zotero-db/storage/3GGYKC63/.zotero-ft-cache

NASA Technical Paper 2835

September 1988

User’s Manual for Interactive
LINEAR, a FORTRAN Program To Derive Linear Aircraft Models

Robert F. Antoniewicz, Eugene L. Duke, and Brian P. Patterson

NASA

NASA Technical Paper 2835
1988

User’s Manual for Interactive
LINEAR, a FORTRAN Program
To Derive Linear Aircraft Models

Robert F. Antoniewicz,
Eugene L. Duke,
and Brian P. Patterson
Ames Research Center
Dryden Flight Research Facility
Edwards, California

NASA
National Aeronautics and Space Administration
Scientific and Technical Information Division

CONTENTS

SUMMARY

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1

INTRODUCTION

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1

NOMENCLATURE

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2

Variables « « 6 6 © © © © © © © © oe we we we ee ew we ew ew ew ew ew

2

SuperscriptS « 6 + « «© © © © © © # © © oe ww oe ee we we ee ee ee et

6

Subscripts «© 6 « © © 6 6 6 © © © © oe 8 ww ew ww ee ew we eh ee ww

6

FORTRAN Variables . . «6 6 © © © © © © © © © © © © ee eo ee ee ew wl

7

PROGRAM OVERVIEW

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EQUATIONS OF MOTION . 6 6 2 6 6 ee eee ee we ee ww ee we ew we ee ee

(4

OBSERVATION EQUATIONS . 2. 2. 2 6 6 6 6 © ew ee ee ee ee ew ew ee eee ee

TT

SELECTION OF STATE, CONTROL, AND OBSERVATION VARIABLES

«. . «© «© «© © © «© © © © «© «© 22

LINEAR MODELS . « 6 « «© © «© © © © © © © © ow ew wh ew ee ww he ew

ew

lt ew we OR!

ANALYSIS POINT DEFINITION . . 2. 6 «© © © «© © © © © © © © © © © © © ee ew hehehehe)

65

Untrimmed . . 2. 2 6 6 te oe ee ew we tw wt wt et tw tt ew

le lle wl le 2G

Straight-and-Level Trim . 2. 6 2 6 ee ee eo ee ee ew ee ew we ee we ee 27

Pushover-Pullup . 2. 2. 2 2. 6 ee ee © © ee eo ee ee ee ee ee ee ee we 7

Level Turn . .. 6 © © © © © © © © © © © oe we we we ee ww lw lw lw hele tl

lhl) lB

Thrust-Stabilized Turn

. « « 2 © «© © © © © © © © © © © oe ew ee ee ee we we

2D

Beta Trim. . «6 «© «© «© © © © © © © © 8 6 wee ee te we et wt tle lw lel wl le RD

Specific Power «© 6 6 6 «© © eo ee ee ee ew ew ew ew ee we eee ee 29

NONDIMENSIONAL STABILITY AND CONTROL DERIVATIVES

~. « « «© «© «© © «© © © ©» © © © © © 31

DATA INPUT

. 2. 2. 2 2 © © © © © @ 8 © 8 6 eo ew we ee ee we

ww et hw we

le ew) 8B

Input Files 2 6 6 6 6 ee 8 ee we we wwe we we ee ew ee ee eee ew ew 83

Title Information .« . 2. 6 «© «© © © © © © © © oe eo ee ew we we tw ee

ee) 83

Geometry and Mass Data

. . . + 6 «© © © © © © ee eo © ee ee ee ee ew ew we)

33

State, Control, and Observation Variable Definitions

... +... + +++...

34

Trim Parameter Specification

. . « « +» « «© © © © © © © © © © eee ee ew we

87

Additional Surface Specification

. « 6 « © « © © © © © © © © © © © © © © we

37

Test Case Specification . 2. 6 6 6 © © © © © © © ew ee we ew ee ew ew

ee 8B

Interactive Data Input . . 6 6 2» «© © © © © © © © ew we we ee te tw el ee 8B

AERODYNAMIC

MODEL

.

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46

OUTPUT FILES

«6 6 6 «© «© «© © © © © © we ee we ww ee ww ww

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«48

USER-SUPPLIED SUBROUTINES . .« 2 2 © 8 © © © © © © © © © © ew ee ww tw ww ew) «4D
Aerodynamic Model Subroutines . . + + 6 © 6 6 © © 8 6 ee ew ew ew ew ew ew ew ee (5D Control Model Subroutines . . «6 « «© © © © © © ee ee ee ew ww ww we ew 8B

Engine Model Subroutines «© + «

6 © 6 © © © © © we ee ee we ew ew ww ew ee «(5B

Mass and Geometry Model Subroutines .« «6 «© 56 6 © © © © © © © © © ee © we ew oe ew (58D

iii

CONCLUDING REMARKS

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APPENDIX A:

CORRECTIONS TO AERODYNAMIC COEFFICIENTS FOR AT THE AERODYNAMIC REFERENCE POINT ..

A CENTER

OF

GRAVITY

NOT

APPENDIX B:
APPENDIX C: APPENDIX D: APPENDIX E;:

ENGINE TORQUE AND GYROSCOPIC EFFECTS MODEL .. a

a

OBSERVATION VARIABLE NAMES RECOGNIZED BY LINEAR

........e.6.,

STATE VARIABLE NAMES RECOGNIZED BY LINEAR

.. a

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ANALYSIS POINT DEFINITION IDENTIFIERS

.

APPENDIX F:

EXAMPLE INPUT FILE ........s.e

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APPENDIX G: EXAMPLE LINEARIZED STABILITY AND CONTROL

APPENDIX H:

EXAMPLE OUTPUT ANALYSIS FILE . .....

APPENDIX I:

EXAMPLE PRINTER OUTPUT FILE

......

APPENDIX J: EXAMPLE USER-SUPPLIED SUBROUTINES

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Mass and Geometry Model... .....6e8e

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REFERENCES

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56 58
66 73 74 76 80
86 90 109 109 113 115 116 117

iv

SUMMARY

An interactive FORTRAN program that provides the user with a powerful and flex-

ible tool
report.

for
The

the linearization of aircraft aerodynamic models is
program LINEAR numerically determines a linear system

documented in
model using

this
non-

linear equations of motion and a user-supplied linear or nonlinear aerodynamic model.

The nonlinear equations of motion used are six-degree-of-freedom equations with sta-

tionary atmosphere and flat, nonrotating
mined by LINEAR consists of matrices for

earth assumptions.
both the state and

The system
observation

model deter-
equations.

The program has been designed to allow easy selection and definition of the state,

control, and observation variables to be used in a particular model.

INTRODUCTION

The program LINEAR described in this report was developed at the Dryden Flight

Research Facility of mented, and verified

the NASA Ames Research Center to tool to derive linear models for

provide a standard, docuaircraft stability analysis

and the

control aircraft

law design. This development was specific linearization programs

undertaken to address common in the aerospace

the need for industry.

Also, the lack of available documented linearization programs provided a strong

motivation for the development of LINEAR; in fact, the only available documented

linearization program that was found in an extensive literature search of the field

is that of Kalviste (1980).

Linear system models of aircraft dynamics and sensors are an essential part of

both vehicle stability analysis and control law design.

These models define the

aircraft system in the neighborhood of an analysis point and are determined by the

linearization of the nonlinear equations defining vehicle dynamics and sensors.

This report describes a FORTRAN program that provides the user with a powerful

and flexible tool for the linearization of aircraft models. with well-defined and generalized interfaces to aerodynamic is designed to address a wide range of problems without the

LINEAR is a program and engine models and requirement of program

modification.

The program LINEAR numerically determines a linear system model using nonlinear

equations of motion and a user-supplied linear or nonlinear LINEAR is_also capable of extracting both linearized engine

aerodynamic model. effects, such as net

thrust, torque, and gyroscopic effects

system model.

The point at which this

and including linear system

these model

effects in the linear is defined is determined

either by completely specifying the state and control variables, or by specifying an analysis point on a trajectory and directing the program to determine the con-

trol variables and remaining state variables.

The system model determined by LINEAR consists of matrices for both the state pTaofnhrduosgt,rhoaebmstehsertpvaratooetvr,iidodenercsoneotfqtrhuoeatlth,iefolnesasnx.ydisbteiomlTbihsteemyrovdpearoltofigoraniaslmlovcwaohiarmnsipgalbeblteaeeelslntyerdtnoeuasntdibeegerneufdsouesrdemtrouliapnctroaionovtnirpsdoaler.toifecausFblyouatrrhtsheelmtreho,ecdetli.tohne
state and observation equations.

LINEAR has several features that make it unique among the linearization programs

common in the aerospace industry.

The most significant of these features is flexi-

bility.

By generalizing the surface definitions and making no assumptions of sym-

metric mass distributions, the program can be applied to any aircraft in any phase

of flight except hover.

The unique trimming capability, provided by means of a user-

supplied subroutine, allows face scheduling, which are

unlimited possibilities of trimming strategies particularly important for asymmetrical vehicles

and and

sur-

aircraft having multiple surfaces affecting a single axis. The formulation

equations of motion permits the inclusion of thrust-vectoring effects.

The

of the ability

to select variables for models, without program

the state, control, and observation vectors of the linear modification, combined with the vast array of observation

variables available, allows any analysis problem to be attacked with ease.

The interactive LINEAR program is an extension of the batch LINEAR program tasdiornteepasubcctirailisniebpf.teuiydtlesTatiohnnedfoDtruhacekoeenrttohpredraoyonlndgbaramatdoimctechhreitrvhpmsaorrtdooiegulv(rg1eah9sm8.c7aa)nf.otreAbelrlmDtihadnetedaafaltifanldeieodgrschactnrfiinobrpbieomtniawgnotmttoedwtdiahoeyffsi:deaaidtinatrecairrnfeaitssfleettter,sacoatfnosidrivmneioatllneyasdfrtiumlflertcnooasmsnieotonhnc-eaasle linear aerodynamic model as used in simulations at Ames-Dryden.

