~LASSICAL FLIGHT DYNAMICS OF A .

VARIABLE FORWARD SWEEP

WING AIRCRAFT

By

BRETT ALAN NEWMAN I[

Bache lor of Science

Oklahoma State University

Stillwater, Oklahoma

1983

Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May, 1985 lh.es·lS lq~~ \'\ 56'lc. C.D\'•~ CLASSICAL FLIGHT DYNAMICS OF A

VARIABLE FORWARD SWEEP

WING AIRCRAFT

Thesis Approved:

r£1?Yr~. r=~~

Dean of the Graduate College

ii 1216348 ACKNOWLEDGMENTS

I want to thank the following people and organizations for their assistance in completing this research effort: Dr. R. L. Swaim for his experience and guidance throughout the research; Dr. F. 0. Thomas for his helpful suggestions and thoughts; Dr. J. K. Good for serving as a committee member; Mr. s. Mohan for his numerical and graphics computer program library and assistance in using the Hewlett-Packard 9000 minicomputer and the Theoretical Mechanics Branch, Langley Research

Center, National Aeronautics and Space Administration, for their vast

resources.

Thanks also go to Mr. and Mrs. G. E. Newman and Mr. M. Maxwell for

their encouragement and support during this research effort and the four years of college leading up to this effort. Finally, much appreciation goes to Mrs. G. E. Newman for typing this thesis.

iii TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION

Variable Forward Sweep 1 Research Scope •• 3

II. LITERATURE SURVEY. 5

Variable Sweep. 5 Forward Sweep • • 7

III. CONFIGURATION DEVELOPMENT. 9

General Description ••••••• 9 Design Method • • 11 Wing Description. • • • • • • • • • 15

IV. ANALYSIS METHOD ••• 18

Equations of Motion Solution. 18 Aircraft Force and Moment Calculations. 26 Element Force and Moment Calculations • 33 Trim Calculations • • • • • • • • • • • 35

v. RESULTS AND DISCUSSION • 38

Preliminary Discussion. 38 Aerodynamic Center and Center of Gravity. 39 Characteristic Root Locus • • • • • • • 43 Mode Shape and Time Response ••••• 46

VI. CONCLUSIONS AND RECOMMENDATIONS. 50

Conclusions ••• 50 Recommendations • 51

BIBLIOGRAPHY. 52

APPENDIXES •• 53

APPENDIX A - CONFIGURATION SPECIFICATIONS • 54

APPENDIX B - AIRCRAFT STABILITY AND CONTROL DERIVATIVES 62

APPENDIX C - ELEMENT STABILITY AND CONTROL DERIVATIVES. 71

! APPENDIX D - GRAPHICAL RESULTS. • • • • • • • • • • • • 78

iv LIST OF TABLES

Table Page

I. SPEED AND ALTITUDE CONDITIONS INVESTIGATED IN THE RESEARCH. 38

v LIST OF FIGURES

Figure Page

1 • Variable Forward Sweep Wing Aircraft at -30° Sweep. • •••••• 10

2. Variable Forward Sweep Wing Aircraft in a Wing's Level Steady State Rectilinear Flight Condition ••••••.• 13

3. Right Variable Forward Sweep Wing. 16

4. Body Coordinate Systems Used During the Research Effort. • • • 20

5. Element Longitudinal Forces and Moments. • • • • • 27

6. Element Lateral-directional Forces and Moments. • • • • • • • • 28

7. Longitudinal Transformation To the Stability Axes • • • • • 30

8. Lateral-directional Transformation To the Stability Axes. • • • • • • •• 31

9. Typical Aerodynamic Center and Center of Gravity Locations Versus Sweep for a Variable Forward Sweep Design 41

1 o. Typical Aerodynamic Center and Center of Gravity Locations Versus Sweep for a Variable Backward Sweep Design. 42

11 • Vertical Stabilizer Plan form Definitions. 57

1 2. Wing Geometry for Center of Gravity and Moments of Inertia Calculations 59

vi LIST OF SYMBOLS

English Symbols:

A aspect ratio or area

b span

X axis aerodynamic force coefficient

y axis aerodynamic force coefficient

z axis aerodynamic force coefficient

drag coefficient

lift coefficient

c1 roll coefficient

C pitch coefficient m C yaw coefficient n CT x axis thrust force coefficient X CT x axis thrust force coefficient x CT y axis thrust force coefficient y CT y axis thrust force coefficient y CT z axis thrust force coefficient z cT_ z axis thrust force coefficient z C side force coefficient y c chord

c mean aerodynamic chord

c airfoil pitch coefficient m D drag

d diameter

vii SYMBOLS (Continued) e aerodynamic efficiency factor

F force

side force

X axis perturbation aerodynamic force f y axis perturbation aerodynamic force A y f z axis perturbation aerodynamic force A z X axis perturbation thrust force

y axis perturbation thrust force

z axis perturbation thrust force g gravity h altitude

I incidence angle or moment of inertia

x axis aerodynamic moment coefficient

K y axis aerodynamic moment coefficient A y z axis aerodynamic moment coefficient

x axis thrust moment coefficient

K x axis thrust moment coefficient T­x y axis thrust moment coeffi:;ient

y axis thrust moment coefficient

z axis thrust moment coefficient

z axis thrust moment coefficient

L lift or roll moment

1 length

M Mach number or pitch moment m mass m x axis perturbation aerodynamic moment A X

viii SYMBOLS (Continued)

y axis perturbation aerodynamic moment

z axis perturbation aerodynamic moment m X axis perturbation thrust moment T X y axis perturbation thrust moment

z axis perturbation thrust moment

N yaw moment q dyn~mic pressure s area s Laplacian variable sm static margin

T thrust force t thickness

wing downwash lag time

vertical stabilizer sidewash lag time v velocity or volume v perturbation velocity

X distance along X

distance along x

distance along x

X • distance along X ' ,, ., X distance along X

X"' distance along XN y distance along y

distance along y

distance along y y' distance along y '

ix SYMBOLS (Continued)

,, y " distance along y

y '" distance along y ~-~

z distance along z

distance along z

distance along z

; z ' distance along z ,, distance along z

z "' distance along z'"'

Greek Symbols:

angle of attack

perturbation angle of attack

i3 perturbation sideslip angle or compressibility factor

control deflection

perturbation control deflection

downwash

damping ratio

n dynamic pr~ssure ratio or percent location of control

surface rout or tip chord

8 pitch angle

8 perturbation pitch angle

K airfoil lift curve slope ratio

sweep angle

taper ratio

p density

a sidewash

time constant

X SYMBOLS (Continued)

perturbation roll angle

perturbation yaw angle

w angular velocity or frequency

Superscript Symbols:

A aileron

B body

E eleyator, engine or exhaust

F flap or fuel

H hori?.ontal stabilizer

I intake

p payload

R rudder

T thrust

v vertical stabilizer

w wing

-w aircraft excluding wing

Subscript Symbols:

A aileron

AC aerodynamic center

Al aileron inboard chord

A2 aileron outboard chord

base base of body

CG center of gravity

DR dutch-roll mode

E elevator

xi SYMBOLS (Continued)

El elevator inboard chord

E elevator outboard chord 2

F flap

Fl flap inboard chord

F2 flap outboard chord

LE leading-edge max maximum min minimum nose nose of body p phugoid mode or wing pivot

R rudder or rright

Rl rudder inboard chord

R2 rudder outboard chord root root chord s sonic, spiral mode or structure stall stall angle of attack

SP short period mode wet wetted area wet wetted area excluding body base ex. 1 .:tse

X along X axis x along x axis y along y axis y along y axis z along z axis z along z axis v partial derivative with respect to v X X a partial derivative with respect to a

xii SYMBOLS (Continued)

. • Cl. partial derivative with respect to Cl.

13 partial derivative with respect to 13 • • 13 partial derivative with respect to 13

oA partial derivative with respect to oA

oE partial derivative with respect to oE

oF partial derivative with respect to oF

oR partial derivative with respect to oR

oT partial derivative with respect to oT

w partial derivative with respect to w X X w partial derivative with respect to w y y w partial derivative with respect to w z z 0 zero lift

1 steady state

Mathematical Symbols:

d ( or • derivative

partial derivative

evaluated at steady state

matrix

{} vector

vector notation

Unit Symbols:

kg kilogram

m meter

N Newton

rad radian

xiii SYMBOLS (Continued) s second

0 degree

xiv CHAPTER I

INTRODUCTION

Variable Forward Sweep

The reemerging forward sweep wing is generating a growing interest in the aerospace community. cne can see from a brief survey of the current aerospace research literature, the advantages and di sad vantages of forward sweep are being throughly investigated for future applications. Research indicates forward sweep is feasible and will he used in future aircraft designs.

A conceivable variation of forward sweep is the merging of forward sweep with variable sweep. Both forward sweep and variable sweep offer significant gains in performance by themselves. The coupling of forward sweep and variable sweep could possibly offer even higher performance levels.

A conventional variable sweep aircraft with backward sweep, or positive sweep angles, is typically designed with minimum longitudinal stability when minimum sweep and subsonic speeds occur. The aircraft's aerodynamic center is at its most forward .location in this flight con di ti on. The aerodynamic center and center of gravity both move backward as the sweep is increased, with the aerodynamic center typically moving farther back than the center of gravity. Excessive longitudinal stability which degrades performance and maneuverability can occur at the maximum sweep angle. Supersonic speeds increase the

1 2

stability even more by the characteristic backward shift of the aerodynamic center. Elevator control effectiveness reductions can also occur at the maximum sweep angle due to large longitudinal wing

moment arms.

The variable forward sweep wing aircraft would logically be

designed with minimum longitudinal stability when maximum sweep and

supersonic speeds occur.. As the sweep decreases or becomes less negative, the aerodynamic center and center of gravity are expected to

both move backward, with the aerodynamic center typically moving farther

back than the center of gravity. The longitudinal stability will

increase with decreasing sweep until subsonic speeds occur. At that

time, the stability will decrease due to the characteristic forward

shift of the aerodynamic center. If this reduction in stability is

significant, the forward variable sweep aircraft will not be hampered as

much by the excess stability and control shortage problems that occur

with backward variable sweep aircraft.

The lateral-directional characteristics will also be affected by

the wing sweep angle. The roll characteristics can be expected to

experience the predominate change among the lateral-directional

characteristics. For the variable forward sweep wing aircraft, the wing

mean aerodynamic chord and aileron moment arm both move inboard as the

sweep in creases or becomes more negative; consequently, the roll damping

and roll response should be reduced, respectively. The inboard movement

of the mean aerodynamic chord with increasing sweep should also be

expected to reduce the yaw damping.

The longitudinal and lateral-directional characteristics will also

be affected through changes in the moments of inertia due to variable

sweep. For the variable forward sweep wing aircraft, increasing or more 3 negative wing sweep should result in pitching and rolling moment of

inertia reductions. These reductions should tend to improve the

pitching and rolling responses.

A variable forward sweep wing aircraft is an interesting and

formidable concept to analyze. For the reasons discussed in this

section, the analysis of the flight dynamics of a variable forward sweep

wing aircraft was selected as the Master Of Science thesis topic.

Research Scope

The level of complexity of the research is a classical aircraft

stability and control analysis such as pre sen ted by Roskam ( 1 ) •

Assumptions include rigid body, linearized dynamics and aerodynamics,

ideal control surface actuators and no automatic flight c'ontiol system.

The flexibility, linearization, actuator dynamics and automatic flight

control system assumptions are introduced because 1) the research is

in tended to analyze the effects

resulting solely from the variable forward sweep concept and 2) to

simplify the analysis.

The equations of motion are uncoupled into a longitudinal set and a

lateral-directional set. Laplace transformation and partial fraction

expansi

motion. Numerical calculations are performed by programming the

equations of moti

of the characteristic equation root loci, mode shapes and time responses

at various wing sweep angles and dynamic pressures. The International

System of metric units (SI) .is used exclusively throughout this thesis. 4

Areas to be addressed in the research include aerodynamic center movement, center of gravity movement, characteristic equatioo root loci behavior, mode shape behavior, time response behavior and dynamic pressure and Mach number effects. The overall goal is to 1) relate the flight dynamics results and trends with the physics of the variable forward sweep concept and 2) indicate any significant advantages or disadvantages associated with the variable forward sweep concept. CHAPTER II

LITERATURE SURVEY

Variable Sweep

Variable sweep is a design which allows an aircraft's wing sweep angle to be varied during flight and is used to achieve acceptable performance levels throughout the flight envelope. Low and moderate subsonic speeds demand a high aspect ratio, lowly swept wing. High aspect ratio, lowly swept wings result in efficient lift curve slopes and low takeoff and landing speeds. High subsonic speeds and transonic speeds demand a moderate aspect ratio, moderately swept wing to achi~ve acceptable lift to drag ratios for cruising flight and to delay compressibility drag. Supersonic speeds demand a low aspect ratio, highly swept wing. Low aspect ratio, highly swept wings result in weaker oblique shock waves rather than stronger normal shock waves. variable sweep is a unique and proven technique which satisfies th~ diversified wing characteristics required over an aircraft's flight envelope.

