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STUDY ON USING & DIGITA_ RIDB QUALITY

&UGD4ENT&TIONSYSYEMTOTRIMANlCNGINE-OUT

IN & ¢ZSSN& 402B

rd-FRL-6132-3

._ by: Kent Z. DonaZdson

Supervised lo_r: Dr. Jan Roskam

..... ,!

f,

UNIVERSITY OF KANS&S CDITER FOR RI_SEKRCH, INC.

Flight Research Laboratory !

Lavrenco, Kansas
STUDY ON USING A DIGITAL RIDE QUALITY AUGMENTATION

SYSTEM TO TRIM AN ENGINE-OUT IN A CESSNA 402B

ABSTRACT

A linear model of the Cessna 402B was used to

determine if the control power available to a Ride

Quality Augmentation System was adequate to trim an

engine-out. Two simulations were completed: one using a

steady-state model, and the other using a state matrix

model. The amount of rudder available was not

,- sufficient in all cases to completely trim the airplane,

:_ but it was enough to give the pilot valuable reaction

time. The system would be an added measure of safety

J

for only a relatively small amount of development.

.I

:X i
TABLE OF CONTENTS

". ABSTRACT ....... i

LIST OF TABLES .... iv

" LIST OF ABBREVIATIONS S O S v

_- 1 • INTRODUCT ION ................ 1

2. RIDE qUALITY AUGMENTATION SYSTEM ...... 2

• _., 3. LINEAR STEADY-STATE MODEL .... 2

'_: 3_ 5

l- 4. SMALL PERTURBATION MODEL ..... I0

i 4.1 E0UAT_OOF.SM_IO.. 10
LIST OF FIGURES

1 Cessna 402B Three-view ........... 3

_' 2 Airplane Coordinate Systems ......... ?

cr

• 3 Rudder required for an engine-out

r

on a Cessna 402B .............. 9

" 4 Time history of an engine-out in a

ROAS controlled Cessna 4C2B ......... 14

:' 5 RQAS Block Diagram ooooooeoooeoe 17

_, A.I Approximate Cessna 402B Lift-Curve ..... A.2

E'-

A.2 Propel!er Efficiencies ........... A.4

1.

:• iii

!. I
LIST OF TABLES

1 Control Surface and Actuator Requirements . . 4

2 Small Perturbation Matrices . . , ...... 11

A.1 Lateral-Directional Non-dimensional

Stability Derivatives . . . . . , ...... A.3

A.2 Variation of Derivatives with Vertical

Tail Size . . ....... . . , ...... A.6

A.3 State Matrices ................ A.7

A.4 Lateral-Directional Dimensional

Stability Derivatives ..... . ...... A.8

A.5 Modified Lateral-Directional Dimensional

Stability Derivatives . . . . . . ...... A.9

iv
i

'" LIST OF ABBREVIATIONS AND SYMBOLS

ABBREVIATIONS

FCl Flight Condition #1

ICAD Interactive Control Augmentation Design

C

_' RQAS Ride Quality Augmentation System

_ SYMBOLS

I"

