Spin-Yaw Lockin of an Elastic Finned Projectile 

by Charles H. Murphy and William H. Mermagen, Sr. 

ARL-TR-3217 August 2004 

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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3217 August 2004 

Spin-Yaw Lockin of an Elastic Finned Projectile 

Charles H. Murphy and William H. Mermagen, Sr. 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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4. TITLE AND SUBTITLE 

Spin-Yaw Lockin of an Elastic Finned Projectile 

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AH80 

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6. AUTHOR(S) 

Charles H. Murphy and William H. Mermagen, Sr. 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

Supersonic finned projectiles carrying very long dense metallic rods have been observed to have significantly large flexing 

motion. The appropriate Lagrangian for a pitching, yawing, rolling, and flexing missile is derived and the powerful finite 

element method used to construct differential equations for finite element parameters. These differential equations for constant 

spin are used to calculate frequencies and damping rates of transient motion and trim response motion to missile-fixed inertia 

and aerodynamic forcing terms associated with inelastic deformations. The results agree with earlier results based on a much 

more difficult iterative integration of a complex differential equation with unusual boundary conditions. Only three elements 

were necessary for reasonable accuracy, although codes for five and seven elements have been prepared and exercised. The 

finite element ordinary differential equations allowed calculation of the time history of missile motion with varying spin and 

demonstrated the occurrence of spin-yaw resonance at the aerodynamic frequency and at the lowest elastic frequency. 

15. SUBJECT TERMS

aeroelastic, spin-yaw lockin, finned projectile 

16. SECURITY CLASSIFICATION OF: 

19a. NAME OF RESPONSIBLE PERSON

Gene Cooper 

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iii

Contents 

List of Figures v

List of Tables v

1. Introduction 1

2. Coordinate System 2

3. Motion of Central Disk 5

4. FEM 8

5. Flexing Motion 10

6. Special Solutions 12

7. Quadratic Roll Equation 13

8. Numerical Results 15

9. Summary 22

10. References 27

Appendix A. Integrals 29

Appendix B. Functions 31

Appendix C. Generalized Forces and Hermitian Polynomials 33

Appendix D. Connector Coefficients for Equations (39–41) 35

Appendix E. Connector Coefficients for Equations (42–44) 37

Appendix F. Differential Equations Coefficients 39
iv

Appendix G. Flexing Motion Finite Element Method (FEM) Quantities 41

Appendix H. Finned Missile Parameters 47

List of Symbols, Abbreviations, and Acronyms 49

Distribution List 52 
v

List of Figures 

Figure 1. X-Z coordinates of cross-sectional disk. .........................................................................3 

Figure 2. X-Y coordinates of cross-sectional disk..........................................................................3 

Figure 3. Sketch of finned missile. ...............................................................................................15 

Figure 4. 1 R φ ω/  vs. σ for finned missile.......................................................................................16 

Figure 5. fj vs. φ R /ω for ΓBF ss R = 0, σ = 5, p = 6ω . ...................................................................18 

Figure 6. R φ

/ω vs. time for ΓBF ss R 0 R = σ= = φ = 0, 5, p 6.0ω , /ω 0, 5.4  . ....................................19 

Figure 7. fj vs. R φ

/ω for ΓBF ss R = 0, σ = 5, p = 0.8ω ...................................................................19 

Figure 8. R φ

/ω vs. time for ΓBF ss R 0 = 0, σ = 5, p = 0.8ω , ψ = 0, –3 s-1. ..................................20 

Figure 9. Angular motion (qb1y vs. qb1z) for ΓBF ss R 0 = 0, 5, p 0.8 σ= = = − ω , ψ 0, 3 s-1. ........20 

Figure 10. Rod forward tip motion (qb7y vs. qb7z) for –1 ΓBF ss R 0 = σ= = = − 0, 5, p 0.8ω , ψ 0, 3 s ...............................................................................21 

Figure 11. Maximum strain ( Mε ) for ΓBF ss R 0 = 0, 5, p 0.8 σ= = = − ω , ψ 0, 3 s-1. ...................22 

Figure 12. ψ0  vs. 0 θ

 showing regions of sse 1e p , p lockin ΓBF ss R = 0, σ = 5, p = 0.8ω .....................23 

Figure 13. fj vs. φ R /ω for ΓBF ss R = 0.02, σ = 5, p = 0.4ω ..............................................................23 

Figure 14. R φ

/ω vs. time for ΓBF ss R 0 = 0.02, σ = 5, p = 0.4ω , θ = 0, 5  s-1.................................24 

Figure 15. Angular motion (qb1y vs. qb1z) for ΓBF ss R 0 = 0.02, σ = 5, p = 0.4ω , θ = 0, 5  s-1. ......24 

Figure 16. Rod forward tip motion (qb7y vs. qb7z) for 

ΓBF ss R 0 = 0.02, σ = 5, p = 0.4ω , θ = 0  s-1.................................................................................25 

Figure 17. Rod forward tip motion (qb7y vs. qb7z) for ΓBF ss R 0 = 0.02, σ = 5, p = 0.4ω , θ = 5  s-1.....25 

Figure 18. Maximum strain ( ) Mε for ΓBF ss R 0 = 0.02, σ = 5, p = 0.4ω , θ = 0, 5  s-1. ...................26 

Figure 19. ψ0  vs. 0 θ

 showing regions of sse 2e p , p lockin ΓBF ss R = 0.02, σ = 5, p = 0.4ω . .............26 

List of Tables 

Table 1. Transient frequencies and damping rates........................................................................17 

Table 2. Equilibrium spin rates.....................................................................................................18 
vi

INTENTIONALLY LEFT BLANK. 
1

1. Introduction 

Very long finned projectiles carrying very dense metallic rods have been observed to be subjected 

to large inelastic deformations during hypersonic flight (1). Spark shadowgraphs of these 

projectiles have shown elastic bending motion with amplitudes as large as the rod’s radius (2). It 

has been conjectured that the cause of the inelastic deformation was the bending loads associated 

with pitching and yawing motion at resonant spin frequencies. These frequencies would be near 

to the aerodynamic frequency of a rigid projectile or the elastic frequencies of the rod. 

Spinning motion at a resonant frequency can be caused by a nonlinear roll moment associated 

with the roll orientation of the fins or by mass asymmetry due to damage or poor construction of 

the projectile. This spin-yaw lockin mechanism has been discussed for a rigid missile by several 

authors (3–7). 

The linear flight motion of an elastic missile has been considered in references (8–13). In 

reference (11), a very simple theoretical model of a projectile composed of three components 

connected by two bent mass-less elastic beams was used to approximate an elastic missile. For 

appropriate selection of beam parameters spin-yaw lockin was demonstrated and large amplitude 

oscillations occurred. In references (12) and (13), the correct linear partial-differential equation 

for a continuously elastic missile was derived, and special solutions for harmonic transient 

motion and trim motion were obtained. A nonlinear roll moment was shown to be capable of 

producing equilibrium spin near resonance. Although the lateral motion of the pitching and 

yawing missile was linearized, it was necessary to retain the quadratic terms in the roll equation. 

A time history of motion going to lockin was not obtained at that time. It is the purpose of this 

report to compute such a time history by use of the finite element method (FEM) (14). 

The lateral motion of a symmetric rigid missile has often been described by complex variables 

(15), and complex variables were used in references (8) and (11–13) to describe the lateral 

flexing motion of the rotationally symmetric rod. The equations of motion of a spinning missile 

are usually derived by vector analysis and Newtonian mechanics, but FEM equations for elastic 

bodies require the use of the more sophisticated Lagrangian mechanics. The Lagrangian of a 

system must be stated in body-fixed coordinates. Separate differential equations in the two 

lateral directions can be obtained from the appropriate Lagrangian. Pairs of equations are 

combined and complex variables introduced to yield half as many complex equations. 

If five elements are used to describe the pitching and flexing motion of the missile, 20 

parameters, usually called connectors, are introduced and 25 second-order differential equations 

and one first order differential equation are required. The introduction of complex angles and 

complex connections reduce this system to 12 second-order complex differential equations and 

two real differential equations. 
2

2. Coordinate System 

The elastic missile is assumed to consist of a very heavy elastic circular rod of fineness ratio, L, 

and mass, m, embedded in a very light symmetric aerodynamic structure that may be longer than 

the rod. The rod’s axial moment of inertia is Ix and its transverse moment of inertia about its 

center is It0. The rod’s diameter can vary over its length, and its maximum diameter will be 

denoted by “d.” All distances will be expressed as multiples of the rod diameter and its length is 

Ld. A nose windshield of length 23 x d may be attached to the forward end of the rod, and the 

fins may extend beyond the end of the rod at a distance of 01 x d . Thus, the rod is located 

between 1 x L2 = − and 2 x L2 = , while the aerodynamic structure extends from 0 1 01 x xx = −

to 3 2 23 xxx = + . 

An earth-fixed coordinate system will be used with the Xe-axis oriented along the initial 

direction of the missile’s velocity vector. The Ze-axis is downward-pointing and the Ye-axis is 

determined by the right hand rule. A nonrotating coordinate system, XYZ is then defined with 

origin always at the center of the rod and the X-axis tangent there. The X-axis pitches through 

the angle,θ , and yaws through the angle,ψ , with respect to the Xe-axis. Body-fixed coordinates 

XYb-Zb are now defined for which the Yb-Zb axes rotate with the missile through the roll angle, 

φ , with respect to the Y-Z axes. 

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

X-axis with thickness, dx. When the rod flexes, the disks shift laterally perpendicular to the 

X-axis and cant to be perpendicular to the centerline of the disks. This canting action neglects 

the shear deformation of the rod, and this constraint is called the Bernoulli assumption (14). The 

lateral displacement of a disk has body-fixed coordinates by bz δ and δ , and the disk is canted at 

angles by bz Γ Γ and . 

by bz

by bz ;

x x

∂δ ∂δ Γ= Γ=

∂ ∂ . (1) 

It important to note that at the central disk 

δ =δ =Γ =Γ = by bz by bz ( ) 0, t 0, t 0, t 0, t 0 ( ) ( ) ( ) . (2) 

Reference (12) used the nonspinning elastic coordinate system with XYZ axes ( ) φ = 0 . The 

lateral displacements of a disk in this elastic coordinate system are shown in figures 1 and 2 and 

can be computed from body-fixed quantities. 

