Effects of dynamic aeroelasticity on handling qualities and pilot rating

Report No. 77-7

PURDUE UNIVERSITY

SCHOOL OF AERONAUTICS AND ASTRONAUTICS

EFFECTS OF DYNAMIC AERDELASTICITY ON HANDLING QUALITIES AND PILOT RATING Wen-Yo Yen and Robert L. Swa1.

Decanber 1977 N 78-U C7 0 .' -1C53~ ~ ) EFFECTS OF ~YNAMI C ( ' A:J~ -C ~ ~ ) .} , 1 ' ~ND

• . ~ _ n " 'L • ~ ~ 1 C I 'l y ON 1: A. N D L Hi G ;) U A LI T :, '..>

J , 1\ :. (' J [ t\ J .L • 1 1 ') P . ~ Uncl d5 PlL O l ~~':'I~r; ( ~'.l Ldue !Jn~v. ) 'J 1 C C::3CL 55d81 G 3/Cd

;;-= ~. ~. (i f " F ,\:) '

Wes' lafayeHe, 'ndiana 47907

Report No. 77-7 _ a PURDUE UNIVERSITY SCHOOL OF AERONAUTICS AND ASTRONAUTICS

•

EFFECTS OF DYNAMIC AEROELASTICITY ON HANDLING QUALITIES AND PILOT RATING Wen-Yo Yen and Robert L. Swaim December 1977 West Lafayette, Indiana 47907

· I ii ACKNOHLEDGEMEN'l'S First and foremost, I would like to thank my advisor I Dr. Robert L. Swaim, for his technical guidance, suggestions, and patience throughout this study. Appreciation is also extended to the members of my advisory committee: Prof. D.W. Alspaugh Prof. Hsu Lo Prof. C.L. Nachtigal - I for their consultation in this effort. In addition, I would like to thank Dr. William Seitz and Dr. James Silverthorn who originally constructed the simulator for their own thesis work.

This research was supporte~ in part by the Vehicle Dynamics and Control Division of the NASA Dryden Flight Re- search Center under grant NSG 4003. Mr. Donald T. Berry, Technical Monitor.

iii T~~LE OF CONTENTS Page LIST OF TABLES ••..••••..•••.••••....•...•.••••••. v LIST OF FIGURES ..................................

vii LIST OF SYMBOLS •••••.••.••.......••.••..•••••••.• ix ABSTRACT ••••••••••••••••••••••••••••••••••••••••• xi i i CHAPTER 1 - INTRODUCTICN ••••••••..•..••.•.••••••• 1 1.1 Handling Qualities and Pilot Rating ••.•.. 1 1. 3 Obj ecti yes ...................... v • • • • • • • • 4 1.4 Thesis Organization .••••.•••••.•••••••••• 5 CHAPTER 2 - EQUATIONS OF MOTION •..•.....•.••••••. 6 2.1 General Description of the Aircraft •••••• 6 2.2 Airframe E~~ations •••.••..•.••••••••••.•. 6 2.3 Flight Director Equation •••.•.••..•••..•• 21 2.4 Pitch Angle Error Equation •.•..••.•••.•.. 23 2. 5 Sununary Equa tions .•••••..•••••.•••••••.•• 26 CHAPTER 3 - DYNAMIC CHARACTERISTICS OF EIGHT CASES •••••••••••••••••••••••••••••••• 28 3.1 General Descriptiort ••••••.••..•••••••..•. 28 3.2 Dynamic Characteristics of Case 1, Case 2, 3.3 Dynamic Characteristics of Case 4 ••••.••. 30 3.4 Dynamic Characteristics of Case 5 and Case 6 ••.•••••••••••••••.•••••••••••••.•• 35 3.5 Dynamic Characteristics of Case 7 and Ca se 8 •••••.••.•..•••••.....•••••••••...• 3.6 Summary of Eight Cases •.....••..•••••..•.

CHAPTER " - EXPERIMENT ••••.•••••.•....•..••••..•• 4. 1 Overview ................................ .

4 • 2 Appar a tus ............................... .

4.3 Methodology .••••.••.•••••••••..•..•.••.•.

iv CHAPTER 5 - RESULTS .............................

SS 5.1 Method of Analysis .......... . ....

5.2 Comparison of Cases ...................

\.

5.3 Time Histories ..........................

i CHAPTER 6 - SUMMARY c •••••••••••••••••••••••••••• 6.1 S \lJDIDAry •••••••••••• . . . ......

6.2 Conclusions .............................

6.3 A Recommendation ••••••• ...

. ....

-

BIBLIOGRAPHY ....................................

9S APPENDIX ' - ANALOG SIMULATION ..................

v LIST OF TABLES Page Table Force and Moment and Elastic Force 2.1 Derivatives as a Function of Stability 2.2 Stability Derivatives for B-1 Bomber in Mach 0.85 Flight Condition ..•.••.•.•.•... 20 Dynamic Characteristics of Case 1 3.1 ........

3.2 Dynamic Characteristics of Case 2 3.3 Dynamic Characteristics of Case 3 Dynamic Characteristics of Case 4 3.4 Dynamic Characteristics of Case 5 3.5 Dynamic Characteristics of Case 6 3.6 Dynamic Characteristics of Case 7 3.7 Dynamic Characteristics of Case 8 3.8 3.9 Natural Frequencies and Damping Ratios 4.1 Order of Presentation of Cases ..•.....•.. 50 5.1 64 Observations of RMS Pitch Angle 5.2 Analysis of Variance •.•••...••..•.•...••• 58 5.3 Summary of Results .•••••••...•......•••.• 62 5.4 Required Difference in Means for Statis- tical Significance •.•..•.........••...•.. 63

vi Table Page 5.5 Summary of Tracking Error, Cooper- Harper Rating, and Pilots' Comments •••••• 70 Appendix Table A.l Normalization of Variables ••••••••••••••. 96 A.2 Potentiometer Settings .••.••••••••••••••• 105 A.l Potentiometer Settings for Eight Cases ••• 107

LIST OF FIGURES Figure Page 2.1 B-1 Bomber and Sign Convention for Body Axe s ••••••••••••••••••••••••••••••••• 7 2.2 Rigid and Elastic Pitch Angles •..••••••••. 23 2.3a Electronic Attitude-Director Display (EADI). 25 2.3b The Airplane Attitude Corresponding to FADI Above ••••••••••••••••••••••••••••••••• 25 3.1 The Locus of Roots of Matrix A, Case 1, Case 2, Case 3, as a Function of Natural Frequency of Elastic Mode 1, wl •••.•.•..•• 29 3.2 The Locus of Roots of Matrix A, Case 4, as a Function of Natural Frequency of Elastic Mode 2, w •••••••••••••••••••••••• 34 3.3 The Locus of Roots of Matrix A, Case 5, Case 6, as a Function of Natural Frequencies of ~lastic Mode - l, w ' l and Elastic Mode 2, w2 •••••••••••••••••••• 37 3.4 The Locus of Roots of Matrix A, Case 7, Case 8, as a Function of Natural Frequencies of Eiastic Mode 1, w ' and Elastic Mode 2, w2 •••••••••••••••••••••••• 40 4.2 Cortll1\and Signal •••.•.••..•••••.••••...•.•.• 52 5.1 Tracking Performance Versus Cases .•..••... 61 5.2 Cooper-Harper Ratings ....••.•....•..••.•.. 64 5.3 Sample Time History - Case 1 · ............ .

5.4 Sample Time History - Case 2 · ............ .

5.5 Sample Time History - Case 3 · . . . . . ..... . . .

viii Page Figure · .......... .

sample Time History - Case 4 5.6 ·.......... .

Sample Time History - Case 5 5.7 Case 6 · .......... .

Sample Time History 5.8 · .......... .

Sample Time History - Case 7 5.9 · .......... .

Sample Time History - Case 8 5.10 Appendix Figure A.l Z~Force Equation .••••••••••..•••..•••••• 97 Pitching Moment Equation •.•.•••••••••••• A.2 ..............

Structural Mode 1 Equation A.3 ..............

Structural Mode 2 Equation A.4 A.5 X-Force Equation and Altitude Equation •.. 101 A.7 Flight Director Equation and Pitch Angle Error Equation .••••.••••••••..•.•..••.•.. 103 A.8 Control Inputs ••..•••••.•••..•..•.••••... 104

x IX,Iy,I z Stability axes mass moments of inertia " JXZ,JYZ,Jxy Stability axes mass cross moments of inertia L,M,N, Roll, pitch, and yaw moments about the stability axes Partial derivatives of x,y,z compon- ents of aerodynamic moments with respect to variable "i" m Aircraft mass P1 ,P2,P3,P4 Pilots (subjects for experiment) P,Q,R Total roll, pitch, and yaw rates about the stability axes p,q,r Perturbation roll, pitch, and yaw rates about the stability axes Generalized force in ith elastic mode °ni On' Partial derivative of generalized force 1.'

J of ith elastic mode with respect to variable "j" RMS Root mean square S Aircraft wing reference area

so

Standard deviation T,t Time u,V,w

Components of V

u,v,w Components of perturbation linear velocity u Control variable Total linear velocity vector Partial derivative of x,y,z components of aerodynamic forces with respect to variable "1" x Vector of state variables

xi Greek a Angle of attack (rad) y Flight path angle - angle between velocity vector and horizontal Incremental aileron deflection - posi- tive for right roll (rad) Incremental rudder deflection - positive for left turn (rad) Incremental elevator deflection - posi- tive for nose pitch down (rad) Incremental thrust variation - positive for increase in thrust (pounds) Incremental flap deflection - positive for le88 flap deflection (rad) Structural damping ratio, mode i Coupled damping ratio of ith mode Generalized displacement of the ith normalized structural mode p Air density (slugS/ft3) Heading, pitch and roll angles ~,e,~ Perturbation heading, pitch and roll angles The ith normalized mode shape, i.e., ratio of local deflection to that at normalizing point (nondimensional); () denotes location on fuselage centerline ~ '() i Slope of the ith normalized mode; () denote8 location Summation operator ....

w Angular velocity vector

xii Undamped natural frequency of ith mode 1,1).