_ii A Continuous State Matrix

i: b Wing Span, ft

tl

B Continuous Control Matrix

'_:i: CI B Variation of Rolling Moment Coefficient with

_._ Sideslip Angle, tad-1

'_

_ CI6DF Variation of Rolling Moment Coefficient with

Differential Flap Deflection, tad-I

_. C16R Variation of Rolling Moment Coefficient with

Rudder Deflection, rad-I

il CnB Vari Sideslip ation Angle of Yawi , tad ng -IMoment Coefficient with

"i_,_ Cn6DF Variation of Yawing Moment Coefficient with

Differential Flap Deflection, rad-I

1 Cn6R Variation of Yawing Moment Coefficient with

i" Rudder Deflection, tad -1

CyB Variation of Side Force Coefficient with

'I Sideslip Angle, tad-I

CY6DF Variation of Side Force Coefficient with

i Differential Flap Deflection, rad -1

"1" V
CY6R Variation of Side Force Coefficient with

Rudder Deflection, tad -1

FYT Side Force due to Asymmetric Thrust, lb

g Acceleration of Gravity, ft/sec 2

LT Rolling Moment due to Asymmetric Thrust,

ft ibs

m Airplane Mass, slugs

NT Yawing Moment due to Asymmetric Thrust, ft ibs

AND Yawing Moment due to Drag of Inoperative

Engine, ft Ibs

p Perturbed Roll Rate, rad/sec

q Dynamic Pressure, ib/ft 2

r Perturbed Yaw Rate, rad/sec

S Reference (Wing) Area, ft2

u Control Vector

VMC Minimum Control Speed, ft/sec 2

VS Stall Speed, ft/sec 2

x State Vector

GREEK SYMBOLS

B Sideslip Angle, deg, tad

Glide Path Angle, deg, tad

_DF Differential Flap Deflection, deg, tad

6R Rudder Deflection, deg, tad

% Bank Angle, deg, tad

vi
h

I

1. IHTRODUCTIOH

Due to the large increase in the commuter airline

industry, with the federal deregulation of major

carriers in 1978, more people are riding in small,

short-haul, propeller-driven aircraf:. This has caused

an increased effort to make riding in such an aircraft

t as pleasant as possible. One projec: undertaken has

been the development of a Ride Quality Augmentation

System (RQAS). This system uses acceleration feedback

to lessen the ef acts of turbulence by counteracting the

undesirable accelerations with appropriate separate

surface control deflections.

It was felt by the developers of this system that

since the control power was available, it would also be

beneficial to use this system to trim the airplane in

[

the event of an engine-out. This was the basis for the

following investigation into using a RQAS to control a

Cessna 402B in an engine-out flight condition. The

investigation was conducted as part of NASA grant NAG1-

345. Mr. Lou Williams is the grant monitor.

< Chap.er 2 of this report describes the proposed

_=c RQAS for the Cessna 402B. The models used to evaluate

the system in an engine-out are then given: the steadystate model in Chapter 3 and the small perturbation
model in Chapter 4. The results of using these models

for _n enginemout are then discussed in Chapter 5.

2. RIDE QUALITY AUGMENTATIONSYSFEM

Reference 1 documents the proposed control surface

.

modifications for the RQAS in the Lateral-Directional

mode (see Figure 1):

1. Replace the outboard section of the split flap

with a differentially deflecting plain flap that

can deflect +15 to -45 degrees.

2. Use the entire existing rudder (limiting the

RQAS range of deflections to _5 degrees).

The outboard flaps were computed to have 67% of the

control power of the ailerons in the nonlinear model.

Table 1 summarizes the control surface and actuator

requirements for the RQAS of reference 1.

). LINEAR STEM)Y-STATE MODEL

A nonlinear simulation model of the Cessna 402B

(Ref. 2) was used to obtain a linear model about its

e

most critical condition for an engine out. That is_

1. sea level_

2. flaps down_

3. full throttle_

• 4. maximum landing weight.

2
_Je
it Table I Control Requirem Surface and Actuator ents

Control Surface Deflection and Rate Limits

!

J Surface Deflection Maximum

_i Rangeldeg) Ratelde_/sec)

SS Elevator +5 50

Flap(inboard) _15 to -45 120

t Differential Flap ,15 to -45 120

_°/I Rudderl _32 50

I Actuator Requirements

"/ Surface Max Load Speeed Stroke

_iI (lbs) (in/sec) (in)

SS Elevator 65 3.50 0.75

t Flap(inboard) 750 8.50 4.25

_:: Differential Flap 380 8.50 4.25

I Rudder 520 3.50 4.50

i

it 1 The deflection is for the standard rudder. The RQAS .:

uses a deflection range of _5 degrees.

o

ii

l '

.i

H_" ,,, 4 ,!
!,

3"

.,li _ This will be called Flight Condition #1 IFCl) and is

|-

defined as=

t_ One Engine Out V - 130 fps .. - 0 ft

Full Throttle _cg " 0.25 W - 6200 Ibs _

__ Full Flaps _

i The stability and control derivatives given in Appendix i

:* A, Table A.I are for the maximum landing weight.

f

• Because the maximum landing weight differs ft'om t_:_,

maximum takeoff weight by less than 2%, the _;. alues

_! were used without correction. _l_qi '

#1 WOUI_

i be the condition in an emerge_.y go-around. 3.1 STEADY-STATE EQUATIONS

The basic assumption made to determine needed

" control surface deflection was that the airplane motion

•l could be modelled about a steady-state point as a set of

_ first-order differential equations, as shown in Eqn. 2.1

t" for an engine-out flight condition.

