Aiaa-1997-3725-958 Optimal External Configuration Design Of Unguided Missiles

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

AIAA-97-3725 OPTIMAL EXTERNAL CONFIGURATION DESIGN OF UNGUIDED MISSILES Omer Tannkulu* & Veysi Ercant

A97-37313

Defense Industries Research and Development Institute (TUBITAK-SAGE) Mechanics and Systems Engineering Research Group (MSMG) PK16, Mamak 06261, Ankara, Turkey

Abstract In this paper a simple optimal external

respectively and two constraint functions related to stability are considered in the case study.

configuration design method is proposed that can be used in conceptual and preliminary design stages of

Nomenclature

an unguided missile development project. Cost and constraint functions are derived from the results of

Drag force coefficient. C,

linear time-invariant aeroballistic theory (constant

Roll damping moment stability derivative.

roll rate and forward speed, de-coupled axial and

Roll moment due to fin cant stability

transverse dynamics); therefore different phases of derivative.

flight are examined separately. Curve-fitting is used to reduce number of trial cases and hence work

c.

Static moment stability derivative.

required to obtain aerodynamic and inertial data.

Transverse damping moment stability

Optimal configurations are determined by using a

derivative.

modified steepest descent algorithm. A case study is presented in which external configuration of an unguided light assault missile is optimized for free

cm

Transverse damping moment stability derivative. Magnus moment stability derivative.

flight at low subsonic speed. Three cost functions Static force stability derivative. related to stability, range and warhead performance * Coordinator. Member AIAA. t Research Engineer, Flight Mechanics Section. Copyright © 1997 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

700 American Institute of Aeronautics and Astronautics

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

d

Gradient vector of F(x).

/.(*)

Cost function.

A

:

Reference length, rocket motor diameter.

p^

:

Free stream density.

H, v

:

Penalty constants.

f

:

Transverse damping factor.

Modified cost function. Inequality constraint function. *,(*)

Equality constraint function.

Hessian matrix of F(X) .

Axial moment of inertia.

Introduction

The main objectives in external configuration

Transverse moment of inertia.

design of unguided missiles are to obtain adequate

Non-dimensional axial radius of

stability in all phases of flight, short minimum range, long maximum range, low dispersion and

gyration. Non-dimensional transverse radius of

large payload mass. In practice it is difficult to

achieve these objectives due to complicated nature of gyration.

unguided missiles as nonlinear, time-varying and m

Total mass. random systems. Significant advances have been Warhead mass. made in analysis and system identification aspects of

Pdyn

Magnus instability roll rate limit.

Yaw-pitch-roll resonance roll rate.

flight dynamics of unguided missiles since World War II. A number of range, dispersion and stability

Dynamic stability factor.

criteria have been determined by using analytical,

Static stability factor.

numerical and experimental techniques. On the other

S

Reference area, ;r/l2/4.

hand, very few studies have been published on

V

Speed of missile. Vector of scaled variable geometrical

external configuration design of unguided missiles1.

The almost untouched synthesis problem is challenging due to three reasons: Firstly, design

parameters. Lower limit of x .

criteria are functions of aerodynamic and inertial parameters which are in turn complicated functions

Upper limit of x . of free stream flow conditions, missile geometry and

a

Adaptive step size of iterations. mass distribution. Secondly, design criteria are often

701 American Institute of Aeronautics and Astronautics

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

contradictory. Thirdly, design criteria are different

flight (subsonic, transonic, supersonic). If M

for different flight phases of a given type of unguided

different

missile. (Naturally, design criteria are different for

aerodynamic data has to be determined MX N3 times.

different types of unguided missiles. Artillery shell,

Separate flight dynamics analysis of all of these

artillery missiles, high kinetic energy projectiles,

configurations is necessary which is difficult and

light assault missiles, anti-tank missiles, sounding

costly. Moreover, such an analysis gives information

rockets and re-entry vehicles all have to be optimized

about configurations with the selected discrete values

with different cost and constraint functions.)

of (/, c, s)only.

flight

speeds are considered, then

Nature of the problem will be discussed by a

Consider an external configuration design

simple example. Consider the unguided artillery

analysis where only s is changed. As s is increased

missile configuration with cruciform tail fins

m, Ia ,

presented in Figure 1:

, C,r, Cls and CD all

increase. An increase in s has both good and bad results in terms of range, dispersion and stability performance. Good results are increase of static and

Warhead

damping dynamic Rocket Motor

Figure 1. Unguided artillery missile configuration.

