{NASA-CR-176610) IHEOBY OF HING BOCK

(Kansas Univ.) 112 p HC A06/HJF A01.CSCL QU

N86-21520

Unclas

63/02 16565

i f *

ThE UNivERsiiy of KANSAS CENTER FOR RESEARC^ INC.

2291 Irving Hill Drive-Campus West Lawrence, Kansas 66045
Theory of Wing Rock

by

Chung-Hao Hsu and C. Edward Lan

Technical Report CRINC-FRL-516-1

February 1984

Flight Research Laboratory

The University of Kansas Center for Research, Inc.

Lawrence, Kansas 66045

Prepared under NASA Grant NAG 1-134

for

Langley Research Center

National Aeronautics and Space Administration
Abstract

A theory is developed for predicting wing-rock characteristics.

From available data, it can be concluded that wing rock is triggered by

flow asymmetries, developed by negative or weakly positive roll damping,

and sustained by nonlinear aerodynamic roll damping. A new nonlinear

aerodynamic model that includes all essential aerodynamic nonlinearities

is developed. The Beecham-Titchener method is applied to obtain

approximate analytic solutions for the amplitude and frequency of the

limit cycle based on the three-degree-of-freedom equations of motion.

An iterative scheme is developed to calculate the average aerodynamic

derivatives and dynamic characteristics at limit-cycle conditions. Good

agreement between theoretical and experimental results is obtained.
Table of Contents

Page

Abstract i

List of Symbols iv

1. . Introduction 1

2. Wing-Rock Phenomenon 2

2.1. Triggering Mechanism 4

2.2. Transient Development 6

2.3. Sustaining of Limit Cycles 7

3. Theoretical Development 10

3.1. One-Degree-of-Freedom Model 13

3.2. Approximate Analytic Solutions of the

One-Degree-of-Freedom Model 17

3.3 Three-Degree-of-Freedom Model 19

3.4. Approximate Analytic Solutions of the

Three-Degree-of-Freedom Model 25

4. Numerical Results and Discussions 30

4.1. One-Degree-of-Freedom Solutions 32

4.2. Three-Degree-of-Freedom Solutions 37

5. Conclusions and Recommendations 40

5.1. Conclusions 40

5.2. Recommendations 41

6. References. 43

ii
Table of Contents (continued)

Appendix A. Derivation of Approximate Analytic Solutions

of the One-Degree-of-Freedom Model 47

Appendix B. Derivation of Approximate Analytic Solutions

of the Three-Degree-of-Freedom Model 52

Appendix C. An Iterative Scheme for Calculating Average

Aerodynamic Derivatives and Limit-Cycle

Characteristics 57

Appendix D. Energy Balance of a Steady-State Wing Rock for

the One-Degree-of-Freedom Model 63

Appendix E. Suppression of Wing Rock with a Roll Damper 69

iii
List of Symbols

Regular Symbols

A amplitude of the limit-cycle in roll oscillations,

deg or rad

AR wing aspect ratio,

a amplitude of roll oscillations, deg or rad

b wing span, ft

Cn induced drag coefficient

CL lift coefficient

C. rolling moment coefficient

C? rolling moment coefficient at zero sideslip

_ 0

\

 Ca

 + c£:

 sin as

P P 3

C roll damping coefficient based on body-fixed axes

A*

P

Cff roll damping coefficient at zero sideslip

PO

C0 dimensionless variation of roll damping derivative

PP

with roll rate, 3C /3p

P

C dimensionless variation of roll damping derivative

P6

with sideslip, 3C /8B

At

P

C roll damping coefficient based on stability axes

PS

C total or effective roll damping coefficient

V

C variation of rolling moment coefficient with yaw rate

£r

based on body-fixed axes

C. variation of rolling moment coefficient with yaw rate

JC

o

based on stability axes

C5 variation of rolling moment coefficient with sideslip

8

iv
C . variation of rolling moment coefficient with rate of

*f»

change of sideslip

C variation of rolling moment coefficient with aileron

5

A

angle

C variation of rolling moment coefficient with rudder

««R

angle

C pitching moment coefficient

m

C yawing moment coefficient

C yawing moment coefficient at zero sideslip

no

C variation of yawing moment coefficient with roll rate

n

P

C variation of yawing moment coefficient with roll rate

n

Po

at zero sideslip

C 3C /3p

n n

PP P

Cn 3Cn 736

PS P

C variation of yawing moment coefficient with yaw rate

r

C variation of yawing moment coefficient with sideslip

ne

C directional departure parameter, C cos aV i ne dyn zz _

-— C sin a

xx *3

C . variation of yawing moment coefficient with rate of

ne

change of sideslip

variation of yawing moment coefficient with aileron

angle

variation of yawing moment coefficient with rudder

.

angle

C side-force coefficient

C side-force coefficient at zero sideslip

0

C variation of side-force coefficient with roll rate

P
C variation of side-force coefficient with yaw rate

r

C variation of side-force coefficient with sideslip

e

C variation of side-force coefficient with aileron

6 A angle

C variation of side-force coefficient with rudder angle

-

c mea \n aerodynamic (geometric) chord of a wing, ft

E energy in roll oscillations, ft-lb

G amplitude ratio of the 3-mode to the <)>-mode at limitcycle conditions

g acceleration of gravity, ft/sec/sec

H amplitude ratio of the ij;-mode to the <j>-mode at limitcycle conditions

i (i -12 /i i r1

XZ XX ZZ

I ,1 moments of inertia about the X-axis and Z-axis

xx' zz

respectively, slug-ft2

I ,1 ,1 products of inertia in the XYZ system, slug-ft2

xy' xz ' yz

 7

L

0 «Sb C*0

/:[xx> sec"2

LpQ qSb^m^V, sec"1

L qSb3 C,, /4 I V2, dimensionless

pp £pp xx

Lpg qSb2 Ct /2IxxV, sec-1

- 2 P ' i

Lr (qSb C0 /2I V)cos a , sec'1

X» XX S

Lfi qSb C r/Ixx, sec'2

A 5A

L, qSb C /I sec'2

R 6R

K system gain of a roll damper (aileron deflection per

unit roll rate), deg/rad/sec

k dimensionless reduced frequency, ub/ZV

vi
M free-stream Mach number CO

m aircraft mass, slug

Nn qSb C /I , sec"2

no

NpQ qSb2 Cn /2IzzV, sec'1

•*

 P 9

N qSbJC A I V , dimensionless

PP

N Q qSb2 C /2 I V, sec"1

pB n . zz

- 2 p6 -1

N (qSb C /2I V)cos a , sec

IT n 2 Z S

r 9

Nn qSb C /I , sec~z

P n0 zz

-

 e -?

N. qSb C /I , sec

6. n, zz

A 5A

N. qSb C /I , sec"2

5

 M ns zz

R °R

P period of the limit-cycle in roll oscillations, sec

p dimensionless reduced roll rate, pb/2V

p roll rate about the X-axis, rad/sec

f\

q flight dynamic pressure, Ib/ft

q pitch rate about the Y-axis, rad/sec

Re Reynolds number

r dimensionless reduced yaw rate, rb/2V

r yaw rate about the X-axis, rad/sec

S wing area, ft2

St area of vertical tail, ft2

t time variable, sec

u,v,w velocity components along X, Y, and Z axes

respectively, ft/sec

V magnitude of flight velocity, ft/sec

X,Y,Z rectangular coordinates of body system, ft

XP moment-center location, ft

vii
YQ qS C /mV, sec"1

° 9 Y qSb C /2mV , dimensionless

P _ P 7

Yr qSb C^ /2mV , dimensionless

r

Yg qS CY /mV, sec"1

Y. qS C /raV, sec"1

A \

Y qS Cv /mV, sec"1

R

 \

Greek Symbols

a angle of attack, deg or rad

a__ angle of attack for vortex breakdown at the trailing

BD

edge of a wing, deg or rad

a_ angle of attack for wing-rock onset, deg or rad

a steady-state angle of attack, deg or rad

s

3 sidslip angle,deg or rad

8 sideslip angle for zero total aerodynamic roll

damping, deg or rad

A net difference

6 phase angle of the ty mode, deg or rad

6. aileron deflection angle, deg or rad

6, differential tail deflection angle, deg or rad

d

6., rudder deflection angle, deg or rad

K

e phase angle of the (3 mode, deg or rad

n amplitude ratio of the i|> mode to the $ mode

8 pitch attitude angle, deg or rad

A leading-edge sweep angle, deg or rad

Li III

X damping of amplitude envelope in roll oscillations,

sec"1

viii
argument of a sinusoidal function, deg or rad

amplitude ratio of the 3 mode to the $ mode

air density, slug/ft^

bank or Euler roll angle, deg or rad

bank angle for zero total aerodynamic roll damping,

deg or rad

heading angle, deg or rad

circular frequency of the limit-cycle in roll

oscillations or angular velocity in a rotary test,

rad/sec

circular frequency of roll oscillations, rad/sec

Superscripts

dot, derivative with respect to time

prime, derivative with respect to amplitude

bar, dimensional aerodynamic derivatives defined in

equation (39)

Acronyms

AFCS

DOF

SAS

VORSTAB

Automatic Flight Control System

Degree-of-Freedom

Stability Augmentation System

a steady-flow aerodynamics computer code for

predicting stability derivatives in edge-separated

vortex flow

ix
1. Introduction

Modern combat airplanes and tactical missiles have been designed to

fly in the regime of high angles of attack to achieve higher maneuverability and a larger flight envelope. Flying at high angles of attack

will often produce lateral-directional instabilities such as wing rock,

wing drop, nose wander, and nose slice. The main topic here is wing

rock.

In this study, a summary based on extensive data is presented to

explain the basic mechanisms of wing rock. Nonlinear aerodynamic models

for one- and three-degree-of-freedom (DOF) motions are developed to

investigate the main aerodynamic nonlinearities causing wing rock. The

Beecham-Titchener method (ref. 1) is applied and extended to determine

the approximate analytic solutions for the one- and three-DOF equations

of motion. Because the interaction between aerodynamics and dynamics of

wing rock is strongly nonlinear, an iterative scheme is developed to

obtain the average aerodynamic derivatives and limit-cycle characteristics.
2. Wing Rock Phenomenon

Wing rock is an uncommanded roll-yaw oscillation dominated by roll

motion oscillating with a constant amplitude (fig. 1). Most flight

records show that the amplitude of the limit cycle in roll is at least

one order of magnitude higher than that of the oscillation in yaw or

pitch (fig. 2). Since wing rock is mainly a result of aerodynamic

nonlinearities at high angles of attack, the characteristic motion must

be described through a nonlinear mathematical aerodynamic model.

However, the response of a lightly damped Dutch roll can be adequately

determined by a linear mathematical aerodynamic model. Therefore, both

mathematically and physically, wing rock is significantly different from

a lightly damped Dutch roll.

The severity of wing rock depends on the amplitude and period of

the limit cycle. It may degrade weapon aiming accuracy (ref. 2),

missile avoidance capability, and turning effectiveness. It may also

cause safety problems during a landing approach or during a dogfight.

To improve the air combat performance or handling qualities, the basic

mechanisms of wing rock need to be understood.

Wing rock strongly depends on the details of the configuration

geometry. It is caused by the intricate flow pattern (ref. 3) at high

angles of attack that exists around an aircraft, especially the wing,

and is very sensitive to small changes in aircraft geometry. For

example, an F-4 when installed with leading edge slats (ref. 4) showed

its onset of wing rock deferred to higher angles of attack and the

amplitude buildup became much milder and more gradual, in contrast to a
basic configuration. The shark nose and wing-root leading-edge

extension (ref. 5) on an F-5 were very effective in suppressing wing

rock. As compared with the Gnat aircraft carrying no fuel tanks

(fig. 3) those carrying fuel tanks closely under its wings (ref. 6)

exhibited wing-rock onset at much higher angles of attack. The early

versions of the Harrier (ref. 7) equipped with two close fences on the

wing suffered from unacceptable wing rock; but a modified Harrier

equipped with two well-spaced fences flew with excellent roll steadiness. Recent wind-tunnel results (fig. 1) disclosed that a delta wing

with 80-degree leading-edge sweep (refs. 8 and 9) would indulge in selfexcited wing rock. On the other hand, another delta wing with 76-degree

leading-edge sweep (ref. 9) did not exhibit such motion at all.

Since the induced increment in angle of attack on a local wing

panel varies periodically during wing rock, the flow pattern differs at

different instants. For a moderately swept wing, unsteady turbulent

boundary layers may alternately separate and reattach on the wing due to

unsteady shock movement and boundary layer interaction. For a slender

wing, leading-edge vortices may periodically vary their lateral and

vertical positions (fig. 4). Furthermore, asymmetric forebody vortices

(fig. 5) may interact with the wing in a nonsteady manner. As a result,

the pressure field changes cyclically during wing rock. This in turn

drives the aircraft to rock back and forth. Hence, the nonlinear interaction between the unsteady flow field and the aircraft motion is

extremely important for wing rock.

