N PS ARCHIVE 1963 BELL, G.

AN INVESTIGATION OF THE EFFECT OF AUTO-THROTTLE DEVICES ON AIRCRAFT CONTROL

IN.THE CARRIER LANDING APPROACH GERALD R. BELL

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AN INVESTIGATION OF THE EFFECT OF AUTO-THROTTLE DEVICES ON AIRCRAFT CONTROL

IN THE CARRIER LANDING APPROACH

* * * * *

Gerald R. Bell

AN INVESTIGATION OF THE EFFECT OF AUTO-THROTTLE DEVICES ON AIRCRAFT CONTROL

IN THE CARRIER LANDING APPROACH

by

Gerald R. Bell

H Lieutenant Commander, United States Navy

Submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE IN

AERONAUTICAL ENGINEERING

United States Naval Postgraduate School Monterey, California

1963

DUDLEY KNOX UBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY CA 93943-5101

AN INVESTIGATION OF THE EFFECT OF AUTO-THROTTLE DEVICES ON AIRCRAFT CONTROL

IN THE CARRIER LANDING APPROACH

by

Gerald R. Bell

This work is accepted as fulfilling

the thesis requirements for the degree of

MASTER OF SCIENCE

IN

AERONAUTICAL ENGINEERING

from the

United States Naval Postgraduate School

"'•■'LPvl I E SCHOOL f.' 'Ttr.tV. CALIrOlmlA 93540

ABSTRACT

An automatic-throttle compensation system has the capability

of removing the problem of aircraft speed instability in the carrier

landing approach. Various systems have been proposed and tested, each

differing in input sensed variables. Two representative systems are in-

vestigated analytically and by the use of digital programmed Myquist

plots and analog simulations. An attempt is made to determine optimum

gain constants by using various forcing functions in the analog simulation.

Comparisons of the responses of the two systems are made using time"

history analog records. A digital computer stability program, applicable

to any aircrafte is included.

The writer wishes to express his appreciation for the assistance

and encouragement given him by Professor E. J. Andrews of the U.S.

Naval Postgraduate School. Gratitude is also due Ling-Temco-Vought

Incorporated, Specialties Incorporated and Bell Aerosystems for providing

the necessary data used in the investigation. In particular„ the author

wishes to thank the Stability and Control Section, Airframe Design

Division , of the Bureau of Naval Weapons for the valuable assistance

supplied in the form of pertinent technical reports „

ii

Section

2.

3.

4.

5.

TABLE OF CONTENTS

Title

Table of Symbols

Introduction

Discussion

Artificial Speed Stability Through Automatic- Throttle Techniques

Experimental Methods and Results

Conclusions

Bibliography

Appendix I Derivation of Transfer Functions

Appendix II Dynamic Stability Fortran Program

Appendix III System Mechanization for the Analog Computer

Pa'

VI11

I

B

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11

PS

iii

LIST OF ILLUSTRATIONS

Table Pay^

I. Computed F-8 Airplane Stability Derivatives and Associated Aircraft Parameters

II. Sample Printout, Fortran Stability Program, LONGSTAB 29

Figure

1. Thrust Required Versus Airspeed 30

2. CL Versus L/D, F-8 Airplane 31

3. Block Diagram, System 1 Auto-Throttle, with Transfer Functions 32

4. Block Diagram, System 2 Auto-Throttle and Engine, with Transfer Functions 33

5. Nyquist Diagram of System 2 Auto-Throttle Open Loop Transfer Function 35

6. Composite Block Diagram of Complete Automatic Throttle-Airframe System 36

7. Angular Definitions and Signs 37

B. Photograph of Analog Setup 38

9. Photograph of Analog Setup 39

10. Time History, 1.3° Up Elevator Deflection Input, System 1-a 40

II. Time History, 1.3° Up Elevator Deflection Input, System 1-b 41

12. Time History, 1.3° Up Elevator Deflection Input, System 2 42

13. Time History, 5 Knots Horizontal Gust Input, System 1-a 43

14. Time History, 5 Knots Horizontal Gust Input, System 1-b 44

iv

15. Time History, 5 Knots Horizontal Gust Input, System 2 45

16. Time History, 4° Step Input Pitch Angle, 9, System 1-a 46

17. Time History, 4° Step Input Pitch Angle, &-, System 1-b 47

18. Time History, 4° Step Input Pitch Angle, &-, System 2 48

19. Time History, 4° Step Input Pitch Angle, &-, Basic Airframe 49

20. Time History, .02 cps. Triangular 1.32° Elevator Deflection Input, System 1-a 50

21. Time History, .02 cps. Triangular 1.32° Elevator Deflection Input, System 1-b 51

22. Time History, .02 cps. Triangular 1.32° Elevator Deflection Input, System 2 52

23. Time History, .06 cps. Triangular 1.32° Elevator Deflection Input, System 1-a

24. Time History, . 06 cps. Triangular 1. 32° Elevator Deflection Input, System 1-b 54

25. Time History, .06 cps. Triangular 1.32° Elevator Deflection Input, System 2

26. Time History, .06 cps. Triangular 1. 32° Elevator Deflection Input, Basic Airframe

Time History, Human Pilot Controlled Flight on Glide Slope, System 1-a

28. Time History, Human Pilot Controlled Flight on Glide Slope, System 1-b

29. Time History, Human Pilot Controlled Flight on Glide Slope, System 2

30. Time History, Human Pilot Controlled Flight on Glide Slope, Sinusoidal Input of 5 Knots Horizontal Gust, System 1-a 60

31. Time History, Human Pilot Controlled Flight on Glide Slope, Sinusoidal Input of 5 Knots Horizontal Gust, System 1-b 61

32. Time History, Human Pilot Controlled Flight on Glide Slope, Sinusoidal Input of 5 Knots Horizontal Gust, System 2 62

33. Time History, Human Pilot Controlled Flight on Glide Slope, Sinusoidal Input of 5 Knots Horizontal Gust, without Auto-Throttle 63

34. Time History, 5 Knots Step Horizontal Gust Input, Anti-Thrust System 64

35. Time History, 5 Knots Step Horizontal Gust Input, Anti-Thrust System with Changed Gain Constants 65

Appendix I, Table I-A Longitudinal Dimensional Stability Derivative Parameters 69

Appendix I, Table I-B Longitudinal Non-Dimensional Stability Derivatives 70

Appendix I, Table II Dimensional Stability Derivatives 71

Appendix II, Table I Program Symbols and Meanings 75

Appendix II, Fig. 1 Sample Data Cards, LONGSTAB 77

Appendix II, Fig. 2 Fortran Program LONGSTAB 80

Appendix III, Table I Potentiometer Settings 90

Appendix III, Table II Scale Factors Used in Analog Simulation 92

Appendix III, Table III Potentiometer Setting Calculations 94

Appendix III, Fig. 1 Analog Schematic, Airframe Equations of Motion 98

vi

Appendix III, Fig. 2 Analog Schematic System 1 Auto-Throttle, Aircraft Engine, Stick System Dynamics 99

Appendix III, Fig. 3 Analog Mechanization, System 2 Auto-Throttle 101

Vll

TABLE OF SYMBOLS

For those symbols not defined in the text„

CD = Drag coefficient, Drag/qS

C^ = Drag coefficient in trimmed condition Do

CL = Lift coefficient, Lift/qS

a = dCy/doc , Lift curve slope

dC /dCT = Rate of change of drag with lift

T = Thrust, lbs.

S = Wing area, sq„ ft.

q = Dynamic pressure, 1/2PV , lbs./sq„ft.

V,U = Forward velocity, reference steady state, ft./sec.

VgL = Stall velocity, ft »/sec.

u = Perturbation forward velocity, ft„/sec.

w = Perturbation vertical velocity j ft ./sec.

o = Flight path angle, measured from the horizontal, radians

©* = Pitch angle, measured from horizontal to FRL, radians

A ( ) = Error quantity, increment, perturbation

nz = Normal load factor, g units

Al^£ = Thrust command increment, lbs.

Cjo = Equivalent propulsion system drag = "c~£äL.

J = Time constant, time in seconds to reach 63% steady state

K^ = Gain constant for CK input

Kna = Gain constant for n input

vm

K« = Gain constant for pitch error input

Kyn Ku = Gain constant for airspeed error input

Ky2 K/ u = Gain constant for integral of airspeed error input

&r\ = Undamped natural frequency, radians/sec«,

3s = Damping ratio

ix

AN INVESTIGATION OF THE EFFECT OF AUTO-THROTTLE DEVICES ON AIRCRAFT CONTROL IN THE CARRIER LANDING APPROACH

1. Introduction

The carrier landing approach speed of current swept wing aircraft

is influenced by many factors. One of the most important of these factors

is often referred to as speed instability. Speed instability can be ex-

plained with the help of Fig. 1. It will be noted that for Aircraft "A" „ a

slight deviation in airspeed from the normal approach speed makes very

little difference in thrust required. This curve is typical of the AF-1E

type aircraft. On the other hand, a deviation towards slower speed for

Aircraft "B" causes a large increase in thrust required» This increase,

if uncorrected, causes a loss of altitude. Thus, in effect^ the normal

approach of Aircraft "B" is made on the "backside" of the thrust versus

airspeed curve„ This curve is typical of the F-8 aircraft,, Essentially

this steep backside of the curve is caused by a high value of Cn ,

drag increase due to angle of attack (or lift).

Still another factor contributing to airspeed instability is slow

engine response. With a comparatively long time lag in engine response,

any hesitation on the part of the pilot in correcting an incipient airspeed

loss leads to wide variations in altitude or airspeed before effective

thrust can be realized.

With an aircraft that displays speed instability tendencies, either

a faster approach speed or an automatic compensating device is required

to prevent undesired altitude loss due to airspeed deviations. Faster

approach speeds are not desirable for carrier approaches. It will be the

purpose of this paper to investigate some methods of applying automatic

power compensation to counteract airspeed instability. A swept wing

fighter,, patterned after the F-8 airplane is used as the vehicle of the

investigation. Both digital and analog computers are exploited in simu-

lating auto-throttle and anti-drag system response to various inputs „

Real-time, pilot controlled, analog simulated mirror approaches are also

used in qualitatively evaluating each system.

This investigation is the first of a continuing series to be conducted

at the U.S. Naval Postgraduate School to evaluate systems of automatic

airplane control.

2. Discussion

The aircraft-carrier mirror-landing approach is made at a constant

speed and a constant angle of glide slope,, This type of approach allows

precision control of the landing, a factor of the utmost importance in

carrier operations. Howeverc the approach is usually made at a minimum

airspeed that is dictated by performance and/or aircraft flying qualities.

Thus, the airspeed must be monitored closely while making flight path

angle adjustments. A simplified equation (given in Ref. 1) shows that

flight path angle, ^ , is approximated by the formula:

EQ. (1) Ö = L W+JT± . ...

Flight path angle, then, is dependent on lift/drag (L/D) ratio, thrust/

weight ratio, and rate of speed change „ In the mirror approach the pilot

can adjust "tf and airspeed by manipulating the throttle and elevator. The

manner in which each of these controls can be used, however, is depend-

ent upon the L/D ratio„ For aircraft with a high L/D ratio, the elevator is

used to control ^6 , while thrust is used to maintain constant airspeed.

With a L/D ratio in the vicinity of 4 - 5 the rate of speed change is large

during maneuvers. If, in addition, the L/D ratio decreases with increasing

CT in the approach speed range then speed instability is accentuated.

