Controlled tumbling of projectiles—I. Theoretical model

Int. d. Impact Engng Vol. 7, No. 1, pp. 101-115, 1988 0734-743X/8853.00+0.00 Printed in Great Britain © 1988 Pergamon Press plc

C O N T R O L L E D T U M B L I N G O F P R O J E C T I L E S - - I . T H E O R E T I C A L M O D E L

OSCAR RUIZ a n d WERNER GOLDSMITH

Department of Mechanical Engineering, University of California, Berkeley, CA 94720, U.S.A.

(Received 1 September 1987; and in revised form 7 January 1988)

Summary~The present investigation is concerned with the generation of controlled tumbling of a blunt-nosed hard-steel projectile and, to some extent, the effect of this tumbling on targets struck subsequently. The mechanism for producing this motion from an initial state of translation in the direction of the projectile axis consisted of the impingement of the bullet on the edge of deflector plates of aluminum, steel or plastic of various thicknesses with various degrees of overlap. These initially plane rectangular plates were rigidly held on three sides and struck at the center of the fourth, or free, edge. This first portion of the study consists of the continuum-mechanical analysis of the impact of the projectile on the plate and the subsequent motion of the projectile. A mode shape analysis is employed to ascertain the deflection of the plate with the impact area modeled as a hexagon. The projectile is considered as rigid and its motion as planar prior and subsequent to the impact with the deflector plate, which is modeled as a rigid-perfectly plastic material. The numerical predictions of the theoretical development, the experimental set-up and the data derived from the tests, as well as a comparison between the analytical and measured values, are presented in a companion paper [Ruiz. and Goldsmith, Int. J. Impact Engng Vol. 7, in press (1988)].

A Ac At

B b C D Dim Dext d

Eo F, H Io K k~j kl L

M~j

Mo m

m* N~j

hi, nj P, /'3 Pc Q, qi q R

N O T A T I O N

width of deflector plate equivalent projectile face area used in mode analysis area of middle plane of deflector plate position of hinge intersection impact area from lower plate edge center opposite clamped edge of the plate proportionality constant obtained from compatibility requirement half-length of equivalent impact area on plate position from free edge to center of impact area; dissipation rate internal energy dissipation rate external energy dissipation rate diameter of the projectile initial energy components of external force on projectile face deflector plate thickness transverse moment of inertia of projectile about its mass center concentrated load factor for )1, equation (43) rate of change of curvature of plate factor defining type of support of lateral edges of plate half-length of deflector plate bending moment per unit length in plate bending moment per unit length at plate boundaries

H 2 plate yield moment per unit length = cr o - -

4 projectile mass dimensionless projectile mass direct (membrane) and shear forces per unit length direct (membrane) and shear forces per unit length at plate boundaries component of unit normal vector in-plane components of external stress on plate, i = 1,2 transverse component of external stress on plate critical or collapsing pressure in plate generalized stress in plate generalized strain in plate correction factor for the projectile-plate contact area radius of projectile

101

102 (). RuIz and W. GOLDSMVfH

R' radial location of external force on projectile face S distance from free edge to center of impact area on plate T half-length of projectile; kinetic energy

T0 initial kinetic energy t time

t* dimensionless time tf duration of impact

t~ in-plane and transverse velocities of plate boundaries

u i in-plane displacements of the plate middle plane, i = 1,2 u* mode solution V o initial velocity of projectile

Vin i initial velocity of plate v in-plane deformation of plate, v = u 2

v* dimensionless in-plane deformation of plate vf final in-plane deformation of plate w transverse deformation of plate

w* dimensionless transverse deformation of plate wf final transverse deformation of plate W maximum transverse deformation of plate

C ratio - - , equation (25)

E ,8 angle of rotation of projectile

fl* dimensionless rotation angle of projectile ~, ratio of radial location of force to projectile radius

D 6 ratio , equation (25)

E A edge offset of plate load, equation (41) e direct or membrane strains in the plate

B ~0 ratio - - , equation (25)

E 41 dimensionless constant in equation (31) 42 dimensionless constant in equation (31) 43 dimensionless constant in equation (31) r/ corrected dimensionless initial plate velocity

q' value of r/if loading is uniform on whole plate area 0 hinge line inclination; rotation angle of projectile