A complete history of variable sweep can be found in the papers by

Polhamus and Toll (2) and Kress (3). The National Advisory Committee for Aeronautics (NACA) conducted the first extensive research an variable sweep for the purpose of achieving sui table performance levels throughout an aircraft's flight envelope ( 2). Early wind tunnel research indicated a fixed wing pivot located inboard the aircraft's

5 6 fuselage forced extreme wing aerodynamic center travel as the sweep angle varied. With backward sweep, excessive static margins and longitudinal stability resulted. Supersonic speeds aggravated the problem even more by the characteristic backward shift of the aerodynamic center.

The X-5 was the first variable sweep aircraft to fly ( 2). The X-5 employed variable backward sweep varying from 20° to 60° with an inboard, translating wing pivot to alleviate the excessive stability problem. Flight tests indicated no significant improvements existed over fixed wing aircraft which was later found to result from a poor design. The XF10F-1 was the next variable sweep aircraft to fly (3).

The XF10F-1 also employed variable backward sweep varying from 12.5° to

42.5° with an inboard translating wing pivot. Flight tests indicated significant improvements in landing speeds, range and maximum speeds relative to similar fixed wing aircraft.

NACA continued research on variable sweep and discovered in the late 1950's that a fixed forewing and a fixed wing pivot located within the forewing would solve the problem of excessive stability without the complexity of a translating wing pivot (2). Many aircraft usinc; this

technique have been produced including the Grumman F-14, General.

Dynamics F-111 and Rockwell B-1.

All variable sweep aircraft produced to date have employed backward sweep. No research could be found an variable forward sweep except for

the mentioning by Polhamus and Toll (2) that variable forward sweep is an unestablished design option. 7

Forward Sweep

Forward sweep is a design where wings a:.:·e swept forward to counter certain high speed aerodynamic characteristics. Backward sweep also counters these characteristics. Sweep is used to delay compressibility drag associated with high subsonic and transonic speeds. Sweep also reduces drag at supersonic speeds by causing the formaticn of weaker oblique shock waves rather than stronger normal shock waves.

The advantages of wing sweep were first realized by Germany in the

1930's (2). The first swept wing aircraft, including both forward and backward sweep, were flown by Germany during World War II. Research from NACA by Diederick and Budiansky (4) in the late 1940's showed the divergence dynamic pressure of forward sweep wings are drastically low compared to the divergence dynamic pressure of backward sweep wings.

Consequently, all feasible swept wing designs throughout the mid 1970's employed backward sweep. The forward sweep concept was ignored up to this time until Krone (5) revealed aeroelastic tailoring with composite rna terials can increase the low divergence limit of forward sweep wings without experiencing the weight penalties associated with the use of conventional metal materials. The solution to the divergence problem has led to the development of the forward sweep X-29 research aircraft which is currently undergoing flight tests conducted by the Defense

Advanced Research Projects Agency, National Aeronautics and Space

Administration and United States Air Force (USAF). Moore and Frei ( 6) give a descripticn of the X-29 and its purpose.

The increased interest in using forward sweep cnce the structural feasibility is verified is due to certain advantages forward sweep has 8 over ba.ckward sweep. Increased lift, reduced drag, improved handling qualities and increased design freedom are some of the major advantages listed by Krone (7). The advantages are difficult to quantify without compariscn to a backward sweep wing with certain parameters constrained to equal the forward sweep wing's corresponding parameters. CHAPTER III

CONFIGURATION DEVELOPMENT

General Description

Figure 1 shows the variable forward sweep wing aircraft studied

throughout this research effort. The aircraft incorporates a variable

forward sweep wing with sweep angles ranging from -42° to -15°. The wing is pivoted about a fixed pivot located directly above the body

centerline. This vertical offset from the body centerline was required

due to the intemal spacing requirements of the engine. The horizontal

stabilizer is a canard while the vertical stabilizer is a single,

con ven ticnal vertical fin. Coo ven tianal aerodynamic can trol surfaces

were selected: elevators on the horizontal stabilizer, rudder on the

vertical stabilizer and flaps and ailerons en the wing. The propulsion

system consists of a single turbofan er gine located in temal to the body

and exhausting out the base of the body. In take for the propulsion

system consists of a single scoop located on the body underside.

The aircraft is a representative of the light weight fighter

category similar to the Northrop F-5 or F-20. The general shape and

size of the aircraft is quite similar to the X-29 research aircraft.

This similarity is deliberate because the X-29 is the cnly existing high

speed aircraft using a forward sweep angle more negative than -15°. The

X-29's basic airframe is unstable; however, by careful selection of the

horizontal stabilizer and wing locations relative to the center of

9 10

Scale: lm=lcm

c:: <>

Figure 1. Variable Forward Sweep Wing Aircraft at -3r:f Sweep 11 gravity the basic airframe shown in Figure 1 is stable.

The aircraft was designed with a static margin of -0.25 for a sweep angle of -30°, Mach number of 0. 75 and altitude of 6000 m. No design iterations or alterations to improve the aircraft's characteristics for other flight conditions were performed after the off design results were obtained. Instead, the changes in the aircraft's characteristics resulting from changes in the flight condition are the results to be analyzed for this research effort. The aircraft is capable of obtaining low supersonic Mach numbers in level flight. The aircraft's aerodynamic performance such as range, endurance, takeoff and landing speeds, rate of climb, ceiling and maximum speed were not considered in the design.

Appendix A contains the specifications for the aircraft.

Design Method

The teclmique of "design extrapolation" is used to determine the aircraft's basic size and geometry. The size and geometry were extrapolated from the X-29, F-5 and F-20 aircraft. Size and geometry includes the lifting surface and stabilizer planform areas, taper ratios, spans and sweep angles. Body length and diameter, mass, moments of inertia and propulsicn are also included in the size and geometry.

The flight condition of -30° sweep angle, Mach number of 0. 75, and altitude of 6000 m was selected as the design flight condition. This flight condition represents a high subsonic cruise condition where an aircraft of this category will typically spend the majority of it's flight time. References (6), (8), (9) and (10) were used in the "design extrapolaticn" calculations.

The fundamental design consisted of locating the longitudinal 12

position of the wing and horizontal stabilizer relative to the center of

gravity so that certain design constraints were satisfied. Consider the

variable forward sweep wing aircraft shown in Figure 2. The aircraft is

in a wing's level steady state rectilinear flight condition. Assuming a

small angle of attack 'i/1 , neglecting the moments due to the drag and

thrust forces and using the derivative and trim results from Chapter

IV, the steady trim equations are approximated by equations (1) through

( 3).

n.""' { Co: i- cO: (V1 -+- Iw- £~)} E T SB 6 v s1/ v T"'"'~ fj. I ( I ) - Co + h sw Coe> 0 + S"" o ~I SW

1-1 H sa B ;- l::.E + s,w CLot. IJI n_Wc.: ( V, + Iw- E~) ~"" {c.. : ( v, +1 11 ) -t Ct. bE I 1 -~= 0 (d.) '!rt sw H -H H - H 6.E + el-l w w ( w w) { I c17, + I H) l\: C. 0< T I - E I + ~ ~wCm~& =-w v, S"" XAc. ""''xA, '""' -H H -13 XJ.\c. sB XAc. 8 + :::-;J c~. ~I:' + =w CL

As mentioned in the discussion of the trim calculations in Chapter F IV, the three unknowns B,) IJ. 1 and q 1 are specified to reduce the total . . e ~~~= number of unkn· -ms to three. Spec1f1cally, 1 and u. 1 have been selected

as zero and q 1 is calculated from a Mach number of 0.75 at an altitude

of 6000 m. The wing and stabilizer airfoil sections have been selected

as symmetric airfoils. Downwash from the horizontal stabilizer onto the

wing and wing and vertical stabilizer dynamic pressure reductions from

the horizontal stabilizer and body, respectively, have been included in

the trim equations (1) through (3). Equations (1) through (3) contain

the three unknowns V,J /J.~ and /:1~. Equations (1) through (3) are used

in the fundamental design. n.ox:i:z.ontal

:z. :z.

Figure 2· variable Forward Wing Aircraft in a Wing'S Levels~eeP SteadY state Rectilinea.X: Flight condition 14

Taking the partial derivative of equation (3) with respect to \l1 and expressing the moment arms in the ~Y~ coordinate system results in the P-xpression for the aerodynamic center location given by equation

( 4). The static margin is given by equaticn ( 5).

W -W SH tl .=: H 8 vJ - ~ cs = n.. CL.O( XAc. + s"" c~...Cl(, xA, + sw '-oc. XI\'- ('f) xAc = s.., H n.w -t- sa B c/; sw c~..« + S"' c~..o(

SM (5)

Equations (4) and (5) are used in the ftmdamental design.

The center of gravity calculaticn is performed by dividing the aircraft up in to the following seven elements: body, wing, horizon tal stabilizer, vertical stabilizer, engine, fuel and payload. Constraining the fuel and payload center of gravities to coincide with the aircraft's center of gravity results in the center of gravity locaticn as equation

( 6). w It v=v c=E s=S w= H ~ + rn XcG + m Xe.G m Xcc; t m XcG + fYI ACG

Equaticn (6) is also used in the ftmdamental design.

The ftmdamental design is now stated. For the flight condition and

~xtrapolated size and geometry mentioned previously, equations ( 4) and

( 5) are used to locate the longitudinal position of the wing and horizontal stabilizer relative to an assumed center of gravity location so that a static margin of -0.25 is attained. Next, equations ( 1) E AT through ( 3) are used to solve for the trim tmknowns 'i/1 > /J. 1 and J.J. 1 for a specified wing incidence 1"" and horizon tal stabilizer incidence I H •

Finally, equaticn ( 6) is used to calculate the center of gravity

location. 15

The fundamental design is iterated until the following design constraints are met: 'i/1 is in the linear angle of attack range, D.~ is reasonably close to zero, A~ is between 0 and 1, II+ is larger than Iw and the assumed and calculated center of gravity locations are approximately the same. I"" is constrained to be larger than Iw so that the horizontal stabilizer will stall before the wing stalls. This constraint is a passive safety factor against stalling for a canard configuration.

Wing Description

Figure 3 shows an enlarged view of the right variable forward sweep wing. At the design flight condi ticn mentioned previously, the wing root geometry and pivot location relative to the wing structural planform were selected to allow approximately the same amount of sweep on either side of -30°. The pivot is selected at the 1/2 root chord location in the -30° sweep condition. Note from Figure 3 that the right aerodynamic planform changes shape as the sweep changes but the right structural planform remains the same shape as the sweep changes.

The wing airfoil sections were selected as the NACA 64A01 0 airfoil. This airfoil is a thin symmetric airfoil with a thickness to chord ratio of 0.1 0. This airfoil selection allows supersonic flight to be feasible. A more realistic airfoil selecticn would be a new supercritical airfoil but limited data en these airfoils prevented this selection. The stabilizer airfoil sections are also the NACA 64A01 0 airfoi 1.

At -30° sweep, the tip chord is aligned with the longitudinal direction. For sweep less negative than -30°, the tip chord becomes a . 16

Scale: lm=2cm

Aw =-42° LE Body Wing Centerline Slot

Aw =-30° LE

\ I \ I Jv

Figure 3. Right Variable Forward Sweep Wing 17 trailing edge and is not directly facing the aerodynamic flow. For sweep more negative than -30°, the tip chord becomes a leading edge and is directly facing the aerodynamic flow. At these off design sweep angles, the wing span is approximated as the average of the leading edge tip chord span and the trailing edge tip chord span. Any adverse aerodynamic flow patterns resulting from the tip chord becoming a leading edqe has been neglected in this research effort.

At -30° sweep, the wing airfoil sections and the aileroo and flap inboard and outboard chords are aligned with the longitudinal direction. At off design sweep angles, the airfoils and control surface chords will be skewed to the longitudinal direction. Any resulting adverse aerodynamic effects have been neglected. The structural f~asibi li ty of the wing root geometry and pi vat is also questionable but is left unspecified at this time. Finally, some type of closure mechanism for the wing slot will be required but is unspecified. These unaddressed areas are felt to be unimportant considerations when regarding the intent and purpose of this research effort. CHAPTER IV

ANALYSIS METHOD

Equations of Motion Solution

A rigorous application of vectorial Newtonian mechanics leads to the uncoupled longitudinal and lateral-directional sets of small perturbation, scalar, differential equations (7) through (15). ~ complete derivaticn is given by Roskam (1 ).

