Aiaa-1997-3491-819 Limit Cycle Behavior In Persistent Resonance Of Unguided Missiles

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

A97-37254

AIAA-97-3491

LIMIT CYCLE BEHAVIOR IN PERSISTENT RESONANCE OF UNGUIDED MISSILES Omer Tannkulu* Defense Industries Research and Development Institute (TUBJTAK-SAGE) PK16, Mamak 06261, Ankara, Turkey

and Kemal Ozgoren1^ Middle East Technical University, 06531, Ankara, Turkey also exhibit limit cycle behavior for some values of

ABSTRACT Flight dynamic behavior of unguided

system parameters and initial conditions. A case

missiles at yaw-pitch-roll resonance has been the

study is presented,

subject of a large number of investigations. Special NOMENCLATURE

emphasis has been given to accurate modeling and

G:

prediction of persistent resonance and catastrophic yaw phenomena which are caused by nonlinear

h:

induced aerodynamic moments and configurational

Nonlinear roll aerodynamics coefficient. H-aT

l-o-

of

H :

Aerodynamic damping coefficient.

persistent resonance problem was performed by

i :

Unit imaginary number.

American and Indian researchers who used a fifth

Kp :

Linear roll aerodynamics coefficient.

order autonomous dynamic system model with linear

p:

Roll rate.

transverse and nonlinear roll aerodynamics. A

pres:

Linear resonance roll rate.

pss:

Steady-state roll rate.

T:

Magnus coefficient.

a:

Angle of attack.

asymmetries.

Recently,

rigorous

analyses

simple graphical method was proposed to locate equilibrium points of the model. Stability of equilibrium points were determined by linearization. In this study, it is shown that the same model can

* Coordinator, Mechanics and Systems Engineering Research Group (MSMG). Member AIAA. 1 Professor, Department of Mechanical Engineering. Copyright © 1997 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

82

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Angle of sideslip.

linear time-invariant theory and dynamic tests led

Scaled complex total angle of attack.

flight

Complex total angle of attack, fi+ia . a :

Ratio of axial to transverse moment of

dynamicists

to

incorporate

nonlinear

aerodynamic effects into their mathematical models to explain such failures. It was soon discovered that the highly nonlinear induced roll moment can cause

inertias.

lock-in of p to pres for certain conditions after a

:

Roll angle.

(/>M :

Asymmetry moment orientation angle.

(/>s :

Steady-state roll angle rate.

linear resonance causes E, to build up. This type of behavior which is known as persistent resonance

was later on associated with the much larger roll T:

Nondimensional transformed arc length.

moment due to lateral offset of center of mass from geometrical center of missile cross section. Another

INTRODUCTION

important discovery was that unguided missiles

One of the important outcomes of linear time-invariant flight dynamics analysis of unguided missiles was the discovery of yaw-pitch-roll resonance

due

to

slight

configurational

which are in persistent resonance can become severely unstable for certain conditions due to the highly nonlinear induced transverse moments. This

type of behavior is known as catastrophic yaw. Many

1

asymmetries : If roll rate p of a missile stays close

researchers have investigated persistent resonance

to its resonance roll rate pres for some time period

and catastrophic yaw characteristics of free fall

during flight, then a resonance takes place and total

bombs, sounding rockets and re-entry vehicles by

angle of attack £ builds up. Despite the fact that

using a large variety of mathematical models and

unguided missiles are lightly damped systems,

methods2"27. Persistent resonance followed by

probability of a destructive linear resonance seems to

catastrophic yaw is one of the most difficult

be low at a first glance since in general both p and

problems in flight dynamics of unguided missiles,

pres change continuously during flight. On the other

not only because of the strongly nonlinear nature of

hand, practical experience has shown that many unguided missiles fail catastrophically when their p and prm values are allowed to match even for brief

the problem but also due to the coupling between yaw, pitch and roll degrees of freedom. One of the important contributions in this

area was made by Murphy28 who focused on time periods. Significant differences of results of 83 American Institute of Aeronautics and Astronautics