Success of any attempt to control wing rock depends on the

understanding of its dynamic and aerodynamic mechanisms. A unified

theory is developed here to interpret the basic mechanisms of wing
rock. After examining a large amount of data, it is concluded that wing

rock can be triggered by some asymmetric flow conditions, developed by

the loss of roll damping, and sustained primarily by nonlinear roll

damping. These are explained in the following.

2.1. Triggering Mechanisms

Wing rock may be initiated either during an asymmetric flight with

a finite sideslip or during a zero-sideslip flight with some flow

asymmetries over the aircraft.

An airplane in asymmetric flight at high angles of attack, such as

uncoordinated turns, asymmetric firing of weapons, and other maneuver

situations, is susceptible to the initiation of wing rock. In addition,

poor lateral-directional stability and 'ineffective control power may

also trigger wing rock. For example, wing rock may be triggered due to

(1) rapid roll-yaw control input for the Gnat aircraft (ref. 10) and the

F-5E (ref. 11), (2) zero Dutch roll damping of the Gnat aircraft (ref.

10) and the Handley Page 115 research aircraft (ref. 12), and (3)

vanishing dynamic directional departure parameter (C ) of the F-4

^8

(refs. 4 and 13) and the F-5 (refs. 5 and 14).

 dyn

Asymmetric flow occurring at zero-sideslip flight is a particular

characteristic of many modern aircraft flying at high angles of attack.

The primary causes are asymmetric forebody vortices, viscous flow

separation of unswept and moderately swept wings, asymmetric leadingedge vortices of slender wings, and the strong interaction between

aircraft aerodynamic components.

At high angles of attack, cross flow dominates for a long pointed

nose so that asymmetric leeward vortices or wakes will be randomly shed
and give undesirable side force. Even small manufacturing asymmetries

or nose misalignment (refs. 15 and 16) could cause a cross-flow boundary

layer to separate unevenly. Such a nonzero side force or moment occurs

not only on the fuselage alone but also on complete airplanes, such as a

one-tenth scale F-5E complete-airplane wind-tunnel model (ref. 17).

Unsteady shock movement and shock boundary layer interaction could

also cause the flow on the upper surface of the outboard section of the

wings to separate first for aircraft with low to moderately sweptback

wings, such as the F-4 (ref. 18), the F-5 (ref. 19), and the Gnat

aircraft (ref. 10). Recent oil-flow studies of a single-engine general

aviation research airplane model (ref. 20) clearly pointed out that

asymmetric flow separation had progressed from wing-root trailing-edge

forward and outboard of the rectangular wing. This resulted in the

earlier stall of one wing compared to the other, consequently producing

fluctuating pressure changes and yielding a rolling moment.

At high angles of attack, aircraft with highly swept wings can

profit from vortex suction lift produced by a pair of strong vortex

cores along the upper surface of the wing, if vortex breakdown or

vortex-sheet contact does not happen on the wing. Once the vortex

breaks down, the vortex pattern is very sensitive to sideslip, resulting

in a large change of the rolling moment due to a small sideslip

disturbance. There is yet another type of asymmetric vortex pattern

without vortex bursting. Vortex sheets from both leading edges of a

slender wing may contact each other at zero sideslip (fig. 6) so as to

expel reattached flow out of the center area on the upper surface of a

wing. Possibly because of hydrodynamic (inviscid) instability (ref. 21)

in the vortex flow field, the initially symmetric vortices are trans-
formed into asymmetric vortex patterns such that one vortex slides up

over the other to produce asymmetric loads. As sweep angles of delta

wings increased from 78 to 84 degrees, induced rolling moment at zero

sideslip was shown (ref. 22) to rise from near zero to a very high

value. Delta wings with leading-edge sweep angles from 80 to 86.5

degrees (refs. 8, 23, and 24) also experienced some asymmetric vortex

structures. Hence, delta wings with sweep angles greater than 78

degrees are prone to having an asymmetric flow field at zero sideslip.

Forebody vortices and shedding vortices from upstream high-lift

devices such as strakes and canards will heavily affect the flow over

the wing and tail. Nonzero side forces or moments due to the strong

interaction between aircraft aerodynamic components are evident on the

F-5 (refs. 11 and 14) and other configurations. Analysis of wing rock

for these configurations should account for the whole aircraft rather

than the wing alone.

2.2. Transient Development

Once asymmetric flow starts, a roll-oscillation amplitude will keep

building up if the roll damping is weak or negative. For example, an

F-4 encountered severe wing rock in flight (ref. 25). Wind-tunnel

force-oscillation tests (ref. 26) later confirmed that the negative roll

damping at small amplitude near the stall angle of attack was

responsible for transient wing rock. Other configurations such as the

Gnat aircraft (ref. 6), the F-5 (refs. 5 and 14), the F-14 (ref. 27), a

recently proposed fighter configuration (ref. 2), a single-engine

general aviation research aircraft (ref. 20), the X-29A forward-swept

wing fighter (ref. 28), a series of flying wings with vertical tails
(ref. 29), and a delta wing of 80-degree sweep (ref. 8), all attributed

wing rock development to the destablizing roll damping at high angles of

attack. For the delta wing with 80-degree sweep, the asymmetric vortex

pattern (fig. 4) became more pronounced when the wing rolled. The

windward vortex appeared to diffuse and shift inboard while the leeward

vortex remained strong and moved outboard, resulting in a propelling

rolling moment at a small roll angle. The transient wing rock due to

roll instability at low roll amplitude will grow gradually over some

oscillation cycles. The decisive factor in developing transient wing

rock is the loss of roll damping at low oscillation amplitude where the

dihedral effect is almost powerless to counteract.

2.3. Sustaining of Limit Cycles

At high angles of attack, the magnitude and sign of roll damping

derivatives may vary with oscillation amplitude, frequency and roll

rate. Several sets of wind-tunnel data, including the F-4 (ref. 26) ,

the F-5 (ref. 5), a recently proposed fighter configuration (ref. 2),

and a delta wing with 80-degree sweep (ref. 8), showed that effective

roll damping was negative at small roll amplitude but positive at large

roll amplitude (fig. 7). For a delta wing with 80-degree sweep, one

reason was that an asymmetric vortex pattern (fig. 4) at large sideslip

during rolling would yield positive roll damping, because the leeward

vortex was displaced above and finally off the wing. Force-oscillation

tests of this delta wing showed only little dependency of roll damping

on oscillation frequency. However, they were conducted at frequencies

well below wing-rock frequencies observed in free-to-roll tests

(fig. 8). On the other hand, the roll damping of the F-4 (refs. 26
and 30) deteriorated quickly as the oscillation frequency was reduced.

Cubic variation of the rolling moment with roll rate seemed to correlate

well with wing rock for the Gnat trainer (refs. 6 and 10). This implied

that the roll damping was a function of roll rate. Other data showing

nonlinear dependency of the rolling moment on roll rate in rotary tests

(refs. 2, 3, and 8) also implied that the roll damping was a function of

roll rate (fig. 9). Thus, nonlinear roll damping depending on amplitude, frequency, and roll rate probably plays the leading role in

sustaining the limit cycle of wing rock.

From the viewpoint of basic fluid mechanics, the nonlinear roll

damping probably results from various vortex asymmetries and time lags

associated with the time history of fluid and induced mainly by the

fluid viscosity. Convective lag is well known to exist because shedding

vortices convect downstream at a finite speed equal to the local freestream velocity and disturbances created at any point propagate at the

local speed of sound. Because the vortex flow over a wing can not reach

its steady-state strength and position at once in plunging motion (ref.

31), the so-called vortex lag is created. During a plunging motion,

pressure changes on a wing for increasing angles of attack are not

simply reversed when angles of attack are decreasing. It was

theoretically demonstrated (ref. 32) that vortex lag would depend on

motion frequency and create a phase lag between the wing motion and its

leading-edge vortex strength buildup. Development of a turbulent

boundary layer and wake must account for the infuence of upstream flow

history on turbulent stresses (ref. 33). Since a finite time is

required to establish or destroy a healthy flow, a time lag in

separation or reattachment was recorded during a boundary-layer wind-
tunnel test (ref. 34). This test revealed that time lag seemed to be

larger in reattachment than in separation under conditions of cyclically

varying pressure gradient.

Though the roll damping is negative or weak over small oscillation

amplitudes, it is positive at larger amplitudes for a sustained wing

rock. Both the effective dihedral effect and positive roll damping

(fig. 7) via aerodynamic nonlinearities at large roll amplitudes will

gradually reduce the roll rate. As restoring moments become stronger,

the aircraft will reach a threshold amplitude and finally reverse the

rolling direction. Thus, wing rock is constrained to a finite amplitude

through nonlinear roll damping.
3. Theoretical Development

The main difficulty in describing the relationship between

instantaneous aerodynamic reactions and dynamic motion variables of an

aircraft lies in the fact that this relationship is not solely

determined by instantaneous values of motion variables but also

determined by the time history of the motion up to the instant in

question. A straightforward approach is to solve the unsteady flowfield equations simultaneously with the equations of motion of the

aircraft. However, to avoid the need for solving both dynamic and

aerodynamic equations at the same time, a proper aerodynamic model

relating aerodynamic reactions and motion variables must be developed.

Currently, there are several aerodynamic models proposed to

describe wing rocking. Besides some disadvantages, each of them is

verified only for a particular configuration.

First, it was shown that by including a cubic term in the roll

damping derivative of a Gnat aircraft (ref. 6), the angle of sideslip

during wing rock could be fairly well predicted by reducing the threeDOF lateral-directional equations of motion to a fourth-order differential equation in 8 and by obtaining solutions from the application of

the Beecham-Titchener method (ref. 1). When similar cubic terms were

introduced in the static yawing moment derivative due to sideslip for

the HP 115 research aircraft (refs. 12 and 35), the estimated amplitudes

of limit cycles were about 40 percent greater than those obtained from

six-DOF nonlinear simulations. However, using the nonlinear stiffness

alone in a one-DOF model can not explain the existence of wing rock.

10
Second, nonzero static lateral-directional aerodynamic forces and

moments at zero sideslip angle, together with aerodynamic hysteresis

were used to reproduce the wing rock of an F-5E in a six-DOF digital

computer simulation (ref. 11). The aerodynamic hysteresis of the

rolling moment with sideslip was also assumed in a simple two-DOF model

including rolling and yawing moments to demonstrate that roll hysteresis

could be a potential cause of wing rock (ref. 36). Mathematically, such

hysteresis (fig. 10) can be accounted for by including a function which

can assign two possible values in the rolling moment for a given

sideslip angle. A single-DOF model with a hysteresis loop is possible

to yield the motion of wing rock. However, limit cycles are obtained

only when an external disturbance is large enough to induce a sideslip

angle to lie outside of the fJ-range in the hysteresis loop. Furtherore, the loop shape must be determined before solving the equation of

motion.

Third, the variation of roll damping with sideslip angle (fig. lla)

such that damping was negative at small sideslip angles but positive at

large sideslip angles could induce wing rock for a delta wing with 80-

degree sweep (ref. 8). A one-DOF nonlinear simulation for this delta

wing produced the motion of wing rock in close agreement with test

results of a free-to-roll model. However, it was also shown that an

approximate analytical solution of the one-DOF equation based on the

Beecham-Titchener method by assuming a linear variation of roll damping

with sideslip angle underestimated the amplitudes of wing rock by 15

percent for angles of attack between 25 and 35 degrees. Figure lib

indicated that in contrast to no hysteresis for static-force-test data,

dynamic hysteresis was recorded during free-to-roll tests.

11
Fourth, by representing the time history effect by a lumped time

lag (ref. 37), limit-cycle amplitudes were predicted by use of

experimental static data (fig. 12) . However, this one-DOF analysis

could only predict the limit-cycle amplitude for delta wings with

leading-edge sweep larger than 74°. Besides, the frequency must be

known in advance.

Finally, the unsteady incompressible inviscid flow equations and

the one-DOF equation of motion were simultaneously solved in reference

38. It resulted in dynamic hysteresis of rolling moment versus bank

angles (fig. 13). However, it failed to predict the maximum limit-cycle

amplitude for an 80-degree delta wing because the vortex breakdown

effects were not taken into account. Further, this method could not

provide any aerodynamic derivatives causing wing rock. In addition to

some numerical simulation problems, it is very time-consuming.

To describe the wing-rock characteristics of any aircraft, the

aerodynamic model developed here will be more general to include all the

essential aerodynamic nonlinearities presented in the aforementioned

current aerodynamic models. In the meantime, the nonlinear roll damping

depending on amplitude, frequency, and roll rate may be a primary factor

in sustaining wing rock and should be accounted for properly.

Since wing rock is dominated by the roll oscillation, three-DOF

lateral-directional equations of motion will be used in the present

investigation, instead of the general six-DOF equations of motion.

Accurate solutions can be achieved through numerical integrations.

However, purely numerical solutions do not provide direct physical

insight into the dynamic mechanisms of wing rock. A useful analytic

approximation to the characteristic motion of wing rock is more suitable

12
for a parametric study of aerodynamic and dynamic causes. The BeechamTitchener method (ref. 1) will be applied and extended here for this

purpose.