( ) i

E Ey Ez by bz i i e φ δ =δ + δ = δ + δ , (3) 

and 
3

V

X

Xe

Z Ze

α

θ

δEz Γz

Missile Centerline

Figure 1. X-Z coordinates of cross-sectional disk. 

δEy

Γy

Y

Y

X

Missile Centerline

V

X

Xe

Y e

β

ψ

Figure 2. X-Y coordinates of cross-sectional disk. 
4

( ) i

y z by bz i i e φ Γ=Γ + Γ = Γ + Γ . (4) 

The earth-fixed coordinates of the central disk are ee e x , y , and z , and the earth-fixed coordinates 

of the other disks are computed in terms of the central disk earth-fixed coordinates, their 

body-fixed displacements, and the Euler angles θ, , and ψ φ . 

( ) {( ) } {( ) } 22 i i

de e by bz by bz x x x 1 2 Re i e Im i e φ φ = + − ψ +θ −ψ δ + δ +θ δ + δ     , (5) 

{( ) }i

de e by bz y y x Re i e φ = + ψ+ δ + δ , (6) 

and 

{( ) }i

de e by bz z z x Im i e φ = − θ+ δ + δ . (7) 

ye, ze, by bz θψδ δ , , , and their derivatives are assumed to be small quantities, but xe and xe  are 

not small. Lagrangian dynamics yields the correct linearized differential equations for these six 

variables when the kinetic energy expansion retains all quadratic terms in these variables. Thus, 

equations 6 and 7 consist of only linear terms, but equation 5, which contains xe, retains 

quadratic terms in these variables. The coefficient of x in equation 5, for example, is the 

quadratic form of the cosine of the angle between the Xe-axis and the X-axis. 

The angular velocity of the central disk in the body-fixed coordinates is 

 p = φ −ψθ   , (8) 

{( ) }i

b q Re i e− φ = θ+ ψ   , (9) 

and 

{( ) }i

br Im i e− φ = θ+ ψ   . (10) 

The angular velocity of any other disk along axes aligned with the disk’s axes of symmetry is 

p p 1 2 q r p Re i 2 i d d by d bz = − ΓΓ + Γ + Γ = + θ + ψ − φΓ + Γ Γ ( ) {( ) }     , (11) 

{( ) }i

d b bz by q q Re i i e− φ = − Γ − φΓ = θ + ψ + Γ      , (12) 

and 

{( ) }i

d b by bz r r Im i i e− φ = + Γ − φΓ = θ + ψ + Γ      . (13) 
5

3. Motion of Central Disk 

The mass of each circular disk is ( ) m L x dx ρ1 ( ) ; its roll moment of inertia is 

( ) ( ) 2

d 2 2a md x dx ρ , and its transverse moment of inertia is ( ) ( ) 2d 2 a md x dx ρ , where d a is 

( ) 1

16L −. ρ1 ( ) x and ρ2 ( ) x describe the variation of mass and moments of inertia along the rod, 

and both become unity for a homogeneous rod with constant diameter. The kinetic energy of a 

disk is therefore 

( )( ) ( )( ) 2 2 2 2 2 2 22 T dx = md d 1 de de de d 2 d d d   ρ 2L x + y + z + a md ρ 2 2p + q + r dx    . (14) 

The total kinetic energy of the missile can be obtained by integrating T over the length of the d

rod: 

( ) ( )( ) 2

1

x

2 2

d R 11 12 21 22

x

T T dx T md T T md 2L T T = =+ + + + ∫ , (15) 

( )( ) ( )

( )

{ } ( ) ( )

{ } ( ) ( )( )

( ) ( ) ( )

( )

2 2

1 1

2 2 22 2 22

R e e e x t0

2

ce e e

11 e c c

12 e e c 6 8

x x

2 2 22

21 Ey Ez 1 d y z 2

x x

22 d

where T md 2 x y z I p 2 I 2

 md x y z x ,

T x Re i i ,

T Re y iz i J J ,

T dx a L dx,

T 2 a L Re 2i

= + + + + ψ +θ

+ ψ − θ − ψψ + θθ    

=− ψ+ θ δ + ψ+ θ δ

= − δ + ψ+ θ +

= δ +δ ρ + Γ +Γ ρ

=φ Γ

∫ ∫

  

     

   

     

   

  ( )

2

1

x

2

x

dx ,

      − φΓ Γρ

    ∫ 

and 

( ) 2

1

x

c 1

x

x 1 L x dx = ρ ∫ . 

The e y and e z components of the central disk velocity can be approximated by linear relations in 

angles of pitch and yaw with respect to inertia axes (θ,ψ) and angles of attack and sideslip with 

respect to the velocity vector ( ) α β, as shown in figures 1 and 2. The magnitude of the velocity 

vector is V. 

y Vd  e = ( ) (β+ψ), (16) 

and 
6

z Vd  e = ( ) (α−θ). (17) 

Equations 16 and 17 can be written as a single complex equation: 

y iz V d q q   e e 1 1e += + ( ) ( ), (18) 

where 1 q i =β+ α and 1e q i =ψ− θ. 

In reference (12), the linear aerodynamic force loading is expressed in terms of three force 

distribution functions, c x , c x , and c x D f1 f2 () () ( ) and the base pressure coefficient, CDbp, plus a 

body-fixed force associated with possible bent fins. Because the lateral motion of the missile is 

quite small, 1 1e q q   ≅ − and the aerodynamic damping force terms in the aerodynamic loading on 

the aerodynamic structure can be combined. This aerodynamic loading in nonrotating elastic 

coordinates is 

( ) x

1 D

dF

gc x dx = − , (19) 

() () ( ) ( )

() () ( ) ( )

i

f1 1 BF E 1 y z

1 i

f 2 1 BF

dF dF c x q x e xq d V

i g dx dx c x 2q i x e d V

φ

φ

   −Γ−Γ + δ −    + =−  

+ − Γ − φΓ  

 

  

, (20) 

and 

F gC xbp 1 Dpb = − . (21) 

The total aerodynamic force acting on the aerodynamic structure is given by the integrals of 

equations 19 and 20 and by adding the base drag of equation 21 to the axial force, 

( ) 3

0

x

x 1 D 1 D Dbp

x

F g C g c x dx C

 

=− =− +       ∫ , (22) 

and 

( ) ( ) ( ) ( ) i F g cq c qd V C e J t J t d V 1 1 1 2 1 NBF 1 2

φ =− + − − −       , (23) 

where various functions are defined in appendices A and B. 

The primary components of drag are head drag and base pressure drag. The third component is 

skin friction drag that is ~15% of the total drag and will be neglected in this report to simplify 

the FEM calculations. 

Similarly, the total aerodynamic moment about the rod’s center can be computed from the 

transverse aerodynamic force and a small axial force contribution: 
7

( ) ( ) () ()( ) ()

y z

i

1 3 1 4 1 MBF 3 4 5

M M iM

i gd cq c qd V C e J t J t d V J t φ

= +

=− + − − − −       . (24)

The motion of the central disk is described by the variables eee x , y , z , , , and φ ψ θ . The motion 

of any other section on the aerodynamic structure located at x with thickness dx is the sum of this 

motion plus its motion relative to the central disk. The work done on any section caused by 

motion of the central disk is 

( ) cd 1x e 1y e 1z e x

2x 2y 1ey 2z 1ez

dW dx = dQ dx + dQ dy + dQ dz

 + dQ d + dQ dq + dQ dq φ , (25)

where the sectional generalized forces are defined in appendix B. 

The total generalized forces can be obtained by integrating the sectional generalized forces and 

adding the work done by the base pressure drag. These are also given in appendix C. 

The linearized Lagrangian differential equation for e x and φ gives the usual drag and spin 

equation, 

mV mx g C ≅ =− e 1D   , (26) 

and 

x x I M φ =  . (27) 

The Lagrangian differential equation for e z can be multiplied by i and added to the Lagrangian 

differential equation for ye to yield a single second-order differential equation in complex 

variables. Next, the Lagrangian differential equation for θ can be multiplied by i and subtracted 

from the Lagrangian differential equation for ψ , 

m V q q Vq d q x d F ( ) 1 2 1 c 2c   + + +δ + =       (28) 

and 

( ) 22 2

t0 c 2 x 2 6 8 c c c I md x q i I q md J J x iM x Fd + − φ + + + δ =− −          , (29) 

where ( ) 2

1

x

c 1

x

x 1 L x dx = ρ ∫ , ( ) 2

1

x

c E1

x

δ = δρ 1 L dx ∫ , and 2 1e q q =  . 

These differential equations for c x 0 = are the same as those derived by Newtonian mechanics in 

reference (12). Equations 28 and 29 contain eight integrals of E δ and Γ . Three of these are 

integrals of dynamics properties( ) c68 δ ,J ,J over the rod (x ,x 1 2 ) and five are integrals over 

aerodynamic properties ( ) 12 3 45 J ,J ,J ,J ,J over the aerodynamic structure (x ,x 0 3 ). 
8

For constant spin and velocity, a rigid unbent missile with its center of mass at the rod center, 

equations 28 and 29 predict a simple epicyclic angular motion: 

11 2 2 ti ti

1 10 20 q Ke Ke λ + φ λ + φ = + , (30) 

where ( ) ( ) 2

m x t 1 t3 x t φ =φ ± − + φ I 2I g d I c I 2I   . 

For zero spin, φ = −φ = ω 1 2R

  , where ω = R 1 t3 ( ) gd I c . 

4. FEM 

The rod is assumed to be represented by the sum of an inelastic bent component rotating with the 

missile and an elastic deformation, 

( ) ( ) ( ) i

E EB E 1 2 x,t x e x,t ; x x x φ δ =δ +δ ≤ ≤  , (31) 

and 

δ =δ +δ ≤ ≤ b ( ) x,t x x,t ; x x x EB b 1 2 ( ) ( )  , (32) 

where ( ) EB ( )

EB

d 0

0 0

dx

δ δ= = . 

Because the aerodynamic structure is rigidly attached to the rod, 

δ =δ + − Γ ≤ ≤ E E1 1 1 0 1 () ( ) x, t x , t x x x , t ; x x x ( ) ( ) , (33) 

and 

δ =δ + − Γ ≤ ≤ E E2 2 2 2 3 (x, t x , t x x x , t ; x x x ) () ( ) ( ) . (34) 

The motion of the elastic component of the rod is controlled by the elasticity of the rod and the 

aerodynamic force acting on it. 