NOMENCLATURE

The units associated with the following quantities are

system. LINEAR will work equally well with any consistent

notable exceptions:

the printed output and the atmospheric

output and the atmospheric model assume English units.

expressed in the English set of units with two
model. Both the printed

Where applicable, quantities are defined with respect to the body axis system.

Variables

A
A‘ a an,i anx Aanx,i any any,i ang

state matrix of the state equation x = Ax + Bu; or, axial force (lb)
state matrix of the state equation cx = A'x + Bru speed of sound in air (ft/sec) normal acceleration not at center of gravity (g) x body axis accelerometer at center of gravity (g) x body axis accelerometer not at center of gravity (g) y body axis accelerometer at center of gravity (g) y body axis accelerometer not at center of gravity (g) 2 body axis accelerometer at center of gravity (9g)

z body axis accelerometer not at center of gravity (g) acceleration along the x body axis (g) acceleration along the y body axis (9g) acceleration along the z body axis (9)
control matrix of the state equation x = Ax + Bu
control matrix of the state equation Cx = A'x + B'u wingspan (ft)

C matrix of the state equation Cx = A'x + B'u; or, force or moment coefficient

mean aerodynamic chord (ft)

dynamic interaction matrix for the state equation x = Ax + Bu + Dv; or, drag force (1b)

dynamic interaction matrix for the state equation cx = A'x + Blu + D'v

determinant dynamic interaction matrix for the observation equation y = Hx + Fu + Ev
dynamic interaction matrix for the observation equation y = H'x + Gx + Flu + E'v
specific energy (ft)
feedforward matrix of the observation equation y = Hx + Fu

feedforward matrix of the observation equation y = H'x + Gx + Flu

fpa

flight path acceleration (g)

G matrix of the observation equation y = H'x +

acceleration due to gravity (£t/sec2 )

altitude (ft) observation matrix

of the

observation

equation

y = Hx + Fu

H'

observation matrix of the observation equation y = H'x + Gx + Flu

angular momentum of engine rotor (slug-ft?/sec)

aircraft inertia tensor (slug-ft2)

rotational inertia of the engine (slug-ft?) x body axis moment of inertia (slug-ft?)

x-y body axis product of inertia (slug-ft2)

x~z body axis product of inertia (slug-ft?)
y body axis moment of inertia (slug-ft2)

y-z body axis product of inertia (slug-ft2)

z body axis moment of inertia (slug-ft2)

total body axis rolling moment (ft-lb) generalized length (ft) Mach number; or, total body axis pitching moment (ft-lb) aircraft total mass (slug) normal force (lb); or, total body axis yawing moment (ft-lb) load factor

specific power (ft/sec)

roll rate (rad/sec)

ambient pressure (lb/ft?) total pressure (lb/ft?)

pitch rate (rad/sec)

dynamic pressure (1b/ft2)

de

impact pressure (1b/ft2)

yaw rate (rad/sec)

Re

Reynolds number

Re'

Reynolds number per unit length (ft71)

wing planform area (£t2)

ambient temperature (°R); or, total angular momentum (slug-ft?/sec2) total temperature (°R)

velocity in x-axis direction (ft/sec) control vector total velocity (ft/sec) velocity in y-axis direction (ft/sec) dynamic interaction vector calibrated velocity (ft/sec)

equivalent velocity (ft/sec) vehicle weight (1b)

velocity in z-axis direction (ft/sec) total force along the x body axis (1b) state vector thrust along the x body axis (1b) sideforce (1b) observation vector thrust along the y body axis (1b)

total force along the z body axis (1b) thrust along the z body axis (1b)

angle of attack (rad) angle of sideslip (rad) flight path angle (rad) displacement of engine from center of gravity along x body axis (ft)

Ay

displacement of engine from center of gravity along y body axis (ft)

Az

displacement of engine from center of gravity along z body axis (ft)

lateral trim parameter

longitudinal trim parameter

incremental rolling moment (ft-lb)

incremental pitching moment (ft-lb)

incremental yawing moment (ft-lb)

directional trim parameter

thrust trim parameter incremental x body axis force (1b) incremental y body axis force (1b) incremental z body axis force (1b) pitch angle (rad)

density of air (slug/ft3) torque from engines (ft-lb) gyroscopic torque from engines (ft-lb)

roll angle (rad)
tilt angle of acceleration normal to the flightpath from the vertical plane (rad)
heading angle (rad)
engine angular velocity (rad/sec)

Superscripts derivative with respect to time transpose of a vector or matrix

Subscripts

ar

aerodynamic reference point

total drag

h L L M m max min n p q r s x y Y z co)
AIX AIXE AIXY AIXZ AIY AIYZ AIZ ALP ALPDOT

altitude total lift rolling moment Mach number pitching moment maximum minimum yawing moment roll rate pitch rate yaw rate stability axis along the x body axis along the y body axis sideforce along the z body axis standard day, sea level conditions; or, along the reference trajectory
FORTRAN Variables inertia about the x body axis engine inertia inertia coupling between the x and y body axes inertia coupling between the x and z body axes inertia about the y body axis inertia coupling between the y and z body axes inertia about the z body axis angle of attack time rate of change of angle of attack

AMCH AMSENG AMSS B BTA BTADOT CBAR CD CDA CDDE CDO CDSB CL CLB CLDA CLDR CLDT CLFT CLFTA CLFTAD CLFTDE CLFTO CLFTQ CLFTSB CLP CLR CM

Mach number
total rotor mass of the engine aircraft mass wingspan angle of sideslip time rate of change of angle of sideslip
mean aerodynamic chord
total coefficient of drag coefficient of drag due to angle of attack coefficient of drag due to symmetric elevator deflection drag coefficient at zero angle of attack coefficient of drag due to speed brake deflection
total coefficient of rolling moment
coefficient of rolling moment due to beta coefficient of rolling moment due to aileron deflection coefficient of rolling moment due to rudder deflection
coefficient of rolling moment due to differential elevator deflection
total coefficient of lift
coefficient of lift due to angle of attack coefficient of lift due to angle-of-attack rate coefficient of lift due to symmetric elevator deflection
lift coefficient at zero angle of attack coefficient of lift due to pitch rate coefficient of lift due to speed brake deflection
coefficient of rolling moment due to roll rate
coefficient of rolling moment due to yaw rate total coefficient of pitching moment

CMA CMAD CMDE CMO CMQ CMSB CN CNB CNDA CNDR CNDT CNP CNR CY CYB CYDA CYDR CYDT DAS DC DELX
DELY
DELZ
DES

coefficient of pitching moment due to angle of attack coefficient of pitching moment due to angle-of-attack rate coefficient of pitching moment due to symmetric elevator deflection pitching moment coefficient at zero angle of attack coefficient of pitching moment due to pitch rate coefficient of pitching moment due to speed brake deflection total coefficient of yawing moment coefficient of yawing moment due to sideslip coefficient of yawing moment due to aileron deflection coefficient of yawing moment due to rudder deflection coefficient of yawing moment due to differential elevator deflection coefficient of yawing moment due to roll rate coefficient of yawing moment due to yaw rate total coefficient of sideforce coefficient of sideforce due to sideslip coefficient of sideforce due to aileron deflection coefficient of sideforce due to rudder deflection coefficient of sideforce due to differential elevator deflection lateral trim parameter surface deflection and thrust control array displacement of the aerodynamic reference point along the x body axis from
the vehicle center of gravity displacement of the aerodynamic reference point along the y body axis from
the vehicle center of gravity displacement of the aerodynamic reference point along the z body axis from
the vehicle center of gravity longitudinal trim parameter

DRS DXTHRS
EIX ENGOMG GMA H HDOT NUMSAT NUMSRF P PpoT PHI PHIDOT PSI PSIDOT Qo QBAR QDOT R RDOT Ss T TDOT THA THADOT THRSTX THRUST
10

directional trim parameter distance between the center of gravity of the engine and the
thrust point rotational inertia of the engine rotational velocity of the engine flightpath angle altitude time rate of change of altitude number of states in the state vector number of recognizable control names roll rate time rate of change of roll rate roll angle time rate of change of roll angle heading angle time rate of change of heading angle pitch rate dynamic pressure time rate of change of pitch rate yaw rate time rate of change of yaw rate wing area time time rate of change of time pitch angle time rate of change of pitch angle thrust trim parameter thrust generated by each engine

TLOCAT
TVANXY TVANXZ
UB Vv
VB VCAS
vboT VEAS WB
x XDOT XYANGL XZANGL
Y YDOT

location of the engine in the x, y, and z axes from the center of gravity
orientation of the thrust vector in the x-y engine axis plane orientation of the thrust vector in the x-z engine axis plane
velocity along the x body axis velocity
velocity along the y body axis calibrated airspeed
time rate of change of total vehicle velocity equivalent airspeed velocity along the z body axis
position north from an arbitrary reference point time rate of change of north-south position orientation of thrust vector in x-y body axis plane orientation of thrust vector in x-z body axis plane
position east from an arbitrary reference point time rate of change of east-west position

PROGRAM OVERVIEW

The program LINEAR numerically determines a linear system model using nonlinear

equations of motion and a user-supplied nonlinear aerodynamic model.

