A study of the effects of Cl-coupling on the lateral stability of aircraft in atmospheric turbulence

ï

A STUDY OF THE EFFECTS OF CL-COUPLING ON THE LATERAL STABILITY OF AIRCRAFT

IN ATMOSPHERIC TURBULENCE rEm~"ç(

E

'~OGESCAOOL

_DM

v ... :.,. TUIG BOUW KUNDI

maUOTHEEK

_by

E. Ryerson Case

A STUDY OF THE EFFECTS OF CL-COUPLING ON THE LATERAL STABILITY OF AIRCRAFT

IN ATMOSPHERIC TURBULENCE

by

E. Ryerson Case

Manuscript received May 1967.

ACKNOWLEDGEMENT

The author wishes to thank Professor B. Etkin for his help and encouragement throughout the course of this work.

The efforts of Professor J. Ham in arranging for this work to be done at UTIAS are appreciated.

The support received from the DeHavilland Aircraft of Canada~ Limited is gratefully acknowledged.

The author would also like to thank Dr. A. J. Ross of the Royal Aircraft Establishment. Farnborough. for her assistance in supplying essential data.

This work constitute s part of the program of re search on flight in turbulence carried out at UTIAS under USAF Contract No. AF33(615}-2305 of the Research and Technology Division. -Control Criteria Branch.

I I

SUMMARY

The effect of a CL -coupling phenomenon on aircraft lateral stab-ility is studied by means of an analog simulation of flight through atmos-pheric turbulence. No instability is observed in control-fixed flight, and the increase in gust response which results from the coupling effect is found to be effectively reduced by introducing synthetic roU stiffness. Correlated side and vertical gust components are considered, as weU as an equilibrium side-slip condition. Both high-:aspect-ratio swept-wing and slender delta-wing jet transports are examined.

TABLE OF CONTENTS

PAGE NOTATION

I. INTRODUCTION 1

H. THE COUPLED EQUATIONS OF MOTION 2

2. 1 The Lateral Cross-Coupling Derivatives 2 2. 2 Flight Through Atmospheric Turbulence 4

,Hl. THE GUST INPUTS 5

3. 1 A Representation of Atmospheric Turbulence 5

. 3. 2 The P0wer Series Approximation 7 3. 3 Power Spectra of the Gust Inputs 9

IV. RESULTS AND DISCUSSION 12

4. 1 Computing Program 12

4.2 Analysis of Data and Presentation of

Results 14

4. 3 Discussion of Results 14

v.

CONCL USIONS 16

REFERENCES 17

APPENDIX A - Aircraft Data 18

-APPENDIXB· - Simulation of the Equations of Motion 20 APPENDIX C·- Simulation of the Gust Inputs 21

APPENDIX D - Measurement Circuit 25

./, ... ' a A al' b 1, b B G C CL ,_{Cx '} Ct • CZI E i k· ₁

,

k· ₁ a2, a3 b2, b3 Cyl C z Cm Cn Cm _'q etc. NOTATION

correlation circuit gain factor rolling moment of inertia .

constant part of lateral. derivative

coefficient of CL -dependent part of later:àl .

derivative wing span, feet

pitching moment of inertia mean aerodynamic chord, feet yawing moment of inertia

lift coefficient

=

Lift/ 1 /2

f

SuJ X, Y , Z, divided by 1/2

f

SuJ L. M, N. divided by 1/2

f

SuJb

'à _{C z}

/afX

I d Cm /o~ etc. In general, a stability derivative is the partial derivative of a non''';dimensionalized force or moment with respect to a non-dimensionalized aircraft variable.

rolling moment cross-coupling derivatives (see Sec. 2. 1)

yawing moment cross-coupling derivatives (,see Sec. 2. 1)

product of inertia Ixz

A, B, C, Emultiplied by 8/

f

Sb 3

[-T

reduçed frequency, k i

=

n

i b/2 reduced frequency cut-offs

iv .'

.

, "

ko L L.

M.

N m p Pg 1\ /\ P. Pg q qg 1\ 1\ q. qg r 1\ r s ;\ s

s

t

1

t* (

.

) Uo u (x) Ui (xl. X2) (ug.Vg. Wg) (xl. x2' x3) x

-(x,

_Y.

z) 1\

"'"

_~} (x.

_Y.

b/2L

scale of atmospheric turbulence external moments acting on aircraft mass of aircraft

roU rate, rad. per. sec.

equivalent roU rate due to spanwise gust gradient t* P. t>:< Pg

pitch rate, rad. per. sec.

equivalent pitch rate due to streamwise gust gradient.

yaw rate, rad. per. sec.

Laplace transform variabIe t* s

wing area

time, seconds (sec. ) t/t>:<, airseconds (asec. ) b/2uo

d( • } / dt

reference speed of aircraft vector velocity field

velocity components of u(x}, in horizontal plane

gust velocity components at aircraft air-fixed co-ordinate system

position vector

body-fixed co-ordinate system (x, Y. z) divided by b/2

x~ y~ Z 0<. cX.g

(3

#g

~o

1l

'

.

9 Ài l t ) . 1 i l i

IJ-1)

.

f

f

ct

fix

"-cr

9'

~

~(D.l) '" <I> (kl)

Px

<.Ch)

external force s acting on aircraft angle of attack

equivalent angle of attack due to vertical gust at C. G.

side slip angle

equivalent side slip angle due to side gust at C. G.

equilibrium side slip angle

ratio of coupled to uncoupled RMS response

(wit h appropriate subscript) .

pitch angle

wave length of ith spectral component

angular frequency W i = U on i

wave number components ..0. i

=

2'it / À i

2m/

f

Sb

(k1)Sp/(k 1) DR

air density (used only under Notation)

correlation coefficient

(ex

g~

g) /

&

2

gust intensity (RMS velocity)

RMS value of variable x

uniformly distributed phase angle

roU angle ~ rad.

one -dimensional gust velocity spectrum . .function

one -dimensional streamwise gradient spectrum function

"-~y(k1)

'f

i/

fi

1.il2) Subscripts s g SP DR c u Superscript

one'-dimensional spanwise gradient spectrum function

b

~y

.(.(ll)

/2u~

two-dimensional gust velocity spectrum function (also without subscripts)

simulated gust Short Period Dutch Roll coupled uncoupled

A prime on a lateral derivative is used to indicate the whole derivative. including the CL - dependent part.

I. INTRODUCTION

Studies of the rigid body motions of aircraft in atmospheric flight are generally based on the so-called 'small-disturbance' equations of

motion. where is assumed that the aircraft motions consist onlyof

small deviations from a steady reference flight condition. The equations relate the six degree-of-freedom motions of the aircraft to the applied

aerodynamic forces and moments, and can be written in non-dimensionalized form (Reference 1) as follows:

.