FEM is a very powerful method for calculating the time history of the elastic flexing motion. 

The rod is divided into nj elements of length L Ln e j = . The shape of the j-th element is given 

by a linear combination of third-order Hermitian polynomials (see appendix C). 

( ) () ( )

4

by bpy p

1

δ = x,t q t N z ∑ ˆ  , (35) 

and 
9

( ) () ()

4

bz bpz p

1

δ = x,t q t N z ∑ˆ  , (36) 

where x L z z ; z x L j 1 0 z 1 = + = +− ≤ ≤ e j j 1e ( ) . 

The coefficients of the polynomials are functions of time and are called connectors. The first 

two are the deflection and slope of the left end of the element and the third and fourth are the 

deflection and slope of the right end. To ensure continuity in deflection and slope at junction 

points, the corresponding pairs of connectors are equal. We will consider only an odd number of 

elements and require that the connecters of the central element satisfy equation 2. 

b3 b1 e b2 q 5q L q ˆˆ ˆ = + , (37) 

and 

1

b4 e b1 b2 q 24L q 5q ˆ ˆˆ − = + , (38) 

where bq bqy bqz q q iq ˆˆ ˆ = + . 

For j n elements, there are j 2n independent complex connectors. It is convenient to let the index 

for the connectors run from 3 to nt, where nt = 2nj + 2. The usual FEM procedure calculates the 

elastic parts of the integrals in equations 28 and 29 by first obtaining integrals over each element. 

These are linear functions of that element’s four connector functions. Due to equations 37 and 

38, the central element has only two independent connector functions, and the next adjacent 

element connector functions are related to them. Integrals for these elements are specially 

computed. The desired elastic integrals are sums of these subintegrals and are linear 

combinations of the j 2n complex connector functions. The three integrals can be written as 

t n

1 i

c 1n n cB

3

L aq e − φ δ = +δ ∑ , (39) 

t n

1 i

6 2n n 6B

3

J L aq Je − φ = + ∑ , (40) 

and 

( ) t n

i

8 d 2n n n 8B1

3

J a b q 2i q i J e φ = − φ −φ ∑    , (41) 

where i

n bn q qe φ = . 

The five integrals have contributions from the aerodynamic structure extensions as well as from 

each element, 
10

() ( ) ( )( )

( )

t n

1 2 1n a1n n 1n a1n n

3

i

1B 2B

J +J d V = f +f q + g +g q d V

+ J +i d V J e , φ

   

  φ  

 ∑ 

 (42)

() ( ) ( )( )

( )

t n

3 4 2n a2n n 2n a2n n

3

i

3B 4B

J +J d V = f +f q + g +g q d V

 + J +i d V J e , φ

   

  φ  

 ∑ 

 (43) 

and 

( ) t n

1 i

5 D 1n an n 5B

3

J Ca L h q J e − φ =− + + ∑ . (44) 

Expressions for cB kB an δ , J , h and all FEM coefficients in equations 39–44 for 1, 3, 5, and 7 

elements are given in appendices A, D, and E. 

Equations 39–44 can be used with equations 28 and 29 to write these two complex differential 

equations in a standard format: 

( )( ) ( ) t n

* *

mn n mn mn n mn mn n m

n 1

R q S i S q T i T q t exp i =

  + +φ + +φ = φ ∑      . (45) 

The coefficients for m = 1, 2 are given in appendix F. 

5. Flexing Motion 

In order to derive the equations for flexing motion, the kinetic energy given by equation 15 must 

be expressed in terms of the rigid-body parameters, plus the connector functions. The usual 

FEM process is used to express T2j in terms of two matrices, two vectors, and two constants. 

t t t nn n

i 2

21 mn my ny mz zn Bm m xB1

33 3

T k q q q q 2Re i e k q I L   − φ = + − φ +φ         ∑∑ ∑      , (46) 

and 

( ) ( ) tt t nn n

i 2

22 d mn m m n Bm m m xB2

33 3

T 2 a L Re b 2iq q q 2ie b q 2i q I L   − φ = φ −φ + + φ +φ    

   ∑∑ ∑   , (47) 

where xB1 xB2 I , I are defined in appendix A and mn Bn mn Bn k ,k ,b ,b are defined in Appendix G. 

The potential energy stored by an elastic deformation is 
11

( ) ( )

2

1

2 2 x 2 2

by bz

2 2

x

P.E. 1 2 EI d dx

x x

    ∂ δ   ∂ δ

= +         ∂ ∂       

∫   , (48) 

where E x I x EI x () () () = ρ 00 3 . 

E and 0 0I are values of Young’s modulus and the area moment of inertia at the rod’s center. 

ρ3 ( ) x gives the variation of their product along the rod. For a homogeneous rod with constant 

diameter, the product is E0I0. The potential energy integral can be replaced by a sum of integrals 

over each of the j n elements. After integrating over each element, the results can be summed to 

yield a linear combination of the j 4n real connectors, 

( ) ( ) t t n n

2 2

0 bmy bny bmz bnz mn

m 3n 3

V md 2L q q q q c = ==ω + ∑∑ , (49) 

where 2 3 ω =0 00 E I md , mn c are defined in appendix G. 

The work done on the rod by this elastic force can be expressed in terms of generalized force 

terms, 

( ) t n

2 2

E 0 Emy bmy Emz bmz

m 3

dW md L Q dq Q dq =

=ω +   ∑  , (50) 

where t n

Emy Emz mn bn

n 3

Q iQ c q =+ =−∑ . 

The Kelvin-Voight elastic damping force (16), which is proportional to the time derivative of the 

elastic shear force, has generalized force terms: 

( ) t n

2 2

D 0 Dmy bmy Dmz bmz

m 3

dW md L Q dq Q dq =

=ω +   ∑  , (51) 

where ) t n

Dmy Dmz 1 mn bn

n 3

Q iQ 2k c q ˆ

=

+ =− ω 

 ∑  , ( )2

1 0 ω = ω 4.730 L . 

The scale factor 1

1 2 − ω is selected so that ˆ

k 1 = corresponds to critical damping of the first elastic 

mode. 

The external aerodynamic force is divided between the force acting on the rod and the force 

acting on the aerodynamic structure extensions at the ends of the rod. The total work done on 

the missile is the sum of the work done by these forces. 
12

( )

j

j

t

n

A Aj an

j 1

n

1 Amy bmy Amz bmz

m 3

dW dW dW

g d Q dq Q dq ,

=

=

= +    

= +    

∑

∑

 (52) 

where ( )( ) ( ) t n

-i

Amy Amz mn amn n mn amn n Bm aBm Bm aBm

n=3

Q + iQ = f + f q + g + g q e + f + f + i g + g φ   φ ∑   

and mn mn amn amn Bm Bm aBm aBm f ,g ,f ,g ,f ,g ,f ,g are defined in appendix G. 

Equations 50–52, in conjunction with the kinetic energy defined by equations 15, 46, and 47, can 

be used to derive the j 4n Lagrangian differential equations for the bmy bmz q ,q ’s for 

m 3, 4,...2n 2 = +j . The equation for bmz q should be multiplied by i and added to the 

corresponding equation for bmy q to yield j 2n complex differential equations for the complex 

variables bm. q . If the bm q are replaced by m q in each equation, the result has the standard form 

given by equation 45, where the 14nj complex coefficients for m 3, 4...2n 2 = j + are given in 

appendix F. 

6. Special Solutions 

A rigid symmetric finned missile flying with constant spin has two natural frequencies: φ1R

 and 

φ2R

 , where φ ≅ −φ 1R 2R  . For zero spin, each of the flexure frequencies would give rise to two 

coning frequencies, ±ωK . The frequencies present in the motion of an elastic projectile would 

form an infinite sequence, where the first two would be related to φ1R

 and φ2R

 , while the later 

ones would evolve from ±ωK , i.e., 2K 1 K 2K 2 K ; + + φ ≅ ω φ ≅ −ω   . The odd numbered modes have 

positive frequencies and are called positive modes, while the even numbered modes are called 

negative modes. For nonzero spin, the pairs of frequencies would no longer be equal in 

magnitude. 

Transient harmonic solutions of the homogeneous constant spin version of equation 45 have the 

form 

( ) iAt

m mk k j q q e A A k 1, 2, 3,...2 2n 1 = == + . (53) 

After substituting equation 53 into equation 45, a set of linear homogeneous algebraic equations 

is derived, which is specified by an t n × t n matrix, 
13

( ) 2 **

mn mn mn mn mn mn u A R A S i S T i T = + +φ + +φ   . (54) 

For nj elements, there are 2nj + 1 pairs of positive and negative frequencies. The lowest 

frequency pair is related to the two aerodynamic frequencies and the other 2nj pairs are 

approximations of first 2nj elastic frequencies of the infinite set of frequencies for the elastic rod. 

The paired frequencies are equal in magnitude when spin is zero but the positive frequency has a 

larger magnitude when spin is positive. For an elastic beam considered by Geradin and Rixen 

(14), the first nj approximations are close to the correct value. This accuracy is determined by 

the ability of nj elements to describe the corresponding mode shape correctly. Geradin and 

Rixen (14) also observe that the approximations are upper bounds of correct value. 

For constant spin, the trim motion in response to the spinning bent missile forces can be obtained 

by assuming a solution of the form 

i t

n nT q qe φ = 

. (55) 

The response parameters, nT q , vary with particular values of the constant spin and are solutions 

of a set of t n linear inhomogeneous algebraic equations, 

t n

mn nT m

n 1

Bq t

=

∑ = , (56) 

where ( )( ) 2* * B R S i S T T mn mn mn mn mn mn = −φ + + φ + +   . 

In reference (12), both these special solutions were computed from iterative solutions of a 

second-order differential equation specified by special boundary conditions. The FEM approach 

obtains the same results and more by much simpler matrix operations. 

