Flight Motion of a Continuously 

Elastic Finned Missile 

by Charles H. Murphy and William H. Mermagen 

ARL-TR-2754 June 2002 

Approved for public release; distribution is unlimited. 
The findings in this report are not to be construed as an 

official Department of the Army position unless so 

designated by other authorized documents. 

Citation of manufacturer’s or trade names does not 

constitute an official endorsement or approval of the use 

thereof. 

Destroy this report when it is no longer needed. Do not 

return it to the originator. 

, 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5069 

ARL-TR-2754 June 2002 

Flight Motion of a Continuously 

Elastic Finned Missile 

Charles H. Murphy and William H. Mermagen 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
Abstract 

The motion of elastic finned projectiles has been analyzed by various 

approximate theories. In this report the exact equations of small amplitude 

motion are derived for a symmetric missile. The aerodynamic and elastic 

symmetries are used to allow the use of complex variables to describe the lateral 

motion in a non-rotating coordinate system. Although the resulting equations 

are both ordinary and partial differential equations, frequencies and damping 

rates of free oscillations are obtained from an ordinary differential equation with 

boundary conditions. Equations for a permanently deformed bent missile are 

derived, and an ordinary differential equation for the forced motion of a bent 

missile is obtained. Sample calculations for a finned projectile with a fineness 

ratio of 20 show resonant motion at the aerodynamic frequency as well as at each 

elastic frequency. The nonlinear roll moment associated with a bent missile is 

computed and the location of possible spin-yaw lock-in is determined. The flight 

motion of an elastic missile is shown to be the sum of two elliptical motions: a 

low frequency pitching motion and a higher frequency flexing motion. The 

induced drag coefficients for both motions are computed as functions of the 

missile’s elasticity. 

ii 
Contents 

List of Figures V 

List of Tables vii 

1. Introduction 1 

2. The Coordinate System 2 

3. Center of Mass Motion 3 

4. Angular Motion of the Projectile 

5. Flexing Motion of the Projectile 

6. Bent Missile 

6 

10 

13 

7. Trim Solution 15 

8. Transient Solutions 18 

9. Flight Motion 

10. Induced Drag 

11. Numerical Results 

12. summary 

13. References 

Appendix A. Integrals 

Appendix B. Functions 

20 

21 

23 

25 

33 

35 

37 

. . . 

Ill 
Appendix C. Relationship Between Boundary Conditions 

Appendix D. Trim Solution Parameters 

Appendix E. Transient Solution Parameters 

Appendix F. Missile Parameters 

List of Symbols 

Report Documentation Page 

39 

43 

45 

47 

49 

53 

. 

iv 
List of Figures 

~ - 

I 

I * 

Figure 1. Sketch of finned missile . .............................................................................. 25 

Figure 2. g$ /mR vs. CT for flare-stabilized rod .......................................................... 26

Figure 3. d,/wR vs. CT for finned missile .................................................................. 26 

Figure 4. &,/CD, ,+5/~z vs. CT for finned missile ..................................................... 27

WF-e 5. 1% 1 vs. CT for k = 1,3,5 ............................................................................. 27 

Figure 6. R(Wr (x)/y~~,) vs. x for Q = 5, k = 1,3,5 . ................................................ 28 

Figure 7. hk/mR vs. a fork = 1,3,5 .......................................................................... 28 

A 

FigweB. A/w, vs. p/w,for a=5,k=0,0.05,0.10 ............................................. 29 

Figure 9. C,/C, vs. 0 ............................................................................................... 29 

Figure 10. & vs. CT for k = 3,5 .................................................................................. 30 

Figure 11. 15, /E,,,l vs. p/w, for bent finned missile at (r= 5 . ........................... 30 

Figure 12. 18E(xz)/~m01 vs. p/oR at CT= 5 for bent finned missile.. ................... .31 

Figure 13. fj vs. p/wR for bent finned missile at c= 5 ......................................... 31 

. 
INTENTIONALLYLEFTBLANK. 

vi 
List of Tables 

Table 1. Values of AkO, N kO, z z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 

Vii 
. . . 

Vlll 
1. Introduction 

Modem antitank kinetic energy projectiles have become very long finned 

missiles carrying very dense metal rods. Because these projectiles have to resist 

very large aerodynamic loads due to their high-velocity flight at sea level, 

designers have been concerned about the possibility of aeroelastic deformations 

[l-7]. There is some evidence that a small number of these projectiles have been 

forced to spin at rates close to their lowest elastic frequency and have been 

subject to large inelastic deformations. M&hail [3, 41 and Murphy and 

Mermagen [6,7] have developed special solutions that showed spin lock-in at the 

lowest elastic frequency. 

Mikhail [3, 41 used an incorrect expression for the angular momentum and 

assumed erroneously that a spinning elastic missile could perform planar-flexing 

motion and a planar-pitching motion in a coordinate system rotating with the 

missile. His numerical calculations, however, showed examples of spin lock-in 

when fin damage produces a roll-inducing moment sufficient to cause a 

steady-state spin greater than the lowest elastic frequency, and initial spin was 

zero. Heddadj et al. [5] approximated an elastic missile by two rigid bodies 

elastically connected and studied the transient motion of these bodies but did not 

consider spin lock-m. 

Murphy and Mermagen [6,7] approximated the continuously elastic missile by 

three rigid bodies connected by two massless elastic beams and derived 

differential equations of motion for this three-body system. The equations 

showed that it is impossible to cause spin lock-in by roll inducing moment and 

zero initial spin alone. Murphy [8,9] showed that the combination of an offset 

center of mass (cm) and a trim pitch moment creates a nonlinear roll moment 

that can produce spin-lock-in at the aerodynamic frequency. The rear beam of 

the S-body theoretical model was given an inelastic lateral deformation and spin 

lock-in at the lowest elastic frequency was observed. 

Because M&hail’s [3,4] approximate equations are far from being theoretically 

correct, their results can be dismissed. Murphy and Mermagen’s [6,7] equations 

are valid, but the use of the 3-body model is a major simplification of the actual 

physical problem. It is the purpose of this report to derive the correct equations 

for a continuously elastic symmetric projectile. This derivation exploits the 

missile’s aerodynamic and elastic symmetry by describing the transverse motion 

with complex variables and using a nonrotating coordinate system with its much 

simpler equations of motion. 

Although the resulting equations form a combination of ordinary and partial 

differential equations containing integrals of the elastic deformation, free 

1 
oscillations and constant-spin trim motion of a bent missile can be computed 

from fourth-order ordinary differential equations. Induced drag for various free 

oscillations are deduced as quadratic functions of their amplitude. Quadratic roll 

moments associated with a bent missile are formulated for trim motion, and 

equilibrium spins that could cause spin-pitch lock-in are located. Numerical 

integration of the complete set of partial and ordinary differential equations for 

varying spin would be necessary to verify the actual occurrence of lock-m. 

Murphy and Mermagen’s 3-body theory [6, 7j would, however, be a good 

indicator of this lock-m behavior. 

2. The Coordinate System 

If we consider a symmetric missile with no elastic flexing, we can define a nonspinning coordinate system with origin at the rigid missile’s cm and X-axis 

directed along the axis of symmetry. The Z-axis is perpendicular to the X-axis 

and initially downward pointing, whereas the Y-axis is determined by the right 

hand rule. 

The elastic missile is assumed to consist of a very heavy elastic circular rod of 

diameter, d, and fineness ratio, L, embedded in a very light symmetric 

aerodynamic outer structure that may be longer than the rod. We will 

conceptually slice the missile into a large number of thin disks perpendicular to 

the X-axis with thickness, dx. For the elastic rod, an elastic nonspinning 

coordinate system is defined so that its origin at the center of the central disk and 

the X-axis is tangent to the rod at this point. When the rod flexes, the disks shift 

laterally perpendicular to the X-axis and cant to be perpendicular to the 

centerline of the disks. 

The angular velocity of the central disk in these nonspinning coordinates is 

defined to be the vector, 

55 = (J&q,+ (1) 

Because the coordinate system pitches and yaws with the missile but does not 

spin, its angular velocity vector is 

Cl = (O,q,r). (4 

A dimensionless vector specifies the location of the displaced center of a disk at X 

with respect to the origin of the elastic coordinate system. 