Longitudinal Set:

( 7)

( g )

• MAy t-tnry = Tyy Wy ( '1 ) • (I 0)

Lateral-directional Set:

( I I )

< 1 a)

( ( 3)

• + ¢ ( ltf)

18 19

(I 5)

Equations (7) through (15) are the linearized equations of moticn for small perturbations about a wing's level steady state rectilinear flight conditicn written in the stability axes coordinate system. The xyz coordinate system shown in Figure 4 is the stability axes. The stability axes are denoted by the x axis coinciding with the aircraft's steady state velocity and the origin coinciding with the aircraft's center of gravity. The stability axes are attached to the aircraft and rotate with the aircraft.

The longitudinal unknowns are the perturbaticn variables 1/x) c< and B which are functions of time. The longitudinal perturbation control - F' :t variables are Sl:'J & andb. Before equations (7) through (10) can be solved, the aerodynamic and thrust forces and moments must be expressed as functions of V¥.;0<, e) ~E)~;:) hT and their derivatives with respect to time. Similarly, the lateral-directional unknowns are the

perturbaticn variables ;3 1 If' and¢. The lateral-directional perturbation R control variables are and b. Before equations (11) through

(15) can be solved, the aerodynamic and thrust forces and moments must be expressed as functions of /3, 4' .J ¢ .J bA, gR. and their derivatives with respect to time.

For a linear analysis, the longitudinal and lateral-directional forces and moments can be sufficiently. represented by the variables P. and and ~ J respectively . . ( 1). Note the use of Wx/'.Jy and u.J.'t inplace of o/ )8 and ¢. The forces

and moments can be expressed as functions of W_x 1 wy and W~ much eas er 20

X X

X

Figure 4. Body Coordinate Systems Used During the Research Effort 21 . . . than directly expressed as functions of C{J) El and¢. Equations (10),

(14) and (15) are then used to transform from Wt.,wy and W1 to tV} Band fp.

Perturbation theory and Taylor series expansion indicates that if

F(X,Y), X, andY can be separated into constant values F(X 1,Y 1 ), x1, and

Y 1 and perturbation values f(x,y), x, and y, respectively, then the perturbaticn value f (x,y) is approximated by equaticn ( 16). (!0) f(X ) ::. ;;>F(CY) I X + ::iJF(f..,Y) I '/ J y -cl X I cl '/ I

As x andy become smaller, the approximaticn to f(x,y) in equaticn (16)

becomes more accurate. Equation (16) extends logically to functions of more than two variables. Applying equaticn ( 16) to the aerodynamic and

thrust forces and moments with the notation Fx for and

using dimensionless coefficients results in the following expressions.

Longitudinal Set:

+

(I 7) fAr= 'f,Sw(CA~vx Vx -t- CA~«cx

+ CAt ~F ~e + CA~[:.f:" ~F) (I 8)

mAt = j 1 Swcw{ KAy Vy.. + KA .:x. ' v,. y <.( + f

.fTx 'if~ Sw ( CTx . = Vx + CTx D<. c{ i- CT • 0( + c "x X ex Tx wywY -r eTA sT ~r) (d.O) . fT-j! =

~~ Swcw( I

+ I

Lateral-directional Set:

+ (;13)

calf)

cas)

. (;I S.,vbw ( KT.x(3 /3 + I

Combining equations (7) through (10) and (17) throuqh (22), taking

the Laplace transformation, and using matrix algebra results in the

longitudinal matrix equation (29). Applying the same process to

equations (11) through (15) and (23) through (28) results in the

lateral-directional rna trix equation ( 30). The Vc ·iables in equation

(29) and (30) are now functions of the Laplacian .rariable s rather than

time. Also note the initial conditions are zero because the equations

of moticn are for perturbations about a steady state rectilinear flight

condition. Both the longitudinal and the lateral-directional matrix

equations can be represented by equaticn (31) with the Laplacian

solution given by equation ( 32).

[AJ{x} ;[B]{s} ( 3 j)

( 3 ~) r-- -..., {IYI}s f-1 1 Sw( CAy..« + CT x~)} S { -'j, s"'"( (.f\tw; + CTXwy) 1 s

+- { S""( ~t..v, t C,-J(vx) +{-9-,Sw(cA~ -t-(yxo<) Vx -tt, J 1 + { mJcos811

{ -1-t:j'( CAlvx + Cr~vx)} {mVx, -1, S~"'(cAt-« -tC-r~~)} S {-m Vx,- ft s(cAl +(~ ) } s L<.}' ~ 0( +{-~,::t·(cAz..oe -t·C-rrc<) 1 + { m'} Sif\81}

{-'ft Swcw(kAy"x + K-ryv)} { -ft Swc~Yt::Ay~ +l

{ 'J-1 s""·c~~ sE} { ~~ SwCA 2 be} ft, s'-"'-'CTz:aT 1 I $/ ~ ( J9) -- < { f' SvvCwkAt?,El 0 w_IVk } { 1r' S c'" t

N w - { m Vxl- ~I s"'(cAy;3 -r('YA)1 s { M Vx, cos 8, ~}- { -~,sv'(cAfu.!f.+Cryu_,) ~~ s f {-c;,-, svv(cAy ~ + CTy;J)} +J,S""'[ (CAYwt.-+C,Iwx) sir~,8 1 t: {-md cos e, ~ ;'3

- (cAYwi. + C-ryu;1 \cos 8 1]} S.

(-a-,J S"'b. "'( KAx; t Kli.;3) 1S f-Ix" siA 8 1 - Ix;;cosEq sd. { Ixx 1Sa

+ f -g I s""'b"'(KA~~ 1-K'Tx/3) 1 f { b' tb""[ ( l

'--- 1. ·"'Lut)cos81]} -

,-- - l Lv } SA { z. 1 Sw CAysA1 L frl s CAy~R I I

( w VJ 1 R { C's wbw KAx~} zCf-,S b KAxsl{ I I ~ I (3o)- '

{ $t SwbWI

N ""' 25

A.s mentioned previously, the numerical results consist of the characteristic equaticn root loci, mode shapes and time responses. The -[ characteristic equation is the denominator of (A] and is given by the determinant of [A]. The longitudinal and lateral-directional characteristic root loci are obtained by solving for the characteristic equaticn roots at various flight coo.ditions. Applying partial fraction expansion to equation (32) and taking the inverse Laplace transformation leads to the time responses. The longitudinal time responses are for

Vx}X and G due to the con tro l inputs The lateral-directional time responses are for !3, 4' and¢ due to the control inputs and ~R. Again, the time responses are obtained for various flight conditions.

The mode shapes are calculated from equation (33) which is obtained from equaticn (31) by setting {S]equal to the zero vector,{o]. [A]{x} ={o} ( 33)

Matrix equaticn ( 33) is three scalar homogeneous equations generally with a rank of two implying infinitely many solutions. In accordance with mode shape theory, only informaticn about the relative magnitudes of the elements of{X}car. be determined. Arbitrarily selectingBequal to 1 L 0° and using the longitudinal equations corresponding to equation (33), Vx andO(can be determined as functions of s. A-rbitrarily selecting !3 equal to 1 L 0° and using the lateral-directional equations corresponding to equation (33), !/' and ¢can be determined as functions of s. The mode shapes for a particular characteristic root are finally obtained by substituting the p:trticular characteristic root of interest in for s. Also the mode shapes are obtained for various flight conditions. 26

Aircraft Force and Moment Calculations

Before equations (29) and (30) can be solved the aircraft stability and control derivatives must be calculated. To accomplish this the aircraft is divided up into the following five elements: wing, horizontal stabilizer, vertical stabilizer, body and propulsion intake and exhaust. The aircraft forces and moments are the summation of the element forces and moments. In this way, the aircraft stability and control derivatives are determined as functions of the element stability and control derivatives. As discussed in this chapter, reference (11) and simple theoretical principles are used to calculate the element stability and control derivatives.

Figure 5 is used to calculate the aircraft longitudinal stability and control derivatives in terms of the element stability and control derivatives. This calculation consists of expressinq the aircraft forces and moments in terms of the element forces and moments. Next,

Taylor series expansion is used to express each force and moment in terms of the longitudinal variables. Finally, the aircraft stability and control deri va ti ves are obtained by factoring with respect to the longitudinal vari< bles. Applying the same process to Figure 6 results in the aircraft lateral-directional stability and control derivatives.

Appendix B lists the results.

The aircraft stability and control derivatives appearing in equations (29) and (30) correspond to the stability axes. However, the aircraft stability and control derivatives resulting from Figures 5 and , 'Al(J 6 do not corresp~d 'to th~:;,~s~bi li ty axes. This result occurs because reference ( 11) and the simple theoretical principles used to calculate the element stability and control derivatives do not correspond to the LH 'J + 0, l

v D H H ~.! 'J l + a. + 1 - eta H D Lw

w w w 'J + Ct + 1 - Ct - s ~Mw l 0 w D B L MB ::;:; ~ 'J + Ct l ){ TE

X

0, v

;(\-­ 'J + Ct l--- 1 1 T­ z :x: z 1 !0 Tz -.l Figure 5- Element LOngitudinal Forces and Moments 28

,, B

H N B N w ~ L w N f3 H ~ L ( FB r1 y

I T- w r y x ~t!

y

v S - a

Figure 6. Element Lateral-directional Forces and Moments 29

stability axes. Therefore, a transformation to the stability axes is

required. Figure 7 is used for the longitudinal transformation and

Figure 8 is used for the lateral-directional transformation. The

transformations are given by equations (34) through (45). Computing the

partial derivatives of the perturbation forces and moments and using

equations (34) through (45) ·lead to the stability and control derivative

transformations which are also listed in Appendix B. There is no

standard notaticn for the thrust derivatives like there is for the

aerodynamic deri va ti ves c0 , c,_ 1 C111 1 Cy 1 c i and cfl • Therefore, the

subscript notaticn xlyl and z is used to denote the thrust derivatives

as determined in the xyz coordinate system. (34) ( 3 5) ( 3 0)

( 3 7)

( 3 8)

( 3 9) (Lj0)

C y. I) ( Lf d) ( 4-3) ( '-1-4-) (LJ-5)

As mentioned in the previous section the longitudinal forces and

moments can be sufficiently represented by the longitudinal variables

' E ~ T Yy. 1 cJ..JC(,J Wy 1 ~ 1 ~ and S • Also, the lateral-directional forces and

moments can be sufficiently represented by the lateral-directional

variables 13,~ rJ; W;t., Ull; ~A and S 1\ • Note the forces and moments do not depend upcn the variable derivatives except for~ and ;i . If the I I I I I I I 30 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Figure 7- LOngitudinal To the I Transfor~ation I stabilitY Axes I I I I I I 31

X

y , y X K T­x

.. c n K A z

z

Figure 8. Lateral-directional Transformation To the Stability Axes 32 . downwash lag theory (1) is used to calculate~~ an<.L(.J, the derivative independence implies the same forces and moments are obtained for the same state no matter how the state was arrived at. The aerodynamic and thrust pressures change instantaneously whenever the state changes.

This assumption which is used exclusively in the force and moment model is called quasi steady flow (1 ).

In order to make the aerodynamic force and moment model more realistic, four interference effects are incorporated in to the stability and control derivative calculations. Downwash from the horizontal stabilizer m the wing, dynamic pressure reductim from the horizontal stabilizer m the wing, sidewash from the body m the vertical stabilizer and dynamic pressure reductim from the body m the vertical stabilizer are included in Figures 5 and 6.

For the purpose of computing moments due to the element forces, the element forces are assumed to lie along the axes of the xyz coordinate system. This assumptim is reasonable because 17; and the perturbation variables have been assumed small. Roskam ( 1 ) also makes this assumption.