7. Quadratic Roll Equation 

The linear roll equation (equation 26) was derived from a kinetic energy function, which 

neglected cubic terms in 1y 1z 2y 2z bny bnz q , q , q , q , q , q and . In order to derive the quadratic roll 

equation, cubic terms involving φ and φ

 must be retained. The resulting roll equation is the same 

as that derived in reference (12) from Newtonian mechanics, 

{ ( ) } 2

x 7 26 28 c x I Re md J iq J iq J iF d M φ+ − + − δ =    , (57) 

where J7(t) is defined in appendix F. 
14

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

of the central disk. 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, 

( ) p ss ( ) ( ) linear C C p dV A A = φ− . (58) 

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 center of the central disk. If we retain only the 

dominant cf1 term in equation 20, the quadratic roll moment has the form 

( ) ( ) ( )

( ) { } ( ) ( ) ( )

x quadratic z Ey y Ez

i

1 f1 1 BF E 1 E

 dM dF dF 

g d c Re i q e xq d V dx φ

=   δ− δ  

= −Γ−Γ + δ − δ       . (59)

The total aerodynamic roll moment acting on the projectile, therefore, 

( ) ( ) () { } 1 M g d C Re Q t ig F x 1 A 1c linear

− = + −δ     A , (60) 

where ( ) ( ) ( ) ( ) ( ) 3

0

x

i

A f1 1 BF E 1 E c

x

Q t i c q e xq d V dx φ = −Γ−Γ + δ − δ −δ   ∫     . 

The nonlinear roll moment from equation 60 can be placed in the spin equation 57 to yield: 

{ ( ) } ( ) 2

X D 26 28 1 A 1 linear I Re md Q iq J iq J g d Q g d C φ+ − + − = A   , (61) 

where the quadratic terms X D I .Q are defined in appendix F. 

For trim motion, { } i t Re Q 0, q i q e D 2 2T

φ φ= = = φ     and equation 61 become a simple equality of 

two functions of p. 

f f 2 1 (φ) = φ( )   , (62) 

where 1 ss f p =φ− , { ( )( )}( )

–1

2 AT 2T 1 6T 8T lp f = -R Q – q md g J + iJ C d V φ

 , and 

( )( )( ) 0

x3

AT f1 1T T BF ET 1T T

x

Q =i c q –   Γ – Γ + i δ – xq d V φ δ dx ∫    . 

ET T δ Γ, are computed from the solution to equation 56 and J6T, J8T are J6, J8 evaluated for 

ET T δ Γ, . For moderate spin QAT is the dominant part of f2. Equilibrium values of spin are 

determined by the intersection of these two curves. Lockin occurs at stable equilibrium spin. 
15

Spin lockin can only occur when the missile has some rigid asymmetry such as that described by 

EB δ or ΓBF . For a rigid missile ET δ and ΓT are EB δ and ΓB . q1T is a function of spin which has 

a large resonant amplitude when spin is near ωR . For an elastic missile 1T ET T q, , δ Γ are all 

functions of spin and have large resonant amplitudes when spin is near φ2k 1−  or near φ2k

 for 

negative spin. 

8. Numerical Results 

In reference (12), calculations were made for a 20-cal. fin-stabilized rod, flying at 6000 ft/s. The 

finned missile has a 1-cal. nose extension and a 1-cal. fin extension (figure 3). The mass and 

aerodynamic parameters for the finned missile are given in appendix H. The bent rod will be 

described by a pair of quartic curves. 

2 4

EB 11 21

2 4

12 22

δ = d x + d x 10 x 0

= d x + d x 0 x 10.

− ≤ ≤

≤ ≤ (63) 

d 10d 10d

2d 2d

d

d/2

2d

Figure 3. Sketch of finned missile. 

The rear of the rod is undeformed ( 11 21 dd0 = = ), and the values of 12 22 d , d are given in 

appendix G. Two values of ΓBF will be considered. These are unbent fins and the rear 1 cal. of 

fins bent uniformly to an angle of 0.02 radians. 

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

missile aerodynamic frequency, σ≡ω ω1 R . Calculations of φ1

 vs. σ have been made for the 1-, 

3-, 5-, and 7-element rod. The results for all but the 1-element rod are practically identical, and 

the curve for a 3-element rod is given in figure 4. At σ = 5, the aerodynamic frequency is 60% 

of the rigid missile aerodynamic frequency; and even at σ =10 , it is 10% less than the rigid 

missile aerodynamic frequency.
16

Figure 4. 1 R φ ω/  vs. σ for finned missile. 

The first six positive frequencies and their damping rates for σ = 5 have been calculated for a 

1-, 3-, 5-, and 7-element rod. These results, together with calculations from the theory of 

reference (12), are tabulated in table 1. We identify frequencies which differ from the reference 

(12) value by >10% as poor values and mark them with x’s. Thus the 1-element rod yields one 

moderately good result while the 3-element and 5-element rods yield three and five good results, 

respectively. For this elastic problem, nj – 1, elastic frequencies are well calculated. For zero 

spin, the negative frequencies are the negatives of the positive frequencies. 

Because the 3-element rod predicts the lower three frequencies quite well, all of the calculations 

given in this report are based on a 3-element rod. Most of these have been repeated for a 

5-element rod, and almost identical results have been obtained. 

In figure 5, 2 f for BF σ= Γ = 5, 0 is plotted vs. φ ωR  for spin near the first elastic frequency. 

Large resonance values occur at R φω ≅ 5.1  and an 1f line for ss R p 6.0 = ω is also shown. This 

1f line has three intersections with the 2 f curve, which are values of equilibrium spin. The 

intersection near ss R p ω is denoted as ss R p ω , while the other two are 1e R 2e R p , p , ω ω and all 

their numerical values are given in table 2. 

1 R φ ω/ 
17

Table 1. Transient frequencies and damping rates. 

σ =5 =0 φ

Code k φ



k R ω k R λ ω

1 element 1 0.645 –0.0530 

3 element 1 0.612 –0.0528 

5 element 1 0.613 –0.0528 

7 element 1 0.613 –0.0528 

reference (12) 1 0.614 –0.0529 

1 element 3 6.146× –0.0855 

3 element 3 5.194 –0.0724 

5 element 3 5.181 –0.0718 

7 element 3 5.180 –0.0717 

reference (12) 3 5.184 –0.0717 

1 element 5 20.31× –0.0644 

3 element 5 13.81 –0.0518 

5 element 5 13.79 –0.0523 

7 element 5 13.76 –0.0518 

reference (12) 5 13.74 –0.0506 

1 element 7 — — 

3 element 7 30.00× –0.1018 

5 element 7 26.97 –0.0914 

7 element 7 26.79 –0.0881 

reference (12) 7 26.69 –0.0825 

1 element 9 — — 

3 element 9 53.37× –0.2060 

5 element 9 44.28 –0.1389 

7 element 9 44.18 –0.1462 

reference (12) 9 43.84 –0.1427 

1 element 11 — — 

3 element 11 100.44× –0.4708 

5 element 11 72.19× –0.1309 

7 element 11 65.88 –0.1256 

reference (12) 11 64.49 –0.1202 

In figure 6, spin histories are plotted for all initial conditions equal to zero except 0 R φ= ω 0, 5.4  . 

For low values of initial spin, lockin occurs at the first elastic frequency, 1e p , and large strains 

result. For larger values of initial spin, lockin occurs at sse p . This behavior was predicted by the 

simple three-body theory of reference (11). 

It is very unlikely that sufficient fin damage occurs to produce a ss p in excess of five times ωR . 

Because it is much more likely that a ss p near ωR can be produced, the observed inelastic 

deformation is probably caused by lockin at the aerodynamic frequency, and this case is 

considered in detail in the following section. 
18

Figure 5. fj vs. R φ

 /ω for ΓBF ss R = 0, σ = 5, p = 6ω . 

Table 2. Equilibrium spin rates. 

σ = 5

ΓBF 0 0 0.02 

ss R p ω 6.000 0.800 0.400 

sse R p ω 5.944 0.788 0.429 

1e R p ω 5.097 0.590 0.564 

ss R p ω 5.393 0.647 0.627 

In figure 7, 2 f is plotted for spins near the aerodynamic frequency, and 1f is shown for 

ss R p 0.8 = ω . The three intersection points are given in table 2. Spin histories are plotted in 

figure 8 for zero initial conditions except for 1

0 0, 3 s− ψ= −  . Here, we see that sse p spin-yaw 

lockin occurs for zero initial angular velocity, and 1e p spin-yaw lockin for an initial yaw rate of 

–3 s–1. The angular motion in body-fixed coordinates is plotted in figure 9. For lockin at sse p , 

the motion is smaller than 0.09 rad, while for lockin at 1e p , it is as large as 0.15 rad. The motion 

of the forward end of the rod in body-fixed coordinates is given in figure 10. Motion for sse p 

lockin is <0.17 (0.7 in), and after 1 s, it is never >0.09 (0.4 in). For 1e p lockin, it exceeds 0.28 

(1.2 in). 

fj
19

Figure 6. R φ ω/  vs. time for ΓBF ss R 0 R = σ= = φ = 0, 5, p 6.0ω , /ω 0, 5.4  . 

Figure 7. fj vs. R φ ω/  for ΓBF ss R = 0, σ = 5, p = 0.8ω . 

fj

R φ

 /ω

psse

p2e

p1e

R φ

 /ω

time(s)
20

Figure 8. R φ

 /ω vs. time for ΓBF ss R 0 = 0, σ = 5, p = 0.8ω , ψ = 0, –3 s-1. 

Figure 9. Angular motion (qb1y vs. qb1z) for ΓBF ss R 0 = 0, 5, p 0.8 σ= = = − ω , ψ 0, 3 s–1. 

psse

p2e 

p1e

R φ

 /ω

 time(s)

- - - 0 ψ 0 =

___ 3 ψ 0 = −

qb1y

• = 1 s 

o = 0 s

qb1z

- - - - 0 ψ 0 =

____ 3 ψ 0 = −
21

Figure 10. Rod forward tip motion (qb7y vs. qb7z) for ΓBF ss R 0 = 0, 5, p 0.8 σ= = = − ω , ψ 0, 3 s–1. 

The maximum strain on the rod is located on the rod’s surface at its center. In terms of 

3-element parameters, the maximum strain is 

( ) ( ) 2

E 2

M e 5 e6 2

0

1 2 L 12q 2L q

x

− ∂ δ

ε= = +

∂ . (64) 

The time history of the maximum strain is plotted in figure 11 for -1 ψ 0 = 0, – 3 s . For most 

metals, yield occurs for maximum strains in excess of 0.0015, and we see that yield does occur 

for resonance lockin. 

The two regions of the initial angular rate plane, i.e., the ψ0 0 − θ  plane, which induces either 

lockin, can be determined by a number of trial and error calculations, and the boundary curve 

between these regions is shown in figure 12. Initial conditions outside this curve will cause 

resonance lockin while conditions inside it induce sse p lockin. 