Longitudinal Force: C _x

=

2fu

"

+ C4,9

A

Vertical Force: C

_z

=

_{2F - 2f-tt}

(1. 1)

Pitching Moment: _Cm

₌

iB~

'

--Side Force: C _y

=

_2f-t.

·

₊ 2

r.r -

"- C f _Lo

A-

r-Rolling Moment: C

=

_{iAP - iE r} (1.2)

.

_•

Yawing Moment: C _n

=

iC r - i

"

_{· E} "-P

The equations are written in body-fixed co-ordinates, with the x-axis initially coincident with the reference flight direction. Horizontal flight is assumed.

It is usually possible to assume that the aerodynamic force and

moment perturbations on the left hand side of the equations are llnear

functions of the aircraft mot ion variables and their derivatives. in .which

case the aircraft c,an be represented by a set of coupled, linear differential

equations with constant coefficients. In many cases of practical interest, due in part to the symmetry of the aircraft. these equations cart be separated into two independent groups, defined respectively by Equations 1.1 and 1. 2.

The first group, represents the longitudinal motions of the aircraft, and

has two characteristic modes, the Short Pe:riod mode and the .:phugoid mode

both oscillatory. The second group describes the lateral motions ofthe '

airC±'aft. There are three characteristic modes in this case:. ~he Dutch

Roll mode (oscillatory), the Rolling Convergencemode, and the Spiral Con-vergence mode (both first order). Separation of the equations in this way

is obviously a great advantage. in aerodynamic studies since it -allows the

longitudinal and lateral motions of the aircraft to be treated separately. However, the assumptions on which it is based must be carefullyreviewed for each newaircraft being considered.

One case where the legitimacy of separating the equations is, open to

question is that treated by Masak (Reference -2), where, for certain slender

configurations at low speed, there is a significant dependence of some of the lateral stability derivatives on the lift coefficient CL, and hence on the

/

/

,,/

1'

,

./t

~le

of

at

1~yk

.

Thus, in this case, the lateral equations are

aerodynami-/ aerodynami-/ cally coupled to the longitudinal equations (the longitudinal equations are

, / / still independent). By parametrically exciting the lateral equations after

an initial disturbance by a CL with a periodically varying component,

Masak found that lateral instability could be induced in what was an

other-wise stabIe system. The instability occurred for excitation frequencies in the neighoourhood of the Dutch RoU natural frequency, and at twice that value. The question arises as to whether or not this lateral instability could exist under normal control-fixed flight conditions.

It is the object of this study to examine the CL -coupling phenomenon

further under the more general and realistic conditions of flight through

turbulent air. Vnder these conditions, the longitudinal equations are

included, and can be visualized as acting as a shaping filter for the random

gust inputs to produce the randomly varying coefficients in the linear lateral

equations (the overall system is, of course, nonlinear). Because the

theory of the stability of linear systems with randomly varying coefficients

is largely undeveloped at the present time, an analytical approach was not

attempted, and analog simulation of the aircraft and the gust inputs was

chosen as the most expedient means for studying dynamic behaviour in

atmospheric turbulence.

Two aircraft were examined at constant speed (250 fps) under a

variety of conditions. The first aircraft was representative of a subsonic

transport with an aspect ratio of about 7, and the other, a supersonic

trans-port with an aspect ratio of about 1. 5. These are the same two aircraft

examined by Masak, except that a more recent set of lateral derivatives

was used in this study for the supersonic transport. The CL -dependent

part of some of these derivatives was not as large as those used by Masak,

which had the effect of weakening the coupling somewhat. They are more

realistic, however, for aircraft typical of this type.

The gust velocity field was assumed homogeneous and isotropic for

the most part, but as a matter of interest, the effect of correlated vertical

and side gust components, such as might be experienced in cross-wind

flight near the ground (Reference 8), was exam'ined briefly. Also,' a

coupling phenomenon reported by Porter and Loomis (Reference 3) prompted

a brief look at the effect of an equilibrium side slip angle on the gust

response.

Il. THE COUPLED EQVATIONS OF MOTION

2. 1 The lateral Cross -Coupling Derivatives:

Expressions for the lateral cross-coupling derivatives can be

derived by considering the lift coefficient CL in two parts, one representing

the steady state component, and the other the perturbation due to changes

=

CL + CL 0/..

o ~ (2.'1)

Substitution of Equation2. 1 into each of the expressions . .for the CL -dependent lateral derivatives will yield two derivatives. The first is equivalent to the usual lateral derivative .. and the second represents the cross-coupling from the longitudinal equations. As an example of the pro-cedure .. consider the total yawing moment;

. where

,..

I "

+

Cl _np P

+

_{Cnr r}

(2.2)

If Equation 2. 1 is substituted into Equation 2 .. 2 .. the following relations are obtained: .

=

a2 + _{b2 CLo} + b2 CLo(.o/.. (2. 3)

=

a3 + b3 C L2 + 2b3CL CL Ql.

o 0 ~

Neglecting the seeond order terms in 0( .. the total y~wing moment can now

be written in terms of the expres sion in Equations

2.

3 as

or Cn

=

(a_l

+

b_l C{o )

P

,..

+ (a2 + b2 C L_o)

P

+ (a3 + _{b 3 CL}2 )

~

o + 2 b 1 _{CLo CLc(.}

cXf

.

,..

+

b2 CLo(.

ocr

,..

,..

Cn = _CnlB

B..

+

_{Cn ...} 6~ .. 8

+

Cn p

+

C -.I P

tIn

....

,"'-1 p no(p ... + C_n r + C_n fX. ~ r ocr (2.4) (2. 5) The corresponding expressions for the rolling moment derivatives are de-rived in the same way .. and the total rolimg moment can be written as

1\

,..

r

+

Cl

.N~r cX.r

"

(2. 6)

The numerical values used for these derivatives are tabulated in Appendix "A", along with other relevant aircraft data.

Note that dropping the second order terms in Equation 2. 3 is tanta-mount to neglecting all third order terms in Equation 2.4 since, if retained, each would be multiplied by a first ord~r term. Neglecting the second order terms in Equation '2. 4 would not be justified, howeve~ because the

relatively large magnitude of the cross-coupling derivatives makes the contributions of the cross-coupling terms to the total yawing moment of nearly the same order as those from the other terms.

2. 2 Flight Through Atmospheric Turbulence:

The metlfod used in this study for calculating the aircraft response to atmospherio turbulence is due to Etkin (References 1, 4 and 5). and will be descr-ibed in ,detail in the next section. For the present, however, it is sufficient to say th at the response can be obtained to a good approxi-mation by a-ssuming that the gust inputs enter the equations of motion (Equations 1. 1 and 1. 2) as aerodynamic perturbations in certain of the aircraft-state 'variables, namely, angle of attack, side slip angle, pitch rate and roll rate. The equations of motion for flight in atmospheric tur-bulence--can _{t-hen be written, using the expressions Cn and} Ct in Equations 2. 5 and 2. 6, and retaining only those derivatives which are of importance for this study (See Appendix "A"), as follows:

1\ -2

f-lJ,

2ff

+2f~-CLo ~

.