Lacking propulsion data, the thrust force at the exhaust location is considered to be solely a functim of b.T". This assumptim implies the inlet conditions do not affect the exhaust thrust force. The thrust force of the in let loca tim is calculated from the fluid mechanics momentum principle given approximately by equation (46)

where V is determined as a function of the perturbation variables. The assumptim that the inlet thrust force occurs in the xyz coordinate

system is taken. Note that equation (46) is applied three times for each axis of the xyz coordinate system. 33

Element Force and Moment Calculations

The fundamental aerodynamic stability derivatives and all aerodynamic control derivatives are calculated from the USAF Stability and Control DATCOM ( 11 ) • The remaining aerodynamic stability derivatives are calculated from the fundamental aerodynamic stability deri va ti ves using simple theoretical principles. The fundamental aerodynamic stability derivatives include the wing and stabilizer

Coo ) Co"( (supersonic)' c'-01. ) Cmo and the body cOo ) CoO( ) CL.oi.J c""o and aerodynamic center location. The body aerodynamic center loca ticn is

where the body Cmcx. is zero and is therefore considered indirectly as a

stability derivative. All thrust stability and control derivatives are

calculated from simple theoretical principles. Appendix C lists the

results.

Reference ( 11) is an extremely extensive handbook for calculating

the stability and control derivatives of rigid aircraft. CKl.ly the

simplest methods of reference ( 11) are selected for the derivative

calculations. Many of the empirical correcticn factors for interference effects between the aircraft elements are neglected. However, the major

interference effects of downwash and dynamic pressure reductions have

been included as mentioned previously. Roskam (1) states that for a

linear analysis this simplificaticn is a reasonable selection. For a

single person using reference ( 11) with hand calculations, this

simplificaticn is also the only practical selecticn possible.

Reference ( 11) was originally developed for aircraft employing

backward sweep. The applicability of reference (11) to forward sweep

questionable. Consequently, USAF is conducting research to determine

the applicability of reference (11) to forward sweep and to develop any 34 necessary modifications for this application to forward sweep (12).

Reference (12) states that CLrx is estimated quite accurately from the

existing methods. The methods for C o0 J Co ex (supersonic), Cm 0 and the aerodynamic control derivatives with regard to forward sweep are not mentioned in reference (12). For this research effort, the existing methods for these derivatives are applied to the forward sweep wing with the question of applicability still mspecified.

The simple theoretical principles used to calculate the element stability derivatives are discussed below. The wing, horizontal stabilizer and vertical stabilizer aerodynamic centers are assumed to be at the l/4 mean aerodynamic chord for subsonic Mach numbers and at the l/2 mean aerodynamic chord for supersonic Mach numbers. The velocity derivatives are estimated from the well known Prandtl-Glauert transformaticn for subsonic Mach numbers ( 1). Using the theoretical lift curve slope for a flat plate at supersonic Mach numbers, Etk.in ( 13) shows the structure of the Prandtl-Glauert transformaticn also applies to supersonic Mach numbers. As mentioned previously, the inlet thrust force is calculated from the fluid mechanics momentum principle.

The angular velocity derivatives are calculated by determinin an induced perturbational velocity, angle of attack or sideslip angle due to the angular velocity perturbation. The velocity, angle of attack or sideslip angle derivatives are combined with these induced perturbations to form the angular velocity derivatives. For the angular velocity derivative calculations, the angular velocities WxJ Wy and Wi are assumed to lie along the axes of the xyz coordinate system since V1 and the perturbation variables are assumed to be small. Roskam ( 1) has a complete development of this computational method for the angular velocity derivatives. 35

The «and ;j derivatives are calculated from the downwash lag theory

( 1). This theory implies there is a finite time delay between the time

the downwash distribution changes and the time this new downwash distribution is propagated downstream and sensed or felt elsewhere. The

downwash and sidewash are given by equations (47) through (50). £w= c,w + 1~woc-J!wtX tE ( 47)

1/ " d c) 11f3 - d a-_" ;3 t d' <'+8) ()" = (J" I + J/3 d /3 - H -w -t E. = XAc. - XAc. Vx, -S -V XAc. - XAc. cso) Vx 1 As mentioned in reference (11), the vertical stabilizer and body v B v B derivatives Cyp, and Cy13 are approximated by - C(./3 and - c~..13 ,

respectively. Che final paint is reference ( 11) does not have a method B B B to predict C;13 and C111J • These derivatives are approximated by Crn«

referenced to the body aerodynamic center location.

Trim Calculations

Equations (29) and (30) correspond to small perturbations about a

wing's level steady state rectilinear flight condition • .3efore thes~

equations can be solved the wing's level steady state rectilinear flight ------· -~------~------. condition or trim condition must be determined. Consider Figure 2 again -----~ which depicts a variable forward sweep wing aircraft in a wing's level

steady state rectilinear flight condition. The corresponding trim

equations are given by equations (51) through (53).

( 5 I)

-mea cos e, ( 5~) 36

0 (53)

The trim equations (51) through (53) are longitudinal equations. For a

steady state rectilinear flight condition, symmetric aircraft and no

lateral-directional ccn trol surface deflecticns, the terms in the

lateral-directional trim equations are identically zero.

Using the stability and control derivative results of this chapter

and the canfiguratioo. results discussed in Chapter III the trim

equations are expanded in terms of the aircraft elements and the trim

unknowns Vx I l ell v,) .6 ~ J /j~) /J. ~ and f' . With an ly three trim

equations, four of the unknowns must be specified. v~., Jel J .b.: and/)

are specified and the trim equations are used to solve for \/1 > ./).~ and~~·

The trim calcula tian is quite similar to how an aircraft is flown. For

up and away flight with zero flap deflection (~~), the pilot desires a

certain speed and direction ( Vx,) e I ) for a given altitude (j)). The

At T eleva tor and throttle can trols ( o 1 J fJ. 1 ) are adjusted with the resulting

angle of attack ( '\/ 1) until the desired conditions are attained. F Specifically for this research effort, e, and /:J. 1 are selected as zero

and Vx, and /) are selected to result in a desirel.. Mach number and

dynamic pressure. The expanded trim equations are given by equaticns

(54) through (56).

(5LJ.)

{ ""(IN ~/~ H n: L + - c,_ « S""' .x -~ - 9.. w + 15 I S 37

+

-"" w -H H -H _Jl { XAc Cw XAc::. :ZAc { -I

v - v II < lAc::. h.v ~ C ( S<'o) S"" c"" Do

Equations (54) through (56) correspond to a subsonic trim speed. For a supersonic trim speed the corresponding wing and horizontal stabilizer derivative must be used.

1. The trim calculatiro is a nonlinear calculatim due to the terms Vj and 'iJ, ll-r; appearing in equations (54) through (56). Even though

linearized aerodynamics are used, the inclusim of the non linear terms is essential in obtaining the correct trim solution because of their magnitude relative to the other terms. The use of linearized aerodynamics implies \/1 is in the linea:t. angle of attack range. The 1.. nonlinear term i71 appears from the wing and horizontal stabilizer LD~ •

CoO( is a function of Cr_ which is a function of VI ; consequently, V, 2 T appears. The nonlinear term V,!::., appears from equatim (37) and (38).

Again for the moment considerations, the element forces are assumed to

lie along the xyz coordinate system. CHAPTER V

RESULTS AND DISCUSSION

Preliminary Discussion

Three different dynamic pressures are selected for the research effort. Each dynamic pressure is separated by approximately 10,000

'l.. N/m • These three dynamic pressures correspond to a mid subsonic speed, high subsonic speed and low supersonic speed. Table I lists h J (), \Is J

Vx M, and Cl 1 for these three speed and altitude conditions. I) .1.

TABLE I SPEED AND ALTITUDE CONDITIONS INVESTIGATED IN THE RESEARCH

3 2 h (m) p (kg/m ) V S (m/s) v (m/s) Ml q 1 (N/m ) xl 6000 0.6597 316.4 158.2 0.5 8255

6000 0.6597 316.4 237.3 0.75 18570

12500 0.2873 442.7 295.1 1.5 28150

At each of the dynamic pressures listed in Table I, the sweep angle is varied through its full range of -42° to -15°. For each sweep angle the aerodynamic center location, center of gravity location, characteristic roots, mode shapes and time responses are calculated and graphed. Appendix D contains these graphs. The scale of each graph in

Appendix D must be carefully noted when viewing the graphs. Some scales

38 39

have been increased or decreased so that the information can be seen clearly. Only the mode shape and time response graphs corresponding to

the sweep angles of -42°, -30° and -15° are contained in Appendix D.

Also, the time response to ~T and ~Fare not included because an aircraft is typically flown longitudinally with the elevator. All the

time responses are for step inputs with each plot indicating the

corresponding control surface and the magnitude of the step.

Before any discussioo. of the graphical results is undertaken,

realize the following paint. Each graph in Appendix D is for me

specific Mach number and dynamic pressure. It would be highly irregular

to fly at a subsonic speed at maximum sweep or at a supersonic speed at minimum sweep. The graphs are generated in this way simply to indicate

the variable forward sweep wing aircraft characteristics as a function

of sweep, Mach number and dynamic pressure.

Aerodynamic Center and Center of Gravity

The aerodynamic center and center of gravity location versus sweep angle graphs verify the expected trends. As the sweep is increased or

becomes more negative, bt.:th the aerodynamic center and center of gravity

move forward with the aerodynamic center movement larger than the center

of gravity movement. In other words, the static margin is decreasing.

~ 4 ~ At the dynamic pressures 8,255 N/m , 18,570 N/m , and 28,150 N/m ,

the static margins range from -0.5830 to 0.01083, -0.6138 to 0.01747 and

-0.6926 to -0.2460, respectively. Both subsonic aerodynamic center

location curves are approximately the same while the supersonic

aerodynamic center location curve is shifted backward due to the 40 characteristic shift of the aerodynamic center at supersonic speeds.

Note the center of gravity locations are independent of the Mach number and dynamic pressure.

Consider a variable forward sweep design with a static margin of approximately zero at the most forward aerodynamic center location and a subsonic Mach number and dynamic pressure of 0. 75 and 18,570 N/m 2 , respectively. If sweep less negative than -30° is used for subsonic speeds and sweep more negative than -30° is used for supersonic speeds,

the aerodynamic center and center of gravity location versus sweep angle would look similar to Figure 9.

If the reduction of the static margins denoted by JlSm1 and ASm3 are much larger than the increase in the static margin denoted by ~Sm~ ,

Figure 9 a applies. The characteristic backward shift of the aerodynamic center will not significantly change the stability. If ~SM 2 is approximately equal to .ll.St"'t and ~s,.., 3 , Figure 9 b applies. The characteristic backward shift of the aerodynamic center wi 11 significantly change the stability. Consequently, the latter case will

reduce the excessive stability problems for a variable forward sweep design. Figure 1 C illustrates the similar graph for a variable backward

sweep design. Clearly, the aerodynamic center movement caused by variable sweep and the characteristic shift are in the same direction causing excessive stability.

Using the graphical results, lls~. and t.S111 3 are approximately 0. 35 and o. 23, respectively, while L\Sft'1 2 is approximately -0. 23. Because

these static margin changes are approximately the same size, the

characteristic shift of the aerodynamic center will indeed be

significant. Consequently, a variable forward sweep design 41

a

XAC == XAC & Llsm2 xc··~'3 ILlsm 1 XCG I

Figure 9. Typical Aerodynamic Center and Center of Gravity Locations Versus Sweep for a Vcriable Forward Sweep Design 42

1 I

w 1\.LE

Figure 10. Typical Aerodynamic Center and Center of Gravity Locations Versus Sweep for a Variable Backward Sweep Design 43 should not experience excessive stability problems caused by variable sweep.

Characteristic Root Locus

At the design condition of -30° sweep, Mach number of 0. 75 and

2 dynamic pressure of 18,570 N/m , the variable forward sweep wing aircraft in Figure 1 has the classical longitudinal and lateral-directional characteristic modes. These modes are the longitudinal short period and phugoid modes and the lateral-directional dutch-roll, roll and spiral modes. Note the spiral mode is stable for this aircraft. The damping ratios, natural frequencies and time constants at the design condi ticn are listed below. u.Jsp = 3.55i:f ra.dfs Longitudinal: ~sp= 0.1<1?.9 Wp = 0.0~1?80 r-~cl/s ~P = o.aa1o

Latera 1-di rectiona 1: ~ OR. :::. 0 . I 'l 4 CD u.J 0 ~ .::: I .l 5 I raJjs

These values yield level 2 handling qualities for this aircraft according to Roskam ( 1). Note the longitudinal and lateral-directional characteristic equations are 4th and 5th order, respectively. One

lateral-directional root is identically zero and corresponds to yaw angle neutral stability.

The short period roots remain stable for each sweep angle and dynamic pressure; however, significant changes in the roots do occur.

At the dynamic pressures 8,255 Njm4 and 18,570 N/m~, increasing or more negative sweep causes the short period roots to approach the real axis

en an almost vertical path and upcn reaching the real axis split in to 44

'2. two real roots. At the dynamic pressure 28,150 N/m , the same trend appears but the roots do not reach the real axis.