For BF Γ = 0 , the first resonance peak of the 2 f curve is negative, and ss p must be greater than 

the aerodynamic frequency for three intersections to occur. 

As is shown in figure 13, bent fins corresponding to BF Γ = − ≤ ≤− 0.02 11 x 10 will produce a 

positive peak in the 2 f curve and a ss p less than the aerodynamic frequency can produce 

resonance lockin. The plots corresponding to figures 8–12 are given by figures 13–19.

- - - 0 ψ 0 =

___ 3 ψ 0 = −

•= 1 s 

o = 0 s

qb7z
22

Figure 11. Maximum strain ( Mε ) for ΓBF ss R 0 = 0, 5, p 0.8 σ= = = − ω , ψ 0, 3 s-1. 

9. Summary 

The application of FEM to the calculation of the motion of a pitching, yawing, and flexing 

missile has resulted in the construction of a family of linear differential equations. 

Computer codes have been developed for 1, 3, 5, and 7 elements and used to obtain the first four 

elastic frequencies. The 3-element code has given excellent values of the first two elastic 

frequencies. The 5-element and 7-element codes provide equally good values for the first four 

elastic frequencies. 

Integration of these differential equations has demonstrated resonant spin lockin at the 

aerodynamic frequency and the first elastic frequency. Lockin can occur near the design steady 

state spin or near resonance with a transient frequency. Calculations have identified regions of a 

yaw rate-pitch rate plane that determine which lockin will occur. 

- - - 0 ψ 0 =

___ 3 ψ 0 = −

εM

time(s)
23

Figure 12. ψ0  vs. θ0

 showing regions of sse 1e p , p lockin BF ss R Γ = σ= = ω 0, 5, p 0.8 . 

Figure 13. fj vs. R φ ω/  for BF ss R Γ = σ= = ω 0.02, 5, p 0.4 .

fj

R φ

 /ω

Region of p1e Lockin

θ0

 (s-1)

Region of psse Lockin

Region of p1e Lockin

ψ0  (s-1)
24

Figure 14. R φ ω/  vs. time for ΓBF ss R 0 = 0.02, σ = 5, p = 0.4ω , θ = 0, 5  s-1. 

Figure 15. Angular motion (qb1y vs. qb1z) for ΓBF ss R 0 = 0.02, σ = 5, p = 0.4ω , θ = 0, 5  s-1.

R φ

 /ω

- - - 0 θ0 = 

___ 5 θ0 = 

time(s)

p2e

p1e

psse

• = 1 s 

o = 0 s

qb7y

qb7z
25

Figure 16. Rod forward tip motion (qb7y vs. qb7z) for BF ss R 0 Γ = σ= = ω θ = 0.02, 5, p 0.4 , 0  s

-1. 

Figure 17. Rod forward tip motion (qb7y vs. qb7z) for ΓBF ss R 0 = 0.02, σ = 5, p = 0.4ω , θ = 5  s-1.

qb1y

qb1z

• = 1 s 

o = 0 s

- - - 0 θ0 = 

___ 5 θ0 = 

- - - θ0 = 0 

___ 5 θ0 = 

εM

time (s)
26

Figure 18. Maximum strain ( ) εM for ΓBF ss R 0 = 0.02, σ = 5, p = 0.4ω , θ = 0, 5  s-1. 

Figure 19. ψ0  vs. θ0

 showing regions of sse 2e p , p lockin BF ss R Γ = σ= = ω 0.02, 5, p 0.4 . 

• = 1 s 

o = 0 s

qb7y

qb7z

Region of p2e Lockin

Region of psse Lockin

Region of p2e Lockin

ψ0  (s-1)

θ0

 (s-1)
27

10. References 

1. Mikhail, A. G. In-Flight Flexure and Spin Lock-in for Anti-Tank Kinetic Energy Projectiles. 

Journal of Spacecraft and Rockets 1996, 33, 657–664. 

2. Guidos, B. J.; Garner, J. M.; Newill, J. F.; Livecchia, C. D. Measured In-Flight Rod Flexure 

of a 120-mm M829E3 Kinetic Energy (KE) Projectile Steel Model; ARL-TR-2820; U.S. 

Army Research Laboratory: Aberdeen Proving Ground, MD, 2002. 

3. Peppitone, T. R.; Jacobson, J. D. Resonant Behavior of a Symmetric Missile Having Roll 

Orientation-Dependent Aerodynamics. Journal of Guidance and Control 1978, 1, 335–339. 

4. Grover, L. S. Effects on Roll Rate of Mass and Aerodynamic Asymmetries On Ballistic 

Re-Entry Bodies. Journal of Spacecraft and Rockets 1965, 2, 220–225. 

5. Price, D. A., Jr. Sources, Mechanisms, and Control of Roll Resonance Phenomena for 

Sounding Rockets. Journal of Spacecraft and Rockets 1967, 4, 1516–1525. 

6. Price, D. A., Jr.; Ericsson, L. E. A New Treatment of Roll-Pitch Coupling for Ballistic ReEntry Bodies. AIAA Journal 1970, 80, 1608–1615. 

7. Murphy, C. H. Some Special Cases of Spin-Yaw Lock-In. Journal of Guidance, Control, 

and Dynamics 1989, 12, 771–776. 

8. Platus, D. H. Aeroelastic Stability of Slender, Spinning Missiles. Journal of Guidance, 

Control, and Dynamics 1992, 15, 144–151. 

9. Legner, H. H.; Lo, E. Y.; Reinecke, W. G. On the Trajectory of Hypersonic Projectiles 

Undergoing Geometry Changes. AIAA Aerospace Sciences Meeting, Reno, NV, 10–13 

January 1994; AIAA 94-0719. 

10. Heddadj, S.; Cayzac, R.; Renard, J. Aeroelasticity of High L/D Supersonic Bodies: 

Theoretical and Numerical Approach. AIAA Aerospace Sciences Meeting, Reno, NV, 10–13 

January 2000; AIAA 2000-0390. 

11. Murphy, C. H.; Mermagen, W. H. Flight Mechanics of an Elastic Symmetric Projectile. 

Journal of Guidance, Control, and Dynamics 2001, 24, 1125–1132. 

12. Murphy, C. H.; Mermagen, W. H. Flight Motion of a Continuously Elastic Finned Missile. 

Journal of Guidance, Control, and Dynamics 2003, 26, 89–98. 

13. Murphy, C. H.; Mermagen, W. H. Flight Motion of a Continuously Elastic Finned Missile; 

ARL-TR-2754; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, 2002. 
28

14. Geradin, M.; Rixen, D. Mechanical Vibrations: Theory and Applications to Structural 

Dynamics; John Wiley & Sons: New York, 1994. 

15. Murphy, C. H. Free Flight Motion of Symmetric Missiles; BRL-TR-1216; U.S. Army 

Ballistics Research Laboratory: Aberdeen Proving Ground, MD, 1963. 

16. DeSilver, C. W. Vibration Fundamentals and Practice; CRC Press: New York, 2000; 

p 351. 
29

Appendix A. Integrals 

A.1 Aerodynamic Terms 

( )

3 3

0 0

3 3

0 0

3 3

0 0

x x

1 N f1 3 M f1

x x

x x

2 Nq N f 3 4 Mq M f 3

x x

x x

NBF FB BF MBF BF BF

x x

f 3 f 2 f1 BF f1 f 2

c C c dx c C c xdx

c C C c dx c C C c xdx

C c dx C c xdx

c 2c xc c c d V c .

α α

α α

== ==

=+= =+ =

=Γ =Γ

= − = +φ

∫ ∫

∫ ∫

∫ ∫

 

 (A-1) 

A.2 Boundary Terms 

( )

( )

( )

( )

1

0

1

0

1

0

1

0

1

0

1

0

1

0

x

1 f1

x

x

3 1 f1

x

x

5 f2

x

x

7 1 f2

x

x

2

9 1 f1

x

11 5 3 1 1

13 7 9 1 3

x

1D D Dbp

x

x

3D 1 D

x

I c dx

I x x c dx

I c dx

I x x c dx

I x x c dx

I 2I I x I

I 2I I x I

I c dx C

I x x c dx

=

= −

=

= −

= −

= −−

= −−

= +

= −

∫

∫

∫

∫

∫

∫

∫ 

( )

( )

( )

( )

3

2

3

2

3

2

3

2

3

2

3

2

3

2

x

2 f1

x

x

4 2 f1

x

x

6 f2

x

x

8 2 f2

x

x

2

10 2 f1

x

12 6 4 2 2

14 8 10 2 4

x

2D D

x

x

4D 2 D

x

I c dx

I x x c dx

I c dx

I x x c dx

I x x c dx

I 2I I x I

I 2I I x I

I c dx

I x x c dx

=

= −

=

= −

= −

= −−

= −−

=

= −

∫

∫

∫

∫

∫

∫

∫

1

0

x

1BF BF BF

x

I c dx = Γ ∫ ( )

1

0

x

3BF 1 BF BF

x

I x x c dx =− Γ ∫

1

0

x

5BF f 2 BF

x

I c dx = Γ ∫ ( )

1

0

x

7BF 1 f 2 BF

x

I x x c dx =− Γ ∫ . (A-2) 
30

A.3 Bent Missile Terms 

( )

( )

( )

2

0

3

0

3

0

x

1B f1 B BF

x

x

2B f 2 B BF f1 EB

x

x

3B f1 B BF

x

J c dx

J c c dx

J c xdx

= Γ +Γ

= Γ +Γ − δ    

= Γ +Γ

∫

∫

∫

( )

( ) ()

( )

3

0

3

0

2

1

x

4B f 2 B BF f1 EB

x

x

5B D EB cB EB 1 cB Dbp

x

x

6B B 1

x

J c c xdx

J c dx x c

J 1 L x dx

= Γ +Γ − δ    

= δ −δ + δ −δ    

= δρ

∫

∫

∫

( ) 2 2

1 1

x x

cB B 1 8B1 d B 2

x x

δ = δ ρ =− Γ ρ 1 L dx J a dx ∫ ∫

( ) ( ) 3

0

x

xB1 1 EB EB 2 d B B

x

I 1 L a L dx = ρδ δ + ρ Γ Γ   ∫  

3

0

x

xB2 d 2 B B

x

I 6a dx = − ρ Γ Γ  ∫   . (A-3) 
31

Appendix B. Functions 

( )

( )

() ( )

( )

2

1

3

0

3

0

3

0

x

c E1

x

x

1 f1

x

x

2 f2 f1 E

x

x

3 f1

x

1 L dx

J t c dx

J t c c dx

J t c xdx

δ = δρ

= Γ

= Γ− δ

= Γ

∫

∫

∫

∫

() ( )

() ( ) ( )

( ) ( )

( ) ( )

3

0

3

0

2

1

2

1

x

4 f2 f1 E

x

x

5 D E c E 1 c Dbp

x

x

6 E1

x

x

8d 2

x

J t c c xdx

J t c dx x c

J t 1 L x dx

J t a 2i dx.