Io,O ,o, ,o.. i

i LCn. ¢"'D,%_ ['_ -_

These equations have been uncoupled from the full 6

t degree of freedom equations by choosing bank angle, 0.

i They are written in the stability axis system (see

t

l•
,q

section 3.2). Their derivation can be found in

reference 3.

3.2 COORDINATE REFERENCE SYSTEM

The body-axis system is an orthogonal, right-hand

set of axes with its origin at the airplane's center of

mass. The X-axis lies along the centerline of the body.

The X- and Z-axes lie in the airplane plane of symmetry,

while the Y-axis is pointed out the right wing of the

airplane. This can be seen in Figure 2.

The non-dimensional derivatives listed in Appendix

I A, Table A.I are given in the stability-axis system.

This system is also an orthogonal, right-hand set of

6 axes with its origin at the center of mass of the

airplane. The difference between the body- and the

"i

.It stability-axis systems is that the stability X-axis is

oriented in the direction of the steady-state velocity

of the airplane on its XZ-plane. They both share the

same Y-axis as seen in Figure 2.

3.3 DEVELOPMENT i

The thrust was calculated us'ng an engine model and

P

its average propeller efficiency. The average propeller

efficiency is greater than the actual propeller

efficiency by approximately 6%t therefore, the actual

pitching moment and yawing moment during engine-out ,i

f

6
_k

Q

¢

i Body Axis Coordlnat- System

Stability Axis Coordinate System

I

FIGURE 2 &irplane Coordinate Systems :

!_

J

7 i

i " ,i
i" t

i

m

would be slightly less• The propeller efficiencies are

Q

given in Appendix A.

By assuming a weight and flight condition, the

,_" angle of attack was obtained from the airplane liftcurve slope and intercept as shown in Appendix A. Some

• of the derivatives are functions of angle of attack.

Once the angle of attack was calculated, the non-

;J dimensional derivatives were obtained from reference 2 •

i' i

They are given in Appendix A.

i

i By Varying the speed and the vertical tail size and

: solving Equation 2.1 as shown in Appendix A, the

!

sideslip, rudder, and differential flap deflections were

i! determined. Figure 3 shows how rudder deflection varies

• !

with flight condition•

T

i 3 • 4 RESULTS

It can be Seen from Figure 3 that at speeds below

approximately 125 fps, there is not enough rudder to i

keep the airplane in straight-line flight. This is the i

_.._ minimum control speed, VMC. This is 15 fps less than _ '_

• e !

,ilj the minimum control speed'given in the operating _

.] handbook; therefore, a minimum control speed of 130 fpa :_

•._ is still conservative At 130 fps, it was found that to ii

i fly straight with a bank angle, _w of -5 ° required 30 °

•.i: of rudder deflection, 7.5 ° of differential flap i

iI 1

J !

i ....................................................................................... I'!
P

+,+

rlGURK 3 Rudder required for an engine-out f

on a Cessna 402B :_

_ ...... =+ _ ==: -+ , +-, + + - - =' :-" = ..... =+ = +:='-" +:i=
i

required was also found to be within federal

regulations.

It can also be seen from Figure 3 that if onefourth of the vertical tail were removed, the airplane

would no longer meet FAR 23 requirements. FAR 23

I requires that the minimum control speed, VMC be greater

than 1.2V S. This model does not account for any

i transient phenomena of the airplane in reaching its steady-state condition.

_i 4. SMALL PERTURBATION MODEL

The non-linear simulation model was also used to

i- develop a linear model to study the dynamic behavior of

the Cessna 402B in open- and closed-l_op simulations.