A variation of lift/drag ratio versus lift coefficient for the F-8 air-

plane is shown as Fig., 2. It can be seen that at the approach speeds

used, ( 1.1 - lo2 VgL ), L/D decreases rapidly with increase in CLo

As reported in Ref. 1, pilots stated that in order to change 16 for an

airplane of this class, increased reliance on thrust rather than elevator

was necessary. In the case of the F-8 airplane, however, height response

to throttle input is slow due to small thrust line angle of attack and

negligible trim changes due to power. The F-8 pilot, therefore, is forced

into using the stick as the primary flight path control„ Then, since the

airspeed mode of oscillation (the phugoid) is lightly damped, the aircraft

will hunt about a new equilibrium airspeed and glide slope angle in res-

ponse to an elevator deflection. These speed changes in the 5 to 10

seconds following elevator input are disturbing to the pilot.

Another way of interpreting airspeed instability is referred to in

Ref. 2. A curve similar to Fig„ 1 can be drawn for thrust required versus

dynamic pressure, q. A definition of a stable speed regime can then be

postulated as that portion of the curve where

Ineq. (2) ^ T"tw"* y 0

clfr

To express this inequality in terms of the lift and drag coefficients,

we use the fundamental equilibrium equations for level flight:

CD'Set. ~ D^acj = TVe^.

CLS<g. = Li-ff = W

therefore "p _ \^/ ^&

and

L

since W and S are constant.

4

Therefore d Ty^ S dl ( ^J

which result is substituted into Ineq. (2). 9 « ( Q£ )

Since S is greater than O, this implies that

Taking the differentials of numerator and denominator:

i~ oKi -old

or

Ineq. (3)1 dp' «(£&' ^ Q

Thus, satisfaction of Ineq» (3) implies speed stability.

The C " term used above is composed of C + C.q , where C is

equivalent propulsion system drag. C.„ is defined as

CAS -"jr^ or for constant altitude and slow speeds,

C.g is seen to be proportional to the negative of the thrust change

with respegt to the velocity change. Inequality (3) may then be written as

I~Q. (4) I ^ ■JTi >0

A more sophisticated approach, based on the suppression of altitude

disturbances by the pilot's use of the elevator, and originating with the

basic equations of motion, would produce essentially the same inequality,

with a small modification factor included,, (See Ref* 2) The resulting

inequality is

Cn + C dt D

JH. - Tel Ct im. > 9

However, the correction term, C , , is of relatively small magnitude D/a

and may be ignored. From this same analysis the time constant, T"~,

of the subsidence of an airspeed error is shown to be proportional to:

Eq. (5) CD + CA? e/Co

T*KCL("^-J%)

Typical values of the parameter I for the F-8 are given below.

At 1.1 VSL, I = .22 - o548 = - .328, showing distinct speed instability.

At 1.2 VSI/ I = „200 - .222 = - .022, a less unstable value.

From an examination of the speed instability parameter, Ineq. (4),

it will be seen that positive stability can be produced if either drag in T

trimmed conditions (CnrJ or C^ is artificially increased.

CDQ could be increased, for example, by extending the speed

brakes during an approach. This would accomplish two objectives; one,

increase C-QQ in the stability formula, and two, cause the engine to

operate in a higher RPM range to produce the added thrust necessary,

thus effectively causing a slight decrease in the inherent engine time

constant»

Remembering the definition for Cjo, the drag coefficient corre-

sponding to the thrust/speed variation, it will be seen that an artificial

means of increasing thrust with decreasing airspeed accomplishes the

desired objective. This is exactly what automatic throttle or approach

6

power compensator devices are intended to do. Still another method of

increasing C is by the use of an anti-drag technique. This may be AS

accomplished by having a drag producing device (such as side extending

speed brakes) extended during the approach. Then, by using an airspeed

controller type device to regulate the amount of drag produced (by opening

or closing the speed brakes) in response to thrust (or anti-drag) required,

airspeed deviations can be closely controlled„ This particular method,

within its limitations of having enough anti-drag available, theoretically

has the advantage of a much shorter time lag than an engine response

scheme „

3. Artificial Speed Stability through Automatic-Throttle Techniques

There is a choice of inputs to an automatic-throttle type controller,

including various combinations of derivatives and integrals of airspeed

error, pitch attitude error, angle of attack error, and normal acceleration.

There are also many different philosophies by manufacturers concerning

the ideal combination for best performance under all conditions „ Two

representative systems, one utilizing angle of attack and normal acceler-

ation, and the other using airspeed error and pitch angle, were chosen for

investigation o

A) Angle of Attack, Normal Acceleration Inputs

The Specialties, Inc. Automatic Power Compensator, (APC), an

auto-throttle in use in the F-8 and F-4 aircraft, is an example of a control

using these inputs. A block diagram, typical of this type of system is

shown in Fig. 3. In this system, constant angle of attack is maintained

in the landing approach using the two sensed inputs of instantaneous angle

of attack and acceleration normal to the glide slope in g units» The normal

g input supplies anticipation whenever the aircraft is not in a one g flight

condition. A thrust command is calculated in the system computer depend-

ent upon error signals and chosen gain and temperature factors,, This

thrust command then becomes an input to the engine which in turn adds

an appropriate thrust to the airframe. This thrust, acting as the forcing

function to the airframe modes of motion, changes the velocity of the

aircraft. Thus the APC computer indirectly controls airframe velocity

through the equations

8

AT= K^AK +Kn Ah* ' AV = KT AT

Eq. (6) ;> £V - ^ax + ^4/^

where the K values are appropriate shaping, filtering and simulation

networks.

Or, looking at it differently, the C of Ineq. (4) has now been Ao

given a greater value ( C/\^ =f ~ ^— )

artifically, thus making the speed stability parameter more positive or

stable.

The transfer function equations for this auto-throttle as supplied

by Specialties, Inc., are also shown in Fig. 3. It will be noted that

the throttle servo is simulated by a . 1 second time constant first order

lag. Angle of attack input is treated by both an integrator and lag net-

work. Normal g input is also integrated»

The results of the analog simulation runs using this type of control

are reported in Section IV of this paper, Experimental Methods and

Results«

B) Pitch Angle, Airspeed Change Input

Flight path angle, 7f , is given by the equation

Eq, (7) V - & - <*

During constant airspeed flight, pitch attitude change (A<9*) and

flight path angle change (&ti) are proportional; but dynamically, A &—

occurs before AX. Thus the means of anticipating flight path angle

changes is available through pitch attitude changes recorded by a

9

vertical gyro mounted in the aircraft. This is the theory underlying the

pitch angle, airspeed change type of automatic-throttle controller.

Through the use of a properly designed network, these inputs make it

possible to minimize transient airspeed changes and to eliminate steady

state airspeed changes resulting from variations in flight path.

The governing equation for this controller may be of the form:

Here the K. term is inserted as a steady state error washout term.

The lag term ( I +7i^-) simulates the airspeed sensor and filter. The

filter attenuates the short period wind gust signals entering the system

and acts as a smoothing influence.

This equation was reduced to a linear open loop transfer function

of the auto-throttle controller. The block diagram for this system, com-

bined with the analytically derived block diagram and transfer functions

of the airframe and engine, are shown in Fig. 4. The stability derivatives

for the F-8 in landing configuration were used for the airframe transfer

functions. The derivation is shown in Appendix I.

The system open loop transfer function was calculated as;

Eq.(9) _*_, to JAT UTJAF

^c t+f±\ tieA /*T\ fu-\ [SejAFi^CMfWMlm. ■Mr

10

where; {J^)AT-^7JJI^JJT1I)

/uA _ .&2+1.Io~s f\ + g7g<L + /ff,1?^ ■* £3.254.0

— ](-^\ ~ ~0,+x Q+37.KL + 176 +i)

0-/^T (l-hi. 16A.)

This equation was programmed on the digital computer to produce a

series of Nyquist plots, each utilizing different gain constants.

Fig. 5 shows the Nyquist plot for the equation using the same gain

constants as were used in the analog simulation. From this figure the

open loop gain margin is seen to be 19.2.

The system closed loop transfer function, —h-r ) was then derived,

This is shown symbolically as follows:

Eq. (10) - = V~ ' ' Hf&r ^)feV<t®F

A digital program was devised to calculate the values of the poles

of the closed loop transfer function for various values of auto-throttle K's,

This procedure would have produced families of Root Loci Plots „ How-

ever, because of lack of time and computer malfunctions, this particular

investigation was not completed. Equation (10) would be used in an

optimizing analysis for gain constants.

In addition the equations of the auto-throttle were set up on the

11

the analog computer. This was incorporated into the complete system

using approximately optimum values of K^. , Ku, and Kyu. All simu-

lation was done in real time. These results are also reported in Sec-

tion IV . Analog mechanization is shown in Appendix III. A composite

block diagram showing the relative positions of the two automatic

throttle systems in the airspeed loop is presented as Fig. 6.

C) Auto-Throttle Controller Linked to Anti-Thrust Device

Another possible approach to the airspeed instability problem is

the use of an auto-throttle controlling an anti-thrust or drag device.

This could be accomplished byflyingthe approach with speed brakes

extended. The output of the auto-throttle would then reduce or in-

crease the drag by closing or opening the speed brakes as required.

Thus, instead of supplying additional thrust when the airspeed drops,

the speed brakes would be closed a proportionate amount, reducing the

drag and in effect accomplishing the same purpose as an increase in

thrust. Note that the time lag of the hydraulically actuated speed brake

system is much less than that of the engine, thus allowing a faster res-

ponse. The time constant of the hydraulic system is of the order of „1

second, while that of the F-8 engine is about 1 „16 seconds. This

approach was also investigated on the analog computer by replacing the

r engine with a first order time lag of . 1 seconds (simulating the hydraulic

system) and using the thrust command output of the auto-throttle com-

puter to control the speed brakes.

12

Obviously the limiting feature to such a system is the amount of

drag the speed brakes are capable of supplying. Of course it goes

without saying that such a device can not be installed on an aircraft

incapable of completing an approach while varying the positioning of

the speed brakes . An airplane in this category is the F-8 which has a

lower fuselage speed brake. Insufficient deck clearance is available

when this brake is extended, thereby prohibiting its use during an

approach.

13

4. Experimental Methods and Results

A) Equations and Analysis

The basic 3-degrees of freedom aircraft longitudinal equations of

motion (in British notation) used in the analysis are shown below. The

equations are based on the small perturbation linear theory. Lower case

letters represent perturbation quantities. Upper case, subscripted o,

letters represent initial steady state conditions. Body axes are used

throughout. The assumptions made are as follows:

1. The airplane has a longitudinal plane of symmetry.

2. The direction of the relative wind, in steady flight, lies in

the plane of symmetry, and in steady state, all angular velocities are

zero. i

3. Initial flight condition is wings level.

4. Products and squares of perturbation velocities are small and

are ignored, and sines of all perturbation angles are approximated by

the angle itself in radians.

5. The longitudinal modes of motion are independent of the

lateral modes.

6. Structural deformations are not considered.

Three degree of freedom non-dimensional equations of motion,

British notation: (See Ref. 3) Figure 7 displays angular definitions and

signs.