.

• A , " IA P - IE r = C Yr

(f

+f

g) =

C,-,.

,,(f.>

A+

f3

g) + C~" (rJ.. + cx..g)(

p

+

f3

f)

+ Cl,.J) ( P + Pg) + C), r r + C),oc r (<<. + Olg) r + C), <t> 4> +

Sto

.

(2. 7)

In these equations, quasi-steady aerodynamics have been assumed, and

the subscript 'g' has been used to designate gust-induced aerodynamic per-turbations. The longitudinal force equation 'has been deleted since speed variations are unlikely to produce any essentially different effects as far

as the cross-coupling phenomenon is concerned. The terms _{Cl o and Cn} represent rolling and yawing moments which are required to maintain an 0 equilibrium side slip angle

pO.

They are given by

C), 0

= -

C).r

f

0

Cno

= -

Cnr

f

0

The Cj, 4> term is introduced to simulate, in a rudimentary way, the effects of roU stabilization due to a pilot or autopilot. Details of the simulation of Equations 2. 7 are given in Appendix "B tI.

lIl. THE GUST INPUTS

3. 1 A Representation of Atmospheric Turbulence:

Atmospheric turbulence can be approximated over regions of limited extent by a homogeneous, isotropic veloc.ity field ~ (~ which is a Gaussian, random process. It is usual to assume that the field is 'frozen' in space during the period of encounter by the aircraft, and can, therefore, be con-sidered to be a function of position only. A reference co-ordinate system OX1 x2 x 3 will be chosen so th at the mean wind relative to it is zero, and it will be assumed that the mean flight path is along the OX1 axis. It wi11 be further assumed (quite reasonably) that only variations over the

hori-zontal plane are important in determining the gust response of the aircraft. One possible representation (References 6 and 7) of the components of such a vector random process is a Stöchastic integral of the form:

00

SJ

co s

Co..

_{1x 1 +!l2 x 2 +}

«f

i )

J

2'0/ ii (!tl'

.ct

_{2) d1l 1 dIl 2 •} -00

i

=

2, 3 (3. 1) where

11

i(1l1'.n 2) is a randomly chosen phase angle uniformly distributed in(O, 2'if), and

i'

ij (IL₁

,n

2) is a two-dimensional pqwer spectral density function of the wave number components !l1 and

11

_2• (It is not necessary to ln"Clude the lil component since changes in speed have been neglected in the equations of motion). This equation can be visualized as representing the superposition of an infinite set of elemental sinusoidal waves of shearing motion, the relative amplitude contribution of each being determined by the spectral density function, and the rehltive phase by the randomly chosen phase angle ~ . Indeed, the above integral represents the whole ensemble of functions defining the random variabIe ui' where a different member is generated or specified for each set of

9>

's choseh. It is a useful repre-sentation from the point of view that it is easy to keep track of the effect on the spectral density of the various mathematical operations which may be performed on the random variabIe.

As a direct result of the isotropic assumption, it can be shown (Reference 5) that the cross spectra

l'

ij LCL l ' 11. 2) , i

f.

j, are all zero, and th at the u2 and u3 gust components have the same spectrum. The form of the two-dimensional spectrum function which is generally accepted, and which will be used in this study, is given by

(3. 2)

where the subscripts have been dropped as being superfluous. The shape of this function will be seen to be independent of the gust intensity (J' ,

and to depend only on the integral scale L of the turbulence. The corres-ponding autocorrelation function .is of the exponential type, which has a discon'tinuity at the origine As a consequence, since continuity of the auto-correlation function at the origin is a necessary condition for differentiability of a random variabIe, there is some difficulty in defining the one-dimensional gradient spectra when using the spe.ctrum of Equation 3. 2. However, the trouble is easily avoided by the use of low pass filtering to remove the excessive energy at high wave numbers.

Consider an aircraft flying along the OX1 axis at a constant speed uo. The 'frozen' gust velocity components can be written as functions of time in aircraft co-ordinates Oxy by making the transformation

Xl

=

U_o t + X

(3. 3)

x2

=

Y

and will be designated by Vg and Wg, for the side and vertical components, respectively.

For example, by substituting Equation 3. 3 into Equation 3. 1, the vertical component can be e~ressed as:

00

w g (t,x,y)

=

55

cos (Wit +!ll x +fi2Y +j)

J

2'Y' (W1,1l2)dw1d1l2

-00

where lOl

=

1l1uo, and

1

(3. 4)

It will be found convenient hereafter to use a non-dimensionalized form of Equations 3. 1 to 3. 4 which wi11 be compatible with the aircraft equations of motion of Section Il. Accordingly, alllengths will be non-dimensionalized by the semi-span b/2, all velocities by Uo and time by t':<

=

_{b/2uo ; non-dimensionalized variables and functions Wi11 usually be} distinguished by a hat symbol ( A ) .

Equation 3. 1 for the gust velocity compon~nts will now be written as: 0)

=

SS

. -00 and since " " A Xl

=

t

+

x (3.5) k3.6)

the side and vertical gust components in aircraft co-ordinates can be writtenj • respectively, as

(3.'7)

~g

(t,x,y)

=

SJ

cos [k_{1(t+i) + k 2}

y

+

fV]

J

2 i (kl,k2) dk1 dk2

-0)

and

~g(t,~,y)

=

_

JJ

_{cos [kl(t+i) + k 2}

y

+~w]J

2

~

_{(k 1,k2)dk1 dk2' (3.}-8) -0)

where k₁ and k

2 are reduced frequencies defined by

k 1

=

-Ul b/2

=

(1)1

t*

k2 =!l.2 b/2

\

Finally,' the two-dimensional spectrum function of Equation 3~ 2,- written

in non-dimensionalized form, becomes

,.. 3

à-

2k (k2

+

k 2)

Y _{(k 1,k2)}

=

0 1 2 (3.9)

.4 ,. (k 2 + k 2 + k 2 ) 5

7

2

1 . 2 .0

The constant ko is a non-dimensional breakpointfrequency corresponding to the characteristic length L, i. e. ,

k _o

=

_{- -}b/2

L

Equatîons 3.7 to 3.9 provide the basis.for determining the gust-

in--puts to the aircraft assutning-homogeneous, isotropic turbulenc·e.

3. 2 The P6wer Series Approximation:

The problem of calculating the aerodynamic forces and mome!lts

acting on an aircraft in atmospheric turbulence can be considerably simplified

by_ approximating the velocity field by a power series in the neighbourhood

interpreted in terms of the usual stability and flutter derivatives (References 1, 4 and 5), and in fact, several turn out to be the same, with a possible

change of sign.