I&(x, t)= (X,Y,Z&’ =(x&&J. (3) 

Because of the mass and aerodynamic symmetry, it is convenient to represent 

transverse displacements, transverse velocities, transverse angles, and transverse 

i 

* 

2 
angular velocities by complex quantities. The transverse disk displacement and 

the transverse angular velocity, for example, are 

6, =6, +iS,, 

(4 

Q=q+ir. 

The lateral location of the cm of the flexing elastic missile can be expressed in 

terms of the missile’s local mass density p,,, (x, y, z) , 

where 

p, (X) = d* jjpm (x, y, z)dyh t and 

m=djp,(x)dx. 

Finally, the transverse location of the center of a disk relative to the elastic 

projectile’s cm is 

6(x, t)=& -6,, 

= 6, +i6,. (6) 

The location of the center of a disk relative to the cm and its velocity relative to 

the cm are specified by the vectors, 

ii, = (x9 $9 s,), (7) 

T& d-’ = (0,8,,8,)+~x8&. (8) 

The transverse components of this relative velocity vector can be written in the 

form 

where 

qx,+=;. 

vdev + iv,, =(chcQ )d, (9) 

3. Center of Mass Motion 

The velocity of a disk is 

3 
where 

Tc = (v, .vy , vz ) is the velocity of the cm. 

The angle of attack, a, and the angle of sideslip, fl, are defined at the central disk 

for the elastic missile. If the transverse velocity components for this disk are 

vEy + iv, =V< , (11) 

where 

<=p+ia,and 

The transverse velocity for cm of the elastic missile is 

v, +iv, =Vlj+6,d. (12) 

The sum of equations (9) and (12) is, therefore, the transverse velocity of a disk. 

vd, +iv, =Vc+8,d+(f$-ixQ)d, 

= vg+(& -i.&. 

(13) 

The aerodynamic force acting on a rigid projectile is assumed to be proportional 

to E, and Q and their derivatives. For an elastic missile, a local aerodynamic 

force needs to be defined in a similar manner. The x coordinate of a point on the 

rod axis lies between X, = -L/2 and x2 = L/2. The aerodynamic structure 

incasing the rod can have a nose windshield of length xz3 d and its fins could 

extend beyond the end of the rod by the distance xol d . Thus the aerodynamic 

force is exerted from x0 = X, -xol to x3 = xz +xz3. If we assume the 

aerodynamic structure to be quite rigid beyond the ends of the rod, then, 

s,(x,t>=s,(x,,t>+(x-xI)[;(X,,t) ; x, lxlx,; (1% 

6,(x, t)= s,(x2,t)+(x-x2)r2(x,,t) ; x2 5 x I x,. (15) 

The transverse aerodynamic force exerted on a disk located at x will be assumed 

to have the form appropriate to a pointed slender body [lo, 111. Thus, the force 

will be proportional to the local angle of attack and its modified time derivative. 

dF = g dx = -g, [cr,q + c/, (li + iQxdlV)b f (16) 

4 
where 

F=F,,+iF,,and 

g, =psv2/2. 

cfn (x) are two normal force distribution coefficients per axial length and 77 is the 

local complex angle of attack. The distribution functions derived for Munk’s 

airship theory [lo-121 are, for example, 

CI, = 43-l ; cf2 = 2AS-‘, 

where A(x) is the local cross-sectional area. 

A general pointed slender body relationship between cf, and cl2 is 

cr2 = -jc&L (18) -9 

In reference [13], computer programs for calculating crI for finned missiles are 

given. 

The complex angle between the missile’s centerline described by &,t) and the 

X-axis will be denoted as I? and is the spatial derivative of S, 

,-=-=E 

ax ax’ 

(19) 

The local angle of attack is the angle between the velocity vector and the surface 

of the missile, which is assumed to be parallel to the centerline. Because the 

complex angle between the velocity vector and the X-axis is given by 

equation (13), the following equation results: 

q=~+(&-ixQ)dlV-r. (20) 

Equation (20) can be differentiated and the small angular acceleration terms in 8 

and & neglected to yield a relation for Q 

Q+-f-. (21) 

The total transverse aerodynamic force acting on the missile can be obtained by 

integrating equation (16) from x, to x3. 

F =-g, lCNII4 + C~~(~~/V)+iC,,(Q~/V)-J,(t)-j,(txdlV)], (29 

where C, are listed in Appendix A, Ji (t) are listed in Appendix B. 

5 
The axial aerodynamic force is directed along the X-axis and can be 

approximated by use of an axial drag coefficient per length and a base-pressure 

drag coefficient cDbP , 

F, = -g,c, E --&CD t w 

where 

The velocity vector of the cm can now be differentiated as 

df 

-= 

a5 ( +,,ri,,+, +fixP. 1 (24) 

The differential equations for the missile’s cm are therefore 

m+x+qvz-rvy EmV=-g,CD; ( 1 (25) 

m(Vi+v{+8cd -iVQ )=F. (26) 

Equations (22), (25), and (26) can be combined to yield a simple relation between 

Q and i. Small terms involving C, and C,, have been neglected. 

(V/d&Q )=-g2CL,J+K 

where 

C La =c,,-c,, 

g, =g,/mdmd 

N is defined in Appendix B. 

(27) 

4. Angular Motion of the Projectile 

The transverse aerodynamic moment exerted on the missile is computed with 

respect to the cm and has contributions from both the transverse aerodynamic 

force and the axial aerodynamic force. 

where 
The total transverse aerodynamic moment acting on the projectile can be 

obtained by integrating equation (28) and adding the base-pressure moment. 

where the CMj are given in Appendix A. 

Each disk pitches and yaws with angular velocity Q + il? and spins with spin 

rate pa. The spin angular momentum about the center of a disk is p,(d2/8)dm 

and the transverse angular momentum about the center of a disk is 

(Q + ZI d “1 ‘/ 16 21 m . Now the angular momentum of a disk about the cm of the 

projectile is expressed in the elastic nonspinning coordinate system and the 

result integrated over all the disks to yield the total angular momentum of the 

missile. In dynamics terms, we will neglect the mass of the aerodynamic 

structure and assume p1 to have the constant value of m/Ld . Because spin-yaw 

lock-in is caused by quadratic terms, quadratic terms are retained in pd and h, , 

where 

pa = p(1 -I-r/2)+ R{Qj?>, 

h, = [2p(l-I’~)+R{(Q-iI’p}~l6,and 

h, +h, = (Q + il? + 2pI’)/16 . 

Equations (7) and (8) can be employed to yield the three components of the 

angular momentum vector. 

H, = I,p + md*R{J, - QJ6); 

H, +%I, = I,Q+imd*(j, + J8), 

(31) 

(32) 

where J j is defined in Appendix B. 

If the missile is assumed to fly at constant pitch angle and its deformed shape is 

rotating as a rigid body (Q = 0,8 = ip6 ), the axial angular momentum is 

where 

7 
i, = I, + (md’lL)$~z -lr/2/16] dx. 

M&hail [3,4] computed I&(fl’ for a flexing missile, neglected the III2 term and 

assumed that equation (33) was valid for the axial angular momentum for any 

motion. This assumption is clearly incompatible with equation (31). 

The derivative of the angular momentum vector can be computed in the usual 

way. 

For a slender missile I, z md2L2/12 and .I8 can be neglected in comparison with 

5,. The differential equations for the missile’s angular motion are 

I,Q-ipI,Q+imd2j, = M. (36) 

Equations (27), (29), and (36) can now be combined to give a simple second order 

differential equation{ and various integrals of S . 

Ittf+(a2 -ipI,E+(a, +ipa,)S = J, + J, +md2 j,, (37) 

where 

a, =- (&M, + g3GA4, II 

a, = -(g,d2/V)z,L 

a2 = -(g,d2/V~CMq + CM, - g,C,)l 

g, = pSd/2m, 

g, = I,/md2 , and 

Jr, JN is defined in Appendix B. 

For a typical missile g, is less than lO+ and the second term in a, can be 

neglected* as well as a similar term in JN. As a general rule, iQ can replace C$ in 

all the aerodynamic force and aerodynamic moment expressions. The rigid 

projectile frequencies and damping rates can be computed from equation (37) for 

*For lighter than air dirigibles, g, is greater than 0.1, and this term makes a statically unstable 

airship stable in flight [14]. 
the right hand side set equal to zero. According to Murphy [15], very good 

approximations for the frequencies are 

4kR = IPI x + J3FGl]/21, ; k = 1,2, 

z +o, + (1 It pI,/41,w,)pIx/21, 

(38) 

where 

O, = ,/a is the rigid projectile zero-spin frequency. 