X Force Equation:

14

■

V

Z Force Equation;

-*~Mä-H£ i<'^^s[^\ Moment Equation:

where symbols are defined as follows:

1 yy]- T* — £Y+ p-^ s air S6CS ,.

cLJ cLt o-* = jt_ real time J %_ air sees

u = perturbation horizontal velocity, ft/sec.

w = perturbation vertical velocity, ft „/sec

'j^ = elevator deflection, positive down, radians

AT = change in thrust required „ lbs „

(ji , angle of attack, radians Ur Vo

$■ = pitch angle , radians

<-i> = ^~n 2 s Inertia Coefficient

T 2

IUVJ = Moment of inertia, slug = ft

fr| = mass , slugs

it = tail length, ft.

/«• = Ac*« = relative density

C = mean wing chord, ft „

S = reference Wing area, square ft.

J\_= Force parallel to x - axis, lbs,

2_ = Force parallel to 2 — axis , lbs .

j^ = Moment about lateral axis, lbs, - ft,

15

h = distance from thrust line to CG, ft„

VX ) 1/2 l-f flft/ non-dimensional Force Stability derivative and similar identities obtained by permuting x, 7 and u, CAT independently.

X/, =

>®

x«, = 1/2 °U oy2* ) non-dimensional stability derivative and . jp \~ similar identities obtained by permuting x,

xT = 1/2 A.prräT ) non-dimensional stability derivative and JK1 similar identify obtained by permuting x

^T and ^_ „ T» and T

Jn_ \ "Mu, = l/2Cjß?U15c/ non-dimensional moment stability derivative

and similar identities obtained by permuting u and Us"

. = 1/2 °[^fV2S/ non-dimensional stability derivative and s &- similar identities obtained by permuting x,

and O^

i/2S^ m- = l/2r- |gyu>JC/ non-dimensional stability derivative

mT = Y 'Jl. non-dimensional stability derivative-h = ' JU distance from thrust line to C.G. position

The necessary stability derivatives were evaluated by making use

of a longitudinal dynamic stability Fortran program, designated LONGSTAB„

The program was compiled while undertaking the course in dynamic air-

craft stability given at the Postgraduate School. This program computed

the necessary British stability derivatives and then used these derivatives

in the aerodynamic equations of motion. The resulting stability quartic

was then solved, by the program, for the phugoid and short period

complex roots. Finally, the periods and times to damp to 1/2 amplitude

16

were computed for both modes. Necessary aircraft parameters used by

the program were taken from F-8 data supplied by Ling-Temco-Vought, Inc.

or calculated. The program-computed stability-derivatives used in the

investigation are shown in Table I, as are the other aircraft parameters.

All program-computed derivatives, after necessary conversions were

found to be in close agreement with those listed in Ref. 6 . The computer

program is explained in Appendix II. A sample printout is shown in

Table II.

From the program printout„ the phugoid period is 33.5 seconds and

that of the short period mode is 6.06 seconds. Times to damp to 1/2

amplitude are 29.9 and 1.73 seconds respectively. Undamped natural

frequency, CO h ,j, is calculated as .189 radians/sec.,, ^T as .123.

Short period Co is calculated as 1,19 rad./sec. , T as .360. n J

Ref. 6 lists a phugoid (Jn of 0,188 rad,/sec. , and a ^ of .12.

Short period results found were listed as Qj = 1.19 rad./sec. , % = n

0.35. The results given by LONGSTAB and those given by Ref. 6 are in

excellent agreement.

Thus, as previously remarked, it is seen that the phugoid is very

lightly damped.

Ref. 6 reports on the results of an investigation conducted along similar lines as the subject of this paper. The two investigations were performed concurrently and independently of each other. After com- pletion, the results of the author's analysis were verified insofar as possible with those given in Ref, 6.

17

These non-dimensional equations of motion were dimensionallzed

for the analog simulation. The engine time lag and a ato-throttle blocks

were next added. Real time stick and throttle simulators were then in-

troduced into the system» together with the F-8 linearized longitudinal

control system dynamics supplied by Ling-Teiaco-Vought. All analog

runs were conducted in real time. The analog computers used were

Donner 3100 and 3400 machines. An eight channel Brush recorder was

used to record the time histories.

The following types of computer runs were then conducted:

(a) Transient analysis runs using step inputs of elevator deflection,

horizontal velocity gusts, and pitch attitude; first with basic airfrarne

alone and then with the two basic types of auto-throttle controllers in-

corporated . Finally sample runs were made with the speed brake device

replacing the engine ,

(b) Optimizing runs using triangular inputs of elevator deflection,

to determine the optimum gain for this type of forcing function.

(c) Sinusoidal gust inputs „

(d) Real time pilot, controlled runs using a simulated glide slope,

with and without the aid of the auto-throttle „ Horizontal turbulence was

also added to determine the controller effectiveness.

Analog mechanization diagrams are shown in Appendix III. Poten-

tiometer settings are shown in Table 1 to Appendix III. Time histories for

these runs are displayed in Figs. 10 to 35»

B) Evaluations of Experimental Analog Runs

The systems referred to are as follows;

System 1, Inputs of angle of attack, and normal acceleration in g units „

System Equation: AT = _L_ f *«** _ KcvAV , KxAKl

System 1-a gain constants? Koc = 1970 lbs./degree K^v = 280 lbs„/knot K-a = 183 lbs./degree ~ sec.

System 1-b gain constants,0

K^ = 3420 lbs./degree KAV = 72 lbs./knot K-^ = 190 lbs./degree - sec.

System 2, Inputs of pitch angle, airspeed error

System Equation: AT - K&A&"- -^_-. V ^^^^±1

Gain constants; K* = 318 lbs./degree YLLC = 360 lbs ./knot Kj«, = 22.3 lbs./knot - sec.

1) Step input of 1. 32° up elevator deflection

Fig. 10 is the analog time history response to a 1.3 step of up

elevator deflection, using the System 1-a auto-throttle. Airspeed dis-

plays a slight oscillatory tendency, reaching an equilibrium speed of

+ .6% of V (.8 kts . above V ). Thrust is applied smoothly in response o ° to demand.

Fig. 11 shows the response of System 1-b. This system had the

gain factors optimized for a step input of horizontal gust velocity. Note

the high airspeed overshoot due to step elevator input.

19

Fig. 12 is the same situation i.sir.,7 Sys*ern ?„ Airspeed is much

more highly damped as compared to Flp „ 1l ., Equilibrium airspeed is

-1% of V (1.39 kts . below V ). Thrust is also applied smoothly,

2) 5 knots of horizontal gust ir:p/j;:f:

Fig. 13 displays the record of Systen ~a in response to this

input. The system is oscillatory wtä\ an approximate damping ratio of

$ = .15,^J = -677 rad/sec. T^e ir.^ia- velocity overshoot is 61% of

initial error. These values of %:ain cons -arr? are less than optimum for

this type of input disturbance „

Fig. 14 displays the response or 3> ste^i b, which has teen

optimized to this input disturbance. -Equivalent ' is now .58, cJ is ■»J IS.

.73 rad/sec. Thrust response is somewhat oscillatory, Final value of

airspeed is +.2% of V .

Fig. 15 is the record oi Systev. f,„ ~ shows a damped response

with final airspeed of about =•£„"% 'V „ T'hrist changes are less than o

that required by System 1.

3) 4° step input of pitch an 5 ie ,.

Fig. 16 shows the response 01 System 1 to this input. Initial

velocity transient is +.8% V or 1/1.2 I i:„ Steady state error is +.4% V .

Thrust application is slightly oscillator/.

Fig. 17 shows System l-b response. Initial velocity transient is

• 6%V . Steady state error is also ,^V , o

Fig. 18 displays the response of Sys'em J c Initial velocity transient

amounts to +1.6% V . Airspeed dan.ps Jo + ,4% VG steady state error in 19

seconds, reaching less than 1% error In 9 seconds.

Fig. 19 shows the response of the basic airframe to this same

input disturbance, without the aid of the auto-throttle. Airspeed

oscillation is +1% V , with a steady state value of about „5% V . o" o

4) Optimizing runs using triangular elevator input

Fig. 20 shows System 1-a under the influence of a .02 ops

triangular elevator deflection (maximum amplitude 1.32 ) forcing

function. Airspeed error is kept, within a range of +2% to -1% of V .

Thrust varies within the range of +1600 to =2400 lbs. about the trim

setting. Thrust application follows elevator movement closely.

Fig. 21 shows the reaction of System 1-b. Note how airspeed

fluctuates + 4% V (+5.56 kts .). Thrust has very noticeable oscillatory ~ o ~~

tendencies and gyrates within rather wide limits.

Fig. 22 shows System 2 under the influence of this same forcing

function. Airspeed is seen to vary +1.7 kts. or + 1.2% of VQ. Thrust

varies approximately 2400 lbs. about trim thrust. Application of thrust,

however, is smoother and displays less hunting characteristics than

shown by System 1-a.

Figs. 23, 24 and 25 show the response of these same systems to

a triangular elevator forcing function of the same magnitude as previously

but of .06 cps frequency.

Fig. 26 shows the reaction of the airplane to this same forcing

function without the benefit of the auto-throttle. Note airspeed deviation

is less than that obtained with System 1-b engaged.

5) Human pilot controlled flight on simulated glide slope

For this series of runs a pilot flew the simulated airplane on the

glide slope, using the auto-throttle. Glide slope reference was obtained

from a zero centered panel meter. The zero voltage level was considered

the correct glide slope. This is shown on Channel I, of Figs. 27, 28 and

29. The pilot was then told to manipulate the stick to fly the aircraft to

a new 4 higher glide slope. This corresponded to a + 14 volt reading on

channel 1, or 14 volts on his reference meter. This simulated a low

"meatball" on a mirror approach and the flight path correction necessary.

This new flight path angle was held momentarily. The pilot was then told

to fly the airplane down to a - 4 glide slope, after which he was to

return to the normal flight path (0 volts).

The response of the aircraft, and speed control is shown in Figs.

27, 28 and 29 for Systems I-a, 1-h and 2 respectively.

It will be noted that Systems I-a and 1 ^maintain airspeed close to

within 1% of V , whereas System I-b has much greater airspeed fluctua-

tions. Thrust application is again much more oscillatory in System 1-b.

6) Human pilot controlled flijnt on glide slope with sinusoidal input of 5 knots horizontal velocity guEi

Figs. 30 and 31 are records ot System 1 runs utilising a pilot

controlled stick with a forcing function sinusoidal input of 5 knots of

horizontal velocity, frequency .05 cps. The pilot was again attempting

to fly the glide slope 4 up and down..

Fig. 32 is the record of the response to this same forcing function

under the influence of the pilot and the System 2 auto-throttle.

Fig. 33 displays the record of the pilot, attempting to maintain

original glide slope while the aircralt was being influenced by a sinu-

soidal u turbulence forcing function, without the benefit of the auto-

throttle .

7) Speed brake anti-thrust: system

Fig. 34 shows the history of the speed brake anti-thrust system

in response to a 5 knot horizontal velocity step input. Mechanically,

the System 2 auto-throttle was used to supply a thrust command, but

this thrust command was used to drive a A second time constant anti-

drag device, simulating the movement of the speed brakes. It will be

noted that in comparison with Fig» 15, (standard System 2 setup) response

and damping is faster. It required 9 seconds for the transient to steady

within 1% of V in the case of Fig. 34, while it required 11 seconds for

System 2.