Consider a first order power series expansion of Equations 3. 7

and 3.8 about the aircraft center of gravity, i. e.;

. A A

~g

(t

,

i,y)

= (vg)o

+

(~) ~

.

+

«lVg

y

(3.10)

o x

0

-ay

0

where the subscript

'0'

denotes the point (t, 0, 0). The first term in each expansion corresponds to a negative perturbation in side slip and angle of attack. respectively, and reacts upon the aircraft accordingly through the

fX. and

f3

derivatives, and therefore:

The second and third terms of each expansion represent gust velocity inputs which vary linearly in the streamwise and spanwise directions, respectively.

The velocity distribution which is produced by these terms is equivalent to

that resulting from angular velocity of the aircraft, and it is reasonable to

expect that their effe cts could be introduced through the aircraft angular velocity derivatives. In fact, only the vertical components allow such a simple interpretation. but for the type of aircraft considered in this study, the major spatial effects of the turbulence are adequately accounted for by the horizontal gradients of the vertical velo city alone, and the gradient

terms become A

( d

':g )

()x 0

=

"

_{qg (t )}"- _.

"

( ~v g) = 0

oy

0

() W

g) = _

p

(t)

() Y

0 g

The expansions in EquatiDns 3. 10 and 3. 11 can therefore be written as:

"

A""

V g (t, x, y)

= -

f3

_{g (t )}

"

(3.12)

and

A " A A ,..

""A

" A A

W g (t, x, y)

=

-

ocg(t) + qg(t) x - Pg(t) y (3. 13)

The variables appearing in these expressions are the gust inputs which occur in the equations ofmotion in Section 2. 2.

It will be evident that the first order approxirn,ation is on1y valid for wavelengths for which a straight line is a reasonable apF>roximation to a sine wave over the lengtp and span of the aircraft~ It is pOEisible to take into account the shorter wavelengths by including second order terms in the expansion (References 4 and 5), but the added complication is not usually necessary unless the elastic modes of the vehicle are being consid-ered In general, for the first order approximation, the wavelength should be limited to

81, = )...'

1 i\. .1'2 _ <. 8b = A "'2':

where ,~ , is the length of the aircraft. The corr,.esponding reduced frequency cut-offs are then given by

, b 2't _ b 1t" k 1

=

2"

~-

i

"8

and

The relationship between the variables in Equatüms, 3. 12 and 3. 13 is re'adily apparent from, the nature of the expansions. The variabIe

p

g is, of course, uncorrelated with the other variables because of isotropy. The variable qg , because of the basic equivalence between

t

and

x,

is simply the negative time derivative of

oc.

g, and since a unique value of Pg is not defihed for a given value of qg (1. e., a section through a two-dimensional' wave cannot defirfe the wave except at that section), the vC;riable

Pg

is statistically independent of both qg and cl.. g. Note that the

OC g input term required in the Equations of MotlOn 2.6 IS just -qg.

It was mentioned eaTlier that the gust velocity field becomes aniso-tropic near the ground due to the shear flow in the boundary layer, This has the effect (Reference 8) of correlating the u1 and u3 velocity components which means that the side and vertical gust inputs are correlated for an air-craft flying cross-wind in a boundary layer (the u1 component becomes a v g component). Since little is known at the pre sent time about the nature of these correla~ions (i. e., the form of the cross-spectra), the simple linear correlator described in Appendix

"e" was used to examine this case.

3. 3 Power Spectra of the Gust Inputs:

The power spectral density functions of the gust inputs are obtained by substituting the expressions in Equations 3.7 and 3.8 for

,

v

_g

and Wg into the series expansions in Equations 3. 10 and 3. 11. To, illustrate the

f3g

(t} =

-(~

g)o : k ' kl. 1 2

= -

J

J

cOS: (kt

t

+PV)J

2

~

_{(k 1.k2) dk 1 dk2 '} -kl _kl · 1 2

The mean square value is given by:

k ' I

Vi

(t~

=

f

I2

(COS

2(kl

t

+ 'f

v~

2

4-

(kj, k 2) dk j dk2 -kl 1 -kl 2

J[t

A

~kj,

k 2 ) dk2 ] = 'Y _{dk 1} 1 2 k '

1

1 A =

P

_{(kl) dk 1} I -k 1 (3. 14)

since (cos 2 (kl t +.

cp

v:~

over g> v is 1/2. The spectrum of

~

g(t)

is therefore equal to

~

( kl) • and Equation 3. 14 becomes:

k '

~g

(t)

= -

f

cos (kjt

+

<pv)

J

2

~

(kj) dk j I

(3. 15) -kl

The other gust inputs are treated in the same way. i. e. 1\ A o<.g (t) = -(w g)o = -

iJ

_{cos (k 1}

t

+

g> w)

J

2

~

_{(kl' k 2) dk 1 dk2} =

-j

c~s

(klt +

~vi)

J

2

~

_{(kl) dk 1 \} (3. 16) ,.." 0" ~ g(t) = ( W g ) . ~x 0 =

SS

sin (kl

t

+

fw)

J

_{2 kr.'Îr (kl. k 2) dk1 dk2 '} = } sin (kl

t

+ fw)

J

2

$

_{x (kl) dk 1} (3. 17)

A "" Pg (t)

I\.

= (

d

Wg )

dy

0

= -

SJ

_{sin (k 1}

t

+ fw)J2

k~ ~

_{(kl' k 2) eik 1 eik2}

= -

S

sin (kl

t

+ fw)

J

2'

i

y (kl) eik 1 , (3. 18) The limits

k~

and .k; have been.. dropped from these integrals for the sake of brevity.

Rreferring back now to the two-dimensional spectrum· in Equation 3.9, the one-dimensional spectra in Equations 3. 15 to 3. 18 can be written (see :Reference 5) as follows:

~

(kl)

---.

~

(k ) x 1 where

,

k 2 I' =

J,

W' (~1. _{k 2) dk2} -k₂ A2 k2 k 2 :: _(J' 0 _X

[~+

₁ 27fko k 2 ₁

+

.

k~

k2 + 1 A 2

cr

21(ko k2 =

5,

-k2 kl. ==

S2

_kl ' 2 = -" 2 (J 21" ko2 k 2 +(/3) _{k5 2} 1 ₎ ( ko ) 2 _{k 2 + k 2}

.f3

1 0 .,... k 2 W 1 (kl' k 2) dk 1 dk2 A.

k~

W (kl' k2)' _{dk 2} , (3 - X2)

J

(3. 19) k 2 0 as k' --. 2 00 (3.20)

, Aside from any' considerations concerning the accuracy of the power series expansions. note' that the finite limits are necessary to insure· convergence of the integrals defining the gradient spectra. A.plot of the vertical and side velocity spectrum function is given in FigJ..lre· J •. Figure:2 shows· a

plot of the spanwise gradient spectrum function.