Exact relations for the damping rates are 

;liup =- a2 ‘m + pa,)/(&.J, -PC)- ( 4 (39) 

The aerodynamic roll moment is the X-component of the aerodynamic moment 

about the projectile’s cm. The linear roll moment coefficient for a rigid finned 

projectile is usually expressed in terms of a roll-damping coefficient and a steady 

state spin. 

( ) c t linear = Gp b - P, Xw) - (40) 

The steady-state spin is usually determined by a differential canting of the fins 

caused either intentionally by the designer or unintentionally by damage to the 

projectile. , 

The roll moment of one of the projectile’s thin disks is the sum of the linear roll 

moment it has as part of a projectile and the quadratic roll moment induced by 

the transverse aerodynamic force acting on its lateral displacement relative to the 

missile’s cm. If we retain only the dominant cfl term in equation (16), the 

quadratic roll moment has the form: 

Therefore, the total aerodynamic roll moment acting on the projectile is 

Mx = kvomrlear +R~J9(t)lL PI 

where J, is defined in Appendix B. 

The nonlinear roll moment from equation (42) can be placed in the spin equation, 

equation (35) to yield 

g,$+R{J, -o& -g,J,}= &&near. (43) 

This equation is nonlinear due to the retention of all three quadratic terms. 

9 
5. Flexing Motion of the Projectile 

Classical beam theory for a circular beam assumes the beam to be slender and 

the elastic moment exerted on the right of a cross-sectional disk, M,, is 

proportional to the second derivative of the displacement of the beam. Although 

this is usually stated in a coordinate system rotating with the beam, the 

rotational symmetry of the beam allows us to state the proportionality in the 

non-spinning coordinates. 

where 

Me = M,, +iM,, 

Id4 = IJy2dydz = [jz2dydz is the area moment of rod, and 

E is Young’s modulus of elasticity. 

A similar proportionality for the elastic shear force exerted on the right can be 

obtained from beam theory. 

(45) 

For a homogenous circular rod with constant diameter, EI is constant and the 

elastic shear force on a disk is 

(46) 

where 

o; = EI/p,d4 . 

&-, has the dimensions of a frequency and appears in the expression for the 

flexing frequencies of a free-free beam with no aerodynamic force present. 

a, = (AK /L)2 @, , 

where 

4 = 4.730,7.853, 10.996, 14.137... 

(47) 

i 

10 
L 

A very common beam-damping assumption is the Kelvin-Voight model [15], 

which requires the beam-damping shear force to be proportional to the time 

derivative of the elastic shear force. This time derivative, however, must be 

taken in rod-fixed coordinates that rotate with the rod. 

where 

6 is a small dimensionless beam-damping coefficient. 

cQ is scaled by 2~;’ so that i = 1 corresponds to critical damping of the first 

elastic mode. 

The aerodynamic shear force is a combination of the aerodynamic force 

distributions cfl (x), cf2 (x) g-i ven by equation (16) and the drag-induced moment 

distribution cD (x)6( x ) a pp earing in equation (28). According to reference [16], 

this aerodynamic shear force is 

(49) 

Althoughc,, andc,, can have a finite number of jump discontinuities, c,, must 

be continuous and have only a finite number of jump discontinuities in its 

derivative. 

The acceleration of a disk relative to the cm can be obtained by differentiating 

equation (9). The linear expression for the lateral components of this acceleration 

is 

‘dcy +iadcz =(a-ixi))d. (50) 

According to equation (26), the lateral component of the acceleration of the cm is 

F/m, and the equation of motion for each disk is, therefore, 

11 
The aerodynamic force distribution and its integral, z, F , are available from 

equations (16) and (22) and the other quantities are provided by equations (46), 

(48), and (49). 

E,g+E,e+ixo-N 

where 

E j (x) is defined in Appendix B. 

Equation (52) is based on the usual assumptions of neglecting the canting inertia 

of the spinning disks and any shear deformation. These small effects can be 

calculated if desired [17J. 

The boundary conditions at x1 are determined by the aerodynamic force 

and moment exerted on the overhanging fins. The elastic force conditions have 

an additional term from the drag-induced moment distribution [17J. 

where 

g, = g&/w:, and 

4, m are defined in Appendix B. 

Similarly, the boundary conditions at xz are set by the aerodynamic force and 

moment exerted on the nose extension 

If beam damping and drag are neglected and the aerodynamic structure does not 

extend beyond the rod (c = cD = xol = xz3 = 0 ), the boundary conditions reduce 

12 
to those for a free-free beam, 

a3b 3’4 -0 .x--x x 

-=-- 

ax3 ax2 

, - 1, 2’ (57) 

A fourth-order partial differential equation usually has four boundary 

conditions. In this case, however, there are two more conditions at x = 0. 

igo)= 0; (58) 

These conditions look like the usual cantilever boundary conditions but actually 

are specified by the motion of the coordinate system which is attached to the 

missile at the midpoint and tangent to it there. These midpoint conditions can be 

easily satisfied for a finite element calculation and need not be used as boundary 

conditions. Calculations of special solutions that assume a single frequency 

harmonic time variation and involve ordinary differential equations have 

difficulty satisfying the midpoint conditions. In this report, these solutions will 

be calculated by pairs of differential equations in x for the aft and fore section of 

the rod (XI I x I 0; OS x I x2). Equations (53), (54), (58), and (59) are boundary 

conditions for the aft part and equations (55), (56), (58), and (59) are boundary 

conditions for the fore part. This use of six boundary conditions implies 

a%, a%, discontinuities in - - ax2 ' ax3 at the central junction point. These six boundary 

conditions for the ordinary differential equations in x are not, however, 

independent, and it can be shown that equations (53) and (54) imply equations 

(55) and (56) an d vice versa (Appendix C). Thus, the calculations should not 

show discontinuiiies in shear force and moment at the junction point. 

6. Bent Missile 

If the rod were inelastically deformed on launch, its shape would be represented 

by the sum of a fixed deformation rotating with the missile and an elastic 

deformation, 

where 

6, = &(x,t)+8m(x)ei+, (60) 

13 
p=qL 

The local inclination of the rod and the lateral location of the cm become 

r=F+r,ei4; 

6, = 5ic +ScBei+, 

where 

F ai& 

=x 

d&B r, =- a!x’ 

g, = (l/Ljf&h, and 

Xl 

6, = clm),~. 

XI 

The location of the centerline of the disks relative to the projectile cm is 

(61) 

(62) 

(63) 

where 

For the rigid aerodynamic structure extension, 

b (4 = 6, (-5 ) + b - Xl )r B (3 ) ; x0 I x 5 x, ; 

s,(X)=~EB(X*)+(X--*)ra(X*); x2 I x 5 x3 . 

(@) 

Now I?, produces an aerodynamic force and moment that rotates with the 

projectile. If the fins are bent with respect to the rod, an additional differential 

force that rotates with the missile is produced and must be added to 

equation (16). 

dFBF =gl+rBF (x)e’(dr , (W 

where 

r,, (x) is a measure of bent fin damage. 

The aerodynamic force acting on the bent missile is 

F=-&lw+(cNti +C,,~~d/V)-J,(t)-j,(tXd/V)-C,e”J, (66) 

where CNBF is defined in Appendix A. 

14 
If the fin damage extends beyond the rear end of the rod, it will exert an 

additional force and moment on the rod at x = x1. This bent fin force term also 

has to be included in the quadratic roll moment appearing in equation (43). 

where 

0 

J 9BF = -ie’+ f c,IBF6dx. x0 

Equation (27) for the cm of the elastic missile becomes 

(V/d)@-iQ)=-g,(C,c-C,,e’+)+N. 

(67) 

(68) 

The aerodynamic moment as given by equation (29) becomes 

M=-i(g,dlC,,g+(C,, +&xEd/V)-J3(t)-J4(t)(d/V)-JS(t)-CMBFei+j, (69) 

where 

Cmr is defined in Appendix A. 

The projectile angular motion equation (36) becomes 

I,Q-ipI,Q = -i(g,d)[C& + (C,, + C, )(ed/V)- CmFeib]- iJ, - imd*J, . (70) 

In the flexing motion equation (51), 6,is replaced by gE in the elastic and 

damping force terms, and the bent fin terms are added to F and F, to obtain the 

partial differential equation for a bent missile. 

= E,c + E,k + ix0 - N - EBFei* 

. 

