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(N&SA-CB- 177272) SIOOX ON OSIIIC i nrnTw.T 

HC A03/HP iOI ' '^ 

CSCI OIC G3/08 



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1186-26342 



Dnclas 
43603 



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ThE UNlvERsliy of Kansas Center For ResearcN, Inc. 

2291 Irving Hill Drive-Campus West Lawrence. Kansas 66045 



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STUDY ON USIH6 K DIGITAL ftlDE QUALITY 
AUGMENTATION SYSTEM TO TRIM AN ENGlNE-OUT 
IN A CESSNA 40 2B 

XU-FtU.-6132-3 



by: K«nt E. Donaldson 
Supervisad by: Dr. Jan Roskam 



THE UNIVERSITY OF KANSAS CENTER POR RESEARCH, INC. 

Plight Raatarch Laboratory 'i 

Lawranca, Kansas 

t 



. .i.;.' 



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 rxidder 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 
for only a relatively small amount of development. 



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TABLE OF CONTENTS 

ABSTRACT i 

LIST OF FIGURES iii 

LIST OF TABLES iv 

LIST OF ABBREVIATIONS AND SYMBOLS v 

1. INTRODUCTION 1 

2. RIDE QUALITY AUGMENTATION SYSTEM 2 

3. LINEAR STEADY-STATE MODEL 2 

3.1 STEADY-STATE EQUATIONS 5 

3.2 COORDINATE REFERENCE SYSTEM 6 

3.3 DEVELOPMENT 6 

3.4 RESULTS B 

4. SMALL PERTURBATION MODEL 10 

4.1 EQUATIONS OF MOTION 10 

4.2 COORDINATE REFERENCE SYSTEM 12 

4.3 DEVELOPMENT 12 

5. DISCUSSION OF RESULTS 15 

r 6. CONCLUSIONS AND RECOMMENDED RESEARCH .... 18 

6.1 CONCLUSIONS 18 

6.2 RECOMMENDED RESEARCH 19 

I REFERENCES 20 

APPENDIX A Cessna 402B Stability Derivatives . . A.l 



[• 



ii 



■4 



LIST OF FIGURES 

1 Cessna 402B Three-view 3 

2 Airplane Coordinate Systems 



•» 



3 Rudder required for an engine-out 

on a Cessna 402B 9 

4 Time history of an engine-out in a 

RQAS controlled Cessna 4C2B 14 

5 RQAS Block Diagram 1"^ 

A.l Approximate Cessna 402B Lift-Curve A. 2 

A. 2 Propeller Efficiencies A. 4 



iii 



LIST OF TABLES 

1 Control Surface and Actuator Requirements . . 4 

2 Small Perturbation Matrices li 

A.l 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 



LIST OF ABBREVIATIONS AM) SYMBOLS 

ABBREVIATIONS 

PCI Flight Condition #1 

ICAD Interactive Control Augmentation Design 
RQAS Ride Quality Augmentation System 
SYMBOLS 

A Continuous State Matrix 
b wing Span, ft 
B Continuous Control Matrix 

Cig Variation of Rolling Moment Coefficient with 
Sideslip Angle, rad"^ 

Ci Variation of Rolling Moment Coefficient with 
"DP 

Differential Flap Deflection, rad"^ 
Cig Variation of Rolling Moment Coefficient with 

Rudder Deflection, rad"^ 
Cnp Variation of Yawing Moment Coefficient with 

Sideslip Angle, rad^^ 

Cn. Variation of Yawing Moment Coefficient with 
"DP 

Differential Flap Deflection, rad"^ 

Cn. Variation of Yawing Moment Coefficient with 
"R ♦ 

Rudder Deflection, rad"^ 

Cyp Variation of Side Force Coefficient with 

Sideslip Angle, rad'^ 

Cy Variation of Side Force Coefficient with 
"DP 

Differential Flap Deflection, rad'^ 



P 



R 



9 



variation of Side Force Coefficient with 
Rudder Deflection, rad" 
Side Force due to Asynunetrlc Thrust, lb 
Acceleration of Gravity, ft/sec^ 
L^ Rolling Moment due to Asymmetric Thrust, 

ft lbs 

Airplane Mass, slugs 

Yawing Moment due to Asymmetric Thrust, ft lbs 

Yawing Moment due to Drag of Inoperative 

Engine, ft lbs 

Perturbed Roll Rate, rad/sec 

Dynamic Pressure, lb/ft 

Perturbed Yaw Rate, rad/sec 

Reference (Wing) Area, ft^ 
u Control Vector 
Vmc Minimum Control Speed, ft/sec 
Vs Stall Speed, ft/sec^ 
X State Vector 