Fig. 35 shows the response of the anti-thrust system to the same

forcing function, but utilizing different gain factors. Response is more

oscillatory.

C) Summary of Analog Results

From the foregoing evaluations it can be seen that the response of

the different systems investigated is dependent upon the sensed variables

and the applied forcing functions. System l~a, optimized to an elevator

deflection input, is marginal to unsatisfactory in performance when

subjected to a horizontal gust input. On the other hand, system I~b,

using the same sensed variables, but with gains optimized to horizontal

velocity input, does not perform well when, subjected to elevator

deflection inputs . System 2 , utilizing different sensed inputs , also

displays varying reactions to each oi the forcing functions.

In general, it can be seen that System 2 has the capability of a

smoother application of thrust in response to an airspeed error „ System 1

has the capability of a more highly damped response, but causes a more

erratic application of thrust. Further investigation as to the optimum

values of gain constants is required to accomplish the desired task more

efficiently.

It is further recommended that additional sensed variables or

integrals of such terms be added to the analog set-up to help optimize

system response. A fruitful area might be to incorporate a normal g

input to System 2 to help the damping characteristics „ It is anticipated

that further investigations along this line will be conducted at the U„ S„

Naval Postgraduate School. These investigations will attempt to utilize

the simplified equations obtained by the elimination of the short period

mode, to develop a more tractable analysis„

J

-N r

t

24

5. Conclusions

The two major approach power compensator systems investigated

appear to have the capability of performing the desired task. The major

difference are degree of anticipation and erratic!sm of thrust application.

With proper choice of gains and input variables, it seems feasible that a

composite system can be designed to provide the advantages of both

individual systems.

The speed brake or anti-thrust system has the capability of pro-

viding faster response, but is limited by dra? area available and the

physical feasibility of approaching with speed brakes extended.

The preceding investigation has firmly convinced the author that

an auto-throttle mechanism is an important contribution to carrier aviation

safety. An aircraft that is difficult to .fly in the approach because of

inherent speed instability problems, can have its whole character changed

by an auto-throttle. By removing the undesired instability an auto-throttle

relieves the pilot of concentrating on airspeed control, thus allowing more

precision in glide slope and line-up control. This ad/antage by itself,

should tend to reduce the frequency of ra.w:p accidents.

In addition, when an automatic landing system is considered, the

pilot-controlled airspeed loop must be replaced by an automatic system

to provide consistency and predictability of response.

Still another advantage of the automatic airspeed control system is

the reduction of minimum approach speed below that recommended by the

pilots when airspeed is controlled by manual throttle manipulation. In

addition the automatic device can he desi.;;r.e.3 4o provide the rinticipatlon

and instant response that can only be cbiainea fror» much actual pilot

experience. Thus it is seen that the a^tc-tLrotlie can act as 'he great

equalizer in carrier approach training „

TABLE I

Computed F8 Stability Derivatives and Associated Aircraft Parameters British Non-Dimensional Notation

Xu = -.2630

X = w .16996

Xq = 0.0

X^ = -.450

?u = -.90

£w- -1.5915

2q = -.5878

£* = -.0640

M = u

.006052

M = w -.15896

fi*= -.1422

Mq = -.5119

A— 54.466

h = .7081

t - 3.2772

CL = .90

CD - .263

CT - .195

Sw = 375 ft2

c^ = -.380

^0 =T 8.1°

.4772

27

C position = 24% y

Weight = 22,000 lbs

I/o = 139 kts = 234 ft/sec

z*.- -.1949

f\= -.349

*T - .2046X10-4

MT = -.635X10-6

INPUT PARANEIERS

TAIL AREA = Ss.'iCOC WINC. ARt A = 375.COOO CL ALFA IJIU 2.3S5CCC tTA TAIL = 1.000000 BCCY AREA= ^E8.50 BOOT LENGTH=52.80COO RHC= .OC227BO GCE=32.17T30X) fcfclGHT* 2;C0Q.CCCC LHS D EPSILCN/U ALFA» .1477200 INITIAL PITCH ANGLE= e.lOCOOO GEGREES THRUST LINE LISTANCO -.l)37CO FEET THRUST CCCF Cl = .19SCCC CMALFA* -.38000C I SOP YY = "56CCO.CCO SLUG FT SCUARFO

STABILITY DERIVATIVES

FORWARD VELOCITY= 23M.0OC T CARAT* 3.277273 G= 65.I0U9

C SUBL» .90000C C SUED* .26300C

XU= -.2630 XW* .16996 XQ= .COOO XTHETA= -.1)5000 11- -. 9C000 Z V. * T1.5915 TOTAL ZQ= -.5B7B

2THETA= -.06U0 fU= .C060S2 PW = -.15B96 MBAR fc'or.T.» - . >U2 2 fC= -.5T19 CUCNE= 5li.li66 I SUB BETA» .7C81

A» 1.00000 8 = 2.776CM C= li).Ott*77 D= 3..C2C2S I- 5.C8U5U

THE VALUE OF SOUTHS UESCRIMINAKT = 69.1)7 THE AIRCRAFt IS STABLE

RUN 3 CG. 2M, V»139 KTSt23i) FPS 3/3/62 LEVEL FLITE

THE COEFFICIENTS CF THE GIVEN PCLYNOCIAL ARE

REAL PART 1.0000000 2.776CUS0 IM.01)67662 3.020290e 5.08US293

IMAGINARY PART .0000000 .ooocooc .ccccooo .occccco .ccccccc

TABLE OF ROOTS

ROOT REAL PART IMAG PART PER1CC TIKE TC CAMP TO HALF AfPL NUMBER OF ROOT OF RCCT

1 -.C75930 -.611*1(1)1) 23.S12E SECS 29.9173 SECS 2 -.C75930 .6|i)MUU 22.512E SECS 29.9173 SECS 3 -1.312092 -3.397553 6.C6C7 SECS 1.7313 SECS * -1.312092 3.397553 6.C6C7 SECS 1.7313 SECS

Table H

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BIBLIOGRAPHY

1. Drinkwater, Fred J„ III, George E. Cooper and Maurice D, White An Evaluation of the Factors Which Influence the Selection of Approach Speeds, Advisory Group for Aeronautical Research, and Development, Report. 230, October,, ISF8„

2. Lean, D. , and R. Eaton, The Influence of Drag Characteristics on the Choice of Landing Approach Speeds, Advisory Group for Aeronautical Research and Development, Report. 122,, May, 1957.

3. Anon., Dynamic Stability of an Airplane in Straight. Flight with Plane of Symmetry Vertical, Royal Aeronautical Society, Aircraft 00.00.02, July, 1947.

4. Anon. , Model F8U-2N Airplane Landing Configuration Data, Chance Vought Corporation, Division of Ling-Temco-Vought Inc., Report 2-53300/2L2Ö48, September, !ö62.

5. Richards, D. B„ , Approach. Power Control System - FBU, Specialties, Inc., Itr of 4 September, 1S62,.

6. Anon. , Comparison of Two Automatic Throttle Controllers for the F8U-2NE Aircraft, U.S. Naval Air Development Center, Engi- neering Development Laboratory Report. No. NÄDC-ED-L6292, 30 November, 1982.

APPENDIX I

Derivation of Transfer Functions

The dimensional derivatives shown in Table I were used to calculate

the analytic transfer functions» Calculated values of these derivatives

using F8 data, are shown in Table II.

The longitudinal Equations of Motion written in Matrix Laplace Form

are (neglecting negligible terms)

to ) X AT

-;e o Mu. -(üAvN^ Ä-I^d\&t+)J IfW [W \Ay

The Transfer function (~r~\ Airframe then can be written as

(in determinant form):

0 -X*. ^ 3

IN/ u-

D,

Where D = determinant of homogeneous equations

Expanded D = /\^ -i-ß^ + CA? -*D*- + E

Where A = 1

B = - (Mf +Xu. +Z«^l/oM^)

67

N.

Expanded /\lu_ - BU~A} + C^-d. + D"-

Where Bu = Zf(t~)(ur

Du= g(/VZc-/1ur?ie)

Substituting values for these derivatives

By a similar procedure ,

(Positive elevator deflection is down)

The determinant for calculating /—— J

Mar -ftAlur+Mur) ^-My.

WiKrVaw«- can l-e written as

4^

Expanded:

Ur).

D, _ AT A) ^BTA* +CT4 +DT

A"T//liViW>i

Where

Aj, - j£ ÖT

BT= -X^T[ Z^th^J^LToMt^

CT - XAT[^^MrUoM^+

MAT[UXUT-3]

DT = M&T 3 i Ur

After substituting in values^ » _,

APPENDIX -X

TABLE J-A LONGITUDINAL DIMENSIONAL STABILITY DERIVATIVE

PAH AMETERS

{STABILITY AXIS SYSTEM)

QUANTITY

rs Tsr.r.is OF BASIC STABILITY DERIVATIVES "

IN TERMS OF NON-

DIMENSIONAL STABILITY DERIVATIVE

PARAMETERS DIMENSIONAL NON-DIMENSIONAL

DEFINITIONS UNIT

X. 1

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69

APPENDIX -][

TABLK I~3

LONCITUDINAL NON-DIMENSIONAL STABILITY DERIVATIVES

(STABILITY AXIS SYSTEM)

BASIC NON-DIMENSIONAL STABILITY DERIVATIVES NON-DIMENSIONAL

STABILITY DERIVATIVE

PARAMETERS

TOTAL A III FRAME THEORETICAL HORIZONTAL .

TAIL CONTRIBUTION DEFINITIONS UNIT

r CRAO I T

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79

Append iy J Table JL

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71

APPENDIX II

DYNAMIC STABILITY FORTRAN PROGRAM

Longitudinal - Program LONGSTAB

This program, once entered with required airplane parameters,

computes the longitudinal stick fixed and/or stick free (when pertinent)

stability derivatives, and the corresponding stability equation coeffi-

cients. It then solves these equations for roots, periods, and times to

damp to 1/2 amplitude for both the phugoid and short period modes. In

addition the value of Routh's discriminant is determined. The sequence

of computations can be repeated for as many different flight regimes as

desired.

There are a total of five data cards required for the stick-fixed

only solutions, with an additional two cards required for a stick=free

solution. Thus for a combination stick-fixed and free solution, there

will be a total of seven data cards required for each flight regime

desired.

The initial data card controls the types of solutions desired

(stick-fixed, free, or both), and the number of consecutive runs to be

made. A list of program symbols with their meanings appear in Table I

to this Appendix.

The second card contains an alpha-numeric run identification or

an arbitrary title.

The third card is the first of the general input data cards. This

card should have entered on its

72

St(tail area), Sw(wing area), CL t, "^ tail, Sß (body area), 1

(body length), 1? , g(acceleration).

The parameters are entered eight to a card, each being allowed 10

columns. There are 80 columns on each card» Each parameter may

be entered anywhere in its assigned ten columns, but must have a

decimal point included in it. This same format is followed for each

of the succeeding cards .

The fourth card should have the following aircraft parameters

entered on it:

W (aircraft weight, lbs.); 4rr ; &o in degrees; h, (distance from

2 e.g. to thrust line, ft0); C (coefficient of thrust); C ; Iw (slug ft .);

t mtx iJL

V (forward velocity ft./sec.).

The fifth card should have the following:

CD; CLQ. ; eJTA ; CL wing; c (wing chord); lt (e.g. to a.c. tail, ft.);

Xc.g. (x distance of e.g., ft.); XAC (x distance of wing a.c. in ft.).

