write

rp/ - ip + ypmrP/G P p

(A-2)

rG aX Xg + Y J

where i- cos e + sin e j(A-3)

m -sin e t + cos 0 3

Substituting equations (A-2) into (A-i), we get

"rp Xpt + Yp g (A-4)

where Xp (x Xp Cos e - y psin e + Xg)(A5

Yp ( p sin e + ypCose + Y )

Equations (A-5) can be differentiated to yield the relative velocities IPand . for the projectile CG as follows.

2p - _x sin e i + i Cos - yp Cos e - sn e + Ip p p 9

(A-6)

tp a xp cos e ; + ip sine - yp sine 4+ pCos e + tg

39

40

APPENDIX B

PROJECTILE DISPLACEMENT AT 9OURRELET

41

4?

APPENDIX B

"PROJECTILE DISPLACEMENT AT BOURRELET

The radial displacement vector of the projectile centerline in the planeof the bourrelet, measured from the bore centerline is (see Figure 22)

r , -.r÷rB rB/*G)G

(./P + rP/G + rG) (rB*/G + rG

arB/P + P/G - rB*/GA * 4

a L pc COS t+ tp sin a m + xpI + yp M- O (B-i)

where P denotes the CG of the projectile and G denotes the CG of the gun, Land m are unit vectors along and perpendicular to the gun axis, and t and Jare unit vectors along the inertial X and Y axes. Here yg, which is theinstantaneous distance of the bourrelet from the gun CG measured along the gunaxis, can be seen to be

t ga x p + Lp cos a (B-2)

Using the above approximation, it can be seen that

r Ai + B63 (B-3)

where A • -(tp sin a + yp) sin e(B-4)

B (Lp sin a + yp) cos e

Contact occurs when

H ;, Rclbc (B-5)

where Rcl,bc Is the radial clearance between the bourrelet and the gun bore.The displacement of the projectile into the bore Is then

abc U 'Rclbc "yp + fsin a I- Rclbc (B-6)

43

CDC

44 4

APPENDIX C

GENERALIZED FORCES ACTING ON SYSTEM

45

r.-.

4 i

46

APPENDIX C

GENERALIZED FORCES ACTING ON SYSTEM

The virtual power developed due to generalized forces through infinites-

imal virtual velocities compatible with system constraints can be represented

--by

6tt k Qk * 8Vk 0 Xa + QYaY9 + Qe h + Qxp axpOtt k*l =lg Xp p

"y+ Q yp 8 p + QU (C-1)

where Qk represents the nonconservative generated force associated with the

generalized velocity vk in the direction of the kth generalized coordinate.

The system virtual power will also be that resulting from the application of

given nonconservative forces acting on the system, which is the sum of the

following contributions.

SP due to force f x(t) generated by the gas pressure on the projectilea f x t) W 4 p

6P due to force f y(t) on projectill -f y(t) 6 p

6P due to moment m(t) on projectile - m(t) 6;

6P due to friction at contact surface between the bourrelet and the bore

3Ff axp sqn (;p) -" kbc kb yi + P Rclbc) 6 sgn (xp)

for small a, upon contact.

SP due to quadratic yaw damping a -(ca• + c2 6 jet)

Thus, 6Ptot { (fx(t) - Ubc kbc ( yp + tp cI-Rcl,bc) sgn ( p )) 6x p

+ {f (t) 8j I + {m(t) - (cl ;~ + C2 ; H)) 6; (C-2)

A comparison of equations (C-i) and (C-2) then yields the following gener-

alized forces

07

*Ql Qyg uQe-O

"QXp afx(t) - Vbc kbc (yp " Rclbd (• •- •-(C-3)

*Yp fy,(t)

Q a.. Qa M(t) - cl-c2&

In the above, m(t) is any external yawing moment acting on the projectile, Ubc

Is the coefficient of friction at the bourrelet and bore contact, c1 and c

are first order and second order damping coefficients for the yawing motion of

the projectile, and f x(t) and f (t) are applied forces on the projectile in x

and y directions given by

ix(t) = p(t) Ap cos a ,(C-4)

f (t) - p(t) Ap sin a ,

where Ap is the area of the projectile cross section on which the gas pressure

p(t) acts.

48

APPENDIX D

COMPUTER PROGRAM LISTING

49

50

:"Vucsd*saysprogsprrogrdm 6iewmark (input, output, keyboard)i

(0 *

(4 NEWMARK ')

(0 Numerical Integration Program 4)

(4 Author: John Baugh, Battelle-Northwest Laboratories(* Date: November 1, 1985(* -)

(0 Abstract: This program determines displacements, velocities, and 4)

accelerations using fhe Newm.ark numerical integration method for 4)

nonlinear, multi-degree-of-freedom systems. Both the constant- 4)

(. average-acceleration and linear acceleration methods are available 4)

(4 to the user. The system matrices (e.g. mass, stiffness, etc.) are *)(e defined by subroutines in the file 'USER.TEXT'. Subroutines are

called at each time step and may be functions of the current(4 displacement, velocity, and time vectors. 4)

Ssearch 'MOPSOX'S (. access matrix operations 4)

import mopboxt$search 'PLOTPAC's (* access plotting routines *)import plotpac;

constMAXDOFS - G6PI - 3.1415926S4i

typestringSG string[Be3;plotrecord -

recorddispi : plotarrayiveloc : plot array;accel : plotarray

end;var

outfile ! textifilename : string80Mndofs : integer,npts integer;ch : char,step-size : real;maxtime : real,time : plot-array;data : array [1..MAXDOFS] of plotjrecordi

(*4444.4444*.44.444444* USER-DEFINED INPUT FUNCTIONS **444444444e**********

$include 'USER.TEXT'S

**********************4********* UTILITIES *

function openoutfile(fileneme : string80) : booleani(* open an output file if possible *)begin

tryrewrite(outfile, filename);

51

y-en..n.outfile -true;

recoverooen-outfile f alse

ends ** open-out~ile *

function ctoi (ch :char) integer;(o convert a character to an integer digit *begin

ctoi :- ord(ch) - odW

ends (# ctoi 4

procedure output..data (npt3 ; integer)i(* output the disple, velocs, end accels to the specified file 4var

i, dof integersbegin

writeln(out~ile, ' do? time (inS) diepi,Iveloc accul' );

writeln(out Pile);for- i :- I to npts dobegin

for do? :- I to ndofz dobeginwrite(outfile, dof:4, ' )write(outfile, timainei: 10'4. ' 9)

write(outfile, datafdof3.displ[L3:10:4, ' %)

writeln(outfLIe, datatdof).accel ii: 10:4)endswri teln( outfilie)

endend; (* output data *)

pr-ocedure display~pluts (dof, npt5 integer)i(.display the displ, veloc, and accal plots at the specified do? *

procedure display (A, y :plot array; title plot string/i

ch :char;begin

if initgraphiC3 thenbegin

plo'v y, , rots,1 title,read(reybosr-d, ch";termgrachics

endends (o display o)

cen-u r!

di~j~inedata[coflhdispl, 'Displacem'ent vs Time');d.-ýplay time, data(dofJ.veloc, 'Velocity v5 Time');dispiay~ti.'e, data(coL~accel, *Acceleration Y5 Tine',

end: " - d; --)a/yPl,ýt 4)

procedure integr-ate (method : char; step, nae.tu'e : reai; vat- i integer);

5?