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

analytical investigation of special cases of persistent

determine the equilibrium points of this system. He

resonance. He used a linear model for transverse

used Lyapunov's linearization method to determine

aerodynamics which included restoring, damping,

stability of the equilibrium points. He found out that

Magnus and slight configurational asymmetry

three types of equilibrium points are possible: steady-

moments. He derived a more convenient form of the

state roll rate (p « p,,), normal persistent resonance

well known linear transverse equation of motion in

(p ~ P™)

an

d

reverse

persistent

resonance

aeroballistic reference frame by transformation of the (P ~ -Pres) •

independent variable (nondimensional arc length) Later Ananthkrishnan and Raisinghani29

and scaling of the dependent variable (complex total

angle of attack). Murphy used a nonlinear model for roll aerodynamics which included linear moments due to fin cant and damping as well as nonlinear roll

orientation

dependent

induced

moment.

He

examined nature of the induced roll moment and expressed it as an infinite series. He then derived the

roll equation of motion by approximating the induced roll moment as the first term of this series.

He showed that the roll moment due to lateral center of mass offset is a special reduced case of series representation of the induced roll moment. Murphy

derived a more convenient form of the roll equation of motion by the same transformation and scaling that he used in case of the transverse equation of

motion. All this analysis led him to two coupled differential equations of motion; one in terms of the complex total angle of attack and the other in terms of the roll angle rate. This corresponds to a fifth order autonomous nonlinear dynamic system model. Murphy developed a simple graphical method to

found out and corrected a simple mathematical derivation mistake of Murphy. This has no significance in terms of location of equilibrium points but it can change stability results. They

performed some of the case studies of Murphy with their revised model. They then focused

on

developing qualitative topological models of normal and reverse persistent resonances and concluded that

the reverse case is much less likely compared to the normal case. They found out through root locus analysis that quasi-steady-state solutions are possible in which unstable angle of attack oscillations are

observed even though design steady-state roll rate is achieved by the missile. (Resonance solution is stable, design solution is unstable.) They tried to construct a topological model of this type of behavior as well and mentioned the possibility of existence of limit

cycles.

Finally,

Ananthkrishnan

and

Raisinghani investigated the problem of catastrophic yaw qualitatively without adding any induced

84 American Institute of Aeronautics and Astronautics

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transverse moments into their model. They explained

Equation (1) is the complex transverse equation of

the phenomenon as a case where resonance solution

motion while equation (2) is the real roll equation of

is unstable due to decreased damping and design

motion.

solution is stable.

(restoring,

Transverse damping,

aerodynamics Magnus

is

linear

and

slight

The authors have carried out extensive

configurational asymmetry moments) while roll

numerical simulations of the analytical persistent

aerodynamics includes both linear and nonlinear

resonance model of the above mentioned references.

terms (fin cant, damping and induced moments).

This study has revealed the fact that limit cycle

Equations (1) and (2) are valid for flight on a

oscillations can be observed in certain cases. In this

straight line with constant forward speed and

paper, firstly the fifth order autonomous nonlinear

negligible gravitational effects.

dynamic system model of persistent resonance is

complex parts of equation (1) can be expanded as,

The real and

discussed briefly. Secondly, a case study is presented where equilibrium solutions and periodic solutions (limit cycles) are observed at steady-state depending

on parameters of the model and initial conditions.

DYNAMIC SYSTEM MODEL

Equations

of

motion

of

a

(3)

slightly

asymmetric unguided missile in free flight are

a+Ha+(2-d)ij>p

presented below.

(4)

= -(1

(1)

Equations (2), (3) and (4) can also be written in

state-space form: (2)

(5)

85 American Institute of Aeronautics and Astronautics

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where

components

of

the

state

vector

The fifth order autonomous nonlinear

x(xj,j = I,---,5) denote ^, f t , a, f$ and a

dynamic system model as defined by equations (6)(10) can theoretically exhibit four types of behavior

respectively. Equation (5) corresponds to five first

at order

coupled

nonlinear

ordinary

steady-state: equilibrium

solution,

periodic

differential

solution, quasi-periodic solution and aperiodic equations:

(chaotic) solution30. Murphy28

(and later on

Ananthkrishnan and Raisinghani29) focused on xl=-Kpxl-2KGx, + Kjs,

(6)

determining location and stability of equilibrium points where, = x.