1.

Coupled undamped natural frequency of ith mode Subscripts Vector written in aircraft coordinates A Flight director pitch command c Vector written in inertial coordinates I Trirr~ed value about which perturbation o variables obtained Phugoid ph Short period SP Matrix transpose T Intermediate coordinate systems used 1,2,3 to obtain Euler angles

xiii ABSTRACT Yen, Wen-Yo. Ph.D., Purdue University, December 1977.

Effects of Dynamic Aeroelasticity on Handling Qualities and Pilot Rating. Major Professor: Robert L. Swaim.

Pilot performance parameters, such as pilot ratings, tracking errors, and pilot comments were determined for a longitudinal pitch tracking task using a large, flexible bomber with parametric variations in the undamped natural frequencies of the two lowest frequency symmetric elastic modes. This pitch tracking task was programmed on a fixed base simulator with an electronic attitude-director display of pitch command, pitch angle, and pitch error. The results of this study indicate that low-frequency structural flexibility can significantly affect the handling qualities and pilot ratings in the task evaluated.

CHAPTER INTRODUCTION Pilot Rating and Qualities Handling 1.1 to control flight responds in airplane The wayan qualities handling as the to is referred the pilot by inputs good, ~re not qualities handling If the vehicle.

of the the to flying attention of his more must devote pilot the as such activities, mission to other and less airplane words, In other combat.

and air-to-air delivery weapon pilot the the greater handles, the airplane the worse to relate qualities handling terms, In simple workload.

control or to maneuver attempting in or difficulty ease the is 100\ qualities handling study of The aircraft.

an pilot oriented.

to contributing factors the individual all Although accept- or satisfactory in the may fall qualities handling composite their boundaries, respective of their ranges able satisfactory a completely produce still not may effect system handling total of the Proof some cases.

in airplane simulation by ground-based required usually is qualities that A tool airplane.

actual of the test and/or flight Cooper-Harper is the evaluations in such used is widely

/ Pilot Opinion Rating Scale (reference 8). It is a ten- point numerical scale whereby the pilot can express his evaluation or opinion of how well the airplane flys or handles.

1.2 Handling Qualities History There were only a few attempts to design an aircraft for good handling qualities before the second world war era. Before then the design of aircraft concentrated pri- marily on aircraft performance goals. Stability and con- trol characteristics were not well understood. It was known, however, that static and dynamic stability character- istics and control system design determined the handling qualities - good or bad - of an aircraft. However, since that era much has been done to analyze the handling quali- ties of existing aircraft and to develop theories that allow the stability and control engineers to design an aircraft that handles well.

Presently, designers mainly consider the rigid air- craft to estimate the handling qualities and pilot rating of the airplane. But large airplanes such as transport category airplanes and bombers need to be as light as possible due to many reasons which include fuel conser- vation. This means a sacrifice in structural rigidity.

Airplane flexure causes additional aerodynamic loads which in turn cause additional flexure, etc. Also coupling occurs between the elastic modes and the rigid body motion

rigid body and the motion flexure sense the gyros as the motion.

as of knowledge lack complete the near point out To handling affect airplane modes aeroelastic to how dynamic their effects to reference the only we quote qualities, - Flying Specification Military MIL-F-8785B, in contained four- The 2).

(reference Airplanes Piloted of Qualities control aeroelasticity, "Since says: merely statement line an important may exert dynamics and structural equipment, effects such qualities, flying airplane on the influence dir- analyses or calculations in be overlooked not should the require- with of compliance investigation toward ected is con- The specification specification."

of this ments on rigid-bcdy values ranges of wit."" desirable onl~· cerned that possible seems quite It parameters.

response dynamic signifi- could be rating and pilot qualities the handling case in particularly modes, elastic by cantly affected at all is not It frequencies.

natural have low some modes by specified be should qualities the handling that clear and elastic rigid when such parameters dynamic rigid-body not be lik~ly will The pilot present.

is interaction mode given pitch of a how much example, for to discern, able and due to rigid-body input is command to response angle These modes.

elastic frequency to low is due how much of assessment pilot's the affected certainly interactions herein.

reported the work in qualities handling the

Therefore, some study of these elastic effects and rev"ised design standards must be inaugurated for large flexible vehicles. To date no design criteria exist for handling qualities in terms of control system specifications. It is hoped that this thesis will be a basic step in that direction.

1.3 Objectives The longitudinal dynamics of the Rockwell B-1 bomber aircraft was simulated on an analog computer. The two lowest frequency symmetric elastic modes were included in the math model. This was sufficient to get an apprecia- tion for the effects on pilot rating, where the elastic mode characteristics were varied between runs.

A fixed base, pilot in the loop simulation was performed to examine the handling qualities and pilot ratings with variations in each of the undamped natural frequencies of the two elastic modes. Specifically, the following questions were addressed: 1. What are the relative pilot ratings in a pitch tracking task as a function of lowering elastic mode frequencies?

2. Which frequency combinations produce the poorest pilot performance?

3. Which of these two elastic modes has the most significant effects on the handling qualities and pilot rating?

Pilot pitch tracking performance, Cooper-Harper rating, and pilot comments were used to answer these questions.

1.4 Thesis Organization Chapter 2 mathematically establishes the task. The aircraft dynamic equations are derived and the simplifi- cations implicit in the perturbation technique and aeroelastic · effects are discussed. In addition, a flight director equation and a pitch angle error equation, including their functions in this study, are presented.

Chapter 3 discusses how and why the eight cases are chosen for this study. The dynamic characteristics of eight cases are presented.

Chapter 4 discusses the experimental objectives, apparatus, and methodology for the fixed base, pilot in the loop simulation.

Chapter 5 gives the results or this simulation study.

The eight cases are compared and the difficulty of some cases is discussed.

Chapter 6 concludes the thesis by reviewing the · results, outlining areas that require additional study, and recommending further activities.

CHAPTER 2 EQUATIONS OF MOTION 2.1 General Description of the Aircraft The aircraft simulated in this study is the B-1 bomber, an American built, large flexible aircraft (Figure 2.1). Rockwell International examined this aircraft which they designed (reference 3) and found they were able to fly a sea level zero degree trim pitch angle at 562.2 knots (949 feet per second airspeed) with the elevators up 6 degrees. It is from these studies that much of the aircraft data (mass, stability deriva- tives, etc.) was obtained.

There is a reason the a-l bomber was used in our studies. This airplane is large and flexible enough to exhibit the desired effects between fuselage elastic motion and rigid-body motion.

2.2 Airframe Equations In order to write equations of motion one must first establish a coordinate system. It is common practice to attach a body-fixed coordinate system to the aircraft and write equations with respect to it. By doing so all moments of inertia will remain constant. The stability

+ >t '0 ~ k o 'W c: o ....

~ c: G) N

~---l ____ :i- ~

o CJ c: 0\ ....

Ul

'g

to ...

U

~

CD .-4 , III + )C

+

axes convention is to let the X axis point forward, the Y axis point out the right wing, and the Z axis point downward.

I. Rigid Body Equations Assuming constant mass:

-

(2.1 )

-

d- I

IS x V) F

=

m<* I + W

ma~ I

B

-

-

dH dH (2.2) H w x

=

r· 1 •

I + -

dt dt

II

B where:

d;\

is the rate of change of the aircraft dt I velocity vector as seen from inertia coordinates.

dvl seen from aircraft is the rate of change as dt B axes.

body of the aircraft.

-

angular velocity is the w momentum vector.

aircraft angular is the H forces.

all external sum of vector F is the moments.

- of all external

sum vector is the M aircraft mass.

is the m

These two vector equations can be written in terms of their components to produce the following six equations: .

G + RV) Fx + QW - • m(U (2.3) x .

G + ... m(V + RU - PW) Fy (2.4) y G + Fz

= m(li + PV - QU)

(2.5) z .

• L + QR(Iz-I ) - RJ - PQJ (2.6) • PI y x xz xz .

(p2_R2)J H :It QIy + RP(Ix-I ) + (2.7) z xz

N ... RI

- ~J + PQ(I -I ) + QRJ (2.8) xz xz y x z where, in body coordinates:

-

U, V, W are the components of V P, Q, R are the components of; G , G , G are the components of the gravity forces x y z F are the components of the aerodynamic Fx' Fy ' z forces L, H, N are the components of the external moments Ix' I ' I are the mass momen : s of inertia y z J is the cross moment of inertia (J and JZy are xz xy zero because the x-z plane is a plane of symmetry).

This body axis frame can be referenced to an inertial coordinate system by a translation of the center of mass and three rotatio~s. Using the subscript "I" to id~ntify the inertial coordinate system, "3" the instantaneous

I I body axes coordinate system, and "1" and "2" intermediate coordinate systems, the transformation takes place in the following order: (1) starting with the "I" coordinate system, rotate about z1 through an angle ~ to establish a new coordinate system xl' Yl' zl" (2) Then rotate about Yl through an angle G to establish a new coordinate system x2' Y2' %2.

(3) Finally, rotate about x2 through an angle ~ to get to the aircraft fixed coordinate system x , Y , 7. • 3 3 From this description one can see that ~, e, 0/ point along x3 , Y2' %1 respectively, whereas P, Q, R lie along x3' Y3' %3· They are related by the equations: P • t - '¥ sin 0

Q a a cos t + ~ cos 0 sin ~

(2.9)

R - , cos e cos t - ~ sin t

The external forces and moments come from two sources: gravity and aerodynamic forces. Gravity, because it acts through the center of mass, produces no external moments.