(. +

(4 Pb.stract: This subroutine carries out the numerical infegretion 4)

(t process. If method - '1'. then the linear acceleration method is•.' used. Otherwise, the constant-average-acceleration method is used. *)(. step specifies the time step to be used and maxtime specifies the(. length of time over which to integrate. i returns the total number *)(0 of points that were integrated over. In addition to using the(. matrix routines provided by mopbox, Iotegrate calls various user- 0)(* defined routines to get information about the particular problem 0)(o at hand. The following routines are Called from the file 0)

(* 'USER.TEXT':get ndofs - returns the number of degrees of freedomget init values - returns initial disple, velocs, and accels ')

(. get maas - returns the mass matrixgeVget stiffness - returns the stiffness matrixget damping - returns the damping matrix 9)getjPorce - returns the force vector 0)

typeint_conrt arrayC(..7? of reall

varT : real;Dp,Vp,Ap vector;Dc.Vc.Ac vector;

mass : matrix;damping : matrix;stiffness, Keff : matrix;F, Feff : vector;a : int const;

vlv2,v3.vAvS : vectorlml,m2 : matrix;

count : integer;npts, nsklp : integer;

procedure save data(T : reall 0. V, A vectori i integer);(* saves the current data 'in the global arrays .)var

dof : integer;begin

timaei] := T * 1000.0ifor dof : 1 to ndofs dobegin

datafdof].displ(i) ;-DOdof);dataldof).veloc[i] := V[dofl]dataidofl.accelli] :- Aldof]

endend; (* savedata *)

procedure getconstants (method : char; step reall var a :nt const);(* returns the integration constants for newmark integration 4)

varalpha, delta : real;

beginif (method - '1') then (o linear acceleration method 4)

alpha := 1.0 / 6.0else (. constant average acceleration t)

53

alpha :- 0.2S6delta := 0.51a([] := 1.0 / (alpha * sor(steo));ael] :- delta ,' (alpha * step);- ['2 .- 1.0 / (alpha * step),J(3 1- 1.0 / (2.0 * alpha) -1.0;

ea4] .- delta / alpha - 1.Ota(S) : step / 2.0 * (delta / alpha - 2.0)1a[C6 :. step * (1.0 - delta);&171 :- delta * stop

end; ( getconetants o)

begin (* integrate t)

ndofs :- get-ndofsi (' get the numb.- of degrees of freedom 0.if set-size(ndofs) then; (- set size of matrix operation* in mopbox 4)

get_init_values(Cp, Vp, Ap); (* get the initial values *)getconstants(method. step, a); (* get newmark integration constants *)

count :- 11 (9 counts each time step taken *)i : 01 (* the current number of plotting pts 4)

T :- 0.01 (. start counting at time T = 0 *)

npts : trunc(maxtime / btap) + I;nskip trunc(npts / MAXPTS) + 1I

while (T <- mextime) dobegin

if ((count-!) mod nskip) - 0 thenbegin

i :- i + 1;savedata(T, Op, Vp, Ap, i) (* for output and plotting 4)

end;

(*4** get the mass, damping, stiffness, and force matrices *o.')

get.mass(Op, Vp, mass);get damping(Op, Vp, damping);getstffrness(Op, Vp, stiffness);get force(Dp, Vp, T, F)i

(#*** form effective rtiffness metriA -440)(* Keff :- stiffness + a[t0*mass + ael ]*damping )

scalem(mass,j[E0,mI)iscaIem(dampLrg,a(l ,mZ);addr,(m1 ,m2 ,ml );addm(stiffness,ml,Keff)i

(-04* Form effective load vector -004)

(. Feff :w F + mass-(a[OJ*[Lp + a[L2]Vp + a[3]*Ap)+ damping*(aL1JoOp + a(4]oVp + aCS]JAp) *o

scelev(Op,"If),vi )I5ca ley(Up ,ar 2 ] v& )!zzalev(p ,a.3] ,v3)

add ( v I , v", v 1 ,

addv(v) ,v3,vl ';

mult(malS5,v ,v4)I

54

scalev(Op 'a[I I VI )Iscalev(Vp a( 41,v2 )iscalev(Ap at5 Slv3)iaddv(vi ,v2,vI )vaddv(vi *v3.vl )Imult(domping~vt vS)i

-addv(F,.v4,Feff)iaddv(Feff,vS,Feff)o

(0-0. solve for displacemente s*(. c t- MefP-I Faff9

if' n(.t solve(Kf.frFoff,Oc) thenwrite~..''can''t solve equation systemt (zero in diagonal)').

(44deter-mine acceleration* 4****(4Ac :- oOlo(Oc - Op) - a[22*Vp -at3).Ap *

subv(Dc,Op.vJ )lscalev(v Ya(01 vl )iscalov(Vp,et22 ,v2)szcalav(Ap,a(31 ,v3)isubv(vl ,y2,vl) )3ubv(vi ,v3,Ac)s

(0-4 determine velocities so**)(* Vc :- Vp + e(S]*Ap + eC7JeAc *

scaklev(Ap,e(6lv) )Iscalev(Ac.a(7) ,v2)eaddv(Vp vJ ,vi )I

addv(v) *vZ.Vc)i

(**to reassign current values to previous variables..)

copyv(Dc Dp~e ( Dc -> Op *)cop yvC Vc ,Vp )copyv( Ac ,Ap )Ip

(4444 Increment time by one time step44)

count i- count + 1;T :-(count -- 1) * stops

end

ends (~integrate o)

begin C.newmark a)

writeln( ' Numerical Integration Program ')irepea t

wr taln;writean('Solutiorn method -- ')iwrlteln( ' 1) Linear- Acceleration Method')iwritali(' 2) Conatent-Averege-Acceleration Method1 )iwrqte( 'Enter the niumber desired ')reed( ch)t