LINEAR is also

capable of extracting linearized gross engine effects, such as net thrust, torque,

and gyroscopic effects and including these effects in the linear system model. The

point at which this linear system model is defined is determined either by specifying

the state and control variables or by selecting an analysis point on a trajectory, selecting a trim option, and allowing the program to determine the control variables

and remaining state variables to satisfy the trim option selected.

Because the program is designed to satisfy the needs of a broad class of users, a wide variety of options has been provided. Perhaps the most important of these
options are those that allow user specification of the state, control, and observation variables to be included in the linear model derived by LINEAR.

Within the program, the nonlinear equations

senting a rigid aircraft flying in a stationary

earth.

Thus, internally, the state vector x is

of motion include
atmosphere over a
computed as

12 states repre-
flat, nonrotating

x=([p qrvVvaBb $6 ph x yi"

11

The nonlinear equations used to determine the derivatives of the states are presented in the following section. The internal control vector u can contain up to

30 controls. The internal observation vector y contains 120 variables including the

state variables, the time derivatives of the state variables, the control variables,

and a variety of other parameters of interest.

Thus, within the program,

where

y= [= x.T ul yT1° yT2 yT3 yT4 yP5 yP“6 yt“7 "|“8 T

y11 = [a x a y a Zz a nx

a ny

a nz

a n a nx,i

a ny,i ‘

nz,i

a n,i

n]*

yo =_ [a

Re

Re '

M q=d Gc

Pa

Qc/Pa

Pt

T Tt

Ve

Vo | T

¥3 = Ly fpa

y. h"* ¢h@ /57.3 | T

y4 = [Eg Pg] T
ys= [Lb Dw a]?

ye=[u vwudev e wi*5T

Y7 == [o,4 Ba A

h° al T

and

yg = [T Ps dg rg]

The equations defining these quantities are presented in the Observation Equations section.

From the internal formulation of the state, control, and observation variables, the user must select the specific vari-

ables desired in the output linear model

(described in the Selection of State, Con-

trol, and Observation Variables section).

Internally, the program uses a 12-state

vector, 120-variable observation vector,

and a 30-parameter control vector.

These

variables can be selected to specify the

formulation most suited for the specific
application. The order and number of

parameters in the output model are com-

pletely under user control. Figure 1

shows the selection of the variables in

the state vector for a requested linear

model.

From the internal formulation on

Output model Parameters
y
a le q 8 |
Figure 1. ables for

f

\

[

Specification of
' state vector or linear model Selection of linear model.

Internal parameters
lp | a q

V

\

Q

B

> 6

L

x
y | 7

state vari-

12

the right, the requested model is constructed, and selected in accordance with the user specification vation variables.

the linear system matrices of the state, control, and

are obser-

aiattsasrhlfreieeliicdtoemteodwistuzsolasTetnilehrhtperolert)yhaoifjwmlaseiiaioactrnnrrthteechbaoauriinarrsavtaysferlettrayvomsrsretoe.iy(rdapasiaelonlllrpyctop.wAoiolindonisdnepatnoirtgrtiitsbTvoi-ihaineolttetnsdnesrocevalweruhbbdlyayedibeaceosnhducaLdsftrlIetliyNafitibsEtighswieeAehnhsdRtei,adcorhnopiinsnotaposiaunatntshtddtiehreeatooittrnvameatdaleimrjrelirlimuenfsenicei,gntanaieorinnsdrdttypoie.uptmohlatoatnetliddruoeyanpl-naTAossnshnp,telataiialisnattoynpelnisarewtsiolovohsnyegicaslrcolabiahenlPsmdlotoiiuatdwtnrhpcLetineocrIosiiNent,nvElhDtueAeese.dRrefo.oarnriutnspiierLazodrtoIeenivNrfssoiEoiptn-dAneoReecs-s

The linear system matrices are determined by numerical perturbation and are the

first-order terms of a Taylor
in the Linear Models section.
user control. The user can

series expansion about the analysis point as described
The formulation of the output system model is under
select state equation matrices corresponding to either

the standard formulation of the state equation

or the generalized equation

“ i ]

Ax + Bu

Cx

A'x + B'u

The observation matrices can be selected from either of two formulations corresponding to the standard equation
y = Hx + Fu

or the generalized equation

y = H'x + Gx + Flu

In
sional

addition
stability

to the linear
and control

system matrices,
derivatives at the

LINEAR also computes the
analysis point. These

nondimen-
derivatives

are discussed in the Aerodynamic Model section.

Aircraft geometry and mass properties and analysis point definition are input

interactively to specified by the

the program or are read from ASCII files of 80-character records

user.

The state, control, and observation vectors are also de-

fined in either manner.

The details of data input are discussed in the Data In-

put section.

The output of LINEAR consists of the following four files: one is intended to batmceianovntaetrlusriysocseleid,ess.xtwripataonhcdiTtnhetefosdoblssleeasotrcewvol-naetodtchniteoefndidalenesabviylegyccnstotiohnsretaanidpunnosaseimarenn,ta.alllyliassnidtTsosfheonpatehnrfedoigrcrsovttanarmtilsmau,oeifsn,sitntwhfoeosasrntedmadaboticftiluhioiemlntee,ysnsttaacttnhoteedhnetacsciopoonannpscttterrioootllnh,sseysdteassenrttmdiaavttaee,-,

13

and observation values, the trim control input positions, and the nondimensional stability and control derivative data. It will also contain the output of the equations of motion if the program fails to attain a trimmed condition. The third file contains all of the information in the second file plus the state space system model matrices. The fourth file contains the stability and control derivatives extracted at the analysis point in a form suitable to be read back into LINEAR as a linear aerodynamic model.
To execute LINEAR, five user-supplied subroutines are required. These routines, discussed in the User-Supplied Subroutines section, define the nonlinear aerodynamic model, the gross engine model, the gearing between the LINEAR trim inputs and the surfaces modeled in the aerodynamic model, and a model of the mass and geometry properties of the aircraft. The control model, shown in figure 2, defines how the LINEAR trim inputs will be connected to the surface models and allows schedules and nonstandard trimming schemes to be used. The last feature is particularly important
for asymmetrical aircraft.

Inputs from ICTPARM/
DES ————_—_> DAS ——————|__ DRS ——————_»|__ THRSTX ———_»

UCNTRL (Gearing)

Outputs to ICONTRL/ 1» DC(1) -——> DC(2}

. ° +——

DC(30)

(Pilot stick, pedal, and throttle)

(Surface deflections and power level setting)

Figure 2. Inputs and outputs to the user-supplied subroutine UCNTRL.

EQUATIONS OF MOTION

The nonlinear equations of motion used in the linearization program are general

six-degree-of-freedom equations representing the flight dynamics of a rigid aircraft flying in a stationary atmosphere over a flat nonrotating earth. The assumption of nonzero forward motion also is included in these equations. Because of this assumption, these equations are invalid for vertical takeoff and landing or hover. These

equations contain no assumptions of either symmetric mass distribution or aerody-

namic properties and are applicable to asymmetric aircraft (such as oblique-wing

aircraft) as well as to conventional symmetric aircraft.

These equations of motion

were derived by Etkin (1972), and the derivation is detailed in the NASA Reference

Publication, “Derivation and Definition of a Linear Aircraft Model," by Eugene L.

Duke, Robert F. Antoniewicz, and Keith D. Krambeer.

The following equations for rotational acceleration are used for analysis point definition:

14

where

p= ((ZL)Iq + (EM)Ip + (EN)I3 - p2(Iyelo - Igyl3)
+ pq(Ixzly - Iyzt2 - DgI3) - pr(Iyyly + DyIg - TyzI3)

+ aq? (Iyzly
~ r2(Iy2Tq

~ IxyI3) - ar(Dyly
- IxzI2)]/det I

- Tyyt2

+ Ixz13)

q = [(ZL)Ig + (IM)Ig + (IN)I5 - p2(ty,I4 - Tyyl5)

+ pq(IxzI2 ~ Iyzly ~ DyIg) ~ pr(IyyIg + DyIg - Iyzts)

+ a? (Iyzl2 ~ Ixyt5) ~ ar(DyIg - IxyTg + Ixzt5)
- r@(IygI2 - IxzI4)]/det I

K e
I

[(2L)I3 + (EM)Is + (IN)Ig - p*(IyzI5 - Iyylg)
+ pq(IxzI3 - TygI5 - Dlg) - pr(IyyI3 + DyI5 - Iyzlg)
+ q*(IyzT3 - Ixylg) - ar(DyI3 - IRxyt5 + Ixztg)
- r2(IyzI3 - IyzI5)]/det 1

det I = IylyIz - 2Ixylyglyz - Ixl*yz - Tyl?xz - Igl@xy

I5 = Ixlyz + Ixylxz Ig = Ixly - I?xy Dy = Iz - ly
Here, the body axis rates are designated p, q, and r, corresponding to roll rate, pitch rate, and yaw rate. The total moments about the x, y, and z body axes are designated IL, =M, and IN, respectively. These total moments are the sums of all
15

aerodynamic moments and powerplant-induced moments due to thrust asymmetries and

gyroscopic torques.

The equations used to determine the change in moment coef-

ficients due to the noncoincidence of the vehicle center of gravity and the refer-

ence point of the nonlinear aerodynamic model are derived in appendix A. The equa-

tions defining the engine torque and gyroscopic contributions to the total moments are derived in appendix B. The body axis moments of inertia are designated I,, Ty.

and Iz:

The products of inertia are designated I xy’ I “x ze and Iyz°

These moments

and products of inertia are elements of the inertia tensor I, defined as

Ix ~Ixy

Ixy

Ixz

Iy -Iyz

~Ixz

~Iyz

Iz

To derive the state equation matrices for the generalized formulation

Cx

Ax + Bu

the rotational accelerations are cast in a decoupled-axes formulation. tions used to derive the linearized matrices are

The equa-

1

Ixy

Ix

Ixz

3)

Ix

Ixy

~Iyz

.

