4,

n%REPORT NO. 1591’

.

GENERALIZED SUBHARMONIC RESPONSE OF A Ml SS ILE

WITH SLIGHT CONFIGURATIONAL ASYMMETRIES

U.saARMY

*

c

by

Charles H. Murphy

June 1972

Approved for public release; distributionunlimited.

ABERDEEN RESEARCH AND DEVELOPMENT

BALLISTIC RESEARCH LABORATORIESABERDEEN PROVING GROUND, MARYLAND

CENTER

-.

. .

Destroy this report when it is no longer needed.Do not return it to the originator.

Secondary distribution of this report by originating orsponsoring activity is prohibited.

Additional copies of this report may be purchased fromthe U.S. Department of Comerce, National TechnicalInformation Service, Springfield, Virginia 22151

+’

d’

fie findings in this repott are not tobe construed asan official Department of the Army position, unlessso designated by other authorized documents.

BALLISTIC RESEARCH LABORATORIES

REPORT NO. 1591

JUNE 1972

GENERALIZED SUBHARMONIC RESPONSE OF A MISSILEWITH SLIGHT CONFIGURATIONAL ASYMMETRIES

Charles H. Murphy

Exterior Ballistics Laboratory

Approved for public release; distribution unlimited.

RDT~E Project No. 1TO611O2A33D

ABERDEEN PROVING GROUND, MARYLAND

BALLISTIC RESEARCH LABORATORIES

REPORT N().1591

CHMurphy/jahAberdeen Proving Ground, lld.June 1972

GENERALIZED SUBHARMONIC RESPONSE OF A MISSILEWITH SLIGHT CONFIGURATIONAL ASYMMETRIES

ABSTRACT

The quasilinear analysis of the angular motion of slightly

asymmetric missiles with a cubic static moment shows the existence

of nonharmonic steady-state solutions. These solutions are described

by sums of three constant amplitude rotating angles. One angle

rotates at the spin rate and the other two rotate in opposite direc-

tions at approximately one-third the spin rate. This generalized

subharmonic motion can be very much larger than the usual harmonic

trim motion and can occur for spin rates greater than eight times the

resonant spin rate. The predictions of the quasilinear theory are

compared with the results of exact numerical integration of the

equations of motion with much better than 5% agreement in all cases.

TABLE OF CONTENTS

..I.

II.

III.

IV.

v.

VI.

VII.

ABSTRACT . . . . . . . .

LIST OF ILLUSTRATIONS. .

LIST OF SYMBOLS. . . . .

INTRODUCTION . . . . . .

NONLINEAR ANALYSIS . . ,

STEADY-STATE NONHARMONIC

APPROXIMATE RELATIONS.

EFFECT OF DAMPING.

COMPUTED MOTION. ●

DISCUSSION . . . .

ACKNOWLEDGMENT . .

REFERENCES . . . .

APPENDIX A . . . .

APPENDIX B . . . .

APPENDIX C . . . .

DISTRIBUTION LIST.

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Figure

LIST OF ILLUSTRATIONS

Page

1

2

3

4

5

6.

7

8

c-1

Nonlinear Harmonic Response 31

kz versus ~ for ma = t 0.2 and No Aerodynamic Damping 32

k3 versus ~ for ma = t 0.2 and No Aerodynamic Damping 33

Maximum Damping Rate as a Function of Spin for 34m = t 0.2a

kz and k~ versus Spin for ma = t 0.2 and for Both 35

Maximum and Zero Aerodynamic Damping

Angular Motion for Parameters of Table I in Non- 36spinning Coordinates

Angular Motion for Parameters of Table I in Missile 37Fixed Coordinates

Difference Between Quasilinear Predictions of kz 38

and k~ and Results of Numerical Integrationof

Exact Differential Equations

Stability Boundaries for Harmonic Response 51

LIST OF SYMBOLS

b

c ,Co 2

CMO

cubic static moment coefficients

coefficients of the transverse components of theaerodynamic moment

aerodynamic moment coefficient due to asymmetry

cMastatic moment coefficient

CM , CM damping moment coefficients& q

H

lX,lY

k.J

k~

k3m

m

axial, transverse moments of inertia

amplitude of the jth mode (j = 1, 2)

amplitude of the harmonic trim motion

maximum value of k3

reference length

mass

ma

Ma

M

P

P

s

s

t

v

LIST OF SYMBOLS (CONTINUED)