~~'~~ n untl (ch in ','') (nc:

writc('Enter ~aOstop size BG:*ireadln( eteQ..size)iatep-size :- 0.001 * stop-..izel

write'('Enter the length of time (m'iec): ')ereadln( nextime )naxtis'i 1- 0.001 0 naxtrimel

repeatwrite('Output data to what file ((enter> ignores) ')

*raeadln(filenome)i-until (filename - '¼) or open..outftle(filonam.)i

if (filename 0> ") thenbegin

writeln('Writirig to file ', filename)ioutpu t~deta ( npt a ) Icloue(outfile, 'gave')

endi

writaln( 'Eriter dof number to view ( ''q" to quit)' )lrep~eat

write( '> 4)read(ch)iws-itelinif (ctoi(ch) in CI..ndofs]) then

dIoplay..plota(ctoi(ch), nptg )luntil (ch in Pq' , '0'])1

end. C'newina,-k

56

module mopboxi

(4' MOPBOX "

Matrix OPerations toolBOX

(e Author: John Baugh, Battelle-Northwest Laboratories 0)

(. Date: November 1, 1985(. 4)

""X(a Abstract: This module provides low-level operations to manipulate-(W matrix and vector entities. In order to use the functions, the(0 calling program must first call at_stize to *define the order of(4 the matrices. For instance, the call .)(4' if set-size(8) then ... ),( from the main program results in all matrices being defined as( W 8x8 and all vectors as Sxi. It should be noted that attempts(* to set the size larger than the maximum declared size (10)(. results in setsize returning false (and the size remaining(. unchanged).(. 4)

(. The calling program must import this module before it can access(t any of its functions, which may be done using the 'import" compiler *)(. directive. The module object code must be online during compilation .)

(0 of the calling program (use the *search* compiler directive or(0 include this module in the system library -- see chapter I of the HP t)(t Pascal 3.0 Procedure Library for more details). Before executing t)(t your program, permanently load this module into memory using the .)(9 p-load command (see page 120 of the HP Pascal 3.0 User's Guide for *)(0 more information).

export

constMAX - 10; (* maximum size of matrices and vectors *)

typevector - array CI..MAXI of reel;matrix a array [I.,MAX, I..MAX] of real,

function set_size (size I integer) : boolean;(* eat the order for ALL matrix/vector operations in the mopbox module *)

function solve (ml ; matrix; v1 : vector; var vY2 vector) : boolean;(o solve up to 10 simultaneous linear algebraic equations *)

procedure mult (ml : matrixi v1 : vector; var vZ : voctor);(. multiply the matrix ml by vector vi: v2-mlmvl a)

procedure zerom (var ml ; matrix);(* zero each element in matrix ml 0)

procedure zerov (var v1 : vector);(. zero each element in vector v1 *)

procedure copym (ml : matrix; ver m2 : matrix);(* copy matrix ml into matrix m2 *)

procedure copyv (v! : vector; var vZ : vector),(o copy vector vi into vector v2 *)

57

--rocedure acalem (ml : matrix; f1 : real; var m2 : matrix)iSIt scale each element in matrix ml by fl, putting it into matrix m2 *)

procedure scalev (vi : vector; f] : real; var v2 vector);(0 scale each element in vector v1 by fl, putting it into vector vZ *)

-- procedure addm (ml. mQ i matrixi var m3 i matrix)l(* add matrix m2 to matrix ml: m3umi+m2 -)

procedure eddy (vl, vZ : vector; var v3 : vector);(* add vector v2 to vector v1: v3-vl+v2 *)

-g)rocedura subm (ml. m2 : matrix; var m3 : matrix)"(s subtract matrix mQ from matrix ml: m3-ml-m2 *)

procedure subv (vW, v2 : vector; var v3 : vector)l

(* subtract vector v2 from vector v1: v3-vl-v2 *)

implement

var

n : integer; (s order of matrix/vector operations *)

function set_size (size : integer) : boolean;(* set the order for ALL matrix/vector operations in the mopbox module *)

beginif (size <- MAX) and (size > 0) thenbegin

set-size :- true;n :- size;

endelse

set-size :- false;end; (* set-size *)

function solve (ml : matrix; v1 : vector; var v2 : vector) : booleant

(, "

(. Abstract: This routine solves up to 10 simultaneous linear algebraic *)(0 equations by Gauss-Jordan Elimination. If all of the diagonal(. terms in the coefficient matrix are non-zero, solve returns true. *)(0 In general, for an accurate solution, the diagonal terms should

(. dominate the coefficient matrix, ml. 4)

input parameters -

(a ml s coefficient matrix a)

(a v1 : right-hand vector 0)

(0 output parameters -

(0 v2 : solution vector(0 solve : returns true if all ter-ms in the diagonal 4)

C. of the coefficient matrix are non-zero

labelI;

varL. J, k : integers

58

hold real;bagin

solve :0 true,for i .v I to n do (' loop over each of the columns of ml *)

beginfor k :a I to n do (t loop over each row except pivot row *)begin

if (k 0 i) thenbegin

if (mll[i,3 0 0.0) then (' be sure diagonal is non-zero *)begin

hold :a -mlCk,±i / mlCI.i~lfor j :- I to n do (. loop over each element in a row *)begin

milk.j] :- ml[k,j] + hold * mi(i,J];if (J Q 1) then

m(kJ] := 0.6end;v1[kl :- viCk] + hold * vili]

endelse (v found a zero diagonal term in coefficient matrix, ml *)begin

solve :* false;goto Ii

endand

end;hole I - mIli,l]for j - I to n do (o this loop is for each element in *)

ml[i,j] :- ml[i,j] / hold; (4 the pivot row a)

ml [(,] :- 1.0;vi[i] :- vili] / hold

end;for I := 1 to n do (0 copy solution into solution into *)vZi] := vili]l (s the solution vector *)

