optimal external configuration design of unguided missiles

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JOURNAL OF SPACECRAFT AND ROCKETS

Vol. 35, No. 3, May–June 1998

Optimal External Conguration Design of Unguided Missiles

Omer Tanrõkulu and Veysi Ercan†

Defense Industries Research and Development Inst itute, Ankara 06, Turkey

A simple optimal external conguration design method is proposed that can be used in conceptual and prelim-

inary design stages of an u nguided missile development project. Cost and constraint functions are derived from

the results of linear time-invariant aeroballistic theory (constant roll rate and forward speed, decoupled axial and

transverse dynamics); therefore different phases of ight are examined separately. Curvetting is used to reduce

the number of trial cases and hence the work required to obtain aerodynamic and inertial data. Optimal cong-

urations are determined by using a modied steepest-descent algorithm. A case study is presented in which the

external conguration of an unguided light assault missile is optimized for free ight at low subsonic speed. Three

cost functions related to stability, range, and warhead performance, respectively, and two inequality constraint

functions related directly to stability and indirectly to dispersion are considered in the case study.

NomenclatureC D = drag force coefcientC l p = roll damping moment stability derivativeC l = roll moment due to n cant stability derivativeC m = static moment stability derivativeC m C mq = transverse damping moment stability derivativesC m p

= Magnus moment stability derivativeC Z = static force stability derivative

d = gradient vector of F xF x = modied cost function f x = cost functiong x = inequality constraint function

H = Hessian matrix of F xh x = equality constraint function

I a = axial moment of inertia,kg m2

I t = transverse moment of inertia, kg m2

k a = nondimensionalaxial radius of gyrationk t = nondimensionaltransverse radius of gyrationm T = total mass, kgm = warhead mass, kg pdyn = Magnus instability roll rate limit, rad/s pres = yaw-pitch-roll resonance roll rate, rad/sS = reference area, 2 4, m2

sd = dynamic stability factorss = static stability factorV = speed of missile, m/s

x = vector of scaled variable geometrical parameters xl = lower limit of x

xu = upperlimit of x= adaptive step size of iterations= transverse damping factor= reference length, rocket motor diameter, m

= penalty constants= freestream density, kg/m3

Introduction

T HE main objectives in external conguration design of un-

guided missiles are to obtainadequate stabilityin all phases of ight, short minimum range, long maximum range, low dispersion,and large payload mass. In practice it is difcult to achieve these

Received Aug. 5, 1997; presented as Paper 97-3725 at the AIAA At-

mospheric Flight Mechanics Conference, New Orleans, LA, Aug. 11–13,1997; revision received Nov. 17, 1997; accepted for publication Nov. 19,1997. Copyright c 1997 by the American Institute of Aeronautics and As-

tronautics, Inc. All rights reserved.

Coordinator, Mechanics and Systems Engineering Research Group, PK16, Mamak 06261. E-mail: [email protected] AIAA.

† Research Engineer, Flight Mechanics Section, Mechanics and Systems

Engineering Research Group, PK 16, Mamak 06261.

objectives due to the complicated nature of unguided missiles asnonlinear, time-varying, and random systems. Signicant advances

have been made in analysis and system identication aspects of ight dynamics of unguided missiles since World War II. A num-ber of range, dispersion,and stability criteria have been determinedby using analytical, numerical, and experimental techniques. Onthe other hand, very few studies have been published on externalconguration design of unguided missiles.1 The almost untouchedsynthesisproblemis challengingfor three reasons.First, design cri-teriaare functionsof aerodynamicand inertialparameters,which are

in turn complicated functions of freestream ow conditions,missilegeometry, and mass distribution. Second, design criteria are oftencontradictory.Third, design criteria are different for different ightphases of a given type of unguided missile. (Naturally, design cri-

teria are different for different types of unguidedmissiles. Artilleryshells,artillerymissiles,highkineticenergyprojectiles,lightassault

missiles, antitank missiles, sounding rockets, and re-entry vehiclesallhaveto be optimizedwith differentcostand constraintfunctions.)

The nature of the problem will be discussed with a simple ex-ample. Consider the unguided artillery missile conguration withcruciform tail ns presented in Fig. 1. The terms c an d s denote thechord and span of tail ns, respectively, while l denotes the length

of the midsection case. These are the geometrical parameters thatare easiest to modify once rocket motor and warhead properties arexed. If N different values of each of these parameters are consid-ered, then the total number of candidate congurationsto be exam-ined becomes N 3. Changing a single geometric parameter changesall aerodynamic and inertial data. The speed of an unguided ar-

tillerymissile variessignicantly during ight (subsonic,transonic,and supersonic). If M different ight speeds are considered, thenaerodynamic data have to be determined M N 3 times. Separateight dynamics analysis of all of these congurations is necessary,which is difcult and costly. Moreover, such an analysis gives in-

formation about congurations with the selected discrete values of (l c s ) only.