GREEK SYMBOLS 

Sideslip Angle, deg, rad 

Glide Path Angle, deg, rad 
5jjp Differential Flap Deflection, deg, rad 
iJ^ Rudder Deflection, deg, rad 

^ Bank Angle, deg, rad 



m 

Nt 

ANd 

p 
q 

r 
S 



If 



vi 



1. UtTRODUCTIQN 

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 aircraft. This has caused 
an increased effort to make riding in such an aircraft 
as pleasant as possible. One project undertaken has 
been the development of a Ride Quality Augmentation 
System (RQAS). This system uses acceleration feedback 
to lessen the efscts 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 NAGl- 
345. Mr. Lou Williams is the grant monitor. 

Chapter 2 of this report describes the proposed 
RQAS for the Cessna 402B. The models used to evaluate 
the system in an engine-out are then given: the steady- 
state model in Chapter 3 and the small perturbation 



model in Chapter 4. The results of using these models 
for an engine«*out are then discussed in Chapter 5. 

2> RIDE QOALITT AUGMEWTATION SYSTEM 

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. 

3. LINEAR STEADY-STATE MODEL 
A nonlinear simulation model of the Cessna 402B 
(Ref . 2) was used to obtain a linear model about its 
most critical condition for an engine out. That is: 

1. sea level; 

2. flaps down; 

3. full throttle; 

4. maximum landing weight. 



X-BOOY AXIS 




SASdOM) 



IJ2 



PIGURS 1 csiana 402B Thr«a-vitiw 



I 



Table 1 Control Surface and Actuator 
Requiremonta 



Conti'ol Surface Deflection and Rate Limits 



Surface 




Deflection 
Range (deg) 


Maximum 
Rate(deg/sec) 


SS Elevator 


±5 


50 


Flap (inboard) 




fl5 to -45 


120 


Differential 


Flap 


♦15 to -45 


120 


Rudder^ 




+32 


50 



■1 



Actuator Requirements 



Surface 




Max Load 
(lbs) 


Speeed 
(in/sec) 


SS Elevator 


65 


3.50 


Flap (inboard) 




750 


8.50 


Differential 


Flap 


380 


8.50 


Rudder 




520 


3.50 



Stroke 
(in) 



0.75 
4.25 

4.25 
4.50 



^ The deflection is for the standard rudder, 
uses a deflection range of +5 degrees. 



The RQAS 



I II I H PIII t !! ]■ 



This will be called Flight Condition »1 ^FCl) and is 

defined as: 

One Engine Out V « 130 fps .. - ft 

Full Throttle Xcg - 0.25 W - 6200 lbs 

Full Flaps 

The stability and control derivatives given in Appendix 

A, Table A.l are for the maximum landing weight. 

Because the maximum landing weight differs from thi' 

maximum takeoff weight by less than 2%, the 7^ alues 

were used without correction. Pl'ni, \iiditior. #1 would 

be the condition in an emerge: ^y go-around. 

3.1 STEADY-STATE EQUATIONS 

The basic assumption made to determiae needed 
control surface deflection was that the airplane motion 
could be modelled about a steady-state point as a set of 
first-order differential equations, as shown in Bqn. 2.1 
for an engine-out flight condition. 



"Cyg Cy« 



DF 



Clg Ci 



DF 



Cl,. 