I:end; (. solve a)

procedure mult (ml : matrix; vl : vector; var v2 : vector);(* multiply the matrix ml by vector vI: VZlmlevl 9)var

i, j :integer;begin

for j :- I to n dobegin

vI'[i] :' 0.01

for j :- I to n dov2(i] : vZi] + ml~i,J] * vIj]

endend; (* nult *)

procedure zerom (var ml : matrix);(* zero each element in matrix ml *)ver

i, j integer,begin

for i := I to n dofor j I to n do

59

-end (.i zerom *)

procedure zerov (var vI vector)l(* zero each element in vector vI *)var

I : integeribegin

for i :- 1 to n do

end; (0 _eroy 4)

procedure copym (ml : matrix; var m2 I matrix)l(* copy matrix ml into matrix m2 o)var

i, j integertbegin

for i I 1 to n dofor J :- I to n do

m2[ij] := ml[i,J]end; (* copym *)

procedure copyv (vI : vector; var v2 vector)i(* copy vector vI into vector v2 *)var

I. : integer;begin

for i :- I to n dov2[i] :- vI~i]

end; (* copyv *)

procedure 5calem (ml : matrix; f1 real; var m2 matrix);(* scale each element in matrix ml by fl. putting it into matrix m2 *)var

i, j integer;bp 'in

for i I to n dofor J : I to n do

m2([ ,j) : fl mlt i,j]end, (* 3calem ')

procedure scalev (vI : vector; fi : reals var v2 : vector);(* scale each element in vec.tor vl by fl, putting it into vector v2 *)var

i : integeribegin

for i - I to n dovZ(] := fl * viceI

endi (a scalev *)

procedure addm (ml, m2 : matrix! var m3 matrix);(* add matrix m2 to matrix ml: m3=ml+m2 *)var

i, J : integer;

60

............................

fori I I I to n dofor I to n do

m31±.JI :- mlEi.jI + mZEi JIends (* addm *)

procedure addv (vy, v2 : vectors var v3 vector)l(a add vector vZ to vector vil v3-vl+v2 *)vet

i I integer;begin

for I :- I to n dov3ti] :a vYI1) + v2il

ends (0 addv *)

procedure subm (ml, m2 : matrix, var m3 I matrix)l(# subtract matrix m2 from matrix mll m3-ml-mZ *)var

L, j integersbegin

for : a 1 to n dofor J :- I to n do

m3Ljl :a m![ij] - mZ•i.J]ends (* subm *)

procedure subv (vl, v2 * vectors var v3 : vector);(* subtract vector vZ from vector vit v3-vl-vZ *)var

i I integersbegin

for i :- I to n dov3(iJ :- vii) - vZil]

ends (s subv 0)

end. (. mopbox •)

61.-- ' e I -

m"dule plotpaciS.. .. (..,....,o..........,.....4.e.4e.eoe.4* *)

P L O T P A CGraphic Plotting Package 9)

(* Author: John Bough, Battelle-Northwest Laboratories(' Date: August 12, 1985 9)

(. Abstract: This is a library for plotting data in the form of line(. graphs. The only information needed by the plotting routine Is 9

.. the errays of points, the number of points; an index for the line(a style (e.g. solid-1, dotted-7, etc.), and a title which, when null, .1(4 causes the current plot to be displayed over the previous one.(f Plots are automatically scaled for convenience. A typical example ).(* is illustrated below: *)

program example (input, output); 1)$e Sceerch 'VOLUME:PLOTPAC'$ 0)

(. import plotpact 0)

c const 0)(4 SOLID =1 1)(" DASHED -2 4)

var 0)CA x, y, z plot_arrayt

keyboard : text; 4)

(' ch : char)begin 41

(0 reset(keyboard, 'console:'); 4)

Ce < define x, y, and z arrays 4)(t from 1 to numpte here > 4)

if initgraphics thenC. begin 4)

plot(x, y, numpts, SOLID, 'y and z versus x')i 4)( plot(A, z, numpts, DASHED, ')1 4*)

C. read( keyboard, ch)o I )•' termgraphics 4)

e andC' and.

(a As in the example, the calling program must initialize the graphics 4)

(0 display by calling "initgraphics' before plotting data withC' procedure "plot. After plotting the data, a "read" or "readln" 4)

•' statement located before a call to "termgraphics" keeps the plot 4)

C' displayed until the user gives an appropriate response. In this•t case, typing any key will remove the plot and return the display to *)(. alpha. 4)

•' The calling program must import this module before it can access 4)

(0 any of its functions, which may be done using the "import* compiler 4)C directive. The module object code mubt be online during compilation o)(' of the calling program (use the "search" compiler directive or 4)

0e include this module in the system library -- see chapter I of the HP *)(' Pascal 3.0 Procedure Library for more details). Before executing 4)(. your program, permanently load this module into memory using the(a p-load command (see page 120 of the HP Pascal 3.0 User's Guide for *)

•. more information).

62

import dgljlibi a* ccess graphics library o)

export

constMAXPTS -ISM;

typeplot...string *string(80J;

plot...rray *array CI..MAXPTS3 of reals

function initgraphics :boolearss(o initialize the device-independent graphics library and display o)

procedure terngraphics I(o terminate the device-independent graphics library and display *)

procedure plot (var x. y :plot-arrayt n : integeri linetype :integerititle :plot..strtng)l

(o plot the input arrays .

implement

constASPECT - 3.01 (o aspect ratio for all characters *

typemodetype - (alpha, graphics)1roundtype ( up, down, near);

varinitetate booleani (o true if graphics initialized *)xmin, xmax reals (* min and max values in x array *)ymin, ymax real; (o min and max values in y array o)

(......4...*.4*.*4**~*4*.MATH ROUTINES*........,.o*..)

funet'Lon log (x :real) : reals(* return log base 10 of xbegin

log :a in(x) / ln( 10.0)ends (* log *)

function power (base, exponent :real) :reals(t return the value of base ^exponent o)beginif (base -0.0) than

power :-0.0

elseif (bane > 0.0) then

power :- axp(ln(base) t exponent)else

if (base < 0.0) thenpower :- - exp(ln(-base) * exponent)

end, (4 power 0)

procedure scinotation (x :reals var mantissa, exponent real);