It does produce an external force that decomposes into: - mg sin 0 Gx • + mq cos 0 sin t (2.10) G • y G cos 0 ~ cos • + mq z along the axes, respecti vel~,..

z3 xl' Y3 '

made have been few assumptions very this point Up to By nonlinear.

are all equations the as a result and perturba- make small will only the aircraft that assuming (a reasonable condition state steady from some tions equations task) the tracking a longitudinal for assumption simplified.

be greatly can Let: + u U - U

o

(2.11) 00 + q

+ v o a

V 1:1 Va + r R - RO and trimmed value to the reters subscript the "0" Where The variables.

perturbation are the letters the small was chosen: condition trimmed following ft/sec U 949

o •

(2.12) degree /:)0 • 0 degrees Q • 3 O condition this flight for settings control steady state The to be: out turn thrust pounds • 41,712 Throttle (2.13) - 6 degrees • Elevator The increment- zero.

out to trimmed controls all other and 0t for by oe' represented are variables control al

elevator and thrust. They represent changes from the above trimmed value.

This same linearization technique can be applied to the aerodynamic forces and moments. This is done by separating them into a steady state term and linear terms due to the perturbation variables. For example: 3F .

x a + .

ae o

o o

o 3F X (2.14) + aT" 5e e 0 • where a ,. W e c q • U o The same is done for F ' F , L, M, and N.

z y The partial derivatives are a function of the charac- teristics of the aircraft and the particular trimmed condi- tion at which they are evaluated. For the trimmed condi- tion selected, Fx' F , and M are independent of v, p, r, z ~, ~, 0a' or while F ' L, N are independent of u, w, q, a, y

0e' 0t' of. As a result, all the corresponding partial

derivatives are zero.

Combining equations 2.3 - 2.14, expanding the trigono- metric functions, and neglecting all second order terms (such as "qw" or "rv"), results in the following nine equations: . .

p • , - \lI sin 00

q • e

(2.15) r - ~ cos 00

• mu = -mgacos00+X u+X a+X q+X~ 6e +X z 6t +X z 6f u a q o~ 0t of

-

This aircraft is considered to be in straight and level unaccelerated flight and then to be disturbed by deflection of the elevator. This deflection applies a pitching moment causing a rotation which eventually causes a change in Fx and F but does not cause a rolling or Z1 yawing moment or any change in Fy~ thus P=R=V=O and the equations 2.4, 2.6, 2.8, may be eli~inated, resulting in the following equations: qse mu--mg8+X U+X a+X. l5e+X~ 0t u a 0e °t (2.16) where X Zs ' Zo ' MIS I and Mo are assumed zero, and 30 =0.

fl t f t f

Mode Equations Elastic II.

the dynamic for accounting of common method The most the of deflection the to represent is effects aeroelastic vibration.

of free its normal modes of in terms aircraft structure, on the any point of deflection The instantaneous in the m&asured as condition steady-state rigid from the of sum as a vector can be expressed frame, body axes directions.

and Z the X, Y, in 'components deflection the to perpendicular and Z direction in the Deflection the planform t), where ~z(x, y, by is indicated XY plane plane.

in the XY plate to a flat been idealized has the to and perpendicular direction in the Y Deflection planfor.m view the side t), where ty(x, z, is XZ plane Deflection plane.

the XZ plate in to a idealized has been on dynamic effect a negligible has X direction in the is neglected.

and stability only need we of motion equations For longitudinal is which direction in the Z the deflection consider to can be deflections instantaneous t). The tz(x, y, of orthogonal number infinite of an as a sum represented as follows: modes vibration normal • (2.17) (t) (x,y) ni

r ~i

~ (x,y,t).

i-l Z

where: +i (x,y) is the ith synmetric normalized mode shape in the XY plane.

ni(t) is the generalized displacement.

Representation of vibration in terms of normal modes results in additional equations of motion of the form (reference 1).

(2.l8) m i is generalized mass defined by where: m(x,y) is the structural mass density per unit area in the respective XY idealized planform.

~~t) is the generalized force defined by

On .( t) • J J f (x , y , t) • i (x, y ) dxd Y

1 z where: f (x,y,t) is the time-dependent aerodynamic pressure z distribution acting in the Z direction.

"i is an additional dependent variable to be intro- duced into the rigid-body equations as well as appeaI ' ing in (2.18). Considering the longitudinal rigid-body equations (2.16), they will contain additional

as follows: due to aeroelasticity aerodynamic terms CD

~ z .]

[u

Tli +

r

ani 11i 311i 1=1 (2.19) CD ~

[llL

"i +

r

~i 1

ani ani i-l In order to make the elastic equations (2.18) com- patible with the rigid-body equations, the Q (t) must be "i expressed in terms of the dependent variables. Thus: (2.20) The numerous partial derivatives are obtained by utilizing the stability der . ~vatives of the aircraft.

Listed in Table 2.1 is the relationship between these partial derivatives and the stability derivatives while Table 2.2 gives the values of these derivatives for the 8-1 bomber at th~ Mach 0.85, sea level flight condition.

Using Tables 2.1 and 2.2 to evaluate the coefficients in equations 2.16 and 2 .18 resl~lts in the following equations,

:1

r -O.025u-25.0a-32.2e+0.000l4l6 U D t

a • -0. 00065u-l. 20Sa+0. 9430 -0. 0090Snl-0. 00021~1

.. .. .

e ~ -O.0026u-7.649a-O.491a-1.S666-0.1999nl-O.00754nl

(2.21) nl+O.272nl+184.69~1=-735.l9a+2.264~-l35.4046+7.263nl ••• •• n2+0.424n2+449.59~2a764.70a+6.l53a+50.l4ge+7.041n1 -0.1206nl-7.962n2-0.426n2+614.966e Additional details of the derivation of perturbation equations for an aircraft simulation are presented in references 6, 7, 14.

I

I

~

~

( I ~ , i - I ,

Table 2.1 Force and Moment and Elastic Force Derivatives as a Function of Stability Derivatives =

-= alc IU

alCx /U alcSnu/u Zu ~ Xu • z

o

O o

u u =- alC

= alCx Za Ma • alcc

z

Xa

a ma a M· C I: a cC

z·

xa =- a2Cx·

• a2 z· · a 2 a

ma

a a X· CC C Ze I: a C Me • a2 m =- a2 • e

2 Ze

xa e

&:I alc -= alcc = alC Z~e MISe X6e m zo xIS e e °e M -= 0 1* Z~t I: X lSt • 1St = alC Mn • alcc nl Xfl • alC Zn m xT'\l znl T'\l z· = M• ,..

X· &:I alCx. IU alC IU al cC • IU

rn z • n

o o o

T'\l T'\l l n· II nl Z &:I &:I alcC .

X :a alC alC Mn m n n2 xn2 zn Mrt &:I

X· Zn I: alcC • IU

• AIC • IU

A1Cxri/UO z

O

o mn

"'2

2 '2 2

*Obtained from reference 3.

Table 2.1, cont.

= :a a c alC 0" 0" 1 "2 Tll 1(1 a a a

0 = a2C C

0 :I a2 Tt2· Tt2· "1_ Tll- a a a a

= a C

0 = a e

Tt2- "1_ Tt2e "1- a

° e

e :a alC :a alC OTt Oq Tll 2 "2 Ttl "1 Ttl Ttl /U = 0 alC C /U :a al O O Tt2_ Tl2_ 'll_

"I_

°

Ttl Tll Ttl "1 :I a C O· Tt2 Tt2 Tt2 "2 /U • a1C OTt O "2- 2 - "2 "2 a c

=

On l Tt26 e e *Obtained from reference 3_

Table 2.2 Stability Derivatives for B-1 Bomber in Mach 0.85 Flight Condition :=It -0.4546 C -1.9659 - -0.08066 C C • mu Zu x u -1.41052 Sn ..

• -3.9367 C C z • -0.OJl500 C1 x C1 C1 =- -11. 005 Cm· • -5 C z• Cx· =- C1 C1 C1 :=It -35.7556 .. 17.8558 C • C • C z• me - 0 e xe = -2.799

C em

• -.9426 ::a 0 C Z5 x e e °e .. -0.0348 C =- 0.02922 C ::a 0 C z m"l n xlli.

..

-1.32169 C .. 0.6592 C c· .. 0 rn~l Zh x". 1 .. 0.0387 C z: -0.015 C z

m"

C • x,, - "2 .. 1.233 .. -0.4733

em·

=- 0 Cz • Cx· "2 "2 "2

Table 2.2, cont.

a: 0.48975 C • -0.06478 C'1 '12 la a C • 0.48779 • 0.02469 C'1 n0 2· 1· a a = 3.97547 .. -1.47658 C C'1 • n 1· 2e e ... 0.00451 .. 0.00064 C C n n '1 1'\1 C .. -0.07243 C = -0.0733 Tl · • n1n1 21'\1 ... -0.00S1 -0.0014 C en • n2Tl 2 1'12 -0.2588 • 0.0765 C • = C - n nl 2Tl o 2 '1

C = 0.3939

.. -0.19635 C nl 6e n25e Note: (1) All coefficients are non-dimensional (2) Data obtained from reference 3 m .. U = 949 ft/sec 7085 slugs

o

slugs/ft p = 0.002378 1946 ft s • Ibs/ft

q = 1071. 754

15.33 ft c • 6 2 5.8916xlO slug-ft Iy • 2.3 Flight Director Equation The total pitch angle time history that the pilot feels and .ees, either on the outside horizon or the attitude indicator display, is given by (reference 4 and 5) •

.....,., _ . .......