0o difference between fl and 0 A dimensionless constant in equation (62), defined in equation (63) ). normalized initial kinetic energy

areal density or mass per unit area of plate Pr ratio of plate to projectile areal densities p density of projectile

a o yield strength of plate t ratio of v- to w-deformation of plate, equation (53)

~ij rate of change of curvature of plate boundaries qJ function of time

I0 f~ normalized inertia moment = -

itAc R2

I N T R O D U C T I O N

In ballistics, yaw describes projectile movement where the axis of symmetry forms an angle, constant or variable, with the tangent to the trajectory of flight path. Tumbling, on the other hand, describes a continuous angular motion of the projectile around an axis perpendicular to the flight plane. The controlled generation and motion analysis of such tumbling projectiles and the influence of this tumbling on their ability to penetrate targets does not appear to have been explicitly treated before. However, a somewhat similar phenomenon involves the study of the mechanics of penetration of rods with a constant yaw [1], but there is no reference to yawing motion here or in other related publications [1-3] .

Controlled tumbling of projectiles 103

S C O P E O F T H E P R E S E N T I N V E S T I G A T I O N

The present work is concerned with the controlled generation of tumbling in cylindrical, blunt-faced, rigid projectiles initially translating at low speeds. The strikers are composed of drill rod cylinders, 38.1 mm long and 12.3 mm in diameter, with a mass of 35.4 g.

After considering several methods that would cause this type of motion in the projectiles, a technique was selected where the bullet impinges on the free edge of a rectangular plate, clamped on the three other sides. The interference or edge offset is quantified in terms of the projectile diameter, and can be varied as desired; it is a parameter governing the subsequent projectile motion. The plates are generally made of steel, a highly strain-rate sensitive material, aluminum 606 l-T6, an almost strain-rate insensitive material, or aluminum 2024- 0. A few exploratory tests were also conducted with Lexan, which exhibited almost completely elastic recovery after the impact.

The projectiles were launched at velocities that do not cause perforation or fracture of the deflector plate, limited here by the plate material and thickness to values of 100 and 125 m s- 1. Interest is focused primarily on determining experimentally the response of the projectiles during and after impact. The final magnitude and shape of the deformed plates were also studied. The projectiles were stopped by a block of clay placed behind the deflector plate. The shape of the crater produced was recorded for different impact conditions. In a few cases, the effect of the impact of such a tumbling bullet on a target plate behind the deflector was also noted. A first-order analysis of the process was executed and is presented in this work; the experimental results and their comparisons with the theoretical predictions are contained in a companion paper.

A N A L Y T I C A L F O U N D A T I O N

In the present analysis, the projectile will be regarded as a rigid body in plane motion and the deflector plate as subjected to impulsive loading when hit by the projectile. Dynamic plastic deformation of the plate will be considered subsequently. The transverse and in-plane deflections, the forces in the plate and the rotational velocity of the projectile will constitute the unknowns of the problem. As part of the solution, expressions for the duration of the impact, final deformation of the plate and the final linear and angular velocities of the projectile will be derived.

The principle of virtual work [4] states that, for a system of loads and stresses in internal and external equilibrium and for any system of strains and displacements satisfying the strain-displacement relations, the rate of work done by the stress field on the strains must equal the rate of external work done by the loads on the displacements

Dim = f (Nij~j + M i j k i j ) d A (la)

D~xt = f [(Pi - #ii,)fi~ + (Ps -/~)~i,] dA + [~7,fiij. + ICl~jb,j], (ib)

Dim = Dext (2)

where the summation convention has been adopted with the indices i andj taking the values 1 and 2. The last term in brackets in equation (lb) represents the work done on the boundaries where N o. = ~7~, Mij = -Mij, ~ij = iq and/~q = q~ij; its meaning is indicated in Fig. 1. The area A extends over the mid-plane of the plate after deformation.