Cards 2 through 5 are repeated for each different run, stick fixed.

If it is desired to compute a stick free solution simultaneously, then

two additional data cards (6 and 7) are required for each run.

Card 6 contains the following parameters:

CT (tail chord, ft.); K (radius of gyration, squared)^^; B ; B ; B';

ZVZV • Card 7 contains the following parameters:

2 m\ ' m\ ' mp ^mass elevator, slugs); S (area elevator, ft.).

73

Thus for one aircraft flight regime (say Sea Level, Mach 0.6),

for both stick fixed and free, there would be required data cards

1 through 7. For each succeeding run, cards 2 through 7 would have

to be repeated with the new parameters entered. Card one would have

to be suitably prepared to reflect the total number of runs and mode of

runs desired. See Fig» 1 this Appendix for sample data cards „ Note

the first card is of format 13 for the number of runs , and II for mode of

operation. All other data cards are of 8F10.0 format (each card holds

8 fields of 10 columns , each, one data number to be entered per field, r

in floating point format; i.e., with a decimal point somewhere in the

number.)

The program prints out the calculated stability derivatives as

well as the computed solutions to equations.

74 r

TABLE I

Program mnenonlc

PROGRAM SYMBOLS AND MEANINGS

Program LONGSTAB

Meaning

NOS

MODE

ST

SW

CLALFAT

ETATAIL

SB

BL

RHO

G

W

EPSALFA

THETA1

YH

CT

CMALFA

YI

V

CSUBD

Number of runs to be made. 13 format, card 1

Decision stick, fixed (l), or stick free (2). II. card 1

St„ tail area, card 3.

S. r, wing area. card 3. w

CL . , Tail lift curve slope, card 3.

>J_T , Tail efficiency, CL/CL. card 3.

Sg, Body area card 3.

1B„ Body length card 3.

Q , Atmospheric density, card 3.

g, Acceleration due to gravity, card 3.

w, aircraft weight, card 4.

Ms card 4.

Q-0\ , initial angle of pitch, degrees, card 4,

h, distance from e.g. to thrust line, card 4.

Ct, coefficient of thrust, card 4,

mcx. per radian. card 4 .

I , moment of inertia. card 4.

V, forward velocity, ft./sec. card 4,

CD, coefficient of drag. card 5.

75

CT , winglift curve slope, card 5. CLALFA

EPIA £TfA, card 5.

CSUBL

G

TL

XCG

XAC

CTAIL

XKE

UE

Bl

B2

B2PRIME

ZETA

ZETADOT

AMETA

AMETADT

XME

SE

CL wing' card 5 *

c, wing chord, card 5

1., e.g. toa.c. tail, card 5.

x , x distance of e.g. card 5. C • y •

x= „ , x distance of wing a.c. card 5.

CL, Tail chord, card 6.

2 k , radius of gyration squared, card 6.

LL^ 5 relative elevator density, card 6.

card S.

card 6.

card 6. B-, 1/2 C. .

■•Lx , card 6 .

"*}■£ , card 6.

>»1* , card 7.

"tolJ , card 7 .

m , mass elevator, card 7. e

S , area elevator, card 7.

76

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/

. 7S

I Fortran Program L0NG5T/4B

„iJCe EELL»LCNGST*E 1.5 MINUTES PROGRAM LCNGSTAB

C PROGRAM FCR THE COMPUTATION OF THE PCCTS CF THE CH ARACTfR IS"! 1C C DYNAMIC LONGITUDINAL STABILITY (STICK FIXEC) EQUATIONS

CI MEN SIGN KAPPA UC)',CPR( 129), CPU 129) ,ROOTR( 1 28),ROOT I ( 126) DIMENSION CRR(129),CRIl 129 ),XR( «4) , XM 5 ) , FXfl Ik )',FXItl4) ,SR (3 ) ,S*:<3 ) READ 66, NOS. MCDE

66 FORMAT «13. II ) DO 777 J «1, NCS ICOUN1 * 0

OCALL INPLT1 (CLAU F A »EP1 A ,C, Tit XCG • XAC »ST» SW.CLAtFATf.ETAT A Hi $e,EL, 1 RHO,G,W, EPSALFA, TFETA1,YH,CT,CMALFA.YI, V.€SU£EtCSUfiL»KAPPA )

C COMPUTATION OF STABILITY DERIVATIVES 0 = .£ »PHO « V«»2 TCARAT « H /(G«RHC*V«SW) THETAC » THETA1 / 57.2957795 XU « -CSLBO XW « .5«CSUBL*( l.C-(2.0«CLALFA/EPIA)) ZW = -.5»(CLALFA ♦ CSUeD ) APRIME « .Ul4 + 2.CM (XAC-XCGJ/C) EPRIME = -.58 - CLALFA » APRIME « {(XAC-XCG)/C) UINGMC * 0.25 *((C/TL)«*2)«BPRIME TAILMC * -.50« CLALFAT * (ST/SW) • ETATAIL BODYMC - -0.01* <SB/SW)*((EL/TL)**2 ) TOTALMQ = WINGMC * TAILMC + EOOYMQ YIB = YI/(U/G)«TL»«2) WINGZC * -.25»CLALFA»(C/TL)*(.Ul4+(XAC-XCG)«2.0/C) TAILZC= -.5«CLALFAT«(ST/SW)«ETATAIL U1 ■ fc/tG*RHO»SW*TL) BARMWCT = -,5*CLALFAT*ETATAIL«{ST/SW)«EPSALFA TOTALZQ = WINGZC + TAILZQ ZU = -C5UBL XTFETA « -.£»CSUBL ZTHETA = -.5»CSUBL«TANF(THETAC) AMU = -CT «YH/TL AMW = .5 * CMALFA • C/TL PRINT 5C5.ST, SW» CLALFAT »ETAT AIL» SB, BL,RHC,G,W^EPS AtlFA,THET Al, YH ,

1 CT.CMALFA,YI 5050FORMAT (1H1 ZHOU INPUT PARAMETERS //

111H TAIL AREA= F8.1| ,2X1 1 H WING AREA= F9.4,2X,14F CL ALFA TML« 2 F9.6,2X»1CH ETA TAIL= F9.6,2X,11H BCDY AREA= F9.2 / 3 13H ECCY LENGTH= Fe.5,2X, SH RHO= F1C.7.2X, 5H GEE= F8.£,2X, M 8H WEIGHT* F 12 .U, 1 X,3HLBS 2X,18H D EPSILCN/D ALFA* F9.6 / 5 21H INITIAL PITCH ANGLE= F9.6, IX »7HGEGREES 2X.22H THRAJST LINE CI 6STANCE* F8.5, 1X,«*HFEET 2X,16H THRUST CCEF CT= F8.6 / 7 8H CMALFA= F9.6,2X,1CH I SUB YY= F 12.3,IX,15HSLUG FT SCUAREC//) WRITE CLTPUT TAPE 5, 101

1C1 FORMAT {/// 47H STABILITY DERIVATIVES //) WRITE OUTPUT TAPE 5, 1C5 ,V,TCARAT,C

105 FORMAT (19H FORWARD VELOCITY* F12.3,1CH T CARAT« F1C.6, Mh C« 1 F 1M.U / ) WRITE CLTPUT TAPE 5, 106, CSUBL, CSL'BC

106 FORMAT ( 9H C SUBL= FlC.6, 9H C SUBD= F1C.6 /) WRITE OUTPUT TAPE 5, 1 C2,XU, XV,, XQ, XTHETA ,ZU,ZW »TCTALZC

1020F0RMAT (5H XU= F7.i|,6H XW = F7.5,6H XC= F7.U,10H XTKETA* F7.5, 1 6H ZU= F7.5, 6H Zfc= F7.«J,12H TOTAL ZC= F7.»4 /) WRITE OUTPUT TAPE 5, 1 C3,ZTHETA,AMU,AMW,BARMWCT, TCTALMC,U1,YIE

1030F0RMAT ( 9H ZTHETA* F7.U, 6H MU= F8.6, 6H MW = F7.5,1UH MEAR W C 10T* F7.U, 6H MC* F7.U, 9H MLONE* F7.3.13H I SUB BETA» F7.U //) OCALL PREROOT (XU,XW,XQ,XTHETA.ZU,ZW.TCTALZCtZTHETA.AMU,AMW,BARMWCT 1,T0TALMC,U1,YIB,TCARAT,Al,Bl,C1,Dl,El, CRI,CRR ) ROUTH = CRR(2)« CRR13)« CRR(U)- CRR(l)« CRK(U)«*2-CRR(2 )*«2«CRR<5) WRITE OUTPUT TAPE 5, 108, ROUTH

108 FORMAT (////35H THE VALUE OF ROUTHS DESCRIMINANT * F10.2 ) IF (RCUTH) 11,11, 12

11 WRITE OUTPUT TAPE 5, 109 1C9 FORMAT (28H THE AIRCRAFT IS UNSTABLE /?/ )

GO TO 13 12 WRITE CLTPUT TAPE 5, 11C

110 FORMAT (26H THE AIRCRAFT IS STABLE 411 ) 13 CONTINUE

N « J* CALL FOLYHUL (CRR ,CRI,N,KAPPA,TCARAT)

APPENDIX II, Fig. 2

80

DECISION FCR LONG CR LATERAL, STICK FIXEC (999 )',OR STICK FREE(2C1) GO TO (777,201,202,2CU),NCDE

2C1 CONTINUE WRITE CLTPUT TAPE 5,301

3010F0RMAT(//// 80H STICK FREE LONGITUDINAL CYNAfUC STABILITY CCEFF 1ICIEN1S //// ) WRITE OUTPUT TAPE 5f, 302, KAPPA

3C2 FORMAT ( 1CA8 ) OCALL INPUT 2 (CTAIL'fXKE^UE,81,B2,B2PRlME,ZETA,ZETADCT,AMET/i 1 AMETADT )

XKE = SCRTF(XKE) ZQ = TCTALZC

AMQ ■ TCTALMQ OCALL PRER0T2 (CTA IL?. EPS AL FA, XKE, YIB ,U 1,UE,TL , EUB2,82PR IME,ZW, 1 ZTFETA,ZQ,ZETA,ZETADCT,AMW,AMCUBARMWDT,AME<TA,'AKETACT, CRR,CRI )

N = 5 CALL FCLYMUL (CRR,CRI,N,KAPPA,TCARAT)

777 CONTINUE END FILE 5 ST0P2

202 STOPU 2C3 ST0F5 20U STOP«

END

O.SUBROLTINE INPUT1 (CLALFA ,EPIA ,C ,TL, >CG ,XAC ,ST ,SK.CL/»LFAT, ETAT AIL , 1 SB,.BL,RFO,G,W,EPSALFA,THETAl , YH, CT ,CMALF/l , Y I .V, CSU8C ,CSUEL'# K AFCA ) DIMENSION KAPPM1C) ,CPR(129),CPI( 129 ) , ROOTR 11 28 ) ,RCET I ( 1 28 ) OIMENMCN CRR(129),CRI( 1 29 ) ,XR ( 4 ), XI ( H ) ,FXP (»4)', FX I (4 ), SR ( 3 ), S 1(3 ) READ £5* KAPPA