(2.22) Figure 2.2 shows the pitch angle response to command input due to rigid-body motion and that due to the two lowest frequency elastic mode responses. Where xn indicates pilot fuselage station and , 41 i (X the slope of the i th syrmetric elastic mode at that p ) station.

The relative contribution ot the elastic terms to the total pitch response increases as the natural fre- quencies of the modes decrease. Now the values of ~i , and 412 ' which are obtained from B-1 data (reference 3), are as follows: ~' = 0.25 ~. = 0.29 So equation (2.22) can be rewritten as (2.23) 2.4 Pitch Angle Error Equation A pitch angle error results trom the reference trajectory which is the commanded signal that the pilot is to follow. Wh~n lhe CRT is implemented, this pitch angle error would be a measured quantity. That is, the CRT would provide the aircraft with its present pitch angle and this would be compared with its desired pitch angle given by the reference trajectory. The difference

Rigid Body Flexure Body Horizon

z

Tangent to Flexure Body at Cockpit Rigid Body Flexure Body Local H.

Figure 2.2 Rigid and Elastic Pitch Angles

between the actual flight direction and the reference trajectory is the pitch angle error (see Figure 2.3).

The equation is ~s foll~ws: (2.24) The pilot's job is to maintain this pitch angle error as close to zero as possible. The electronic attitude dir- ect , or is the display presently on the simulator built by William Seitz when the simulator was located in the M.E.

school at Purdue (reference 13). A drawing of the dis- play is shown in Figure 2.3. The dashed medium length line is attached to the face of a cathode ray tube (CRT) and represents the wings of the aircraft. The single long line moves up and down due to aircraft motion and represents the horizon. When the aircraft is pitched up, the horizon line is below the fixed aircraft symbol. The two short lines on the EADI are often referred to as a flight director. They move up and down to give pitch commands. The pilot's job is to keep the aircraft symbol centered in the two short lines. This display can be configured in two ways. The first is called a control director. With the control director, movement of the horizon line is directly related to movement of the control surface (control stick). When the pilot pulls the stick back (up elevator) the horizon line goes down. The displacement of the horizon line is related to control movements by a constant. As a result th~ two short lines

2S Fued Aircraft Symbol

-

e C01IIIII&nd

-

Horizoll L1n.

llllbt Director (A1rcraf t :U Pitched Up) Figure 2.3& Electronic Attitude-Director Indicator (£ADI) Local H.

Fiqure 2.30 The Airplane Attitude Corre.pondinq to Above £AD I

configuration The other deflection.

elevAtor command two And the line The horizon flight director.

is A are just they mode Also, in this present Are short linea A change director.

control thAn the differently driven of displacement a vertical produces C6i) pitch angle in short the two of movement As a result, line.

horizon the The electronics angle.

in pitch change commands lines as A flight configured are presently display the driving it used we have director, control that rather director, work.

in this director A flight AS Equations 2.5 Summary in form in matrix written can be 2.21 Equations Table 2.3 the of dynamics the longitudinal 2.21 are Equations condition level flight sea mAch 0.85, at B-1 bomber operating.

syatem mode control structural the without and equation director the flight 2.23 is Equation equation.

error angle is the pitch 2.24 equation on ~re aimulated linear and are all These equations in us~d simulation the piloted in computer an analog study.

this '.

N ...,J 0 0 0 1 .00015 .00697 .91534 -.8491 0 0 0 0 .00465 .2078 -15.414 -465.523 0 0 1 138!? 0 -.00021 -.00744 -1. -.1219 0 0 0 0 -.00905 -.1845 -177.447 6.985 Form 0 1 0 0 .943 -2.0195 -133.268 55.951 Matrix T in • 0 0 0 0 0 0 0 -32.2 • Motion of • T

0 0 0 o o o o

o o o

.000141 -25 -1.205 -7.0694 -737.92 757.29 Bu (u,(l,e,e,n1,n1,n2,n2) (oe,Ot) + Equations = ..

Ax x U 0 0

0 o o 0 o

= 2.3 x -.025 -.00065 -.00228 -.00147 -.00399 -.288 -15.0541 -2229.054 613.183 r Table A=I B • Where: ~

CHrlPTER CASES EIGHT OF DYNAMIC CHARACTERISTICS Description General 3.1 in as shown of motion the equation this study, In natural The modes.

two elastic includes (2.21) equation be parametrically modes will elastic the two of frequencies were which eight cases by represented They are reduced.

characteristics Dynamic pilots.

by several each flown by four be specified cases will eight of these for each mode 1, elastic mode, phugoid period mode, short modes: will cases eight of these dynamics 2. The mode and elastic sections.

the !ollowing in be described and Case Case 2, Case 1, of Characteristics 3.2 Dynamic elastic of frequency the natural only section, In this of frequency the natural and was reduced 1 (wl) mode unchanged.

2 (w2) was mode elastic to 13.59 rad/sec from varied of wl was The value of this range in were chosen ca~es Three rad/sec.

4.24 3.1.

in Figure shown plot is root-locus The wi.

is the original it rad/sec; wl is 13.59 of The value 1.

as Case chosen It was of wi.

value

~ ~

I

.8 I 2, of .6 Case Axi8 l 1,

.. ••

.4 a Frequency CaBe .2 A,

•••

~

I

-.0

$ -

Natural • ..

"8 a

Matrix of 2• -.2 w

•

• ...

of 1w2 I -.4 WI ~ l. .

~ w.

) 1• Roots Function 1, I

18 1w

~ a ;. -.6 • of as Mode -.8 3, Locus

~

-1.0 Elastic The Case (~1·11.S9.W2-21.1a)

--'\

I 2 (wI· 9.17;w2-21.18) 1 1 (wI· 6. 16;w2- -1.2 3.1 •• •• •• '" ca -1.4

--

o ca A ca a

--

P p iqure ' ~ s -1.6

ot·

-5 -10 -IS -20

As the value of wl was varied from 13.59 rad/sec to 9.21 rad/sec, all modes (phugoid, short period, elastic mode 1, elastic mode 2) were stable.

When the value of wl is less than 9.21 rad/sec, the phugoid mode became unstable.

A value of wl at 9.17 rad/sec was chosen as Case 2.

A value of wl at 6.16 rad/sec was chosen as Case 3, and the phugoid mode is unstable and non-oscillatory. This value of wl was the lowest value that could be chosen for the pilot to fly. When the value of wl was less than 6.16 rad/sec, the pilot could not control the unstable phugoid mode due to the mode interaction with elastic mode 1.

The dynamic characteristics of Case 1, Case 2, and Case 3 are shown in Tables 3.1, 3.2, and 3.3, respectively.

3.3 Dynamic Characteristics of Case 4 In this section, only the natural frequency of elastic mode 2 (w~, was reduced and the natural frequency of elastic mode 1 (wl) was unchanged.

The value of w2 was varied from 10.14 rad/sec to 2. 83 rad/sec, and just one case was chosen in thi ~ range of w2' The root-locus plot is shown in Figure 3.2.

The value of w2 at 4.79 rad/sec was chosen as Case 4.

This value of w2 was as low as could be chosen as previously discussed. If the value of w2 is larger than

~ "" Mode ...

0.29 1.49 0.0215 21.

21.354 - Elastic :!:.21.349j Mode 1.0~ 0.47 0.0494 0.6583 13.296 13.312 - :!:.13.295j Elastic 0.0108 0.01913 0.01081 0.00708j 0.00139 Phugoid 88.7 0.424 494.7 0.212 Case - :!:.

-

=

of w 2'2 2tlw1 Period 2.648 0.46 2.373 case): 0.5339 2.806 rad/seci rad/sec; -1.498 :!:.2.373j Short Characteristics 21.18 13.59 or = a (original aroped oynamic W2 1 Wi hal-f damped freq. damping ur (rad/sec) AJilp.

to 3.1 Case (sec) (sec) Period freq. twice (rad/sec':) Time ratio coupled natural coupled Coupled Roots Parameter Tablp

W tv l Mode 0.0213 1.5169 0.2943 Elastic -4.5676xI0- 21.356 21.3512 !.21.35l1j I Mode 8.7891 0.08769 8.7552 0.8953 0.7176 Elastic -7.7072xlO- !.8.7552j j Phugoid Case 0.424 0.057305 0.057304 0.183 +3.4536xlO- !.5.7305xlO- -0.00060267 = of

=

2 109.6447 W 19979.149 2t 2tlwl Period 2.5724 0.5235 2.19 0.5124 2.8667 -1.3468 !.2.l917j rad/sec; rad/sec: Characteristics Short 9.17 21.18 = = or Dynamic 2: WI W undampea freq.

damping damped half 3.2 (rad/sec) Case Amp.

to Table (rad/sec) Parameter Roots natural Coupled ratio (sec) Coupled Coupled freq. twice (sec) Time Period

w w Mode 0.2943 1.5142 0.021337 4.5568x10- 21.

21.357 - :!:.21.352j Elastic Mode 1.0930 5.7485 0.5883 0.1999 5.8669 -1.1728 Elastic :!:.S.148Sj Phugoid 0.424 Case 0.123 -1.6123x10- +9.0978x10- =

=

of 2~2w2 2~lw1 Period rad/sec; 4.1631 1.5093 0.7476 0.5217 1.1691 rad/seci -9.2283x10- Short :!.1.S093j Characteristics 21.18 6.16 or

=

=

Dynamic W2 3: W damped freq. damping ball uridamped (rad/sec) Amp.

3.3 to Case lp:'ed riod ~ :l ' twice (s~c) (sec) Pe (rad/sec) freq.

rdlia Coupled T1me natural ~ Coupled Parameter Roots Table

w ..

i I

, .4 of ., .2

••

Function a

o

as A""'., 4, w I , ~ 2, -.2 2e Case Cal2e Cal R • R I A, Mode -.4 I Matrix -.6 Calle Elastic Calle R R of of +6- ....