stability, and

decrease of

dispersion due to thrust misalignment. Bad results

are decrease of range and increase of dispersion due to wind. Increase of s can have good or bad results

c and s are chord and span of tail fins

in terms of Magnus dynamic stability since Ia , It,

respectively while / denotes length of mid-section

Cm
case. These are the geometrical parameters that are

easiest to modify once rocket motor and warhead

good or bad results in terms of the likelihood of yawpitch-roll resonance since Ia , It, Cm^ , C, and Clg

properties are fixed. If N different values of each of

these parameters are considered, then total number of candidate configurations to be examined becomes N*. Change of a single geometric parameter changes all aerodynamic and inertial data. Flight speed of an unguided artillery missile varies significantly during

all increase.

One must also note that cost and constraint functions are different for boost and coast phases of

flight of an unguided artillery missile. (As an example, most of the dispersion takes place during boost phase. Hence, dispersion criteria may not be

702 American Institute of Aeronautics and Astronautics

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

considered among cost and constraint functions in a

data by using curve fitting. This approach reduces

first order analysis of coast phase.)

number of trial cases significantly and it also creates

In the next section, details of a method that

the possibility of analysing configurations with x

can be used in optimal external configuration design

data other than the original sparse set. Aerodynamic

of unguided missiles will be presented This will be

data at a given Mach number and inertial data are

followed by a case study about a light assault

usually smooth functions of geometrical parameters;

unguided missile.

hence, polynomial functions give satisfactory results in terms of curve fitting.

Optimal External Configuration Design

The proposed method for optimal external

• Step#4:

Determination

ft (x)(i = 1,2,3, ...,Nf),

of

cost

inequality

functions

constraint

configuration design has four basic steps:

functions

gl(x^i = l,2,3,...,Ng^

and

equality

• Step#l: Investigation of characteristics of the

unguided missile that is to be optimized.

constraint functions hl(x%i = 12,3,...,Nh) by using

Determination of cost and constraint functions for

the approximate aerodynamic and inertial functions

different phases of flight. Determination of lower and

that were obtained in Step-#3. The optimization

upper limits for variable geometrical parameters with

problem is formulated as follows:

special

emphasis

given

to

producibility

considerations.

• Step#2: Determination of aerodynamic data (at

Minimize:

//(*)

Subject to:

g,(x)^0,(i = 1,2,3,...,Ng)

several Mach numbers depending on flight speed range of the unguided missile) and inertial data for a sparse set of variable geometrical data by using

ht(x) = 0,(i = 1,2,3,...,Nh) x,<x<xu

experimental, numerical and theoretical methods. • Step#3: Determination of approximate functional

In above expressions x, and xv are lower and upper

relationships between aerodynamic data at different

limits of x that are usually dictated by producibility.

Mach numbers and geometric data by using curve

Optimal configurations for different cost and

fitting. Determination of approximate functional

constraint functions are determined by using the

relationships between inertial data and geometric

703 American Institute of Aeronautics and Astronautics

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

following

iterative

modified

steepest

descent

algorithm2:

iteration, a is the adaptive step size of iterations wmle

j

is the

gradient vector and H is the Hessian

matrix:

a = vr * d.I ~ ——— s+

ak = _ T

-

(2)


(3)

Case Study

9

(5)

(6)

The subject matter of this representative case study is optimal external configuration design of an unguided short range light assault missile, schematic (*/),.'/(**+>),

drawing of which is shown in Figure 2. The one man

portable missile is shoulder launched and there is no

rocket motor propulsion once the missile leaves its tube launcher. The original anti-armour shaped (A: = 1,2,3,...).

charge warhead of the missile is to be replaced by another warhead as a part of a product improvement

In above expressions F is the modified cost function which is obtained by combining cost and

constraint functions

to turn the constrained

optimization problem into an unconstrained one. x is the vector of variable geometric parameters which has a dimension of n. // > 0 and v > 0 are penalty

constants of very large magnitude for inequality and

equality constraints respectively. Magnitudes of //

project. No modifications will be performed in terms

of solid propellant rocket motor since mass of the new warhead is constrained to be less than or equal to mass of the old one. Free flight Mach number of

the missile is 0.32 with the old warhead. The missile will have a low subsonic flight speed with the new warhead as well. The missile has six planar tail fins with no dihedral which means that it has six-gonal

and v are increased in every couple of steps of 704 American Institute of Aeronautics and Astronautics

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

rotational symmetry and six planes of mirror

section. Lower and upper limits of these parameters

symmetry3.

that were determined by taking producibility and warhead mass constraints into account are presented in Table 1. Five equally spaced values of each parameter were used to obtain a sparse set of 125 configurations.