3.1 . One-Degree-of-Freedom Model

A one-DOF rolling equation is to be studied first to describe the

idea. By ignoring any pitching or yawing effects, the principal motion

variable in pure rolling motion is the Euler roll angle, ij>. Flight path

properties such as flight velocity, altitude, Mach number, Reynolds

number, and mass distribution are all assumed to remain constant

throughout the motion. Effects of elastic, gravitational, and

propulsive forces are all excluded. For a rigid aircraft with the

observer fixed in the geometric body-axis system (fig. 14) whose origin

is at the aircraft center of mass, the equation of motion is given by

I-V-V •

— ^-P = C, (t) (1)

q S b

where Ixx is the rolling moment of inertia, q is the dynamic pressure, S

is the reference wing area, b is the wing span, t is the time variable,

•

p is the angular acceleration about the longitudinal axis, and C.(t) is

• Jt

the total aerodynamic rolling moment coefficient. Equation (1) simply

states that the inertial rolling moment is balanced by the timedependent aerodynamic rolling moment. Since classical stability

derivatives in Taylor series expansions alone are not adequate to

account for the aerodynamic nonlinearities at high angles of attack, the

concept of a simplified nonlinear functional (ref. 39) is used to

describe the total aerodynamic rolling moment.

13
After examining the wind-tunnel data of a flat-plate delta wing

with 80-degree sweep (ref. 8) subjected to static-force, forcedoscillation, rotary, and free-to-roll tests, the characteristic

variables of C (t) are selected to be a (the steady-state angle of

Xt S

attack), g (sideslip angle), p (the reduced rolling velocity about the

X-axis), and 6 (aileron deflection). Hence, the total aerodynamic

rolling moment coefficient at fixed a can be expressed in the form of a

S

functional as

C^Ct) = C^(as, B(t), p(t), 6 £t)) (2)

Practically, the functional will be replaced by a multivariable

function which depends on a few characteristic parameters rather than on

continuous functions. Furthermore, the multivariable function will be

expanded into a form of component buildup as

Cl(t) = \ + V + C

* ~ 0 B pt

P

 6

 + \ 'A

where

C. = C + C |B| + C |p| (4)

pt pO pg pp

The total aerodynamic rolling moment coefficient in equation (3) is

composed of (1) C. (a , B = 0), the aerodynamic rolling moment

*0

 S

coefficient arising from asymmetric flow at zero sideslip angle, being

highly dependent on aircraft configurations; (2) C, (a , B), the

*f»

 S

dihedral effect, being usually negative (statically stable), but

possibly positive (statically unstable) for some configurations (refs.

40 and 41) at high angles of attack; (3) C, (o , B, p), the total or

X> 5

pt

14
effective aerodynamic roll damping coefficient, being a strong nonlinear

function of a ,3, and p; and (4) C. (a , 6, p, 6 ), the aerodynamic

S x* f. S A

6A

roll derivative due to aileron deflection.

Equation (4) represents the present proposed aerodynamic model to

describe the aforementioned aerodynamic nonlinearities. This expression

is expected to give negative roll damping for small roll amplitudes and

positive roll damping for large amplitudes in accord with experimental

data. If C is plotted against 3 or p, the resulting curve is assumed

P

to be symmetric with respect to 3 = 0 or p = 0 respectively. The

absolute signs in equation (4) will cause the values of C at

V

different instants to be the same if | (31 or |p| have the same values at

the corresponding instants in a limit-cycle oscillation. At a

fixed 3, C. may have two values if the sign of p is different.

Furthermore, at a fixed p, C may also have two values if the sign

J6

of 3 is different. Thus, 3- and p-dependent roll damping is able to

produce discontinuous and double-value rolling moments with respect

to 0 or p. In other words, it may result in a hysteresis loop of

rolling moment with respect to 3 due to the cyclical change of total

roll damping.

Equation (1) together with the kinematic relationships

p = 4> (5)

3 = sin'1 (sin a sin <|») (6)

S

form a nonlinear second-order differential equiation in <j>, with

coefficients being also functions of time. To simplify equation (3),

equation (6) is approximated by

15
3 = <j> sin a (7)

This is a good approximation, because 3 of equation (7) is about 1

degree higher than 3 of equation (6) for most cases of wing rock.

Substituting equations (3), (4), (5), and (7) into (1) yields

I

qSb

 £0 *<SA A S 6

+ (C + sin ag C£ |*| + C£ ||£|) || (8)

pO p6 pp

For convenience, equation (8) is rewritten in terms of dimensional

derivatives as

= Lo + 6A + sin

( L pO+ sina sL

P 6 1*1

 +L

P P

where dimensional derivatives are defined by:

q S b „ , -2, . q S b -2,

Jn

 = T Co (sec

 ^ L^i

 = T Co (sec

 ^

W n A xx Is

A A 1J tX j^A U!

. = iAJ> C^ (sec'2) L = 4f&L C£ (sec'1) (10)

xx 3 xx pO

* R

 } LPP = TT-

 C*

xx prg 4 V I pp vv

 r "

16
Equation (9) is the nonlinear equation of motion to be solved for the

one-DOF model.

3.2. Approximate Analytic Solutions of the One-Degree-of-Freedom Model

To solve equation (9) so that the amplitudes and periods of wing

rock can be determined, the Beecham-Titchener method (ref. 1) appears to

be the most appropriate. This method is basically an extension of the

classical Krylov-Bogoliubov method. It is shown to be a combination of

the averaging principle and the method of variation of parameters (ref.

42). For a damped linear system (ref. 1), the first-order approximation

of the Beecham-Titchener method gives the exact solution. For such a

system, the Krylov-Bogoliubov method yields an incorrect undamped

natural frequency. Other examples (ref. 1) involving nonlinear terms

are solved with excellent accuracy by using the Beecham-Titchener

method. This method is also capable of obtaining the higher-order

approximations. Therefore, this method is applied here to obtain an

approximate analytic solution to equation (9).

By analogy with the solution for a linear second-order differential

equation, the time-dependent solution of equation (9) is assumed to be

4>(t) = a(t) cos v(t) ' (11)

That is, a product of an amplitude function a(t) and a sinusoidal

function in terms of an argument v(t) is assumed. Both o> and X are

defined in such a way that

17
(o = v (12)

X = a/a (13)

To the first-order approximation (appendix A), equation (9) without

control deflection is reduced to

X

2

-o>2 = sin a !,„ + XL + -?- aX sin a L+|-aXa> L (14)

s 8 pO 3ir s pB Sir pp

2Xa) = u> L _ + 4— a u) sin ct L „ + •=- a (X2 + 2o>2) L (15)

pO 3u s p8 3ir pp

Equation (9), a formidable nonlinear second-order differential equation,

is thus reduced to two coupled nonlinear first-order differential

equations. Both X and o> turn out to be functions of the amplitude.

Equations (14) and (15) can be further integrated by a numerical

technique. All the aerodynamic derivatives in equations (14) and (15)

should be evaluated at some average conditions according to the BeechamTitchener method.

If ft, P, and A are the circular frequency, the period, and the

amplitude respectively for a steady-state limit-cycle wing rock, then

they can be obtained by setting X to zero in equations (14) and (15).

The closed-form solutions can be shown (appendix A) to be

Q - (-sin a L0)

1/2

s p

, q S b . . .1/2

(- *- sin a C )

L S *

(16)

18
(17)

sin a

. V + 2 '

\o

I L

PP

(18) s

_ pO

4 . _ nb „

Sin as

 C£ „ + — C£

P0 PP

They must be real and positive to be physically realistic. As can be

seen from equations (16) and (17), other than inertial properties, the

limit-cycle period depends only on C . The limit-cycle amplitude is a

*e

function of C. , C , and c. . Equation (18) can be reduced to the

TM f\ * n ** po pg pp

expression used in reference 8 if C is ignored.

PP

3.3. Three-Degree-of-Freedom Model

To obtain a better understanding of wing rock, a three-DOF model

will be developed. Since the angle of attack for a sustained wing rock

remains essentially constant, the effects of the longitudinal mode on

lateral-directional modes can be neglected in the theoretical analysis.

Most aircraft are symmetric with reference to a vertical plane (X-Z

plane) aligned with the longitudinal axis (X-axis) so that Ixv = 0 and

Ivz = 0 are satisfied. If inertial coupling moments associated with pq

or qr are ignored, the equations of motion for three-DOF lateraldirectional modes (ref. 43), based on body-fixed axes, are

— (v - wp + ur - g sin 4> cos 6) = (^(t) (19)

qS

ITM. • I

3 - p - -

qSb qSb

r = c(t) (20)

19
qSb qSb

= cn(t) (21)

where m is the airplane mass and g is the acceleration of gravity; u, v,

and w are velocity components along X, Y, and Z axis; p and r are roll

and yaw rates about X and Z axes respectively; 9 is the pitch attitude

angle; Cy and Cj, are the total aerodynamic side-force and yawing moment

coefficients respectively. CyCO, C£(t), and Cn(t) can be written in

forms of functional as

Cy(t) = CY (og, B(t), p(t), r(t), 5A(t), 6R(t)) (22)

C,(t) - C, (a , 0(t), p(t), r(t), 6.(t), 6_(t)) (23)

Jt JC S A K

C (t) = C (a , 6(t), p(t), r(t), 6.(t), 6_(t)) (24)

n n s A K

where r is the reduced yaw rate and 5 is the rudder deflection angle.

K

Similar to the one-DOF analysis, Cy(t), C,(t), and (^(t) are further

expanded into forms of component buildup as

Cv(t) = C + C 3 + CL p + CY r + C 6 + C 5 (25)

Y Y0

 YS p Yr

 Y6.

 A Y6B *

(26)

Cn(t) ' C

nn

 + Cnft ^ + (°n

 n + Cn J*'

 + Cn l?|) ' +

0 3 p O p 6 p p

C r + C 6A + C 6n (27)

n_ n A n. R

5 6

20
Because the motion of wing rock is primarily rolling around the longitudinal body axis, the total aerodynamic yawing moment due to roll rate

in equation (27) is expected to be as important as the total aerodynamic

rolling moment due to roll rate in equation (26). Both roll and yaw

damping terms associated with sideslip and roll rate are thus assumed to

have similar forms.

To simplify the equations of motion, the following approximations

are applied.

sin ()>«<)>

cos <j) » 1

u/V » cos a (28)

s

v/V « S

w/V « sin a

Equations (25) - (27) are now rewritten as

6 - p sin a + r cos a - f <j> cos 9 = -Si (C + C B + C p + C r

s s v mv XQ YB Yp Yr

+

 V 5 A

+ C Y , V (29)

5A 6R

I . - ,

p . JE£r =S^ [C + c a + (c +c |6| + C |;|) p + c r

^•xx

 LKK *0 *g *pO pg pp r

+ C £.

 6 A+

\

 6R^

 (30)

6A 6R

p = > [c + C 6 + (C + Cn Jel + Cn |p|)p + Cn r

zz zz 0 8 pO p6 pp r

+

 V 6A

 + cn. 6R]
Rewriting equations (29), (30), and (31) in terms of dimensional

derivatives yields

3 - p sin a + r cos a - ~ d> cos 8 = Y«+Y 8+Y n+Y r+Y,, 6 +Y 6 (32}

s V 0 0 p^ r 6AA 6RR

I

xz '

P * j— r = LQ + L&8 -l- (L 0+L Je|+L |p|) p +_Lr£_+ Lfi ^A + LS <5R (33)

xx

 p cosa A R

s

*

 JX2 '

zz

 F *^^ cosa A R

s

where

f

C (dimensionless) y = - c (dimensionless)

p

 r SraV^ Yr

(sec"1}

— ^ — 7 _Ot__l (^l_ 1

Lnn =

 ~^ - T C? (diraensionless) L - ^T • C, cosa (sec" )

4IxxV pp xx r

22
C

n

zz pO zz

(sac'1)

_ 3 _ 2

N = — - =- C (dimensionless) N = 5= — 77 C cos a (sec )

ppv , T ..i n r LL V n s 41 V pp zz r T ..

1 V pp

zz

"

2

 Sb '2

N -§C (sec") N =5Sb c (sec'2)

6A J

zz °6

 6R J

zZ

 n5

The equations of motion for the three-DOF model are finally obtained by

rearranging equations (32) - (3A) as

Y

0

 + V + V + Yrr + V + Y6 6A + Y6 6R

 (36)

V

(5pO + V|B|

 + 5 ppM>

 P +Jrl + N5 «A + N5 6R (38)

g A R

where

Y = Y Y = Y o o 3 e

Y = Y + sin a Y =(Y - cos ct )cosa

p p s r vr s s

cos 9 Y6A = Y A A

Y = Y

6 6 R R xxz z

23
XX

PO

PP

xx PP

+ **!

r

 T- *p)

a^

and

PO

*

zz

. r

cos

is e

<7uiv,

/-_

» fr.f.
3.4. Approximate Analytic Solutions of the Three-Degree-of-Freedom Model

The Beecham-Titchener method was originally developed for a single

nonlinear second-order differential equation only. This method will be

extended here for the three-DOF equations of motion to obtain the

approximate analytic solutions for wing rock. The solutions to

equations (36) - (38) are assumed to be

3(t) = a(t) 5 (t) cos [v(t) + e(t)J (42)

<|>(t) = a(t) cos v(t) (43)

<Kt) - a(t) n(t) cos [v(t) + 5(t)J (44)

where £ and ri are the instantaneous amplitude ratios of 3 and ty modes to

the $ mode and e and 6 are the phase angles of 3 and >p modes with

respect to the $ mode.