In the boundary conditions specified by equations (53)-(56), 6E is replaced by 

gE in the partial derivatives on the left side of the equations. The conditions at 

x1 are modified by force or moment contributions from any fin damage between 

x0 and x1 that may have occurred. 

7. Trim Solution 

The general solution for the motion of a bent missile requires the numerical 

integration of one partial differential equation (equation [71]) and three ordinary 

15 
differential equations (equations [67-691). A simple special solution is that for 

the steady state motion of a bent missile with a constant spin. This motion is 

called trim motion and has the form 

6 = l&e+ ; (72) 

gE = &eipt ; (73) 

F = fpf ; (74) 

SE = (JEB + gET )P’ = amepr ; (75) 

r = (rB + FT )P = lITeiPt. (76) 

For this motion, beam damping is zero ( & = 0 ). 

Equation (68) can now be used to eliminate Q and Q from equations (70) and (71) 

and equations (72-76) inserted to yield: 

b& = J, + J, + J,, -md*p*J,,; (77) 

d4g, dc 

-- E d&r 

-+ E&r - s4G dx4 4dx 

L=E&r - 

dx 

N; -E;r +E,, (78) 

where J ar,J,,J,,b,,Ej,E,,Nt,Eir are defined in Appendix D. 

Boundary conditions as given by modified equations (53)-(56), (58), and (59) 

become 

d’% h) 

dx3 + g4CJX$%(X*) = -g4&; 

d2%T (xl ) = g4mlT ; 

dx* 

(79) 

(80) 

d’%r (x2) = -g4m2T ;

dx* 

&Jo)= 0; 

DIET = 0 

a5 ’ 

where fjT , mjT are defined in Appendix D. 

(82) 

(83) 

(84 

16 
These boundary conditions can be satisfied by expressing gET as the sum of’ 

three functions of x, 

Wd= wt(x)+~2wz(x)+~3w3(~). (8% 

The first function is the solution of the following differential equation and initial 

conditions 

d4w, 

-- E dw1 

- + E,w, = E,& - dc, 

dx4 4dx 

-N; -E;r +E, +g,6,-- 

dx’ (86) 

where 

w(,)~dw,o~d’w,(0)~~3w,(0)~0 

1 

a!x a!r2 ak’ - 

The other two functions are solutions of the homogenous equation with different 

initial conditions. 

d4W, ‘E dw, -+EE,w, =O; 

dx4 4dx 

m = 2,3, 

where 

w (0) dw,(o) d2w2@) = 0 

2 z-z 

a5 dx2 ’ 

w (,)=dw3(o)=d3w3(0)=o,~d 

3 

dx’ 

d3w2(0) =‘w3(o) = 1. 

dx3 dx2 

Equations (86) and (87) can be integrated from zero toward negative x or toward 

positive x. w,,,~(x) denotes w,,, for negative x, and wm2(x) denotes w,,, for 

positive x. B2r, B 31 are values of B2,B3 which satisfy equations (79) and (80), 

and B,, ,B32 are those values which satisfy equations (81) and (82). The values 

should be close to each other and their difference is a measure of the accuracy of 

various approximations and the numerical integration process. 

The presence of the constants NT,&,& in equations (77) and (86) and the 

boundary conditions present some difficulty because they involve integrals of 

$& and its first derivative. Our numerical technique is to replace these 

constants with guessed values N,, , &,, ,gCo and use an iterative threedimensional Newton’s Method to converge on the proper values. The partial 

derivatives used in Newton’s Method are computed by the Chapman-Kirk [X3] 

17 
technique. If drag is neglected, the need to know gG vanishes, and this reduces 

to a two-dimensional process. This process, of course, depends on reasonably 

good first guesses. 

For trim motion i, = j, =O,a=ip*& and equation (67) become a simple 

equality of two functions of p. 

f2b)=.txP)~ (88) 

where 

f2 = -R&‘*/& ): &iT + J,, j&d/V)’ , and 

x0 

Equilibrium values of spin are determined by the intersection of these two 

curves. The stability of an equilibrium spin can be found by the integration of 

equations (67), (68), (70), and (71), for nonconstant spin near an equilibrium spin. 

The stable equilibrium spins locate spin lock-in possibilities. 

8. Transient Solutions 

A rigid symmetric-finned missile flying with constant spin has two natural 

frequencies, & and d2R, where & z -d,, . Each of the flexing frequencies 

given by equation (47) would give rise to two coning frequencies, 31 w, . The 

frequencies present in the motion of an elastic projectile would form an infinite 

sequence where the first two would be related to & and d2R, whereas the later 

ones would evolve from f wK, i.e., (d2,+, EW, , 4 2K+2 = --CO,). The odd 

numbered modes have positive frequencies and are called positive modes, 

whereas the even numbered modes are called negative modes. 

Transient solutions of equations (70) and (71) are solutions of the homogeneous 

form of these equations. (6, = lY, = C,, = C,,, = 0). Special transient solutions 

of equations (70) and (71) have the form 

~5, =T,e*“; (89 

6, = wk (x)TkeAkt . (90) 

where, 

18 
Tk = Kkoeibko. 

Equation (68) can be used again to eliminate Q and 0 from equations (70), (71), 

(89), and (90) inserted to yield 

b Ik = J, + J, +md’A:J,,; (91) 

where b,, J,, J, ,E,k,v,,N; are defined in Appendix E. 

These equations and their boundary conditions can be solved in a manner quite 

similar to that used for the trim solution by the introduction of three auxiliary 

functions. 

wktx)= W4(X)+B5W5(X)+ B6w6(X)s (93) 

w4 is the solution of the following differential equation and initial conditions: 

E d4W4 E dw4+)3 w d% -- - 

6k &4 4k dx 

5k 4= 

-E 

3k -N; +g,wkcdx' (94) 

where 

w (())- dw4@) 

4 --= 

d2Wq(0) = d3Wq(0) = o 

dx dx2 dc3 - 

The other two functions satisfy the homogeneous equation with different initial 

conditions 

E d4W, E dw, +E w 

-- - 

6k &4 4k dx 

5k m =O; m=5,6, 

where 

w (o)- dw6(o) 

6 --= 

d3w6to) = 0 and 

a5 dc' ' 

d3w5(0) = d2w6(0) = 1 

dx3 dx2 * 

(95) 

The parameters B,, , B,, are values of B5, B, that satisfy the boundary conditions 

at x,: 

19 
d3Wk 

-+&EicC, \vk -vkc = dx3 ( ) -&E,:f,k; 

d2vk 

- = g,E,Lm,, . 

dx2 

Similarly, B,, , B 62 perform the same function at x2. 

d3Wk 

- + g,E,tcD dx3 - Wkc) = hE6kf2k; 

d2Wk _ 

- - -&E;;m2k. 

dx* 

(96) 

(97) 

(98) 

(99) 

The presence of the parameters N; , A,, wkC complicates these differential 

equations. N; involves A, and N, . The definition of N, involves integrals of 

vk and its first derivative and Ak must satisfy equation (91) that contains N, 

and integrals vk and its first derivative. Guessed values of these parameters 

NkO, Ak,, , vkc,, are used in equations (94)-(99) and another three-dimensional 

Newton’s Method is used as well. For frequencies greater than 30,~ a good 

estimate for z,, is - N,A;i. 

9. Flight Motion 

The complete flight motion of an elastic bent symmetric projectile with constant 

spin is described by the sum of a trim mode and an infinity of transient modes. 

If we limit ourselves to the aerodynamic modes and the first two elastic modes, 

the pitching motion and the lateral motion of the forward end of the rod can be 

expressed by two sums of five terms. 

5 = cTeiPt + T,e*I’ + T2eAzt + T3eAJt + TqeA4’; (100) 

where 

‘k = vktx2)T, = \vkZTk’ 

Because w12 andv2, are usually less than 0.5 and the angular motion is usually 

less than 0.1 radian, the motion described by the aerodynamic modes is primarily 

a pitching motion and is best characterized by (K, ,K,) II+J~~( and Iv421 are, 

20 
however, usually greater than fifty and the motion described by these flexing 

modes is primarily a flexing motion. For this reason we will characterize this 

flexing motion by the modal amplitudes of the rod lateral motion (- K,, g40). , 

For a nonspinning missile, equations (91) and (92) imply an important pairing of 

solutions. If A,, vk is a solution, Ak ,vTk is a solution. 