39 215

50

50

Table 1. Upper and lower limits of variable

geometrical data.

Figure 2. Unguided light assault missile. All

dimensions are in millimeters.

Lower Limit

Upper Limit

/ [mm]

30

60

There are three reasons while such a missile

c [mm]

10

15

was selected as the subject of this case study: Firstly,

s [mm]

50

100

there is only one flight phase to be considered (short

duration free flight during which the missile has a

Closed

form

expressions

that

relate

relatively straight mean trajectory). Secondly, the

aerodynamic stability derivatives to geometry were

flight takes place at low subsonic Mach numbers

obtained by using Bryson's method which is

where aerodynamic characteristics are almost

restricted to incompressible potential flow. Drag

constant, and hence only one set of aerodynamic data

coefficient data of the 125 configurations were

has to be determined. Thirdly, it is possible to derive

determined by Missile DATCOM data base for a

approximate closed form expressions for a large

Mach number of 0.32. CD was assumed to be related

number

of

aerodynamic

stability

derivatives

to x by the following functional relationship:

3

(including Cm !) by using the method of Bryson for

flight at low subsonic Mach numbers.

CD(x) =

Three geometrical parameters of the light assault missile were selected to be variable (7)

parameters of the optimization problem: Span s and chord c of tail fins and length / of warhead mid-

705 American Institute of Aeronautics and Astronautics

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

x data were scaled by using xl = l/lmax, x2 = c/cmax

asymmetries.) Closed form expressions for

and x3 - s/smm to improve quality of curve-fitting.

pres are given below:

and

c,(/= 0,1,...,9) coefficients for CD were determined by using curve-fitting utility of Sigma Plot software.

M

2V I,

(8)

A Turbo Pascal program was prepared to

determine inertia! and aerodynamic data of the light V

assault missile based on a simplified solid model,

(9)

results of Bryson's method for aerodynamic stability derivatives and curve fitted Missile DATCOM CD

where,

data. Values of the following cost and inequality constraint functions were calculated for the 125 kt2

configurations by using the program: •

(10)

f i : The first cost function is related to Magnus

instability and roll-pitch-yaw resonance. Linear time-

_ 2T

(11)

invariant aeroballistic theory has two important results related to roll rate p of a statically stable (12)

slightly asymmetric unguided missile4. Firstly, if

magnitude of p exceeds a certain limit denoted by pdyn during flight, then a Magnus instability takes

1 T = ——=

-cmpp'-cz;-cD\

(13)

place. Secondly, if magnitude of p coincides with a

certain value denoted by pres then a yaw-pitch-roll resonance takes place. (Non-linear aeroballistic

(14)

ka , =

theory predicts that a linear resonance can be

followed by lock-in of p to pns which in turn can be followed by a severe instability known as catastrophic yaw due to nonlinear induced aerodynamic

moments

and

configurational

706 American Institute of Aeronautics and Astronautics

2m

(15)

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

Magnitude of piyn is usually three to five

• /3: The third cost function is related to warhead

times larger than magnitude of pres. In external

performance. A very simple criterion in terms of

configuration design, it is desirable to have a large

wea

difference between

*"***

tries to meet

Pdyn

P°n Wstem effectiveness is the ratio of warhead

and Pres since one usually

Pns
condition by using tail

m

«

to total mass

Hence

-

>

the thirf cost

functionis selectedas

'

fin cant. Hence, the first cost function is selected as, /s=—-•

m

/,=-

2V

(20)

(16)

•

g[: The first inequality constraint function is

related to the static stability factor sf which is • / 2 : The second cost function is related to range defined as, performance. Linear time-invariant aeroballistic theory predicts variation of speed of an unguided s. =

missile in free straight flight to be,

*

(17)

Cz« '

(21)

c

Magnitude of ss should be larger than a certain limiting value (which is usually taken as 1) for

where 5 is the non-dimensional arclength:

adequate static stability. In case of the light assault missile a survey of s3 data of 125 configurations (18)

showed that there were no problems in terms of static stability, ~ 4.5 < ss <~ 5.5. Nevertheless, the first inequality constraint function was selected as,

Hence, the second cost function is selected as, 81 = ^D

m

(22)

(19)

The reason for this selection is related to dispersion of the light assault missile which is due to aiming 707

American Institute of Aeronautics and Astronautics

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

errors and configurational asymmetries. Nothing can

of cost and constraint functions with 5 and c at the

be done to reduce dispersion due to aiming errors in

maximum value of / are shown in Figure 3.

terms of external configuration design. On the other

Optimal external configurations for the three

hand, dispersion due to configurational asymmetries

cost functions were determined by using a Turbo

can be reduced by increasing s,.