For small strains and moderate transverse deflections or rotations of a rectangular plate element as shown in Fig. 1, the relations

1 % = ~ (ui, J + uj, i + w lw j ) (3a)

kij = W,i j (3b)

104 O. Rtqz and W. GOLDSMITH

/Vf I

I /VI2 @ Q2

Q2 ®

®

Q~

Nrz

N~z

/Vii

( a )

i _ _

x 2

M22 I MI2

g ~ 2 d X z

dXt

x I

g f l

~ ( b )

L _

~2

(c)

~s, ~ ~'

FIG. 1. Forces (a) and moments (b) acting on the middle plane of a plate, and moments (c) acting on the boundaries.

apply [5], where e u are the direct strains and k u the curvature changes; this is quite similar to the von Karman plate approach. Substituting equation (3) into equations (1) and (2) yields

f (Nij, i "]- Pi - - ] 2 i ~ i ) U i dA + f [(N,jw,j + Q,),, + P3 - dA = 0, (4)

with Qi = Q_.~, No = Nu and M u = Mu at the boundaries. In order for equation (2) to be valid, the coefficients of fi~ and wi, which are virtual displacements, must vanish in equation (4). Hence,

with

Nu. ~ + Pj - p//j = 0 (5a)

(Qi + Nijwj),i + P3 - / t ~ = 0 (5b)

Qi =- - Mji,j (5c)

In terms of the generalized strains, equation (3) and stresses N u and M u, the yield condition for a rigid-perfectly plastic material of the plate can be written as [6]

f (elj, k u) ~ O, (6)


while the flow rule can be expressed as

if ) < 0 :

if ) ' = 0 :

e0 =/~0 --- 0 (7a)

f f eu = b ONi---- ~ (7b)

and /~0 = b ~ ' ~ (7c) 8Mif

where b is a positive constant determined by compatibility. Equation (7) represents the constitutive relations.

MODAL ANALYSIS

The method of mode approximation was proposed originally in [7] to provide reasonably accurate solutions for problems involving impulsive loading of rigid-plastic structures in the small deformation range. It resembles the normal mode solution technique used in vibration problems in linear systems, but with significant differences. For example, no superposition of modes is possible in the plastic case. The basic hypothesis is that the deformed structure maintains the same deformed shape throughout the process, varying only in amplitude. Thus, the problem is treated as a deformation having a single degree of freedom. It is assumed that the mode solution can be written in separated form as

fi*(s, t) = adp*(s)T(t) (8)

where 4~* is a function of only the space variable s, T is a scalar function of the time alone and a is a constant to match initial velocities. In the case of plates, s refers to the coordinates xi and ds is dA.

Consider now for a particular problem the actual solution given by fib(s, t), (1, Qi and the mode solution given by fi*(s, t), (t* and Q*, where (1i and Q~ are the strain rate and stress fields, respectively. The initial velocity distribution is v(s). Using the principle of virtual velocities, equation (2), the difference between the two solutions is

- fs/~(//,-//*)(fi,- fi*) dS = fs (QJ- Q*)(qi-q*)dS, (9)

where/~ is the mass per unit magnitude S, which is area in the case of plates. Furthermore, considering the normality of plastic flow, it is necessary that

d d t (~) "< 0, (10)

where

~ = ~ /~(fi~ - fi*)(~i - ~ ) dS (11)

is the first term of equation (9). In equation (11), ~ is a positive function of time, and from equation (10) it can be

concluded that it decreases with time. Since ~ represents a difference between the velocities of the two solutions, they approach each other with time. From equation (9), they are equal only if Q~ = Q~' or if q1 = q~'" Replacing the given initial velocity V~,~ and the mode form, equation (8), in equation (11), the value of ~ at t = 0 is obtained as

1 ~" , [//o = ~ js~/(Vini __ at~i )(Vi.i - adp*)dS (12)

The best value for a is the one that makes A a minimum. This is given by

d~0 ° =0. (13)

da

106 O. Rulz and W. GOt.DSMIrH

Solving for a, the resultant expression is

j's# V~.,q~* dS a - * *d (14) ~ , ~ ,~, s

Substituting this value of a in equation (12) gives q~° as

0 ° = #ViniVinidS-2a 2 #~b*~b*dS = T O - To* (15) S J

which is a difference between the initial kinetic energies. The choice of the best mode shape can be based on the largest lower bound to the response

time

/) * j's# i. i¢i dS t~ D(~') ~< tf, (16)

where t~ and tr are the mode and the actual response time, respectively, and

D(~b*) = ( Q*0* dS (17) ds

is the energy dissipation rate corresponding to the mode ~b*. The response time is that taken by the structure to come to rest after impulsive loading. When the external applied loads are concentrated, the numerator of equation (16) is almost constant for different types of mode shape and the problem reduces to minimizing the dissipation rate, so as to maximize t*, which will be done here.