Consider an external conguration design analysis where onlys is changed. As s is increased, m I a I t C m C mq

C m p C l p

C l ,and C D all increase. An increase in s has both advantageous anddetrimental results in terms of range, dispersion, and stability per-

formance. Advantageousresults are an increasein static and damp-ing dynamic stability and a decrease in dispersion due to thrustmisalignment. Detrimental results are a decrease in range and an

increasein dispersiondue to wind. An increasein s can have advan-tageous or detrimentalresults in terms of Magnus dynamic stabilitybecause I a I t C mq

C m p C l p

, and C l all increase. An increase ins can have advantageouso r detrimental results in terms of the like-lihood of yaw-pitch-rollresonance because I a I t C m C l p

, and C lall increase.

Also notethat costand constraintfunctionsare differentfor boostand coast phases of ight of an unguided artillery missile. (As anexample, most of the dispersion takes place during the boost phase.

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314 TANRIKULU AND ERCAN

which is restricted to incompressible potential ow. Drag coef-

cient data of the 125 congurations were determined by using theMissileDATCOM databasefor a Mach numberof 0.32.The term C Dwas assumed to be related to x by the following functional relation-ship:

C D x c0 c1 x1 c2 x2 c3 x3 c4 x21

c5 x 22

c6 x23

c7 x1 x2 c8 x1 x3 c9 x2 x3 (7)

The x data were scaled by using x1 l lmax x2 c cmax ,and x3

s smax to improvethe qualityof curvetting.The ci i 0 1 9coefcients for C D were determined by using the curvetting utilityof Sigma Plot software.

A Turbo Pascal program was prepared to determine inertial andaerodynamic data of the light assault missile based on a s impliedsolid model, the results of the method of Bryson for aerodynamicstability derivatives,and the curvetted Missile DATCOM C D data.Valuesof the followingcost andi nequalityconstraintfunctionswerecalculated for the 125 congurations by using the program.

The rst cost function denoted by f 1 is related to Magnus insta-bility and yaw-pitch-roll resonance. Linear time-invariant aerobal-

listic theory has two important results related to the roll rate p of a

statically stable, slightly asymmetricunguided missile.4

First, if themagnitudeof p exceedsa certain limit denotedby pdyn during ight,thena Magnusinstabilitytakes place. Second,if the magnitude of p

coincideswith a certainvalue denotedby p res, then a yaw-pitch-rollresonance takes place. (Nonlinear aeroballistictheory predicts that

a linear resonancecan be followed by the lock in of p to pres , whichin turncan be followedby a severeinstabilityknown as catastrophicyawdue to nonlinearinducedaerodynamicmoments and congura-

tional asymmetries.) Closed-form expressions for pdyn a nd pres aregiven next:

pdyn

2V I t

I a

M

sd 2 sd

(8)

pres V M (9)

where

M 1 k 2t

C m

(10)

sd 2T H (11)

H C z 2C

D 1 2k 2t

C mq

C m

(12)

T 1 2k 2a

C m p

C z

C D

(13)

k a t I a t mT 2 (14)

C S 2m T C (15)

The magnitudeof pdyn is usuallythreeto vetimeslargerthanthemagnitude of pres . In externalcongurationdesign, it is desirable tohave a large difference between pdyn and pres b ecause one usuallytries to meet the pres p pdyn condition by using tail n cant.Hence, the rst cost function is selected as

f 1 pdyn pres

2V (16)

The second costfunctiondenotedby f 2 is relatedto rangeperfor-mance. Linear time-invarianta eroballistic theory predicts the vari-

ation of speed of an unguided missile in free straight ight to be

V V 0e C D

s (17)

where s is the nondimensionalarc length:

s 1

t

t 0

V dt (18)

Hence, the second cost function is selected as

f 2 C D m T (19)

The third cost function, denoted by f 3, is related to warhead

performance. A very simple criterion in terms of weapon systemeffectiveness is the ratio of warhead mass m to total mass. Hence,the third cost function is selected as

f 3 m m T (20)

The rst inequality constraint function, denoted by g1, is relatedto the static stability factor ss , which is dened as

ss C m C Z (21)

The magnitude of ss should be larger than a certain limiting value(which is usually taken as 1) for adequatestatic stability. In the case

of the light assault missile, a survey of s s data of 125 congura-tions showed that there were no problems in terms of static stabilitywi th 4 5 ss 5 5. Nevertheless, the rst inequality constraintfunction was selected as

g1 5 ss (22)

The reason for this selection is related to the dispersion of the lightassault missile, which is due to aiming errors and congurationalasymmetries.Nothing can be done to reduce dispersion due to aim-

ing errors in terms of external conguration design. On the otherhand,dispersiondue to congurationalasymmetries can be reducedby increasing ss .