To the first-order approximation (appendix B), equations (36) -

(38) without control deflections can be shown to be

C(Xcose - wsine) = Y0 Scose + Y X + Y TI (Xcos6 - u)sin6) + Y\ (45)

P p r <p

C(Xsine -I- wcose) = Y0 Csine + Y u> + Y n (Xsin6 + oicosS) (46)

P p r

L A + - L „ a?X + - L

pO 3n p3 3ir pp

+ L n (Xcosfi - oisin6) (47)

25
2Xu> = L0 ?sine -f- L nu> + 4~ L 0 a?u> + 4~ L a(X2

 + 2u)2)

p pO JIT pfi JIT pp

+ L n (Xsin6 + <i>cos6) (48)

n[(X2-u

2

) cos 6 - 2Xujsin<5] = N0 Scose + N nX + •=- N „

p pO JIT pp

+ .§— N aXto + N n(Xcos6 - o)sin6) (49)

JTT pp r

n[(X 2 - co2) sinS + 2Xu)cos6] = N0 ?sine + N rto) + 4~ N .

P pO JTT pp

+ 4—N a(X2 + 2to2) + N n (Xsin6 + tocosS) (50)

JTT pp r

Again, all the aerodynamic derivatives in equations (45) - (50) should

be evaluated at some average conditions according to the BeechamTitchener method. These six coupled nonlinear first-order differential

equations can be integrated numerically to find a, 5, n, e, v.and 6 as

functions of t for known initial conditions.

To suppress the wing rock effectively, the main aerodynamic

nonlinearities causing the self-induced limit cycle must be understood

and justified first. An approximate theory, such as the one developed

here, is an efficient tool for recognizing the aerodynamic nonlinearities important to the amplitude and frequency of a limit cycle. The

amplitude of a limit cycle is the primary concern as a handling-quality

or safety problem. Therefore, the main effort here is to find out the

effects of amplitudes of (5 and fy modes on that of the $ mode for a

steady-state limit-cycle oscillation.

26
To obtain the limit-cycle amplitude of a sustained wing rock, X is

set to zero. In addition, the phase angles, e and 6, are ignored since

they are found numerically to be insignificant for a steady-state

constant-amplitude wing rock. Equations (45) - (50) are then reduced to

0 = Yfl G + Y (51)

p <p

Y + Y H (52)

P

 r

= -LBG (53)

L-+4-L 0 AG+|- L Aft + LH (54)

pO Sir p0 3ir pp r

Hfi2 = -N0G (55)

P

N r i +4-N f l AG+|- N Afl + NH (56)

pO Sir pB Sir pp r

where G and H are the amplitude ratios of 3 and ty modes to the <j> mode at

limit-cycle conditions.

It Is assumed that the frequencies of <j>, (3,and v|> modes are the same

in equations (45) - (50). Because equations (51) and (55) are related

to the frequencies of 6 and fy modes respectively, they can be ignored in

order to find the effects of Q-ty amplitudes on the <j> amplitude for a

steady-state wing rock. The limit-cycle conditions are then obtained in

appendix B by rearranging equations (52), (53), (54), and (56) into the

forms

27
3ir

 LpO

A - - 4 ^

GL . + 2J2L

PB PP

gC£ + fCn

3ir_ _ pO pO

4 + fC

fl n

pg pg pp pp

where

*«

 C£ + Z

xZ

 C

_

n

 r _ r

I C. + I C

xz £ xx n

r r

zz

I I

XX / XZ

ZZ XX

-

GL3

I C. + I C

zz I xz n

-GqSb 2 5-E- (58)

II - I

XX ZZ XZ

G = Y + HY

P

= sin a - H cos2a + - C + H - C cos a (59)

3 s 2mV2 Yp 2mV2 Yr s

H = - - [SpQ + A (G Npe + 2. Npp)

r

c. + I C +-A[G( I Cn +1 C )-»( I C +1 C )]

xz I „ xx n 3ir xz £ xx n V xz £ xx n

gO gO £g £§ ££ pp

(I

xz

C

£

 +I

x x

C

n > c08 ° s

(60)

23
The roll amplitude of a limit cycle is affected not only by C ,

xp

C2. ' Ci ' anc* C£ as in the one

~

DOF mo<*el but also by c , Cy , Cy ,

pO pS pp g p r

C , C ,C , C , and C as shown in equation (57). Equations

npO np3 "pp r nr

(57) - (60) are coupled nonlinear algebraic equations in terms of A, G,

H, and fl. They can be solved through simple iterations started with G

= sin a . For a steady-state wing rock, all solutions must be real and

s

positive.

29
4. Numerical Results and Discussions

It would be of great interest if the predictions of the present

theoretical results could be shown to agree with measured wing-rock

characteristics of aircraft, such as the F-4, F-5, F-14, X-29A, Gnat,

Harrier, HP 115, and some reentry vehicles. Unfortunately, not all

aerodynamic derivatives needed in equations (57) - (60) are available.

In addition, it is difficult to estimate the high-a aerodynamic

derivatives for a full-scale airplane especially when effects of vortex

breakdown, viscous flow separation, and the aerodynamic hysteresis on

the wing become significant.

A recent theoretical study of slender wings (ref. 44) based on the

quasi-vortex-lattice method (ref. 45) and the suction analogy (ref. 46)

showed that the longitudinal and lateral-directional aerodynamics with

vortex-breakdown effects at high a can be reasonably well predicted.

Therefore, the updated steady-flow aerodynamics computer code

(VORSTAB)** based on reference 44 will be used here to calculate all

required aerodynamic derivatives for slender wings for the present

purpose.

To test the one-DOF theory, a delta wing with 80° leading-edge

sweep (ref. 8) is chosen. To verify the three-DOF theory, eight flying

wings with vertical tails (ref. 29) are selected.

Having formulated the mathematical model on body-fixed axes,

•

g derivatives must be included in all p and r derivatives. However, no

•

existing theoretical methods are available for predicting 8 derivatives.

**The VORSTAB computer code is available through the NASA Langley

Research Center.

30
Since the aerodynamic nonlinearities of wing rock mainly arise from the

nonlinear roll damping of the wing, it is decided to correlate the roll

damping derivatives (C ) based on stability axes and calculated from

Ps

the VORSTAB code with the roll damping derivatives (C + C . sin a )

P B S

based on body axes and obtained in forced-oscillation tests. The reason

is that the effect of C.. on C0 based on body axes is similar to that

*

of C. on C. based on stability axes. The results of four delta

r P s s

wings with 80°, 76°, 74°, and 60° leading-edge sweep (refs. 8, 22,

47,48, and 49) are presented in figures 15-18, Two conclusions can be

made with regard to comparison between the calculated

C0 and data of C C . sin a . First, the agreement is reasonably

-C At T Xt Q S

P Q p

s

good if the amplitude in forced-oscillation test is not too small and

the vortex-breakdown effect is not significant. Second, the agreement

is better if data of the forced-oscillation technique are compared

instead of those of the free-to-damp-oscillation method.

To simulate a one-DOF free-to-roll test or forced-oscillation test

based on body axes, an appropriate location of moment center must be

chosen if the VORSTAB code is to be used to calculate the aerodynamic

derivatives on stability axes. It should be noted that the VORSTAB code

is to simulate the rotary-test results. The moment-center locations

used in the present study are set at the aerodynamic-center location

taken from wind-tunnel data (refs. 22 and 47). Figures 15-18 confirm

that the aerodynamic-center location is a good choice for the moment

31
center if the aerodynamic derivatives are to be calculated on stability

axes. Furthermore, numerical results also show that the aerodynamic

derivatives change very little if the moment-center location moves +0.10

c from the aerodynamic-center location.

Knowing that the roll-dominated wing rock is mainly a lateral

problem caused by the wing, the VORSTAB code is now used to calculate

C. and cn for C. (actually ca +€„. sin a) and * JC Jt JC Jo - S Ps rg p p 0

C, (a tually C0 -C . cos a ) respectively. The other aerodynamic

r *r

 ng

 S

derivatives are calculated on body axes.

4.1. One-Degree-of-Freedom solutions

To verify the one-DOF theory, a flat-plate delta wing with 80°

leading-edge sweep is chosen because its wind-tunnel data of staticforce, forced-oscillation, rotary, and free-to-roll tests have been well

conducted and published (ref. 8).

According to the Beecham-Titchener method, all aerodynamic

derivatives should be evaluated at some average dynamic conditions.

Since the interaction between aerodynamics and dynamics in wing rock is

strongly nonlinear, an iterative scheme (appendix C) is developed to

calculate the average aerodynamic derivatives and dynamic characteristics for a steady-state self-induced limit cycle. In other words, the

aerodynamic derivatives are calculated at some average dynamic

conditions (i.e. certain average values of 3 and p) which in turn depend

on the aerodynamic characteristics.

32
The calculated longitudinal aerodynamic derivatives of the test

configuration are presented in figure 19. The predicted results agree

quite well with wind-tunnel data even at a beyond a. (aTM = 38°).

BD BD

The limit-cycle amplitude is the main concern in studying the wingrock problem. The calculated results are plotted in figure 20. To the

first-order approximation, the prediction is generally good up to a of

38°. Beyond that a, a converged solution is difficult to obtain.

Besides the strong vortex breakdown effects at higher a, the viscous

flow separation will probably also become important. The latter effect

is not included in the VORSTAB code. The predicted limit-cycle period

is reasonably good as shown in figure 21.

For a limit-cycle oscillation, the values of C are positive and

PO

those of C are negative; while those of C. may be either positive

Xf _ x

P3 PP

or negative (fig. 22). The effect of C is to decrease the predicted

PP

limit-cycle amplitudes for a below 30° and above 36°; while it increases

the predicted amplitudes for a in between. Without C , the predicted

x»

PP

amplitude is 54.4° (overestimated) at a of 25° and is 31.1° (underestimated) at o of 32°. The predicted maximum amplitude with C is

Xf

PP

40.7° at a of 32° and is very reasonable when compared with available

data (fig. 20). Thus, the roll-rate dependency of roll damping plays an

important role in determining the limit-cycle amplitude.

The theoretical variations of all terms in equation (4) for a = 27°

and 32° are plotted in figures 23 and 24 respectively. It is seen that

magnitudes of C. and C |p| ar° much higher than those of p |&|

 at

pO pp pg

lower bank angles while C |3|is dominating at higher bank angles.

V

The shape of C and its values are affected by the sign of c.

pt pp

33
The histograms of C0 versus $ for one limit cycle of wing rock

J6

based on equations (3) and (4) without C or the control term at ct of

*0

27° and 32° are presented in figures 25 and 26 respectively. The loops

give evidence of dynamic hysteresis during the wing rocking. Agreement

of theoretical contours with the data derived from free-to-roll tests

(ref. 8) is reasonable. The theoretical loops are symmetric against

positive and negative bank angles, but the experimental loops depict

some asymmetries.

To understand the loop asymmetries, it is known that the induced

rolling moment at zero sideslip C increases with increasing a for a

*0

highly swept delta wing. The existing data of C (refs. 8, 9, and 22)

£

0

vary with the model and the wind tunnel used in the static-force tests

for a delta wing with 80° sweep. The vortex asymmetry is the main

factor to cause the induced rolling moment at zero sideslip. The net

difference of experimental C between clockwise and counterclockwise

*0

loops (i.e. AC at <|> of 0°) determines which direction the wing will

Xr

tilt and how the dynamic loop will shift with respect to the theoretical

prediction. The VORSTAB code can not predict C0 . At a of 27°, AC. is

*0 *0

+0.01 (fig. 25). The dynamic loops shift up. The positive amplitude is

5° greater than the negative amplitude. At a of 32°, AC is -0.05

0

(fig. 26). The dynamic loops rotate counter-clockwise. The positive

amplitude is 5° less than the negative amplitude. In addition, during

the oscillation the effect of time lags strongly affects the vortex

patterns, causing the dynamic hysteresis. It is the combination of

vortex asymmetry and time lag which produces the asymmetrical histogram

in experiment.

34
The total aerodynamic energy exchanged over a limit-cycle

oscillation is given by

AE = qSb / C (t) <f>(t) dt (61)

C(t) *

or

AE = qSb / C (<(>) d<(> (62)

C(c|>)

where C(t) and C(<f>) are the contours obtained by plotting C versus t

x»

and <j> respectively. For a steady-state wing rock, C(<j>) is a closed

curve over one oscillation cycle as shown in figures 25 and 26, where

the arrows indicate the direction of increasing f'me. As derived in

appendix D, equation (62) can be reduced to

AE = 2qSb2J2/V / C (A2 - <f>2)

1/2 d<() (63)

0 pt

This equation points out that the net energy transferred is

exclusively determined by the total aerodynamic roll damping. In other

words, to the first-order approximation it has nothing to do with

C and C . The areas within the clockwise loops are positive or £o *e

dynamically undamped due to a positive C so that energy is added to

V

the aircraft. On the other hand, the areas within the counterclockwise

loops are negative or dynamically damped due to a negative C so that

Pt

energy is consumed by the aircraft. It is shown in appendix D that AE

is theoretically zero, or the area of the destabilizing loops equals

that of the stabilizing loops. Therefore, the net aerodynamic energy

added to the aircraft and extracted from it is theoretically zero over

an ideal limit-cycle oscillation. Numerical results (figs. 25 and 26)

35
further testify that the energy balance between the work input and the

energy consumed is required to sustain a steady-state wing rock.