When the mode solution is applied to a problem involving large deflections--as in the present case--the stress fields that originate during motion will generally violate the yield condition at some points. Therefore, the convergence property expressed by equations (10) and (1 i) will not be explicit. One way of avoiding yield violations is to choose instantaneous modes in such a way that the yield condition is never violated during the deformation process [8, 9]. This technique requires the use of numerical methods since the equations to be solved are rather complicated. An alternative is to use a fixed mode shape along the deformation process with mode shapes resembling small deformations of the structure in the early stage of the response. In the present case, the best mode choice is based on the maximization of equation (7). This is equivalent to minimizing the static collapse load of a rigid-perfectly plastic structure under small or null deformations which are similar to the dynamic deformation of the plate in the early stage of the process. Therefore, the deformation pattern of the structure, a plate with a plastic hinge pattern, under the 'best' static collapse load, will be taken here as the best mode shape.

SIMPLIFICATION OF THE EQUATIONS

To apply the mode approach to the present problem, the mode form given in equation (8) can be substituted into equation (5b), assuming an arbitrary mode shape, and an expression depending only on the variable t can be obtained. Solving for t and substituting it again into equation (1) will provide the complete solution. Effects like strain hardening could also have been included in the expressions for N u and M u. However, this procedure leads to complicated expressions even if strain-rate independence were to be assumed. Neglect of bending terms would mean consideration only of membrane effects [10].

An alternative procedure is to apply the mode solution to the expression that results after the virtual work principle has been applied. This procedure was suggested in [11] for the static case and extended in [12-14] to the dynamic range.

First, recalling the expression for the external work rate, equation (lb)

De = [" [(el - lziii)ui + (Ps - ~ ) w ] dA, (18) JA


where i takes the values 1 and 2 that refer to the plate in-plane deformation in the directions X and Y respectively, and P and w are the external transverse applied stress and transverse deformation, respectively. Substituting equations (5a) and (5b) into equation (18), making use of the Green's theorem and assuming that the area A of the plate, with boundary of length C, can be decomposed into l smaller areas Am, each with boundaries of length G, within which Nijti~ is constant, the following expression is obtained:

fA [(Pi - I'tfii)fii k- (Pa - I't~)~] dA = - ~m= , fc N'ja n' dcm

+ N~jfij,~ dA + Mj~,~i,n~ dCm = 1 m

-- ~ fc Mjiw,lnjdCm+fAMjiffdi dA m = l m

-- ~ fc Nijw.jffnidCm m = l m

q- ~ ;C Nijw'l'v inj dCm - f (Nij'jwl~'i "l'- Nijwff'Ji) dA' m=l m " JA

(19)

where n~ and n~ are the components of the unit normal. The subdivision of the plate into smaller areas does not yet imply that only plastic hinges

are allowed, since the areas can be deformable. It can now be observed that the deflection w and velocity ~ must be continuous

throughout the plate. In a perfectly plastic material, w,~ is continuous, but the velocity of the slope ~,~ can be discontinuous at travelling plastic hinges. Furthermore, when P~ is small compared to Pa, u~ and fit can be neglected. For the case of the deflector plate interacting with the projectile, it can be argued that this is true when the edge offset is small and the in-plane change of momentum of the projectile during impact is small compared with the transverse change. However, the same cannot be said when the edge offset is large and large in-plane momentum changes ensue. For the first case, using equation (5a), equation (19) becomes

(P3 -/~l~)ff dA ( N i j w - M~j)~,inj dCm + (Mij - Nijw)~,~ i dA. (20) m = l ,.