Handom signals representing the gust inputs OC-_{g ,}

<X.

_g, qg,

f3 '

and

Pg were simulated from Gaussian, white noise sources by linear fiftering.

For the sake of convenience in treating the various cases, it was necessary

to make certain' approximations and simplifications in the filter design,

mainly involving the cut-off frequencies, k

1

and k

_2.

To begin with, the

same form of the filters was used for both aircraft. which invol ved a

compromise in the values used for the cut-off frequencies. Also,to

facili-tate changing from one scale to another, the cut-offs we re made

propor-tional to the characteristic reduced frequency ko, which caused the cut-offs

to decrease with increasing scale. The effect of these approximations on

the results is minimal, however, because the frequency range of significant

aircraft response is located almost a decade below the chosen cut-off

regions. Details of the shaping filters are given in Appendix "C".

IV. RESULTS AND DISCUSSION

40 1 Computing Program:

, The functional arrangement of the analog computer for studying the

CL -coupling phenomenon is shown in the block diagram in' Figure 3. From

the diagram, it can be seen that provision was made for varying the scale

L of the turbulence, center of gravity position C_Inot ' the degree of

correlation

f

between ~g and

f3g,

the ron stiffness

ct,.

and the

equili-brium side slip angle

f3

00 Provision was also made for switching back

and forth between the couplèd and uncoupled modes of operation.

The aircraft speed uo was kept constant at 250 fps. throughout the

program, and the gust intensity cr' was taken as 10 fps., which corresponds

to very rough turbulenceo

The bulk of the runs were made for various combinations of the

parameters Land Cm to determine the effect of the CL-coupling on the

~

lateral stability under control-fixed flight conditions. For each value of

scale, Cmelt. was chosen so that the ratio of the Short Period natural

fre-quency to the Dutch Ron natural frequency varied between 1 and 2, the

region of instability found by Masak. This ratio win be defined by

1) = (kl>sp

(kl>DR

and will be used as a fundamental parameter instead of Cm • Theyare

related by the equation (see Reference 1) ' Q(.

\ \

\

A program of the runs made is given in the accompanying table. In, general, two runs were made for each case, one uncoupled and the other

couplèd, with the sam~ sample of gust inputs in each mode. In this way,

it was possible to obtain a reasonably precise comparison between the uncoupled and the coupled modes of operation even though the particular samples of gust inputs may not have been altogether typ ic al because of the

finite 'length of the runs. Each run corresponded to about six minutes

real time; which was a compromise between excessive computing time on the one hand, and the stability of the mean square response estimates on

the other. In one critical case, that of Aircraft 2 with L

= 500 ft. and

-V

= 2.0, an extremely long run (over one hour re'al time) was made in

order to establish additional confidence for this: case.

The effect of correlating CXg and

f3

g was el'afUined for Aircraft 2, .

again for the critical case mentioned above. Since each value of finvol ved different samples of the gust inputs (see Appendix "C"), the uncoupled

run was repeated for each

f

to maintain a valid comparison.

TABUE -PROGRAM OF RUNS No. of

-V

Cl+

~

Runs Aircraft L

L

4 1 ,500 1. 73 2.00 2.76 0 0 0 4 1 1000 1.73 2.00 2.76 0 0 0 4 1 2000 1. 73 2.00 2.76 0 0 0 2 1 500 2.76 0 -0.05 0 5 2 500 .74 1.0 1.5 2.0 0 0 0 5 2 1000 .74 1.0 1.5 ·2.0 0 0 0 5 2 2000 .74 1.0 1.5 2.0 0 0 0 2>'.< 2 500 2.0 0 0 0 2 2 500 2.0 0 -0.05 0 2 2 500 2.0 0 0 O. 1 10 2 500 2.0 0, 0.25 0 0 0.5, 0.75, 1.0 ~< Extra-length runs

Finally, the effect of roU stiffness on the results was examined

for each aircraft for the L= 500 ft.,

1>

= 2.0 case,' using the value

. Clf '= ":0.05 per radian. Despite the fact that the pure stiffness does

not take into account the usual pilot-control system dynamics, it represents a somewhat more realistic condition than that with no roll stabilization at all.

Recorded time histories of the aircraft variables for a typical run

are shown in Figure 4. Figure 5 shows comparison of the uncoupled and

coupled pitch rate respo~se taken from a section of the extra-length run

referred to above.

4. 2 Ana1.ysis of Data and Presentation of Results :

The aircraft response variables were analyzed using the averaging

circuits described in Appendix "D". The averaging time constant T was

chosen equal to 100 seconds~ which was about one quarter the length of an

average run. The first 100 seconds was thus weighted very lightly, and

the initial aircrafi; transients had a negligible eff.ect on the results. Both

the mean value and the variance we re calculated for each of the variables

oc

,

(3 ,

f

and ~, and we re recorded continuously throughout each

run. The mean values serve principally as an indication of slow divergence,

whereas the variance measureménts mainly provide an indication of

oscilla-tory instability. (For finite duration runs with a highly oscillatory response,

measurements of the total mean square might tend to obscure incipient

divergent instability.) Typical time histories of the mean and variance are

shown in Figure 6.

Comparison of the uncoupled and coupled lateral response win be

do ne in terms of the ratio,

_ (cr) coupled , - (cr) uncoupled

where the appropriate subscript is used depending

on

the variable being

considere d. This ratio is plotted against

.v

in Figure s 7, 8 and 9, and

Figures 10, 11 and 12~ for Aircraft 1 and 2, respectively, for the variables

~

,

4> '

and

r.

The actual RMS response values shown in these

figures apply to the uncoupled case only.

In order to provide a relative measure of the intensity of the

para-metric exqitation by the angle of attack, a plot of the RMS angle of attack

response against .,) , with L as a parameter. is shown in Figure 13 for Aircraft 1 and 2.

The eÏfects of introducing ron stiffness and an equilibrium $ide

slip angle are shown by separate symbols for the 1) = 2.0 case in Figure

11, and the comparison between the coupled and uncoupled response for the

case of correlated side and vertical gust components is given in Figure 14. In the latter figure, the ordinates have been normalized by dividing through

by the values oÎ

1l.

obtained for the uncorrelated case.

4. 3 Discussion of Results :

The most important observation which can be made from the results of this study is that there was no indication of lateral instability in either

of the two aircraft under any of the conditions examined. Since the qu.estion

here is really one of stochastic stability, the conf~dence level which can

reasonably be placed in any conclusions based on this observaiion is that

associated with the extra-length run made for one of the worst cases

.' (Aircraft 2,

-V

=2.0, L . = 500 . ft. )-, where substantially the same results

were obtained as for the shorter runs made under the _, same conditions.