The intersection of stabilizing and destabilizing loops represents

a turning point of roll damping because C is zero at that point. The

V

critical bank and sideslip angles, <j> and {3 , are defined as the angles

of zero total aerodynamic roll damping. The aircraft is dynamically

undamped below <(> and damped above <jj . The solutions of <j> are (appendix D)

~

 4rd) 1/2,

if C > 0

PP

where

pO

2r ~ sin a Cn s £

P6

if

= 0

PP

PP

(64)

t • r - (sin «

PS PP

s=2 sin a C C.

s *

(65)

_2 ,bfl _ ,2

= C£ ~ (2v

 A c£ )

pO pp

The predicted 4> and 8 are shown in figure 27. It is fairly good

as compared with the available data (ref. 8).

36
4.2. Three-Degree-of-Freedom Solutions

Simple flying wings with a vertical tail but without a horizontal

tail or fuselage were extensively tested in the NACA Langley free-flight

tunnel (ref. 29). Four models having delta wings with 53°, 63.5°, 76°,

and 82.9° leading-edge sweep and four cropped delta wings with a taper

ratio of 0.5 are used in verification of the three-DOF theory.

The iterative scheme used in the one-DOF theory (appendix C) is

also applicable here. The theoretical results of two flying wings are

presented in figures 28 and 29. For both configurations, a 76°-delta

wing and a 76°-cropped-delta wing with a vertical tail, the predicted

lift coefficients agree well with data up to a of 30°.

The three-DOF theory overpredicts the starting a of wing rock by

about 5° as shown in figures 28 and 29. The shaded areas in both

figures represent the approximate a range in which the steady-state wing

rock was observed. Both a ranges extend over 5° of a approximately.

At a below that range, no wing rock was observed. At a above that

range, a constant-amplitude roll oscillation kept increasing rapidly

as o was increased further. As a result, the flying model rolled

completely over out of control. The predicted a ranges of wing rock for

both configurations are less than 10° of a. The solid lines in both

figures can not be extended to higher-a values because the iterative

scheme (appendix C) yields no converged solutions or equations (57) -

(60) gives negative amplitudes or imaginary frequencies. Since no

limit-cycle amplitudes had been recorded during the flying tests, it is

impossible to verify the theoretical amplitudes for both flying models.

Before the flying model reached the a of maximum limit-cycle

amplitude, it had probably rolled off because of the great reduction in

37
the eleven effectiveness at high a. Furthermore, the above theoretical

results are calculated for free oscillations without control deflection

while the elevons had been used during the free-flight-tunnel tests. At

any rate, additional data on these configurations are needed to

ascertain the accuracy of the present theory.

Only five flying models with an aspect ratio equal to or less than

1 were reported to exhibit wing rock. The theoretical prediction (fig.

30) has confirmed the same trend. Agreement between the predicted

onset o of limit cycle and data is within ± 5°. Since the free-flighttunnel models simulate the more realistic cases as compared to one-DOF

f ree-to-roll models, figure 30 can serve as a preliminary design guide

for aircraft design engineers.

The predicted period of a limit cycle decreases as ct increases.

The calculated limit-cycle periods are generally greater than four

seconds for those five flying models exhibiting wing rock. No data of

frequency or period were reported. Hence, no comparison is made.

The amplitude ratio of @ to <|> modes at the limit-cycle condition G

varies from 0.01 to 0.30 for the above five flying models at

different a. On the other hand, the amplitude ratio of ty to <j> modes at

the limit-cycle condition H remains approximately 0.11 for most cases.

Therefore, 3 and fy modes have some effects on the roll amplitude of

limit cycle in the three-DOF theory.

It is found in the above numerical analyses that in determining the

wing-rock characteristics the importance of C,, , C_ , and C is next

X* Tl ^T*

r pO

 r

to that of C , C , and C ; while the effects of r , CL, , C ,

**/•***/ ! * " ^^^ «

pO pg pp p r pg

and C are almost negligible. In the one-DOF theory, it is required

"PP

that C be positive (undamped) to cause a limit-cycle motion. In

38
contrast to the one-DOF theory, wing rock is possible in the three-DOF

theory even if C is negative (damped) but weak. The reason is that

PO

the combined effects of C. , C , and C may make the aircraft more

]L ti _ n

r pO r

unstable in roll oscillations.

In general, there are three possible approaches to suppress or

prevent wing rock: (1) airframe reshaping, (2) maneuver limiting, and

(3) employing a stability augmentation system (SAS) or an automatic

flight control system (AFCS). The first approach requires the detailed

aerodynamic reshaping of the basic airframe configuration to provide

inherently wing-rock-free capability. The second approach gives a fast

solution eliminating wing rock by simply adopting an a-limiter.

However, it may degrade usable maneuverability. Thanks to tremendous

advances in avionic technology, the third approach employing SAS or AFCS

has become the most effective method for attaining strong resistance to

wing rock without degrading high maneuverability.

Because wing rock is mainly due to the loss of roll damping at

high a, it can be effectively suppressed by using a roll damper SAS to

artificially augment roll damping. For example, a 76°-delta wing with a

vertical tail is predicted to exhibit a limit-cycle amplitude of 50°

at a of 20°, but the limit cycle of this flying wing can be eliminated

completely by incorporating an ideal roll damper as shown in appendix

E.

39
5. Conclusions and Recommendations

5.1. Conclusions

A unified theory based on extensive experimental data is proposed

to elucidate the basic mechanisms of wing rock. It is concluded that

wing rock Is triggered by flow asymmetries, developed by negative or

weakly positive roll damping, and sustained by nonlinear aerodynamic

roll damping.

A new and more general nonlinear aerodynamic model that includes

all essential aerodynamic nonlinearities causing wing rock has been

developed. The key element is the total aerodynamic roll damping which

depends on oscillation amplitude and roll rate and is formulated to

encompass the aerodynamic hysteresis implicitly.

The Beecham-Titchener method has been applied and extended to solve

the one- and three-DOF equations of motion. To the first-order

approximation, the equations of motion are reduced to coupled nonlinear

algebraic equations at limit-cycle conditions. These equations are then

solved through iterations for the limit-cycle amplitude and frequency.

On the other hand, closed-form solutions are obtained for the one-DOF

theory.

To determine the numerical values of the limit-cycle amplitude and

frequency, the important aerodynamic derivatives such as C , C , C ,

•** s\ * n ** pO pB pp

C , C , C , C. , and C , are required. Because of the strong

A, n n. Jt _ n.

r pO r e B

interaction between aerodynamics and dynamics during wing rocking, these

derivatives are evaluated with a steady-flow aerodynamics computer code

at some average dynamic conditions in an iterative manner.

Theoretical results show that to sustain a steady-state wing rock,

the total aerodynamic roll damping must be negative at small bank angles

40
in the one-DOF case but it may be negative or weakly positive in the

three-DOF case. On the other hand, at larger bank angles it must be

positive in both cases. Furthermore, the one-DOF theory indicates that

the energy balance between the work input and the energy consumed during

motion is another reason for a self-induced wing rock.

Good agreement between theoretical and experimental results has

been obtained for many slender wings, of which all but one are equipped

with a vertical tail.

5.2. Recommendations

Good comparison of theoretical results with data has been shown for

some simple aerodynamic configurations. However, additional comparisons

should be done for more configurations if data are available.

More experiments should be conducted to understand some fundamental

dynamic phenomena. First, systematic forced-oscillation and rotary

tests should be made to investigate the effects of amplitude, frequency,

and roll rate on the roll damping for simple configurations at high

angles of attack. Second, curved-flow and rolling-flow and pure

lateral-translation-oscillation techniques should be applied to determining the pure derivatives such as C , C , C , C , C ., and C . .

J6 J6 tl Tl J6 _ Tl n p r p r 3 8

Finally, since the one-DOF free-to-roll test seems inadequate to predict

wing-rock characteristics in free flight, more free-flight tests should

be conducted.

At present, it is very difficult to predict theoretically the

hlgh-a aerodynamic derivatives for a complete airplane. The effects of

fluid viscosity, time lag, asymmetric forebody vortices, asymmetric

leading-edge vortices, viscous flow separation, shock boundary layer

41
interaction, and other unsteady aerodynamic interferences should be

somehow accounted for. At least, a theoretical method for predicting

•

the important (3-derivatives for slender wings should be developed.

It is shown that wing rock can be effectively suppressed if a roll

damper SAS is incorporated to augment roll damping. However, more

research should be done on high-ct control law to provide a fast and

reliable AFCS for future fighters.

42
6. References

(1) Beecham, L.J., and Titchener, I.M., "Some Notes on an Approximate

Solution for the Free Oscillation Characteristics of Non-Linear

Systems Typified by X + F (x,x) = 0," British Aeronautical Research

Council, R & M No. 3651, Aug. 1969.

(2) Chambers, J.R., Gilbert, W.P., and Nguyen, L.T., "Results of

Piloted Simulator Studies of Fighter Aircraft at High Angles of

Attack," Paper 33, AGARD-CP-235, May 1978.

(3) Orlik-Ruckemann, K.J., "Aerodynamic Aspects of Aircraft Dynamics at

High Angle of Attack," Journal of Aircraft, Vol. 20, Sept. 1983,

pp. 737-752.

(4) Burris, W.R. and Lawrence, J.T. , "Aerodynamic Design and Flight

Test of U.S. Navy Aircraft at High Angles of Attack," Paper 25,

AGARD-CP-102, Apr. 1972.

(5) Skow, A.M. and Titiriga, A., Jr., "A Survey of Analytical and

Experimental Techniques to Predict Aircraft Dynamic Characteristics

at High Angles of Attack," Paper 19, AGARD-CP-235, May 1978.

(6) Ross, A.J., "Lateral Stability at High Angles of Attack,

Particularly Wing Rock," Paper 10, AGARD-CP-260, Sept. 1978.

(7) Bore, C.L., "Post Stall Aerodynamics of the Harrier GR1," Paper 19,

AGARD-CP-102, Apr . 1972.

(8) Nguyen, L.T., Yip, L.P., and Chambers, J.R., "Self-Induced Wing

Rock of Slender Delta Wings," AIAA Paper 81-1883, Aug. 1981.

(9) Levin, D. and Katz, J., "Dynamic Load Measurements with Delta Wings

Undergoing Self-Induced Roll-Oscillation," Journal of Aircraft,

Vol. 21, Jan. 1984, pp. 30-36.

(10) Ross, A.J., "Flying Aeroplanes in Buffet," Aeronautical Journal,

Vol. 81, Oct. 1977, pp. 427-436.

(11) Cord, T., "Hysteresis-Induced Wing Rock," AFFDL-TM-75-76-FGC, June

1975.

(12) Ross, A.J., "Investigation of Nonlinear Motion Experienced on a

Slender-Wing Research Aircraft," Journal of Aircraft, Vol. 9, Sept.

1972, pp. 625-631.

(13) Johnston, D.E., "Identification of Key Maneuver-Limiting Factors in

High Angle of Attack Flight," Paper 36, AGARD-CP-235, May 1978.

(14) Titiriga, A., Jr., Ackerman, J.S., and Skow, A.M., "Design

Technology for Departure Resistance of Fighter Aircraft," Paper 5,

AGARD-CP-199, Nov. 1975.

43
(15) Fldler, J.E., Schwind, R.G., and Nielsen, J.N., "Investigation of

Slender Body Vortices," AIAA Journal, Vol. 15, Oct. 1977, pp. 427-

436.

(16) Clark, W.H., Peoples, J.R., "Occurrence and Inhibition of Large

Yawing Moments during High Incidence Flight of Slender Missile

Configuration," Journal of Spacecraft and Rockets, Vol. 10, Aug.

1973, pp. 510-519.

(17) Kao, B.C., "Side Forces on Unyawed Slender Inclined Aerodynamic

Bodies," Journal of Aircraft, Vol. 12, Mar. 1975, pp. 142-150.

(18) Ray, E.J., McKinney, L.W., and Carmichael, J.G. , "Maneuver and

Buffet Characteristics of Fighter Aircraft," Paper 24, AGARD-CP102, Apr. 1972.

(19) Hwang, C. and Pi, W.S., "Some Observations on the Mechanism of

Aircraft Wing Rock," Journal of Aircraft, Vol. 16, June 1979, pp.

366-373.

(20) Chambers, J.R., DiCarlo, D.J., and Johnson, J.L., Jr.,

"Applications of Dynamic Stability Parameters to Problems in

Aircraft Dynamics," Lecture 17, AGARD-LS-114, Mar. 1981.