A, =A, v2 =uI, 

A, =A3 w4 =i&. PO4 

Thus for zero spin, the negative modes have the same damping as the positive 

modes, and their waveforms are conjugates of the positive waveforms. For small 

spin, the magnitudes of the waveforms are essentially equal, 

1~,21~lu1221;Iw~21~Iu142I- (103) 

The transient angular motion and the rod lateral motion are both expressed by 

the sum of two damped elliptical motions with frequencies, $, ,$, . The initial 

semi-major axes of the angular modes are K,, + K,, and ( l& + K,, 1 Iv,,l -1 

, 

whereas the initial semi-minor axes of the angular modes are lKIo -K,( and 

&I - lq l\11121-1. Similarly, th e initial semi-major axes of the rod lateral’modes 

al-e (K,, + K,&J,,( and (Go + rz,), whereas the initial semi-minor axes are 

I &I -~2llllv12l ancl IGO -q - 

If either motion goes through zero, the elliptical motion would be planar. For 

gun launch, both the angular and the flexing motion are forced to be near zero 

initially, and “near planar motion” is expected after launch. Guidos et al. [19] 

have analyzed the flight motion of long elastic finned projectiles in the 

U.S. Army Research Laboratory (ARL) Transonic Experimental Facility’s nonrotating coordinates and have observed near planar nose-tip motion with an 

amplitude in excess of 0.5 diameter. 

10. Induced Drag 

t The aerodynamic drag force is directed along the velocity vector and the linear 

normal force on a disk is perpendicular to the disk axis, which is canted at an 

angle of c - II with respect to the velocity vector. Thus, the normal force has a 

quadratic contribution to the drag. This contribution is called induced drag and 

can be computed for a pitching and flexing missile as 

21 
g,c,, = x[(~-ry~+(cx-r,,a]dx. 

x0 

Pw 

Although the nonlinear axial flow along the rod has a quadratic contribution to 

drag, this contribution is usually significantly less than the induced drag. 

The normal force distribution will be approximated by its dominant term, 

Cfk -9 

where 

J, = ~c,,I%lx . 

The first term in equation (105) is the well-known expression for the induced 

drag of a rigid missile. The induced drag for a particular angular mode is 

C DIk = ‘kKh PO6) 

where 

‘k = ‘F&x - JIk -Jlk + J,, and 

The average induced drag for a missile whose motion contains all five modes of 

equations (100) and (101) can be computed from the sum of the pure-mode 

induced drag contributions and the trim induced drag. 

where 

22 
11. Numerical Results 

In references [6] and [7j, numerical results were obtained for two specific 

missiles. The first was a flare-stabilized 25-cal. rod flying at 18,000 ft/s 

considered by Platus [l] and the second was a fin-stabilized 20-cal. rod flying at 

6000 ft/s. The flare on the Platus missile was 7.5-cal. long and the flare base 

diameter was 4 cal., i.e., 4 rod diameters. The finned missile has a l-cal. nose 

extension and a l-cal. fin extension (Figure 1). The necessary parameters for 

these missiles are given in Appendix F and converged values of A, ,N, ,\v~,-, for 

the finned missile are given in Table 1. The slender missile values of the 

moments of inertia are given in parenthesis are used in the numerical calculation 

of the transverse motion. The actual value of IX should be used in an integration 

of the roll equation (67). 

A measure of the flight flexibility of a missile is the ratio of the first elastic 

frequency to the rigid missile aerodynamic frequency, cr = o, /wR . The 

equations for the transient solution derived in section 8 have been solved for the 

first aerodynamic frequency, &, and the results plotted versus o for both 

missiles. In Figure 2, it is shown that this frequency for the flare-stabilized 

missile is 98% of the rigid missile value for CT = 10 and decreases as CT decreases. 

We see that the theory of this report falls between the 3-body theory and the 

Platus theory. 

The remainder of the numerical results was obtained for the finned missile. In 

Figure 3, this report’s theory is compared with the 3-body theory, and we see 

that it has a similar variation but predicts lower frequencies. Note that the 

frequency for 0 = 5 is near 60% the rigid missile value. The first two positive 

flexing frequencies 4, /oI ,$, / o2 are plotted versus cr in Figure 4. For small cr 

both frequencies are slightly larger than the free-free beam values, but they 

approach these values as o grows. 

The magnitude of the forward end motion is plotted against cr in Figure 5. At 

CT = 5, we see that 0.1 radian of the positive aerodynamic mode causes 

0.18 diameter of flexing motion. For the flexing frequencies, one diameter of 

flexing motion is associated with 0.018 radian (1.0”) of angular motion for the 

first positive mode and 0.072 radian (4.1”) of angular motion for the second 

positive mode. 

The forward end of the rod bends away from the velocity vector for an 

aerodynamic mode (UC 12 5 0) while it bends toward the velocity vector for both 

the first positive symmetric mode and the first positive antisymmetric mode. 

The flexing shape of the entire rod is very close to planar for each mode. Thus 

23 
RhJk (xhk2 I cl escribes the shape of the rod during a modal motion and is 

plotted versus x for k = 1,3,5 in Figure 6. For the first two positive frequencies, 

the shape is quite similar and the rear end of the rod is deflected about 15% less 

than the forward end. The third positive mode is antisymmetric and rear end 

deflects about 5% more than the front end and in the opposite direction. 

The damping rates of the three modes for no beam damping (i( = 0) are plotted 

versus o in Figure 7. For o = 5, the first aerodynamic damping rate is 85% of the 

rigid body damping rate, while the rates for the second and third positive mode 

damping rate are 75% and 35%, respectively. In Figure 8 1,/o, is plotted 

versus p/o, for two non-zero values of beam damping and CT = 5 . & /oR for 

this value of CT is 0.6 and beam damping reduces the damping rate when p/~a 

exceeds this value. The behavior was first observed Platus [l] and predicted by 

the 3-body theory [6,7]. 

The induced drag coefficient for the aerodynamic mode, C/C,, , is plotted 

versus o in Figure 9. It is 12% less than that for a rigid middle at o=5. 

Although the nose of the elastic missile bends to a larger angle of attack, the fins 

bend to a lesser angle of attack and the net effect is less induced drag. The 

induced drag coefficients for the two flexing modes are plotted versus o in 

Figure 10. Here we see that for the same amplitude motion the induced drag for 

the symmetric mode is more than 50% greater than that for the antisymmetric 

mode. Because both ends of the missile bend to a lesser angle of attack for the 

antisymmetric mode, this mode should have much smaller induced drag. 

For the bent missile calculations we will assume no fin damage (Ia, = 0) and the 

bent rod will be described by a pair of quartic curves. 

6, = d,,x2 + d2,X4 

= d,,x2 + d2p4 

Equation (62) provides a relation for 6,, . 

-101x50 

01x510 (108) 

(109) 

We wilI assume the rear of the rod is undeformed d,, = d,, = 0 and the values of 

d,, ,d22 are given in Appendix F. 

In Figure 11, the magnitude of the ratio of the trim angle of attack to the zero 

spin trim angle of a rigid missile is plotted versus spin for o = 5. The zero spin 

trim angle of the elastic missile is about four times that of a rigid missile. The 

first resonant peak is near p/o, = 0.6 and shows a five-fold magnification of the 

zero spin elastic trim. A smaller resonant peak occurs at the elastic frequency 

near 5 and much larger resonant peak near 14. The magnitude of the rod 

24 
deflection is plotted versus p/o, in Figure 12. The trim deflection for the first 

elastic mode is about eight times that for the aerodynamic mode while the trim 

deflection for the second elastic mode is three times that for the first elastic mode. 

In Figure 13, f2 is plotted versus p/o, . A sample fr line for pss/aR = 6.5 is also 

shown. The intersections of these curves are loci of possible spin-yaw lock-ins. 

The actual occurrence of lock-m would be shown by integration of the partial 

differential equation for flexing motion. 

12. Summary 

1. A system of equations has been derived for the motion of a symmetric elastic 

missile. This system consists of three ordinary differential equations and one 

fourth-order partial differential equation. Each of these equations contains 

integrals of the elastic deformation of the missile. 

2. A fourth-order ordinary equation with boundary conditions has been 

obtained to determine natural frequencies and damping rates of the motion. 

3. A simple special ordinary differential equation for the trim motion of a bent 

missile has been obtained. 

4. Equations for nonlinear roll moments for a bent missile have been obtained 

and a condition for possible spins-yaw lock-m stated. 