^DPr 

«R 



^- (mqsin»cosY ♦ FyT^ 



qSb 



(2.1) 



- (Nt » ANp) 
qsD 



B ©DP 6^ 
These equations have been uncoupled from the full 6 
degree of freedom equations by choosing bank angle, ^. 
They are written in the stability axis system (see 



p ' 



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 center line of the body. 
The X- and Z-axes lie in the airplane plane of symmetry, 
while the Y-axis is pointed out the right ving of the 
airplane. This can be seen in Figure 2. 

The non-dimensional derivatives listed in Appendix 
A, Table A.l are given in the stability-axis system. 
This system is also an orthogonal, right-hand set of 
axes with its origin at the canter of mass of the 
airplane. The difference between the body- and the 
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 Figurf 2. 

3.3 OEVfiLOPMENT 

The thrust was calculated using an engine model and 

» 

its average propeller efficiency. The average propeller 
efficiency is greater than the actual propeller 
efficiency by approximately 6%; therefore, the actual 
pitching moment and yawing moment during engine-out 




Body Axis Coordinate System 




Stability AmU Coordtnat* Syttam 
FIGURE 2 Airplane Coordinata Systams 



I ■ 



'\ 



would be slightly less. The propeller efficiencies are 
given in Appendix A, 

By assuming a weight and flight condition, the 
angle of attack was obtained from the airplane lift- 
curve 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- 
dimensional derivatives were obtained from reference 2. 
They are given in Appendix A. 

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 
determined. Figure 3 shows how rudder deflection varies 
with flight condition. 
3.4 RESULTS 

It can be Seen from Figure 3 that at speeds below 
approximately 125 fps, there is not enough rtidder to 
keep the airplane in straight-line flight. This is the 
minimum control speed, Vmc* This is 15 fps less than 
the minimum control speed' given in the operating 
handbook; therefore, a minimum control speed of 130 fps 
is still conservative. At 130 fps, it was found that to 
fly straight with a bank angle, ^, of -5* required 30* 
of rudder deflection, 7.5* of differential flap 
deflection, with a sideslip of -3*. The pedal force 



8 



CESSNA 402SC9AS/C) 
Ws 4500 It X..3»o.i5 

SEh LEVEL 

FULL THROTTLE 
0N£ ENC^INEaur 




/WAX. RUDDER ^EFLfCT/ON 
FACTbfi. TlfhES 

EKivriN^ y.T SIZE 
' o.so 



MO fho ISO 



200 




Sy«42.0ft2 



PI6URS 3 Ruddtr rsquirtd for an angi no-out 
on a Caasna 40 2B 



■.A 



required was also found to be within federal 
regulations. 

It can also be seen from Figure 3 that if one- 
fourth of the vertical tail were removed, the airplane 
would no longer meet PAR 23 requirements. PAR 23 
requires that the minimum control speed, Vmc be greater 
than 1.2Vs. This model does not account for any 
transient phenomena of the airplane in reaching its 
steady-state condition. 

4. SMALL PERTt»BATlON MODEL 

The non- linear simulation model was also used to 
develop a linear model to study the dynamic behavior of 
the Cessna 402B in open- and closed-loop simulations. 
This was only done for FCl. 
4.1 EQUATIONS OF MOTI(»f 

From the assximptions made in the steady-state 
model, a new set of matrix equations can be written: 

X - A X ♦ B u (3.1) 

where 

X' - {B» P, r, ♦}, and 

u' - {6dp» «r}. 
Derivations of this equation can be found in 
reference 4. These matrices are defined in Appendix A. 
The matrices used for FCl are given in Table 2. 



10 



ir''" 



Table 2 small Perturbation Matrices 



Cessna 402B (PCI) 



f 



I 



B 



•f 



0.105 


-0.000867 -0.9899 


0.245 


1.329 


-1.752 


0.483 


0.0 


1.408 


-0.0428 


-0.299 


0.0 


0.0 


1.0 


0.149 


0.0 


0.0 


0.030r 






•0.968 


0.206 






0.0603 


-0.835 






0.0 


0.0 __ 






0.0 " 








-0.0468 








0.44S 








0.0 









11 



Equation 3.1 is written with the following assumptions: 