The effects of the CL - coupling are, nevertheless, distinctly

noticeable, and can result in increases in the RMS response _; which are as

high as 1 ~ percent, although such values mostly occur for C. G. positions

which ar~ rather far forward. Examination of the actual time histories

(see Figure 5, for example) reveals that on some occasions the oscillatory

peaks of the response can be up to 50 percent greater than the corresponding

peaks for the uncoupled case; on other occasions, the response is' actually

less. The net result is that the ratio of the RMS responses does not

necessarily reflect the true' magnitude of the coupling effect~ . and rhay

'indicate, moreover, that a different kind of data analysis. such as mea

sur-ing the peak statistics ,would be more appropriate for purpos.es of assessing

. the effect further. These comments may be somewhat academic, however,

in view of the further observation that the CL -coupling effect is easüy

controlled . (in fact, almost eliminated) by the introduction of a roll stiffness

term which approximates the practical case of lateral contrpl by a pih>t

or autopilot (see Figures 7 and 10). .

The following additional observations can be made concerning the

results :

. (a) Aircraft 2 is more sensitive to the CL-coupling than Aircraft 1.

This substantiates Masak's results, and is.probably due

t

o th

e considerably

higp.er values for some of the cross-coupling derivatives (particularly

C. ) for Aircraft 2, and its slender configuration.

N~f

(b) The ratio

"l

generally increases with

V

This is pa

rticu-lady noticeable for

f3

and

+ ;

~ appears to be relativelyinsensi~ive

to

-V

,especially at the smallest scale. The RMS value of the angle of

attack also increases with 1> (see Figure 13), and may partially account

for the increase of"1l with

V

;

.

most of the increase, however, is

probably due (see (c) ) to the increase inthe high frequency content of

( OC

+

d.

g)

with increasing

1>

'

.

The .dip in

-rt

between .

V

=

1. 0 and 2. 0

which would be expected from Masak's re~ults can be detected in the

cf

anti

r

results for Aircraft 2 (Aircraft 1 does not go down to 1>

=

1. O).

(c) The ratio

"l

generally decreases with in creasing scale. This

is consistent with Masak's results since increasing the scale shifts .the

parametric excitation to l0wer frequencies and away from the sensitive

region ·around.2(kl> DR. Note that it is the frequency content· of the pàra

-metric excitation, rather than the intensity .. which predominatés Ejince

Figure 13 shows the RMS angle of attack to be relatively independent of

(d) The ratio

1L

increases with equilibrium side slip angle. This

result is expected since a

f3

0 will introduce first order cross -coupling

terms which depend on ()( only, as well as the usual second order terms

which depend on a product of cJ.. with one of the lateral variables. This

case gave, the highest value of

1l.

olptained, but no sign of instability was

observed from either the RMS or average values, and the responses appeared

typ ic al in all respects to the other qases except for magnitude. The effect

of a C,t4' was not tried for this case, but it se ems reasonable to assume

that the effect could be controlled as easily as for the case with no

p

o.

, , (e) Effect of correlated

<x'g

and

f3

g. The results of this case

are summarized in Figure 14, and mdicate a rather slow rise in 11... with

increasing

p .

There is a general flattening beginning at

f

=

O. 5 for

f}..

,and at O. 75 for bQth

tf

and ~ • The slight tendency to peak at

0.75 for

fo

and

f

is not believed ~o be significant. Again, no sign of

lateral instability was observed. It' should be stressed h~re that the

re-sult$ for this case are tentative, anH are intended only to indicate the possible effects of correlated gust inputs. Further work in this area will

have to await specification, by theory or measurement, of the various

cross-spectra in boundary layer turbulence at low altitudes.

It is unfortunate that a combination of correlated inputs with an

equilibrium side slip angle was not tried, but, in view of the remarks made

above, and, because of the relatively small values of 7l obtained in each

case, it is unlikely that the combined result would be substantially different from that obtained for each case alone.

v.

CONCLUSIONS

, The results of this study indicate that the CL-coupling phenomenon

does not cause lateral instability under normal control-fixed flight

condi-tions for the two types of aircraft examined. Increases in the RMS value

of the late.ral response resulting from the CL-c.Qupling. are of the order of

10 percent, and can be reduced to negligible values by roll stiffness approximating that normally provided by a pilot or autopilot.

It was found that both an equilibrium side slip angle and correlated

gust inputs increased the effect of the CL-coupling, but in neither c~se

was instability observed. The overall increases are small, and it is

tentatively c onc1uded that they can likewise be effectively reduced by

roll fee.dback.

It is likely that arealistic lateral control system (human or

other-wise) would be relatively less effective in reducing the coupling effect than a simple roll stiffness, and might conceivably even exaggerate the effect. This could provide the basis for further research in this area.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Etkin, B. Masak, M. Porter, R. F. Loomis, J. P. Etkin, B. Etkin, B. Rice, S. O. Press, H. Tukey, J. W. Etkin,· B. Bendat,' J. Mazelsky, B . . AmeY7 H. B. Newton, G. C. Gould, L. A. Kaiser, J. F. REFERENCES "Dynamics of Flight" Wiley & Sons, Ln.c., 1959.

"On the Lateral Instabilities of Aircraft Due

to Parametric Excitation"

UTIàS Technical Note No. 86. January 1965.

"Examination of an Aerodynamic Coupling Phenomenon "

Journal of Aircraft, Volume 2, No. 6, 1965.

"A Theory of the Response of Airplanes to Random Atmospheric Turbulence"

UTI1\. . Report No. 54, May 1958.

"Theory of the Flight of Airplanes in Isotropic Turbulence; Review and Extension"

AGARD Report 372, April 1961.

"Mathematical Analysis of Random Noise ", in Wax, Selected Papers on Noise and Stochastic Processes, Dover Publications. "Power Spectral Methods of Analysis and Their Application to Problems in; Airplane Dynamics"

AGARD Flight Test Manual, Volume 4C, 1~56~\

"Dynamics of Aerospace Vehicles - A Quarter Century of Change and Two Current Pro bI ems" CASI Journal, Volume 11~ No. 8, Oct. 19650

"Principles and Applications of Random Noise Theory"

Wiley & Sons, Ine., 1958.

"On the Simulation of Random' Excitations for

Airplane Response Investigations on Analog Computers"

Journalof Aeronautical Science, Volume 22, September, 1957.

"Analytical Design of Linear Feedback Controls"

APPENDIX "A"

Aircraft Data

The numerical values used for the aircraft parameters are given below in Table I.