M 3>~ TO

aerodynamic moment due to asymmetry

Y

d+’011 ‘ate’ m

(1/1)~,x y

gyroscopic spin

.+angular velocity components along Y, Z axis

dimensionless distance along flight path ft V/i dtto

reference area

time

. .Y, z components of the velocitj’vector

magnitude of the vclocit~r\’ector

nonrolling Cartesian coordinate axes

1~1 , sine of the total angle of attack

Nla/llo)trim angle for zero spin

10

LIST OF SYMBOLS (CONTINUED)

% o

“2ICI

ej

effective squared yaws (j = 1, 2, 3)

i.J

damping rate of the j-modal amplitude (j = 1, 2)

.v+iwT

air densityP

*(-M) S

o-r

roll angle

roll rate, (g)

roll rate at which the maximum harmonic response occurs

j-modal phase angle, (j = 1, 2)

trim mode phase angle ($ + $~0)

phase angle between the fast mode and trim mode, $ - $31

phase angle for the magnitude of the angular motion forsymmetric missiles, $ - @

1 2

Y

11

LIST OF SYMBOLS (CONTINUED)

Superscripts

(-) components in a nonrolling coordinate system

()’

●

()

(a)

(-)

Subscripts

o

av

derivatives with respect to s, the arc length in calibers

go

() (-M)-*o

complex conjugate

evaluated at -r= 0

average over a distance that is large with respect to thewavelength of either @r or @s

12

I. INTRODUCTION

The theory of the linear motion of a “slightly asymmetric” missile1*

was first developed by Nicolaides. The assumed aerodynamic force and

moment have the same form for an asymmetric missile as for a symmetric

missile with the exception of the presence in the asymmetric case of

constant amplitude force and moment terms whose orientations are fixed

relative to the missile. These terms induce a trim angle of attack

that rolls with the missile (tricyclicmotion). For a constant roll

rate, the amplitude of this trim angle of attack is a function of t-he

roll rate with a maximum angle occurring when the roll rate equals the

natural pitch frequency (resonance). Recently, asymptotic relations

have been derived for trim motion when spin varies through a constant2

natural frequency and when the natural frequency varies through equal-3

ity with a constant spin rate.

In 1959 Nicolaides4extended his analysis of asymmetric missiles

to include nonlinear roll-orientation-dependentterms and thereby

introduced the concepts of “spin lock-in” and “catastrophic yaw.”-.!3-6

Since then, spin lock-in has been studied by a number of authors.

The angular trim produced by a nonlinear static moment has been treated

by Kanno.7

The quasilinear analysis technique, which has been most8-9

successful in describing the angular motion of symmetric missiles,

has recently been extended to the general angular motion of a slightly10

asymmetric missile. The detailed nonlinear behavior near resonance

has been discussed by Clare.11

Finally, Nayfeh and Saric12

have shown

that the results of References 10-11 can be obtained by the more so-

phisticated method of multiple scales.

The complex differential equation for the two-dimensional angular

motion of a slightly asymmetric missile with a cubic static moment is13-14

quite similar in form to Duffingfs equation. The harmonic

response curve has the same form as for the one-dimensional Duffingfs

equation. The response curve is a multivalued function of spin with

*Referencesare Lzsted on page 39.

13

an in-phase solution and two out-of-phase solutions existing near

resonance, and it can be shown that the intermediate size solution7is unstable. Since the harmonic response for the two-dimensional

motion is a constant amplitude coning motion, the mathematical

analysis is much simpler than for Duffing’s equation and does not

require any approximations.

As a result of this similarity with Duffingfs equation, the

question of the existence of subharmonic solutions naturally arises.

In the one-dimensional case, these are obtained by requiring the

quasilinear transient frequency to be one-third the forcing function

frequency. It is then shown that for sufficiently small damping, sub-

harmonic solutions exist for spin rates three times the resonance spin

rate.

The situation is somewhat more complicated for the two-dimensional

case. There are two transient frequencies present and, therefore, the

condition for subharmonic response is not clear. In this paper we will

obtain a condition for subharmonic response and derive the equations

for the amplitudes of the three modes of angular motion. These the-

oretical predictions will then be compared with the results of numerical

integration of the fourth order equations of angular motion.