I -.8 4.79) Roots ~ of Cal2 '" -1.0 Frequency ..

Locus 13.59.

= I: -1.2 The Natural (Call I 3.2 -1.4 RSp~ A Case

I

5 o Figure

20r·------------------------------~~------ 15 10 -5 -10 -15'

4.79 rad/sec, the situations would be the same as Case 1 and Case 2.

The dynamic characteristics of Case 4 are shown in Table 3.4.

3.4 Dynamic Characteristics of Case 5 and Case 6 Both natural frequencies of elastic mode 1 (wl) and elastic mode 2 (w2) were reduced.

The values of wl and w were varied from 13.26 rad/sec to 4.89 rad/sec. Case 5 and Case 6 were chosen in this range for wl and w2' The root-locus plot is shown in Figure 3.3.

When the values of wl and w were 11.66 rad/sec, the phugoid mode was unstable and oscillatory. This was chosen as Case 5.

When the values of wl and w were 6.93 rad/sec, the phugoid mode was unstable and non-oscillatory. This was chosen as Case 6. These values of w and w are as low l as could be chosen in Case 6, as previously discuss~d.

The dynamic characteristics of Case 5 and Case 6 are shown in Tables 3.5 and 3.6, respectively.

3.5 Dynamic Characteristics of Case 7 and Case 8 The natural frequency of elastic mode 2 (w ) was varied from 10 rad/sec to 6.32 rad/sec while the natural frequency of elastic mode 1 (wl) was varied from 10 rad/sec to 12.61 rad/sec. The root-locus plot is shown in Fig- ure 3.4.

W Q\ Mode 5.9702 0.1137 5.7544 1.0165 1.0919 Elastic -6.7877xlO- :!.:5.9315j l Mode 7.0119x10- 0.05284 9.8405 4.7415 13.270 13.251 Elastic - !.13.251j 1 1 ~hu~oid Case 0.272 4654x10- 0.096

=

of - +1. -1.3167xlO-

I

2~lw1 2'2w2 Period rad/sec; 1.5745 0.6872 1.1438 0.6377 5.4931 rad/se~; -1.

!.1.1438j Characteristics Short 13.59 4.79

=

=

or Oynamic 2 4: WI W undamped freq.

damping damped halt 3.4 (rad/sec) Amp.

Case to Table (rad/sec) Parameter Roots natural Coupled ratio twice (sec) Coupled fre...!!.

Coupled T~me lsec) Period

w "

,

I

I .6 Mode I .4 ~ Axis Function a Elastic , Real as .2

8'

and 6, n wl'

P

Case 1, S, ?

I I • Mode 2e Cal Case 2e R w R A A, , Elastic of r I Matrix

I'

of n I Calle 11.66) R 6.93) Roots

=

z Frequencies W2 of -.

Cal2 I.; 11.66; 6.93; Locus Natural w2

= =

I

2, The of (WI (wI 5 6 " 1.1 ~ "'""'---- Case Case 6 o sp Sp R R Figure -5 20.

-10 -15 I

w CD Mode 0.01621 1.8770xl0- 3.6778 0.5429 11.574 11.5725 - +11.573j Elastic Mode 9.1264xl0- 0.077338 0.7560 0.5340 11.801 11.7657 - Elastic +11.765j 6 2 j Phugoid 0.233 Case 0.233 0.053745 0.053745

= D -0.00011217

+6.0288xlO- !.5.3748xl0- 116.9073 of 114454.8772 2C)wl 2t2w2 Period 2.5819 0.54359 2.1671 0.4916 2.8993 rad/sec; rad/sec: -1.4035 :t2.1671j Characteristics Short 11.66 - 11.66 or

= =

-- Dynamic 5: Wl w2 undamped freq. half damping damped (rad/sec) 3.5 Amp.

Case to (rad/sec) (sec) (sec) twice natural freq.

Table Parameter Roots Coupled Coupled ratio Tl.me Period Coupled

W \D Mode 6.9178 0.0075997 6.9176 0.9083 Elastic -5.2984x10- 13.1246 :!:.6.97l6j Mode 7.3305 0.19196 7.1942 0.4903 0.8734 -1.4072 Elastic :!:.7.l942j Phugoid Case 0.139 0.139 +1.758lXIO-~ -l.5307xlO-

of =

=

l 1 j 2tlw1 2t2w2 Period rad/sec; 0.70279 0.9721 0.7185 6.4634 rad/sec; 1.3665 Characteristics Short -9.0635xl0- :!:.9.72l2xlO- 6.93 6.93

= z

or Dynamic 1 6: W w2 undamped freq. half damping damped (rad/sec) 1.6 Amp.

to Case (rad/sec) (sec) (sec) twice Parameter Roots natural ratio freq. Period Table Coupled Coupled Coupled Time

o ..

.4 a wl' Real Axis as 1, .2 8, Mode

o

Case 1, 2e 2e C11 Elastic C11 R -.2 R Case of A, -.4 Matrix -.6 Frequencies w2 of 2, ~ .--.

9.75) 9.27) -.8 Roots Natural Mode

= •

of le W2 W2 of le w CII R R -1.0 Locus Elastic It 10.25. 10.68; ~ c • c.

.. and

• The Function

sp -1.2 R Rsp (wI (WI 1 8 ,...

...

3.4 -1.4 Case Case o x Fiqure 5 o 20 15 10 -5 -10 -15

When the values of wl and w2 were 10.25 rad/sec and 9.75 rad/sec, the phugoid mode and elastic mode 2 were unstable and nscillatory. This was chosen as Case 7.

When values of w and w were 10.68 rad/sec and 9.27 l rad/sec, only the phugoid mode was unstable. This ' ... as chosen as Case 8.

The reason that both Case 7 and Case e were chosen

-

was that their dynamic characteristics were quite similar, yet elastic mode 2 in Case 7 was a little unstable and that in Case 8 was stable. What ~'IOuld the handling qualities and pilot ratings in these two cases be?

The dynamic characteristics of Case 7 and Case 8 are shown in Tables 3.7 and 3.8, respectively.

3·6 Summary of Eight Cases All these eight cases included most of the situations in which the handling qualities and pilot rating would be affected differently by the two elastic modes included in the mode 1.

Each of the eight cases programmed in the analog computer driving the simulator displays was investigated to answer the questions posed.

Case 1, Case 2, ~nd Cas£ 3 were chos~n when only the natural frequency of elastic mode 1 was reduced.

Case 4 was chosen when only the natural frequency of elastic mode 2 \ ... as reduced.

2 3 Mode 0.6388 9.8345 0.00042775 4.2338xl0- 9.B978j 9.8978 162.9748

-

Elastic + + -

Mode 0.5972 0.6179 0.1129 1.1554 10.1666 10.234 - +10.168j Elastic j 3 2

.'

Phugoid 0.205 Case 0.02814 0.028172 0.195 -0.048305 +1.3608xl0- !.2.8139xl0- ~ 223.2901 507.0366 • of 2tlwl 2t2w2 Period ,...

3.1473 0.5225 2.3937 0.55174 1.9964 rad/sec; -1.3207 :tl.9964j rad/sec; Short Characteristics 9.75 10.25 or •

=

-

Dynamic 7: W2 Wl freq. half undamped damping damped (rad/sec) Amp.

3.7 to Case (sec) twice (sec) (rad/sec) Period ratio freq. T"Tme natural Coupled Coupled Roots Coupled Parameter Table

~ w 2 3

-

Mode 0.6426 9.7781 0.00053066 9.7781 5.1889xlO- 9.778lj 132.9715

- + -

Elastic Mode 0.6109 0.6051 0.11021 1.1404 10.2839 10.347 - :!:.10.284j Elastic j 3 2 id Phu9_o 0.02553 0.214 Case 0.02557 0.185 -0.054096 +1.3833xlO- :!.2.S533xl0- 246.0852 498.8307 - of 2tlwl 2t2w2 feriod 3.1611 0.5204 1.9817 2.3893 0.55493 rad/sec; -1.3259 :tl.9876j rad/sec; Short Characteristics 9.27 10.68 or - ..

-

Dynamic w 8: w half damped freq. damping undamped (rad/sec) Amp.

•• to 3.8 Ca (sec) (sec) twice Period (rad/sec) freq.

ratio coupled T1.me coupled natural Roots coupled Parameter Table

Case 5 and Case 6 were chosen when both the natural frequencies of elastic mode 1 and elastic mode 2 were reduced.

Case 7 was chosen when elastic mode 2 was unstable.

Case 8 was chosen with its dynamic characteristics close to that of Case 7 but with elastic mode 2 stable.

Table 3.9 shows the coupled undamped natural fre- quencies and dampinq r(! , t . ~:.os of the rigid aHd elas , tic modes for eac~ of the eiqht cases.

.... VI

~ 5.9702 6.9178 21.354 21.~ 21.357 11.574

r~sec

t;2e 0.0005306 9.7781 0.1137 0.0162 0.007599 0.0213 0.0213 0.0215 -0.0004277 9.8978 7.3305 8.7891 5.8669 10.347 11.801 10.234 13.270

r~sec 13.312

Cases C;le 0.1919 0.1129 0.11021 Eight 0.1999 0.05284 0.0773 0.0494 0.08769 of 0.0256 0.0537 0.0282

r~sec 0.0708 0.0573

ltx)ts Ia)t.s ltx)ts Ratios Real +0.17581 -0.15307 Real +0.090978 -0.076723 Real +0.14654 -0.13167 l:ph Damping 0.0197 -0.0541 -0.0483 -0.00060267 -0.0001122 and 2.3893 2.3937 2.5819 1.3665 2.806 2.5724 1.7691 1.5745

~

C;sp 0.5549 Frequencies 0.5436 0.7028 0.5517 0.5217 0.6872 0.5339 0.5235 6.93 9.75 9.27 4.79 11.66 21.18 21.18 21.18

Natural ~sec

3.9 6.93 9.17 6.16 10.25 10.68 0.59 11.66

r~sec 13.59

6 7 8 3 4 5

• 1

Table case

CHAPTER 4 EXPERIMENT 4.1 Overview A pilot in the loop simulation was performed to answer the following questions. Of the eight cases exam- ined, which one results in the worse pilot performance?