('return x in the form of scientific notation *

63

"If (x = 0.0) thenbegin

mantissa := 0.0,exponent :a 0.0

end- a1t0

if (x < 0.0) then

beginmantissa in -power(10.0, Iog(-x) - trunc(log(-x)));exponent :a trunc(log(-x))

end-else

it (x > 0.0) thenbegin

mantissaa power(10.0, log(x) - trunc(log(x))),exponent :u trunc(log(x))

end;if (abs(mantissa) < 1.0) then (o normalize the result *)beginmantissa : mantissa * 10.0iexponent = exponent - 1.0

endend; (* scinotation *)

function absmax (x, y : reel) : real,(* return the maximum of the absolute values of x and y *)

beginif (abs(x) > abs(y)) then

abamax . abs(x)else

absmax := abs(y)end; (* absmax *)

function round2 (n, m : real; mode : roundtype) real;(. round n to the nearest m, according to mode )

const

ROUNDOFF I IE-100l (i roundoff error fudge factor *)

var

rounded real; (* temporary holding arec *)negative boolean; ( flag: "is it negative?" *)

beginnegative := (n < 0.0)i (* is the number negative? *)if negative thenbegin

n :- abs(n); (l work with a positive number *)

if mode - up thenmode : down

elseif mode = down then

mode : up

end;case mode of

down : rounded - trunc(n / m) *m

up : beg.nrounded :w n / m iif abs(rounded - round(rounded)) > ROUNDOFF then

rounded :- (trunc(rounded) + 1.0) * m

64

else

rounded :a trunc(rounded) * mend i

near: rounded :a trunc(n / m 4 m 0 0.5) *.+end iif negative then (' reinstate the sign *)

rounded :- -rounded;round2 r- rounded

'end; (* roundZ *)

(******************* MACHINE-DEPENDENT GRAPHICS ROUTINES *.*....*....*..)

function setmode (mode modetype) : boolean;(* set the display mode to alpha or graphics *)const

ALPHAOISP - 1051; (. alpha display address .)6RAPHDISP - 10501 (. graphics display address *)

varerror : integer; (a outputasc error (0 if ok) *)on, off : integer; (a on/off switches *)x : real; (i ignored *)

beginsetmode :- true;on :- 1;off :- 0;if (mode - alpha) thenbegin

outputesc(GRAPHOISP, 1, 0, off, x, error); (o graphics off *)if (error 0 0) then

setmode :- false,outputeac(ALPHAOISP, 1, 0, on, x, error); (a alpha on *)if (error 0 0) then

setmode :- falseendelsebegin

output-esc(ALPHAOISP, 1, 0, off, x, error); (a alpha off *)if (error 0> 0) then

setmode :- false,outputesc(GRAPHDISP, 1, 0, on, x, error); (. graphics on *)if (error 0 0) then

setmode :- falseend

end; (* setmode a)

function initgraphIcs : booleant(* initialize the device-independent graphics library and.display *)const

XPIXEL - 4001 (0 pixels in x dir (HP9816) *)YPIXEL - 300; (o pixels in y dir (HP9816) 0)DEVICE = 3; (* address of screen *)CONTROL - 01 (a ignored by screen a)

varerror : integer; (i diplay init error (0 if ok) *)

begingraphics initjdisplayinit(DEVICE, CONTROL, error);if (error - 0) and satmode(graphics) thenbegin

• •.... • =Initatate .'u true;

oetaspect(XPIXEL - 1, YPIXEL - 1) (t use entire screen *)andelse

initstate :- falseiInltgraphics :a initstate

end; (o initgraphics *)

procedure termgraphics;(* terminate the device-independent graphics library and display 4)begin

if initatate thenbegin

if se3tmode(alpha) thenbegin

graphics_tarml ( terminate graphics device *)initstate :- false (o uninitialize graphics *)

endend

end; (* termgraphic5 *))

(*4*ae**4e***** 4****4******* * PLOTTING FUNCTIONS *********************e**

procedure setcharaize (width real);(0 set the character size using ASPECT as the aspect ratio *)begin

set-char_size(width, ASPECT * width)endi (* setchersize *)

procedure getjormat (x : reall var numlen, fraclen : integer)l(o determine the required string length (total and fraction) of x *)begin

if abs(x) > 0 thennumlen :- trunc(log(abs(x) + I

elsenumlen : t

fraclen := 4 - numleniif numlen < I then

numlen :- Iiif fraclen <a 0 thenbegin

?racleni 0=1

numlen :- numlen + fraclen + 1;endelse

numlen :- numlen + fraclen + 2;end; (* get format *)

procedure writenum (x : reel; numlen, fraclan integer);it write x on the graphics display at the current location *)var

slen ; integeris : plotstring;

beginstrwrite(s, I, slen, A:numlen:fraclen);setstrlen(s, slen- ;gtext( s)

66

"ana i writanum *)

procedure writestr (s : Plot-string)t(# write string a. centered and at the top of the screen *)

constWIDTH - 0.041

beginseatcharsize(WIDTH);move(-strlan(s) 9 WIDTH / 2.0, 0.9);gtaxt(s)

and; (* writestr *)

procedure drawbox (x, y : real)h(* draw a box through points (x,y) and (-x,-y) *)begin

move( x, y)Jline( x,-y);l ine( -x ,-y )ilina(-x, y)i

line( x, y)and; (* drawbox *)

procedure findmaxmin (var a : plot array; n integer; var max, min real);(* determine the max and min values of array "a* relative to zero 4)

vari : integer;

beginmax : 0.0;min :- 0.0;

for i :- 1 to n dobegin

if atil >- max thenmax :- amil;

if a[i] <- min thenmn : aill;

end;if (max 0.0) and (min 0.0) then

max :- 1.01end; (* findmaxmin *)

procedure findspacing(var max, min, spacing real);(* determine the spacing and adjust the max and min values appropriately *)

varmantissa, exponent : real; (i mantissa and exponent of absmax(x) 0)

a integer; (* a & b are used to determine spacing *)

b realsbegin

5cinotation(absmax(max, min), mantissa, exponent);a :- trunc(mantissa) + Itb :- power(10.0, exponent);case a of

I : spacing :h 0.20 * tn2 spacing :- 0.215 0 bi3,4 spacing : 0.50 * b;

5,6,7,8 spacing :, 1.00 * bi

9,10 spacing : 2.00 # b

max .m round2(max, spacing, up);min rounad(min, spacing, down)

and; l. findapacing *)

function glx ixglobal : real) * real;(o convert global x coordinates to local x coordinates *)

beginglx :- 2.0 * (xglobal - xmin) / (xmax - xmin) - 1.0

endi (t gix *)

function gly (yglobal : real) : real;(* convert global y cocrdinates to local y coordinates a)

begingly :- 2.0 * (yglobal - ymin) / (ymax - ymin) - 1.0

end; (I gly *)

procedure dra oxesi(* drew the x and y axes 0)begin

move(gl<( xr.in , p . (0.0))