1. perturbations are small, and 

2. initial condition is a straight-line 
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 instrxjments 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. 
I x-Ax + Bu-fDw (3.2) 

where 

w » {0 or 1} 
The open-loop response of the airplane with the 
addition of the disturbance matrix was calculated using 



12 



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[■ l lip I I II lyipTjppUpWWi^PW^WI I I "'f "" - ^" "" " ■■■ .!,. ■,. . ^ .^ m ... i m ^pi p nn n i^ ^ M " ipw 



the Interactive Control Augmentation Design program 
(ICAD)[5]. This was done on the University of Kansas 
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 
written, tdeally, bank angle, sideslip, and yaw rate 
should be driven to -5®, 0, and 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 
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 




Time t, (Seconds) 



Time t, (Seconds) 



FIGURE 4 Time history of an engine-out in a 
RQAS controlled Cessna 402B 



14 



,,j ♦ 



only slows the rates and could not trim the airplane 
below its "never exceed" speed. 

5. DISCUSSION OP RESULTS 

The control deflections obtained for FCl were: 

Model B 6d 6nF 

(deq) (deq) (deq) 

Steady-State -2.9 30.4 7.5 

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 
two models is due to the fact that in the small 
perturbation model, yaw and pitch rate are not exactly 
zero. The bank angle is also not quite -5". It can 
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 
assuming the servo is modelled as a first-order lag. 
The slowing of the yaw rate due to only 5* of 

rudder deflection would give the pilot valuable time to 

» 

react to the increasing sideslip. In addition, the 
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 
pilot has to respond to the yaw. 



15 



The RQAS has two obvious advantages. First, if the 
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 wii.l introduce digital 
con^uters 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 would 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 control power to the system for an 
engine-out could be done in two ways. First, the RQAS 
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 1 which is for 32». Secondly, an engine-out 



16 






Q^ 




Hi 


-J 


-4 


5 


o 


h 


0^ 


^^ 


g 


«.« 


Q 


a 


VJ 



i 

u 



U 

o 



s 

3 



in 



17 



n^ 



I 



■•.***■ 



sensor could trigger a change In the amount of rudder 
I allowed so that the RQAS Is still only allowed 5* while 

j the englne-out Is allowed more. 

' NO additional software will be required If gain 

\ scheduling Is not needed or Is already Incorporated In 

' the RQAS. The components needed for this system are a 

hydraulic pump to be mounted on one of the engines, an 
•: accumulator, and a set of electro-hydraullc actuators. 

:' The design of the actuator and Its Implementation will 

n be contracted to Cessna, in addition, a set of englne- 

out sensors will be required. Reliability, ease of 
[ Implementation, and cost of these should be 

r Investigated. 

1 

T fi. CONCLUSIONS AND RECOMI^IDSD RESBARCH 

6.1 CONCLUSIONS 
j The following conclusions were reached: 

1. The RQAS with control of all 32- of rudder 
) deflection can trim an englne-out in any steady- 

f state flight condition.^ 

/' 2. With only the proposed 5* of rudder deflection, 

• i the RQAS can slow the divergence of the airplane 

i. ■' 

\. significantly. 

{■'' 3. using only the proposed control surface 

V 

i modifications. Implementing the englne-out 

i i 



■li 

n 



18 



i 

': 1: 

■.it" 



capability would require a set of englne-out 
sensors and the Incorporation of gain scheduHng. 
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 RECOMMENDED RESEARCH 

Further development of this project must Include 

the following steps: 

1. Perform a preliminary failure analysis for 
rudder deflections greater than 5 degrees. 

2. Use the gains required at the most 
critical condition to see their effect In other 
conditions, 

3. Examine and evaluate options available to 

sense an englne-out, 

4. Investigate the possibility of Increasing 
the amount of rudder deflection available to 
the RQAS, 

5. Design and build the necessary hardware, 

6. Perform a flight test of the system on the 
Cessna 402B. 



19 



Referenceat 

1. Downing, D.R.j Davis, D.J.; Llnse, D.J.? Bntz, 
D.P.: "Prjllmlnary Control Law and Hardware 
Designs For a Ride Quality Augmentation System 
For Commuter Aircraft", NAGl-345, Feb 1986. 