(7)

(11) (8)

Equation (11) is a set of nonlinear algebraic i4 = -Hx4 + (2 - a-)*

equations in terms of xe? and it can be solved

iteratively by using the Newton-Raphson method. Murphy proposed a simpler graphical method to

[(l-a)h-Kp]Xlx3

determine locations of equilibrium points. Equations (1) and (2) simplify as follows at equilibrium: -(l-cr)hcos<j>M,

(9) iV.,-

=-/»«*'*,

(12)

x5 = -Hx5 - (2 - a)xl (13)

Locations of equilibrium points can be easily found

(10)

by plotting the functions /, \0eg j = IGa^ &,-&.

and

determining

and their

intersections. Stability of the system around a given 86 American Institute of Aeronautics and Astronautics

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equilibrium point xe
can be investigated by

examining the time evolution of a small perturbation

are stable in each case. (Stability results of Murphy28 were the same

qualitatively despite his derivation

£x from xe<7 (Lyapunov's linearization method):

Table 1. System parameters for numerical case

(14)

#

(15)

G

rM

A

3.0

5.0

90°

B

3.0

5.0

270°

studies of Ref. 28 and Ref. 29.

(16)

In case A there are two stable equilibrium points: A

Hence, stability of xe? can simply be determined by

normal persistent resonance with <j>eq =1.011 and a calculating eigenvalues of the Jacobian matrix J at x e?

• fa > G , <j)M and h influence location of

normal steady-state with (/>eq = 2.861. In case B there are also two stable equilibrium points: A reverse

equilibrium points while <j>s , G , <j>M, h , H , Kp

persistent resonance with ij>eq = -0.976

and a

and a all affect stability properties. Murphy28 (and later on Ananthkrishnan and

normal steady-state with ^ = 3.115. Ref. 29 gives

Raisinghani29) performed numerical case studies to

the values of eigenvalues of the Jacobian matrix J

verify the above procedure by using system

both at stable and unstable equilibrium points.

parameter values presented in Table 1, (h , H , Kp

Different initial conditions can lead to different types of dynamic behavior. In Figure 3 and Figure 4

and a are all equal to 0.1). f\\^eq] and /2(^«J

response of the system to the four sets of initial

graphs for cases A and B are presented in Figure 1

conditions that are given in Table 2 are shown for

and Figure 2 respectively. In each case there are five

cases A and B respectively. (Numerical simulations

intersections

different

that are presented in this paper were performed by

and

using fourth order Runge-Kutta integration routine.

Raisinghani29 carried out linear stability analysis and

Step size was Ar = 0.1 in all simulations.) In case A,

found out that only two of these equilibrium points

the first and second initial conditions lead to the

equilibrium

corresponding points.

to

five

Ananthkrishnan

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normal steady-state solution (eq = 2.861), while the third and fourth initial conditions lead to the normal

persistent resonance solution (<j>eq =1.011J. In case

of this case study are presented in Figure 5 where

regions of <j>M that lead to different types of steadystate behavior are shown for each initial condition by using pie-charts. (Note that in these charts $M

B, the first, third and fourth initial conditions lead to increases in counter-clockwise direction. If the

the normal steady-state solution (d>eq = 3.115], while missile is viewed from behind in aeroballistic the second initial condition leads to the reverse

persistent resonance solution (0eg = -0.976].

reference frame, then 0M increases in clockwise direction.) An interesting result was obtained for the

Table 2. Initial conditions used for simulation of

fourth initial condition where a limit cycle (periodic

cases A and B.

solution) was obtained at steady-state for <j>M = 330° .

#

0(0)

*«>)

o(0)

*«>)

,(0)

Figure 6 shows variation of ^ with T while Figure

1

0

0

0

1

0

7 shows variation of ft with a at steady-state for

2

0

0

0

0

1

this particular case. Figure 7 shows that if a limit

3

0

0

0

-1

0

cycle takes place, then mean value of total angle of

4

0

0

0

0

-1

attack is non-zero indicating possibility of large dispersion similar to normal and reverse persistent

LIMIT CYCLE BEHAVIOR

A numerical experiment was performed in which the effect of asymmetry moment orientation

resonance cases. /i^

and / 2 ^

graphs for M = 330°

is presented in Figure 8. The number of equilibrium

angle (/>M on dynamic response of the system model

points is three. (Note that cases A and B of Table 1

was examined systematically. Value of <j>M was

each had five equilibrium points.) Equilibrium

increased with A^M = 10° steps in the 0°-350°

points and eigenvalues of corresponding Jacobian

interval and at each step, steady-state response to the

matrices are presented in Table 3 and Table 4

four initial conditions given in Table 2 were

respectively which show that the only stable

determined. System parameters except than 0M

equilibrium point is the normal steady-state solution

were the same as those presented in Table 1. Results

(4= 3.0561).