II. NONLINEAR ANALYSIS

Although a missile-fixed coordinate system is very commonly used

for the analysis of the flight of missiles and aircraft, a related non-

spinning coordinate system which pitches and yaws with the missile is

most useful for treating the motion of symmetric missiles. For this-.

coordinate system with axes X,Y,Z, the X-axis lies along the axis of

symmetry of the missile and the ~- and ~-axes are so constrained that

the ~-axis is initially in the horizontal plane and the angular veloc-

ity of the coordinate system has a zero X-component. The angular

motion of the symmetric missile can be described by a complex angle of

attack, ~~ and a complex angular velocity, ~, where

(1)

(2)

~ is a complex number whose magnitude is the sine of the angle between

the velocity vector and the missilets axis of symmetry (the total angle

of attack) and whose orientation locates the plane of this angle.

By definition, a slightly asymmetric missile has a transverse

moment expansion of the same form as for a symmetric missile with an

added constant amplitude term that rotates with the missile. This type

of term can be introduced by a small control surface deflection or a

small manufacturing inaccuracy. If we limit the nonlinearities under

consideration to a cubic static moment and neglect any Magnus moment

contribution, the transverse moment expansion for a slightly asymmetric

missile assumes the form

(3)

where

11For this moment the differential equation* of motion is

*The effects of lift and drag have been neglected for simplicity.These effects plus that of a linear Magnus moment cm be easilyadded. Small geometric angles (6 c 0.2) are also assuwed so thatcertain geometric nonlinearities can be omitted.

-It

C + (H - ip)<’ - (MO + M2 ~2)~ = - Maei$

(4j

where H, P, MO~ M2 and Ma are defined in the list of symbols. For

constant spin, the asymmetric moment forcing function induces an aero-

dynamic trim angle that rotates with the missile. If both the spin

and the cubic static moment terms vanish (P = M2 = O), this trim angle

becomes the linear trim angle for zero spin and is given by the rela-

tion

6To = Ma/MO (5)

; can now be scaled with respect to this zero-spin trim angle by the

substitution

A further simplification is possible by use of a new independent

variable ~, where

/!-c = (-MO) S (7)

These changes of variable reduce Equation (4) to a much simpler form:

where

(A)= ( ) (-Mo)-*

(8)

16

d(“)=~()

m is the ratio of the cubic part of the static moment to the lineara

part when the angle of attack is equal to the linear zero-spin trim

angle. Thus it is a measure of the nonlinearity as well as of the

amplitude of the asymmetric moment.

Duffing’s equation for one-dimensional motion could be written

in the form

● ☛

x + hi + (1 + ax2)x = cos ~ -c (9)

For constant roll rate, small ratio of the moments of inertia

(i ~ O) and planar motion, the left side of Equation (8) takes on the

form of Duffingfs equation but the forcing function will not reduce to

Duffing’s forcing function. Indeed, for this circular forcing function,

planar motion is impossible. If we take the real part of Equation (8),

the forcing function has the same form as Duffing?s but the nonlinear

term contains the imaginary component of the angle as well as the real

component. Thus Equation (4) is similar to Duffing’s equation but is

not a generalized form of it.

The harmonic response solution of Equation (8) for constant roll

rate can be easily obtained by assuming a solution of the form

where k and $ are constants.3 30

Direct substitution of Equation (10) in Equation (8) yields:

i+k3e 30 = [1 -(l- Ix/Iy)~2 + mak~ + i~~]-l

(lo)

(11)

17

This cubic equation

negative ma. These

in k is plotted in Figure 1 for both positive and3

7curves and the instability of the intermediate

size solution are very similar to corresponding features of Duffing’s

equation. The maximum value of the harmonic response, k , and the3m

spin rate at which it occurs, ~m, can be easily estimated from

Equation (11) by letting $30 be - 7T/2and assuming the moment of in-

ertia ratio to be small.

‘_2 * ~“42 = (1/2) {1 + [1 + 4maH 1 } : ma2/H (12)m

k = [ii$ml-l +&4~-+ (13)

3m

The approximate forms of Equations (12-13) are, of course, valid only“-zwhen maH is large.

The form of the solution to the linear version of Equation (8) is

the usual tricyclic equation

i+ i$ i+1 2 3

t = kle + kze +ke3

(14)

where

i. = i k.J JJ

j=l,2

The first two terms in Equation (14) are modes of free oscillations

induced by initial conditions and for constant ;. their amplitudesJ

are exponentially damped.