At this stage there are two alternatives. Ifa continuous deformation field or mode shape is assumed, the first term on the right hand side of equation (20) drops out and only the term corresponding to the energy dissipated in continuous dissipation fields is retained. However, in a non-symmetric geometry like the deflector plate, it is difficult to determine precisely where yield is occurring and to include the effects of N~j and M~j in those yield zones.

The second choice is to retain the term corresponding to the internal energy dissipated at travelling plastic hinges only and drop the other one. In this case, the mode shape will be a set of rigid plane regions separated by plastic hinges. Thus, the pattern of plastic hinges will be crucial in the solution of the present problem.

Additionally, it is assumed that the hinge pattern of the dynamically deformed plate resembles that of the static collapse load [ 11]. Therefore, the plastic hinge pattern is time- independent and the plate is composed of rigid plane regions separated by a number p of straight hinges of length 1 k each. The angular rotation rate of the flat regions across the hinges is O k = ~,~n~; thus,

f,4 (Pa - l.t~)~ dA = m~= l fq (Nw - M)Ok dl k (21)

and the dissipation rate is

D = (Nw - M)O k. (22)

108 O. Rulz and W. GOLDSMITH

YIELD CONDITION

The Tresca yield condition, used here, is linear in the direct stresses and qbadratic in the moments. This yield condition has been linearized in three different ways. In the present case, a so-called two-moment limited interaction surface approach, originally proposed for axisymmetric shells [15], is utilized. The hoop and longitudinal moments, as well as interactions between force and force and between moment and moment, are maintained, while interactions between force and moment are disregarded. This results in a linear surface in stress space in four dimensions. When applied to the dissipation relation, equation (22) becomes

4w • D = M 0 ( 1 +~-)0 k. (23}

This surface entirely contains or circumscribes the exact yield surface, as shown in Fig. 2.

It can be proved that, if it is multiplied by 1/2xf5-- 1 = 0.618, the outer bound becomes wholly interior to the exact surface. Then, if P* is the collapse load according to this yield surface, the actual collapse load, Pc, is bounded by

0.618P* ~< Pc ~< P*. (24)

DEFLECTOR PLATE DEFORMATION

The dynamic deformation of the deflector plate will be analyzed using a given mode shape, the plate equations (21) and the two-moment limited interaction yield surface. The dimensions and loading of the plate are given in Fig. 3. The loading is uniform in the area of diameter d; the edge offset is A = S/d.

Following the suggestions given in [16] for the lay-up of hinge lines in static loading of

-Q6~8 I

M

Mo I

I . . . . . T ~ I C i rcumscr ib i r~ so~jore yield

io.A,

-I curve

FIG. 2. Two-moment yield interaction surface.

I

L L

FIG. 3. Overall dimensions of the deflector plate. The load is applied on a circle of diameter d.


m --------~ x. j?

~ \ ~ . I"I" ~ Hinge I / \ / ',<-, /

\ . . . . - - • I ,~ ~,

\ a / ~ .-. k /

S ' S S

L L

/9

FIG. 4. Mode shape hinge line pattern adopted here.

plates, and after evaluating a geometrically simpler alternative consisting of a Y-shaped hinge pattern [17], the mode shape shown in Fig. 4 was adopted for the calculations. It consists of a total of seven regions comprising three sets of symmetrical pairs and a central triangle. The original circular loading area has been replaced by a hexagon which has an analogous shape as the deformation mode, in order to simplify the algebra. Since the loading area is small compared to the plate area, such replacement does not notably alter the results. Both regions, the actual circular and the assumed hexagonal, have the same area and the resultant forces are therefore the same.

The following notation has been used:

B ~o=~; D C 6 = --" (25) L ' 0~=~.

F u r t h e r m o r e , A L represents the total area of the plate and A c that of the hexagon; W is the maximum plate deflection, S is the distance from the center of the impact area to the free edge, 0 is the hinge line inclination and the other terms are defined in the figure or notation.