(21) Keener, E.R. and Chapman, G.T., "Similarity in Vortex Asymmetries

over Slender Bodies and Wings," AIAA Journal, Vol. 15, Sept. 1977,

pp. 1370-1372.

(22) Shanks, R.E., "Low-Subsonic Measurements of Static and Dynamic

Stability Derivatives of Six Flat-Plate Wings Having Leading-Edge

Sweep Angles of 70° to 84°," NASA TN D-1822, 1963.

(23) Wentz, W.H., Jr. and Kohlman, D.L., "Vortex Breakdown on Slender

Sharp-Edged Wings," Journal of Aircraft, Vol. 8, Mar. 1971, pp.

156-161.

(24) Bird, J.D., "Tuft-Grid Surveys at Low Speed for Delta Wings," NASA

TN D-5045, Feb. 1969.

(25) Chambers, J.R. and Anglin, E.L., "Analysis of Lateral-Directional

Stability Characteristics of a Twin-Jet Fighter Airplane at High

Angles of Attack," NASA TN D-5361, Aug. 1969.

(26) Grafton, S.B. and Libbey, C.E., "Dynamic Stability Derivatives of a

Twin-Jet Fighter Model at Angles of Attack from -10° to 110°," NASA

TN D-6091, Jan. 1971.

(27) Nguyen, L.T. and Gilbert, W.P., "Application of High - a Control

System Concepts to a Variable-Sweep Fighter Airplane," AIAA Paper

80-1582, Aug. 1980.

(28) Murri, D.G., Groom, M.A., and Nguyen, L.T., "High Angle-of-Attack

Flight Dynamics of a Forward-Swept Wing Fighter Configuration,"

AIAA Paper 83-1837, July 1983.
(29) McKinney, M.O., Jr., and Drake, H.M., "Flight Characteristics at

Low Speed of Delta-Wing Models," NACA RM L7K07, Jan. 1948,

(30) Schiff, L.B. and Tobak, M., "Some Applications of Aerodynamic

Formulations to Problems in Aircraft Dynamics," Lecture 16, AGARDLS-114, Mar. 1981.

(31) Lambourne, N.C., Bryer, D.W., and Maybrey, J.F.M., "The Behavior of

the Leading-Edge Vortices over a Delta Wing Following a Sudden

Change of Incidence," British Aeronautical Research Council, R & M

No. 3645, Mar. 1969.

(32) Lan, C.E., "The Unsteady Suction Analogy and Applications," AIAA

Journal, Vol. 20, Dec. 1982, pp. 1647-1656.

(33) Green, J.E., Weeks, D.J., and Brooman, J.W.F., "Prediction of

Turbulent Boundary Layers and Wakes in Compressible Flow by a LagEntrainment Method," British Aeronautical Research Council, R & M

No. 3791, Jan. 1973.

(34) DePonte, S. and Baron, A., "Experiments on a Turbulent Unsteady

Boundary Layer with Separation," Paper 12, AGARD-CP-296, Sept.

1980.

(35) Ross, A.J. , and Beecham, L.J., "An Approximate Analysis of the NonLinear Lateral Motion of a Slender Aircraft (HP 115) at Low

Speeds," British Aeronautical Research Council, R & M No. 3674, May

1970.

(36) Schmidt, L.V., "Wing Rock due to Aerodynamic Hysteresis," Journal

of Aircraft, Vol. 16, Mar. 1979, pp. 129-133.

(37) Ericsson, L.E., "The Fluid Mechanics of Slender Wing Rock," AIAA

Paper 83-1810, July 1983.

(38) Konstadinopoulos, P., Mook, D.T., and Nayfeh, A.H., "Numerical

Simulation of the Subsonic Wing-Rock Phenomenon," AIAA Paper 83-

2115, Aug. 1983.

(39) Tobak, M. and Schiff, L.B., "Aerodynamic Mathematical Modeling -

Basic Concepts," Lecture 1, AGARD-LS-114, Mar. 1981.

(40) Skow, A.M., Titiriga, A., Jr., and Moore, W.A., "Forebody-Wing

Vortex Interactions and Their Influence on Departure and Spin

Resistance," Paper 6, AGARD-CP-247, Oct. 1978.

(41) Johnson, J.L., Jr., Grafton, S.B., and Yip, L.P., "Exploratory

Investigation of the Effects of Vortex Bursting on the High-Angleof-Attack Lateral-Directional Stability Characteristics of HighlySwept Wings," AIAA Paper 80-0463, Mar. 1980.

(42) Simpson, A., "An Algorithm for Autonomous Nonlinear Dynamical

Equations," Aeronautical Quarterly, Aug. 1977, pp. 211-234.

45
(43) Roskam, J, "Airplane Flight Dynamics and Automatic Flight

Controls," Roskam Aviation and Engineering Corporation,

Lawrence,Kansas, 1979, Chapter 2.

(44) Lan, C.E. and Hsu, C.H., "Effects of Vortex Breakdown on

Longitudinal and Lateral-Directional Aerodynamics of Slender Wings

by the Suction Analogy," AIAA Paper 82-1385, Aug. 1982.

(45) Lan, C.E., "A Quasi-Vortex-Lattice Method in Thin Wing Theory,"

Journal of Aircraft, Vol. 11, Sept. 1974, pp. 518-527.

(46) Polhamus, E.G., "Prediction of Vortex-Lift Characteristics by a

Leading-Edge Suction Analogy," Journal of Aircraft, Vol. 8, Apr.

1971, pp. 193-199.

(47) Tosti, L.P., "Low Speed Static Stability and Damping-in-Roll

Characteristics of Some Swept and Unswept Low-Aspect-Ratio Wings,"

NACA TN 1468, Oct. 1947.

(48) Boyden, R.P., "Effects of Leading-Edge Vortex Flow on the Roll

Damping of Slender Wings," Journal of Aircraft, Vol. 8, July 1971,

pp. 543-547.

(49) Johnson, J. L., Jr., "Low-Speed Measurements of Static Stability,

Damping in Yaw, and Damping in Roll of a Delta, a. Swept, and an

Unswept Wing for Angles of Attack from 0° to 90°," NACA RM L56B01,

Apr. 1956.

46
Appendix A. Derivation of Approximate Analytic Solutions of the

One-Degree-of-Freedom Model

The Beecham-Titchener method (ref. 1) is basically an extension of

the Krylov-Bogoliubov asymptotic method. This method is shown to be a

combination of the averaging principle and the method of variation of

parameters (ref. 42). For a damped linear system (ref. 1), the firstorder approximation of the Beecham-Titchener method gives the exact

solution. For such a system, the Krylov-Bogoliubov method yields an

incorrect undamped natural frequency. Other examples (ref. 1) involving

nonlinear terms are solved with excellent accuracy by using the BeechamTitchener method. This method is also capable of obtaining the higherorder approximations. Therefore, this method is applied here.

Equation (9) is rewritten as

*

 = Lo + Ls

 6A

 + sin as

 Le * + (L

Po

 + sin as A

The solution of equation (A.I) is assumed to be

4>(t) = a(t) cos v(t) (A. 2)

u> and X are defined as

u> = v (A. 3)

a/a (A. 4)

47
where the superscript dot means a time derivative. In addition, let

differentiation with respect to the amplitude a be denoted by a

• • ••

superscript prime. Then 01, X, and a can be expressed as

•

u = a X io? (A.5)

X = a X X' (A.6)

a = a X(a X1 + X) (A.7)

On differentiating equation (A.2) with respect to time, one obtains

•

<)> = a(X cos v - oj sin v) (A.8)

$ = a {[X(a X' + X) - o>2] cos v - X(au)' + 2u) sin v} (A.9)

The Beecham-Titchener method assumes that <u and X do not vary

greatly during one cycle of oscillation. In other words, a, X, <D, and

amplitude derivatives are interpreted as having fixed values and are

assigned the values at midcycles. Equation (A.I) without the control

deflection term is rewritten as

<j> = L- + sin a L0 a cos v + [L _ + sin a L ., sgn (41) a cos v

u s p pu s pp

+ L sgn (<j>) a(Xcosv - cosinv)] a(Xcosv - usinv)

PP

= F (v) (A.10)

48
Equations (A.9) and (A.10) are combined into

F(v) = a[X(aX' + X) - ui2] cos v - aX(ao)' + 2o>) sin v (A.11)

Regarding equation (A.11) as a Fourier expansion for F(v), the Fourier

coefficients can be determined to be

7 l 2u

X(aX' + X) - u = ~ / F(v) cos v dv (A. 12)

ira J

i

 ir

X(au>' + 2u) = - ±£ / F(v) sin v dv (A.13)

Equations (A.12) and (A.13) embody the averaging principle. They are

similar in form to the Krylov-Bogoliubov first-order-approximation

2

equations, but include new terms in X and amplitude-derivatives. The

amplitude derivatives are neglected in the first-order approximation.

Mathematically, this is in accord with the method of variation of

parameters (ref. 42).

Before equations (A.12) and (A.13) can be evaluated, the following

three groups of nonzero integrals have to be estimated in advance.

Group 1

2ir

/ sin v sin v dv = TT

0

2ir

/ cos v cos v dv = IT

0

49
/ sin3v sin v dv = |* (A.14)

0

2:T 3 3

Jf cos v cos v dv = yn

 4

0

2ir 2 2 1

/ sin v cos v dv = -TIT

0

Group 2

/ sgn (<j>) cos v dv = 4

0

/ sgn (<)>) sin v cos v dv = — (A.15)

0

2* 2 8

/ sgn (<(>) cos v cos v dv = —

Group 3

/ sgn (<(>) sin v dv = -4

0

/ sgn (<fr) sin2 v sin v dv = - -| (A. 16)

0

|

2ir

0

' 2 4

sgn (<|>) cos v sin v dv = - —

After lengthy integration, equations (A.12) and (A.13) are reduced

to

X

2 - ^ - sin a L + X L + aX sin L + aX u L pp

50
2Xoi au>L-+4-aa i Sin a L . + 4- a(X2 + 2u>2) L (A. 19)

pU JIT s pp JIT pp

All the aerodynamic derivatives in equations (A. 18) and (A. 19) should

then be evaluated at some average conditions.

Equation (A.I), a nonlinear second-order differential equation is

then converted into two coupled nonlinear first-order differential

equations, ft and A are defined as the circular frequency and amplitude

for a steady-state wing rock. Limit-cycle characteristics are obtained

by setting X to zero in equations (A. 18) and (A. 19). Equations (A. 18)

and (A. 19) are then reduced to

-nz = sin ct L (A.20)

s p

and

0=fiL n +4-A n (sin a L 0 + 2J2 L ) (A.21)

pO 3ir s p3 pp

The closed-form solutions of ft and A are

Q = (-sin ct LQ)

1 / 2

S p

since C, )1/2 (A.22)

s

 ^

rt 4 sin a L „ + 20 L

s p6 pp

C,

3TT pO ,

4 flb „ sin a CV + rr— C.

s H „ V £

P3 PP

It is noted that at limit-cycle conditions equations (A.18) and (A.19)

have been reduced to the frequency and the amplitude equations

respectively.

51
Appendix B. Derivation of Approximate Analytic Solutions of the

Three-Degree-of-Freedom Model

The equations of motion, equations (36), (37), (38), (40), and (41)

are rewritten without control-deflection terms as

V * V * V + V (Ba)

(EpO + £pf5|31 +Splp

'

) P + V (B'2)

r = NQ + Ng3 + (5pQ + Spe|0| H- Npp|p|) p + 5rr (B.3)

Solutions to the above equations are assumed to be

(B.4)

(B.5)

S(t) = a(t) C(t) cos [ v(t) + e(t)] (B.6)

<j>(t) = a(t) cos v(t) (B.7)

a(t) n(t) cos [v(t) + 6(t)J (B.8)

Again, let oj and X be defined as

•

w = v (B.9)

•

X = a/a

52
Also, all time derivatives are to be expressed in terras of amplitude

derivatives as

•

a) = a X 01*

•

X = a X X'

a = a X (a X' + X) (B.10)

•

n = a n' X

n = a X(ari"X + an'X' + n'X)

•

5 - a 5' X

C = aX(aC"X + a?'X' + 5'X)

Because the oscillation frequencies of <j>, i|>, and g modes are assumed to

be the same, e and 6 are virtually independent of time. To the firstorder approximation, the amplitude derivatives are all neglected in

equation (B.10). It follows that the time derivatives of equations

(B.6) - (B.8) can be shown to be

•

0 » a 5 [(Xcose - ojsine) cosv - (Xsine + oicose) sinv] (B.ll)

•

<j> « a(Xcosv - ojsinv) (B.12)

<j> « a[(X2 - u)2) cos v - 2Xwsinv] (B.13)

•

ty " an[(Xcos6 - tosin6)cosv - (Xsin6 +• cocos6)sinv] (

2 2

fy <* an{[X - a) )cos 6 - 2Xusin6]cosv

- [(X2 - u2)sin6 + 2Xo) cos6]sinv} (

Substituting equations (B.4) - (B.15) into (B.I) - (B.3) and averaging

over one oscillation cycle according to Beecham-Titchener technique

53
yield the following first-order-approximation equations without control

deflections as

£(Xcose - cosine) « Y. 5 cos e + Y X + Y r\ (Xcos6 - u>sin6) + Y, (B.16)

p p r <f>

£(Xsine + oocose) » Y. E, sin e + Y u) + Y n (XsinS + oicos6) (B.17)

$ p r

O O _ ,_ Q _ Q _

X -w * LQ?cose+L -X + -z- L a a?X + ^— L aXu>+L n(Xcos6-tosin6) (B.18)

P p(J JIT pp JIT pp r

2Xto«L 5sine -t- L .0) + -_ L a^oo + 4~ L aX2 + -r- L ao>2

g pO 3ir pg 3ir pp 3n pp

n (Xsin6 + ojcos6)

ri[(X2-w

2

)cos6 - 2Xo)sin6] =• N0 C cose + N^X+-|-N n

P pO JIT pp

N aXw + N n(Xcos6 - O)sin6) (B.20)

pp r

n[(X2-o)2)sin6 + 2Xo)cos6] » NQ Csine + N _oj + -|- N

p pU JIT

-|- N aX2 + -|- N aoj2 + N n (Xsin<5 + oicos6) (B.21)

3ir pp 3ir pp r

During this integration process, a, C, n, X, to, e, and 6 assume fixedvalues at midcycles for slowly varying motions. In the meantime, \Q\ is

approximated by |C<(>| because <j> and Q modes are in phase in most cases.