18

These free oscillation modes are so numbered that 1~11~

For a statically stable missile, the larger frequency, which is com-

monly called the notational frequency by ballisticians, has the same

sign as the spin while the smaller frequency (the ballistician’s pre-

cessional frequency) has a sign opposite to that of the spin. If the

actual solution to the nonlinear Equation (8) can be approximated by a

solution with the form of Equation (14), the nonlinear coefficient of

~ can be approximatedby

+ 2 klk~ COS@r + 2 k2k3 COS ($S - @r)

where

$~ is the phase angle for the magnitude of the angular motion for

a ~mmetrlc missile and is constant for a gyroscopic stability factor

of unity. @r is the phase angle between the fast (notational)mode. ●

and the trim mode and is constant for resonance ($1 = 0). The—

parameters (;., ~j, k; $30) of Equation (14) can be obtained asJ

functions of ki by an averaging process. If the spin rate is not nearJ

10zero or 41 these relations take the form:

(16)

i2 = - [~i2‘i2‘ma k; k3 k;l (sin Y)av][2$2 - ;]-l (17)

19

‘2

ij(ij - ‘1 =1 + ma ICIej j =1,2

.2k3[l -(l- Ix/Iy)~2 + mal~le~] = Cos +30

;; k~ + ma $k2 [sin Y]av = - sin $30

(18)

(19)

(20)

where

Y= +s+4)r=24)l -@2-4)3

“2Icl = k; + 2 k2 + 2 k; + 2 k2k3 [Cos ‘Iav

el 2

2l&le2 = k; + 2 k; + 2 k: + k; k3 k;l [cos ‘Ylav

20

III. STEADY-STATE NONHARMONIC MOTION

For a dynamically stable missile the transient motion induced by

initial conditions damps out and usually only the harmonic trim motion

represented by k~ remains. In the case of Duffing’s equation, another

steady-state solution is possible under certain conditions. This

solution is a subharmonic one with a frequency of ~/3. As we shall

see, Equation (4) can have a somewhat more complicated steady-state

solution which is a generalized subharmonic response.● ☛

For a steady-state motion, @j is zero and Equation (14) will.

yield a nonharmonic steady motion for three combinations of k. and A.:J J

A

(l)k=O, ~2 =()1

(2)k2=0, ;l=0

or (3) ; = ~z = O. A quick inspection of Equations1

(16-17) shows that the third case is the only possibility and then

only if the average of sinY is nonzero. A condition for generalized

subharmonic response is, therefore,

Equations (16-17) for zero damping now provide a condition for Y

and a relation between the modal amplitudes

sin Y = ~~ (2mak2 k~)-l1(22)

k; =2bk; (23)

where b = - ;2/;1

21

— —

Equations (18-23)now provide a set of seven equations in seven

unknowns, (kl, k2, k3, ;ls i2, +30) y)’

Equations (18-20) can be greatly simplified if the ratio of

moments of inertia is neglected and Equation (23) used to eliminate

kl . The results of this simplificationare given in Table I. Under

the further simplification of no aerod~amic damping (R = O), Equation

(22) reduces to Y = O, n and the fourth equation of Table I requires

4 =O,?T. For the four possible choices of $ and Y , solutions30 30

of Equation (21) and the remaining three equations of Table I were

sought. The results are plotted in Figures 2-3. These figures give

kz and k as functions of ~ for various values of m Two solution3 a“

sets of k~ and k3 are showno In Figure 2, the two sets of values of

k are separated by plotting one as positive up and the second as2

positive down. In Figure 3, k goes through zero for one set of3

solutions. At this point, say at ~ = A, $30 jumps from ~ to O and

this is shown by plotting k3 as positive down after the shift in phase.

The solutions identified by the dashed line have the values Y = O,

4 = n for all values of ~ while the solutions identified by a solid30

line have the values Y = ~ , $ = n for ~ less than A and the values30

Y =0,+ . 0 for ~ greater than A.30

As a result of numerical integrationsdescribed later in this

report, it is conjectured that the dashed solution is unstable and

only the solution identified by the solid curves can be realized by an

actual missile in flight.

As can be seen from Figure 2, generalized subharmonic exist when

$ < 3 for negative m and when ~ > 3 for positive m The precisea a“

value for the appearance of a subharmonic can be computed by setting

k2 equal to zero in the frequencies as given by Table I and inserting

the result in Equation (21):

22

*i =3[l+2mak2]

3(24)

Since the amplitude of a pure harmonic, k3, at $ = 3 is quite close

to 1/8, we see that subharmonic appear slightly to the left of

i = 3 for negative ma and slightly to the right of this point for

positive ma’

Table I. Frequencies and Harmonic Response

for Ix/I =Oandk~=2bk~Y

;1 = {l +ma[2(b+l)k~+2k~+ 2k2kqcosY]l *

i=- {l +m[(4b+l)k2+2k2+ 2bkk3cos Y]}k2 a 2 3 2

ii (; k2 - ;2 k;)/k3 = - sin $3 30

23

IV. APPROXIMATE RELATIONS

A simple approximate relation for k can be obtained by neglecting2

k~ in the frequency equations of Table I and approximatingb by 1.