What are the pilot comments on each case? How well can pilots perform in these cases compared to the original case (Case l)?

The eight cases previously discussed were used to answer these specific questions. They are: (1) Does Case 1 provide good handling qualities and pilot rating?

(2) Are the handling qualities and pilot rating for Case 2 and Case 3 acceptable, in which only the natural frequency of elastic mode 1 is changed? The natural frequency of elastic mode 1 is changed to a greater extent in Case 3 than that in Case 2.

(3) Are the handling qualities and pilot rating for Case 4 acceptable, in which only the natural frequency of elastic mode 2 is changed?

(4) Are the handling qualities and pilot rating for Case 5 and Case 6 acceptable, in which both the natural frequencies of the elastic modes are changed?

(5) Are the handling qualities and pilot rating for Case 7 and Case 8 acceptable? Elastic mode 2

-

is slightly unstable in Case 7, but it is stable in Case 8.

4.2 Apparatus A fixed base simulator was adapted to answer these questions. A block diagram of the apparatus involved in this test is shown in Figure 4.1. Two Applied Dynamics analog computers comprising 60 amplifiers were used to simulate the equations of motion, flight director equation, and command signal. The mathematical forms for these equations were presented in Chapter 2. A schematic diagram of this simulation is shown in Appen- dix.

The cockpit mockup contained a control stick, a throttle, vertical velocity indicator, airspeed indicator, angle-of-attack indicator, altimeter, and the electronic attitude-director indicator (EADI).

A control stick provides elevator input to the aircraft simulation. Springs produce a restoring force on the control stick to return it to a neutral position.

... CD

• State Variable I I I I I I I I I I I -1

---

- - - - Data Sianal aecorder Aulol Coalputer Command Aircraft S1IIulation - - - -.

---

,- • I I I I I • I I I

I .

Apparatus of I I I I I I I I

- -

-~- Diagram

-

-

Control.

-

Block

-

rto

-

-

4.1

-

PUot Cockpit

-

Figure

-

~ t----

-

---

-

-

---

-

SituatioQ Dbplay Command --- -- I I r- I I 1- I I

The spring stiffness was suitable for pilots to fly (40 pounds for full elevator deflection). The resulting gradient provides good centering of the stick but does not require excessive force. The stick neutral position can be adjusted by usinq the elevator trim crank on the cockpit floor.

A throttle provides thrust input. Both of these inputs, elevator and throttle, are measured by the wiper of potentiometers mechanically connected to the movable controls. These measurements are then electrically connected to the input 6f operational amplifiers for use in the simulator.

Initially the aircraft is flying at 949 ft/sec at constant altitude with 0 degree flight path angle and aO = 3°, 00 • 0°, and up elevator 6 = -6 degrees.

e The electronic attitude-direction indicator (EADI) vertical velocity, airspeed, altimeter, and angle-of- attack indicators are all visible. Description of the instruments is provided in reference 9.

The vertical velocity and angle-of-attack indicators are driven by miniature direct current (DC) servo systems that convert an analog computer DC voltage into an anqular rotation of the needle within the instrument.

The EADI '.s produced by drawing special s~~bols on cathode ray tubes (CRT). Details of the electronics are provided in reference 9.

4.3 Methodology Data was recorded on four pilots flying each of eight cases during two separate approaches (replications). As a result, a total of 64 data runs were performed. This does not include all the time each pilot spent learning the simulator and practicing each case. This section describes the details of how the experiment was made including the cases arranged, instructions given, and data recorded.

To minimize the possibility of learning trends affecting the results of this experiment, the eight cases ", , . ,.. .

were presented in a different order to each pilot. The / ' . ,. ." .f / (/ \ .

I . , order they were presented in by means of random table I I (reference 12) is shown in Table 4.1.

Table 4.1 Order of Presentation of Cases Pilo ,

~ 2 3

5 6 Pl C2 C6 C5 Ca C4 Cl C3 C7 C C P2 C C C, C C C l 2 a s 3 P C C C C, C C C J C s 2 3

s l

P C C C C C C C C l 3 6

4 a

s Each of th~ four pilots was instrument rated.

Their experience varied from private, military, to commercial. The averzlge total flight hours was more

simulator, the of description A detailed hours.

than experiment the of purpose the and task, the display, each was task The pilot.

to each distributed first was error tracking the maintain ·continuously as described '~hile as possible to zero close ee as angle pitch of " value.

airspeed trim the to maintain throttle using getting hour to one an hour half from spent pilot Each the learning involved This the simulator.

with acquainted dynamics the and instrument each of operation and location had each time this By aircraft.

the of limitations and and what was experiment the of understanding good a required.

instru- the with acquainted been had pilot each After the for flight of simulated session hour a two ments, were minutes Fifteen conducted.

was cases eight first minutes fifteen of the minutes Ten case.

each for used being case particular the with practicing in spent were at flew pilot the minutes ten this During examined.

another provided, signal command no for one - twice least signal command This provided.

signal command for one angle of pitch deviation The 4.2.

Figure in shown i.

of a means as acts this EADI; the on visible was error his When situation.

lea~ning pilot's the determining

_.-

....

-

pe::c==:a.-:ce they finished, w~re cases eight t~ese A!ter taken.

w~s sixteen Thus, order.

same in the repeated were

S2 o fa rot CI c: CJl

- til

..

IQ .., u U c:

•

-

...

I

C"oI •

0 •

.-

G k ::I CJl ..

..

o

fit 1ft 1ft ..

cD fa I

--- _ ..... ~

actual data runs were performed.

Each run beqins with the aircraft at 562.2 knots at 6000 feet altitude. Durinq the first 10 seconds, the pilot flies at a commanded 0.85 deq/sec pitch up. Durinq the 10 second to SO second period, the pilot makes any trim and power adjustments to keep the aircraft pitched up at 8.5 deqrees. At 50 second, the pilot is qiven a command siqnal to start pitchinq down. After 20 seconds elapsed, the reference pitch angle is chanqed to -8 .. ') deqrees. From the 70 second to 110 second period, the pilot makes any trim and power adjustments to keep the aircraft pitched down at -8.5 deqrees. At 110 seconds, the pilot flies a commanded 0.85 deg/sec pitch up. At 120 seconds the experiment is completed.

In computinq wor~ the tracking error, which was recorded by FM recorder, was sampled ten times per second and converted to diqital words. The root mean square CRMS) of the tracklnq error was then computed off-line.

The analysis was done later off-line by using a mini- computer. Because each pilot by observing the EADI knew how much the aircraft symbol was off the command siqnal, he was always aware of how well he was doing. Also time histories of six pertinent parameters were recorded by three two-parameter strip chart recorders.

Each pilot was gi7en a que.tionnai=e at the comple- tion of the experiment. It requested he list comments

....... _0 _° 0 ~

"

on each case and assign a Cooper-Harper rating to each case. A tape recording was also made of verbal comments.

I

CHAPTER 5 RESULTS 5.1 Method of Analysis This chapter compares the performance in each of the eight cases, including an analysis of the mean tracking pe.formance, a Cooper-Harper rating, and comments by the pilots. Root mean square (RMS) error of some quantity is frequently used to evaluate tracking performance. In this study, RMS pitch angle error is used. The magnitude of the RMS pitch angle error is an indication of how well the pilots followed the trajectory . ~he task as described to the pilots was to "maintain the pitch angle error as small as possible." As a result, average RMS error i.

the most appropriate measure of performance.

One must be careful when comparing the mean perform- ance because of the experimental nature of obtaining the results and variability of the pilots. For example, suppose one obtained results where the mean performance of system A is one unit better than system B but the t standard deviation for each was 10 ~nits. One could not • say with much confidence that system A was better than system B. With such large variance in the results the mean of A could have accidentally been better than the

mean of B. The technique of Analysis of Variance is a method for determining the probability that the results were due to chance.

Sixty-four obs~rvations of the RMS pitch angle error, Cjj, are shown in Table 5.1. The notation Cij is used to denote the RMS pitch angle error for each observation.

Subscript i is the row number corre~ponding to pilot performance while subscript j is the column number corre- sponding to case number. Two sets of observations w~re made for each pilot, thus there are two rows corresponding to each pilot. Cj denotes the mean of RMS pitch angle error in jth case; C is the overall mean. Analysis of variance (reference 10 and 11) was perfor~ed on the tracking error of pitch angle and is shown in Table 5.2.

Degree of freedom means freedom to vary. Degrees of freedom for total cases are the number of observations in total minus 1 or 63; degrees of freedom for "between" casel are the number of cases minus 1 or 7; degreesof freedom for "within" cases are the sum of the number of observations within each minus 1 or 56. Table 5.2 indicates that the differences in performance among the cales were statistically significant at the 0.01 level.

F ratio i. a non-dimensional quantity that reflects the probability that the indicated results are due to chance.

The F ratio is interpreted by use of the F t~ble (refer- ence 11). This table is entered with the number of

V1 "

I I I I I C =0.0202 0.0185 0.0179 0.0052 0.0232 0.0272 0.0199 0.0160 10.0339 (1258 C 0.

c =, 0.00851 0.0575 0.0220 0.0141 0.0369 0.0220 0.0261 0.0190 6 6 C =0.1321 C C 0.1119 0.1488 0.0651 0.0796 0.1033 0.1748 0.1791 0.1943

I

C 0.0205 0.0509 0.0274 0.0226 0.0382 0.0214 0.0180 0.0114 ~5=0.