The deflection mode of Fig. 4 is defined by

W x I : w = ~ , dA = D dx

II : w = W L sin---~' dA = (L sin 0 - x) dx

W x 2S' III: w = ~ , dA = ~ - (B - x) dx

m 2 1: 0, = W - -

L

L - S I 2: 02= W - -

BL

3: B

1 a: 0 , = W - -

L sin 0

1 5: 0s = W -

D

(26)

1 10 O. Rtqz and W. GOLDSMITH

The following dimensionless quantities have also been employed in the equation of motion :

H 1 1 ......... v¢,* = ~'-- ; w * = w (27)

f~'* = g: Vi 2' V i H

P3 H P * - /~ V~ 2 (28)

p, Vii2 L 2 - (29) Mo H "

The quantity 2 is a dimensionless form of the initial kinetic energy. Substituting equations (23), (25) and (26) into equation (21), the following dimensionless equations are obtained:

where

# . 2441 12¢2 + ~ W* + ~ - - ~3 P * = 0 , (30)

A A

26 + 2 cot 0 + cot 28

Ca = 46 + sin 20 + 4o cos 20

42- (1 + kl)(6 + 2 cot 0) + 2 cot 20

46 + sin 20 + ¢o cos 20

2u2136(2 - c~) + (sin 20 + 4o cos 20)(3 - 2e) ¢ 3 - 46 + sin 20 + ~o cos 20

1 Ire- arctan (~o2- 1)'/2]. 0=2

(31)

(32)

The static collapse load for this mode shape is

6Mo (1 + kl)(6 + 2 cot 0) + 2 cot 28 Pc = L 2 3e26(2 _ e) + (sin 20 + ¢o cos 28)(3 - 2c~) (33)

and for a uniformly distributed load over the whole surface of the plate, the collapse load is

6M o (1 + k l ) ( f + 2 c o t O ) + 2 c o t 2 0 P~ - L 2 36 + sin 28 + ~o cos 20 (34)

The factor kl accounts for the fact that during experimentation, the edges 3 were not restrained completely from displacement, resulting in an intermediate condition between being clamped and simply supported. It is probable that the actual condition is closer to a simple support and, therefore, k 1 < 0.5. The edge 4 was completely restrained during impact.

For static loading, the internal dissipation energy rate is

D i = 2 ( 6 -t- 2 cot 8)(1 + k l ) -t- 4 M o W (35)

which varies between 5.67 M o W and 7.49 M o W for k I = 0.5 and a value of 6 that varies between 0 and 0.25. This mode dissipation, equation (35), for the Y-shaped hinge pattern is less than that for other cases considered [17] and, therefore, provides a better pattern for the deflector plate.

Based on conservation of momentum and inelastic impact, and recalling that the plate is at rest before impact, the initial velocity V~ni of the plate is

1 Fin i - - Vo, ( 36 )

1 +/[/r

where #r is the ratio of areal density of the plate to that of the projectile and V 0 is the initial velocity of the projectile. Under impulsive loading, the initial velocity V~n~ of the plate is


constant in the area under loading. Thus equation (14) becomes

a = Vinir/, where

J'A/J~ dA

q - ~A/~q~*qg* dA'

(37)

(38)

In the present case of a blunt-faced projectile of mass density p, length 2T and face area Ac, impinging perpendicularly on the deflector plate on an area A~, the value of q is given by

SAc pc/p* dA¢ + Sac 2pT49" dA¢ = ~A L ]~(~,z dA L + ~a¢ 2pTc~., dA c ,

where p is the mass per unit area of the plate. The edge offset can be expressed as

(39)

D + ~ B 3 A - - - - +40 (40)

~(D + B)

which yields 1 when 6 = 0. The relation between a and the actual face area of the projectile is

~ B rt d = (4o+ 3) L q' (41)

where d is the actual diameter of the projectile and q is a correction factor for the projectile plate contact area.