Figure 3 shows that k3 is less than .15 for the region of interest and

the relations for the frequencies show the b approximation to be good.

If we make these approximations and substitute the frequency equations

from Table I in Equation (21), a quadratic equation in k; results.

One of the roots vanishes near ~ = 3. If we approximate the radical

in the relation for this root, a very simple expression results

ma kz =(J2 - 9)(;2 - 1)

26(7 ;2 - 11)

(25)

Equation (25) matches the exact curves f Figure 2 over the range1

2 < ~ < 10 with an accuracy of t .02 m; .

In the vicinity of ~ = 3, Equation (25) can be simplified to

ma k2 =2(4 - 3)/13 (26)2

The corresponding approximations for the frequencies are

?1=1 + 4 (; - 3)/13

;2 = - [1 +5 ($ - 3)/13]

(27)

(28)

24

If Equations (27-28)predicted the true subharmonic frequency present

for Duffingts equation, i.e ~/3 , they would have the form

=1+(;.-3)/3 (29)

The similarity between

to call the results of

Equations (27-28) and Equation (29) leads us

this paper generalized subharmonic response.

v. EFFECT OF DAMPING

In order for subharmonic motion to be observable in practice,

some damping must be present to eliminate the transient motion.

Equation (22) indicates an upper bound on fisince the sine cannot

exceed one. As fiis increased from zero, the two pairs of solutions

for k~, k~ approach each other and coalesce for the maximum value of

;.

In Figure 4, the maximum value of ; is plotted versus ~ for

m= .2. Since ; = .01 induces an amplitude decrement of 3% pera

cycle of the linear frequency, the restriction on damping is seen to

be quite severe. The corresponding curves for kz and k~ are compared

with those for the solid lines (stable solution) in Figure 5. We

see that damping has little effect on k~ although it does smooth out

the variation in k~. The jump in phase angle $~0 as k~ goes through

zero is replaced by a continuous variation in phase.

VI. COMPUTED MOTION

The theory of this report gives us sufficient information to

verify the existence of generalized subharmonic motion. For the

values of ;, ma and ~ given in Table 11, values of Y, k~, k~, k

$Is +2 and $30

3’

can be computed; these values are also listed in Table

II. All of the parameters of the tricyclic motion described by

Equation (14) are available except for ~10 and $20, although the

25

Table II. Parameters for Sample Subharmonic Motion

A

m= 1.0 “~ $ = 3.5 ; H= .0230a

Y

k2

‘H.

2.6180 (150°)

.4281

.2976

.0890

1.1534

- 1.1931

.1000 (5.73°)

.6892 (39.49°)

3.1760 (181.97°)

.5667

.2289

.5631 (32.26°)

.3745

.2613

.0952

1.1563

- 1.1875

.1000 (5.73°)

2.7513 (157.64°)

3.1688 (181.56”)

.0358

.1342

.1872 .0839

●

ivo

- .0941 .3846

26

values of Y and $30 specify 2 $10 - $20 . A simple choice of $10 = .1b

allows us to compute a set of initial values of ~ and c for the.

generalized subharmonic motion.

Equation (8) was numerically integrated using both sets of

initial conditions. No steady-state motion was found for the set of

initial conditions associated with Y = 32° , but a steady-state

motion for the other set of initial conditions was observed and is

shown in Figure 6. In all of our other numerical integrations, the

solutions associated with the smaller Y were never observed although

most of those associated with the larger Y have been. It is, there-

fore, our conjecture that the motions associated with the smaller Y’s

are unstable.

The motion shown in Figure 6 appears in a much simpler form if we

transform Equation (14) to missile-fixed coordinates by multiplying by

exp (- i $) and make use of Equation (23) and the definition of Y:

i$r i(2 @r - Y) i$~O

i= {k2[(2b)ke + e ]+k}e3

(30)

The two terms involving @r represent an epicyclic motion with a

notational frequency that is twice the precessional frequency and a

notational amplitude that is approximately 40% of the maximum of the

motion. The maximum value of the epicycle occurs for $ = Y . Thisr

epicyclic motion appears in the catalogue of such motions on page 63

of Reference 16 and is a closed curve with a single loop. The points

of the curve plotted in Figure 6 were transformed to the non-rotating

coordinate system and plotted in Figure 7. As can be seen, they fall

precisely on a closed curve with a single loop.