1CU FORMAT t 8F10.C ) READ lOl4,ST,SW,CLALFAT,ETATAIL.SB,BL,RFa,G READ 1C»4, W,EPSALFA,THETA1,YH,CT,CMALFA,YI,V READ 1C4,CSUBD,CLALFA,EPIA,CSUBL,C,TL,XCG,XAC

65 FORNAT ( 1CA8) RETURN END SUBRCLTINE PREROCT ( XU,XWf XC , XTHETA. ZU, ZW'.CTCTALZQ , ZTHETA, *KU, ANW , ieARMWCT«TOTALKQTUl.YIBTTCARATfA1,B1fC1t01f.E1t CRI, CRR) DIMENSION KAPPA(10),CPRt 1 29 ) ,CPI ( 129 ) , PCCTR ( 1 26 ) .RCCTK 128) DIMENSION CRR(129),CRI(129),XR(U),XI(li),FXR(i4 WFXI(»4),SR(3), SI (3)

1 fi^ = l.C ZQMU = (1 .0 +ITCTALZC / Ul))

2 El = -liXU + ZW)-<TCTALMQ/YIB)-(ZQMU * EARMKCT/YIB ) 3CC1 » (XL« ZV> - XW« ZU ) ♦ (TCTALMC/YIE)»(XU* ZW) +( XU»ZCMU - ZU « KXQ/U1) - ZTHETA)« I BARMWDT/Y IB ) - ( Ul «/»MW/YIB )* ( ZQMU + XG/U1)

. UCD1 = -(TCTALMC/YIE)« (XU»ZW - XW * ZU) + (-*THETA» ZU ♦ ZTHETA « 1 XU JMBARMWDT/ Y ie ) ♦ (U1*AMk/ YIB )« { XU* ZCNU -ZU« XC/U1 -ZflTHETA)* 2 (Ul»/»MU7YI8)«(-XTHETA -XW* ZCMU + ZW «XQ/U1 )

5CE1 = (Ul«AMk/YIB)*(-XTHETA«ZU *ZTHETA*XU)-{Ul«AKU/YIB)«i-XTHETA« 1ZW + ZTHETA« XW )

WRITE OUTPUT TAPE 5,101, AltBl,C1,01.El lClCFGRMAT (/3H A* 1X.F13.5,»4H B* 1X.F13.5, **H C« 1X.F13.5, UH C«

1 1X,F13.5,UH E« 1X,F13.5 //) CRR(1 ) = Al CRR(2) * Bl CRR(3) * Cl CRRC4) = Dl CRR(5) * El CRK 1 ) = 0.0 CRK2) = C.C CRK3) = C.C CRI(»4) = O.C CRI (5) = 0.0

RETLRN END

0 SUBROUTINE INPUT2 (CTA IL *XKE ,UE,B1 , B2, B2FR IM£ , ZET A, ZiET ACCT , AM ETA', 1 /METADT ) SUBROUTINE BRINGS IN STICK FREE FACTORS

OREAC l,CTAIL,XKE,UE,Bl,82,B2PRIME,ZETA,ZE7ACCTCflAMET;A,*KETACT,XME,

1. FORMAT ( 8F1C.C /UF1C.0)

81

WRITE OUTPUT TAPE 5!,2 2CF0RMAT(V///1CX70H INPUT PAR/METERS ETICK FREE

1 ) END

SUBROUTINE PREROT2 (CTAI L , EPS ALFA, XKE , VIB .m.UE.TL.e 1 ,E2, E2PP IKE, Z 1h,ZTHETA,ZQ,ZETA, ZETACOT, AMW, AMC, 8ARMWOT ,A>ETA, AMETADT,A, E )

; SUBROUTINE CALCULATES STABILITY STICK FREE GU-INTIC COEFFICIENTS CIMENSICN AM29), B1129) UFACTCR = 2.0» UE /(U1*«2) UFACT = LFACTOR FACTORK =(XKE / TL)««2 XKFACT ■ FACTCRK

CTFACT = CTAIL / TL ZFACT = (l.C + <ZC/U1 )) EPSFACT = 1.0- EPSALFA WRITE OUTPUT TAPE 5,2C01, UFACT,XKFACT.CTFACT,ZFACT*EPSFACT

20G10FORMAT (M9H FACTOR CALCLLATICNS // 1 9H L'FACTOR = Fl»4.7, 9H KFACTOR= F14.7, ICH CTFACTCR* FU.7, 210H ZCFACTCR= Fl ^. 7, 16H EPSALFA FACTCR= FH).7 771

; A CCEFFICIENT CALCULATION A(1)=LFACT »XKFACT « YIB

: B CCEFFICIENT CALCULATION 0A(2)=LFACTOR« XKFACT»( BARMWDT • (ZETACCT«CTFACT/U1-ZFACTJ-AMC-ZW* 1 YIB + AMETADT« CTFACT ) - B2PRIHE • CTFACT • VIE / Ul

: C CCEFFICIENT CALCULATION CA(3)=LFACT »XKFACT «( ZW «AMC + AMW»( ZETACOT «CTFACT- U1«<ZFACT) 1 ) + EARfWDT «(ZETA - ZTHETA1+ Ul * AKETA - APETADT * CTFACT «ZW ) 2 + B2FRIME • CTFACT / Ul*( AMC + ZW«YIE + ZFACT « BARMWDT ) - E2 » 3 YIB -Bl«( AMETADT «CTFACT/U1 ♦ ZETAOCT «CTFACT/L'1 «( YIE«EPSFACT 14 + BARKUCT /Ul))

; D CCEFFICIENT CALCULATION CAU) = UFACTOR » XKFACT *( U1«AMW«(ZETA-ZTHETA)- U1«AMETA«2W ) 1 -B2PPIME * CTFACT /LI «( ZW *AMQ - Ul«AMfe» ZFACT - ZTHETA • 2 BARMfcCT )+B2«(AMG + ZW « YIB + BARMWDT * ZFACT)-Bl•<AKETADT«CTFACT 3 »(EPSFACT « ZFACT - ZW/UD + ZETAOOT * CTFACT/U1»( AKW - AMC « U EPSFACT )+ ZETA »(BARKWDT/Ul + EPSFACT * VfB ) + AMETA )

: E CCEFFICIENT CALCULATION CA(5) = B2PRIME * CTFACT /Ul *U1 « AMW» ZTHETA ♦ E2«( Z'THETA*EARMWO IT - ZW • AMG + Ul » AMW • ZFACT ) + El«( ZETA «(EPSFACT * AMC ■" 2 AMW) + AMETA «( ZW -U1*EPSFACT * ZFACT )- AMETADT * CTFACT * 3 EPSFACT « ZTHETA )

I F CCEFFICIENT CALCULATION 0A(6) = Ul* ZTHETA» (B2* AMW - AMETA « B1«EPSFACT ) CO ICC J=l,129

100 B(J) = CO WRITE CLTPUT TAPE 5 ,3,CTAIL,XKE,UE,B1,E2

3CF0RMAT (/// 13H TAIL CHORC = F7.3,1X, 8HK SUB E» F9. 5, IX, 9KMU .SUE 1E= F9.5*IX,8HB SUE 1= F9.5,1X, 8HB SUE 2= F9.5 ) WRITE OUTPUT TAPE 5 ,»4 ,B2PR IME ,ZETA, ZETADOT , AMET A , AMETADT

4CFCRMAT(/1UH B SUB2 PRIME= F9.5,1X, 11H Z XUB ETA= F9.5,lX, U5H Z SUE ETA COT= F9.5,1X,11H M SYB ETA= F9.5,1X,16H K ISLE ETA 2 DOT= F9.5 ///) Cl = UFACT » XKFACT »( Ul » AMW « (ZETA -ZTHETA)- *Jl«AMETA« ZW) C2 = -B2PRIPE /Ul * CTFACT*(ZW *AMQ-U1«AMW «ZFiACT -ZTHETA,«EAPMWCT) C3 = E2«(AMC +ZW»YIB + BARMWCT« ZFACT)

004 = -B1»(AMETADT«CTFACT«<EPSFACT*ZFACT-ZW/UI)+ZETACCT/Ul«CTFACT»< 1 AMW - AMC»EPSFACT) +ZETA»( BARMWDT/U1 +EPSF AJCT«YI B) +AMETA) C = D14D2 +C3 + DU A(U) * C

60F0RMAT (// 4H Dl= F15.8, 4H D2= F15.8, Uh D3 = F15.8,4H DU- F1S.8 1 , 9H 0 TCTM.= F15.8, // ) El =E2PRI>r 'CTFACT »(U 1«AMW»ZTHETA) E2 =B2 •«.'. - BARMWDT -ZW«AMQ +U1 «AMW« ZFACT )

0E3 =B1 »(ZE t««{EPSFACT»AMQ -AMW l+AMETA«(ZW-U1*EPSFACT«ZFACT)- 1 AMETACT»CTFACT» EPSFACT « ZTHETA) E = El + E2 +E2

70FORMAT {// 4H E1= F15.8,UH E2=? F15.8,l<H E3= F16.8, ,9H E TCTAL* 1 F15.E, //) 0C1 =UFACT «XKFACT *(ZW «AMQ+AMW« ( ZETACCT«CTFACT-U1 »ZFACT HEARMWCT 1 «(ZETA-ZTHETA)+U1«AMETA-AMETADT «CTFACT« ZW ) C2 »B2PRIKE/U1»CTFACT«(AMQ +ZW*YIB +ZFACT« €ARMWCT)

CC3 * -B2 «YIB -B1«(AK£TADT/U1 «CTFACT* ZETJSC0T/U1*CTFACT« Of IE *

82

1 EPSFACT ♦ BARMWCT/Ul) ) C = C1 ■* C2 ♦ C3 WRITE OUTPUT TAPE St 8, C1, C2, C3, C WRITE OUTPUT TAPE 5,6,D1,C2 ,03,04,6 WRITE OUTPUT TAPE 5,7, E1,E2,E3,E

8CF0RMAT (/// UH C1 = F15.8, UH C2» F15.8, 4H C3 = f15.8,9H C TOTAL« 1 F15.8, // ) RETURN END SUBROUTINE POLYMUL 1CRR,CRI,N,KAPPA,TCARAT)

CIMENSICN KAPPM1Q)tCPR(129)tCPI( 129) .ROOTRJ 128 ) ,RCCT I M2§ ) 01 MEN SIGN CRR(129),CRI(129),)<R<M»XlU>,FXRi»4T,FXlU),SR(3)iSI(3) IMAX »50 DEL * Cl NUM « 5 RATIO =5.0 ALTER »1.C0CC01 EP1 =.CCCOOOOC00OCCCCC00C01 EP2 =.CCCOO0C1 EP3 =.0CCCGC1 EPU =.CCCCCC01 SR( 1)» -.5 SI(1)= CO SR(2)» C.5 SI(2)= CO SR(3)= CO sim« CO MODE * 1 NNN = N+1

749 IF (CRP(D) 750,751,750 751 CO 753 L=1,NNN

M = L +1 CRR(L) = CRR(M)

753 CONTINUE 752 N = N-l

GO TO 749 750 CONTINUE 65 FORMAT (1CA8)

DO 5 1=1,1C KON=£F IF(K/>PPMI)-KCN)2,5,2

5 CONTINUE GO TC 999

2 PRINT 65,KAPPA 66 FORM*T(2X,2HN=I4,12X,5HIMAX=I3,10X,UHNUK=I3,1lX,m-DEL=E8.2{ 661 6X,6hRATIC=E8.2,4X,6HALTER=F10.8/ 662 2X,UFEPl=E8.2,6X,4HEP2=Ee.2,6X,4HEP3=E8.2,6X,4HEP4«E8.2/>