Iilneý glx( xma;K) gly( 0.0 )1

move(glx( 0.0) gly(ymin))iline(glx( 0.0;, gly(ymax))

er.d; (* drawaxes *3

function in range(x, y : real) boolean;(* determine whether or not point (xy) lies inside the display area *)

beginif (x <a xmax) and (x >- xminn and (y <- ymax) and (y ?- ymin) then

in_range . trueelse

in-range f- falseend; (* in_range *)

procedure drawlines (var :,, y plot array: n : integer);(* draw lines connecting the points in the x and y array5 *)var

S: integer;count integer;

begincount : I;while (not inrange(A[count], y[countJ)) and (count < n) do

count :- count + Ii

move(glx(xfcount]), gly(y(count]))iif count -. n then

for i•: count + 1) to n do

if in range([fi , yf1)) thenl.ne(gIA(AC i ), gly( y( iI

end; (* drawlines ')

procedure labelxaxls (xspacing : real)iý* put tick marks and scale numbers on the x axis *

canst

68

-..... WIDTH = 0.03; (' width of a character (local) *)SEMI u 0.021 (* half length of a major tick mark *)

vat

xg : reali l global x coordinate 6)

xl, yl : real; (0 local x and y coordinates *)numlen, fraclen integer; (* number and fraction lengths *)

height : realt Ce character height *)

beginsetcharsize(WIDTH);height :- ASPECT * WIDTH;get_format(absmax(xmax, xmin), numlen, fraclan);yl :- giy(0.0);xg :- xmini

while (xg <- (xmax + 0.S # xspacing)) dobegin

if abs(xg) < (0.S * xspacing) then (* x is zero *)if (ymin < 0.0) and (ymax > 0.0) then (e y axis crosses x axis *)

xg :- xg + xapacingo (. skip to the next label *)xl :- glx(xg); (o convert x from global to local *)move(xl, yl - SEMI); C* move to the proper location *)line(xl, yl + SEMI); C. and draw the tick mark *)

if (ymax <- 0.0) ( x axis at top of the screen a )move(xl - nu, - WIDTH / 2.0, yl + O.S * height)

else (o x axis is somewhere bolow top *)MOVeAi - numlen * WIDTH / Z.0, yl - I.S * height);

writenum(xg, numlen, fraclen);xg :- Xg + xspacing

end

end; (* labelxaxis 4)

procedure labelyaxis (yepacing : real)l(o put tick marks and scale numbers on the y axis *)

cometWIDTH - 0.03; (1 width of a character (local) *)SEMI = J.021 (o half length of a major tick mark *)

varyg : reali (* global y coordinate *)Al, yl reel; (. local x and y coordinates *)numlen, fraclen : integer; (o number and fraction lengths *)

height real; (o character height a)begin

setcharsize(WIDTH);height :a ASPECT * WIDTH,get.format(absmax(ymax, ymin), numlan, fraclen);xl : glx(0.0);

yg ymiln;

while (yg <a (ymax + 0.5 * yspacing)) dobegin

if abB(yg) < (0.S * ys5pacing) then (* y is ?ero *)ill (,min ( 0.0) and (xmax > 0.0) then (. x axis crosses y Ale *)yo :- yg + yspacing; (* skip to the next label >

yl - gly/yg); (o convert y from global to local 4)move(xl - SEMI, yl); (# move to the proper location *)line(xl + SEMI, yl); (* and draw the tick mark 4)

69

if (xmax <a 0.0) then .(A y axis in at the right edge *)move(xi + WIDTH, yl 0.25 height)

else (o y axis somewhere left of it *)move(xl - WIDTH * (numlen + 2), yl - 0.2S * height);

writenum(yg, numlen, fraclen),yg :- yg + yspacing

end

andi (* labeiyaxis *)

- procedure plot (var x, y plotarrayl n : integer; linotype I integer;title : plotstring)i

(.44...... ** *0 ****0 ****4o0,40*o*0*** 40044**64**4eO*#400e0e00*,ee**e 0ee0**O0e )

(4 Abstract: This routine plots the input arrays as a line graph. The 4)

(0 x array should contain the horizontal coordinates and the y array *)(9 should contain vertical coordinates. n is the total number of a)

(9 points, which must be greater than zero. linetype is an Integer '1specifying the desired line style - 0)

(* I - solid line( 2 - dashed line(. 7- dotted line

(sea page 32S of the HP Pascal 3.0 Graphics Techniques Manual for *)(o more styles). title specifies the 4 itic for the plot, unless it 4)

* s declared a null string, in which'case the current plot is(0 displayed over the previous one. 4).

(.o...e.~eeeoeooeeeeeeeeeeoeesto~eeil ..**.s***************************.*.**** )

var

xspacing, yspacLng : real; (o spacing for tick marks *)begin

if initstate and (n > 0) thenbegin

if (title <> '') than (. a new plot *)begin

5et_l ins_style(1I)i

writestr(title );drawbox( 1 , I ),setviewport(O.12, 0.98, 0,07, 0.65)1

findmaxmin(x, n, xmax, xmin)lfindmaxmin(y, n, ymax, ymin)lfindspacing(xmax, xmin, xspacing);findspacing(ymax, ymin, yspacing);drawaxaeslabelxaxis•(.pacing)ilabelyaxis( yapacing)

end;set_linestyle(linatypa);drawlines(x, y, n)

endend; (i plot 4)

end. (4 plotpac *)

70

* .. e.. .*..q.......eee *.. .. *******************

USER OATA FOR NEWMARK INTEGRATION PROGRAM* (. (SIX-DOF IN-BORE PROJECTILE MODEL)

(.o*eo..e..e......ee... GLOBAL VARIABLE DECLARATIONS ".....e...........e)

g : real { gravitetional acceleration constant }mp : real; ( amess of projectile )mg : reel. i mes of gun )1I : reali < mass moment of inertia of projectile >Ig : real; { mass moment of inertia of gun )R : real; ( radial clearance between gun barrel and projectile >Mu I reali { coefficient of friction at bourrelet (zero) }L : real# ( distance between obturator and bourrelet )Lb real; 4 distance between projectile c.g. and bourrelet )Lo realf I distance between projectile c.g. and obturator >Xe I real; i angular stiffnees at the obturator )Kb real; i bourrelet stiffness (linear) >Ko real; nylon obturator stiffness (linear) )Kol real; < aluminum stiffness at obturator >C1 real; a ngular damping coefficient )C2 real; nonlineer angular damping coefficient >P real maximum applied pressure to projectile >area : real; ( cross-sectional area of projectile >