27

The result of the exact numerical integrationof Equation (8) as

given by Figure 6 has been fitted by the usual tricyclic data

1,15analysis used for ballistic range tests and the various tricycle

parameters compared with the quasilinear prediction of Table II. The

agreement is quite good. This process has been repeated for values of

~ up to 10 with equally good results, which are given in Figure 5.

Thus the predicted generalized subharmonic response does occur for

quite large values of ~ . Since it is not possible in the numerical

work to use the maximum values of ~, lesser values were used as indi-

cated. In Figure 8 the results for kz and k~ are compared with their

predictions for the appropriate values of ~ and it can be seen that in

most cases the agreement is better than 5% .

VII. DISCUSSION

In Figure 5 the harmonic response (k = O) is shown as a function2

of spin rate. At a spin rate of 9 the harmonic response is .012 and

the harmonic component of the subharmonic motion is .14. The t~(o

generalized subharmonic amplitudes, however, are much larger

(kl = 4.2, k2 = 3.0) and steady-state motion of maximum amplitude 7.2

can occur. This steady-statemotion is 600 times the harmonic stead>r-

state motion. If it occurs, it would have an important unexpected

effect of the flight of the missile.

It should be noted that there are rather severe limitations on

the occurrence of special steady-statemotion at high spin rates

(; > 3). They are:

10 small aerodynamic damping, ; (less than 6% decrement per

pitch cycle);

2. nonlinear static moment which becomes more stable at larger

angles (C2 < O);

.

3. initial conditions near the generalized subharmonic motion.

28

Thus generalized subharmonic motion should trouble the missile designer

very infrequently. When it does occur, it can have a very large and

puzzling effect since it occurs for spin rates which most designers

consider to be very safe. ~

ACKNOWLEDGMENT

The author is indebted to Professor I. D. Jacobson of the

University of Virginia for several stimulating discussions which

initiated the work of this report.

29

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34

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35

10.

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-lo.

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+

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I

1

—. +

+

-8

4

.—-—.- ——..—.- ..

,. . ./-

.,. ,,,, ,

.,.!

,,,

—.

6..—— ..

. 6 -4 -2 0 .2 4.. . .

10‘a= ●

; +=3.5 ; R=.023

0 10.

6. Angular Motion for Parameters of Table II in Nons Dinning Coordinates

36

10.

●8

6.

4●

.2

0

-2.

-4..

-6.

-lo.-lo●

L .———.

-8.. -6 }-. 4● .-2

ma=l.O ; + =

——

c H

I

IL —. . 1 J

4. 6. .8 10.0 .2

3.5 ; ii= 023

Figure 7. Angular Yotion for Parameters of Table II in Missile Fixed Coordinates

37

A

r s!!

Lcl)Ez7 +o

?3WI

2ucm

f-

ucro

co 11-0

vls

II m

CIJ

0E

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+

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38

REFERENCES

1.

2.

3.

4.

5.

6.

7.

8.

9.

10●

11.

J. D. Nicolaides, “On the Free Flight Motion of Missiles HavingSlight ConfigurationalAsymmetries,“ Ballistic Research Labora-tories Report No. 858, AD 26405, June 1953; also Institute ofthe Aeronautical Sciences Preprint 395, January 1953.

C. H. Murphy, “Response of an Asymmetric Missile to Spin Varyingthrough Resonance,“ AIAA Journal, Vol. 9, No. 11, November 1971,pp. 2197-2201.

R. J. Tolosko, “Amplification Due to Body Trim Plane Rotation,”AIM Paper 71-48, January 1971.

J. D. Nicolaides, ‘TWO Nonlinear Problems in the Flight Dynamicsof Modern Ballistic Missiles,“ Institute of the AeronauticalSciences Report 59-17, January 1959.

D. A. Price, “Sources, Mechanisms, and Control of Ro1l ResonancePhenomena for Sounding Rockets,” AIAA Sounding Rocket VehicleTechnology Specialist Conference, February 1967, pp. 224-235.

J. J. Pettus, “Slender Entry Vehicle Roll Dynamics,” AIAA Paper70-560, May 1970.

J. Kanno, “Spin Induced Forced Resonant Behavior of a BallisticBody Having Stable Non-Linear Aerodynamic Properties,” GeneralResearch in Flight Sciences, Vol. III, Flight Dynamics and SpaceMechanics, LMSD-288139, Report 11, Lockheed Aircraft Company,Sunnyvale, California, 1960.