Pascal program that implemented the modifed

•

steepest descent algorithm discussed in the previous

g 2 : The second inequality constraint is related to

section. Optimal (scaled) x values are presented in

transverse damping factor which is defined as,

Table 2:

H

(23)

Table 2. Optimal x values for ft(i = 1,2,3). /I

ft

/3

*1

0.997

0.996

0.997

X2

0.993

0.813

0.667

*3

0.715

0.711

0.707

Magnitude of £ should be larger than a certain

limiting value for adequate dynamic stability. In case of the light assault missile a survey of ^ data of 125

configurations showed that there were no problems in terms of dynamic stability, ~0.12<£<~0.19. Nevertheless, the second inequality constraint

Values of the cost functions at the optimal x values are

function was selected as,

/, = -0.82108,

/2 = 1.59146

and

/3 = -1.43706 respectively. (ft values are scaled in (23)

such a way that the difference between maximum and minimum values and mean are both equal to

The reason for this selection is also related to the

one.)

desire for keeping dispersion due to configurational Discussion

asymmetries below a certain level. Polynomial functional relationships similar to (7) were assumed to exist between each /, g and x , coefficients of which were determined by using curve fitting utility of Sigma Plot software. Variation

• An examination of Figure 3 shows that the correct optimal value were obtained in each case.

• In all cases xl =~ 1, / =~ lmm for the optimal configurations.

708 American Institute of Aeronautics and Astronautics

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

• In all cases

x2 =~ 0.7

for the optimal

configurations. Value of optimal span is specifed by

mission, rather than the best configurations for

different phases of the mission.

the damping factor inequality constraint g 2 .

References

• The static stability inequality constraint g, is overruled by the damping factor inequality constraint

^ayzac,

R.

and

Carette,

E.,

"Parametric

Aerodynamic Design of Spinning Finned Projectiles g2 fOTXl=l,l

= lmaic.

Using a Matrix Interpolation Method," AIAA

• No convergence problems were observed in Journal of Spacecraft and Rockets, Vol. 29, No. 1,

iterations to find the optimal x in all cases. January-February 1992. Conclusion

2

Leblebicioglu, K., EE-553 Optimization Course

There are a large number of problems that Lecture Notes, Electrical Engineering Department,

remain to be investigated in terms of external Middle East Technical University, 1994. configuration design of unguided missiles:

• Extension of the current method so that a number

3

Nielsen, J. N., Missile Aerodynamics, Nielsen

of cost functions can be optimized at the same time.

Engineering & Research, Inc., Mountain View, (Multiple objective constrained optimization.) California, 1988. • Optimal external configuration design analysis of

other types of unguided missiles (artillery shell, artillery missiles, high kinetic energy projectiles, anti-tank missiles,

sounding rockets,

4

Murphy, C. H., "Free Flight Motion of Symmetric

Missiles", Ballistic Research Laboratories, Aberdeen

re-entry

Proving Ground, Report No. 1216, July 1963.

vehicles). • Optimal external configuration design analysis

with a much larger number of variable parameters. • Optimal external configuration design analysis of cases where nonlinear aeroballistic phenomena are observed. • Development of a method that can determine the

best configuration for the whole of a specified 709 American Institute of Aeronautics and Astronautics

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

f, (Magnus Instability-Resonance)

9, (Static Stability)

,.-0.8-''

0.70

0.75 0.80

0.85 0.90 0.95 1.00

x, (Chord)

f2 (Range) -—0.3-

—-Q.2—

1.0 -T————1.8-

0.70

0.75 0.80 0.85 0.90

0.95 1.00

X2 (Chord)

g2 (Damping Dynamic Stability)

o.°S

0.70

0.75 0.80

0.85 0.90 0.95 1.00

i, (Chord)

o.*f, (Warhead)

-o-*

0.70

0.75 0.80 0.85

0.90

x, (Chord)

I x"

0.70

0.75 0.80

0.85 0.90 0.95 1.00

Xj (Chord)

Figure 3. Variation of cost and constraint functions with c and s for / 710 American Institute of Aeronautics and Astronautics

.

0.95 1.00