Again, all the aerodynamic derivatives should then be calculated at some

average conditions.

54
Define G and H as the amplitude ratios of the 8 and ty modes to

the <(> mode at limit-cycle conditions. The phase angles, e and 6, are

numerically negligible at limit-cycle conditions. In addition, X is

zero for a steady-state wing rock. Equations (B.16) - (B.21) are thus

reduced to

0 = YQG + Y. (B.22)

P 9

G = Y + Y H (B.23)

P r

n

2

 = - L.G (B.24)

P

0 = So +1? S* AG +

 IT SP ^ + V (B-

25)

-HJ22 = -NQG (B.26)

P

0 = N-+4-N.,AG+|- N Afl + NH (B.27)

pO Sir p8 3ir pp r

It is implied that the frequencies of <f>, 8, and fy modes are the

same in equations (B.16) - (B.21). Because equations (B.22) and (B.26)

are related to the frequencies of 8 and ip modes respectively, they will

be ignored in determining the effects of 0 - 4» amplitudes on

the <|>-amplitude for a steady-state wing rock. Thus, equations (B.23),

(B.24), (B.25), and (B.27) are reduced to

G = Y + Y H

P

 r

= sin a - H cos2a + ^- Cv + H 5§L_ c cos a (B.28)

s S 2mV2 Y

P 2mV2 Yr

Q

2 = - LgG

- - G qSb - 2 - _£. (B.29)

I I - I

XX ZZ XZ

55
A . - 3-

G L 0 + 2« L PS pp

f Cn

(B.30)

pp pp pp pp

where

I C. + I C

zz I xz n

h = r r

I C. + I -C

xz i xx n

r r

g = 1 - h

zz

I I

f =

 xx

 ( xz

zz xx

and

H = - — [N . + 4~ A(G N Q + 2Q N )]

P°

 3lT

 PB PP

I C, +1 C 4-A[G(l Ca +1 C )+^<I C. +1 C )]

x z I xxn^Si r xzil ^ xxn . V xzJl xx n

pO pO pg p6 pp pp

(I

xz

 CZ + Z

xx

 Cn )c

°s a

r r s

(B.31)

56
Appendix C. An Iterative Scheme for Calculating Average Aerodynamic

Derivatives and Limit-Cycle Characteristics

For a given angle of attack, it is necessary to calculate three

values of C at different 6 and p in order to determine C0 and

** Jw — P P6

C. by the VORSTAB code. The three values of co are

Jt Jo

pp p

called C. , C. and c. respectively. Due to the nature of

pO pi p2

nonlinear interaction between aerodynamics and dynamics, the most

suitable g and p for calculating C , C and C are obtained

*P0 V p2

through iterations.

•

<J> and <f> are defined as

<(> = a cos v (C.I)

and

• •

<|> = a cos v - aui sin v = a(Acosv - uisinv) (C.2)

By applying the limit-cycle condition (X = 0) or for slow-varying

amplitude with a » 0, equations (C.I) and (C.2) can be combined into

4L.= A2 (C.3)

or

(C.4)

where A and n are the limit-cycle amplitude of bank angle and circular

frequency respectively.

57
Equation (C.3) represents the larger ellipse in the phase plane as

shown in figure C.I. The smaller ellipse in figure C.I having one half

of the limit-cycle amplitude of the larger ellipse can be defined by

=

 TA (c'

5>

sr

From equation (C.4), the maximum roll rate is

p « U 1 - QA (C.6)

max ' max'

The average bank angle <j»ave and the average roll rate pavg are

calculated from equation (C.5) to be

*ave - ¥ (C'

7)

and

- * - J8A - <;•• Prr__ (C.8)

The maximum and average sideslip angles, & and 0 , are obtained as max ave

and

5 = A sin a (C.9)

max s

t =4 sin a = ~r A sin a (C.10)

ave ave s 2 s

The dimensionless maximum and average roll rates are defined by

P » P w (CU)

^ max 2V

and

58
Pave

 = Pave fv (C'12)

Now C is calculated at g = Q and P - P • C. , is calculated

At y. 3V 6 Ar , po pi

at 6 = 8 and p = 0. In actual calculation, this is taken to be

ave

 v

at pK = 0.05. C. is calculated at 8 = g and p = p i „ ave

 v Kmax

p2

Finally, C and C are calculated by

V PP

C£ ~ S

-E 20 (C.13)

£ - £ ,

^2_ El

and

C

PP

After C0 , C. , and C. are obtained, the roll amplitude of wing

*/%*•/ * « po pe pp

rock based on the one-DOF model has been shown (eq. (18)) to be

C

_ , bu _ -

£ .

 +

 V-

 Ct

pe PP

Define & and p as

= Afsin a g (C.16)

and

p = ^Afb/2V (C.17)

59
To start the iteration (figure C.2), a proper value of 8 is

chosen, say 10 degrees. The corresponding p and p are calculated

'

 Kave 'max

from equations (G.6) - (C.12). ft is obtained from equation (16).

C , C , and C are then properly calculated by using the VORSTAB

So p6 pp

code. The predicted C. , Cp , C. , and C. are used to

*B pO p$ pp

find AF, Bf ,and p . If 8 and 6 (or p and p ) do not agree within a

r r r j .

required accuracy, say 0.5 degrees, a second 0 is selected to be the

* max

average of the first 8 and 8C. Two or three iterations are normally

max f

needed to get a final converged solution.

In case that p is less than 0.05, C will be calculated

ave ' £ npO

at 6 = 0 and p =0.05 and C. will be calculated at 8 = B and

ave £ . ave

P

2

p =0.10. In this case, only Qf and 8 are iterated to be within the

ul£L3v I

required accuracy.

60
-•e-

-ec

•ro

Q.

X

-•eCM

<— I

O

0

0

CM

•a:

I—I

>

<

3

• -eCM

«=c

CM

X

<

s

-•eCM

Ico

CM

<c

fj

CM

id.

a.

o.

03

ns

o

•ra.

<u

s.

O)

61
0

!£<U

-Cuooa; 13S. (Uc<ou

4->0) CMat

62
Appendix D. Energy Balance of a Steady-State Wing Rock for the OneDegree-of-Freedom Model

The total aerodynamic energy exchanged over a limit-cycle oscillation

is given by

AE = qSb / C (t) <t>(t) dt (D.I)

C(t)

or

AE = qSb / C (<J>) d<(> (D.2)

where C(t) and C(<()) are the contours obtained by plotting C. versus t Xt

and 4) respectively. For a steady-state wing rock, C(<)>) is a closed curve

over one oscillation cycle. Since the contours for positive and negative

bank angles are theoretically the same, equation (D.2) can be simplified

to be

AE = 2qSb / C (<j>) d<j> (D.3)

ABCBD

where the integration path ABCBD is shown in figure D.I. The arrows in

figure D.I indicate the direction of increasing time.

The total aerodynamic roll damping coefficient is defined as

= c£ + ca I3' + cn IP'

 (D'4)

pt pO pg pp

The total rolling moment coefficient C can now be rewritten as

63
C + C 3 + C p (D.5)

*o *B V

<f> derived in appendix C is given by

Q (A2 - <j)2)

1/2 (D.6)

When <j> is increasing, e.g. along the path ABC,

p - <fr = n(A2 - <|)2) (D.7)

When ij> is decreasing, e.g. along the path CBD,

>

2)1/2 (D.8)

Thus equation (D.3) can be rewritten as

or

A

AE - 2qSb / (C - C ) d(fr (D.lO)

0 ABC CBD

where

|| CA2 - *2)

1/2

 (D.ll)

pt

 ZV

and

64
(A2

0 0 pt

so that

9 O 9 91/ 9

AE = 2qSt> ± I C (A* - O1' d* (D.13)

0 pt

The physical mechanism of a steady-state wing rock shows that the

dynamic system is undamped over small bank angles and damped over larger

bank angles. This can be explained by the dynamic loops in figure D.I

with the help of equation (D.13). The loop ABD is clockwise and the area

within this loop is positive. It results in a positive AE and implies

that C is positive, or undamping. Thus, a clockwise loop is

Pt

dynamically destabilizing because the aerodynamic energy is added to the

dynamic system. On the other hand, a counterclockwise loop, say the loop

BCB, is dynamically stabilizing due to a negative C so that the

Pt

aerodynamic energy is consumed by the dynamic system.

Substituting equations (D.4) and (D.6) into (D.13) yields

AE=2qSb2!/ {[C +sinot C. <M~| C. (

V

 0 %0

 S %3

 2V

 pp

where

A

 C. (AV)1/2d<|> = AC UU2-*2)

172

 + A2 sin'1 1}

0 pO ^ pO

 A 0

jC A2 (D.15)

pO

65
/ sinct C. <(.(A2

-<t.2)

1/2d4, = -I sina C. { [(A2

-*2)

3

]

 1/2}

0

 S p( 5 3 s £ p g

1 sin a C A3 (D.16)

J S

r

A

 bfl _ ,.2 .2. _ bfi .2. 1 ,3,A

n 2V

 C£ (A

 ~*

 )d<t>

 - 2V

 C

4

 {A

 * * I * }

n

0 pp pp 0

pp

Finally, the aerodynamic energy exchanged over an ideal limit cycle is

AE = 2qSb2 |{1 C^ A2 + isinag C£ A3 + i H C£ A3}

po pe PP

- f ^2 A2 | tf C * A (Sin% C^ * ^ C, )}

pO p$ pp

2 - ,2 .2 n ,3ir „ 3ir _ ,

- 3 qSb A - {— c - - C }

pO pO

= 0 (D.18)

The intersection of stabilizing and destabilizing areas, i.e. the

point B in figure D.I represents a turning point of roll damping

because C is zero at that particular point. The critical bank and

V

sideslip angles, <j> and 3 , are defined as the angles of zero total

c c

aerodynamic roll damping. To find <(> , C. is set to zero to obtain

C Jw

pt pO pB pp

= 0

66
or

pQ PP

-sin

(D.20)

Squaring both sides of equation (D.20) and rearranging give

r dr T > + s d> +d (D.21)

where

PS

(C, )

PP

s = 2 sin a Cn Cn s So

(D.22)

-

( A

pO p S >p

Solving the quadratic equation (D.21) and comparing its solutions with the

numerical solutions obtained from equation (D.20), <f> can be shown to be

ijC-s+Cs2 -' "1/2' if C£ > 0

PP

I , I = S pi

|9c' ~ 2r ~ sin ct C,,

if C. =0 (D.23)

PP

C-s - (s2 - 4rd)1/2) if C£ < 0

PP

67
i- <0

flfc ^

C7) ^t

e

•»- 01

•T* •»•

i— M

oo

S_

•T*

O

S

-

o

•eu

o

of

O)

o

u c ^ §* ° I

O> 10

m »~ * S g 2

<u

68
Appendix E. Suppression of Wing Rock with a Roll Damper

A conventional roll damper is usually designed to quicken the roll

response and augment roll damping. It is also a simple control device for

suppressing wing rock. An ideal roll damper consists of a rate gyro which

senses the airplane roll rate p and then feeds it back to actuate the

lateral control unit 6. in proportion to p but such as to oppose p. The

lateral control deflection angle is then

A6A = Kp (E.I)

A

where K is the system gain (control surface deflection per unit roll rate)

of a roll damper. The corresponding increases in C , C , and C due

i Xf n p p P

to A<S are expressed as

A

AC = C • A6 = AC p (E.2)

p (E.3)

P

AC = C • A6. = AC p (E.4)

n nr A n

6A P

where

P|V

 (E

-

5)

C 1

S A

- K^|f (E.6)

P

C

£

( E

-

7)

69
n

SA

= K^ (E.8)