I This was only done for FCI.

4.1 EQUATIONS OF MOTIO_

From the assumptions made in the steady-state

! model, a new set of matrix equations can be written:

i - A x u (3.1)

t• where 0

x* - {B, P, r, _}, and

I

_I U* - {6DF, 6R}.

Derivations of this equation can be found in

reference 4. These matrices are defined in Appendix A.

1.1 The matrices used for FC1 are given in Table 2. I

ii

N
C1

i!_ Table 2 Small Perturbation Matrices

Cessna 402B (FCI)

, !

; o

, ---0.105 -0.000867 -0.9899 0.245"

!: -1.329 -1.752 0.483 0.0

i ! 1.408 -0.0428 -0.299 0.0

', 0.0 1.0 0.149 0.0 u

iT -0.968 0.0 0.0306 0.206

i:i o.o,o - 0.0 0.0

" 0.0

;!I_ -o.o_68

D =

O.445

il o.o

ill"

'! _ 11
Equation 3.1 is written with the following assumptions_

1. perturbations are small, and

2. initial condition is a straight-llne

trimmed flight condition.

4.2 COORDINATE REFERENCE SYSTEM

All of the dimensional and non-dimensional

derivatives were calculated in the stability-axis

system of Figure 2. The instruments onboard the

airplane will sense the body-axis motion. This can be

simply transformed to the stability-axis by rotating

about the Y-axis by the airplane angle of attack.

4.3 DEVELOPMENT

To simulate an engine-out situation using the state

matrices, a disturbance matrix, D, was added to the

state and control matrices in Equation 3.1 yielding

Equation 3.2. This disturbance matrix was made up of

the constant angular accelerations imparted on the

airplane in the pitch and yaw directions due to the

engine-out. In all the cases, this matrix was commanded

to "turn on" at one second into the simulation.

X " A x + B u + D W (3.2)

where

w = {0or i}

The open-loop response of the airplane with the

addition of the disturbance matrix was calculated using

12
the Interactive Control Augmentation Design program

(ICAD)[5]. This was done on the University of Kansas

r

School of Engineering's Harris computer system. The

closed-loop response was also done on ICAD.

As allowed by FAR 23, a bank angle of -5° was used

to lower the rudder required as much as possible. This

turned out to be a difficult task on ICAD as currently

i written. Ideally, bank angle, sideslip, and yaw rate

should be driven to -5", 0, and 0 simultaneously. To

get these, only bank angle was commanded and all the

variables except roll rate were weighted heavily. This

gave large transient values for control positions and

e:

i

i rates, but in an actual system this would not happen.

Figure 4 shows an example time history. The average

values of control deflections and control rates have

been drawn to show an approximation of what would

actually happen.

It can be seen that to trim the airplane requires

all 32 ° of the rudder to be available to the RQAS. As

proposed, only 5 ° of rudder deflection is available to

_ the RQAS_ therefore, time histories were also simulated

allowing only 5 and 15 degrees of rudder deflection.

With 15 ° of rudder, the RQAS would be able to trim an

engine out at speeds above about 170 fps. Five degrees

13
•¢.i i ' I ' i J ' i ' i "" ,

Time t, (Seconds) Time t, (Seconds)

°e4 _p

-4' _ 0 "-- ----------- ....

n • u ii n

"6 ' I ' l ' I ' I ' l -_ " I ' ! ' I ' | i

Time t, (Seconds) Time t, (Seconds)

LN- I-.

" I__ _ "'°"

"t_ ' I ' I ' i ' i ' I _i _ I ' I ' i " I ' I

O _ 4 6 e _ i 4 i i I

Time t, (Seconds) Time t, (Seconds)

FIGURE 4 Time history of an engine-out in a

RQAS controlled Cessna 402B

, 14
J

d

only slows the rates and could not trim the airplane 1

below its "never exceed" speed.