(.....*...e.......0......e e*.. INPUT FUNCTIONS eee .aeeeeeee~eeeeeeeoee)

function getndofs I integeri(o set the number of doft *)aegin

get-ndofs :- Bend; (* getndofe *)

procedure getAnitvaluel (var dispi, veloc, acce'! vector')I(* specify the initial values *)ver

yo : realibegin

zerov( di spl ) Izerov( valoc )izerov( accel )I

g :- 386.4; ( in/saec^2Mp ;- 16.0 I go ( lb-sec'2/in )mg - 4000.0 / go < lb-seec^/in )Ip :- 0.531S lb-in-sec^2'Ig 150.0; ( lb-in-secaZR := 0.005Z i4 nLp S.{ inrKa 1" 8.6ESt 4 in-lb/rad dXo = S.OMES 4 lb/in >C1 la 71,0; < lb-in-sec (about S percent of critical)C2 :-0.0P : S .0E3 4 ib/ln'* >area 1- 17.53; 4 in'2 }

71

riarto('Enter the distance fr-om the obturator to the c~.(in.)readln(Lo)iwrite( 'Enter the Initial projectile angle (radians) : ')treadln( di spi 16 1)warite('Enter the initial vertical displacement at the obturator (in.)readln(yo)iLb :- LP - odispltS] l- yo + Lo * sin(disrIES])i

and# (4 getlinit..valuea *

procedure get._,meas (dispi. veloc zvector; ver mass m atrix)i(o specify the maet matrix( *)var

Xg, Xgdot :realiYg, Ygdot : real;0, Odot : real;xp, xpdot : real;

yp. ypdot :real;a. adot :real;

beg.uXg -dispiri]; Xgdot :~veloc[12;Yg z-displC22; Ygdot :-vuloct21i

o0: displ133i Odot 1-velocE33i

xp diaplE4]; xpdot :-veloc(41;

yp :-diplES~i ypdot :-voloc[Sli

a 3-dlapl(6); adot :-veloc[63;

zerom(maea)i

inasstl,1J :,w Mg + opiaSOCi ,3) -mp * ixposin(0) + ypocos(0))lmass(1,4J :-pp * cos(O);maas*El,5 --pp * ain(O);

maes[2,&` m-rg + mpipmaas[2,31 mp * (xpecos(O) - yposin(0))iinasz[2,4J : mp * sin(0)ipiass[2,5 3- mp * cos(O);

naess3,1] : -mpp (xpeoin(O) + yp~cos(O))lnasst3,2) :- p 4 (xp*coa(0) - yp*S3fl(O))imaaat( ,] : Ig + mp * (sqr(xp) + sqr(yp))smass[3,43 -mpi ypipiaO513,S) M- m*XP;

nasaa4,IJ :-pspnass(4,2) m-pipnaass(4,3J :- p* (xp*(cos(O) - sin(O)) - yp*(uin(0) + coa(0)))maess[4,42 mp 0(sin(O) + cos(0))fnaI15f4,5]3m MP (cos(O) - sin(O,));

maess5.1 3 I- -pip s ir,(O);piaSSES.2) Mpp a coOS;~masa(5,31 :- p * pM8361t5,5) M-PI;

nmOSt6,61 I-end ( getnmass a)

72

procedure get..damping (dispi, voloc vectori var damiping matrix)i(0 specify the damping matrix *var

Xg, Xgdot :reali--Yg, Ygdot : reeli0, Odot :r-ealixp, xpdot :realiyp, ypdot :real;a, adot :raeal

begin:- d isp I II Xgdot 2wvelactH;i

Ygo di~pIE23i Ygdot :uvelocC~li

0J: displ(3li Odot :velocE33hxp :dispj[41e xpdot -velocl42i

yp :dLSPIE53; ypdot :.veloctS];a :-displt6]; adot :*velocE61i

zerom( damp ing);

damping[1,41 :-20*mp * sin(O) # Odotldempingf).5) :-2.0 *mp 0 cosCO) * Odoti

dampingE2,3J : 2.0 *mp *(xpdotecoe(0) - ypdotosin(0))o

damping(3,1J -mp *((xpdot~oin(O) + ypdotocos(0))+ Odot*(xp*cos(O) - yptsin(O)))i

dampingf3,21) :-mp * ((xpdot*cos(0) - ypdot'sin{0))- Odoto(,xposin(0) + yoo())

dampingf3,3J : -mp * ((Xgdoteyp - Ygdotoxp) * sin(O)- (Xgdot*xp + Ygdotoyp) * cos(0))i

dampingI3,4] mp * (Xgdotoein(O) - Ygdot'cos(0))idamping[3,52 : mp * (Ygdotesln(0) + Xgdot~cos(0))i

demping[4,1) Pip * sin(O) *Odoti

dam'ping[4.23 -mp * cos(0) *Odotl

dampingj[4,31 2w-mp * (xp'Odot + ypdot)idamping[4,4] : 2.0 * mp *Odot *(cosCO) - sinCO))damping(4,5J : -2.0 4 mp *Odot *(sin(Q) + cosCO))i

dampingf5,1) 2 -mp *cos(0) * Odot;damping(5,2] -rip *sin(O) * Odotldamplng(5,33 m-rp * (Xgdotocos(0) + Ygdot~sin(0'))odaripring[,41 2.0 # rip *Odoti

dampingf6,6J :- C + CZ *ab3(adot)end; (* get~daripir'g t)

procedure get~ati.fness (dispi, veloc vector; var stiffness matrix,:(0 5pecify the stiffness matrix(*var

Xg, Xgdot real;Yg, Ygdot real;0, Ociot :realtAP Apdot 2real;

yp, ypdot '-eal;a, adut 2real;Ro, Rb ret.i;

begin

73

Yg i dispi~lIi; Xgdot za velooc( IIYg i- displCZ3; Ygdot 3m valooc(2]o : displt3); Odot i- velocC3]ixp disol[4]i xpdot :-veloc141i

yp 2- displtS3; ypdot YS OcCES);a :- displCG); edot :-veloc[63,

zerotr( stiffness h

Ro z.yp - Lo # sin(a)iRb 'ayp + Lb # uin(a)o

if (abs(Ro) >- R) then ('contac~t at obturator *Kol a .OES

elseKol 3m0.0;

if (ebs(Rb) >- R) then (.contact at bourrelet *begi±n

Kb :-S.OE7#

mu : .endelsobegin

Kb 3-0.0imu : .