Error

--

C =O.0331 Angle 0.025' 0.0246 0.0309 0.0291 0.0381 0.0360 0.0395 0.0396 C Pitch C 0.0711 0.1082 0.1061 0.1077 0.0824 0.0784 0.1121 0.1258 C)=0.0990 RHS of c2 0.0234 0.0052 0.0038 0.0256 0.0284 0.0190 0.0240 0.0166 C2=0.0183 Cl .0481 =0.0201 a 0.0113 0.0186 0.0192 0.0191 0.0177 0.0285 0.0053 0.0408 observations C C

-

.

i 2 ) 5 6 7 1 4 8 5.1 l 2 P P P4 P Pilot Table

~i

......... _ ... . : .. - Table 5.2 Analysis of Variance RMS Pitch Degree of Source of Anqle Error Freedom Variation Sum of Mean square F-ratio square 0.01518 3S.20f* 0.10637 Between cases 0.00043 0.02425 Within cases 0.13042 Total level at .01 *·significant where a _ e}2

(C

square: "Between" sum of

I L

j i=l j-l C ) 2 (Cij

"Within" sum of square: -

r I

j j=l i-l 8 8 _ ~ )2 (C

"Total" sum of s~~are: I j

L

ij a j 1 1-1 "mean squares" are obtained by dividing each of the sam of squares by its respective number of degrees of freedom mean square for "between" cases "F-rat1o" • ------------------------------ mean square for "within" cases

t degrees of freedom for the greater mean square across the top and with the number of degrees of freedom in the lesser mean square on the left-hand side. For this problem, we go over to 7 and down to 56. Ir. that loc~ tion we observe that the value of F needed for signifi- cance at 1 percent point is 2.98. Since our obtained F is greater than this, this means that the chance that these means of cases are significantly different is 99 percent.

It should be emphasized that statistical significance and engineering significance are two different concepts.

Statistical significance refers to the probability that the results are due to chance whereas engineering signifi- cance refers to the benefits due to increased performance.

Statistical significance can be well quantified whereas engineering benefit is frequently very subjective.

Unfortunately, Analysis of Variance tells only if the differences amor.g Cl C21 C31 C4' Cs, C6' C" and ' Ce are statistically significant. What is of interest is a comparison among the cases. This can be done by using the "Newman-Keuls Test" (reference 12). This test determines how large a difference is required between two meanS for this diffe=ence to be statistically significant.

The difference reflects the magnitude of the means and variances and the nunber of pilots and replications performed. That is, as the means or variances increase,

so must the difference between the means to maintain the same level of significance.

5.2 Comparison of Cases Table 5.3 gives the means and standard deviations for each of the different cases among all pilots. For example, the average RMS pitch angle error for the four pilots

-

over two replications Cl is .0201 radians. These results are plotted in Figure 5.1. The diamond represents the mean RMS pitch angle error and the dashed lines extend to the one sigma deviations.

Table 5.4 is the result of the "Newman-Keuls Test" discussed previously. It presents the required differ- ences between means for various levels of significance.

Comparing the difference seen in Table 5.3 with those required as presented in Table 5.4 indicates that Cases C6 and C3 were each significantly worse than C4' C ' S C,' C ' Cl' C . In between C6 and C3, C was signifi- S 2 6 cantly worse than C). While C4 appears to be worse than Cs, C" Ce, Cl C ' the difference was not statistically I significant. The above indicates that the pilots do equally well for cases C4' C ' C7, Cs, Cl' and C2 in S controlling the pitch angle error. In eight cases, C6 is the worst, C3 is the se~ond worst.

-

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Versus Ca ••• Tracking Performance Figur. 5.1

Table 5.3 Summary of Results RMS pitch Angle Error Radian (Degree) Mean Std. Dev Case C1 .0201 (1.1516) .0107 (.6131) Case C .0183 (1. 0485) .0093 (.53~9) Case C3 .0990 (5.6723) .0192 (1.1001) Case C4 .0331 (1.8965) .0060 (.3438) Case C5 .0263 (l.5069) .0126 (.7219) Case C6 .1321 (7.5688) .0488 (2.7960) Case C7 .0258 (1.4782) .0153 (.8766) Case Ca .0202 (1.1574) .0084 (.4813)

Table 5.4 Required Difference in Means for Statistical Significance Level of RMS Significance Pitch Angle Error Cases radian (degree) 0.01 level 0.0311 (1.7S) The pilots were asked to assiqn to each case a Cooper- Harper rating for the tracking task and the aircraft as simulated. The means and standard deviations of these ratings are shown in Figure 5.2. CI C , CS, and Cs were I good; C7 required minimal pilot compensation; C required minimal to moderate pilot compensation: C3 required con- siderable to maximum compensation: C6 required extensive to unacceptable compensation. This result is consistent with the tracking performance just presented. That is, C6 is worse than C3 which is much worse than C , C , C , 4 7 s CS' C2 , and Cl- Here, C4 is worse than C which is I worse than CS C5' C2 and Cl' CI C , C ' and C show I S s little difference.

Another important criterion for determining the "worst" case is the pilot's preferences and criticisms about the cases. In response to the request, "compare the difficulty of flying Case 1 with the difficulty of the other seven cases," the pilots' answers centered around:

PILOT COMPENSATION PILOT RATING REQUIRED UNACCEPTABLE T I 7 MAXIMUM

-

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Cl • easy

C2 = about same as Cl

C3 .. difficult C4 • slightly difficult

Cs = about same as Cl

-

C6 - more difficult C, • slightly difficult Ca • about same as Cl In addition, the pilots gave the following co~ents on the eight cases: (1) Nothing objectionable on the pitch a~titude tracking at all. Very nice case.

(2) It was very easy to hold it precisely. ~o control effort at all.

(3) I felt this was the easiest case to con- trol.

(4) I find no problem with the dynamic charac- teristics of this simulator.

(1) Fairly easy one to fly-it was not diffi- cult to hold the command symbol right with the airplane symbol. There was very little oscillation in it; very little control motion is required to follow the command.

(2) Pitch-wise a little more difficult than Cl situation, but still able to maintain pretty much within the limits.

(3) The slight continuous oscillation plus the slight lagging control response would be acceptable in a long flight in the event

-

that it did not get worse.

(1) It did appear to be unstable in rigid- body dynamics. It was fairly difficult to hold the airplane symbol in the command box. It took fairly strong amounts of con- centration to keep the symbol in the com- mand box and if you got away from it a little way, it was very difficult to get it back.

(2) Unable to maintain within the pitch limits and had a problem of pilot-induced oscilla- tions due to large elastic amplitude oscillations.

(3) This case was very objectionable due to the extreme lag in the change of pitch of the aircraft following a control move- ment.

(1) Not quite as easy as some of the others, but it was not difficult at all either.

Did seem to be a little unstable in rigid- body dynamic motion. It took a little more forward pressure to do it and stick has a very stiff spring in the forward motion.

(2) The pitch control was a little bit annoy- ing. There did not seem to be quick enough response with the cyclic stick.

Cs: (1) That was about exactly the same difficulty as Case 2: that means its very easy to fly and was exactly the same in all charac- teristics as the Case 2.

(2) Handling characteristics seem to be pretty good there - not too much excursion on pitch.

(3) Has very good pitch control.

(1) That wab tremendous amount of oscillation in it. The control motions were very exaggerated and it took large displacement on the control to try to follow up.

(2) Difficulty maintaining within the limits itch pitch wise, probably out on r .

(3) The cyclic response is not real good.

Abrupt immediate large changes produces di3astrous effects.

(4) With the severity of these oscillations, caused in this case, it would be virtually uncontrollable.

(1) It was not terribly difficult to track the command box. There was noticeable oscillation, due to the elasticity; but again, it was not too difficult to ignore that and to fly simpLY the rigid portion of the pitch profile.

(2) That was easier to control. The oscilla- tion due to elasticity was of main annoy- ance.

(3) In this case, tha aircraft would be quite controllable without an augmentation.

(1) I believe that this case is the same as Case 2 and Case 5 - not difficult ~t all to fly the command profile. This one I can hold almost at the cente~ all the ti~e precisely. There was a little more

oscillation involved due to elasticity apparently, but it was high enough frequency that it was easier to ignore that and simply to fly the rigid-body of the profile that is coming from the command.

All three performance measures, tracking error, Cooper-Harper Rating, and pilots' comments agree that C6 was generally worse than C3 which was worse than C4' C7, CS' CS' C2' and Cl.

Table 5.5 shows the summary for each of eight cases in three performance measures.

5.3 Time Histories A sample time history of six pertinent variables (8i' ee' nl~l" n2;2', a, 0e) is examined for eac~ of the eight cases to see the how and why of the differences in pitch angle error. Al l the sample time history plots did not come from one pilot.

Figure 5.3 is a sample time history plot for Case 1.

During the 120 seconds flying time, 6 is almost the same i

as a, due to the :act that 1l¢l' and T)2 2' are small and

Q do not contribute much to 9i. This case approaches that of a rigid-body. ea was easy to zero out; that is, the dynamic characteristics are good .

..