88 American Institute of Aeronautics and Astronautics

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Table 3. Equilibrium points for
different ^0 orientations are shown in the form of a pie chart. (Note that orientation angle increases in the counter clockwise direction in this chart as well.)

1

0.8284

-0.218788+/0.217013

2

1.0487

0.662572+/0.195644

3

3.0561

0.010593-/0.005603

Figure 9 shows that for 0M = 330° a significant

proportion of initial conditions do lead to limit cycle oscillations.

Table 4. Eigenvalues of Jacobian matrices of DISCUSSION AND CONCLUSION equilibrium points for <j>M = 330°.

Limit cycle behavior in persistent resonance #1

#2

#3

was first observed experimentally by Nicolaides5 in

-0.0595+; 1.745

-0.0569±/1.9415

-0.0586+/3.864

1966 during dynamic supersonic wind tunnel tests of

0.0043±/0.4712

-0.8676

-0.1044

Aerobee sounding rocket. Nicolaides also observed

-0.1895

0.7414

-0.0392±(1.9386

aperiodic (chaotic) oscillations of Aerobee but did

-0.0601

not perform any theoretical or numerical analysis of periodic

As can be seen from Figure 5 periodic behavior is observed for a very narrow range of (j>M values. A numerical case study was performed in

which value of asymmetry moment orientation angle was taken constant as (f>M = 330° and all other system parameters were the same as those presented

or

aperiodic

behavior.

Since

this

experimental study of Nicolaides, periodic, quasi-

periodic and aperiodic oscillation phenomena in persistent resonance have remained as an untouched problem.

In this study it is shown that the fifth order autonomous nonlinear dynamic system model

developed

first

by Murphy28

and

later

by

in Table 1. Magnitude of the rate of change of angle

Ananthkrishnan and Raisinghani29 can also exhibit of attack initial condition £"„ was taken as 1.0 while

limit cycle (periodic) behavior for some values of its angle was increased with 10° steps in the 0°-350°

interval. (All other initial conditions were taken as zero.) Results of this case study are shown in Figure 9 where the type of steady-state behavior reached at

system parameters and initial conditions. The limit

cycle that is discussed in this paper has nothing to do with the limit cycle suggested by Ananthkrishnan

and Raisinghani's topological model of quasi-steady-

89 American Institute of Aeronautics and Astronautics

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state oscillations. Persistent resonance solution is

only. Time domain analysis (similar to or more

stable while design solution is unstable for quasi-

extensive than the one presented in this paper) of a

steady-state oscillations. Linear stability analysis of

model with more general aerodynamics can be

the three equilibrium points of the limit cycling

expected to yield new phenomena (periodic, quasi-

system presented in this paper shows that its design

periodic, aperiodic) such as the one discussed in this

solution is stable.

paper. Research work in this area is also in progress.

Obviously obtaining limit cycle oscillations with a certain model for a certain set of system

REFERENCES

parameters and initial conditions is not an end in

'Nicolaides, J. D., "On the Free Flight Motion of

itself. The physics behind the phenomenon must be

Missiles

understood. This requires development of a method

Asymmetries," U. S. Army Ballistic Research

for determination of location and stability of periodic

Laboratories, Aberdeen Proving Ground, MD, BRL

attractors in state space of the model without

Report 858, 1952, AD 26405.

Having

Slight

Configurational

numerical integration of equations of motion. Research in this area is in progress with particular

2

emphasis given to quasilinearization30.

Astrodynamics," Bureau of Naval Weapons, TN 100-

Nicolaides,

The only aerodynamic nonlinearity that is

J.

D.,

"Missile

Flight

and

A, 1961.

included in the analysis of this paper is associated to the roll moment due to lateral offset of center of

3

mass from geometrical center of missile cross

Aerospace Engineering Department, University of

section.