5. DI SCUSS l ON OF RESULTS

The control deflections obtained for FC1 were:

Model B 6R 6DF

(deq) (deg) (deq)

Steady-State -2.9 30.4 7.5

l Small Perturbation -5.0 30.0 5.0

These values were obtained for a steady-state velocity

of 130 fps. The difference in the values between the

I two models is due to the fact that in the small

i perturbation model, yaw and pitch rate are not exactly

zero. The bank angle is also not quite -5 °. It can

i also be noted from Figure 4 that the RQAS has the

airplane trimmed in a fraction of a second, much faster !

than a human pilot could react. This result is obtained

i

l assuming the servo is modelled as a first-order lag.

The slowing of the yaw rate due to only 5" of i

'

i give the pilot valuable to

rudder deflection would time

react to the increasing sideslip. In addition, the

J differential flaps have far more control power than is

needed to control roll and bank angle. The quicker the

airplane reaches a bank angle of -5 °, the longer the

r

t pilot has to respond to the yaw.

l 15
The RQAS has two obviou_ advantages. First, i£ _be

dynamic pressure is high enough and the available rudder

is adequate, the RQAS could control an engine-out

without the pilot touching the controls. Second, the

RQAS would give the pilot more time to react to the

engine-out even if the RQAS was not able to trim the

airplane entirely.

In addition, the RQAS will introduce digital

computers into commuter airplanes which could then be

used for many other jobs which of themselves do not

warrant the expense of a computer.

If only 5° of rudder deflection is deemed adequate,

no modifications need to be made in addition to those

proposed for the RQAS. The RQAS block diagram is shown

in Figure 5. This system vould treat the engine-out as

!

a large disturbance. The gains chosen in the normal use

of the RQAS, however, might not be suitable for the

engine-out. In that c&se, engine-out sensors would be

required to tell th_ system when to change gains.

Giving more c_nt_ol power to the system for an

engine-out could be done in two ways. First, the R_S

could be given the authority to move the rudder more

than 5". This would require a proportionately stronger

actuator, but it would not change the stroke given in

Table I which is for 32". Secondly, an engine-out

16
r 111
" sensor could trigger a change in the amount of rudder Q

allowed so that the RQ_S is still only allowed 5 ° while

it the engine-out is allowed more.

No additional software wilI be required if gain

scheduling is not needed or is already incorporated in

o_

the RQAS. The components needed for this system are a

4

- hydraulic pump to be mounted on one of the engines, an

, accumulator, and a set of electro-hydraulic actuators.

The design of the actuator and its implementation will

_ be contracted to Cessna. In addition, a set of engine-

__ out sensors vilI be required. Reliability, ease of

implementation, and cost of these should be

'-f-I investigated.

6. CONCLUSXONSaND _CO__, Us_._cH

"" 6.1 CONCLUSIONS

-/I The following conclusions were reached:

t 1. The RQAS with control of all 32 ° of rudder

deflection can trim an engine-out in any steadyi state condition. flight

2. With only the proposed 5 ° of rudder deflection,

I_" the RQAS can slow the divergence of the airplane

i

i_ significantly.

i 3. Using only the proposed control surface

modifications, implementing the engine-out

18
i,I,

I capability would require a sot of engine-out Q

sensors and the incorporation of gain scheduling.

I 4. If it is deemed necessary to use more than the 5

degrees of rudder needed by the RQAS, the actuator

would need to be proportionately stronger.

6.2 RECOMMENDEDRESEARCH

:"i!'" the followingFUrther deve inclopment Udesteps_ of this project must

1. Perform a preliminary failure analysis for

rudder deflections greater than 5 degrees.

i!I 2. Use the 9ains required at the most

i critical condition to see their effect in other

conditions,

3. Examine and evaluate options available to

sense an engine-out,

_t 4. Investigate the possibility of increasing

:..! the amount of rudder deflection available to

t

_'_ the RQAS, "i

i 5. Design and build the necessary hardware,

,_it Cessna 6. PLrfor402B. m a flight testthe, of the system on

• ¢

i

.i

• l:

f

?
t

F

t

References:

. i

1. Downing, D.R.; Davis, D.J.; Linse, D.J.; Entz,

D.P.: "Pr_)liminary Control Law and Hardvare !

