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

ABSTRACT also exhibit limit cycle behavior for some values of

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

emphasis has been given to accurate modeling and

prediction of persistent resonance and catastrophic

yaw phenomena which are caused by nonlinear

induced aerodynamic moments and configurational

asymmetries. Recently, rigorous analyses of

persistent resonance problem was performed by

American and Indian researchers who used a fifth

order autonomous dynamic system model with linear

transverse and nonlinear roll aerodynamics. A

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

G:

h:

NOMENCLATURE

Nonlinear roll aerodynamics coefficient.

H-aTl-o-

H : Aerodynamic damping coefficient.

i : Unit imaginary number.

Kp : Linear roll aerodynamics coefficient.

p: Roll rate.

pres: Linear resonance roll rate.

pss: Steady-state roll rate.

T: Magnus coefficient.

a: Angle of attack.

* 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

American Institute of Aeronautics and Astronautics


a :

Angle of sideslip.

Scaled complex total angle of attack.

Complex total angle of attack, fi+ia .

Ratio of axial to transverse moment of

inertias.

<t> : Roll angle.

(/>M : Asymmetry moment orientation angle.

(/>s : Steady-state roll angle rate.

T : Nondimensional transformed arc length.

INTRODUCTION

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

asymmetries1 : If roll rate p of a missile stays close

to its resonance roll rate pres for some time period

during flight, then a resonance takes place and total

angle of attack £ builds up. Despite the fact that

unguided missiles are lightly damped systems,

probability of a destructive linear resonance seems to

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

pres change continuously during flight. On the other

hand, practical experience has shown that many

unguided missiles fail catastrophically when their p

and prm values are allowed to match even for brief

time periods. Significant differences of results of

linear time-invariant theory and dynamic tests led

flight 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

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

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

moment due to lateral offset of center of mass from

geometrical center of missile cross section. Another

important discovery was that unguided missiles

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

researchers have investigated persistent resonance

and catastrophic yaw characteristics of free fall

bombs, sounding rockets and re-entry vehicles by

using a large variety of mathematical models and

methods2"27. Persistent resonance followed by

catastrophic yaw is one of the most difficult

problems in flight dynamics of unguided missiles,

not only because of the strongly nonlinear nature of

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

83American Institute of Aeronautics and Astronautics


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

aeroballistic reference frame by transformation of the

independent variable (nondimensional arc length)

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

(p ~ P™) and reverse persistent resonance

(P ~ -Pres) •

Later Ananthkrishnan and Raisinghani29

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



transverse moments into their model. They explained

the phenomenon as a case where resonance solution

is unstable due to decreased damping and design

solution is stable.

The authors have carried out extensive

numerical simulations of the analytical persistent

resonance model of the above mentioned references.

This study has revealed the fact that limit cycle

oscillations can be observed in certain cases. In this

paper, firstly the fifth order autonomous nonlinear

dynamic system model of persistent resonance is

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 slightly

asymmetric unguided missile in free flight are

presented below.

= -(1 (1)

(2)

Equation (1) is the complex transverse equation of

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

motion. Transverse aerodynamics is linear

(restoring, damping, Magnus and slight

configurational asymmetry moments) while roll

aerodynamics includes both linear and nonlinear

terms (fin cant, damping and induced moments).

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

straight line with constant forward speed and

negligible gravitational effects. The real and

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

(3)

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

(4)

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

state-space form:

(5)



where components of the state vector

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

respectively. Equation (5) corresponds to five first

order coupled nonlinear ordinary differential

equations:

xl=-Kpxl-2KGx, + Kjs, (6)

= x.

i4 = -Hx4 + (2 - a-)*

[(l-a)h-Kp]Xlx3

-(l-cr)hcos<j>M,

x5 = -Hx5 - (2 - a)xl

(7)

(8)

(9)

(10)

The fifth order autonomous nonlinear

dynamic system model as defined by equations (6)-

(10) can theoretically exhibit four types of behavior

at steady-state: equilibrium solution, periodic

solution, quasi-periodic solution and aperiodic

(chaotic) solution30. Murphy28 (and later on

Ananthkrishnan and Raisinghani29) focused on

determining location and stability of equilibrium

points where,

(11)

Equation (11) is a set of nonlinear algebraic

equations in terms of xe? and it can be solved

iteratively by using the Newton-Raphson method.

Murphy proposed a simpler graphical method to

determine locations of equilibrium points. Equations

(1) and (2) simplify as follows at equilibrium:

i V . , - =-/»«*'*, (12)

(13)

Locations of equilibrium points can be easily found

by plotting the functions /, \0eg j = IGa^ and

&,-&. and determining their

intersections. Stability of the system around a given



equilibrium point xe<! can be investigated by are stable in each case. (Stability results of Murphy28

examining the time evolution of a small perturbation were the same qualitatively despite his derivation

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

(14)

(15)

(16)

Hence, stability of xe? can simply be determined by

calculating eigenvalues of the Jacobian matrix J at

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

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

and a all affect stability properties.

Murphy28 (and later on Ananthkrishnan and

Raisinghani29) performed numerical case studies to

verify the above procedure by using system

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

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

graphs for cases A and B are presented in Figure 1

and Figure 2 respectively. In each case there are five

intersections corresponding to five different

equilibrium points. Ananthkrishnan and

Raisinghani29 carried out linear stability analysis and

found out that only two of these equilibrium points

Table 1. System parameters for numerical case

# G rM

A 3.0 5.0 90°

B 3.0 5.0 270°

studies of Ref. 28 and Ref. 29.