2. Hoh, R.H.; Mitchell, D.G.; Myers, T.T.: 
"Simulation Model o£ Cessna 4028". NASA CR 152176, 
July 1978. 

3. Roskam, J.j Airplane Flight Dynamics and Automatic 
Flight Controls, Part I . Roskam Aviation ana 
Engineering Corp. 1979. 

4. Roskam, J.: Airplane Flight Dynamics and Automatic 
Flight Controls. Part II . Roskam Aviation ano 
Engineering Corp. 1979. 

5. Hammond, T.A.; Amin, S.P.; Paduano, J.D.; Downing, 
D.R. ! Design of a Digital Ride Quality Augmentation 
System For Commuter Aircraft, NASA CR 172419, Oct 
1984. 

6. Davis, D.J., "A Comparison of Two Optimal Regulator 
Design Techniques for the Weighting of Output 
Variables Which are Linear Combinations of States 
and Controls", M.S. Thesis, The University of 
Kansas, Lawrence, KS, 1986. 

7. Hoak, D.B. et al? USAF Stability and Control DATCOM 
Wright Patterson Air Force Base, Ohio, 45433, 
April, 1976. 



20 



; 1 



APPENDIX A Cessna 402B Stability Derivatives 

This appendix gives the method used for estimating 
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: 

W - Cl q S 
Prom this lift coefficient, the AC^ due to flaps from 
reference 2 was subtracted. This lift coefficient was 
then found on Pigure A.l and its corresponding angle of 
attack was read. The airplane angle of attack was then 
used to obtain the non-dimensional derivatives from 
reference 2. These are listed for an angle of attack, 
a, of 8.5* in Table A.l. 

The engine model was taken from reference 2. This 
gave the maximum power of a C402B engine as 300 brake 
horsepower. The propeller efficiencies, average and 
actual, are given in Pigure A. 2. Thrust was then 

calculated using the follbwing equation: 

t 

T - 550 BHP np (A.l) 

V 

where 

np " Propeller Efficiency 

V ■ Airplane Speed 

BHP * Engine Brake Horsepower 

A.l 



i : 




Figure A.l ApproxinaU Caisna 402B(Tail-offf) Lift-Curv« 



A. 2 



Table A.l Lateral-Directional Non-dimensional 
Stability Derivatives 



: Li 



One Engine Out 
Pull Throttle 
l^ill Flaps 



Cessna 402B 
V - 130 fps 



teg 



0.25 



h • ft 

W » 6200 lbs 



! I 









Derivatives 




Cyg a -0.670 


rad~' 




Cyp » -0.00063 


CyfiR ' 


• 0.195 


rad"^ 




Cyr = 0.42 


Cy6DP ' 


• 0.0 


rad~^ 




Cnp = -0.084 


Cn0 


» 0.129 


rad"^ 




Cnr = -0.170 


CnfiR - 


• -0.0795 


rad"^ 




Clp • -0.81 


CnfiDP " 


• 0.0057 


rad"^ 




Clr « 0.216 


Cl0 - 


. -0.0888 


rad"^ 






ClfiR - 
C1*DP ■ 


> 0.0146 

> -0.0685 


rad"^ 
rad-^ 







I; \i 



A. 3 







soo 



tta* AifSpMtf (tt/«M) 



Propeller Efficiency at Several Plight 
*^ Conditions 




I 







/ _^ Cstrepolotim fram Pig. IS4 
in Oi«M 



/-"^ in owa 

■ f4h 

'lOO So 20 



ISO 200 280 300 

TrMAirapttdtft/Me) 







Assumed Average Propeller Efficiency for All 
Plight Conditions 

Plgure A.2 Propeller Efficiencies 



A.4 



I 

") 






r From Figure 1, 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 feet* The orientation 

i 

of the thrust line of one of the engines is given in 
I reference 2. By finding the X and 2 components of the 

thrust y the moments, % and L^, can be found from: 
' % « 7,50 Txf and 

f Lt « 7.50 T^. 