"2. The phugoid root loci at the dynamic pressures 8,255 N/m and

18,570 Njm4 are quite similar. As sweep increases or becomes more negative, the roots move en an almost vertical path away from the origin which gradually changes to a nearly clockwise arc about the origin.

Starting from a stable location, the roots travel a significant distance

'2. in to the unstable regioo.. At the dynamic pressure 28,150 N/m , the phugoid roots consist of one real stable root and one real unstable root, each moving away from the origin as sweep increases. Note the instability is quite small though. Also, if low sweep is used for subsonic speeds and high sweep for supersonic speeds, the significant

'2. 2 phugoid instability at the dynamic pressures 8,255 N/m and 18,570 N/m can probably be avoided.

4 '2. For the dynamic pressures 8, 255 N/m and 18,570 N/m , the phugoid roots become unstable at approximately the same sweep where the aerodynamic center moves forward of the center of gravity. This is also noted by the well known stability derivative KAyat or CmO( changing from negative to positive values. Obviously, the aerodynamic center forward of the center of gravity is causing the instability. Some other

2 occurrence is causing the instability at the dynamic pressure 28,150 N/m because the aerodynamic center is always behind the center of gravity.

A closer examination of the stability derivative I

Positive values for I

2 initial forward speed perturbation. At the dynamic pressures 8, 255 N/m

4 and 18,570 N/m'2. I

'2 phugoid instability at the dynamic pressure 28,150 N/m is due to the characteristics of supersonic flow.

The dutch-roll root loci do not indicate any significant changes with respect to sweep or dynamic pressure. Increasing or more negative sweep does lead to a slight movement of the roots an an almost vertical path away from the origin. The dutch-roll mode consists predominately of sideslip motions. The wing contribution to the sideslip stability derivatives is quite small; consequently, the dutch-roll mode is a relatively weak function of wing sweep.

The roll and spiral roots remain stable for each sweep angle and dynamic pressure; however, significant changes are indicated. As sweep in creases or becomes more negative, the roll root moves toward the origin and the stable spiral root moves away from the origin. At the

'2. dynamic pressure 8,255 N/m and very high sweep angles, the roll and spiral roots have moved enough to eventually meet and combine to form a pair of complex conjugate roots. The roll and spiral modes are then coupled in to a decaying, oscillatory mode. Again, if low sweep is used at subsonic speeds and high sweep is used at supersonic speeds, this coupled roll-spiral mode can te avoided.

The wing con tributian to roll damping is a major portion. Wing roll damping decreases as sweep increases or tecomes more negative due 46 to the mean aerodynamic chord moving inboard; consequently, the roll stability is reduced and the roll root moves toward the origin. As expected, the wing subsonic and supersonic yaw damping decreases and increases, respectively, as the sweep increases because of the inboard movement of the mean aerodynamic chord and the sign of the

Prandtl-Glauert transformaticn at subsonic and supersonic speeds.

However, the numerical results indicated the aircraft yaw damping increases with increasing sweep regardless of the Mach number. This result is a little unexpected but there is a reason. As sweep increases, the wing and hence the aircraft center of gravity moves forward. Consequently, the vertical stabilizer's yaw moment arm increases. The vertical stabilizer's increase in yaw damping apparently masks any change in the wing or body yaw damping as sweep increases.

The aircraft yaw stability increase causes the spiral stability to increase and the stable spiral root moves away from the origin.

Mode Shape and Time Response

2. The longitudinal mode shapes at the dynamic pressures 8,255 N/m and 18,570 N/m2 indicate the phugoid mode is the dominate mode in the Vx motions for -15° and -30° sweep. The short period mode is the dominate mode in the 0{ motions and the short period and phugoid modes both contribute to the G moticns for -15° and -30° sweep. At -42° sweep, both the phugoid and short period modes con tribute to the Vx ct and e J motions. The time responses indicate these same results.

The V'l. responses to the ~/ step for -15° and -30° sweep are slow oscillatory responses (phugoid). The« responses to the ~£step for

-15° and -30° sweep are fast oscillatory responses (short period). 47

The$ responses to the f:/ step for -15° and -30° sweep are a combination of slow and fast oscillatory responses (phugoid, short period). At -42° sweep, the v)(J 0( and e responses to the ~€ step are a combination of oscillatory and exponential responses (phugoid, short period) even though the time responses look like purely exponential responses for the first 5 seconds. Realize the short period roots at -42° sweep and

'2 J. dynamic pressures 8,255 N/m and 18,570 N/m are two real roots leading to two exponential modes. Also note at -15° and -30° sweep the time responses are stable, but at -42° sweep the responses are significantly unstable because of the significant phugoid instability discussed previously. 2 At the dynamic pressure 28,150 N/m , the longitudinal mode shapes again indicate \/x motions are dominated by the phugoid mode, 0( motions are dominated by the short period mode and emotions are a combination of the phugoid and short period modes. This is true for all sweep angles. The time responses verify these same results.

The Vx responses to the 'bE step are exponential responses

(phugoid). The 0( responses to the ~£ step are fast oscillatory responses (short period). The e responses to the $/ step are a combination of exponential and fast oscillatory responses (phugoid, short period). Realize the phugoid roots at the dynamic pressure 28,150

1.. N/m are two real roots leading to two exponential modes. Also note that even though the phugoid roots are slightly unstable at this dynamic pressure, the time responses are well behaved with regards to stability. lh other words, the phugoid instability is insignificant as discussed previously. This is a well known fact that when the characteristic roots become slightly unstable, there is not a sudden change from good flight dynamics to bad flight dynamics ( 1). 48

The 1a teral-directional mode shapes at all the dynamic pressures and sweep angles are all quite similar. The /3 motions are dominated by

the dutch-roll mode while the r.p and ¢ motions are dominated by the roll and spiral modes. The time responses verify these same results. The A A R responses to the S and ~ steps are oscillatory responses

(dutch-roll). The If responses to the ~A and ~R steps are a combination

of exponential and linear responses (roll, spiral, tfl neutral

stability). The wiggle in the If responses which look like an

asci lla tory response is actually caused by the combination of the exponential and linear responses. The appearance of the linear terms is

discussed later. The ¢ responses to the bA and ~R steps are exponential

responses (roll, spiral). Note the coupled roll-spiral mode at the

2.. dynamic pressure 8, 255 N/m and -42° sweep is insignificant or not

visible in the 1/J and¢ responses. The /3 and ¢ responses are stable even

though the¢ response looks unstable. The~ responses are unstable.

This response stability is also discussed below.

The v.,.., 0{ and e Laplace transform numerators are complete

polynomials. Complete means that the zero power of s coefficient in the

polynomial is not zero. The characteristic roots lead to the

characteristic modes. The control step inputs (zero root) lead to a

constant mode because the numerator polynomials are complete and no

cancellation of the zero root occurs. The time response stability thus

depends upon the characteristic roots. The /3 and¢ Laplace transform

numerators are not complete polynomials. Remembering that one

lateral-directional characteristic root is identically zero,

cancellation occurs and the characteristic and constant modes appear

just as in the longitudinal case. The 'I' Laplace transform numerator is 49 a complete polynomial. Cancellation does not occur, and the double zero root results in an extra linear mode.

Since all the lateral-directional characteristic roots are stable, the /3 and ¢ responses are also stable even though the ¢ response looks like an unstable response. The 1/J response is an unstable response because of the linear mode. This means that an ai ler

?JA step, will result in a constant¢ after the transient response has decayed. This roll stability indicates a "rna thema tical" maneuver restriction occurs because the responses indicate the steady state ccnstant ¢is only 1.6 complete rolls at the dynamic pressure 8,255 N/m'2. and -42° sweep. At higher dynamic pressures and lower or less negative sweep the restricti

lateral-directional analysis because this would result in the responses all returning asymptotically to zero and the small perturbation equations would then still be valid. CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

Conclusions

A conceivable variation of the revived forward sweep concept is the merging of forward sweep and variable sweep. Both forward sweep and variable sweep offer significant performance gains over fixed backward sweep wing designs. The coupling of forward sweep and variable sweep might result in even higher performance levels. The goal of this research is to 1) study the flight dynamics of the variable forward sweep concept and 2) indicate any significant advantages or disadvantages associated with the variable forward sweep concept. Goal

1 is accomplished by conducting and discussing this research. Goal 2 is accomplished by stating the following conclusions.

1. Variable forward sweep designs should have "-ar fewer problems with excessive longitudinal stability and elevator cuntrol effectiveness reduction than variable backward sweep designs.

2. Significant changes occur in the short period and phugoid modes over the variable forward sweep range and dynamic pressures. A fixed forewing and fixed wing pivot located in the forewing, artificial longitudinal stability or center of gravity control might be required to reduce these effects.

3. The dutch-roll mode is affected very little by the variable forward sweep; however, the roll and spiral modes are significantly

50 51

affected by variable forwar~ sweep. Artificial lateral-directional

stability might be required to reduce these effects.

4. A "mathematical" roll maneuver restriction occurs if all the

lateral-directional modes are stable. However, this restricticn breaks

down as the small perturbation assumption fails. An unstable spiral

root may be advantageous to the "mathematical" roll performance.

5. The flight dynamics of a variable forward sweep wing aircraft

present no major difficulties for the variable forward sweep concept to

become a feasible design option.

Recommendations

Many unaddressed areas of the variable forward sweep concept need

to be studied further. These unaddressed areas include the effects of

·flexibility, non linear dynamics and aerodynamics, nonideal control

surface actuators and automatic flight control systems. The forward

sweep wing is a flexible structure and its affects upoo. the linear

dynamics and aerodynamics assumptions need to be analyzed. Any high

performance aircraft that is designed today employs some type of

automatic flight control system to squeeze ·.he maximum possible

performance out of an aircraft. The variab.L.e forward sweep wing

aircraft wi 11 not be excluded; hence, the incorporation of automatic

flight control systems should be addressed. Finally, to insure the

accuracy of the research, nonlinear dynamics and aerodynamics and

nonideal control surface actuators should also be used. BIBLIOGRAPHY

( 1) Roskam, J. Airplane Flight Dynamics And Automatic Flight Controls. Lawrence: Roskam Aviation and Engineering Corporation, 1979.

( 2) Polhamus, E. c. and T. A. Toll. "Research Related To Variable Sweep Aircraft Development." NASA-TM-83121, May, 1981.

( 3) Kress, R. W. "Variable Sweep Wing Design." AIAA-80-3043, March, 1980.

(4) Diederick, F. w. and B. Budiansky. "Divergence of Swept Wings." NACA-TN-1680, 1948.

(5) Krone, N. J. "Divergence Elimination With Advanced Composites." AIAA-75-1009, August, 1975.

(6) Moore, M. and D. Frei. "X-29 Forward Swept Wing Aerodynamic OVerview." AIAA-83-1834, July, 1983.

(7) Krone, N. J. "Forward Swept Wing Flight Demonstrator." AIAA-80-1882, August, 1980.

(8) Nicolai, L. M. Fundamentals Of Aircraft Design. Fairborn: E. P. Domicone Printing Services, 1975.

(9) Jane's All The World's Aircraft 1984-85. London: Jane's Publishing Company Limi L··d, 1984.

(10) "Specifications." Aviation Week & Space Teclmology, Vol. 120, No. 11 (12 March 1984), pp. 131-173.

(11) Finck, R. D. USAF Stability and Control DATCOM. Wright-Patterson Air Force Base: Air Force Flight Dynamics Laboratory, April, 1978.

( 1 2) Sharpes, D. G. "Qualification of the Da tcom for Sweptforward Wing Planforms-A Summary of Work to Date." AIAA-83-1836, July, 1983.

(13) Etkin, B. Dynamics of Flight-Stability and Control. New York: Jolm Wiley & Sons, 1982.

52 APPENDIX A

CONFIGURATION SPECIFICATIONS

The specifications of the aircraft shown in Figure 1 are listed in this appendix. The appendix is divided into the following sections: body, wing, horizontal stabilizer, vertical stabilizer, propulsion, mass and center of gravity and moments of inertia. Some sections include a discussion of the methodology used in calculating the specifications.