Ty

ly

q

aTxL

_y P

> I x

XI Z
Iz,

y“I z
I,

1

Fae

pq —— + rq aI + (q@ - r2) x

HI + rq IPL - pg I HE + (2-?52)

ly

y

y

y

aI=,N +

—TILx, = qr ITyxz + pr II,yz + (p 2 - q*2)

—

The translational acceleration equations used in the program LINEAR for both analysis point definition and perturbation are

Vv

[-D cos B + ¥ sin 8 + Xp cos a cos B + Ym sin 8 + Zp sin a cos B

~ mg(sin 0 cos acos 8 - cos 0 sin $ sin 8B - cos Q cos ¢ sin a cos £B)]/m

16

a= [“L + Zp cos a - Xp Sin a+ mg(cos 8 cos ¢ cos a
+ sin 60 sin a)]/Vm cos 8B

+ q- tan B (p cos a+ r sin aq)

B = [D sin 8 + Y cos B - Xp cos a sin 8 + Ym cos B - Zp sin a sin 8B

+ mg(sin 6 cos a sin 8 + cos 8 sin ¢ cos 8

- cos 8 cos $¢ sin a sin £)]/Vm

+ p sin a- r cos a

The equations used to define the vehicle attitude rates are

$ =p t+q sin $? tan 8+ r cos ¢ tan 6

8

q cos ¢- r sin 6

v

q sin ¢ sec 6 + r cos » sec 6

The equations used to define the earth-relative velocities are

W l

r e

Vi(cos 8 cos a sin 8 - sin 8 sin $¢ cos 6 - cos B sin a cos ¢ cos 9)

K

X e

V[cos 8 cos a cos 6 cos ~ + sin B (sin > sin 6 cos ~ - cos $ sin jp)

+ cos 6B sin a (cos » sin 6 cos p+ sin @ sin »)]

y = Vi[cos 8B cos a cos 8 sin ~ + sin 8 (sin ¢ sin 6 sin p+ cos ¢ cos jp)

+ cos 8 sin a (cos $¢ sin 8 sin p - sin $ cos })])

OBSERVATION EQUATIONS

The user-selectable observation variables computed in LINEAR represent a broad

class of parameters useful for vehicle analysis and control design problems. These

variables include the state, time derivatives of state, and control variables.

Also

included are air data parameters, accelerations, flightpath terms, and other miscel-

laneous parameters. The equations used to calculate those parameters are derived

from a number of sources (Clancy, 1975; Dommasch and others, 1967; Etkin, 1972;

Gainer and Hoffman, 1972; Gracey, 1980). Implicit in many of these observation

equations is an atmospheric model. The model included in LINEAR is derived from

the U.S. Standard Atmosphere (1962).

The vehicle body axis accelerations constitute the set of observation variables that, except for state variables themselves, are most commonly used in the aircraft control analysis and design problem. These accelerations are measured in g units

17

and are derived directly from the body axis forces defined in the previous section

for translational acceleration.

The body axis acceleration equations used in LINEAR

are

ay = (Xp - D cos a + L sin a - gm sin 9)/g ym

(Yp + Y + gm cos 8 sin $)/gym

ag

(Zp - D sin a - L cos a+ gm cos 6 cos $)/gm

The equations for body axis accelerometers that are at vehicle center of gravity are

an, = (Xp - D cos a + L sin a)/gom

any = (¥p + ¥)/gym

an, = (Zp - D sin a- L cos a)/gG om
an = (-2p + D sin a+ L cos a)/gGom
For orthogonal accelerometers that are aligned with the vehicle body axes but are not at vehicle center of gravity, the following equations apply:

Nyx,i ~ 4nx 7 [(q? + x?) xy - (pq - LY x ~ (pr + D2x1/5, any i = any + [(pq + r)xy - (p2 + r)yy + (qr - p)Zy]/g,

angi = an, + [(pr - qx, + (qr + Py, ~ (q2 + p*)z2]/9,

an,i = ay - [lpr - q) x, + (qr + P)y, - (q2 + p*)zz]/9,

where the subscripts x, y, and z refer to the x, y, and z body

and the symbols x, y, and z refer to the x, y, and z body axis

sors relative to the vehicle center of gravity.

The symbol Jo

axes, respectively, locations of the senis the acceleration

due to gravity at sea level. Also included in the set of acceleration parameters
is the load factor n = L/W, where L is the total aerodynamic lift and W is the vehicle weight.

The air data parameters having the greatest application to aircraft dynamics and control problems are the sensed parameters and the reference and scaling parameters. The sensed parameters are impact pressure q,, ambient or free-stream pressure pa, total pressure p;,, ambient or free-stream temperature T, and total temperature T,.

The selected reference and scaling parameters are Mach number M, dynamic pressure qe

speed of sound a, Reynolds number Re, Reynolds number per unit length Re', and the

Mach meter calibration ratio Ic/Pa>

These quantities are defined as

1/2 a= (1.4 PPoolo n)
18

rae a

R, € =

Ve u

Rg e' u= p&v - _ pv 4 2
[(1.0 + o.2m2)3°5 - 1.0]p,

qd. Cc = 4
( 1.2m2

(5.—62M2:2e~ 2M0e.8 2.5 - 1.0]

p a

ae

( (1.0 + 0.2mM2)"°°3?.5 ~ 14.0

Pp a

[ t.2m2

{5.—62M2.:76- 2M0*.8 2.5

- 1.0

(Mm < 1.0)
(M > 1.0)
(M < 1.0)
(mM > 1.0)

Tt T(1.0 + 0.2M2)

where p is the density of the air, yw is the
script O refers to sea level, standard day stream temperature, and the coefficient of ard Atmosphere (1962).

coefficient of viscosity, and the

conditions.

Free-stream pressure,

viscosity are derived from the U.S.

sub-
freeStand-

Also included in the air data calculations are two velocities: speed Ve and calibrated airspeed V,, both computed in knots. The

that internal units are in the English system. speed is

The equation used

equivalent aircalculations assume
for equivalent air-

Ve = 17.17Vq

which is derived from the definition of equivalent airspeed,

Ole

where Po = 0.002378 slug/ft3 and Ve is converted from feet per second to knots. ibrated airspeed is derived from the following definition of impact pressure:

Cal-

19

3.5 fc =~ Po[\'*°1.0 *+=-To~p2, Yey?

1] (Vy c *§ Fo ag)

1 i t

Vo 2

5.76

2.5

°

of “915.6 = 0.8 (ac/Vg)*

°

For the case where Vo * ag, the equation for Vg is

Vo = 1479.116

(2q + 1.0) 2/7 - 1.0 (Vy < ag) Po

Calibrated airspeed The equation

is

found

using

an

iterative

process

for

the

case

where

Vo

? age

Vo = 582.95174

(= + 1-0) Po

1.0

7.0

1:0 (Vo/ao)

7

2.5

(Ve > ag)

is executed until 0.001 knots.

the

change

in V,

from

one

iteration

to the

next

is

less

than

Also included in the observation variables are the flightpath-related parameters (described in app. C), including flightpath angle y, flightpath acceleration fpa,

vertical acceleration h, flightpath angle rate Y and (for lack of a better category

in which to place it) scaled altitude rate f/57.3. these quantities are

The equations used to determine

Jo
h = a sin @ - ay sin ¢ cos 6 ~ a, cos » cos 8
-_ovh - iv
Y=
v Vv2 - £2
Two energy-related terms are included in the observation variables: energy Eg, and specific power P,, defined as
Eg =h+ V42
2g
Ww s g Po =ht—

specific

20

The set of observation variables available in LINEAR also includes four force parameters: total aerodynamic lift L, total aerodynamic drag D, total aerodynamic normal force N, and total aerodynamic axial force A. These quantities are defined as
L = qSCy,
D = gSCp

N= L cos a+ D sin a

A = -L sin a+D cos a

where Cp and Cy, are coefficients of drag and lift, respectively.

Six body axis rates and accelerations are available as observation variables. These include the x body axis rate u, the y body axis rate v, and the z body axis

rate w. Also included are the time derivatives The equations defining these quantities are

of these quantities,

u, vy, and We

u

i H

V cos a cos £6

v= V sin £ w = V sin acos B

.

Xm — gm sin 6 - D cos a+ L sin a

u =

m

+ rv - qw

°

Yp + gm cos 6 sin ¢+ Y¥

v=

m

+ pw - ru

°

Zp + gm cos 8 cos 9 - D sin a-~ L cos a

The final set of observation variables available in LINEAR is a miscellaneous collection of other parameters of interest in analysis and design problems. The first group consists of measurements from sensors not located at the vehicle center of gravity. These represent angle of attack a,i, angle of sideslip B,i, altitude his and altitude rate hi measurements displaced from the center of gravity by some x, y, and z body axis distances. The equations used to compute these quantities are
ai — = o — ( {AX_v—_by )
Bhi = B+ (>)
h,i = h+ x sin 6- y sin $¢ cos 6 = 2 cos ¢ cos 6

21

hoi =h+ (x cos 6 + y Sin » sin 6 + z cos q¢ sin 6)

- oly cos 9 cos 8 - z sin » cos 6)

The remaining roll rate pg, as

miscellaneous parameters are total angular momentum T, stability axis stability axis pitch rate q,, and stability axis yaw rate r s’ defined

T =-12 (Ixp 2 - 21.xyP4 - 21,,,Pr + Ta 2_-. 2tyzar + I,ré2)

Ps = pcos a+r sina ds = q Yg = -p Sin a+r cos a

SELECTION OF STATE, CONTROL, AND OBSERVATION VARIABLES

The equations in the two preceding sections define the state and observation

variables available in LINEAR. The control variables are defined by the user in the

input file.