5. Numerical results have been obtained for the natural frequencies, flexing 

waveforms, damping rates, and induced drag coefficients. 

Figure 1. Sketch of finned missile. 

25 
Platus lhmy -_--_ 

Thraefhdynlaoly - 

Cmtinuous Elastic llxoty .-.-.-.-.- 

Figure 2. 4, /wR vs. o for flare-stabilized rod. 

1 

0.9 I0.E 

0.7 

0.6 

0.5 

0.4 

0.3 

, I I I I I I I I 

1 2 3 4 5 6 7 a 9 10 

Figure 3. 4, /wR vs. (r for firmed missile. 

26 
1.07 

1.06 

1.05 

1.04 

Oj 

1.03 

1.02 

1.01 

1 

0 2 4 6 8 10 12 14 16 18 20 

Figure 4. &,/o, ,4,/o, vs. CT for finned missile. 

I Y12 I 

E 

i, - 

(b5--- 

3- 

2.5 - 

2- 

1.5 - 

l0.5 - 

O0 

k=l 

k=3 ---_ 

k=5 ------- 

2 4 6 8 10 12 14 16 18 20 

Figure 5. I%2 1 vs. 0 for k = 1,3,5. 

27 
oRh bhk2 I *' / 

/- 

/' 

/' 

-0.5 - /' 

/' 

/' k=l 

/' k=CJ ---- 

/' 

-1 -I 

k=5 ------- 

I 

a 

-0.01 

-0.02 

-0.02 

h,/o, -O-O4 

-0.05 

-o.oE 

-0.07 

-0.OE 

-1.5’ I I I I I I 

-IO -8 -6 -4 -2 0 2 4 6 8 

Figure 6. R(Wk (x)/w,,) vs. x for r~ = 5, k = 1,3,5. 

I- 

, _ 

IIiirIO 

I 1 I I I I I I 

k=l 

kc3 ---- 

k=5 ------- 

-----__-___--_______________________ ---_- 

_/-- _/---___ 

.- 

__-- 

/ / / 

/ 

L I I I I I I 

2 4 6 8 10 12 14 16 18 20 

Figure 7. h, /oR vs. CT for k = 1,3,5. 

10 

i 

i 

28 
0.04 

0.02 

0 

-0.02 

t = 0.0 ,-’ 

I 

,/’ 

.’ 

-H 

4l% 

-0.04 

/ 

*/’ 

A’ 

/ ,’ 

/- I’ 

,/- ,’ 

,’ I’ 

,’ .’ / 

/’ ,I A’ 

/’ I ’ I ,’ I 

-0.06- // 

,‘/’ 

/’ 

, / 

-0.08 1 I I I I 

0 0.5 1 I.5 2 2.5 3 3.5 4 4.5 5 

Figure8. A,/@, vs. p/mR for CT =5,i=O,O.O5,0.10. 

‘5 

0.86L I I I I I I I I 

0 2 4 6 8 10 12 14 16 18 20 

Figure 9. Cl/C, vs. CT. 

29 
0.25 

0.2 

0.15 

c 

0.1 

0.05 

70 I , I I I 

6C 

50 

40 

30 

20 

10 

0 IL 

0 2 4 6 8 10 12 14 16 

I I I I I I 1 

4 k5 ---- 

\ 

i--\- 

----.---__------- 

0 2 4 6 6 10 12 14 16 18 20 

Figure 10. & vs. CT for k = 3,5. 

i 

Figure 11. lb/hol vs. P/ CD, for bent-finned missile at CT= 5 . 

30 
goo; 800 

t 

700 

600 

500 

I 

lad5TROl 4oo I 

300 

t 

1 

0 0 2 4 6 8 IO 12 14 16 

Figure 12. I~E(d~mol vs. P/ wR at D= 5 for bent-finned missile. , 

0.5, I I I , I 

-0.5 - 

-2.5 - 

-3 

-3.5 

-4’ I I I I I I I 

0 1 2 3 4 5 6 7 -8 

Figure 13. fj vs. p/wR for bent-finned missile at O= 5 . 

31 
Table 1. Values of Ak,, , NkO, z,, . 

G &I NkO Z 

5 -3.18 + 30.6i 2082 -14.6i -024 

10 -3.55 + 48.63: 434 -10.5i - 0.17 

20 -3.65 + 52.4i 101 - 3.3i - 0.04 

-- - 

5 -2.78 + 280i (13.1 +3.6i)W 18 + 4.8i 

10 -2.44 + 543i (3.5 +7.9i)l@ 12 +27.3i 

20 1 -2.33+10773: 1 (1.0 +18.5i)lW 1 0.9 +15.8i 

k=5 

5 -1.34 + 747i -55729 +10586i -0.22 -0.02i 

10 -1.38+14853: -53957 +20861i -0.05-O.OOSi 

I 20 1 -1.39+2963i 1 -53374 +41579i 1 -0.01-0.004i I 

32 
13. References 

1. Platus, D. H. “Aeroelasiic Stability of Slender, Spinning Missiles.” Journal of 

Guidance, Control, and Dynamics, vol. 15, pp. 144-151, January/February 1992. 

2. Legner, H. H., E. Y. Lo, and W. G. Reinecke. “On the Trajectory of 

Hypersonic Projectiles Undergoing Geometry Changes.” AIAA Aerospace 

Sciences Meeting, Reno, NV, AIAA 94-0719,10-13 January 1994. 

3. M.&hail, A. G. “In-Flight Flexure and Spin Lock-in for Anti-Tank Kinetic 

Energy Projectiles.” Journal of Spacecraft- and Rockets, vol. 33, pp. 657-664, 

September-October 1996. 

4. M.&hail, A. G. “In-Flight Flexure and Spin Lock-in for Anti-Tank Kinetic 

Energy Projectiles.” ARL-TR-882, U.S. Army Research Laboratory, Aberdeen 

Proving Ground, MD, October 1995. 

5. Heddadj, S., R. Cayzac, and J. Renard. “Aeroelasticity of High L/D 

Supersonic Bodies: Theoretical and Numerical Approach.” AIAA Aerospace 

Sciences Meeting, Reno, NV, AIAA 2000-0390,10-13 January 2000. 

6. Murphy, C. H., and W. H. Merrnagen. “Flight Mechanics of an Elastic 

Symmetric Projectile.” Journal of Guidance, Control, and Dynamics, vol. 24, 

November-December 2001. 

7. Murphy, C. H., and W. H. Mermagen. “Flight Mechanics of an Elastic 

Symmetric Projectile.” ARL-TR-2255, U.S. Army Research Laboratory, 

Aberdeen Proving Ground, MD, April 2000. 

8. Murphy, C. H. “Some Special Cases of Spin-Yaw Lock-In.” Journal of 

Guidance, ControZ, and Dynamics, vol. 12, pp. 771-776, November-December 

1989. 

9. Murphy, C. H. “Some Special Cases of the Spin-Yaw Lock-In.” BRL-MR3609, U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, 

MD, August 1987. 

10. Wood, R. M., and C. H. Murphy. “Aerodynamic Derivatives for Both Steady 

and Nonsteady Motion of Slender Missiles.” Journul of Aeronautical Science, 

vol. 22, pp. 670671, December 1955. 

11. Wood, R. M., and C. H. Murphy. “Aerodynamic Derivatives for Both Steady 

and Nonsteady Motion of Slender Missiles.” BRL-MR-880, U.S. Army 

Ballistic Research Laboratory, Aberdeen Proving Ground, MD, April 1955. 

12. Munk, M. M. “The Aerodynamic Forces on Airship Hulls.” NACA-TR-184, 

1923. 

. 

33 
13. Devan, L., and L. A. Mason. “Aerodynamics of Tactical Weapons to Mach 

Number 8 and Angle of Attack 180: Part II Computer Program and Users 

Guide.” Naval Surface Weapons Center, DahIgren, VA, NSWC-TR-81-358, 

September 1981. 

14. Durand, W. F. “Aerodynamic Theory.” Reprint ed., Pasadena: California 

Institute of Technology, pp. llO-112,1943. 

15. Murphy, C. H. “Free Flight Motion of Symmetric Missiles.” BRL-MR-1216, 

U.S. Army Ballistic Research Laboratory, Aberdeen Proving Ground, MD, 

July 1963. 

16 De SiIva, C. W. Vibration: Fundamentals and Practice. Boca Raton: CRC Press, 

p. 351,200o. 