In case A there are two stable equilibrium points: A

normal persistent resonance with <j>eq =1.011 and a

normal steady-state with (/>eq = 2.861. In case B there

are also two stable equilibrium points: A reverse

persistent resonance with ij>eq = -0.976 and a

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

the values of eigenvalues of the Jacobian matrix J

both at stable and unstable equilibrium points.

Different initial conditions can lead to different types

of dynamic behavior. In Figure 3 and Figure 4

response of the system to the four sets of initial

conditions that are given in Table 2 are shown for

cases A and B respectively. (Numerical simulations

that are presented in this paper were performed by

using fourth order Runge-Kutta integration routine.

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

the first and second initial conditions lead to the



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

B, the first, third and fourth initial conditions lead to

the normal steady-state solution (d>eq = 3.115], while

the second initial condition leads to the reverse

persistent resonance solution (0eg = -0.976].

Table 2. Initial conditions used for simulation of

cases A and B.

#

12

3

4

0(0)

0

0

0

0

LIMIT

*«>)

0

0

0

0

CYCLE

o(0)

0

0

0

0

*«>)

10

-1

0

,(0)

0

1

0

-1

BEHAVIOR

A numerical experiment was performed in

which the effect of asymmetry moment orientation

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

was examined systematically. Value of <j>M was

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

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

four initial conditions given in Table 2 were

determined. System parameters except than 0M

were the same as those presented in Table 1. Results

of this case study are presented in Figure 5 where

regions of <j>M that lead to different types of steady-

state behavior are shown for each initial condition by

using pie-charts. (Note that in these charts $M

increases in counter-clockwise direction. If the

missile is viewed from behind in aeroballistic

reference frame, then 0M increases in clockwise

direction.)

An interesting result was obtained for the

fourth initial condition where a limit cycle (periodic

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

Figure 6 shows variation of ^ with T while Figure

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

this particular case. Figure 7 shows that if a limit

cycle takes place, then mean value of total angle of

attack is non-zero indicating possibility of large

dispersion similar to normal and reverse persistent

resonance cases.

/ i^ and /2^ graphs for </>M = 330°

is presented in Figure 8. The number of equilibrium

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

each had five equilibrium points.) Equilibrium

points and eigenvalues of corresponding Jacobian

matrices are presented in Table 3 and Table 4

respectively which show that the only stable

equilibrium point is the normal steady-state solution

(4= 3.0561).



Table 3. Equilibrium points for <pM = 330°

1 0.8284 -0.218788+/0.217013

2 1.0487 0.662572+/0.195644

3 3.0561 0.010593-/0.005603

Table 4. Eigenvalues of Jacobian matrices of

equilibrium points for <j>M = 330°.

#1 #2 #3

-0.0595+; 1.745 -0.0569±/1.9415 -0.0586+/3.864

0.0043±/0.4712 -0.8676 -0.1044

-0.1895 0.7414 -0.0392±(1.9386

-0.0601

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

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

of attack initial condition £"„ was taken as 1.0 while

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

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.)

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

proportion of initial conditions do lead to limit cycle

oscillations.

DISCUSSION AND CONCLUSION

Limit cycle behavior in persistent resonance

was first observed experimentally by Nicolaides5 in

1966 during dynamic supersonic wind tunnel tests of

Aerobee sounding rocket. Nicolaides also observed

aperiodic (chaotic) oscillations of Aerobee but did

not perform any theoretical or numerical analysis of

periodic 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

Ananthkrishnan and Raisinghani29 can also exhibit

limit cycle (periodic) behavior for some values of

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-



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

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for determination of location and stability of periodic Laboratories, Aberdeen Proving Ground, MD, BRL

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Research in this area is in progress with particular 2Nicolaides, J. D., "Missile Flight and

emphasis given to quasilinearization30. Astrodynamics," Bureau of Naval Weapons, TN 100-

The only aerodynamic nonlinearity that is A, 1961.

included in the analysis of this paper is associated to

the roll moment due to lateral offset of center of 3Nicolaides, J. D., "Free Flight Dynamics," Text,

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section. Other nonlinearities in transverse NotreDame, 1961.

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induced moments) and roll aerodynamics (damping 4Glover, L. S., "Effects on Roll Rate of Mass and

and induced) can be easily incorporated into the Aerodynamic Asymmetries for Ballistic Re-Entry

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isolated to a single type of aerodynamic moment



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6

41

2 - j

°

,

-4

-4 -3 0 1

Roll Rate2 3 4

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

-2H

- 4 H

-6H

- 4 - 3 - 2 - 1 0 1 2 3 4Roll Rate

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



Initial Condition #1

20 40 60 80 100

Nondimensional Distance T


20 40 60

NondmensiCTial Distance T


—]—————|————}————|———20 40 60 80 100

Nondimensional Distance T


20 40 60 80 100

Nondimensional Distance t

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




20 40 60 80

Nondimensional Distance i


3 -

2 -

|

0 -

\

0 20 40 60 80 100

Nondimensional Distance i

4 -j

3 -

2 -

1 1 ~K

0 -

-1 -

-2 -C


/^

i

J

20 40 60 80 100

Nondimensional Distance i:


20 40 60 80

Nondimensional Distance *

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



180


90

270


90

270


90

180

270


90

180

270

HI Steady-state solution.|_| Normal persistent resonance solution.IH Reverse persistent resonance solution.H Limit cycle solution.

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



2.0 —

11.0 Hi

o •a:

0.5 -

0.0100 200 300 400 500

Nondimensional Distance t

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

-0.4 -0.2 0.0 0.2 0.4

Scaled Angle of Sideslip p

0.6

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



10.0

-4 -3 -2

Figures. f\^ and graphs for 0M =330° (limit cycling system).

90

180 0

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

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


aiaa-1997-3491-819 limit cycle behavior in persistent resonance of unguided missiles

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