The value of q defined by equations (38) and (39) is obtained as

where

and

with

and

q = Kq', (42)

2TL l + - -

q# (43) K = 2TL a2 46(3 - 3~ + ct 2) + (sin 20 + 4o cos 20)(6 - 8a + 3at 2)

1+ q~ 46 + sin 20 + 4o cos 20

2~2133(2 -- ~) + (sin 20 + 40 cos 20)(3 -- 2a)] q' = , (44)

45 + sin 20 + 40 cos 20

~/4~ -- sin q~ cos 4) q = , A < I

7Z

(45) =1 A > I

~b = cos-1(1 - 2A). (46)

The relation between a and the projectile face area is then

= (sin 20 + 4o cos 20 + 26) q' (47)

To summarize the results, the deflector plate mode solution for dynamic loading is given by equations (25)-(33) and (42)-(47). For a given plate material and initial projectile velocity, the deflection w--equation (27)--depends only on the edge offset A and on the force P3 that the projectile exerts on the plate. Thus far, this force is unknown-- i t will be determined in the next section. On the other hand, the value of the edge offset is part of the definition of the problem. The derivation of equations (30)-(35) is detailed in Ref. [17].

112 O. Rulz and W, GOLDSMITtt

A P P R O X I M A T E D Y N A M I C A N A L Y S I S O F T H E P R O J E C T I L E

The dynamic behavior of the rigid projectile during and after impact with the deflector plate will be modelled so as to predict the motion of the projectile after impact. The rather complicated external force acting on the projectile during impact is affected by the deformation of the plate. It is spread over the contact area between the plate and the projectile face that alters its shape and extension during impact; the resultant force also varies concurrently in magnitude, direction and point of application. At the same time, as observed experimentally, there is relative motion of the projectile face with respect to the plate and there is rotation of the projectile as a rigid body with respect to the contact area. Therefore, the contact area has relative motion with respect to the plate.

Since the deflector plate is symmetric to an in-plane axis that passes through the middle of the plate and is perpendicular to its free edge, the force acting on the projectile has only two components. One is parallel to the initial velocity (w-direction) and the other parallel to the transverse axis of symmetry (v-direction). Therefore, the resultant force moves during impact along a line on the axis of symmetry. The length of this line may be longer than the projectile diameter because of the displacement of the contact area. In the present model, only the resultant force will be considered and it will be assumed that the point of application of this force remains fixed. The magnitude of the force components will be determined from the analysis of dynamic deformation of the plate.

The external forces acting on the projectile are shown in Fig. 5. The coordinates of the point of application of the external force are (w, v). The mass center

is C with coordinates (w c, Vc). The following relations apply if the effects of drag and gravity are neglected:

1 ~,Fw: fv + R/~'+--F3 = 0 (48)

m

~ F~:

Mc:

with the initial conditions

w=v=O=O ff = Vini and

/~- T p ' + I F 2 = 0 m

F3R - F z T - Io1~ "= 0

(49)

(50)

b = 0 = 0

at

at t = (51)

Ig

½

II ~ . . . . . . . . . . . . .

FIG. 5. Forces acting on the projectile during impact.

F I

I


Based on the experimental observation that the contact between the plate and the projectile occurs while the angle 0 is small, the following approximations have been used:

fl=O + Oo ] p sin fl ~ R '

pcosf l~ T

/ ~ o

(52)

The distance R' is given by R' = 7R. (53)

PLATE PROJECTILE INTERACTION

If the plate behaves rigid-plastically, the forces F 2 and F 3 can be calculated from the mode solution. Using a dimensionless representation similar to equations (27) and (28), the projectile equations (48)-(50) may be written as

w* + 7//.* + #r P~' = 0 (54)

Rr__ .. /~*- /~* +urP~ = o (55)

P* - ~-~ P~ - f~7//'* = 0, (56)

where the following dimensionless variables have been used:

p , _ Fi H //., . . R H p . I o /~A V02' i=2 ,3 ; =fl Vo*; P r = P -~' f l=#AcR 2. (57)

In order to retain an uncoupled and linear system of equations, the deflection in the v- direction is assumed to be related to that in the w-direction by:

v* = - r w * . (58)

It was found experimentally, as indicated in the companion paper [ 18], that when the final deformations of the plate are considered, the factor z varies between 0.31 and 0.55. If it can be assumed that the factor z remains constant during the impact, equation (58) relates the deformations w and v during this interval. This is not far from the actual situation, as film records of the impact show that the transverse and in-plane deformations of the plate exhibited a nearly constant ratio. The complete problem can then be described by the set of equations (30), (54)-(56) and (58) in the five variables v, w, r, P2 and P3, The equation for w* is given by