Figure 4 shows the body coordinate systems used in the research effort. The xyz stability axes were mentioned previously. The xyz coordinate system is attached to the aircraft's center of gravity, xy2 coordinate system is attached to the aircraft's nose' xyz' coordinate system is attached to the wing pivot, 11£ coordinate system is attached to the wing's center of gravity and :.f/z"' coordinate system is attached to the aircraft minus the wing center of gravity. All the coordinate systems shown in Figure 4 with the exception of the stability axes have their x axis parallel to the body centerline. 13 Body: m 13 1 -== 1'+.~3 J nose -== i.f.3~9 m ""-z. d'3 = I. 8d.9 m sa a. eo a 7

8 (Y'\1.. 2 .sb~se =a. G, ,;n sawet -- 7~.0(o m el(.b .. )ie vB :::- 3 3.31 I'Y\'3

The volume is based on a conical nose shape with a nose length of

20% of the body length. However, the nose length listed is 30% of the body length. These lengths were selected because the nose taper ratio

54 55 is only significant in the leading 20% of the nose. The volume is used in the body subsonic aerodynamic center calculation. The exact nose profile is unspecified but a parabolic profile is assumed when the methods of reference ( 11) are used.

Wing: NACA t;,l.fAOIO airfoil

xp = -ta. 5 s m

~ F = - 0 . '+ Y.Y 5 M

"" 0 .,u " 0 0(0 = 0 ex Stell! = I Ol

e""(.subSot\lc.) = 0. 8 5 )1_\/IJ = 0 . 9

E,w = 0.0° JEw;do( = 0.1

s"" = /8.5fi m 1 S~e+ = ~8 .00 m "2. cw = ~- 3'-+ I m bw = 8.d30 m

C~oi =3.o 10 fY\ t.w = o.5 Cpwjc"" =0. a. ct/cw=o.a w h. F1 = 0. 'j hA,"" = o.l w hF2 - o.l )1~2 -::-0.~

The flaps and ailerons are plain flaps. There is no available w '/II w dE~'~ method to calculate lL , e , E, and dot. for a canard forward sweep wing configuration. Therefore, values for these variables were selected as reasonably conservative values compared to typical values for a ccnventional configuration. As a first degree estimate, ot:' and o<.~-IQII are set equal to the airfoil ct0 and otst~U·

Horizon tal Stabilizer: 56

-H c = /.17/ m

H ]\1-1 C rooi = I. 5o 5 rn = 0.5

CEH/-H C. ::: 0. "'COl

IH = ~0

H H H The elevators are plain flaps. e , O(o and c< 5t~ll are estimated by the same method as stated in the wing section.

Vertical Stabilizer:

N ACA ~4Ao 10 air-\ol\ --1-L.E" :: 3 0 0

11 S == 3.7/(o m' S~e+ = 7. Lf 3'J. f'Y) '2. -v C :1.8Ji.fm b" =d..l34 fY'\

1/ Croo-t :: d. 3 57 m "A" :: o.L.f77Lf

h.R-z" ::: 0. ~

!3 IJ = 00 II "<> 0 !3s-tQII =/d. € 11 (.sub.sonic) = 0. 8 5 hv' = 0.9

dcJ''/ci/3 = o. 1

The rudder is a plain flap. ·}\'r is selected so that the trailing

1/ 'I v J~'l ~II l?ll edge sweep angle is zero. n: ... e ) 6, J cJ/3 .J /-:, and f.).S-h:&ll are estimated by the same method as stated in the wing section. Figure 11 shows the defini tioo of s" and b" • 57

Figure 11~ Vertical Stabilizer Planform Definitions 58

Propulsion: Gef\efal E led·,. i c. F'fOLf -Gi;- 4D0 +vrbot4t'\

t: AL 5,., 2 Tmax =7/J/70 N o. j€ = 4.o'3CJ m dE =o. ggCJo m =I =I 0 X -7.3/5/')'\ y =E r=I - J.4o a m J( -1'1.~3fV1

=t= =Ey =- 0 e - 0

The engine centerline coincides with the body centerline and the engine base coincides with the body base.

Mass and Center of Gravity:

iDOOO m/1'\11;<. = I aI ooc Kc:a m"';l\ = K~

tYl"" = ~ 5 ~ 5 /(, mE = 785 I<~ =-1/11 - 7. '15CJ m o ~ ( mP+ m,:) ~ c.oc~ Kd XcG = --'tV =-""' M \jc(!, =0 ~e.G - .-o.o'J-14-'1

At 1\'E =-3o : xcell = - 8 • octo m YcG:: 0

icc& ::: - o • I rn

The aircraft•s center of gravity location will vary depending upon the sweep angle. Therefore, the analysis required a method to estimate the wing•s center of gravity location as a function of sweep. Combining this method with the stationary center of gravity for the remaining elements of the aircraft, the aircraft•s center of gravity location can be calculated as a functim of sweep.

For this method, the wing is modeled as a solid homogeneous panel.

Equatim (57) and the geometry shown in Figure 12 are used to calculate 59

x'

y'

Figure 12. Wing Geo~etry for Center of Gravity and Moments of Inertia Calculations 60

the wing center of gravity location. (57) x~ = "Xp + ~: ~\\ x'dV

The aircraft center of gravity location is then calculated from equation

(58). -w= -w rn XcG -r (58) m

- w = w Note YCG and %,G and hence YcG and lc.G do not vary with sweep.

The wing density in equation (57) is given by equation (59). mw (59) - s~ i.w Assuming the wing to be thin' the z integrals with the limits of -{.WI a tol"/a_will cancel the-twin equaticn (59). Therefore, a value for-t.~~" is not required. After examining Nicolai's ( 8) weight estimations, the wing mass was selected as 30% of the aircraft's minimum mass excluding the engine mass. reG. was designed to be -0.1m because the wing and vertical stabilizer are located above the body centerline.

~w "t -w 1 Moments of Inertia: I ,, 111 1111111'::"4 ~70 J

Similar to the previous section, a method to estimate the wing moments of inertia as a function of sweep is required. Combining this method with the constant moments of inertia for the remaining elements of the aircraft, the aircraft's moments of inertia can be calculated as a functicn of sweep. The wing is again modeled as a solid homogeneous panel. 61

Equations (60) through (63) and the geome-trY of Figur~ 12 are used

((.I )

( (., 3)

The aircraft moments of inertia are then calculated from equations (64)

through (67). I -w( = ;;-wj'2. -.L-w + mw( ;cG- icwG )"2. + I X~J" ii = m ~ c

-IN :- : -w '2. -w W = = W '2. VII Iii =m (x,G-Xc:G) + Il'i''' + m (x,G-XcG) + Iz"~" (G,,)

Iii=- n1w(xC6-x;:)(2c15 -i;;') + I~~z"' +mw(x,G -x:)(~,G-i,';) +rx":i'' (G.7)

The thin wing assumption is used again.

As sho'Wl1. by Roskam ( 1 ) , equations ( 68) through ( 71 ) are finally

used to transform the moments of inertia from the xyz coordinate system

to the xyz coordinate system sho'W11. in Figure 4.

(G,8)

Iyy = I -yy ((.,9)

Ii~ =: 5t., 2 v, Ixx + cos'l.l71 Iii + J. ,os 17, stn 111 Ixi (70)

1 I x~ = cos 17, 5in 171 (Ixx. -I~.£) + (,os 17,- s,t,'Z.17,) Iii ( 7 I)

This is required because the moments of inertia appearing in the

equations of motion correspond to the xyz coordinate system. APPENDIX B

AIRCRAFT STABILITY AND

CONTROL DERIVATIVES

The aircraft stability and control derivatives in terms of the element stability and control derivatives as mentioned in Chapter IV are contained in this appendix. The stability and control derivative transformations to the stability axes as mentioned in Chapter ~V are also contained in this appendix.

62 63

Aircraft Aerodynamic Derivatives: I

""' w <)1·1 H sB B .S v 'I ~' Co-= h: Co i- s,wCo + S"' c[) i- ~ SN Co / /j / c - l w "" s~-+ c ~ ~ B v sv v . r/"' Dvx- 1_ Co"x +- sw Ovx t- 5"" Co"x + )l. s"" Covl( \ \ C w '"' \ . Cowt = Y~C~ 1 C .s ~ 1-1 o sc- = S"'" c. o ~ E" ------Co~~= = )LvJ c~F

H - t-l 1-1 sB(_p8 [j + XAc. CH - l!k C0 ) + 5w =w C.... c"' L. c"' c v Cy -= 'h.

c v s" " ss s 'lwx == ll S"' Cywx -1 SW Cyt.>J( Cv....s_,._ - :h_v ~ C v +- .sB C B ' -.:o - S"" ywr: s w 'I wr Cy~A =. 0

5s( 1acB- l!cs) + s"" b"" :i b"" Y B 1/.8 13 - B B + 2-- {- c A - rA<. c1 ) S"" 6"" ~/.l b"" ;3 Aerodynamic Derivative Transformations:

CAt.wy ::- - Cowy

CAx~~ ::: - Co 5f"

-~~-,.,c:-._AxsF -= - Co:,F

CAl; -::: - CL. '•p ., ,,. '' ' ~";': "'v"l. CA CL. VI( - L 'lvx -:::- - a c ' '·,~'./ ~l.,f'r) Vxl >/ CA:r ot ::- - C .. O(. - C.o \,/ CA~Oc :::-CL~ '\,, ./ ...... ) }""f / / '',.f / CA ~""t = - cl..uJy CAi;:;~r ::::- -CLSF

----- ... ~Ai sF =- - CL.&F """ I

l

KAy.;_ :: C "1~

K;.'Ywy -= Cf)\L.Jy

I

~

CAy= Cy

CAy;3 :=. Cy~

CAy;j = CyA

CAy.vx =Cywx 66

CAy~A =cfsA

CAysR =CyDR ---.. KAx ::: C;_ + V, Cn

K.A:t:/3 =c'-/3 + ~ cf'\!3

KAt;3 ::: C 1;3 -+ V, Cn;j

1'-Af.wx ::: Cfl.wx -t V'l Cnwx

f(A~< lJ -e = CQuJ't + V, Cf'\w.r

,I(AJ(sA =C.e_':JA + v 1 c;'\ ~A KA,..sR ::: CQ~R + \/, C"~R

I

KA ZA =Cr.;J - 17r c1_;f

KAl,..;J( ::: C"wx- V1 CL..Jx

KAluJl =C(l.._..;r- 171 Ci.wl:

kA taA = Cn sA - v, C,t?J A

f(Al-$,R. ::: cl\~~ - v, c~ sR 67

Aircraft Thrust Derivatives:

r CT X lfx -:: c,lvx c,- .. I Xex :::: L 7 xcx Crxc( -=Cix«

Cr 1 t.>Jy :::: cfxwy ( E ~-~~X sr ;:: CT X ?JT E Crz ~ CT-z:'5r t.i~ -:r: CT::- = C-ri lvt.. Vx

c.,~ 0( -= cTi "'< r Crio<:;; C7 i..:i(

I Crlwy::: Cr~LL}y C £ T;! ~T :: CTj ~y £ t

l

I Cr- -- CT- Ywf..- Ywx 1: Cryt.ve::::: CTywi

I<---I K - c 68

:I: Krxij ::: I

I

"I: KTi_,1 =kri/3 .X KTi;i-= I

Crx Vx ::. Crx v;. + V1 CT~ ""

c,xo(-= Cr;Q( -1- V1 Cr2 o<

Cr I(«. :::: Crx,;, +- V1 Cr l ~

CT X wy ::= C.r/. •.:y + V1 ( T j uJ y

, "~-Crl. 'ZJT :::: Cr; ?JT + VI Cr z ~ T

CT;c :::: c,i - VI Cr-;

Cr 211~,::: Cr~ltx - V1 Cri. vx

Cr2

( T ~ d,. :: C T J 0< - f 1 C r X Ot;

Cr'luJy = c.,.iwy- VI CTx wy

.. Cr :r ~r ::: Cr1. sr - V1 C r; '1?. l

Kryvx = i

l

I

I

Kry ~/ ::: KTy ?>T

Cry .::. CTy

CTyf3 ::: Cr:y/3

Cry;3 ::: C7y;i

CTfwx:::: CTYwx ---CTyw"t =CrYw-e l

l<-rtj3 = KTx13 + \1 1 l

t

KTX uJx = t-<:rx wx +- \11 I-<.Ti w X

..... , ...... , l

Kr~ = Kr~- V1 Krx

J

t

KTz.w:: = KTiuJ:t - \/ 1 1<-rxw-z APPENDIX C

ELEMENT STABILITY AND

CONTROL DERIVATIV~S

The element stability and control derivatives as mentioned in

Chapter IV are contained in this appendix. The notation of w, H, Band V before each equation indicates the equation is applicable to the corresponding element. Some derivatives are obtained from graphs contained in reference ( 11) rather than a closed form expression. For these derivatives, the notation "see reference (11 )" is used. The element aerodynamic derivatives are divided into a subsonic and supersonic category. If an element aerodynamic derivative does not appear in the supersonic category, the supersonic expression is identical to the subsonic expression.