Reference 2 also gives the ACp due to an engine-out. 
ANd in Equation 2.1 is the yawing moment due to drag on 
the inoperative engine and is: 

ANd * 7.50 ACd q S 
Equation 2.1 was then solved to obtain the sideslip, 
differential flap deflection, and rudder deflection 
required. To account for changing the vertical tail 
size, the non-dimensional derivatives were recalculated 
and are sxommarized in Table A. 2. 

The dimensional derivatives were then calculated as 
shown in Table A. 3. These were calculated using: 

Ixx • 11100 slug ft^ 
Izz « 14900 slug ft^ 
1x2 ■ -5B3 Slug ft^ 
To use these in the state matrices. Table A. 4, they must 
be in the form shovn in Table A. 5, where, 

Ai - Ixz/Ixx and Bi « Ixj/Izz • 



A. 5 



J^ 



Table A. 2 variation of Derivatives witft 
Vertical Tail Size 



Cessna 402B (FCl) 



Relative 
V.T. Size 


="6 
deg"^ 


dec"* 


1 ^"*^. 
i dec - 


! ^ 


0.50 


0.00085 


Q.CCiaE 


; -C. 000-0 


i I * jQ^iSr 


0.75 


0.00165 


0.0024E 


i -C. 00104 


■ i . 000232 


1.00 


0.00240 


C.0023C 


1 -0.0013? 


C-00C3I: 


1.25 


0.00216 


0.00413 


1 -0.00174 


C. 000286 


Note: C^ ' 

*^ tJP 


« -0.00062 deg~i vas 


calculated 


froxr ref. 



A. 6 



Table A. 3 state Matrices 



:1 



',1 

B 



L*J 



I 



^i 


V Yr' y/ 







LS' 


Lp' Lr 


X " 


P 


Nfi' 


Np Nr 




r 





1 tanQi 




♦ 






7 



B 







6df 



'sr 



i i 



A.7 



I «i 



Table A. 4 Lateral-Directional Dimensional Stability 

Derivatives [31 



q.SC ^l I. 

^ ^& -2 *A -•> 

Y, • ' (ft sec *) L. • = — ~ (sec * or 

* ' ^ ^xx . -2 . -I 



L. ■ , (sec " or 



Ng--j— i (sec 2) 



^0. (ft sec" or 

"^i'—r- ftsec-^deg'h ^l^^S 

N- - -: ' (sec ^) 

^ ^ij^ (ft sec ^ or 

^6j^ ' S ft sec'^ deg'b ^l^^^^n 

«p--2I-uf ^"^"'^ 

^ 22 1 



q^SbC 

3 , -2, 



h ' TT' ^"*'" ^ q,Sb2c 



XX 



q,Sb2c 



1 % -1 

22 1 



^ \ ,-..-i. 






'A -5 






L • -T (sec" ) 

r 21 U, ^'*^ ' 



iiSbC 



^xx"l ^1-" ng 

N. - —7 ^ (sec*^ or 

'^ '« sec-2 deg'S 



A.8 



Table A. 5 Modified Lateral-Directional Dimensional 
Stability Derivatives [6] 



Yb - Yg/Ui Lj^j • (AiN6^j+Lfi^j)/(l-AiBi) 

Yp - Yp/Ui 



Yp - (Yp/Ui)-1 
Y^ » gcos9i/Ui 



^«df " ^«df/"l 



^«sr • ^«s/"l 



Lg - (AiNj+L3)/(l-AiBi) 
Lp' • (AiNp+Lp)/(l-AiBi) 
Lr' - (AiMr+Lr)/(l-AiBi) 



Wgr - <AiN5^^*Lfi^^)/(l-AiBi) 



Ne • (BiLg*Kg)/(l-AiBi) 
V* (BlLp+Np)/(l-AiBi) 
Kp » (BiLr^Nr)/(l-AiBi) 
N«^f - (BiLfi^^*Nfi^^)/(l-AiBi) 



Nfi^^. - (BlL6^^+N«^^)/(l-AiBi) 



A.9