NotreDame, 1961.

Other

nonlinearities

in

Nicolaides, J. D., "Free Flight Dynamics," Text,

transverse

aerodynamics (restoring, damping, Magnus and 4

induced moments) and roll aerodynamics (damping

Glover, L. S., "Effects on Roll Rate of Mass and

and induced) can be easily incorporated into the

Aerodynamic Asymmetries for Ballistic Re-Entry

model. Many researchers have investigated effects of

Bodies," AIAA JSR, Vol. 2, No. 2, March-April

these nonlinearities by using models of varying

1965, pp. 220-225.

complexity. Most of the time quasilinear analysis methods were employed and nonlinearity was isolated to a single type of aerodynamic moment 90

American Institute of Aeronautics and Astronautics

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

5

12

Progress in Understanding Catastrophic Yaw,"

Persistent Roll Resonance on Re-Entry Vehicles,"

AGARD Report 551, 1966.

AIAA J, Vol. 6, No. 6, June 1968, pp. 1030-1035.

6

13

Resonance," AIAA Paper No. 66-49.

Studying the Anomalous Roll Behavior of Re-Entry

Nicolaides. J: D., "A Review of Some Recent

Pettus, J. J., "Persistent Re-Entry Vehicle Roll

Vaughn, H. R.,

"Boundary

Conditions for

Barbera, F. J., "An Analytical Technique for

Vehicles," AIAA JSR, Vol. 6, No. 11, November 7

Daniels, P., "Fin-Slots vs. Roll Lock-In and Roll

1969, pp. 1279-1284.

Speed Up," AIAA JSR, Vol. 4, No. 3, March 1967, 14

Nayfeh, A. H., "A Multiple Time Scaling Analysis

pp. 410-412.

of Re-Entry Roll Dynamics," AIAA J, Vol. 7, No. 8

Chadwick, W. R., "Flight Dynamics of a Bomb with

11, November 1969, pp. 2155-2157.

Cruciform Tail," AIAA JSR, Vol. 4, No. 6, June 15

Daniels, P., "A Study of the Nonlinear Rolling

1967, pp. 768-773.

Motion of a Four-Finned Missile," AIAA JSR, Vol. 9

Platus, D. H., "A Note on Re-Entry Vehicle Roll

7, No. 4, April 1970, pp. 510-512.

Resonance," AIAA J, Vol. 5, No. 7, July 1967, pp. 16

Price, D. A. Jr., Ericsson, L. E., "A New Treatment

1348-1350.

for Roll-Pitch Coupling for Ballistic Re-Entry 10

Vehicles," AIAA J, Vol. 8, No. 9, September 1970,

Control of Roll Resonance Phenomena for Sounding

pp. 1608-1615.

Price, D. A. Jr., "Sources, Mechanisms, and

Rockets," AIAA JSR, Vol. 4, No. 11, November 17

1967, pp. 1516-1525.

Nicolaides, J. D., Clare, T. A., "Non-linear

Resonance Instability in the Flight Dynamics of "Migotsky, E., "On a Criterion for Persistent Re-

Missiles," AIAA Paper No. 70-969.

Entry Vehicle Roll Resonance," AIAA Paper No. 67137.

18

Clare, T. A., "Resonance Instability for Finned

Configurations Having Nonlinear Aerodynamic

91 American Institute of Aeronautics and Astronautics

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

Properties," AIAA JSR, Vol. 8, No. 3, March 1971,

25

pp. 278-283.

"Analysis of the Nonlinear Rolling Motion of Finned

Cohen, C. J., Clare, T., and Stevens, F. L.,

Missiles," AIAA I, Vol. 12, No. 3, March 1974, pp. 19

Bootle, W. I, "Spin Variations in Slender Entry

303-309.

Vehicles During Rolling Trim," AIAA J, Vol. 9, No. 26

Pepitone, T. R., lacobson, I. D., "Resonant

4, April 1971, pp. 729-731.

Behavior of a Symmetric Missile Having Roll 20

Nayfeh,

A. H.,

Saric,

W. S.,

"Nonlinear

Resonances in the Motion of Rolling Re-Entry

Orientation-Dependent

Aerodynamics,"

AIAA

JGCD, Vol. 1, No. 5, September-October 1978.