Internally, the program uses a 12~state model, a 120-variable obser-

vation vector,

anda 30-parameter control vector.

These variables can be selected

to specify the formulation most suited for the specific application.

The order and

number of parameters in the output model are completely under user control. Figure 1

shows the selection of variables for the state vector of the output model.

However,

it should be noted that no model order reduction is attempted.

Elements of the

matrices in the internal formulation are simply selected and reordered in the for-

mulation specified by the user.

Specific state, control, and observation variables for the formulation of the

output matrices are selected by alphanumeric descriptors in the input file. The use of these alphanumeric descriptors is described in the Data Input section. Appendix C

lists the observation variables and their alphanumeric descriptors.

Appendix D lists

the state variables and their alphanumeric descriptors.

The alphanumeric descriptors

for the selection of control parameters to be included in the observation vector are

the control variable names defined by the user in the input file, as described in the

Input Files section.

LINEAR MODELS

The linearized system matrices computed by LINEAR are the first-order terms of

a Taylor series expansion about the analysis point (Dieudonne, 1978; Kwakernaak and

Sivan, 1972; NASA RP by Duke, Antoniewicz, and Krambeer) and are assumed to result in

a time-invariant linear system.

The validity of this assumption is discussed in the

Analysis Point Definition section. The technique used to obtain these matrices numerically is a simple approximation to the partial derivative, that is,

dg £ (xq + AK) - E(x ~ dx)

Ox

2 dx

where f is a general function of x, an arbitrary independent variable. The Ax may be
set by the user, but defaults to 0.001 for all state and control parameters, with the
single exception of velocity V, where Ax is multiplied by a, the speed of sound, to
obtain a reasonable perturbation size.
From the generalized nonlinear state equation

and the observation equation

Tx

£ (x, x, u)

Y

g(x, x, u)

the program determines the linearized matrices for the generalized formulation of the system:

Cc éx = A' 6x + B' éu

oy H' 6x + G éx + F' éu

where

Q I B t
|

At =—_—

23

with all derivatives evaluated along the nominal trajectory defined by

point

(Xo, Xo, Uo).

The state,

expressed as small perturbations

time derivative of state, and about the nominal trajectory,

control so that

the analysis vectors can be

KX = Xo + 6x

Xs = Xeo + 6xe

u=Uo+ du

of

In addition to the matrices for this requesting linearized matrices for a

generalized system, standard formulation

the user has the of the system:

option

é&xk = A 6x + B 6u

where

éy = H 6x + F fu

a=|r r-2 7-1 2

a

ox |

*

B= r- pie

a

ox

n= 2842p[ ahr 7)- at

a

ox|

°*

r= 2“3,a%x 1-2Lox]13

with all derivatives evaluated along the nominal trajectory defined by the analysis

point

(Xo, Xor Uo).

LINEAR also provides two nonstandard matrices, D and E, in the equations

xX = Ax + Bu + Dv

or D' and E' in the equations

Y

Hx + Fu + Ev

Cx = A'x + Blu + D'v
y = H'x + Gx + F'u + Ey

24

These dynamic interaction matrices include the effect
acting on the vehicle. The components of the dynamic mental body axis forces and moments:

of external interaction

forces vector

and moments v are incre-

is éY 6Z éL 6M
L ON
Thus,

and

E' = 2g

I s l e

[

an of

ov

Oxe T - ~a~x

noTv

These matrices allow the effects of unusual subsystems or control effectors to be easily included in the vehicle dynamics.

The default output matrices for LINEAR are those for the standard system formulation. However, the user can select matrices for either generalized or standard state and observation equations in any combination. Internally, the matrices are computed for the generalized system formulation and then combined appropriately to accommodate the system formulation requested by the user.

ANALYSIS POINT DEFINITION

The point at which the nonlinear system equations are linearized is referred to
as the analysis point. This can represent a true steady-state condition on the spec~ ified trajectory (a point at which the rotational and translational accelerations are zero; Perkins and Hage, 1949; Thelander, 1965) or a totally arbitrary state on a tra-
jectory. LINEAR allows the user to select from a variety of analysis points. These analysis points are referred to as trim conditions, and several options are available to the user. The arbitrary state and control option is designated NOTRIM, and in selecting this option the user must specify all nonzero state and control variables. For the equilibrium conditions, the user specifies a minimum number of parameters,
and the program numerically determines required state and control variables to force

25

the rotational and translational accelerations to zero. are described in detail in the following subsections.

The analysis point options

For all the analysis point definition options, any state or control parameter

may be input by the user. Those state variables not required to define the analysis

point are used as initial estimates for the calculation of the state and control

conditions that result in zero rotational and translational accelerations.

As each

state variable is read into LINEAR, the name is compared to the list of alternative

state variable names listed in appendix D. All state variables except velocity

must be specified according to this list. Velocity can also be defined by specify-

ing Mach number (see alternative observation variable names in appendix C). Appen-

dix E lists analysis point definition identifiers that are recognized by LINEAR.

It should be noted that the option of allowing the user to linearize the system equations about a nonequilibrium condition raises theoretical issues (beyond the
scope of this report) the potential user should be aware of. Although all the analysis point definition options provided in LINEAR have been found to be useful in the analysis of vehicle dynamics, not all the linear models derived about these analysis points result in the time-invariant systems assumed in this report. However, the results of the linearization provided by LINEAR do give the appearance of being time invariant.

The linearization process as defined in this report is always valid for some time interval beyond the point in the trajectory about which the linearization is done. However, for the resultant system to be truly time invariant, the vehicle must be in a sustainable, steady-state flight condition. This requirement is something more
than merely a trim requirement, which is typically represented as x(t) = 0, indicating that for trim, all the time derivatives of the state variables must be zero. (This is not the case, however: Trim is achieved when the acceleration-litekrems are identically zero; no constraints need to be placed on the velocity-like terms in
Ke Thus, for the model used in LINEAR, only PB, qe r, V, Oy and é must be zero to satisfy the trim condition.) The trim condition is achieved for the straight-andlevel, pushover-pullup, level turn, thrust-stabilized turn, and beta trim options described in the following sections. In general, the untrimmed and specific power analysis point definition options do not result in a trim condition.

Of these analysis point options resulting in a trim condition, only the straight-

and-level and level turn options force the model to represent sustainable flight con-

ditions.

In fact, only in the special case where the flightpath angle is zero does a

truly sustainable trim occur.

As previously stated, the linearization of a nonlinear model and its represen-

tation as a time-invariant system are always valid for some time interval beyond the

analysis point on the trajectory.

This time interval is determined by several fac-

tors (such as trim and sustainable flight conditions) and ultimately by accuracy

requirements placed on the representation.

Thus, in using the linear models provided

by this program, the user should exercise some caution.

Untrimmed For the untrimmed analysis point, the user specifies all state and control variables that are to be set at some value other than zero. The number of state var-
26

iables specified is entirely at the user's discretion.

If any of the control var-

iables are to be nonzero, the user must specify the control parameter and its value.

The untrimmed option allows the user to analyze the vehicle dynamics at any flight

condition, including transitory conditions.

Straight-and-Level Trim

The straight-and-level analysis points available in LINEAR are in fact wings-

level, constant-flightpath-angle trims.

Both options available for straight-and-

level trim allow the user to specify either a flightpath angle or an altitude rate.

However, since the default value for these terms is zero, the default for both types

of straight-and-level trim is wings-level, horizontal flight.

The two options available for straight-and-level trim require the user to specify

altitude and either an angle of attack or a Mach number.

If a specific angle of

attack and altitude combination is desired, the user selects the “Mach=-trim" option,

which determines the velocity required for the requested trajectory.

Likewise, the

"alpha-trim" option allows the user to specify Mach number and altitude, and the trim

routines in LINEAR determine the angle of attack needed for the requested trajectory.

The trim condition for both straight-and-level options is determined within the following constraints:

p=q=re=od0

¢= 0

The trim surface positions, thrust, angle of sideslip, and either velocity or angle

of attack are determined by numerically solving the nonlinear equations for the

translational and rotational acceleration.

Pitch attitude 8 is determined by itera-~

tive solution of the altitude rate equation.

Pushover-Pullup

The pushover-pullup analysis points result in wings-level flight at n # 1. For

n> 1, the analysis point is the minimum altitude point of a pullup when h=0.

For

n < 1, this trim results in a pushover at the maximum altitude with h = 0. There

are two options available for the pushover-pullup analysis point definition:

(1) the

“alpha-trim" option in which angle of attack is determined from the specified alti-

tude, Mach number, and load factor, and (2) the "load-factor-trim" option in which

angle of attack, altitude, and Mach number are specified, and load factor is deter-

mined according to the constraint equations.

The analysis point is determined at the specified lowing contraints:

conditions,

subject to the fol-

p=r=0

q e Yme co1s 8 [mg(n - cos(6 - a)) ~ Zp cos a + Xp sijn a]

¢= 0 27

The expression for q is derived from the a equation by setting a = 0 and ¢ = 0; Ois

derived from the h equation. The trim surface and either angle of attack or load factor are nonlinear equations for the translational and

positions, determined rotational

thrust, angle by numerically accelerations.

of sideslip, solving the

Level Turn

The level turn analysis point definition options result in nonwings-level,

constant-turn-rate flight at load factors greater than one. The vehicle model is

assumed to have sufficient excess thrust to trim at the condition specified.

If

thrust is not sufficient, trim will not result, and the analysis point thus defined

will have a nonzero (in fact, negative) velocity rate.