= Γ− δ

= δ −δ + δ −δ    

= δρ

= Γ − φΓ ρ

∫

∫

∫

∫   (B-1)
32

INTENTIONALLY LEFT BLANK. 
33

Appendix C. Generalized Forces and Hermitian Polynomials 

( )

( )

x

1x 1 D

y z x

1y 1z

x

2x

y z xx

2y 2z E

dF dQ dx g c x dx dx

dF dF dF dQ idQ i i dx dx dx dx

dM dQ dx dx

dF dF dF dM dQ idQ x i dx dx dx dx dx

= =−

  + = + + −ψ + θ    

=

    + = + −δ −θ        

( )

( ) 2z

1x x

1y 1z y z x

2x x

2y y z x c x

Q F

Q iQ F iF i F

Q M

Q iQ i M iM iF M

=

+ = + + −ψ + θ

=

+ =− + − δ −θ

( )

( )

( )

2 3

1

2

2 e

2

3

2

4 e

N 1 3z 2z

N Lz1 z

N z 3 2z

N Lz z 1

=− +

= −

= −

= −

( )

( )

( )

( )

/ 2

1

/ 2

2 e

/ 2

3

/ 2

4 e

N 6z z

N L 1 4z 3z

N 6z z

N L 3z 2z .

= −

= −+

= −

= − (C-1)
34

INTENTIONALLY LEFT BLANK. 
35

Appendix D. Connector Coefficients for Equations (39–41) 

D.1 Rod Parameters 

( ) () ( )

( ) () ( )

( ) ()

1

1

qj e 1 q e j

0

1

2

qj e 1 q j 1 e

0

1

2 '

qj 2 q

0

a L x N z dz x L z z ˆ

a L x x N z dz z x L j 1 ˆ

ˆ

b x N z dz q 1, 2, 3, 4.

=ρ = +

= ρ = +−

=ρ =

∫

∫

∫ (D-1)

For central element j n 1

r

2

+ =

1 1 1 11 1 1 1 1

1c 1r 3r e 4r 2c 2r e 3r 4r a a 5a 24L a a a L a 5a ˆˆ ˆ ˆ ˆˆ ˆ ˆ − =+ + =+ +

For adjacent element j n 3

r

2

+ =

1 1 11 1 1 1

1a 1r e 2r 2a e 1r 2r

11 11

3a 3r 4a 4r

a 5a 24L a a L a 5a ˆ ˆ ˆ ˆ ˆˆ

a a a a, ˆˆ ˆˆ

− =+ = +

= =

222 2

qc qa qc qa ˆ ˆ a , a , b , and b ˆ ˆ are computed from 2 2qj qj ˆ a and b ˆ in same manner as previously shown. 

11 12 aa0 = =

t e

1

13 14 2c

1

1c

n4 LL

a a aˆ aˆ

= =

= =

t e

1 1 111 11 1 1 1

13 11 14 21 15 31 1c 1a 16 41 2c 2a 17 3a 18 4a

n 8 L L3

a aˆ a a a aaa a aaa a a a a ˆ ˆˆˆ ˆˆˆ ˆ ˆ

= =

= = =++ =+ + = =

() () ()

t e

1 1 11 11

13 11 14 21 15 31 12 16 41 22

1 11 1 1 1 1 1

17 32 1c 1a 18 42 2c 2a 19 3a 15

11 1 1

1 10 1 11 1 12 4a 25 35 45

n 12 L L 5

aa aa aaa aaa ˆ ˆ ˆˆ ˆˆ

a aaa a aaa a aa ˆˆˆ ˆˆˆ ˆˆ

a aa a a a a ˆˆ ˆ ˆ

= =

= = =+ =+

= ++ = ++ = +

=+ = =
36

() ()

() () () ()

1 1 11 11 11

13 11 14 21 15 31 12 16 41 22 17 32 13

1 1 111 1 1 1 1 1

18 42 23 19 33 1c 1a 43 2c 2a 3a 16 1 10 1 11

1 1 11 1 1

1 12 1 13 1 14 1 15 4a 26 36 17 46 27 aˆ

n 16 L L 7 t e

aa aa aaa aaa aaa ˆ ˆ ˆˆ ˆˆ ˆˆ

a aa a aaa a aaa a aa ˆˆ ˆˆˆ ˆˆˆ ˆˆ

a aa a a a aa a ˆˆ ˆ ˆˆ

= =

= = =+ =+ =+

= + = ++ = ++ = +

=+ =+ =+ = ( )

1 1

37 47 1 16 a aa ˆ ˆ =

2n 2n a and b are computed from 22 2 22 2

qj qc qa qj qc qa ˆˆ ˆ a , a , a , b , b and b ˆˆˆ , in the same manner as previously 

shown. 
37

Appendix E. Connector Coefficients for Equations (42–44) 

E.1 Rod Terms 

( ) () ( )

( ) () ( ) () ( )

( ) ()

( ) () ( ) ()

1

1 /

qj f1 q e j

0

1

1 /

qj f 2 q f1 q e j 1 e

0

1

2 /

qj f1 q

0

1

2 /

qj f 2 q f1 q e

0

ˆ

f c x N z dz x L z z

g c x N z c x N z L dz z x L j 1 ˆ

ˆ

f c x N z xdz q 1, 2, 3, 4

g c x N z c x N z L xdz, ˆ

= =+

= − = +−    

= =    

= −    

∫

∫

∫

∫ (E-1)

1n 2n 1n 2n f , f , g and g , are computed from 1 21 2

qj qj qj qj

ˆ ˆ f , f , g and g ˆ ˆ , in the same manner as in appendix D. 

E.2 Nonzero Aerodynamic Extension Terms 

() ()

t

1

a13 e 2 a14 1 2

1

a 23 e 18 a 24 17 18

1

a13 1 2 e 6 4 a14 5 3 2 e 6 4

1

a 23 17 18 e 16 a 24 15 18 e 16

1

a3 ID 2D e 4D a 4 3DE e 2D 4D

n 4

f 24L I f I 5I

f 24L I f I 5I

g I 5I 24L I I g I I I L 5 I I

g I 5I 24L I g I I L 5I

 h I 5I 24L I h I L I 5I

−

−

−

−

−

=

= =+

= =+

=− − + − = − − + −

=− − + = − +

=+ + = + +

( )

( )

( )

t

t

t t

t t

t t

t

a14 1 a1n 2

a 24 17 a 2n 18

a13 1 a14 5 3 2 a1n 6 4 a1 n 1

a 23 17 a 24 15 18 a1n 16 a1 n 1

a3 ID a 4 3D 2D an 4D an 1

n 4

fI f I

fI f I

g I g I I g I g I I

g I g I g I g I

 h I h I h I h I .

−

−

−

≠

= =

= =

=− = − =− = −

=− = =− =

== = = (E-2)
38

INTENTIONALLY LEFT BLANK. 
39

Appendix F. Differential Equations Coefficients 

F.1 Equation 44 m = 1, 2 

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )( ) ( )

( )( )

2 2

1n 1n 2n 2n d 2n c 1n

2 2

11 1 2 12 c

2 2

1n 1 1n a1n 21 1 4 c 2

22 2

22 t c 2n 1 2n a 2n c 1n

11 1 1 D 12

1n 1 1n a1n 21

R md L a R md L a a L b x a

S md V g d V c S x md

S g d V g g S g d V c x c

S I md x S g d V g g x S

T g d c C T md V

T g d f f T g

= = ++    

=+ =

=− + = −

= + =− + −

=− =

=− + = ( )( )

( ) ( )

( )

() () ( )

( ) ( )

1 3 c1

1

2n 1 2n a 2n D 1n an c 1n

*2 *

2n d 2n 22 x

2 2

D1 cB D2 6B 8B1

A1 1B 2B A2 3B 4B

d c xc

T gd f f C a L h xT

S 2 md a b T I

t i t i J J

t J i d V J t J i d V J

−

−

=− + − + −

=− =−

= φ −φ δ = φ −φ −

= + φ = +φ

   

 

 

F.2 Equation 44 m = 3, 4...nt

( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )( ) ( )

( )

2

mn mn

2

m1 m 1 m1 am1 m2 2m

2 22

mn 1 mn amn 0 1 mn

1 1

m1 1 m1 am1 D 1m m2 1m

2 2 22

mn 1 mn amn 0 mn d mn

* 2 *2

mn d mn m2 d 2m

R md L k

 S md Vd g d V g g S R

ˆ S g d V g g md L 2 kc

 T g d f f C a L T mdVa L

 T g d f f md L c 2 md a b

 S 4 md a b T 2md a b

− −

=

=− + =

=− + + ω ω

=− + + =

=− + + ω + φ

=− =



( ) ( ) * 22

mn 0 1 mn ˆ T md L 2 kc −− ω ω

( )( ) 2 2

mD Bm d Bm d Bm t i k L 2a b 4i a b = φ −φ + − φ   

( ) ( )

( ) ( )

3A B3 1B B3 5B

4A B4 3B B4 7B

t f I i dV g I

t f I i dV g I

= + +φ +

= + +φ +




40

 `

( )

() () ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

tt t

tt t

mA Bm Bm t

n 1A Bn 1 Bn 1 2B 2 6B 2

n A Bn 4 B 2 Bn 8 B 2

t f i dV g 4 m n 1

t f I x i dV g I x

t f I x i dV g I x

−− −

= +φ <<−

= +Γ +φ +Γ

= +Γ +φ +Γ







( )

() ()

( ) ( ) () ()

() ()