24¢1 1242 #* + w* -~ = 0, (59)

).(1 + ~3 k) 2(1 + ~3 k) where

k =

Expressing the other variables in terms of if*, there result

1 [ T 1 /~* PJ'= -k~* ; P*=~rr z + ~ (p r -1 ) i f • ; =--z~/,*;

(60)

//., = _ 1 (1 - / ~ k ) ~ * . Y

(61)

114 O , R u I Z a n d W . GOLDSMITtt

The solution to equation (59), subject to the initial mode conditions at t = 0, w* = 0 and ~* = r/, is

1 ~z ~z w* = t /S sin At* + ~ - i cos At* - 2~-~, (62)

where

t* V°t and A = V / ( 24~1 = H- 1 + k~3)2' (63)

In equation (62), t* is the dimensionless time corresponding to equation (27). The end of the impact occurs when the velocity is zero and the time is

t* = s t a n -1 ( 1 + k ( a ) J , r / 2 . (64) \V6~ z

Substituting this last expression into equation (62), the equations for the maximum deflection of the plate are

w* = ~ 1 + (1 + k~3)2r/2 - 1 and v* = - r w * . (65)

The final velocities of the projectile are obtained as follows. The angular velocity/~* is obtained by integration of the last equation (61) as

/~. = 1 (1 - i~rk)( q - ~*) (66) 7

and the final angular velocity fl* occurs when ~* = 0, i.e. when

/~, = _1 (1 -/~rk)t/. (67)

The change of linear momentum in the w-direction is

where

m*A~¢* = P{' dt = k q - q cos At* + ~ A sin At* , (68)

m m* = - - . (69)

A/x

At the end of the impact, the velocity of the mass center of the projectile is

k ~ * = 1 m~q. (70)

The change of linear momentum in the v-direction is

m*At~* = P ] d t = - - - z + (/~rk - 1) r / - r/cosAt* + A sin At* (71)

and the final velocity of the mass center is

t3* = - - "t" n t- (fir k - 1) r / . ( 7 2 ) .r ;)X

The total energy of the system is given by the sum of the kinetic and potential energies of the plate and projectile. At the end of the impact, the kinetic energy of the plate is zero. Disregarding changes in the potential energy of the projectile (height variations), the final energy of the system is

1 -2 t;~f) 1 1 "2 Ef = JA Q,cfidA + ~(Wcf "1- + ~ O]~f, (73)


where a dimensional form has been used. The first term on the right hand side of equation (73) represents the plastic work in the plate in terms of the generalized stresses and strains. The remaining terms express the kinetic energy of the projectile. If Eo - 1 2 - ~m V~ is the initial energy of the system, the plastic work done on the plate is

fAQiql dA = - Tf (74) E0

where Tf is the final kinetic energy of the projectile. The left hand side of equation (74) is equivalent to equation (35).

C O N C L U S I O N S

A study of the generation of controlled tumbling motion in rigid projectiles using a deformable deflector plate has been performed. It was assumed that the velocity of the projectile is low enough so as not to cause failure of the plate. Employing a mode approach for the plastic deformation of the plate and rigid body dynamics for the projectile, several expressions that describe the motion of the plate and projectile were obtained. Finite deformations were considered for the plate. However, the material was considered rigid-perfectly plastic and strain-rate independent, which although convenient for computations may need to be reconsidered for better correspondence with experiments. Boundary effects such as the actual clamping of the plate and localized effects such as the indentation of the projectile were not considered. The validity of the model is tested in the sequel, where predictions are compared with corresponding experiments.

Acknowledgements--This work was abstracted from the M.S. thesis of the first author, written.in partial fulfilment of the requirements for the Master of Science degree at the University of California, Berkeley. The activity was supported by the Army Research Office, Triangle Park, N.C. under Contract No. DAAG29-84-K-0021.

R E F E R E N C E S

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Impact Engn 9 (In press).

Controlled tumbling of projectiles—I. Theoretical model

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