TABLE I

Quantity Aircraft 1 Aircraft 2

Airc·raft Type Subsonic, Swept Wing Supersonic Delta

'WIS 40 40

b/2 ft 65 37. 5

A 7 1. 45

Uo fps 250 250

CLo 0.5 O. 5

CLoe. per rad 4.0 2. 6

}J- 8. 15 13. 9 C ft 20 66 iB 2. 16 13. 3 CZo( -4. 5 -2. 9 Crn..c. Varied Varied Cffi;( -0.086 -0. 30 ~mq -0.47 -0. 60 IA 1. 35 2. 7 iC 3. 5 13. 7 iE 0 -3. 1

C~~

-0. 02 -0. 1 CL -0. 34 CL Ct.. r -0.01 +0.23 CL O. 18 CL

C~f

O. 04 +0. 05 CL 2 0.06 C· _{-0. 12 CL -0. 3 CL} C_nr pp -0. 1 -0.02 CL 2 _{-0. 3} C_YP -0. 15 -0. 66 CYp 0 -0 CYr 0 0 CLp -0.40 -0. 18 Ctf -0.07 -0. 17 Ct.tp -0.40 -0. 88 C~r O. 125 0.090 CJ,oe.r -0.92 0.47 Cn" 0.0525 0.06 Cn~ 0.200 0 Cnp -0.060 -0. 150 C~p -0.480 -0.780 Cnr -0. 105 -0. 30

Quantity Cn IX. r

t*

Aircraft 1 Aircraft 2

o

kOm (L~ = 500 ft. ) -0.080 0.262 O. 130 O. 168 0.075

The lateral characteristic equations and roots for control-fixed flight are as follows:

Aircraft 1 .

Aircraft 2

Aircraft

i

2

(lOkü4 + 3.35 (lOk1)3 + 2.84 (lOk1)2 + 7.01 (lOk1) + 0.0513

=

0 (k1)l = -0.00073 (k1)2

=

-0. 315 (k1>3' (k1)4 = -0. 00815 ~ i. 148 (k1h = -0. 011 (k1)2 = -0. 1025 (k 1) 3, (k 1> 4

= -

O. 0 115 ~ i. 163

The characteristics of the modes are given below in Table Il.

Spiral t 1/2 944 asec. 246 sec. 63 asec. 10.6 sec. TABLE Il Roll t 1/2 2. 19 asec. O. 57 sec. 6.7 asec. 1. 13 sec. Dutch Roll (k1>DR O. 148 rad/ asec. O. 565 rad/ sec. 0.163 rad/asec. O. 97 rad/ sec. t 1/2 84.8 asec. 22. 2 sec. 60 asec. 10. 1 sec. The values of CIl1çt corresponding to the values of ']) . which were used in the study are given in Table III beIow.

TABLEIII Aircraft 1

.v

emoe.

%c

1. 73 -0.012 1.0 2.0 -0.059 4.7 2.76 -0. 230 18. 3 APPENDIX "B" 0.74 1.0 1.5 2.0

I

}

"

.. _{1\ .}

Airîaft 2

_Î

Illcr-J

o

~

13 -0.1.

19

-0. ,3 -1.

!?

'1 2. 8 6.4 16.0 29. 6

Simulation of the Equations of Motion

The following scaling was used in sinulating the equations of

notion:

(a) Amplitude

Variable Est. Maximum Value 'Machine VariabIe

0(, t ~ , 9 ,

4>

1 rad. (l00DC. ), etc . - volts

.

,..",...

Öl

,p

1 q,

P

,r 1 rad/asec.

.

(1000{ ), _etc. - volts

.

. .

A A A. 2

q,

P

,r 1 rad/ (asec. ) (lOoct)A " etc. - volts

(b) Time

If t = real (problem) time - sec.

"

_{t = Non-dimensionalized time - asec.}

1;

=

machine time - msec.

"

then t = t/t* t':<

=

(see Table I)

A

t

= t/tG I<,

=

10

t

=

(t>:< K- )t

The scaled equations of motion are given below, and the

corresponding analog computer diagrams are shown in Figure 15 and 16.

2

IA

[100~] = _ ( _

Cz G(. ) [100 (0{+

cx.g)]

-Cmcx:

= -( 10

(100 (<Î+qg>]

2

ia

[lOOp]

=

_(21~)

10

1

100

i-] -

(-~&P

)

l100(P+

f

g)] +

(C~o

) [ 50f]

,

iA(100~]

=

_(~E

)

19[50~]

- (-C).r) [100(f+fg)] - (-CJ,d.f) [100 (ot + oC._g> (f+fg)]- (-C.t

_{p )}

[100 (F+pg)] + (Ctr)

[100~]

+(Ctd.r)[100(O(+olg)~]

-(-

2C

lf

)[504>]

ic[100

~

1

=

_(i~)

10 (

100~]+

(C nt ) [100

(f

+

f

g)] + (Cnd.p [100 (lJ.+ Gt_g>

(f

+

f

g)] - (-Cnp) [100

(p

+ Pg)] - (-Cnoe.p ) [100 (oC..+o<.g)

(~+pg)1-

(-C_nr)

[100~]

_{- (-Cnotr)} [ 1 00 (oL + eX. g>

~].

APPENDIX

"c"

Sirnulation of the Gust Inputs

Methods for sirnulating random signals with prescribed spectral and statistical characteristics are described in the literature (References

9 and 10), and are commonly used in analog computer practice. When

the required signals are Gaussian, as is the case here, the procedure is straightforward, and involves shaping the output of a Gaussian noise

generator with a linear filter. It is based on the fflct that the output spectrum

of a linear filter F(iW) is related to the input spectrum ~ i( (J» by:

Since the spectrum of most noise generators is uniform over the frequency range of interest in analog computing, the shaping filter alone determines the shape of the output spectrum.

The basic arrangement of the shaping filters for simulation of

the gust input signals is shown in block diagram form in Figure 17. The

three uncorrelated, white noise sources required were obtained from an

FM tape recording of the output of a random noise generator recorded on

,

The choice for the cut-off k2 presented a problem because the

same shaping filter was to be used for all three scales. Rather than

aUow the cut-off to interfere with the response with changing scale, a

compromise value k

2

=

_{50ko was chosen. This is ab6u}t a decade higher

(it is less for the other scales) than would ordinarily be chosen for the

L = 500 ft. case for Aircraft 2, but the effect on'the rigid body modes was

smaU because of their relatively low characteristic frequencies.

The expression for the spectrum~of 0( g and

f3

g is given by

Equation 3. 19 in Section lIl. When k

2

=

_{50ko, th}is spectrum can be appro

x-imated reasonably weU (see Figure 1) by

=

&2

21' k o

ko 2

( 2 )

k:₁

+

ko

Thus, for a noise source with aspectral density No volts 2/rad• / asec.,

the desired shaping filter wiU have the foU0wing transfer function:

,.. ko A. (J' ~+ff F(s)

=

/2 koN; ko

J3

= 100

f f

~

k )2

N~

0

S

3 + __ k....;o=---_)"- --::,.-1 O_O_k~o~ 1\ A s + ko s

+

100 ko k~/, ko

~

+

;7'/3

s

+ 100

~~

where s

=

_{ik 1. The analog diagram for this fil}ter is shown in Figure 18.