( ) ( ) () ()

j t

2

mD Bm

1

3A B1c 1B 2 B 2 e 4 B 2

1

B1c 5B 6 B 2 e 8 B 2

4A B2c 3B e 2 B 2 4 B 2

B2c 7B e 6 B 2 8 B 2

For n 1 n 4 m 3, 4

t i k L

ˆ t f I 5I x 24L I x

i d V g I 5I x 24L I x ˆ

ˆ t f I L I x 5I x

i d V g I L I x 5I x . ˆ

−

−

===

= φ −φ

= ++Γ + Γ

+φ + + Γ + Γ

= + + Γ +Γ

+φ + + Γ + Γ

 



 (F-2)

F.3 Quadratic Spin Equation (61) 

( ) ( ) 2

1

x

7 1EE 2d 2

x

J i L a L 4i iq dx = − ρ δ δ − ρ Γ + φΓ + Γ   ∫     

() ( )

( ) ( ) ( )

( ) ( ) ( )

t t

t

tt t

n n

D mn d mn m n d mn m n

m 3n 3

n

i 2

Bm d Bm m m d Bm n m

m 3

nn n

2 i

X x xB d mn m n Bm d Bm m

m 3n 3 m 3

2

xB xB1

Q i L a a Lb q q 8i a Lb q q

i L e a a Lb q q 8i a Lb q i q

I I I md L Re 4a L b q q e a 3 a L b q

I md I

= =

− φ

=

− φ

== =

=− − + φ    

− − +φ + φ − φ        

 

=+ − + −    

= +

∑∑

∑

∑∑ ∑

  

   

( ) xB2 I . (F-3)
41

Appendix G. Flexing Motion Finite Element Method (FEM) Quantities 

G.1 Kinetic and Potential Energy Coefficients 

( )

( )

( )

1

pqj e 1 p q

0

1

1 p q

pqj e 2

0

1 2 2

3 p q

pqj e 3 2 2

0

a L x N N dz ˆ

dN dN ˆ

b L x dz

dz dz

dN dN

c L x dz. ˆ dz dz

−

−

= ρ

= ρ

= ρ

∫

∫

∫ (G-1)

2 × 2 central element matrix j n 1

r

2

+ =

( ) ( ) ( )( )

( )

( )

2 1 12

11c 11r 13r 31r e 14r 41r 33r e 34r 43r 44r e

1 1

12c 12r e 13r 14r 32r e 42r e 33r 34r 43r e 44r

1

21c 21r 23r 41r e 24r

a a 5 a a 24L a a 25a 120L a a 24 a L ˆ ˆ ˆˆ ˆˆ ˆ ˆˆ ˆ

a a L a 5 a a 24L a 5L a 25a 24a 120L a ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ

a a 5 a a 24L a ˆˆ ˆˆ ˆ

− −−

− −

−

=+ + + + + + + +

=+ + + + + + + +

=+ + +

( )( ) ( )

1

e 31r e 33r 34r 43r e 44r

2

22c 22r e 23r 32r 24r 42r e 33r e 34r 43r 44r

L a 5L a 24a 25a 120L a ˆ ˆˆˆ ˆ

a a L a a 5 a a L a 5L a a 25a ˆ ˆˆ ˆˆ ˆ ˆˆ ˆ

− ++ + ++

=+ + + + + + + + H

4 × 4 adjacent element matrix j n 3

r

2

+ =

( )( )

( )

2 1 2

11a 11r e 12r 21r 22r e

1

12a e 11r 21r 12r e 22r

1

21a e 11r 21r 12r e 22r

2

22a e 11r e 12r 21r 22r

1

13a 13r 23r e 31a

a 25a 120L a a 24 a L ˆ ˆ ˆˆ ˆ

a 5L a 24a 25a 120L a ˆ ˆˆˆ ˆ

a 5L a 25a 24a 120L a ˆ ˆˆˆ ˆ

a L a 5L a a 25a ˆ ˆ ˆˆ ˆ

a 5a 24a L a 5 ˆˆ ˆ ˆˆ

− −

−

−

−

= + ++

= +++

= +++

= + ++

=+ = 1

31r 32r e

1 1

14a 14r 24r e 41a 41r 42r e

23a e 13r 23r 32a e 31r 32r

24a e 14r 24r 42a e 41r 42r

33a 33r 34a 34r

43a 43r 44a 44r

a 24a L ˆ

a 5a 24a L a 5a 24a L ˆˆ ˆ ˆˆ ˆ

a L a 5a a L a 5a ˆ ˆˆ ˆ ˆˆ

a L a 5a a L a 5a ˆ ˆˆ ˆ ˆˆ

aa aa ˆˆ ˆˆ

aa aa ˆˆ ˆˆ

−

− −

+

=+ =+

=+ =+

=+ =+

= =

= =

pqc pqa pqc pqa pqj pqj

ˆˆ ˆ b , b , c , and c are computed from b and c in the same manner as previously shown. ˆˆ ˆ
42

t n 4 =

e mn ( )( ) m 2 n 2c L L a a m,n 3, 4 ˆ − − == =

t n 8 =

e

33 111 34 121 35 131 36 141

43 211 44 221 45 231 46 241

53 311 54 321 55 331 11c 11a 56 341 12c 12a 57 13a 58 14a

63 411 64 421 65 431 21c 21a 66

L L3

aa aa aa aa ˆˆ ˆ ˆ

aa aa aa aa ˆˆ ˆ ˆ

aa aa aa aa aa a a aa aa ˆ ˆ ˆˆˆ ˆˆˆ ˆ ˆ

aa aa aa a a a ˆ ˆ ˆˆˆ ˆ

=

== = =

== = =

= = =++ =++ = =

= = =++ = 441 22c 22a 67 23a 68 24a

75 31a 76 32a 77 33a 78 34a

85 41a 86 42a 87 43a 88 44a mn

a a a aa aa ˆˆ ˆ ˆ

aa aa aa aa ˆˆˆ ˆ

a a a a a a a a other a 's are zero ˆˆˆ ˆ

++ = =

=== =

=== =

t n 12 =

L L5 e =

33 111 34 121 35 131 36 141

43 211 44 221 45 231 46 241

53 311 54 321 55 331 112 56 341 122 57 132 58 142

63 411 64 421 65 431 212 66 441 222 67 232 6

aa aa aa aa ˆˆ ˆ ˆ

aa aa aa aa ˆˆ ˆ ˆ

aa aa aa a aa a aa aa ˆ ˆ ˆˆ ˆˆ ˆ ˆ

aa aa aa a aa a aa a ˆ ˆ ˆˆ ˆˆ ˆ

== = =

== = =

= = =+ =+ = =

= = =+ =+ =

( )

( )

8 242

75 312 76 322 77 332 11c 11a 78 342 12c 12a 79 13a 14a 7 10

85 412 86 422 87 432 21c 21a 88 442 22c 22a 89 23a 24a 8 10

97 31a 98 32a 99 33a

aˆ

a a a a a a a a a a a a a a a a ˆ ˆ ˆˆˆ ˆˆˆ ˆ ˆ

a a a a a a a a a a a a a a a a ˆ ˆ ˆˆˆ ˆˆˆ ˆ ˆ

aa aa aa ˆ ˆ ˆˆ

=

= = =++ = ++ = =

= = =++ =++ = =

= = =+ () () ( )

() () () () () ()

() () ()

115 34a 125 135 145 9 10 9 11 9 12

10 7 10 8 10 9 10 10 10 11 10 12 41a 42a 43a 215 44a 225 235 245

11 9 11 10 11 11 315 325

a a a a a a a a ˆˆ ˆ ˆ

a a a a a a a a a a a a a a ˆ ˆ ˆˆ ˆˆ ˆ ˆ

a a a a a ˆ ˆ

=+ = =

= = =+ =+ = =

== = ( )

() () () ()

335 345 11 12

12 9 12 10 12 11 12 12 415 425 435 445 mn

a a a ˆ ˆ

a a a a a a a a other a 's are zero ˆˆ ˆ ˆ

=

== = =
43

t n 16 =

L L7 e =

33 111 34 121 35 131 36 141

43 211 44 221 45 231 46 241

53 311 54 321 55 331 112 56 341 122 57 132 58 142

63 411 64 421 65 431 212 66 441 222 6

aa aa aa aa ˆˆ ˆ ˆ

aa aa aa aa ˆˆ ˆ ˆ

aa aa aa a aa a aa aa ˆ ˆ ˆˆ ˆˆ ˆ ˆ

aa aa aa a aa a a ˆ ˆ ˆˆ ˆˆ

== = =

== = =

= = =+ =+ = =

= = =+ =+

( )

( )

7 232 68 242

75 312 76 322 77 332 113 78 342 123 79 133 143 7 10

85 412 86 422 87 432 213 88 442 223 89 233 243 8 10

97 313 98 323 99 333 11c 11a

a aa ˆ ˆ

aa aa aa a aa a aa a a ˆ ˆ ˆˆ ˆˆ ˆ ˆ

aa aa aa a aa a aa a a ˆ ˆ ˆˆ ˆˆ ˆ ˆ

aa aa aa aa a ˆ ˆ ˆˆˆ

= =

= = =+ =+ = =

= = =+ =+ = =

= = = ++ () () ()

() () () () () ()

() () () () ()

9 10 9 11 9 12 343 12c 12a 13a 14a

10 7 10 8 10 9 10 10 10 11 10 12 413 423 433 21c 21a 443 22c 22a 23a 24a

11 9 11 10 11 11 11 12 11 13 31a 32a 33a 116 34a 126 13

aaa a a a a ˆˆˆ ˆ ˆ

a a a a a aaa a aaa a a a a ˆ ˆ ˆˆˆ ˆˆˆ ˆ ˆ

a a a a a aa a aa a a ˆ ˆ ˆˆ ˆˆ ˆ

=++ = =

= = =++ =++ = =

= = =+ =+ = ( )

() () () () () ()

() () () () () ()

() ()

6 146 11 14

12 9 12 10 12 11 12 12 12 13 12 14 41a 42a 43a 216 44a 226 236 246

13 11 13 12 13 13 13 14 13 15 13 16 316 326 336 117 346 127 137 147

14 11 14 12 416 426

a aˆ

a a a a a aa a aa a a a a ˆ ˆ ˆˆ ˆˆ ˆ ˆ

a a a a a a a a a a a a a a ˆ ˆ ˆˆ ˆˆ ˆ ˆ

a aa aa ˆ ˆ

=

= = =+ =+ = =

= = =+ =+ = =

= = () () () ()

() () () ()

() () () ()

14 13 14 14 14 15 14 16 436 217 446 227 237 247

15 13 15 14 15 15 15 16 317 327 337 347

16 13 16 14 16 15 16 16 417 427 437 447 mn

ˆ a a a a f a a a a ˆˆ ˆ ˆ ˆ

a aa aa a a a ˆˆˆ ˆ

a a a a a a a a other a 's are zero ˆˆˆ ˆ

=+ =+ = =

=== =

=== =

mn mn pqj pqc pqa pqj pqc pqa ˆˆˆ b and c computed from b , b , b , c , c , and c in the same manner as previously ˆˆ ˆ

shown.

k a aL b mn mn d mn = + ( )

G.2 Kinetic Energy FEM Vectors 

( ) ( ) () ( )

( ) ( ) () ( )

1

Bpj e 1 EB p e j

0

1

'

Bpj 2 B p j 1 e

0

a L x x N z dz x L z z ˆ

ˆ

b x x N z dz z x L j 1.