C. H. Murphy, “The Prediction of Nonlinear Pitching and YawingMotion of Symmetric Missiles,“ Journal of the AeronauticalSciences, Vol. 24, No. 7, July 1957, pp. 473-479.

C. H. Murphy, “Quasi-Linear Analysis of the Nonlinear Motion ofa Nonspinning Symmetric Missile,“ Journal of Applied Mathematicsand Physics (ZAMP), Vol. 14, No. 5, September 1963, pp. 630-643;also Ballistic Research Laboratories Memorandum Report No. 1466,AD 410213, April 1963.

C. H. Murphy, “Nonlinear Motion of a Missile witl~Slight Con-figurational Asymmetries,” Journal of Spacecraft and Rockets,Vol. 8, No. 3, March 1971, pp. 259-263.

T. A. Clare, “Resonance Instability for Finned ConfigurationsHaving Nonlinear Aerodynamic Properties,” JouPnaz of spacecraftand Rockets, Vol. 8, No. 3, March 1971, pp. 278-283.

39

12. A. H. Nayfeh and W. S. Saric, “Nonlinear Resonances in the Motionof Rolling Reent~y Bodies,” AIAA Paper 71-47, .Tanuary1971.

13. J. J. Stoker, Nonlinear Vibrations, intersciencejNew York, 1950.

14. C. Hayashi, “Forced Oscillations in Nonlinear Systems,” Osaka,Japan, Nippon Printing and Publishing Company Ltd, 1953.

15. C. H. Murphy, “Free Flight Motion of Symmetric Missiles,”Ballistic Research Laboratories Report No. 1216, AD 442757,July 1963.

16. S. J. Zaroodny, “A Simplified Approach to the Yawing Motion of aSpinning Shell,” Ballistic Research Laboratories Report No. 921,AD 58529, August 1954.

40

APPENDIX A

DERIVATION OF EQUATION (25)

If we assume k = O and b = 1, the frequency equations of Table I3

become

4 =. [1 + 5 m k2]*2 az

Substituting Equations (A-1--A-2) in Equation (21), we have

i= 2[1 + 4 ma k;]*+ [1 + 5ma k~]*

Equation (A-3) can now be simplified by squaring it twice.

121 m: k; + 6(11 - 7 ~2) ma k; + (;2 - 9)(;2 - 1) =0

(A-1)

(A-2)

(A-3)

1

“k;=- 3(11 - 7 ;2) f [9(11 - 7 ;2)2 - 121(;2 - 9)(;2 - l)]Z

,.121 (A-5)

According to Equation (A-3), $ is 3 when k vanishes. Thus the root2

with the minus sign in Equation (A-5) is extraneous. We therefore

take the root with the plus sign and approximate the radical with the

first two terms of the binomial expansion.

“2:. mak2~ ($ ‘9)(~2 -1)

26(7 42 - 11)

. (A-6)

Near ~ = 3, we can approximate j with 3 in every factor of

Equation (A-6) except (~ - 3).

“ ma k; = 2(~ - 3)/13,, (A-7)

42

APPENDIX B

EFFECT OF LIFT, DRAG AND MAGNUS MOMENT

A linear Magnus moment can be easily introduced into Equation (3)

i$C;+i C;=- iCMe -i(co+c2~2)E

o

iCM&

j+cMld

q

,&c:() Mpa

With this linear Magnus moment, a linear lift force and constant drag11

coefficient, Equation (4) becomes:

-?1c + (H - iP)c’ - [(MO +Mo 82) + iPT)~ = - M ei~

a

psdwhere H = —

21v

I

(Lc -c - c +C

La D )( M M6m%2 q

)

[

Ixpst

T=m —CL+CMx m$2 u pa1

(B-1)

(B-2)

and the other symbols are unchanged.

43

The changes of dependent and independent variables yield

(B-3)

The harmonic response solution to Equation (B-3) is quite similar to

Equation (11):

i$k3 e 30 = [1 - (1 - lx/Iy)~2 + ma k:

(B-4)

The presence of ~ affects the quasilinear analysis through the

damping equations (Equations (16-17))and one of the harmonic

component relations (Equation (20)). For constant frequencies and

10constant y~ the equations for the damping exponents become:

(B-5)

‘1 k siny][2;2i2=- [H;p - ~~+mak~k~ ~ - 6]-1 (B-6)

The revised version of Equation (20) is:

(B-7)

44

We now require i. = O and obtain revised versions of EquationsJ

(22-23):

sinY = (H? - ;?)(2 ma kz kJ-l1

k; = 2bk;

(B-8)

b.