71 72

Element Aerodynamic Subsonic Derivatives:

C,.,H / ~~ If 1 H H) .., =L.Oo + CoO( c.. 17, - o

C~ =Co! + Co! ( V1 ) v v Co =Ct~o

W) H,B} V re {eref\ce (If) M,, Cp W; It; BI v Cov = X Vs ( 1-Mf)

see re(er-eAce (It)

see re..ferer1ce (I I)

w ct.ot.. :::

13 CLo( -- w CL. 0( = 73 IN C .. ~F = H eLsE =see re.feve"'ce (LI)

1/oJ CMW .:: + Cmo< (. \7 1 - rxt +- :r ""- E., w) CmH = -+ Cn,~ ( r;1 - o<} + r r1 ) B c~ = + Lf\1 C( ( \J 1 )

W;H A casL. ...t1_c14

A -t dCos ..f\.cl'+

C-'" "x -

vS CfYI csF :::- See re{e.rel'\c.e C. ll) H- Cm~f == :)ee re-fe.rer~ ce.. ( ll)

v) 13 Cy = o

8 Cy/3 v Cy/3· =

v) B C.'lwx = 2Ac. Cy Vx 13 I ~B C.yw:z =- XAc... Cy Vx 1 13 v Cy~R. - see re-\= e r e V\ c e ( I J )

w,H)3 Cfl-:::: o 74 vJ_. H) B c~.13 -= o

YAcR YAcr< w) H c.R.wx. = -- c_L~ 13 b"" thl Cfl. =- B B u.Jx -= ~c - - x, {)!3 W)H Cf.~l. --- YA

w)H I ~v c(\ =- o

W1 H) 13 .IV C('l.fi :::- 0 c" v dcJ-..- a (\A -= c"13 d/3 -t

y - v· cf\wl. ::: xi-\L c 11 -Vx f\;3 l 13 -c B c(ju);;: = XJ\c C -vl(, "A w c ... ~A -- see re. -\=e.revtce ( 11) y ( "&R - see. re +er-e.\c.e.. ( tl) 75

Element Aerodynamic Supersonic Derivatives: w,H) 13, V re terence (I I) Co a -= ~ee

W;l-l Coo< == see re fe re A(e ( II ) w Co~~= = see re-\ereV\c.e ( I I) Co~~; = 5ee re-\-erence (1[) Wli-IJ 6 (II) c~... C( -- see re-fereV~c.. e w CL&F - see. re. -\'e.r e ,'1 c e (I I ) H CL&E" - see re{ere(\ce (I l)

w)H)B c""o::: See re.feren u:? ( ll) = 8 XA<. = see re tere ,, r e. (II)

IN c""?;F ::: See re .f e.l... e-~~.c e ( ~ l J H c (Y\. ~£ -:-:: See... re {:e 1 e " c -e ( { I J

'{

Cy/3 :: see r e ter-e.,,-, c e ( J I )

II Cy?JR :::: se.e re-f erer\c.e (I I ) w C_a_~:,A -= see. reference (!1) vV c(\~A ::;:: .s.ee re-f'evenc.e ( Ll) v C A~R :::: see.. re. -f erertc e (I L) 76

Element Thrust Derivatives:

:r c,_ = f?AI Vt, xvx t' sw I cT, - 0 0(

X Cr.x. - 0 0(

I f?AI ~~~I CT- -::. Xwy Jl sw

E E T IVIIAj, CTX~T -- 'jr yJ r Cr- 0 :z vx - ~-2. I (J A..L. x., Cr- l« g.-~ s"'

E CT i

r f?ArVx, ~r Kr- Y "x ~( sv-~zw "2..- ('AT. Vx, Xr

rg, 5 ,J c_w 0

r Cr- . 0 Y/3 I. j)AI Vx, ~I 77 Cy- - Ywx ~I s w J. f?ArVx, XI CT-y w:z. - tj, s~~'~ r lAI Vx, Z za. J<.Tx w.x = g.,swbw r 1<-r- - fJAL Vx XL ji Xwi! - 'jt S"'"' h vJ r fl A -r Vx~ xr. l

r l

GRAPHICAL RESULTS

The aerodynamic center and center of gravity location, root locus, mode shape and time response graphs for the various flight conditions mentioned in Chapter V are contained in this appendix. Each figure describes which variables are being graphed and the corresponding flight condition.

78 79

-10------~

-9

-8 x (m) x CG CG

-7

-6 0 -10 -20 -30 -40 -50 w 1\ (deg) LE

Aerodynamic ce~ter and center of gravity location vs. sweep angle 80

2 q 1=18,570 N/m

-10

-9

.. .. • & -8 x X CG (m) CG

-7

-6 0 -10 -20 -30 -40 -50 w /~LE (deg)

Aerodynamic center and center of gravity location vs. sweep angle 81

2 q 1=28,150 N/m

-10

-9 •

X AC

& -8 x ~ (m) CG CG

-7

-6 0 -10 -20 -30 -40 -50 w ALE(deg)

Aerodynamic center and center of gravity location vs. sweep angle 82

4 Jl.w I.E -15' • -3011 2

.. Cl -42~ r -42 Im s(rad/s) 0

-z -

~ j

-4 I I -4 -2 0 2 4

Re s (rad/s)

Short period root locus with varying sweep 83

. 5 ,,w n LE c -41 - .25 / - 0 -42 -40 t

-30:"\-15- Im s(rad/s) 0 I \ -.25 ---

-.5 I I -.5 -.25 0 .25 . 5

Re s(rad/s)

Phugoid root locus with varying sweep 84

2 a =8,255 N/m ~l

2 . v: 1- LE

ll -42 -30° 1 - -15° ll

Im s (rad/s) 0

-1 - 1

-2 I l -2 -1 0 1 2

Re s (rad/s)

Dutch-roll root locus with varying sweep 85

. 5 ,W !~ LE

.25 -

0 -42

Im s (rad/s) 0 I ~ ~0 -15 - 1 :J -30" _;5: \ -30° Spiral Roll -35°

-.25 -

-.5 I I -.5 -.25 0 .25 . 5

Re s (rad/s)

Roll and spiral root locus with varying sweep 86

2 q =18,570 N/m l

]0 Aw LE

-15. - 5 .. -30-~

- 4 .c.~0 -42° / Im s(rad/s) C

~ -5 - J

-10 T I -10 -5 0 ·5 10 Re s (rad/s)

Short period root locus with varying sw2ep 87

. 5

.25

Im s (rad/s) 0

-.25

-.5 -.5 -.25 0 .25 . 5 Re s (rad/s)

Phugoid root locus with varying s~eep 88

q =18,570 N/m2 1

2 -42'' Aw -30° LE -lSa l

1 -

Ims(rad/s) 0 I I i I I I -1 - I I I -2 -2 -1 0 1' 2 Fe s(rad/s)

Dutch-roll root locus with varying sweep 89

2 q 1=18,570 N/ro

l Aw LE

• 5

-15° -30d -42° Im s(rad/s) 0

-.5 -

-1 I -1 -.5 0 . 5 l Re s(rad/s)

Roll root locus with varying sweep 90

2 q 1=18,570 N/m

.1

Aw LE

.05

)

-30" 0 0 -15 -42 I Im s(rad/s) 0 /

-.05

-.1 I -.1 -.05 0 .05 .1 Re s (rad/s)

Spiral root locus with varying sweep 91

2 q 1=28,150 N/m

8

w 0 ]\ LE -15-30° \

0 -42 4 -

Im s (rad/s) 0

-4 - 1

-8 l l -8 -4 0 4 8

Re s(rad/s)

Short period root locus with varying sweep 92

2 q 1=28,150 N/m

.04 w 1\.LE

.02

Iw s(rad/s) 0

-42. -30° -15~ -15° -30G -42°

-.02 -

-.04 I I -.04 -.02 0 .02 . 04 Re s(rad/s)

Phugoid root locus with varying sweep 93

4 w 1\.LE

-420 2 -30° l -15°

Im s(rad/s) 0

-2 l

-4 I -4 -2 0 2 4

Re s (rad/s)

Dutch-roll root locus with varying sweep 94

2 q 1=28 ,150 N/ m

• 4 .f\ w LE

• 2 -

Im s (rad/s) 0 - 15° - 30° - 42°

-. 2 -

-. 4 ' - .4 - .2 c . 2 .4

Re s (rad/ s)

Roll root locus with varying sweep 95

.05

Aw LE

.02 5-

Im s(rad/s) 0 -42° -3o· -15°

-.02!~

-.05 I -.05 -.025 0 .025 .05

Re s(rad/s)

Spiral root locus with varying sweep 96

2

Aw LE

1 -

a.,-30° 0 v /V ,-30 __;:;;;: x x 1 it. -:(_ o., 15° 0 / e 0 v /V ,-15 not visible x x 1

-1 -

-2 I I -2 -1 0 l 2

Shor~ period.mode shape w~th vary~ng sweep 97

2 q 1=8,255 N/rn

8:=1 L Oc

2

f.w LE

1

v /V ,-42° x x 1 ...... 0 / /' , I e a,-42" I

-1

-2 t I -2 -1 0 1 2

Short period(l) mode shape with varying sweep 98

2 q1=8,255 N/m

8=1L o"

l .. w '·LE

. 5 -

/ ...... 0 " / """ a,-42° 8 v /V '-42° X xl

-.5 -

-1 I -l -.5 0 .5 l

Short period(2) mode shape with varying sweep 99

2 g1=8,255 N/m

8=1 L o·'

2 ~------~------,

v /V , -15" X X_ 1

1 V /V -30c X X ' 1

e o.,-15~-30'

-1

-2~------.------+------.------~ -2 -1 0 1 2

Phugoid mode shape with varying sweep 100

4

w ALE

2

cp,-15~

/ ~¢,-30° 0 / / " 6 ¢,-42° 1)!,-15,-30,-426 0 <

-2 -

-4 I I -4 -2 0 2 4

Dutch-roll mode shape with varying sweep 101

.&=1 L o"

250 w ALE

125 -

¢,-30"' cjJ, -30c

\../ ~ .... 0 / / I '\ I 1~,-15~ ¢· ,-15"

not visible -125 - B

-250 I ' -250 -125 0 125 250

Roll mode shape with varying sweep 102

200

~w " LE

100)-

;;_,,-30° ¢,-30''

0 / / "'/ / ' \ ' ¢,-l5c \(J,-15°

B not visible

-10 0-

(l -20 I -200 -100 0 100 200

Spiral mode shape with varying sweep 103

2 q 1=8,255 N/m

S=l L 0°

200

w ALE

100 -

0 .,.- / ¢,-42° 1];,-42° (.; j..) not visible -100

-200 -200 -100 0 100 200

Roll and spiral mode shape with varying sweep 104

2 q1=18,570 N/rn

8=1 L Oe

2

/,w LE

1

o.,-30° __;z .:. a,-15° 0 ,. ---- e

v /V X not visible l xl

.:: -2 -1 0 1 2

Short period mode shape with varying sweep 105

2 q 1=18,570 N/rn

8=1 .c.. 0 c

2 w I\ "LE

1-

v /V '-42 ° x x 1 0 ', 'r

(• IJ o.,-42"

l_

i r '

') I I I I -2 -1 0 1 2

Short period(1) mode shape with varying sweep 106

2 q 1=18,570 N/m

8=1 L Oc

2

,~w LE J I

8

/ --"'- 0 , ;" ! ' ' I a,-42"' I v /V X x 1 ,-42°

L

')._ I -2 -1 0 l 2

Short period(2) mode shape with varying sweep 107

2 q 1=18,570 N/m

1.------~------~ v /V ,-15~ x x 1

e

-.5

-1~------,------~------.------; -1 -.5 0 .5 1

Phugoid mode shape with varying swe2p 108

5

w f ' 1LE

2.5 -

cp,-15° cp,-30° J ~ 0 " 7 B

-2.5 -

-5 I -5 -2.5 0 2.5 5

Dutc'1-roll mode shape WJth varying sweep 109

2=1LO"

400 w II "LE

200 - 0 ~,-42~ cp,-30° cp,-15\ \ \ I ...... 0 ~ ', ~.-42: {!/ \jJ,-30 \jJ ,-15/)

-200 - S not visible

-400 I -400 -200 0 200 400

Roll mode shape with varying sweep llO

2 q 1=18,570 N/m

500

r~w LE

250 -

0 ,-42

lJJ,-30° ::=if~\ ./ , ..., 0 ...... , ,, I ' y, -15° ~J,-42°

-250 - S not visible

-500 I I -500 -250 0 250 500

Spiral mode shape with varying sweep 111

2 q 1=28,150 N/m

8=1 L 0"