The level trim options available in LINEAR require the specification of an alti-

tude and a Mach number. The user can then use either angle of attack or load factor

to define the desired flight condition. These two options are referred to as “"alpha-

trim" and "load-factor-trim," respectively.

For either option, the user may also

request a specific flightpath angle or altitude rate. Thus, these analysis point

definitions may result in ascending or descending spirals, although the default is

for the constant-altitude turn.

The constraint equations for the coordinated level turn analysis point definitions are derived by Chen (1981), and Chen and Jeske (1981). Using the requested load factor, the tilt angle of acceleration normal to the flightpath from the vertical plane 47, is calculated using the equation

dy, = ttan™1 E

=

ccooss?

wt)
Y

where the positive sign is used for a right turn and the negative sign is used for a left turn. From 9$,, turn rate can be calculated as
b= deana

Using these two definitions, the state variables can be determined:

1 Q t l

y sin? g

|-sin y sin B + (ss02 y sin? g - sin? Y- cos

sin2

or,

:) 1//2

Le S = tan @7, cos B

= -¥SiIN Y _ tan

P.

cos Bp

4

8

P= Ps, COS O- Lg sin a

r= P, sina + Ys cos a

28

o c I

sin71(-p/p)

>

tan7l(g/r)

The trim surface positions, thrust, angle of sideslip, and either angle of attack or
load factor are determined by numerically solving the nonlinear equations for the translational and rotational accelerations.

Thrust-Stabilized Turn

The thrust-stabilized turn analysis point definition results in a constant throt-

tle nonwings~level turn with a nonzero altitude rate. The two options available in

LINEAR are "alpha-trim" and "load-factor-trim."

These options allow the user to spe-

cify either the angle of attack or the load factor for the analysis point. The alti-

tude and Mach number at the analysis point must be specified for both options.

The

user also must specify the value of the thrust trim parameter by assigning a value to

the variable THRSTX in the input file after the trim has been selected.

The constraint equations for the thrust-stabilized turn are the same as those for
the level turn. However, for this analysis point definition, flightpath angle is
determined by LINEAR.

Beta Trim

The beta trim analysis point definition results in nonwings-level, horizontal

flight with the heading rate d equal to zero at a user-specified Mach number, alti-

tude, and angle of sideslip.

This trim option is nominally at 1 g, but as 8B and

vary from zero, normal acceleration decreases and lateral acceleration increases.

For an aerodynamically symmetric aircraft, a trim to £8 = 0 using the beta trim option

results in the same trimmed condition as the straight-and-level trim.

However, for

an aerodynamically asymmetric aircraft, such as an oblique-wing vehicle, the two trim

options are not equivalent.

The constraint equations used with the beta trim option are

The trim surface positions, thrust, angle of attack, and bank angle are determined by
numerically solving the nonlinear equations for translational and rotational acceleration. Pitch attitude 8 is derived from the equation for flightpath angle y with y = 0 and is defined as

6 = tan-1 (2

B sin » + cos B sin =)
cos 8 cos a

Specific Power

The specific power analysis point definition results in a level turn at a user-

specified Mach number, altitude, thrust trim parameter, and specific power.

Unlike

29

the other trim options provided in LINEAR, the Specific power option does not, in

general, attempt to achieve the zero velocity rate Vv.

In fact, because the altitude

rate h = 0 and specific power are defined by

the resultant velocity rate will be

v= 82 Vv

However, the other acceleration-like terms (p, qe r, Oy and B) will be zero if the requested analysis point is achieved.

The constraint equations used with the option can be derived from the load factor analysis point definition with y = 0:

specific power analysis point definition tilt angle equation used in the level turn

OL = ttan71(n2 - 1) 1/7? (where the positive sign is used for a right turn and the negative sign is used for a left turn),

yy= 7Fd tan qy,

q = p sin dy, cos £
Ys = tan 91S,. cos £ Ps = -q tan B
P= Pg coS a- rg sin a Y= Pg sin a + Yg COS o

mf5nH.f —_>
“4a -<e fU
“ee”

i l

@ D

I l

o e

tan71 qxar

The analysis point surface positions,
sideslip are determined by numerically tional and rotational acceleration.

load factor, solving the

angle of attack, and angle of nonlinear equations for transla-

30

NONDIMENSIONAL STABILITY AND CONTROL DERIVATIVES

The nondimensional stability and control derivatives computed by LINEAR from the nonlinear aerodynamic model assume broadly formulated linear aerodynamic equations. These equations include effects of what are normally considered exclusively lateral-
directional parameters in the longitudinal force and moment coefficient equations,
and, conversely, effects of longitudinal parameters in the lateral~-directional equa-
tions. The reason for this is two-fold: application to a larger class of vehicle types, such as oblique~wing aircraft, and computational ease.

The nondimensional stability and control derivatives assume the following equations for the aerodynamic force and moment coefficients:

n
Ce = Cog t Cysat Cyoh + Cy bh + Cy, bv + 2 Cys. 63
L=

+ Cy ep + Cyd + Co

ft

a
Cara

+

a
Cye8 e

n
Cm = Cao + Smg% + Cmg8 + Cm, Sh + Coy SV + 2 Cng, 64
a
+ CoP + Cmd + CmF + Cngo + Cmge

Ch = Cy + Cage + Cyph + Cy bh + Cyy ov + 2 Cg, 64
1=

+ CnpP + Cngt + Cnt + Cyta t Cn g8

n
Cy = Chg + Cho + Cy, 28 + Ch éh + CLy év + i=>1 CLs; rt

a

n e

+ Crop + Crd + CLF + Cyga + Cr 38

n
Cp = Cpg + Cpga + Cr gb + Ch, dh + Cp, ov + 2 D6; 83
1=

+

Cp.B

+

Cpud

+

Cp f

+

a
Cpta +

a
Cp g8e

31

Cy = Cyg + Cya

t+ CygB

+ Cy,

St

cy

vt

n
i=SY1

cy 6;

ok

aA

+

Cy ¥pPB++

CCyy

aG++ CyCry,t+tcoYtQ ai”t+c?y

e
38

where the stability and control derivatives have the usual meaning,

Cex FT

with C, being an arbitrary force or moment sional variable. The rotational terms in of the corresponding state variable with

coefficient the equations

and are

x an arbitrary nondimensional

nondimenversions

P- ™. b2Vpv

a4 *= £o&vJ
p= b2eV

2V *

_

ca

: _ bB
BR 5

The 6; in the summations are the n control variables effects of altitude and velocity are included in the those parameters and in the incremental multipliers
6h =h = ho

defined by the user. The derivatives with respect to

" < t < fe)

6V

where the subscript zero represents the current analysis described in the Analysis Point Definition section.

point

(xo,

Xo,

Uo),

All stability tude and velocity units used in the Mach and velocity

derivatives are computed as nondimensional terms except the

parameters. The units of the control nonlinear aerodynamic model. LINEAR derivatives. Using the equation

derivatives is capable of

correspond handling

altito the
both

Cey = Cey 4h

(where a, is the Mach derivatives

speed of from the

sound at velocity

the reference altitude), LINEAR will compute the derivatives or vice versa. Internally, however,

32

LINEAR uses velocity derivatives in the computations of the total force and moment

coefficients.

Derivatives with respect to angle of attack and angle of sideslip can

be obtained in units of reciprocal degrees. These derivatives are simply the cor-

responding nondimensional derivatives multiplied by 180/n.

DATA INPUT
The interactive linearizer allows the user to input data in various manners: 1. All input data can be loaded from a single file (either a file assigned to logical unit 1 or any file named with 10 characters or less); 2. Data can be separated onto different files, each containing different sec-
tions of data; 3. Data can be typed in from the terminal; or 4. A combination of (2) and (3).

Input Files

The interactive LINEAR input file (defined in table 1) is separated into six

major sections:

vehicle title information; geometry and mass data for the aircraft;

state, control, and observation variable definitions for the state-space model; the

trim parameter specification; additional control surfaces that may be specified for

each case; and various test case specifications.

All the input data can be written

on one file or various files according to the divisions specified at the beginning of

this paragraph and according to the input format defined in table 1. An example

input file is listed in appendix F. All the input records to LINEAR are written in

ASCII form.

Title Information

There is one title record name of the specific vehicle.
title appears on each page of

that needs to be specified for interactive LINEAR: the This record is read with a 20A4 format. The vehicle
the line printer output file.

Geometry and Mass Data

The geometry and mass data consist of four records that either can follow the

vehicle title, exist as a file on its own, or be input interactively from the ter-

minal.

If it is input from the terminal,

then it will be stored either on the same

file as the vehicle title, or on a separate file.

The geometry and mass data records

define the geometry, mass properties, location of the aerodynamic reference point

with respect to the center of gravity, and angle-of-attack range for the vehicle

model being analyzed.