Equations (18-19, 21, B-7--B-9) once again provide a set of seven

(B-9)

equations in the seven unknowns (kl, kp, kg , ~1 & $30) y) .

45

-.

APPENDIX C

STABILITY OF HARMONIC MOTION

The harmonic solution to Equation (8), given by Equation (11),

can be multivalued. It is shown in Reference 7 that the inter-

mediate size solution is unstable and this solution is, therefore,

identified by a dashed line in Figure 1. The stability argument is

quite simple and will be repeated here for the convenience of the

The similar stability argument for Duffing’s Equation13

reader.

is much more complicated and involves a detailed discussion of

Mathieu?s Equation.

We first perturb the harmonic solution of Equation (8) by a

small complex function n .

(c-1)

Equation (8) then takes on the form

~+[fi+i(2- Ix/Iy);l ~ + [1 - (1 - Ix/Iy)$2 + 2 ma k; + ifi~ln

(c-2)

if all higher order terms in rIand its derivatives are neglected.

47

—

Next we assume an exponential solution for n and separate into

real and imaginary parts

{X2 +h + [1 - (1 - I /1 )~z + 3m k~]}n10XY a.

- [(2- Ix/Iy);A + fi;]~20= O

(C-3)

(C-4)

+ {A2 + h + [1 - (1 - Ix/Iy)~2+ma ~@20 = o (c-5)

The characteristic equation for A can now be obtained by setting the

determinant of the coefficients of q. in Equations (C-4) and (C-5)Jo

to zero. For simplicity we will neglect the ratio of the moments of

inertia in comparison to one.

A4 + aA3 +b A2 + CA + d= O (C-6)

48

—

where

a =Zi

b= 2(l+~2)+4mak2+fi23

d= (1 - ~2+3mak$) (l- 42 + ma k2) + ;2$23

The usual analysis can now be employed on the coefficients of this

quartic equation to determine the existence of a positive real part of

A. For small damping, this analysis can be considerably simplified

since a and c can then be neglected and Equation

quadratic equation in A2 with solution

A2=- b/2 t i(b/2)2 - d

For positive ma, b is always positive and A will

part only when

d= (1 -~2+3mk2)(l-~2+maj a

(C-6) becomes a

(c-7)

have a positive real

k:) < 0 (C-8)

For negative ma, the situation is more complicated since b can change

sign. The analysis can, of course, be performed but will not be

given here.

49

Under our approximations Equation (11) for the harmonic amplitude

and phase becomes

k3 [1 - ~2+mak~]=tl (c-9)

Thus the curve

1 -$2+mak~=() (c-lo)

is the asymptote of k . As can be seen in Figure C-1, it also can be3

called the “backbone” of the harmonic response curve.

If we differentiate Equation (C-9),we have

(C-n)

Thus the first term in d vanishes when d~/dkQ is zero. The curve.)

1- ~2+3mak~=0 (C-12)

is also shown in Figure 9. d is negative between these curves and the

intermediate size solution for k~ is, therefore, unstable.

50

,

..

-.

7

6

5

4

3

2

I

o[

ma=ooz

Ro =

/

Figure c-1 ● Stability Boundaries

51

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GENERALIZEDSUBHARMONICRESPONSEOF A MISSILE WITH SLIGHT CONFIGURATIONALASYMMETRIES

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~harlesH. Murphy

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3. A9STRACT

The quasilinearanalysisof the angularmotion of slightlyasymmetricmissileswitha cubic staticmoment shows the existenceof nonharmonicsteady-statesolutions.These solutionsare describedby sums of three constantamplituderotatingangles.one angle rotatesat the spin rate and the other two rotate in oppositedirectionsat approximatelyone-thirdthe spin rate. This generalizedsubharmonicmotion canbe very much larger than the usual harmonic trim motion and can occur for spin ratesgreaterthan eight times the resonantspin rate. The predictionsof the quasilinear~heory are comp&edof motion with much

with the resultsof exact numericalintegrationof the equationsbetter than 5% agreementin all cases.

3DRE’..1473m8PLACCS DO PORM 1478. 1 JAM ●4. WHICH ISOOSOLET8 ●on AmUv US8. Unclassified—

UnclassifiedSmtrlty gasdgcation

14.KaY Wonos

ROLR WT

Spin-YawResonanceAsymmetricMissilesVarying Roll TrimSubharmonicResonance

[Unclassified

LINK ●

ROLE WT

— —

LINK C

ROLC

&cllrity cls*slficatlon