P

By replacing <5 , C , C. , and C in the equations of motion, equations

A

 p pO

 npO

(25) - (27), by AS , C + AC , C. + AC , and c + AC , the

A i L XT A x ii- n P P pO P pO p

modified solutions at limit-cycle conditions are obtained to be

g(C£ + AC4 ) + f(Cn + ACn )

_ 3Ti_ _ pO _ p _ pO _ P

PS PP PP

-GqSb

XX ZZ XZ

G = sinct - H cos2a + -^r (C + AC ) + H -^^ C Cos a (E.ll)

s

 2mV p p •>"•" " s

I (Cn +ACn )+I (C +AC )-hr-A[G(I Cn +1 C ")+rr<I Ca +1 C )

x z I n I x x n - n3irxz£ . xxn.VxzJ l xx n

pO p pO p pg pp pp pp

^^•7C9

 + I-x*C

r, )COS a

xz x,r xx nr g»

(E.12)

where

+ I

x z

C

n

xz Jl xx n

r r

T

xz

g = 1 - h-— (E.14)

ZZ

f = ^2L (_2E£_ h) (E

ZZ XX

70
Equations (E.9) - (E.12) are then used to calculate the system gain

needed to alleviate the limit-cycle amplitude. A delta wing of 76°

sweep with a vertical tail (ref. 29) is chosen to demonstrate the

concept. Without a roll damper, C and AC, and the limit-cycle

JC r\ Xf

pO p

amplitude are predicted respectively to be 0.0071 and -0.0410 and 50°

at a of 20°. In contrast to that, the limit cycle can be eliminated

A Ci

completely by installing an ideal roll damper with K (K = -^- —*—?-) less

\

than -0.15°/rad/sec. Note that C « 0, C = 0.0046 (per degree)

S 5

A A

and C = 0.0008 (per degree) calculated from wind tunnel data in

ns

reference 29 have been used in equations (E.6) - (E.8). The value of K

seems small because the wing span of the wind-tunnel model is only 1.63

ft and the test velocity is 50.2 ft/sec. It is shown that the effect

of AC is negligible and augmenting roll damping via AC, is the most

JL Jt

p p

efficient. Therefore, wing rock can be suppressed by using a roll

damper to artificially augment roll damping.

71
i

n

COO) £_ O

r*v

CM

ii0

1ccu OI OO00OQ. 3COOonIO) s.

72
o

JT CM

c

<

<

c

c

1

1 1

'

>

>

N

2

>

>

1

1 1

0 00 0 0

60

OJ

o

.*•

U

1)

05

CM

60

&0

oo oo o o o w <

CM fl in in i

<*~* --H

O U

0) OJ

to to *o <"

>4 ^, o.^. «o -a

so oo •-'

3> U

•a TJ

OJ

u

o

Q£

I

cn

*>

?v

~ '

\

1

» (

1

*

i

**",

•

"7*,

i *

i *

• ;

• '

i '

' / M*

n

,*>.

•v '

i ^i

oc

5

o

C£

//

/r

ao

C

a>

60 TM

t5 m

•a

73
0.2 r

- 0.2

-0.4

-0.6

M=0.8

deg

8

10

c<,deg

WING-ROCK ONSET

TANKS OFF

TANKS ON

WING-ROCK REGION

I I

0 0.6 0.7 0.8 0.9 1.0

Figure 3. Wing Rock of the Gnat (ref. 10)

74
C

O

ooIIca

OV

ai s_CDcOJ ai oooo

oo

s_o II«/ a toOJus_o oo A<^ ai

cr-

-oUJiCDca; ai £S>>in

qon

*3-

<U

oII 75
LOW-cx

HIGHFigure 5. Asymmetric Forebody Vortices at High

Angles of Attack (ref. 15).

76
>

-

OC

*h

oen

M

oOD

2O•aGJS_ caXO) i. —V O <£> •ocs- -u o cuE U(0

<s s

 M.

 S

o re W) 0) 0) O s. ro cu C LU 3 I O Q

.

CD S- m

_i (/>

—I C3

cu

s_3O)

Oun

OCM

77
.0

CL

-.002

-.004

-.006

-.008

-.010

.8

.6

A

-.2

Static-force test

O

o o

O Q0

Forced-Oscillation test

Oscillation amplitude

O ±5°

Q t!0°

O ±29°

o

a

o

a

O

a a

o a 9 O

10 20 30 40 50

Figure 7. C». and C. versus a for an 80°-Delta Win g (ref.8 )

78
10enIo4JIO) (US- 00<ui.3Ol

•r-

U_

79
0.04 r~

-0.08 __

-0.12

-0.4 -0.2 0. 2

Qb

2V

0.4

Figure 9. Rotary Data of an 80°-Delta Win g at cts = 30° (ref. 8)

80
•=r

CT»I

I/)O) s_<u

UJ

uni

IQ

O) £O2:IOJ3O»

>

81
<I—_l LU

—e-

ca.

--e-

CO.

00.

QQ.

OJ •a

1/1

oo

5O)

•t—a

.

EO,_oo: M-Oco•w(O

•I—5-

OJ s- — • Q.

•*~O) ooen^toa»

4-JCO) £O

CU r— C i—

•t- O _j a: cu3

CT)

82
r-

.

CO

u

UJ

(=>I Oooo

reS- u

o

•e-

+•> CO

•f— • s "«-o>

c/) ^

u

•e-

i

S(OC>> •OOS- 0) <cO3CCOoin

C.

QJenIO) a;Q.

<Vs_

83
L

Uai

<3

00COUoa:enocooo<Us3a; s_

84
Y q

r V

Figure 14.The Body System of Axes.

85
0.8

0.6

OF POQR QUALITY

0.4

Roll-Damping

Derivative

-0.2

-0.4

-0.6

-0.8

-1.0

_:_ EL.

,0' ' --

- Theory (VQRSTAB)

-T ty •" Axf 5-r^Z.^E-

- "

Data of . Force'd-Qs'ci 11 ati on Tes t

' ~'''"~ "

- ( ref :8)

10 20 30 40

o , deg

Figure 15.Damping-in-Ron Derivatives of a Delta Wing with 80-Degree

Sweep (XCQ=0.4c) at Mj- 0.1.

86
0.4

0.2

Roll-Damping

Derivative 0.0

-0.2

-0.4

-0.6

-0.8

-1.0

- Theory (VQRSTAB)

— C^ (Body Axis)

P

C. (Stability Axis) • **n \

Data

-A—

Forced-Oscillation

Tests about Body Axis (ref.22) A

~-O—

^o Steady-State Roll Tests

'•'s

about Stability Axis (XCG=0.25c)

(ref.47)-

10 20

a , deg

30 40

Figure 16.Damping-in-Roll Derivatives of a Delta Wing with

76-Degree Sweep (XCQ=0.40c) at ^=0.1.

87
0.2 r

Roll-Damping 0.0

Derivative

-0.4

10 20

a ,deg.

30 40

Theory (VORSTAB)

— — C£ (Body Axis)

P

Ct (Stability Axis)- (XCG=0.39c)

p

s

Data of Forced-Oscillation Tests (ref.48)

C» + Cfl.sin a (Body Axis)

*P - (3

k = 0. 200. 12 , A* = + 2.5°

Figure 17. Damping-in-Roll Derivatives of a Delta Wing with 74-Degree

Sweep at M^.2.

88
Roll-Damping

Derivative

-0.2

-0.4

-0.6

-0.8

-l.OL

Theory (VORSTAB)

C. (Body Axis)

P

C. (Stability Axis) (Xr

pH

s

0.34c)

Data of Free-to-Damp Oscillation Tests (fef. 49)

--A-- C» +C». sin a (Body Axis)

k = 0.13, 30

--O-- C£ (Stability Axis) (XCQ=0.25c)

p

s

k = 0.14 * 0.10, A* = * 30°

Figure IS.Damping-in-Roll Derivatives of a Delta Wing with 60-Degree

Sweep at M =0.1.

89
ORIGINAL PAGE IS

OF POOR QUALITY

10

a , deg.

20 30 40 50

m

0.2

0

-0.2

Langl ey:;30 !x60' tunnel '~

1.4 ,. . A LangTey. 12'

OWentz (ref.23)

Theory (VOftSTAfe)

1.2 L

1.0

0,8

0.6

0.4

0.2

Tunnel (ref:8)7 ^-:-"' " :

1KD

-"- ^-without" d^l

with a-ifv Effect-;- •=.»";: .

1.4

1.2

1.0 -

0.8 -

0.6 .

0.4 -

0.2

0

. •./

/.

10 20 30

a , deg.

Figure 19.Longitudinal Aerodynamic Characteristics of a Delta Wing

of AR=0.705 with Sharp Leading Edges at M^O.l. XCQ=0.4c;

Re=1.4xl05.

90
One-DOF Theory

Data of Free-To-Roll Tests (ref.8)

Data of Free-To-Roll Tests (ref.9)

50 iA

(deg.)

40

30

20

10

i n» n ' '

) 10 20 30 40

o , deg.

a

PvJnJ

50

i

60

Figure 20.Steady-State Amplitudes of Wing Rock for a Delta Wing

of 80-Degree Sweep (AR=0.71).

91
p

(sec.)

2.0

1.6

1.2

0.8

0.4

0.0

One-OOF Theory

O Data of Free-To-Roll Tests (ref.8)

0 o ©

10 20 30 40

a , deg.

50 60

Figure 21. Steady-State Periods of Wing Rock for a Delta Wing of

80-Degree Sweep.

92
0.2

'PO

0.1

0.0

0.0

-1.0

-2.0

2.0

1.0

0.0

-1.0

-2.0

One~DOF Theory for

Liroit>Cycle Conditions

10 20 30 40

a , deg.

Figure 22.Theoretical Variations of G£ and C

50

and G£ with Angle of Attack .

PP 93
—r o

o

— o

CM

O

o

d o

O) (/)

C -t->

f- C

a. aj

O ir— ttl

O O

0)

o

r»

CVJ

11

s

OJ

V

0>

T

o

C

<U

O

to

o

0)

o

0)

o

£

O

CO

i

o

I

o

I

evi

s_

en

94
iis4J03a

.

VVina>£01

en

01

•a

a. ai

fO U

r— <0

O O ex u

tooas-o0)

•»•Ua>oO)

•r-a

. iooa:L. IBaicoa»

95
(fl

oID

— o _ o

iHoI

O

fio

>enc«Ocas. 0) <u

«^»uOJo0)£

•e- i—f—ou

onio^ri

CO* O

o• o

CM•OCO• OI

201co *Jo

«*-o(Qs- o>o

•*•»

(/I

O)

in.

CVJ

to01O) oCM

<*- OU(Oo0) •uid

cr>o<u

96
oinO•a- oCO

a; o<cnccVI 3

(/I

S_ai s. 0) a; 3

— O

<u

OJ

cn

OJ •o

o

ac.

t*.o

£toI J

2

CO•o

CM» O

o•o

C\4« OI

oCOt o

«3-

I Oini

o

5

<+- o

c/i

•i~0)

10CM0) a>

uo

Of.

o>

97
20

16

deg.

12

a

12

deg.

8

One-DOF Theory

h-OH Data of Free-To-Roll Tests (ref.8)

10 20 30

a , deg.

40 50

Figure 27.The Critical Bank and Sideslip Angles for a Delta Wing

of 80-Degree Sweep during Steady-State Wing Rock.

98
A

(deg.)

1.6

1.2

0.8

0.4

0.0

100

80

60

40

20

O Data (ref.29)

— theory (VORSTAB)

Approximate Angle-of-Attack Range of

Observed Wing Rock (ref.29)

Three-OOF Theory

Observed Roll-Off Range

10 20 30 40

a , deg.

Figure 28. Lift Coefficient (Data Obtained at Re=696,000) and

Steady-State Amplitude of Wing Rock for a Delta Wing

(AR=1.0) with a Vertical Tail at M^O.l. XC(,=0.30c.

99
A

(deg.)

1.6

1.2

0.8

0.4

0.0

200

160

120

80

40

I 7K A St A

Data (ref.29)

Theory (VORSTAB)

Approximate Angle-of-Attack Range of

Observed Wing Rock(ref.29)

Three-OOF Theory

Observed Roll-Off Range

10 20 30 40

o , deg.

Figure 29. Lift Coefficient (Data Obtained at Re=812,000) and

Steady-State Amplitude of Wing Rock for a Cropped Delta

Wing of 76-Degree Sweep (AR=0.33)witn a Vertical Tail

at 1^=0.1. XCG=0.27c.

100
"Onset

(deg.)

30

20

10

A Data (ref.29)

—O Three-DOF Theory

1 1

50 60

<\

X

X

& ^0

i i

70 80

i

90

AL£fdeg.

(a) Delta Wing with a Vertical Tail.St/S=0.10.

30

"Onset

(deg.)

20

10

O Data (ref.29)

••O Three-DOF Theory

Q

50 60

1LE

70

, deg.

80 90

(b) Cropped Delta Wing with a Vertical Tail.

St/S=0.134.

Figure 30. Comparison of the Onset a with Data at M^O.l.

Moment-Center Location is at Forward GravityCenter Position.

101