=ρ δ = +

= ρ Γ = +−

∫

∫ (G-2)

1

Bm Bm Bpj Bpj 1n nj ˆ a and b. are computed from a and b in the same manner as a was computed from a , ˆ ˆ

k a aL b Bm Bm d Bm = + ( ) . 
44

G.3 Rod Aerodynamic Force Terms 

( ) () ( )

( ) () ( ) () () ( )

1

/

pqj f1 q p e j

0

1

/

pqj f 2 q f1 q e p j 1 e

0

ˆ

f c x N N z dz x L z z

g c x N z c x N z L N z dz z x L j 1 ˆ

p,q 1, 2, 3, 4.

= =+

= − = +−    

=

∫

∫

 (G-3)

mn mn pqj pqj mn pqj ˆ f and g are computed from f and g in the same manner as a was computed from a . ˆ ˆ

( ) () ( )

( ) () ( )

( )

1

pj1 e f1 p e j

0

1

pj1 e f 3 p j 1 e

0

f3 f2 f1

ˆ

f L c x N z dz x L z z

g L c x N z dz z x L j 1 ˆ

c x 2c xc p 1, 2, 3, 4

=− = +

=− = + −

=− =

∫

∫

( ) ( ) ( ) ()

( ) ( ) ( ) ( ) ()

1

Bpj e f1 B BF p

0

1

Bpj e f 2 B BF p

0

ˆ

f L c x x x N z dz

g L c x x x x N z dz ˆ

= Γ +Γ    

= Γ +Γ    

∫

∫

m1 m1 Bm Bm pj1 pj1 Bpj Bpj 1n

1

nj m1 m1 Bm Bm

ˆ ˆ f , g , f , and g are computed from f , g , f , and g in the same manner as a was ˆ ˆ

computed from a . All other f , g , f , and g 's are zero. ˆ

G.4 Nonzero Aerodynamic Extension Terms 

t n 4 =

( )

( ) [ ]

( ) ( )( )

() ( )

1 11

a31 1 2 e 4 a33 e 2 2D e 4

1

a34 1 2 2D e 4 4D a 41 3 e 2 4

1

a 43 Dbp 2 2D e 4 4D a 44 3 e 2 2D 4 4D

aB3 1 B 1 2 B 2

f I 5I 24L I f 24L 5 I I 24L I

f I 25I 24I 120L I I f I L I 5I

f c 24I 25I 120L I I f I 5L I I 25 I I

f I x 5I x 24

− −−

−

−

=− + + = + +      

= + + + + =− + +    

= + + + + =+ + + +        

=Γ + Γ + ( ) () ( ) 1 1 L I x I f I x 5I x 24L I I e 4 B 2 1BF aB4 5 B 1 6 B 2 e 8 B 3BF

− − Γ + = Γ + Γ + Γ+
45

t n 4 ≠

( ) ( ) ( ) ()

() () ( ) () ()

t t t tt t t t

t t

a31 1 a34 1 a 43 Dbp a 41 3 a 44 3

a n 1 1 a n 1 n an n 1 a n 1 2 2 2D 4 an n 4 4D

aB3 1 B 1 1BF aB4 3 B 1 3BF 2 B 2 aBn 4 B 2 aB n 1

Amn

f I f I f c f I f I

f I f I f I f I f II

f I xI f I xI f I x f I x

other f 's are zero

−− −

−

=− = = =− =

=− = = =− = +

=Γ + =Γ + =Γ =Γ

t n 4 =

( )( ) ( )

( )( )

[ ] ( ) ()

( )

2 2 12

a31 11 12 e 14 a33 1 2 e 4 6 e 10 8

a34 3 5 e 2 4 6 10 8

1

a 41 13 e 12 14 a 43 3 e 2 4 6 e 10 8

2

a 44 9 7 e 2 e 4 6

g I 5I 24L I g I 25I 120L 2I I 24 L I I

g I I 5L I 24 2I I 25 I I

g I L I 5I g I 5L I 24 2I I 120L I I

g I I L I 5L 2I I

− −−

−

=− + + =− + + − + −        

=− − + + − + −    

=− + + =− + + − + −    

=− − + + − ( )

() ( ) ( )

() ( ) ( )

10 8

1

aB3 5 B 1 6 B 2 e 8 B 2 5BF

aB4 7 B 1 e 6 B 2 8 B 2 7BF

25 I I

g I x 5I x 24L I x I

g I x L I x 5I x I

−

  + −  

=Γ + Γ + Γ +

=Γ + Γ + Γ +

t n 4 ≠

( ) ( )( ) ( )( )

( ) ( )( ) ( )( )

() () ( )

n nn nn

t tt tt

t

a31 11 a33 1 a34 5 3

a 41 13 a 43 3 a 44 7 9

a n 11 a n 1 n 1 a n 1 n 12 2 6 4

an 1 an n 1 an n 14 4 8 10

aB3 5 B 1 5BF aB4 7 B 1 7BF aB n 1

g I g I g I I

g I g I g I I

g I I g I I

g I g I g I I

g I x I g I x I g

− −− −

−

−

=− =− = −

=− =− = −

=− =− = −

=− =− = −

=Γ + =Γ + () () t 6 B 2 aBn 8 B 2 =Γ =Γ I xg I x

amn other g 's are zero . (G-4) 
46

INTENTIONALLY LEFT BLANK. 
47

Appendix H. Finned Missile Parameters 

123 c ρ = ρ = ρ =1 x =0 

 L = 20 V = 6000 ft/s 

 d = 0.35 ft. ρ = 0.002 slugs/ft3

 m = 3.50 slug 01 23 xx1 = =

 x I = 0.054 slug-ft2

s a = 1100 ft/s 

 t I = 14.318 slug-ft2 ( )

1

2

L s a 2 Va 1

−

  = −    

( )

( )

( )( )

( )

f1

2 x 10

L

L

c 4 11 x

e

2 15 3x a

12 a

−

= −

=

=− π +

= π

10 x 11

5 x 10

7x 5

11 x 7

< ≤

−< ≤

− < ≤−

− ≤ ≤−

 

( )

( ) ( )

( )( )

( )( )

2

f 2

2 x 10

2

L

L

c 2 11 x

2 0.5 1 e

2.5 1 3 15 3x a

2.5 12 6 x a

−

= −

=+ −

=+π +

=− π +

10 x 11

5 x 10

7x 5

11 x 7

< ≤

−< ≤

− < ≤−

− ≤ ≤−

 

c 0.30 11 x D ( )( )

0

= −

=

10 x 11

x 10

< ≤

≤ 

C 0.14 Dbp = C 18 Ap = −

3

12 d 10− = 522 d 0.25x10− = −

N M

Mq M

C 9.7 C 34.4

C 980 C 190

α α

α

= =−

=− =− 
48

INTENTIONALLY LEFT BLANK. 
49

List of Symbols, Abbreviations, and Acronyms 

c x f j( ) aerodynamic force distributions functions 

d rod diameter 

E Young’s modulus 

F F iF

y z

+ complex transverse aerodynamic force 

d a ( ) 1

16L −

g1

2

1 g VS2 ρ

I () () 4 4 2 2 d y dydz d z dydz = ∫∫ ∫∫ , area moment of rod 

Ix axial moment of inertia of projectile 

It transverse moment of inertia of projectile 

ˆ

k beam damping coefficient 

L rod length / rod diameter 

m projectile mass 

j n number of rod elements 

t n j 2n 2 +

p projectile spin 

1 q i β+ α , complex angle of attack of central disk (nsc) 

1e q ψ− θi complex yaw and pitch of central disk (nsc) 

2 q i ψ− θ  complex yaw and pitch rate of central disk (nsc) 

n q t n 3.4 n = ... FEM connectors (nsc) 

bn q t n 3.4 n = ... FEM connectors (bfc) 

L e Ln j

S 2 πd 4
50

T total kinetic energy 

T kinetic energy of disk at x with thickness dx d

V magnitude of projectile velocity 

1 2 x , x location of beam ends 

01 23 x , x dimensionless length of fore and aft aerodynamic extensions 

c x axial location of center of mass 

α angle of attack of central disk 

β angle of sideslip of central disk 

Γ E

x

∂δ

∂ , complex cant of disk 

Mε maximum strain of rod 

2

1

x

1

c E

x

L dx − δ δ ∫ , lateral location of missile’s cm 

E Ey Ez δ δ +δi , lateral displacement of disk (nsc) 

φ roll angle 

φk

 frequency of k-th mode 

λk damping of k-th mode 

ρ air density 

ρj axial variation of mass, moment of inertia, elasticity 

σ ω ω1 R

( ) 2 ω0 00 = E I L md

ω1 lowest elastic frequency of beam in vacuum 

ωR rigid projectile zero-spin frequency 

F F ,F ,F = ( ) xy z

G , aerodynamic force exerted on missile (nsc) 

M M ,M ,M = ( ) xy z

G , aerodynamic moment exerted on missile (nsc) 
51

Re z{ } real part of z 

Im z{ } imaginary part of z 

Carat superscript denotes quantity for a single element 

Tilde superscript denotes elastic parameter for bent missile 

B subscript denotes parameter for bent projectile 

BF subscript denotes bent fin parameter 

E subscript denotes an elastic coordinate parameter (nsc) 

T subscript denotes trim motion parameter 

b subscript denotes an elastic coordinate parameter (bfc) 

(bfc) body-fixed coordinates 

(efc) earth-fixed coordinates 

(nsf) non-spinning coordinates 
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