! Designs For a Ride Quality Augmentation System i

For Commuter Aircraft", NAG1-345, Feb 1986.

2. Hob, R.H.; Mitchell, D.G.; Myers, T.T.:

"Simulation Model of Cessna 402B". NASA CR 152176,

July 1978.

3. Roskam, J.: Airplane Flight Dynamics and Automatic

Flight Controlsf Part I. Roskam Aviation and

Engineering Corp. 1979.

4. Roskam, J.: Airplane Flight Dynamics and Automatic

Flight Controls r Part II. Roskam Aviation and

Engineering Corp. 1979.

5. Hammond, T.A.; Amin, S.P.; Paduano,.J.D.; Downin_

D.R.: Design of a Digital Ride Qualxty Augmentation

i System For Commuter Aircraft, NASA CR 172419, Oct

1984.

i 6. Davis, D.J., "A Comparison of Two Optimal Regulator

i Design Techniques for the Weighting of Outpu_

Variables Which are Linear Combinations of States

and Controls", M.S. Thesis, The University of

Kansas, Lawrence, KS, 1986.

7. Hoak, D.E. et al; USAF Stability and Control DATCOM

Wright Patterson Air Force Ease, Ohio, 45433,

April, 1976.

" J_,t 20

•
- APPENDIX A Cessna 402B Stability Derivatives Q

This appendix gives the method u_ed for estimating 1

i the airplane angle of attack and determining its

stability and control derivatives.

! Because weight, wing area, and dynamic pressure are

known, the airplane lift coefficient can be found from

the equation_

WsCLqS

From this lift coefficient, the &CL due to flaps from

reference 2 was subtracted. This lift coefficient was

_ then found on Figure A.I and its corresponding angle of

i. attack was read. The airplane angle of attack was then

_ used to obtain the non-dimensional derivatives from

4

reference 2. These are listed for an angle of attack,

i a, of 8.5 ° in Table A.I.

The engine model was taken from reference 2. This i

gave the maximum power of a C402B engine as 300 brake

i horsepower. The propeller efficiencies, average and

actual, are given in Figure A.2. Thrust was then

! calculated using the following equationz

, 1

T m 550 Bllp np (A.I) _

V

i

where

np - Propeller Efficiency 

' _ V - Airplane Speed

i,[ BlIP - Engine Brake Horsepower

i,. i Aol

[j-:
i1

1

. i_ I_ ....... _Z., ' CLw_ ,

_4

1 I.O ....

,I

I

"i!

i

m

: •

J

• !

i"

.4 i i ii

i

, , [ ,

r ....

I r ' -'

' -S ' S I0 IS 20 25

! a(deg)

!" !

1

i.t!

_ ,

. _

;_" I! Figure A.1 &pproxtmate Cessna 40211(Tall-Off) Lift-Curve

:T

,:, , -, = r .... :- = - " . = = c= , : ,= ..... ' ": :
I

; Table A.1 Latera Stability l-DirectionalDerivatives Non-dimensional .

!

1

Cessna 402B ._

One Engine Out V - 130 fps h I 0 ft

Full Throttle _cg = 0.25 W - 6200 Ibs .

I Full Flaps

•I! DerivatiVes

i . _ !

i Cy B _ -0.670 tad-I Cyp -0.00063

CY6R - 0.195 tad-I Cy r i 0.42

I! CY6D F - 0.0 rad-I Cnp - -0.084

Cn B - 0.129 tad-I Cn r - -0.170 1

li Cn6R - -0.0795 tad -1 Clp - -0.81 i

--, Cn6D F - 0.0057 rad -1 C1 r s 0.216

4

Cl B - -0.0888 tad-I 1

C16R - 0.0146 tad -1 _ I

:i

Cl6D F I -0.0685 tad -1 _ i

I

.t

• _
Propeller Efficiency at Several Flight

Conditions

e_ i , , ,, i i

|

L /

• J

IO0 ISO 20O 210

Trim AlrspNd (ftAllle)

,i

Assumed Average Propeller Efficiency for All ,_

* Flight Conditions '

Figure A.2 propeller Efficiencies ,

Ii A,4
[| From Figure I, it can be seen that the thrust from one

engine creates a moment about the airplane center of

! gravity with a moment arm of 7.50 fee_. The orientation {

of the thrust line of one of the engines is given in

jl reference 2. By finding the X and Z components of the

.... _ thrust, the moments, NT and LT, can be found from: !