33

TABLE 1. — INPUT FORMAT FOR LINEAR

Input record

VEHICLE TITLE

Ss

b

c

Ix

ly

I,

DELX
Omin
NUMSAT STATE 1 STATE 2 STATE 3

DELY
Oma x
EQUAT DRVINC DRVINC DRVINC

Weight

Tye

Ixy

DELZ

STAB 1 2 3

LOGCG

Format
(20A4)
(4F13.0) (6F13.0)
(3F10.0,12A4)
(2F13.0)
(110,A4,11X,A4) (5A4,F10.0) (5A4,F10.0) (5A4,F10.0)

NUMSUR
CONTROL 1 CONTROL 2
CONTROL 3

LOCCNT 1 LOCCNT 2 LOCCNT 3

CONVR 1 CONVR 2 CONVR 3

CNTINC 1 CNTINC 2 CNTINC 3

(110)
(5A4,110,A4,6X,F10.0) (5A4,110,A4,6X,F10.0)
(5A4,110,A4,6X,F10.0)

NUMYVC
MEASUREMENT
MEASUREMENT
MEASUREMENT

EQUAT
1
2
3

PARAM 1 (1-3)
PARAM 2 (1-3)
PARAM 3 (1-3)

(I110,A4)
(5A4,3F10.0)
(5A4,3F10.0)
(5A4,3F10.0)

Semin Semax NUMXSR
ADDITIONAL ADDITIONAL
ADDITIONAL

Samin Samax
SURFACE 1 SURFACE 2
SURFACE 3

Srmin Srmax
LOCCNT 1 LOCCNT 2
LOCCNT 3

Sthmin

Sthmax

(SF 10.0) (1X,12)
(5A4,110,A4) (5A4,110,A4)
(5A4,110,A4)

ANALYSIS ANALYSIS VARIABLE VARIABLE VARIABLE

POINT POINT 1 2 3

DEFINITION DEFINITION
VALUE 1 VALUE 2 VALUE 3

OPTION SUBOPTION

(20A4) (A4) (5A4,F15.5) (5A4,F15.5) (5A4,F15.5)

34

TABLE 1. — Concluded

Input record

NEXT
ANALYSIS
ANALYSIS VARIABLE
VARIABLE
VARIABLE

POINT
POINT 1
2
3

DEFINITION
DEFINITION VALUE 1
VALUE 2
VALUE 3

OPTION
SUBOPTION

Format
(A4)
(20A4)
(A4) (5A4,F15.5)
(5A4,F15.5) (5A4,F15.5)

END

(A4)

35

The first and second records describe the vehicle geometry, mass, and mass dis-

tribution.

The first record defines the wing planform area S in units of feet

squared; the wingspan b in units of feet; the mean aerodynamic chord of the wing c

in units of feet; and the sea level weight of the vehicle Weight, in units of pounds. The second record defines the vehicle moments and products of inertia in units of

slug-feet squared. The third record defines the offset of the aerodynamic reference

point with respect to the vehicle center of gravity in the normal right-handed body

axis reference system with the positive x axis forward. The format of the data can

be seen in table 1 with DELX, DELY, and DELZ representing the x, y, and z body axis

displacements of the aerodynamic reference point with respect to the center of gra-

vity in units of feet (see app. A). The fourth variable of the third record is an

alphanumeric variable, read using a 12A4 format, to specify if corrections due to a

center-of-gravity offset are to be computed in LINEAR or in the user-supplied aerody-

namic subroutine CCALC. The variable LOGCG defaults to a state that causes the aero-

dynamic reference point offset calculations to be performed by subroutines within

LINEAR.

Any of the following statements in the LOGCG field will cause LINEAR not to

make these corrections:

NO CG CORRECTIONS BY LINEAR

CCALC WILL CALCULATE CG CORRECTIONS

FORCE AND MOMENT CORRECTIONS CALCULATED IN CCALC

However, LINEAR will read only the first four characters of the string.

The final record of this geometry and mass data set defines the angle-of-attack range for which the user-supplied nonlinear aerodynamic model (CCALC) is valid. The parameters Omin and max specify the minimum and maximum values of angle of attack to
be used for trimming the aircraft model. These parameters are in units of degrees.

State, Control, and Observation Variable Definitions

The state, control, and observation variables to be used in the output formula-

tion of the linearized system are defined in records that either follow the last of

the previously described sets of records on the same file, are stored on a separate

file, or are input through the terminal.

If they are input through the terminal,

they will be stored in a file specified by the user. The number of records in the

state, control, and observation variable definition set is determined by the number

of such variables defined by the user.

The states to be used in the output formulation of the linearized system are

defined in the first set of records in the state, control, and observation variable

definitions.

The first record of this set, as shown in table 1, defines the number

of states to be used (NUMSAT), the formulation of the state equation (EQUAT), and

whether the nondimensional stability derivatives with respect to angle of attack and angle of sideslip are to be output in units of reciprocal radians or degrees. The

variable EQUAT is read using an A4 format and is tested against the following list:

36

NONSTANDARD NON~STANDARD GENERALIZED EXTENDED
If EQUAT matches the first four characters of any of the listed words, the output formulation of the state equation is

Cx = A'x + Btu

If EQUAT is read in as STANDARD or does not match the preceding list,
fault standard bilinear formulation of the state equation is assumed,
put matrices are consistent with the equation

then the deand the out-

x = Ax + Bu

The variable STAB is also read using an A4 format and is compared with the following
list:

DEGREES DGR

If STAB matches the first four characters of either of these words, the nondimen-
sional stability derivatives with respect to angle of attack and angle of sideslip are printed in units of reciprocal degrees on the printer file. Otherwise, these derivatives are printed in units of reciprocal radians.

The remaining records of the state variable definition set are used to specify
the state variables to be used in the output formulation of the linearized system and
the increments to be used for the numerical perturbation described in the Linear Models section. The state variable names are checked for validity against the state variable alphanumeric descriptors listed in appendix D. If a name is not recognized, the variable is ignored and a warning message is written to the printer file. The increment to be used with any state variable (in calculating the A' and H' matrices)
and the time derivative of that state variable (in calculating the C and G matrices)
can be specified using the DRVINC variable. The DRVINC specified for velocity will
be multiplied by the speed of sound within LINEAR in order to scale up the pertur-
bation size to a reasonable value while keeping DRVINC on the same order of magnitude
as for the other states. If DRVINC is not specified by the user, the default value
of 0.001 is used,

The next set of records in the state, control, and observation variable defini-
tions are those defining the variables to be used in the control vector of the output model. The first record of this set defines the number of control parameters to be
used (NUMSUR). The remaining records define the names of these variables (CONTROL),
their location (LOCCNT) in the common block /CONTRL/ (see the User~Supplied Sub-
routines section), the units associated with these control variables (CONVR), and
the increments (CNTINC) to be used with these variables in determining the B' and F' matrices.

Because LINEAR has no default control variable names, the control variable names
input by the user are used for subsequent identification of the control variables. Therefore, consistency in the use of control variable names is extremely important,

37

particularly when the user attempts to establish control variable initial conditions when using the untrimmed analysis point definition option.

The CONVR field in the control variable variables are given in degrees or radians. compared to the following list:

records is used to specify if the control CONVR is read using an A4 format and is

DEGREES DGR RADIANS RAD
If CONVR agrees with the first four characters of either of the first two names, it is assumed that the control variable is specified in units of degrees. If CONVR agrees with the first four characters of either of the last two listed names, it is assumed that the control variable is specified in units of radians. No units are assumed if CONVR does not agree with any of the listed names. When it is assumed that the control variable is specified in units of radians, the initial value of the control variable is converted to degrees before being written to the printer file.

The variable CNTINC can be used to specify the increments used for a particular

control surface when the B' and F' matrices are being calculated.

It is assumed that

the units of CNTINC agree with those used for the surface, and no unit conversion is

attempted on these increments.

If

a default value of 0.001 is used.

CNTINC is not specified for a particular surface,

The final set of records in the state, control, and observation variable defini-

tions pertain to the specification of parameters associated with the observation vec-

tor, observation equation, and observation parameters.

The first record of this set

defines the number of observation variables (NUMYVC) to be used in the output linear

model and the formulation of the output equation (EQUAT). The remaining records in

this set specify the variables to be included in the observation vector (MEASUREMENT)

and any position information (PARAM) that may be required to compute the output model for a sensor not located at the vehicle center of gravity.

The variable used to specify the formulation of the observation equation (EQUAT)

is compared with the same list of names used to determine the formulation of the

state equation.

If it is determined that the generalized formulation is desired, the

observation equation

y = H'x + Gx + F'u

is used. Otherwise, the standard vation equation used is

formulation

is assumed,

and the form of the obser-

y = Hx + Fu

The records defining the observation variables to be used in the output formulation of the linear model contain a variable that includes the parameter name (MEASUREMENT) and three fields (PARAM) defining, when appropriate, the location of
the sensor relative to the vehicle center of gravity. The parameter name is compared with the list of observation variables given in appendix D. If the parameter name is recognized as a valid observation variable name, that observation variable is

38

included in the formulation of the output observation vector. If the parameter name is not recognized, an error message is printed and the parameter named is ignored. The three variables represented by PARAM(1), PARAM(2), and PARAM(3) provide the x-
axis, y-axis, and z-axis locations, respectively, of the measurement with respect to
the vehicle center of gravity if the selected observation is one of the following:
The unit associated with these variables is feet. If the selected observation var-
iable is not in the preceding list, the PARAM variables are not used. The sole
exception to this occurs when Reynolds number is requested as an observation variable. In that case, PARAM(1) is used to specify the characteristic length. When no
value is input for PARAM(1), the mean aerodynamic chord ¢ is used as the character-
istic length.
Trim Parameter Specification
There is one record in the trim parameter specification set that is associated with the subroutine UCNTRL (described in the User-Supplied Subroutines section). This record specifies the limits to be used for the trim parameters 64, Sar Sy, and
Sth representing the longitudinal, lateral, directional, and thrust trim parameters,
respectively. The units associated with these parameters are defined by the implementation of UCNTRL.
Additional Surface Specification
The first record of the set of additional surface specifications defines the
number of additional controls to be included in the list of recognized control names
(NUMXSR). The purpose of defining these additional controls is to allow the user to set such variables as landing gear position, wing sweep, or flap position. The
controls are only defined in the additional surface specification records; actual
control positions are defined in the analysis point definition records. If such controls are defined, the records defining them will have the format specified in
table 1. The control variable name (ADDITIONAL SURFACE), location (LOCCNT) in the common block /CONTRL/, and the units associated with this control variable (CONVR) are specified for each additional control.
39

Test Case Specification

The test case specification records allow the user to define the flight condi-

tion, or analysis point, at which a linear model is to be derived.