Bodies," AIAA Paper No. 71-47. 27

Bennett, M. D., "Roll Resonance Probability for

21

Nayfeh, A. H., Saric, W. S., "Nonlinear Motion of

Ballistic Missiles with Random Configurational

an Asymmetric Body with Variable Roll Rate,"

Asymmetry," AIAA IGCD, Vol. 6, No. 3, May-June

AIAA Paper No. 71-932.

1983, pp. 222-224.

22

28

Rolling Motion of Finned Missiles," AIAA Paper

Lock-In," AIAA IGCD, Vol. 12, No. 6, November-

No. 72-980.

December 1989, pp. 771-776.

23

29

Rate of a Launch Vehicle Under the Influence of

and Quasisteady Resonant Lock-In of Finned

Random Fin Misalignments," AIAA I, Vol. 10, No.

Projectiles," AIAA JSR, Vol. 29, No. 5, September-

3, March 1972, pp. 324-325.

October 1992.

24

30

Cohen, C. J., Clare, T., "Analysis of the Nonlinear

Madden, R. G., "A Statistical Analysis of the Roll

Nayfeh, A. H., Saric, W. S., "An Analysis of

Asymmetric

Rolling

Bodies

with

Nonlinear

Aerodynamics," AIAA J, Vol. 10, No. 8, August

Murphy, C. H., "Some Special Cases of Spin-Yaw

Ananthkrishnan, N., Raisinghani, S. C., "Steady

Parker, T. S. and Chua, L.O., Practical Numerical

Algorithms for Chaotic Systems, Springer-Verlag, New

York,

1972, pp. 1004-1011.

92 American Institute of Aeronautics and Astronautics

1989.

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

6

4

1

2-j

° ,

-4

-4

-3

0

1

2

3

4

Roll Rate

Figure 1. /jf^J and / 2 ^J graphs for case A of Table 1.

-2H

-4H

-6H

-

4

-

3

-

2

-

1

0

1

2

3

4

Roll Rate

Figure!. /,^J and f2(0eg) graphs for case B of Table 1.

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Initial Condition #2

Initial Condition #1

20

40

60

80

100

20

40

60

Nondimensional Distance T

NondmensiCTial Distance T

Initial Condition #3

Initial Condition #4

—]—————|————}————|——— 20

40

60

80

Nondimensional Distance T

100

20

40

60

Nondimensional Distance t

Figure 3. Variation of with T for case A and initial conditions of Table 2.

94 American Institute of Aeronautics and Astronautics

80

100

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

Initial Condition #2

Initial Condition #1

3 -

2 -

|

0 -

\

20

40

60

0

80

20

40

60

80

Nondimensional Distance i

Nondimensional Distance i

Initial Condition #4

Initial Condition #3 4 -j

3 -

/^ 2-

i 1 K

1

~

0-

J

-1 -2 C

20

40

60

80

Nondimensional Distance i:

100

20

40

60

Nondimensional Distance *

Figure 4. Variation of with r for case B and initial conditions of Table 2.

95 American Institute of Aeronautics and Astronautics

80

100

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

Initial Condition #1

Initial Condition #2

90

90

180

270

270

Initial Condition #3

Initial Condition #4

90

90

180

180

270

HI |_| IH H

270

Steady-state solution. Normal persistent resonance solution. Reverse persistent resonance solution. Limit cycle solution.

Figure 5. Effect of <j>M on response to initial conditions of Table 2.

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2.0 —

1

1.0 Hi

o •

a:

0.5 -

0.0 100

200

300

400

500

Nondimensional Distance t

Figure 6. Variation of <j> with r for M = 330° and initial condition #4 of Table 2.

-0.4

-0.2 0.0

0.2

0.4

0.6

Scaled Angle of Sideslip p

Figure 7. Steady-state ft versus a graph for M = 330° and initial condition #4 of Table 2.

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10.0

-4

Figures. f\^ and

-3

-2

graphs for 0M =330° (limit cycling system).

90

180

0

270 Igjgl Steady-state solution. ^H Limit cycle solution.

Figure 9. Effect of ^0 orientation on response for^M = 330°

98 American Institute of Aeronautics and Astronautics