end;

stiffnass(1,41 :--mp * cos(O) * eQr(Odot)istiffnaess(1,] m-p *sin(O) *sqr-(Odot);

atiffnese(2,42 a -mp * sin(O) 0 sqr(Odot)istiffnasaE2,S] -mp * coo(O) 0 oqr(Odot)t

stiffness[3,4J : 2.0 * mp* Odot * xpdotostiffneast3,5J : 2.0 * mp a Odot * ypdoti

3tiffneass4,4] a -mp * 5qr(Odot) # (coaCO) + sin(O))iatiffnes5E4,S] 3-MP * 3qr(Odot) 4 (ain(O) -cos(O))i

stiffnass(S,S5 : -mp # sqr(Odot) + Kb + Kc + Kolistiffnaaa[5,61 : Kb *Lb -(IKo + Kol ) * Los

5tiffnaee(6,S1 -Kb *Lb -(Ko + Kol) * Lolstiffnea[s6,61 -Kb * sqr(Lb) + (Ko + Kol) * oqr(Lo) + Ka

endi (* get~itiffnas3 0)

pr-ocddure gdtjforce (diepl, valoc vector; time 3real; var- force va ctor);(* 5pecify tha force vector 0)var

Xg. Xgdot :reallYg, Ygdot : reel;0, Odot : realsxp, xpdot areelsyp. ypdot realsa, edot arealiload, F 3real;

function agn(x : real) areal;

74

-- itirri the sign of x

ifn x < 0.0 then

S~~sgn u.-~else

son j- I.9end (* son ')

beginXg :- diasi[1]e Xgdot I- veloc[1l;Yg ;- disDl[2]# Ygdot i- vYloc(2]o0 "- dlspl[3]i Odot : velocl3]ixp le d1spl(431 xpdot 3- v~looc4];yp I- displ[S]i vpdot 3w veloc(5];a I- dispIE6S; adot i- veloc[S]l

zarov( force) !

if (time < 0.0025) thenload :a - 0.1 + 508.0 0 time

else

load :- 1.5 - 180.0 * time;

if (load > 1.0) thenload :- 1.0

elseiP (load < 0.0) than

load 0 8.0;

F :- load * P * area;

Porce[2] :" -g # (mp + mg)iforce(4l :- -pip * g * sin(O) - mu Kb * (abs(yp + Lboa) - R) a .gn(xpdot)

+ F * cos(s);force[S] :- Kb * R a sgn(yog + Lb a a) - * g * cos(O) + F * sin(a)

+ R * Kol * agn(yp - Lo & a)iforce(S] :- Kb * R * Lb * ogn(yp + Lo a)

- R K Kol * Lo # ugn(yp - Lo * a);endi ¼ getJorce *)

75

USER DATA FOR NEWMARK INTEHRATION PROGRAM

~*...*e************.*...eo*** * INPUT FUNCTIONS ........ e...............u...)

function getndofs : integers(* set the number of dofs *)begin

get-ndofs :- 27 -ndi (* getndofs *)

procedure getinit.values (vat displ, veloc, accel vector);(t specify the initial values *)begin

zerov(displ)izerov(valoc);zerov(accel)iaccel[2] := 10.0

ends (* get..nitsvalues *)

procedure getmess (dLpl, veloc vectors var mass : Matrix);(t specify the mess matrix 4)

beginzerom(mass);mass[ l1l1 1 Z.0messC2,2] :. 1.0

ends (t getmas5 *)

procedure getdamping (displ, veloc vectors var damping : matrix);(* specify the damping matrix *)begin

zerom(damping)ends (" getdamping )

procedure getstiffness (displ, veloc : vectorl var stiffness matrix)i(4 specify the stiffness matrix .)begin

zerom(stiffness)istiffness[I,1] := 6.0; stiffness 1,21 -- 2.0;stiffness[2,1J :-2.01 stiffness(2,2J :- 4.0

end; (o gatstiffness 5)

procedure get fcrce (diepl, veloc vector; time reel; var force vector);(* specify the force vector *)begin

zerov(force)iforce[2] :- 10.0

end; (* get_force *)

76

LIST OF SYMBOLS

an coefficient of nth term in power series representation of thenonlinear foundation moment

Ap area of projectile cross section on which gas pressure acts

c1, C2 first and second order damping coefficients of projectile resistanceto yawing motion

fCJ damping matrix in system equations of motion

Ix fy applied forces on projectile in x and y directions

Ff friction force on projectile at contact between bourrelet and gun-bc bore

(F} column vector of forces in system equations of motion

g acceleration due to gravity; also denotes gun

G denotes gun CG

i, J unit vectors in X, Y directions

19 mass moment of inertia of gun *about an axis through its CG perpen-dicular to plane of motion

Ip mass moment of inertia of projectile about an axis through its CGperpendicular to plane to motion

kbc spring constant for bourrelet/gun-bore contact spring deflectionmodel

ko stiffness of obturator plastic band in obturator transverse springmodel

Ik. stiffness of metallic part in obturator transverse spring model

[K] stiffness matrix in system equat1•ns of motion

1g instantaneous distance of bourrelet from gun CG measured along gunbore axis

I p distance between projectile CG and plane of bourrelet

1 0 distance between projectile CG and plane of obturator

e, m unit vectors along x, y directions

77

L Lagrangian function

tf mg mass of gun

mp mass of projectile

m(t) applied yawing moment on projectile

(M] inertia matrix in system equations of motion

n number of terms in power series used to represent the nonlinearfoundation moment

p denotes projectile

P(t) time history of gas pressure on projectile

P denotes projectile CGSP virtual power

qr rth generalized coordinate

{q) column vector of generalized coordinates In system equations ofmotion

Qrnc nonconservative generalized force associated with rth generalizedcoordinate qr

r radial displacement vector of projectile centerline in plane ofbourrelet measured from the centerline of the gun bore

Rcl, bc radial clearance between bourrelet and gun bore

Rcl,o radial clearance between obturator and gun bore

t time

T kinetic energy

vk generalized velocity in direction of kth generalized coordinate

V potential energy

x, y, z gun body-fixed axes of coordinates

XP, yp coordinates of projectile CG relative to gun body-fixed axes ofcoordinates

X, Y, Z inertial frame of reference

73

Xg, Yg displacement of gun CG relative to inertial frame of reference alongX, Y axes

angular (yawing) displacement of projectile

6bc displacement of projectile into the gun bore at contact betweenbourrelet and gun bore

60 projectile displacement at obturator

. . angular displacement of gun in direction of positive Z axis

"Ibc: coefficient of friction for bourrelet/gun-bore contact

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