~ o C C2 difficult than than Comments oscillation of control Pilots' oscillation, to oscillation oscillation and oscillation oscillation tude amount i easy more more Conunents ampl little very annoying little little pilots' Good Large control Not Tremendous Very A An A Cooper-Rating, - Error, 2.0 5.9 2.0 2.3 1.6 3.1 6.7 1.9 Pilot Rating ---~----" Tracking of 8965 4782 ~----~ Degree 1.1516 1.0485 5.6723 1. 1.5069 7.5688 1. 1.1574 Summary Error 'l'racking - .0201 .0183 .0990 .0331 .0263 .1321 .0258 .0202 Radian - 5.5 -- of -- 2 r: J l 6 7 a - C C C) C4 C C C C - Tabie No. Case ~

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TIME (sec) Figure 5.38 Sample Time History - Case 1

' \ Future work should use a full state feedback control law to place the roots of the characteristic equation at precise values for each case. The rigid- body dynamics could then be maintained at their nominal values and the elastic mode coupled frequencies placed where desired. This would ensure that the pilot ratings would be based on the relative amplitudes of rigid and elastic pitch angle responses as presented on the EADI and not on poor rigid-body dynamics.

..... ~ ' .. .

BIBLIOGRAPHY 1. Bisp1inghoff, R. L., Ashley, H., and Halfman, R. L., Aeroelasticity, Addison-Wesley, 1955.

2. Chalk, C. R., et al, "Background Information and User Guide for MIL-F-8785B (ASG), 'Military Specifi- cation - Flying Qualities of Piloted Airplanes,'" AFFDL-TR-69-72, Air Force Flight Dynamics Laboratory, Wright-Patterson AFB, Ohio, 1969.

3. Wykes, J. H., "B-1 Flexible Vehicle Equations of Motion For Ride Quality, Terrai.n Following, and Handling Qualities Studies," Internal Document TFD- 71-430-1, North American Rockwell, Los Angeles, CA, 1971, revised 1973.

4. Swaim, R. L., "A Proposal for Research on Handling and Ride Qualities of Large Flexible Control-Config- ured Air~raft," Purdue University, July 1, 1974.

s. Crother, C. A., GabeL~an, B., and Langton, D., "Structural Mode Effects on Flying Qualities in Turbulence," AFFDL-TR-73-88, August 1973.

6. Swaim, R. L. and Fullman, D. G., "Prediction of Elastic-Airplane Longitudinal Dynamics from Rigid- Body Aerodynamics," Journal of Aircraft, Vol. 14, No.9, September 1977, pp. 868-873.

7. Roskam, J., Flight Dynamics of Rigid a~d Elastic Airplanes, The Un~versity of Kansas, Lawrence, Kansas, 1973.

8. Swaim, R. L., "Aircraft Handling Qualities and Stability Augmentation Systems," unpublished notes.

9. Silverthorn, J. T., "An Aircraft Simulator for the Study of Pilot-Display Interactions," School of Aeronautics and Astronautics, Purdue University, March 1975.

10. Carlborg, F. W., Introduction to Statistics, Scott, Foresman and Company, 1968, pp. 208-223.

' \ ,

I

11.

Dowine, N. M. , Heath, R. W., Basic Statistical Methods, Harper and Row, 1965, pp. 215- 2 22 12.

Anderson, Virgil L., and McLean, Robert A ., Desi~n of Experiments, Marcel Dekker, Inc., New York, 1 74.

13.

Seitz, W. R., "Flight Director Design for an STOL Aircraft," Ph.D. thesis, Purdue University, August 1971.

14. Blakelock, J. H., Automatic Control of Aircraft and Missiles, John Wiley and Sons, 1965.

I • APPENDIX

9S APPENDIX ANALOG SIMULATION The equations presented in Table 2.3 and equation (2.23) and (2.24) were normalized by the scale factors given in Table A.l and implemented on two Applied Dynamics analog computers. Patching diagrams of these equations are shown in Figures A.l - A.B. Included are the longitudinal equations of motion, flight director equation, pitch command input, and control inputs.

These differential equations were used to obtain the t:acking error of pitch angle, angle-of-attack, airspeed, vertical speed, and altitude. Table A.2 gives the poten- tiometer settings used for this simulation, and Table 2.3 gives the potentiometer settings for each of the eight cases.

Figure 5.4 is a sample time history plot for Case 2.

In this case n l ~ l' is slightly greater than that in Ca~e 1- However, its contribution to a· is still small.

ea ~an still be zeroed out easily.

Figure 5.5 is a sample time history for Case 3. In _ ..

, this is much greater than that 2, case nl4ll in Case causing greater contribution to ai. Significant changes in 6i are observed. The pilot has difficulty zer o ing out ee: ee is large. The difficulty in operating this case can be seen from the great movement of the control stick (as shown by 5 plot). Dynamic characteristics are poor e in this case.

Figure 5.6 is a sample time history plot for Case 4.

In this case the major contribution to ai is n2~2" How- ever, the effect of n292' is less than that of nlol' in Case 3; ai in this case is smaller than 6i in Case 3.

Even though the natural frequency of elastic mode 2 is reduced to a greater extent than that of elastic mode 1 in Case 3, the effect on 8i is less. eS is slightly greater than that in Case 2. The pilot has to work harder to control the pitch angle (as shown by 5 plot).

e Figure 5.7 is a sample time histor y plot for Case S.

Both ~l~l' and n2~2' contribute to 8i' However, their magnitudes are small. The pilot still can zero it out; ea is not quite 35 large. e e has an oscil13tion of small magnitude due to the interaction betw~en nl~l' and n2~2"

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8 2 This oscillation causes annoyance to the pilot but is o f no majo~ concern due to its small magni t ude.

Figures 5.8 is a sample time history pl ut for Case 6.

that in Case 5, causing greater contribution to 6i.

6· is significantly increased. The pilot can not zero it out: ee is large. The difficulty in operating this case can be seen from the greater m ovement of the control stick (as shown by 0e plot). Dynamic characteristics are the worst among the eight cases examined.

Figure 5.9 and Figure 5.10 are sample time histories for Case 7 and Case 8. nl~l' and n2~2' in Case 7 have greater contribut~on to 6i than that in Case 8. The phenomena of oscillation in nl ~ l' and n2~2' is present in both cases. Elasti~ mode 2 is slightly unstable in Case 7, while it is stable in Case 8. Howeve r , in both cases, the pilot can operate the aircraft with ease: ee is small in both cases .

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CHAPTER 6 SUMMARY, CONCLUSIONS, AND RECOMMENDATION 6.1 Summary The longitudinal tracking task was evaluated using a fixed base simulation of the aircraft equations of motion. Pilots' comments on the task revealed the effect of low frequency dynamic aeroelasticity on the haudling qualities and the pilot rating.

In Case 3 and Case 6 there is a large over shoot in the pitch angle error. These two cases received worse Cooper-Harper ratings indicating significant difficulty due to mode interaction. In Case 5, Case 1, and Case 8 there is not too much error in the pitch angle and the Cooper-Harper ratings are fair, but the pilots' co~~ents are that it was annoying due to the oscillation from aeroelasticity. For the rest of the cases, the aero- elasticity has little effect on handling qualities and pilot ratings because of the minimal interaction with the rigid-body dynamics.

6.2 Conclusions All conclusions are restricted to a constant altitude trim flight condition, which is the only flight condition which was investigated.

The most obvious conclusion is that the natural frequencies of elastic modes that were investigated repre- sent good to very bad pilot ratings, primarily because of adverse aeroelastic mode interaction effects with rigid- body dynamics.

Elastic mode 1 affects handling qualities and pilot rating more than elastic mode 2 when the natural frequen- cies of the two elastic modes are reduced to the same numerical value.

From the point of view of handling qualities of rigid-body dynamics, Case 3 seems to have better handling qualities than Case 4. However, it is not true. So we , , have to look at ~l~l and "2$2 in pilot time history for Case 3 and Case 4, respectively. The 8 was affected i more in Case 3 than in Case 4 since "l$~ contributes , more to the value of 8 in Case 3 than "2 ~ 2 contributes i to the value of 8 in Case 4.

i As seen in Case 7 and Case 8, the pilot rating is the same as the original case (Case 1). From the pilot time history and commeHts we know the small elastic oscillation can be visually separated from rigid-body \ I (

dynamics by the pilot. The elasticity is only slightly annoying to the pilot since the elastic modes do not significantly affp.ct the rigid-body dynamics.

In Case 6 both natural frequencies of the two elastic modes were reduced to a still greater extent. This is the worst investigated and resulted in the poorest pilot ratings.

The results of the study indicate that handling qualities and pilot rating are functions of the natural frequencies of the elastic modes. The lower the value of natural frequency of an elastic mode, the worse the handling qualities and pilot ratings. This was due to adverse coupling effects on the phugoid dynamics and a larger contribution from the elastic mode amplitudes to the total pitch angle 6i' 6.3 A Recommendation In this study, the natural frequencies of the elastic mod~s were lowered from their nominal values to values close to the nominal short period frequency. However, this resulted in mode interaction which lowered the short period frequency and caused the phugoid mode to split into positive and negative real roots. Much of the pilots' difficulty in tracking on the worst cases was thus due to the positive phugoid root.

Table A.l Normalization of Variables Variable ~ .... ro . aliza tion 10v volts • u 400 tt/sec a rad 0.2 6 0.6 rad • 0.2 rad e 0.6 rad Tl1 • SO Tl1 Tl2 • SO Tl2 h 6000 ft 1. 0 rad e 50,000 lba cSt

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Settings Potentiometer Table A.2 , .123 2B1 2Al .144 J .; .500 2B2 2A2 .029 , .032 2B3 2A3 .130 2B4 .310 2A4 .121 .224 2B5 2A5 .943 .140 2B6 2A6 .453 .122 2B7 .053 2A7 .913 2B8 .233 2AS .849 2B9 2A9 .038 .048 2B10 .013 2A10 .753 .446 2C1 202 .333 .500 2C2 .456 .012 2C3 .707 2C4 .295 .202 .533 2C5 .923 .355 2CG .186 .114 2C7 .104 .308 2C8 .174 .915 2C9 .025 2C10 .052 lFl .625 lA1 .052 1F2 .725 1A2

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