M0CE=MCDE+1 67 F0RM/JT16H SRI« E12.5.6H SI 1= E12.5,6H SR2 = E12.5,6K SI2 = 671 E12.5,6H SR3= E12.5.6H SI3« E12.5)

NP1=N+1 710F0RMAT (46HCTHE COEFFICIENTS OF.THE GIVEN PCLYREMIAL APE ///

1 11H REAL PART ) 501 WRITE OUTPUT TAPE 5,71 681CF0RMAT(11CH A B C E £

1 F G H /) 687 CONTINUE

WRITE OUTPUT TAPE 5, 680,(CRR(J),J=l,NP1) 680 FORMAT ( 1H 8F15.7 / )

WRITE OUTPUT TAPE 5, 682 682 FORMAT (16H IMAGINARY PART )

WRITE OUTPUT TAPE 5, 680,(CRI<J),J=1,NP1) 5C2 DO 6 1=1,NP1

CPR(I)=CRR(I) 6 CPI(I)=CRim

CALL COMAGtCRRM),CRI(1),C1,KE) DO 3C5 K=1,N GO TC(9,609),MODE

53 FORMAT!1F02X4HRC0T5X7HITERANT6X2(10HREAL PARTICXlOFIMAG PART10X) 1,10HRC0T IS IN/2X6HNUMBER4X6HNUMBER7X2(9H CF -ROOT 1 IX J ,2( SV. CF R( 2Z)11X),11HA RADIUS CF) :(M 1AM I PRINT 53

83

6C9 1=1 NP1=N+2-K NN = NP1-1 IF{K+1-N)13.12.1 1

11 CALL DIVD(-CRR<2),-CRI(2),CRR<1),CRI( 1),XR.(1)VXI< 1>,KE) ■ K1 = 1 K2 = l GO TC 16

12 AR = CRRM) AI=CRI (1) BR=CPR(2)

CR'CPRII) . CI=CPI(3) Kl = 1 K2=l MU = 2 GOTO 123

114 XR(1)=CEARR XI(1)=CEARI GO TC 16

13 DO 15 J = l,3 XR(J)=SR(J)

15 XI(J)=»SHJ) Kl »1 K2 = 3

16 M1=l M2=1 M3=1 MU*1

700 DO 717 L=K1,K2 701 ZR=XP(L) 7C2 ZI=XIU)

CALL PCLYNCM(NP1 ,ZR,ZI*CRR,CRI,RR,RI,KE ) 710 FXR(L)=RR 711 FXI(L)=PI

CALL CCNAG(RR,RI»PFAG.KE) 713 RA0=ALTER»CPMAG/C1)«*(1.O/FLCATF(NN)) 71U GO TC <715,19,17S ) ♦f3 715 GO TC(716,718),MODE

55 FORM/T< 2(16, MX), ME2G.11, 5X, E1C>» ) 716 PRINT 55, K, I,ZR,ZI,RR.RI,RAC 718 POLYC=PCLYN

POLYN=PPAG IF(PKAG)717,300,717

717 1=1+1 IF(K+1-N)17,30C,3C0

17 GO TC ( 18,200 ,M 1 18 VAL=CEL*PCLYN

DBARP=RAD D8ARI=0.0

19 K1=U K2=U M1*2 M3*1

101 ABARR = XR(1)-XR(3 ) 1C2 ABARI-XI I1J-XH3 ) 103 BBARR=XR(2)-XR(3) 1C14 B8ARI=XI(2)-XI(3) 105 APIBR = XR{1 )-XR(2 ) 106 AMIBI=XI(1)-XI(2 ) 1C7 CALL MULT(ABARR,ABARI,BEARR,eeARI,TA,Te,K£l) 1C8 CALL KLLKTA.TB, AN I ER,ANIBI,DENR,DENI,KE2)

CALL CCNAG(DENR.DENI,TA,KE) CALL CCNAGIXR(3) ,XI (3)tT«4,KE) IF(T/-EP1»TU) 110,1 10,111

110 CALL CERIV(NPUXR(3),XI(3),CRR,CRI,DR,0I,K£0) CALL CCPAG(RR,RI,TR,KE) CALL CONAGtCR.CI ,TC,KE) IF (TR-EPi4«TC) 192,171,171

111 DELAR=FXR(1)-FXR<3) 112 DELAI=FXI(1)-FXI ('3) 113 . DEL8R=FXR<2)-FXR<3)

84

11U DELBI = FXH2)-FXI<3) 115 CALL fULTfEBARR, EEAR I,DELAR,CELA I ,TA,TB, KE5) 116 CALL NULT( ABARR, />E/>R I ,CELER , CELB I, TC .TD.KIE6) 117 CALL CIVD(TA-TC,TE-TC.DENR,DEN I ,AR,A I,KE7) 118 CALL PULTtABAPR, />E*RI,TC,TD,T1 ,T2,KEE> 119 CALL MULT(EBARR,EBAR I , TA ,TB ,T3 ,T»» , KES) 120 CALL CIVD(Tl-T3,T2-Ti4,OENR,DENI,BR,BI,KElO)

CR=FXR(3) CI-FXK5)

123 CALL PULT(ER,BI,ER',EI,T1,T2,KE11 ) 12U CALL NULT(AR,AI, CR,CI,T3,T1*,KE12) 1*1 TA=Tl-U.C»T3 132 TB»T2-»».C»T1»

CALL CSCRT(TA»TB,TC,TD) U7 T1=-ER + TC U8 T2*-EI + TD 11*9 T3=-ER-TC 15C Ti4=-EI-TD

CALL CCPAG(T1,T2,TA,KE1»4) CALL C0NAG(T3,TU,TB,KE15)

153 IF(TA-Te) 15*» * 168,168 15U TA=TE 155 Tl«T2 156 T2*TH 168 GO TC ( 157,159),MU 157 IF(TA)161, 161, 156 158 CALL COMAG(2.0*CR,2.0«CI,TB,KE16)

IF(TE-RAD«TA)159,159,18C 159 CALL CIVD(2.0«CR,2.C*CI,T1,T2,DBARR,DBARI,KE17)

GG TC ( 161.Hi) ,M»4 161 XR(1*) = XR(3) + CBARR 162 XI(l*) = XI(3) + DBARI

TR = AESF(XR(i4) ) TI»AESF(XI(1*)) IF(TR)167. 167,1UC

11(0 IF(TI)U7,167t169 169 IFUR-TI )16U, 167,162 163 IF(TI-EP2»TR)165,167,l67 165 XI(i4) = C.C

GO TC(5C3,50»O,MCDE 56 FORf/!T(U0HCITERANT ALTERED TO BE PURE REAt NUMeER.)

503 PRINT 56 5C1» GO TC 167 16U IF(TR-EP2*TI)166,167,167 166 XR(i4)=0.0

GO TC(5C5, 167),MCDE 57 F0RM/»T(i*5HCITERANT ALTERED TO BE PURE IMAGINARY NUMBER.)

5C5 PRINT 57 167 GO TC 7CC 180 CALL DIVD(T1,T2,TA ,C.C,T1,T2,K20)

CALL DIVD(CR,CI, TB , CO ,CR,CI, K2 1 ) CALL MULT(CR,CI, RAC,C.0,CR,CI,K22) GO TC<5C6,507) ,MCCE

58 F0RP/1T(87H ITERANT IS CUTSICE CIRCLE WHICH BOUNDS A ROOT. INTERP 5810LATE ITERANT TO ECGE OF CIRCLE.)

5C6 PRINT 58 507 GO TC 159 171 CALL CGMAGtABARR,AE0R I ,T 1 ,KR1)

ENA(l) STA(ITA) ENA(3) STA(ITB) IF(T1-EP3«T1*)17U,171*,172

172 CALL CCNAG(B8ARR,eE/>RI,T2,KR2) ENA(2) STA(ITA) ENA(3) STAUTB)

IF(T2-EP3«Ti()175,175,173 173 CALL CCF AGUMI8R , £M I 8 I,T3,KR3)

ENA(l) STA(ITA) ENA(2) STA(ITB) IF(T2-EP3»TU)17U,17»4,in

171* ENA(l) STA(Kl) STA1K2) ENAC2) STACM3) SLJC176) 175 ENA(2) STA(Kl) STAIK2) ENA(3) STAIM3) 178 XRi(Kl)=XR(Kl )»(1 .C+2.0«EP3)

XIKK1 ) = XI(K1 )«<1 .C+2.0*EP3) GO TC(5C8,5C9) ,MCDE

6C FORMATUlHCITERANTS XII,6H AND XII,HlH ARE <TOC CLOSE TOGETFER ALT 1ER ITERANT X I1',1H.)

85

5C8 PRINT 6C,ITA,ITB, ITA 5C9 GO TC 7C0 179 PCLYC=PPAG

GO TC 19 2CO IF<PCLYN-VAL)201,2C1,210 2C1 VAL=CEL«POLYN 2C2 LIM=I+NLM 203 M2=2

CALL CERIV<NP1,XR(4),XI (4),CRR,CRI,DR,01,KdO) CALL CCKAG(RR,RI ,TR,KE) CALL C0NAG(DR.0I ,TC,KE) IF<TR-EP4*TC)192,21C,21C

210 DLT«RATIO»PQLYO/PCLYN 211 IF(1.0-CLT)22C,22G,212 212 CALL MULT(CBARR,DeARI,0LT,0.C,DBARR,DeARI,K3C) 215 LIM=LIM+1

GO TC (510,5111,NCCE 72 FORMAT«120H0PCLYNCMI AL HAS INCREASEO IN MAGNITUDE TCO KUCH WITH CU 721RRENT STEP. THEREFORE REDUCE CURRENT STEP. )

510 PRINT 72 511 GO TC 161 220 GO TC (221.231>iM2 221 IF(I-INAX)250,250,222 222 PRINT 62

CO 68l< J=1,NP1 684 CRR(J) = ICC * CRR<J)

ICOUNT = ICCUNT ♦ 1 IF ( ICCUNT - 3 1 665,685,777

685 fcRITE OUTPUT TAPE 5,686 6860FORMAT ( 55HCTHE REVISED COEFFICIENTS CF TEH <UVEN POLYNOMIAL ARE

1 ///> K = 1

GO TO 6£7 62 F0RMAT169H0MAXIMUM NUMBER OF ITERATICNS R£»0€B WITHOUT REDUCING 621P(Z) EY DELTA.)