The second inequality constraint function, denoted by g 2, is re-lated to the transverse damping factor, which is dened as

H 2 M (23)

The magnitudeof should be largerthan a certainlimiting value foradequate dynamic stability. In the case of the light assault missile,

a survey of data of 125 congurations showed that there were noproblems in terms of dynamic stability with 0 12 0 19.Nevertheless,the secondinequalityconstraintfunctionwas selected

as

g2 0 15 (24)

The reason for this selection is also related to the desire to keepdispersion due to congurational asymmetries below a certain

level.Polynomial functional relationships similar to Eq. ( 7) were as-

sumed to existbetween each f g, and x, coefcients of which weredetermined by using the curvetting utility of Sigma Plot software.Each cost function was scaled in such a way that the difference be-

tween its maximum and minimum values and its mean value wereboth equal to 1. The optimal external conguration for each costfunction was determined separately by using a Turbo Pascal pro-gram implementing the modied steepest-descent algorithm thatwas discussedin the precedingsection.[Four differentoptimizationproblemswere solvedwith f i i 1 2 3 f 4 f 1 f 2 f 3 g1

0 g2 0, and xl x xu .] Optimal x values for roll dynamic

stability performance f 1 , range performance f 2 , warhead per-formance f 3 , and composite cost function f 4 are presented inTable 2 with the corresponding values of the cost and inequalityconstraint functions.

The variations of f 1 g1, and g 2 with x 2 and x3 at x1 0 5 areshown in Figs. 3–5. The variations of f 2 f 3 f 4 g1, and g2 with x 2

and x 3 at x 1 1 are shown in Figs. 6–10. Figure 3 shows that the

lowest f 1 value is located at the upper-left-handcorner of the f 1 vs x2 and x 3 graph at x 1 0 5. Figures 4 and 5 show that inequalityconstraints g1 0 and g2 0 are both satised at that particularpoint. Similar visual inspection of Figs. 6–10 and an examinationof the cost and constraint function data of the 125 congurationshave shown the authors that correct optimal solutions were foundfor all fourof the optimization problems.No convergenceproblems

were observed in iterations with different initial conditions, whichresulted in the same optimal solutions.

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TANRIKULU AND ERCAN 315

Fig. 3 Variation of f 1 cost function with x2 and x3 for x1 = 0.5; f 1

(Magnus instability resonance).

Fig. 4 Variation of g1 inequalityconstraint function with x2 and x3 for

x1 = 0.5; g1 (static stability).


x1 = 0.5; g2 (damping dynamic stability).

Fig. 6 Variation of f 2 cost function with x2 and x3 for x1 = 1.0; f 2

(range).

Fig. 7 Variation of f 3 cost function with x2 and x3 for x1 = 1.0; f 3(warhead).

Fig. 8 Variation of f 4 cost function with x2 and x3 for x1 = 1.0; f 4(composite).

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316 TANRIKULU AND ERCAN


x1 = 1.0; g1 (static stability).

Fig. 10 Variation of g2 inequality constraint function with x2 a nd x3

for x1 = 1.0; g2 (damping dynamic stability).

Table 2 Optimal x values for f i (i = 1 2 3 4)

f 1 1 570 f 2 1 592 f 3 1 437 f 4 1 021g1 0 239 g1 0 363 g1 0 446 g1 0 682g2 0 045 g2 0 000 g2 0 000 g2 0 043

xopt

0 5000 6671 000

xopt

0 9960 8560 712

xopt

0 9970 6670 707

xopt

1 0000 6671 000

Conclusion

In this paper an optimal external conguration design methodfor unguided missiles was presented. The method is based on lin-ear time-invariant aeroballistic theory, and hence different phasesof ight are analyzed separately. Cost and constraint functions are

determined by using curvetted aerodynamicand inertial functionsbasedon a sparse set of geometricaldata.Optimal externalcongu-rationsare obtainedby using a modied steepest-descentalgorithm.The method was demonstrated through a representative case studybased on a light assault missile with subsonic free ight. Differ-ent co st functions that were s ubject to inequality constraints were

minimized successfully in a three-dimensionalgeometrical param-eter space. The method can be used as a simple and reliable tool in

the preliminary design stage of an unguided missile developmentproject.

There have been signicant advances in the analysis and systemidentication of unguidedmissiles since World War II. On the other

hand, the external conguration design of unguided missiles hasbeen neglecteddespite its obvious importance.Hence, a number of problemsthat require further investigationremain: optimalexternalcongurationdesign based on nonlinear time-invariantaeroballistictheory and the developmentof a method that can determinethe bestconguration for the whole of a specied mission rather than thebest congurations for different phases of the mission.

References1 Cayzac, R., and Carette, E., “Parametric Aerodynamic Design of Spin-

ning Finned Projectiles Using a Matrix Interpolation Method,” Journal of Spacecraft and Rockets, Vol. 29, No. 1, 1 992, pp. 51–56.

2 Pierre, D. A., OptimizationTheory with Applications, Dover, New York,

1986, pp. 264–366.3 Nielsen, J. N., Missile Aerodynamics, Nielsen Engineeringand Research,

Inc., Mountain View, CA, 1988, pp. 349–431.4 Murphy, C. H., “Free Flight Motion of S ymmetric Missiles,” Ballistic

Research Lab., Aberdeen ProvingGround, Rept. 1216, Aberdeen, MD, July

1963.

J. R. Maus Assoc iate Editor

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