17. Geradin, M., and D. Rixen. Mechunicul Vibrations: 7’heoy and Applications to 

Struchtrul Dynamics. New York: Wiley & Sons, 1994. 

18. Chapman, G. T., and D. B. Kirk. “A New Method for Extracting 

Aerodynamic Coefficients from Free Flight Data.” AMA ~oumul, vol. 8, 

pp. 753-758, April 1970. 

19. Guidos, B. J., J. M. Garner, J. F. NewiII, and C. D. Livecchia. “Measured InFlight Rod FIexure of a 120-mm M829E3 Kinetic Energy (KE) Projectile Steel 

Model.” U.S. Army Research Laboratory, Aberdeen Proving Ground, MD, to 

be published. 

34 
Appendix A. Integrals 

A.1 Aerodynamic Integrals 

C Na = jfcfldx 

x0 

&q = i’(‘f, - “,,b 

X0 

c, = )sf2dx 

X0 

C NBF = jfCflrBFdX 

X0 

A.2 Boundary Conditions Integrals 

Xl 

I, = I c,,dx 

X0 XI 

I I3 = 21, -I, - x,1, 

Xl 

I 113 = I ‘Ddx + ‘Dbp 

X0 

Xl 

.I 1BF = I 

x0 

C Ma = jfCf,Xdx 

C Mq = it’,, - XCfl )xh 

X0 

c jfCf2Xdx Mdr = 

x0 

C mF = ycJBFxdx 

X0 

7.3 

I, = I c&c 

I, = 1(x - X2)cf2dx 

I ,* = 21, - I, - x*1, 

I 14 = 21, - I,, - x,1, 

x3 

I 2D = 

‘3, = I(’ - xl )cflrBFdx 

x0 

35 
INTENTIONALLYLEFTBLANK. 

36 
Appendix B. Functions 

J,(t) = jf@dx 

x0 

J,(t)= ~c,6dx + 6(x, )cDbp 

X0 

J*(t)= 7@-%1%)dX J6(t)= (l/Ljjh,dx 

x0 XI 

J,(t) = ~c+ix J, (t) = -(i/Lf@% + fmr)ax 

X0 XI 

J, (0 = 7(Crzr -CA )xh J 8 (t ) = -(l/l 6L)7(l? - 2ipT)dx 

J,(t)=F/c,,[c(!-r)+(& -ixQ)(d/V$dx 

Xl 

JE(t)= ;g,dxJ, + j,(d/V)+ J5] 

J,(t) = [I$ - (ipI, )Nkd/V) 

Gr 

=&g-l 8 a - 

- --ip6, 

t 1 ‘ax’ at 

N(t) = g, [J, + & (d/V)]- 8, 

E,(x)=&, -h,J 

~,(x)=&,i +c, -wskcvv) 

E BF = &NBF -+f,T,,) 

Cf, (4 = 2Cf, - XCfl 

f’,,, (x, t) = (I’ - 2ipT + iQ)/16 

f,(t)=I1[5(t)-r(x*Yt)l+[I,,~(t)+I,8,(x,,t)+(I, -oI%t)l(d/V) 

fZ(t)=I2C~(f)-r(x*,tll+[I,,~(t)+I,~,(x,Yt)+(I, -m(X2Yt)kd/V) 

m,(t)=I,[s(t)-r(x,,tll+[I,,~(t)+I,s,(x,,t)+(I, -I,N%0~@+% 

m~(t)=I,[S(t)-r(x,,tll+[I,,~(t)+I,8,(x,Yt)+(I,, -ww)~w+m2D 

m,,(t)=I,,[G,(x,,t)-6,(tll+I,,r(x,,t) 

mm(t)= I*D[~E(XZYt)-8E(fll+I4Dr(XZrf) 
i 

. 

38 
Appendix C. Relationship Between Boundary Conditions 

If we make the slender missile assumptions (I, z md2L2/12; 1,/I, =I,/& z 0), 

the boundary conditions at x, and x2 are directly related. This can be shown by 

first integrating equation (51) from x1 to x at fixed time. 

iXdx-iQ62 -x:)/2-g,[C,5-J, +[(& +C.g~-j2kdlV)kx-x*)/L= 

XI 

(C-1) 

Equation (C-l) is now evaluated at x2 using the integrals of C.l. 

where fj is defined in Appendix B. 

According to equation (C-2), boundary condition (53) implies boundary 

condition (55). 

If equation (C-l) is integrated from x1 to x2, five single integrals and six double 

integrals result. Each of the double integrals can be simplified by integration by 

parts and are given in section C.3. 

Under our stated assumptions, equation (36) becomes 

( I ) L2 12 Q+i& =-ig2[CMac-J3 +((C& +CMq)$-j4)cl/V)-J,]s (c-3) 

Equation (C-l) is now integrated and simplified by use of sections C.l-C.4 and 

equations (53) and (A-3), 

a2Ux2) 

ax2 +i;cd2(x2)= “‘6~~1)+icd2(x,)+g4(ml +m,) , (C-4 

where mj is defined in Appendix B. 

Boundary condition (54) is equivalent, therefore, to boundary condition (56). 

39 
C.1 Integrals in Equation (C-2) 

x2 x2 

I 8dx = 0 I c&x = c,, -1, -1, 

Xl Xl 

x2 x2 

I c,,lYx = J, -IIF, -I& I c&x = c,, -1, -1, 

XI XI 

x2 

I( Cf2 - XCfl b = c, - (4, - 15 I- 012 - 16 1 

XI 

x2 

I( Cf& - Cf2f b =-j2 -I& -(I3 -I$, -I,&, -(I4 -I,)F2 

Xl 

C.2 Integrals in Equation (C-4) 

(1 )I( 1 2x2 x2-x$x=-L3/12 L-IX/(x-x&x=1/2 

XI XI 

x2 a3s,(x) 

t 

a3s,(x,) dx 

XI ax3 - ax3 1 = 

d26,(x2) ~2~E(x,) a3%(dL 

&2 - &2 - &3 

Xl 

x2 

f 

c&lx = J, -I,, -I,, 

XI 

C.3 Multiple Integrals in Equation (C-4) 

x2 7. 

If hxdx = -L& 

XI XI 

40 
J J(ct2 --xc&~ =&2)cN4 -(I,, -Q]-[c,, -(L -I~)-(114 -IdI XI XI 

x2 x JJ( c$, - c,,F)lx dx = L[- (l/Z)& -I& - (I3 - I$,] 

=I XI 

- -&-I& -(I9 -I#, -I& -(I,, -I&] 1 

41 
42 
Appendix D. Trim Solution Parameters 

b, =a, +ip(a, +a3)-p2(It -1,) 

J ET = -(gldXJJT + J5T +ibd’vb4Tl 

JNT = i(It - I, )(pd/V)N, 

J =i&(It -u.PwwNBF 

i: =&[h +J,~(~p~/~)]+p*~, 

J, for j = 1,2,3,4,5,6 is Jj with 6, ,lY replaced by 6,) rT 

E = E2 + g*wvN, 

Er= (E, + ipE,, - p*x)/o~ 

E4 = g4 [c,, + (kpd/Vh* - %I 

E, =-p*/w~ +g, c,(ipd/V)+$$ 1 

XcT = (l/Lflf,dx 

N; = [l + &d,V)~T~,2 

JGJF = [EBF + 4Pd/v)cNBFb,2 

E, =g, 

[ 

cd', +(crzTB -Gdipd/V)- 

4CDSB) 

dx 1 

fiT ='&T -rT(xl))+[1116T +11sET(x1)+(13 -l,~T(x,IXipd/V)-l,, 

f2T ='2kT -rT(xl))+[1125T +12s,(x2)+(14-16)rT(X11Xipd/V) 

mlT =‘3kT -rT(xl))+[113~T +1SsFI.(x1)+(19 -I,~T(x,h/VbmlDT -hBF 

m2T =I4kT -rT(X1))+[1146T +14sET(x2)+(IIIl -18)T,(x,)bd/V)-m2DT 

mlDT = l*D[s,(x,)-sT~l+l,DrT(x,) 

m2DT =12D[6,(x2)-sTcl+14DrT(x2) 

c 

43 
44 
Appendix E. Transient Solution Parameters 

b lk = a, + ipa, + (a2 -ipI,)A, + I,A: 