2

Aw LE

l

a,-15,-30,-42" 0 0

~ ~ 0 r I I e I

v /V not visible X xl :t

I i ..j -2 -1 0 1 2

Short period mode shape with varying sweep 112

8=1 L o"

4

fl.w LE

2 - c v /V ,-30 X xl

v /V ,-42° x x1 \ \ ...... 0 / 7 ,. / e I v /V ,-15" X xl

-2 - a. not visible

-4 I I -4 -2 0 2 4

Phugoid(l) mode shape with varying sweep - 113

2 q 1=28,150 N/rn

2 ~------.------~ I I v ;v ,-15 1 x x 1 0 v /V ,-30 r,vx /Vx1 ,-42 j j x x1

0 +------~~~~J~j~------+------~'------~ '' "' / 8

a not visible

-1-

-2~------~------~------~~------~l -2 -1 0 1 2

Phugoid(2) mode shape with varying sweep 114

4 ,,w ll LE

2 -

0 ¢,-15 ~ -" 0 '" 7 s ¢,-30°

I ' 0 0 0 lp,-15,-30,-42 ¢,-42° I -2 ~

-4 I I -4 -2 0 2 4

Dutch-roll mode shape with varying sweep 115

2 q1=28,150 N/rn

S=lL.O"

800 ,w .tLE

400 -

~

cp,-30"

I .... 0 , ,'- ,' ¢,-15" ¢,-42.. ~>,-421\)!,-30'" l)!,-15" 6 not visible -400 -

I I j -800 I -800 -400 0 400 800

Roll mode shape with varying sweep 116

2 q 1=28,150 N/m

1000

~w 1'LE I ! 500 - I ¢,-42°~

6 not visible -500 I

-1000 I -1000 -500 0 500 1000

Spiral mode shape with varying sweep 117

2 q =8,255 N/m 1 ~E

OE=So step -42° L3 ~ -30° G El -15"' ~ ~

7.500

.000

-7.500 v (m/s) X

·-15.000

·-22.500

.000 ·1.000 2.000 3.000 4.000 5.000

t (s)

v time response X 118

2 q =8,255 N/m 1 {E -42" A t:. OE=So step -30., 0 El -15" 0 0

.800

.600 /

.400 a (rad)

.200

.000

·- .200

.000 ·1.000 2.000 :..'3.000 4.000 ~.000

t (s)

a time response 119

2 ql=8,255 N/rn ~E -42° d i::, oE=So step -30° 0 0 -15" 0 0

·1.500 l

1.200

.BOO -

8(rad) I .400 l

JJOO

·- . 4 0 0 -~------.,.------.,.------,.------.,.------,

.000 ·1.000 2.000 3.000 4.000 '5.000

t (s)

e time response 120

2 q1=8,255 N/m ~E -30° 6 6 E=So 0 step -15° G El

.000

-.500

·-·1.200 v (m/s) X ·-1.800

-2.400

·-3.000

.000 ·1.000 2.000 3.000 4.000 5.000

t (s)

v time response X 121

~E o u =5 step -30° t,,.------,/). -1Soo 0

.030

024

.0·18 a(rad)

.0·12

.006

.000

.000 ·1.000 2.000 3.000 4.000 '5 .000

t (s)

a time response 122

2 q =B,255 N/m 1 ~E E o =5° step - 30o A.----b. -15° 0 0

.080 l

.060

.040 e (rad)

.020

.000

·- .020

.000 ·1.000 2.000 3.000 4.000 5.000

t (s)

e time response 123

2 q 1=B,255 N/rn fl.w LE OA=5° step -15D J:...------c6 -30" 0 0 -42" Q C>

.160

.120

.080

S(rad)

.0.40

.000

·- .040

.000 •1.000 2.000 3.000 4.000 5.000

t (s)

S time response 124

2 q =8,255 N/m 1 ~E oA=5" step -15° Cs e. -30° 0 El -42° <> ~

.300

.225

. 150

1/J (rad)

075

.000

-.075

.000 1.000 2.000 3.000 4.000 5.000

t (s)

1/J time response 125

2 q =B,255 N/m 1 ~E OA=So step -15" 6 -A -30°0 0 -42° <> \)

2.400

1.800

1.200

(rad)

.600

.000

·- .600

.000 1.000 2.000 .3 .000 4.000 5.000

t (s)

time response 126

2 q 1=8,255 N/m ~E -15° b t::. oR=So step -30" G El -42° <) 0

.020

.000

·- .020

S (rad)

-.040

·- .060

·- .080

.IJOO 1.000 2.000 :.'l.OOO 4.000 5.000 t (s)

S time response 127

u~R =5 o step -15° &er-----,8 -30° 0 0

.100

.ORO

.060

1/J (rad)

.040

.020

.000

.000 1.000 2.000 3.000 4.000 5.000

t (s) 1/J time response 128

~E

R o -15" 6,-----.!:::. o =5 step -30° 0 0 -42" <) 0

.BOO

.600

.400 cp(rad)

.200

.000

-.200

.000 1.000 2.000 3.000 4.000 5.iJOO

t (s)

cp time response 129

2 q1=18,570 N/m ~E c oE=S"' step -42 .6 ~ -30° 0 0 -15° <) \;)

.000

i I -20.000 l I ·- 40.000 l v (m/s) I X . --60.000 -1 I I -80.000 I

- •10 0. 0 0 0 -+-1------,------.------.------.------

.000 "1.000 2.000 2.000 4.000 '5.000

t (s)

v time response X 130

Aw LE -42" 6 h. E -30° 0 0 o =5° step -15°0----~

2.'500 l / 2.000

·1.'500 a(rad)

1.000 -

.500

.000

.000 •1.000 2.000 3.000 4.000 s.rJoo t (s) a time response 131

~E

-42° 6 -!::::. -30"' G --El sE o u =5 step -15° 0 0

5.000

4.000

3.000

8 (rad)

2.000 ~

' 1.000 l 000 -i

.000 ·1.000 2.000 3.000 4.000 '5.000

t (s)

e time response 132

2 w q1=18,570 N/m J\r..E -30° 6 e. l=so step -15" 0 El

.000

·-. 750

·-1.500 v (I'l/s) X II -2.250 1

·-3.000

·-3. 750

.000 ·1.000 2.000 3.000 4.000 5.00G

t (s)

v time response X 133

2 q 1=18,570 N/m w 1\LE

OE=S" step -30" 6 -6 -15(> 0 0

.030 I I I I I .022 ~ I I _j .0·15 I a (rad) .007 ~ i

.000 v I I -.007 ---~---

.000 1.000 2.000 3.000 4.000 5.000

t (s)

a time response 134

Aw LE -30° & t. E o o =5 step -15" [3 El

.100

.080

.060

6(rad)

.040

.020

.000

.000 1.000 2.000 3.000 4.000 5.000

t (s)

e time response 135

2 q 1=18,570 N/m Aw LE

OA=5° step -15° f:. ~ -30° 0 D -42" 0 0

.100 l

.080 l

.060

S (rad)

.040 l

.020 1

.000

.000 ·1.000 2.000 3.000 4.000 5.000

t (s)

S time response 136

2 w q 1=18,570 N/m 1\LE -15° b. -A OA=So step -30° 0 El -42" 0 ~

.400 I .300 l J .200 I

\jJ(rad) I _j .100 I

.000

·-. 100

.000 ·1.000 2.000 3.000 4.1JOO 5 .IJOO t (s)

lj! time response 137

w 2 i\LE q 1=18,570 N/m -15a 6 6. oA=S" step -30" 0 El -42D () 0

B 000 i I I I I 6.000 -j

4.000 J I cp (rad) I I ~ I ;-..:.ooo l I

.000

·- 2. 0 0 0 ~~------.,---

.000 ·1.000 2.0!JO 3.000 4.000 :=s .000

t (s)

cp time response 138

2 q 1=18,570 N/m -15.. 8------,8_ R o 0 =5 step -30° 0 0 -42° () 0

.020 l

.000

·- .020 s

·- .040 -

·- .060

- .oeo

.000 ·1.'JOO 2.000 3.000 4.000 s.:.JOo t (s)

S time response 139

2 q1=18,570 N/m ~E -15° 6 6 oR=S., step -30(1 0 El -42° 1> () .125 -1 I! I jl I , I j .100 -j jl .075 b \j;(rad) /~

.0'50 J I

.025

.000 ~~------~-----~

.fJ 00 ·1.000 2.000 3.000 4.000

t (s)

\jJ time response 140

2 q1=18,570 N/m ~E -15" 6 1::. oR=So step -30" 0 El -42° .)

1.000 l

I . 750

.'500 I qJ (rad) I I 250 J I I I .000

·- .250 -,----·~----

.000 l.fJOO 2.000 4.000 5.1JOO

t (s)

qJ time response 141

Aw LE 2 q 1=28,150 N/m -42'' .&r------6 -30"' 0 0 .rE o u =5 step -15"' 0 0

.000

·- .400

-.BOO v (m/s) X

-1.200

-- ·1.500

--2.000

.000 ·1.000 2.000 3.000 4.000 5.000 t (s)

v time response X 142

2 q 1=28,150 N/m ~E -42° & £::, OE=So step -30° 0 El -15" -o ~

8 OE-03 i

B.OE-03

4.0E-03 a()ad)

2.0[·-03

O.OE+OO

·- 2. 0 E- 0 3 -+------,------,,------,.---·----,------:

.000 •1 000 2.000 2.000 4.000 ~.000

t(s)

a time response 143

2 q =28,150 N/m 1 ~E OE=5o step -42° b. 6 -30° 0 0 -15° 0 ~

.020

.0·16

0•12

8(rad)

.008

.004

.000

.000 ·1.000 2.000 3.000 4.000

t (s)

e time response 144

2 q1=28,150 N/m ~E -15"' tJ e. oA=So step -30"' G El -42° ~ "' .100 1

.080

.060

8(rad)

.0 40

020 I

.000

.000 ·1.000 2.'JOO 3.000 4 .'JOO '5.000

t (s)

8 time response 145

2 q 1=28,150 N/m fl.w LE s:A o u =5 step -15 ° 6,------,£, -30° 0 0 -42" <)-----~

.300

.225

.150

1jJ (rad)

.075

.000

·- .075

.000 1.000 2.000 3.000 4.000 'JJJO'J

t ( s)

1jJ time response 146

2 q 1=28,150 N/m ~E oA=So step -15° A 8 -30° G E1 -42° ~ ~

10.000

8 000

6 000

4.000 -

2.000

.000

.000 1.000 2.000 3.000 4.000 5.'JOO

t (s)

step -15° 6:------,8, -3o· o o -42° <) 0

.000

!

-.040 -

-.080 i3(rad)

·-. 120 -

-.160

·- .200

.000 1.000 2.000 3.000 4.000 5.000 t(s)

i3 time response 148

2 q1=28,150 N/m ~E .i'R o u =5 step -15" 6:------.,D, -30"' 0 0 -42° <) <)

. .375

.300 -

.225

1jJ(rad)

.150

.0 75

.000

.000 1.000 2.000 4.000 5.000 t (s)

1jJ time response 149

2 q 1=28,150 N/m Aw LE oR=5° step -15" fl,----'------:A -30" 0 0 -42° ()

4.000

3.000

2.000

1.000

.000

·-·1.000

.OOr:J '1.000 2.000 3.000 4.1JOfJ 5.000

t (s)

¢ time response (\ VITA d-'

BRETT ALAN NElvMAN

Candidate for the Degree of

Master of Science

Thesis: CLASSICAL FLIGHT DYNAMICS OF A VARIABLE FORWARD SWEEP WING AIRCRAFT

Major Field: Mechanical and Aerospace Ehgineering

Biographical:

Personal: Born in Oklahoma City, Oklahoma 23 October 1961. Single. Ehjoy sports and outdoors.

Education: Graduated from Putnam City High School, Oklahoma City, Oklahoma, May, 1979. Received Bachelor of Science Degree in Mechanical and Aerospace Engineering from Oklahoma State University, May, 1983. Completed requirements for the Master of Science Degree at Oklahoma State University in May, 1985.

Experience: Research Fellow at Theoretical Mechanics Branch, Langley Research Center, NASA, during summer 1984. Engineering Technician at Materiel Management Engineering Analysis Reliability, AFLC, Tinker AFB during summer 1982. Teaching Assistant at School of Mechanical and Aerospace Ehgineering, Oklahoma State University duri g 1983-1985.

Organizations: Member of National Society of Professional Ehgineers and American Institute of Aeronautics and Astronautics.