513 GO TC 777 777 RETURN 231 IF( I-LIK)250,250,300 250 DO 251 L = l,3

XR(L )=XR(L+1) XKD-XI (L + l ) FXR(L)=FXR{L+1)

251 FXI(L)=FXI(L+1) GO TC 1C1 GO TC(514,200),MODE FORMAT119H0FIRST DERIVATIVE ( E17.9,E20.9,S5H) INDICATES THAT ITE

1RANT IS SUFFICIENTLY CLOSE TC ROOT.) PRINT 69,DR,CI DO 3C2 J=2,NN CALL FULT(ZR,Zi;CRR(J-l),CRI(J-l),TR,TI,K4C) CRR(J)*TR+CRR(J) CRI(J)*TI+CRI(J) FORMATI59H0THE COEFFICIENTS OF. THE REDUCED POLYNOMIAL ARE AS FELL

ICWS ) R00TP(K)=ZR R00TI(J<)=ZI GO TC(3C3,305),MCCE PRINT 63 F0RMATJ1H 6E18.9)

PRINT 68,(CRR(J), J=1,NN) PRINT 66, (CRKJ ), J-1.NN) CONTINUE GO TC 1515,516),MCDE PRINT 75 PRINT 8C

8CCF0RMAK 1HC42X14HTABLE OF RCGTS//3X4HRCCT8X ICHREAL PART10X lGKiMA IG PART 10X 6HPERI0C 10X 26HTIME TO DAMP JO HALF AMPL 2 /2X6HNUMEER7X2(9H CF R00T11X) )

DO 3C6 1*1.N CALL PCLYNCM(N+1 ,R00TR{ I ) ,RCOT I ( I), CPR, GP'ItRR t* It KtE ) THALF = -.6931«47ieC6/R00TR(I)« TCARAT PERTOC » 6.2831853C72 /ABSF(RCOTI(I)) « TCARAT

3C6 WRITE OUTPUT TAPE 5, eU, I, RCOTR( II , RCCTIH > »eERIGC, THALF

86

192 69

514 300

302 63 63

3C3 68

304

3C5

515 516

8U FORKAI (I6,»4X,Fl6.6,3X,FU.6,7X,FlC.I|t8H S-ECS 5X',F IG.U.SH SECS) 83 FORMATU6, MX, 4E2C.11)

PRINT 75 75 FORMATMHl)

IMAX = 50 NUM = 5 CEL = C.l RATIO =5.C ALTER =1.C0C001 EP1 =.CCCCCC0000OCCCCCCGCC1 EP2 =.CCC0C001 EP3 =.0000001 EPM -.0CCCCCC1 SRI 1)= -.5 SI I 1)* 0.0 SR(2)= 0.5 SI(2)= C.C SR(3>= CO SI(3)= 0.0 tfOOE = 1 GO TC 1

1 RETURN S99 CONTINUE

END SUeRCUTINE DER IVCN,ZR,ZI,CR,CI,OR,DI,KER) DIMENSION CR(129),CI(129)

ENA(C) STA (OR) STA(DI) STA<RR> ST*<RI )ENA<1 )STA (KER ) . DO 2 J=1,N CALL KULT(ZR,ZI,RR.Rl.TRR.TRI,Kl) CALL KULTtZR.ZI,CR,CI,TCR.TO I,K2)

RSC(Kl) AJPKL + 2) RSC<K2) AJP ( 1 ) ENA<2) ST AKKER 1SLJ (3 } . 1 DR=TCR+RR

DI=TCI+RI RR=TRR+CR(J)

2 RI = TPHCI(J) 3 CONTINUE

END SUERCUTINE PCLYNOKfN.ZR.Zl,CR,CI,RR,RI,KER) DIMENSICN CR(129)',CI(129)

ENA(O) STA(RR) STA(RI) ENA(1) STAIKER) DO 2 J=1,N CALL NULT(ZR,ZI,RR?RI,TR,TI,K1)

RSC(Kl) AJP(l) ENA<2) STA(KER) SUÜ(3> 1 RR=TR+CR(J) 2 RI«TI4CI(J) 3 CONTINUE

ENC SU8RCUTINE CSCRT(XP.XI,YR,YI )

CCN(SQ2=1.MU2135621*>. LCA(XR) LDQ(XI) AJP2U + 1) LAC(XR) CJP21L+1) LCC(XI) STA(A) STQ(B) CJPK1) STG(C) SLJM18) +STA (P )SLJ (£ ).

1 AJPK2) LDA(B) SLJU18) +FDV(SC2) STA(P) SSKOI) SLJ(L*2) LAC(P) +STA(C) SLJ(5)

2 THS(E) SLJ<3) STC(S) FCV(fi) STA(T) STA(R)SLJH») 3 STA(S) LDA(8) FCV(A) STA IT) LCA(I.C) ST A (R 1 SLJ (*i) 7 Y'SQPTF(X) 8 SLJ(*> STA(X) SLJ(7) U LCA(T) FMU(T) FACH.O) SLJM8) +F/SDIR) FCV(2;C)

SLJM8) +STAIT) LDA(S) SLJ<4(6) + FJ't<T) STA(P) LCA(XI) FCV12.C) FCV(P) STA(C)

5 SSKJXR) SLJ(6) LDA(Q) LCC(P) AJP21L+1) LAC(C) SSK(XI) LDQtP) STA(YR) STC(YI) S1UU+3)

6 LCA(P) LDQ(C) STA(YR) STQ(YI) END SUßRCUTINE CCKAG(XR.XI»ZtKER)

LCA(XR) LDQ(XI) AJP21L+1) LAC(XR) QJP2(L*1) LCC(XI) STQ(T) +THS(T) LLS(*48) CJP(2) STC(T) FCV(T) STA(H) FMU(H) FAO(l.O) STA(H)

Y-SQRTF(H) -FfUtT) +EXF7(lHlB)SLJ<L+2) ENC (1 ) SLMi3) ?ENC(2)

3 STQ(KER) STA(Z) END

.SUBRCUTINE DIVCUR.XI , YRt YUZR#ZI.KER )

f

87

CALL PPCD(XR,XI,YR^-YIfBl,B2fPRfPI,DR1DI)' LCA(B2) AJPKl) ENA<3) SLJ(3>

2 Ei\A<2) SLJ<2> 1 T*OR«DR*CI«CI

LCAlEl) -FDV(B2) +EXF7<141B)SLJ<2) STA(Bl) LCA(PR) FDV(T) -FNU(El) + EXF7 U «41 E ) S LU< 2 ) StfA(;ZR) LCA(PI) FCV(T) -FPU(El) + EXF 7 < 14 IE )SL\J 12) STA<21)ENA{1)

3 STA(KER) ENC SUERCUTINE PUL T (XR ,X I. YR, YI, ZR.ZI . KER ) CALL PRCOlXRiXIiYRtYI,8 1 ,B2»PR ,P 1,01,C2 )

LCAIB2J -FHU B] ♦EXF7{141B ) SLJ(1) STAIE1) LCA(PR) -FNU<E1> + EXF7M41BJSLJ (1 ) 5TACZRJ LCA(PI) -FMU(EI) ♦EXF7(l4lB)SLJ(1) ST*(ZI) ENA(l) STA(KER) SLJIL + 2)

1 ENA(2) STACKER) ENC SUERCUTINE PROC<XR,XI,YR,YI,B1 ,82,PRtPitCR,CI) CALL NCRK(XR,XI,81f,AR,AI) CALL NGRP(YR,YI,B2,CR,DI) PR=AP«DR-AI»CI PI=AI»DR+AR»CI ENC SUBROUTINE NCRN(A 1,A2,61,SI,S2>

1A SLJ(l) +SEV7(70000B) ZRC(C) + ZPC< iiOOOE )ZRO(C) 1 LOAMA+l) LDQ(Al) 0JP2CL+1) LGC(Al) STL(E) LCCCA2J

QJP2(L+1) LQC(A2) LCL<1A+1)+THS<E> SLJtL+2) LCA(E) ♦ AJPKL + 2) STA(Sl) STA(El) SLJ(L*5) «ACCMA+2) STA(El) LCA(AI) FDV1B1) STA(Sl) LDAIA2) FDV(BI) +STAIS2)

END END

■03 1 RUN 1 " CG 26 V 224 FPS F8U 1.29.63 BELL

93.14 375.C 2.393 l.C 288.5 ■21CC0.0 .1)772 8.1 -.»»37 C.195

.195 2.759 9.3645 .902 11.78 F8U LCNGITLCINAL RUN2 STICK FIXED M=.209,V=234 93.4 375.0 2.392 .95 232.C ■22COO.O .4772 5.6 -.«437 .0511 El9 2.59 9.3845 .90 11.78

RUN 3 CG. 24, V=139 KTS,234 FPS 3/3/63 LEVEC 93.4 375.0 2.393 1.0 288.5 22000.0 .«4772 8.1 -.»437 0.195 '.263 2.920 9.3845 .90 11.78

KLNBERS 52.8 .002278 22. -.2927 S22CC.C 22*4 u.ce 29.625 29.!

CG24VV-1, 52.8

.13.VSL .CC227E 32.'

-.380 S6CCC.C 234, U.C8 29.375 29.!

FLITE 52.8 •.CG227E 32. -.380 96CCC.C 234. 14.C8 29.375 29.!

88

APPENDIX III

SYSTEM MECHANIZ ATION FOR THE ANALOG COMPUTER

Fig. 1 to this Appendix displays the analog schematic- for the

equations of motion of the airfrarae. Fig. 2 includes the schematic

for the System 1 auto-throttle, aircraft engine, and stick system

dynamics simulations« Fig« 3 shows the System 2 auto-throttle

schematic.

Table I lists the potentiometer settings for ail systems .

Table II lists the scale factors used In computing these settings.

Table III describes the calculations used in arriving at these values

8s

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31

TABLE II

SCALE FACTORS USED IN ANALOG SIMULATION

Scale Factor Value

t^^. = „005 radians/volt = „286 degrees/voli:

t^bfr = 005 radiars s/Volt - „286 degrees/ voii

b(<* = „002 radians/volt = „ 115 degrees/voit

^«Dot, = „005 radiarrs/volt = „28S dejrees/"'-ck

fcCu. = .002V =*.2%ofV = „278 Kis/Wt o o

C^Xe_ = „002 räGi;-..ns/VoIr-= „115 degrees/voh:

°(r = 200 Ibi/volt

<^TCi = 40 lbs/radians - volt

Ö(TC2 = 50 lbs At - volt

b<Tcj = 12 5 Ibe/railariS -sec - volt

^bfci = 2:0 .i>v:/:£.:2is.r..3 - sec - volt

2 0(OTC2- = 30 LbS/~E.dl«:r?.3 ~ sec - volt

^0TC3 = 300 lifi/racl^ans - sec^ - volt

C^^v = 2„5 r>s/vo.<>

W.TCT = 10Ö It i/volt for System I „ £0 Lb^/vol*. tor System 2

^repiiot = 50 Ibs/voi?

C^FI = 2 Ibs/VcV:

(Xp3 = „01 radians/volt

o<DF3 = - i lb/sec - volt

&RWF = o75 lbs/volt

fio

CX« = .286 degrees/volt

C^Ah^, = .02 g/volt

Kw = 1>5

.01 ft/ft/sec - volt

93

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W. fry^K^ I? - .0141 I Al

94

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Appencliy HI F/.'g.--l

Analog ScAernat/c . Air'frame. E«u&Wj of Motion

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System 1 Auto-Thrattle Ejozti'ons

Ar = _L_ f K«AtX _ K^vAV + fry

AH2 ■= cha-*iqe o-f normal acceleration^ q 4.

ty0 r ^eferenc^ att<j/e o-f attack

Loncu/udihAl Cö^tVoi ^MsTeinry J)whäiYi*ci Ljnea/'/iec/

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T - '/*?.r See -£ - ?.27 /£s /^dJ/sec2- f f. ^ 2,5t It*/ *

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Appendiy HE . FiVJ 101

hi- lhesB3614 An investigation of the effect of autoth

r Ulli pin i

3 2768 002 12973 6 DUDLEY KNOX LIBRARY

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