J Ek = -6&d]J3k + J5k + (Akd/V)J4k 1 

J, = CA,‘, - blx xd/v)Nk 

Nk = g2iJlk + J2k(Akd/V)I-A:Wkc 

Jjk for j = 1,2,3,4,5,6 is Jj with 6,,lY replaced by v,,z 

E 3k = (E, + AkEZL + A:Y+;~ 

E 4k = g4 [‘fl + tAkd/%2 - ‘D 1 

E Sk = A;/$ + g, cfl (A,d/V)+ 2 1 

E 6k = 1 + 2iiW;’ (Ak - ip) 

wkc = (1/Llj2vkdx 

N; = [l+ x&kd,V)Nko,2 

fik = I,(,- dv;f”)+~l, +IIyfk(xI)+(13 -&)dv$l)](Akd/V) 

f2k =I,(l-dv$2))+[I,2 +12Wk(x2)+(14 -16)dv$2)](Akd/V) 

mlk =I,(l-dv$))+l,3 +13yJk(x,)+(I, -I,)dv$‘)](Akd/V)-m,, 

m2k = 14(1-dv$2’)+[I,4 +14\vk(x2)+(11,, -&)dv$2)](Akd/V)-m,, 

mlDk = 11D[~k(XI)-%c]+13D dv$l’ 

m2Dk = 12D[~k(X2bWkcl+14D dvf2’ 
46 
Appendix F. Missile Parameters 

F.l Flare-Stabilized Missile Parameters 

L=25 

d = 0.20ft 

m = 2.40 slug 

I, = 0.0132s1ug - ft* (0) 

I, = 5.006slug - fi* (5.000) 

Cfl =o 

=-1.14(x +5) 

Cf2 =o 

C, = 32 

F.2 Finned Missile Parameters 

L= 20 

d = 0.35 ft. 

m = 3.50 slug 

I, = 0.054 slug&2 (0) 

1, = 14.318 shg-ft2 (14.292) 

Cfl =4(11 -x) 

= 4e7(X-‘0) 

= -(2/x)(15 + 3x)a, 

= @/4% 

Cf2 =2(11-x)2 

= 2 + 0.571(1- e7(x-Io)) 

=2.571+(1/37cXl5+3x)2a, 

=2.571-(12/~)(6+x)a, 

CD = (0.30X1 1 - x) - 0.295e-‘+-‘0) 

= 0.0050 

= -0.1 O(5 + x) + 0.005 - 0.1 98e’0(x+7) 

= 0.0070 

CD& = 0.14 

hare = 0.8Ofi 

S = (x/4)(d) 

V = 18000 fi/sec 

p = .002 slug/fi * 

-5.~~112.5 

-12.55x1-5 

C Ma = -320 

V = 6000 ft/s 

p = 0.002 slugs/W 

xO1 = x23 =I 

a, = 1100 ft/s 

a, = 2[Ji$j7’ 

lO<xlll 

-5<xr10 

-7<XI-5 

-111x<-7 

lO<xlll 

-5<XIlO 

-7<xS-5 

-lllxI-7 

lO<xlll 

-5<x110 

-7<XI-5 

-lllxI-7 

Ccp =-18 

47 
d,, = 1O-3 

C Na = 9.7 

C Mq = -980 

d,, = -0.25~10-~ 

C = -34.4 

c:; =-190 

48 
List of Symbols 

L 

63 

g2 

g3 

g4 

gx 

gt 

I 

IX 

It 

ii 

L 

m 

Me 

P 

Q 

S 

local cross-sectional area 

aerodynamic force distributions functions 

rod diameter 

Young’s modulus 

complex aerodynamic shear force on rod at x 

complex beam damping shear force on rod at x 

complex beam elastic shear force on beam at x 

pv2s/2 

dmd 

pVSd/2m 

g,L/d 

I, /md2 

I, /md2 

(dr JJy’dydz = (d)4 f]z’dydz, area moment of rod 

axial moment of inertia of projectile 

transverse moment of inertia of projectile 

beam-damping coefficient 

dimensionless length of rod 

projectile mass 

complex elastic moment on rod at x 

projectile spin 

q + ir , complex transverse angular velocity of body 1 

71d2/4 

49 
magnitude of projectile velocity 

location of beam ends 

dimensionless length of fore and aft aerodynamic 

extensions 

angle of attack of central disk 

angle of sideslip of central disk 

a6 

2 cant of disk 

6, - 6, lateral displacement of disk relative to missile 

center of mass (cm) 

~-1x~~E&, lateral location of missile’s cm 

XI 

6, +i&, lateral displacement of disk relative to central 

disk 

roll angle 

frequency of k-h mode 

damping of k-th mode 

complex angle of attack of disk 

air density 

m/M, linear density of rod 

(%/% 

EI/P,d4 

lowest elastic frequency of beam in vacuum 

rigid projectile zero-spin frequency 

p + ia, complex angle of attack of projectile 

aerodynamic force exerted on missile 

angular momentum of missile 

i 

50 
G=(M,,M,,M,) aerodynamic moment exerted on missile 

~=(v,,vy,v,) missile velocity 

?& =(v,,v tiy Y vdcz ) velocity of disk relative to missile’s cm 

:, = (x,&y,&J &l-n ensionless location of disk relative to central disk 

ii A (x,6, ,6,) dimensionless location of disk relative to missile’s cm 

6 = (p,q,r) angular velocity of central disk 

fi = (O,q,r) angular velocity of non-spinning elastic coordinates 

&I real part of 2 

44 imaginarypartofz 

Circumflex superscript denotes a beam shear force 

Tilde superscript denotes elastic parameter for bent missile 

B subscript denotes parameter for bent projectile 

BF subscript denotes bent fin parameter 

E subscript denotes elastic coordinates at central disk 

T subscript denotes trim motion parameter 

51 
i 

52 
REPORT DOCUMENTATION PAGE Form Approved 

OMB No. 07044188 

,b,,c mpo,ling burdm for this colladlon of ,nfomut,on is .sUmated to avo”ge , hour par mspome. IneludIng the time for -g Ir~~f~~tlons, swmhlng UlSUnP dab SOW-C-. 

“,,dng and ,mlntafnlng fh, data “.qdH. qnd coqq,f.f,ng and ““fwlng fh. mlf.cfbn of Infq”“adon. S.nd commnts mrding fhh burdm es”mafe or Uy ofher UD.Cf Of fhb 

IktIon of Infomuflon. lncfudlng su&om for mducln~ this bunbn. lo WashIngton HUdquattws Smvicm~, Dlnctomfe for InfomUffon OpmUonr and ffNIDt% 1215 - 

‘light Motion of a Continuously Elastic Finned Missile 

iberdeen Proving Grotmd , MD 21005-5069 

I 

1. SUPPLEMENTARY NOTES 

12a. DlSTRlBUTlONlAVAlLABlLlTY STATEMENT 12b. DlSTRlBUTlON CODE 

4pproved for public release; distribution is unlimited. 

13. ABSTRACT(Maximum ZOO words) 

lhe motion of elastic finned projectiles has been analyzed by various approximate theories. In this report the exac 

:quations of small amplitude motion are derived for a symmetric missile. The aerodynamic and elastic symmetries an 

used to allow the use of complex variables to describe the lateral motion in a non-rotating coordinate system Althougl 

he resulting equations are both ordinary and partial differential equations, frequencies and damping rates of fret 

oscillations are obtained from an ordinary differential equation with boundary conditions. Equations for a permanentl] 

deformed bent missile are derived, and an ordinary ditferential equation for the forced motion of a bent missile i! 

obtained. Sample calculations for a finned projectile with a fineness ratio of 20 show resonant motion at tht 

tierodynamic frequency as well as at each elastic frequency. The nonlinear roll moment associated with a bent missile i! 

computed and the location of possible spin-yaw lock-in is determined. The flight motion of an elastic missile is showr 

to be the sum of two elliptical motions: a low frequency pitching motion and a higher frequency flexing motion. Thf 

induced drag coefficients for both motions are computed as functions of the missile’s elasticity. 

14. SUBJECT TERMS 15. NUMBER OF PAGES 

flight mechanics, elastic missile, spin-yaw lock-in, aeroelastic motion 53 

16. PRICE CODE 

17. SECURITY CLASSlFlCATlON 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LlMlTATlON OF ABSTRACT 

OF REPORT OF THIS PAGE OF ABSTRACT 

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL 

NSN 7540-01-2806500 53 Standard Fon 298 (Rev. 2-89) 

Prescribed by ANSI Std. 239-18 298-102 
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