;i

l NT = 7.50 Tx, and

Jl LT = 7.50 T z.

Reference 2 also gives the AC D due to an engine-out.

"!! AN D in Equation 2.1 is the yawing moment due to drag on

the _noperative engine and is:

[

l AND = 7.50 ACD q S

Equation 2.1 was then solved to obtain the sideslip,

__.i

_:. differential flap deflection, and rudder deflection

_ F required To account for changing the vertical tail

F size, the non-dimensional derivatives were recalculated

and are summarized in Table A.2.

_i The dimensional derivatives were then calculated as

shown in Table A.3. These were calculated using:

i

Ixx " Iii00 slug ft 2

Izz - 14900 slug ft 2

Ixz - -583 slug ft 2

i! To use these in the state matrices, Table A.4, they must

i

be in the form shown in Table A.5, where,

_I A1 = Ixz/Ixx and B1 =

Izz/Izz
Table A.2 var_atio_ of Der_vative_ wi_n

Ver_=al Tai_ S_ze

Cessna 4023 (FCI)

V.T. Size

i

" 0.50 0.0008_ O.Cr;l_5 -O.O0J'_ _.DO_iS_

0.75 0.00165 0.0026E _r 0_ _ L.O002_:

1.00 0.00240 0.0033C -0.0013_ C.0003i_

1.25 0.00316 0.00413 -0.001_ C.OOO3B_
6

Table A.3 State Matrices "

P

i=

1"

i

m

m

" L_" _" L/ o p

A = X =

.. Ni N "_" o [J i. 0 1 tan81 0

B s o

0 0

;[1 -

A.7
Table A.4 Lateral-Directional Dimensional Stability

Derivatives [3]

qlSbCr'6A (sec -2 or

;lscy

" ¥_ =-- (fC sec "2) L6A- m Z

xx sec " de8 "l)

• qlSbC

¥p • 2mU1YP (fc sec"I) qlSbCg-6R

- Csec"2 or

L_K Ixx __

sec " de_'')

qlSbCy r

¥r = 2mU1 (fc sec "1) _lSbCn s

NS = I (see'2)

qlSCY_A zz

(fc sec "2 or

I

¥6A m £t: sec"2 des"l) qlSbC"

. = (sec -2)

NTs Izz

qISC7_R (fusec "2 or

Y6R • " m fc sec"2 des-1) qlSb2Cnp S = (sec -I)

P 21_zzU1

qlSbCg'_ (sea-2)

L_ • _XX 11r

_lSb2C

N = (se¢"1)

r 21zzU 1

. qlSb2C_.

P (sec-l) qtSbCn6_

+ II •

Lp 2[xxUl

• • ", (sec"2 or

_lSb2Cg" N6A Izz sec-2 deg "1)

tr . 2IxxO1r (eec-l) qlsbc

n6R (sec"2 or

N =- t -2

+R zz sec de_"t)

A°8

+
Table k.5 Modified Lateral-Directional Dimensional

Stability Derivatives [6]

P

0

:: YB = YB/UI L6df = (AIN6df*L6df}/(I-AIBI)

:" y_ - Yp/U 1

L_'sr = (AIN_sr 

#

Yr = (Yr/Ul)-i

, Ns"= (BILB 11 

Y¢ = gcos81/U I

, NpS = (_ILp+Np)/(I-AIBI)

Y_df = Y6df/Ul

:, Nr" = (BILr 

0

Y_sr = Y&sr/U1 • = ( )/(I_AIBI )

: L_ = (AINB+L_)/(I-AIB I)

: N_sr = (BIL6sr+N_sr)/(I-AIBI)

L_ - (AINp,Lp)/(I-AIBI)

Lr - (AINr+Lr)/(I-AIB I) •

"i"