Note that the

ti:

g and

qg

signals are readily available before the last

integrator. For convenlence in changing the integral scale, all integrator

gains are multiplied by the factor ko/kom , where kOm is the value of ko

for L

=

500 ft.

The ratio between the simulated RMS gust intensity

(T

2 and the

desired value

0-

2 is found by calculating the area under the simu1.ated

spectrum. That is,

.-\.2 0' _s

=

ioo

J

-i 00 A.. J\. ' " A. 2 F (s ) F ( -s ) d s

= •

98 tf

Integrals of the above form are easily evaluated with the aid of the table

in Reference 11. The ~MS value of the derivatives is folind similarly

•

A

=

12. 2 ko 0"

This last relation was useful for checking the filter calibration by actual measurement of the mean square value of the

eX

g signal (a better s"tatis-tical estimate, for a given measurement time, is obtained by measuring

ei.

g instead of

rx.

g)'

Design of the Pg filter was based on an empirical fit, using Bodé plots, to the square root of the <Py(k 1) spectrumgiven in Equation

3. 21 in Section lIl. Noting that

/ '

;:F.. " 2

't' Y ( 0 )

=

1. 56 ko (J"

the following filter was found to be areasonabIe approximation ( see Figure 2 ) to the desired function:

Fy(~)

=

1. 81d

J

3 \ ko 5 ko

S

+ 9 ka ( 40 ko )2

2 t-No

s

+ 5 ko 9 ko

S

+

40 ko A

S + 100 ko _{280 ko}

100 ko

"

s

+

280 ko

The computer diagram for this filter is shown in Figure 19. Because of the extreme sensitivity of roll angle to any residual Pg resulting from tape recorder drift, a high-pass filter with a break at 10-3 ko was used in series with Fy. The RMS is given. again using the tables of Reference

11, by

/ ' ~ / '

a'Pg

=

6.94~ko(j, which was used for calibrating the Fy filter.

Table IV below sets out the RMS values used for the gust inputs assuming the following reference conditions :

Quantity h/2 kom cr' = 10 fps L = 500 ft/rad. Uo = 250 fps. TABLE IV Aircraft 1 65. ft. O. 130 . Aircraft 2 37. 5 ft. 0.075

Quantity Aircraft 1 Aircraft 2 Jkom' 0.361 '0.274 A A _{0.04 0.04} (Jo('

CfJ

0.0634, 0.0366

& '

_~ (JIJ,

_O.

100 0.076

(l~8

G-ot,r]

4.00 volts 4.00 volts,

[1008::-"1 _«..,\- 6. 34 volts 3.60 volts

[100 o-~] 10.0 volts 7. 60 volts

Values corresponding to the other scales (1000 and 2000. ft. ) we re dete r-mined automatically by the appropriate multiplier (see Figures 18 and 19L

The simple circuit shown in Figure 17 was used to study the effect of various degrees of correlation between the gust inputs eX g and

,IJ

go Assuming n1 and n2 are two noise sourees such that

o .

_•

then the varianee of n4 is

<

n~

>

= 1 \ a2

< (

n1 + an2)2

>

Simîlarily:

1

2

=

0' n

The covariance of n4 and n5 is

(n4 n 5 )

=

_{1 +} 1 _a2

<

(nI + an2) 1 (ni

'>

=

_{1 + a 2} 1 - a 2

()'~

=

- a 2 1

=

<n

2₂) _'

=

0"2 _n (nI - an2)

>

a2

<

n~

1 )

and so the correlation coefficient (normalized covariance) is given by

or 1 - a 2 1 + a2

f

, =

[ L I '

a

={Q+jJ

o~ a ~ 1

Negative values are obtained by simply reversing the sign of n2. Since

n1 and n2 are assumed uncorrelated. there are no cross-spectral

com-ponents in the output.

APPENDIX "D"

Measurement Circuit

Analysis of the aircraft response variables was done with the

averaging circuits shown in Figure 20. which consists basically of two low-pass filters. one arranged to measure the average value. and the other, along with a multiplier, to measure the variance. or mean square about the mean. The simple low-pass filter was chosen because it has

the desirabIe property of providing a visual indication. when the outputs

are recorded, of the stability of the mean and mean square estimates obtained.

True, or perfect. averaging of a variabIe x (t) is defined by

x = x (t)

=

lim _1_ JT / 2 x (t) d t

T-. ro T

- T/2

In practice, the average is taken over a finite time.

true value is obtained which is a function of T. i. e ••

1 T/2

S

x (t) d t

=

T

-T/2

and an estimate of the

The low-pass filter gives an exponentially weighted average. which is a

function of the averaging time constant T.

1 T

, -(t-t')/T

x (t)

e

dt

It is simple to determine from the records when the transient terms

o 10

-

I 10

-

2 10 --4-10

...

~

,

~

\

'~

\

_,

~

\

~

~l,

I /

I~

R2 =ISOko SIt.AUJUED SPECTRUM ko !to 10lto

10

REDUCEDFREQUENCY - kl FIG: 1 SPECTRUM FUNCTION

-VERTICAL AND SInE VELOCITY COMPONENTS

I RZ:oo

I

'

~,

_\

v

,.

_,

lOOlto

10 o 10

-,

'0

I -t '0 -4

'0

_Ro

1000

r--...

~

_~

"

,

_\

~~

\

_~

~

\

,

~

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SIMULATEO SPECTRUM-Ro Ro \00 10 REDUCED FREQUENCY k,

FIG: 2 SPECTRUM FUNCTION

-SPANWISE GRADIENT OF VERTICAL VELOCITY

I

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, t'-• Î Noise Sources S P E C T S R H U A M p I F N I G L T E R S

f

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Gust Inputs 0'. g CIIIo. (t»

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Aircraft Outputs

.

LONGlTUDINAL 0'. g _0'.

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qg EQUATIONS

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Coupled

--

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P Pg EQUATIONS ,., _r

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~o M E A N S Q U A R C E I R C U I T S

FlG: 3 ARRANGEMENT FOR STUDY OF COUPLING PHENOMENON

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FIG: 5 COMPARISON OF COUPLED AND UNCOUPLED ROLL RATE RESPONSE

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7

COMPARISON OF COUPLED AND UNCOUPLED RESPONSE FOR

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FIG: 9 COMPARISON OF COUPLED AND UNCOUPLED RESPONSE FOR

AmCRAFT 1 WITH L

=

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-FIG: 10 COMPARISON OF COUPLED AND UNCOUPLED RESPONSE FOR

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FIG: 11 COMPARISON OF COUPLED AND UNCOUPLED RESPONSE FOR AffiCRAFT 2 WITH L . 1000 FT.

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