Multiple cases

can be included in the test case specification records.

The final record in each set

directs the program to proceed (NEXT) or to stop (END) execution.

The first record of a test case specification set determines the analysis point,

or trim, option to be used for the current case.

The ANALYSIS POINT DEFINITION

OPTION parameter is read in and checked against the list of analysis point defini-~

tion identifiers described in appendix E. The second record of a test case specifi-

cation set, defining an analysis point definition suboption (ANALYSIS POINT DEFIN~

ITION SUBOPTION), will be read only if the requested analysis point definition option

has a suboption associated with it. These suboptions are defined in the Analysis

Point Definition section.

The valid alphanumeric descriptors for these suboptions

are described in appendix E.

The remaining records in a test case specification set define test conditions or

initial conditions for the trimming subroutines.

These records consist of a field

defining a parameter name (VARIABLE) and its initial condition (VALUE). ‘These

records may be in any order; however, if initial Mach number is to be defined, the

altitude must be specified before Mach number if the correct initial velocity is to

be determined.

The parameter names are checked against all name lists used within

LINEAR.

Any recognized state, time derivative of state, control, or observation var-

iable will be accepted. Trim parameters also can be set in these records.

In general, setting observation variables and time derivatives of the state vari-

ables has little effect. However, for some of the trim options defined in_the Anal-

ysis Point Definition section, Mach number and load factor are used. The thrust trim

parameter only affects the specific

power trim.

For the untrimmed option, the

initial values of the state and control variables determine the analysis point com-

pletely. For all other trim options, only certain states are not varied; all con-

trols connected to the control and engine models are varied.

Interactive Data Input
Upon starting program execution, LINEAR will ask the user:
WHAT IS THE TITLE OF THE CASE BEING RUN?
The user should answer with some meaningful title followed by a carriage return. This title, along with the vehicle title discussed later, will identify this interactive session on all of the output data files.
The program will then display the following menu:
PROGRAM LINEAR READS ALL INPUT DATA FROM 1 UNLESS A DIFFERENT FILE NAME IS SPECIFIED.
1) ALL INPUT DATA IS ON 1
2) DATA IS ON ONE OTHER FILE

40

3) DATA IS ON VARIOUS FILES
4) INPUT DATA FROM A FILE(S) AND FROM THE TERMINAL

5) INPUT ALL DATA FROM THE TERMINAL

The user must answer with the appropriate number followed by a carriage return.

If the user answers with a "1", the program will read all the input data from the

file assigned logical unit one. If the user selects choice "2", the program will ask

for the file name that contains all the input data required by LINEAR.

If the user

selects "3" or "4", the program will prompt the user for each file name pertaining to each section of input data. If option "5" is chosen, the program will prompt the

user for the input data values.

The data is separated into the six sections described previously:

1. Vehicle title information,

2. Vehicle geometry and mass properties,

3. State, control, and observation variable definitions,
4. Limits for the trim parameters,

5. Additional control surface definitions, and

6. Test case specifications.
The program will prompt the user for the name of the file that contains each section
of data. An example prompt follows:

WHAT IS THE NAME OF THE FILE CONTAINING THE TITLE INFORMATION? (TYPE 0 TO INPUT THE TITLES FROM THE TERMINAL AND STORE ON A FILE)

If the user already has created a file containing the appropriate data needed for the

section, type the

a 10-character (maximum) file name may be entered. data from the terminal and have the program format

If the a file,

user wishes to a zero may be

entered and the program will prompt

WHAT WOULD YOU LIKE THE FILE TO BE CALLED?

At this time the user must input a file name (10 characters maximum) on which all

data will

typed from the terminal will then prompt the user for the

be stored for specific data

that section it needs for

of input data. this section.

LINEAR

If the user wishes to input all the data from the
be chosen and the program will prompt for a data file input data. The program will prompt the user for all

terminal, selection "5" should
name on which to store the the input data required,

41

When acters. floating

inputting the title data, the program reads character strings of 80 charWhen prompting for vehicle mass and geometry data, the program reads a point field. An example data prompt is as follows:

INPUT THE VEHICLE GEOMETRY AND MASS PROPERTIES WING AREA (FT**2) =

The user would input the vehicle wing area in square feet followed by a carriage return. The remainder of the vehicle geometry data is input similarly.

When inputting the state vector from the terminal, the and formulation of the state equation as well as the units attack and sideslip stability derivatives will be output.

user must define the size in which the angle of The program will prompt

HOW MANY STATE VARIABLES ARE IN YOUR MODEL? (MAXIMUM 12)
The program will read an integer variable and then prompt

WHICH FORM OF THE STATE EQUATION DO YOU WISH TO USE?

1)

DX/DT = A*X + B*U + D*V

2) C * DX/DT

A*X + B*U + D*V

If the standard formulation is being used, the user should input "1" and for the

generalized equation, "2". The LINEAR program uses the generalized formulation for

internal calculations and then performs any transformations necessary for the output

formulation requested.

The program will then prompt for the units of the angle of

attack and sideslip stability derivatives.

The program uses radians for all inter-

nal calculations and transforms the stability derivatives to degrees for output

if desired.

The variables used in the state vector are input next and are read in strings

20 characters in length. Each variable name entered is checked against legal names

for state variables.

If a variable is incorrect, the program will respond

VARIABLE so~and-so

IS AN INVALID STATE PARAMETER

PLEASE CHOOSE ANOTHER OR TYPE HELP FOR MORE INFO.

If the user types HELP, the program will list all the valid names for the state

variables.

The names are given in appendix D.

After the entire state vector has been entered, the program will prompt

THE STEP SIZE USED FOR DERIVATIVE EXTRACTION

IS INITIALIZED FOR ALL VARIABLES TO 0.001.

DO YOU WISH TO INPUT A DIFFERENT VALUE

FOR ANY OF THE STATE VARIABLES?

(Y/N)

42

To change the perturbation step size for any of the state variables, the user should enter "Y" and, after being prompted, the variable name to be updated and the desired step size.
For the interactive input of the control variables, similar prompts are given with one major difference: after each variable name is input, the program prompts
for the location of the variable in the common block /CONTRL/ (the named common block /CONTRL/ reserves 30 locations for control surface variable values used as interface
to the user-supplied aerodynamic model). The program also prompts for the units to
be used in calculations involving the control surface:

WHAT UNITS ARE USED IN CALCULATIONS WITH THIS CONTROL SURFACE?
1) NONE
2) DEGREES

3) RADIANS

The units should be consistent with the calculations in UCNTRL. The program performs

transformations based on the surface type for the printer file only.

No other con-

trol surface transformations are performed.

Again, the derivative step size can be

changed for individual variables specified by the user.

For the interactive input of the observation vector, similar prompts are given
with a few variations. The forms available for the observation equation are

1) Y¥

H*X + F*U + E*V

N
K |

H*X + G*DX/DT + F*U + EB*V

For variables defined in the observation vector as being offset from the vehicle cen-

ter of gravity, the program will prompt for the offset in the x, y, and z stability

axes.

For any variables that are calculated based on some reference value, such as

Reynolds number, the program prompts for the correct value when the name is entered.

Again, the step size for derivative extraction can be modified for any observation

variable. Valid observation variable names are listed in appendix Cc.

If selected to be input from the terminal, the program will prompt the user for the limits to be used for the trim parameters d., 53, Sy, Stns Which represent the

longitudinal, lateral, directional, and thrust trim parameters, respectively.

The

units associated with these parameters are defined by the implementation of UCNTRL.

(See app. J of this report for details.) An example prompt is as follows:

WHAT IS THE MAXIMUM THRUST PARAMETER?
Again, the program reads a floating point field.

After the trim axis parameter limits are defined, the program prompts

HOW MANY ADDITIONAL SURFACES DO YOU WISH TO DEFINE?

43

At this time, the user can define additional controls to be included in the list of

recognized control names. The purpose of defining these additional surfaces is to

allow the user to set such variables as landing gear position, wing sweep, or flap

position.

The control surface name, location in common /CONTRL/, and units are

defined here for each surface; actual control surface positions are defined during

analysis point definition.

The program will then prompt the user to indicate what aerodynamic model will

be used for this analysis. this report.

Aerodynamic models are described in the next section of

If a nonlinear aerodynamic model is used, the program will prompt the user to

determine if there are additional control surfaces for which derivatives are desired.

These controls will not become part of the control vector. This option of inter-

active LINEAR does not allow these control variables to be read from or saved to a

file, and is not available on the batch version of LINEAR.

However, the stability

and control derivative output file will contain these surfaces and their derivatives.

The point at which the nonlinear system equations are linearized is referred to

as the analysis or trim point. This can represent a true steady-state condition on

the specific trajectory (a point at which the rotational and translational accelera-

tions are zero) or a totally arbitrary state on a trajectory.

LINEAR allows the user

to select from a variety of analysis points.

Within the program, these analysis

points are referred to as trim conditions, and several options are available to

the user.

If the user is defining the trim conditions interactively, data from a formatted file, the program will prompt:

instead of reading the

WHAT TYPE OF TRIM DO YOU WISH TO RUN?

1) NO TRIM
2) STRAIGHT AND LEVEL FLIGHT 3) LEVEL TURN 4) PUSH~OVER/PULL-UP 5) THRUST STABILIZED TURN 6) BETA 7) SPECIFIC POWER

The user must specify the number corresponding to the type of trim desired.

The

arbitrary state and control option is designated as NO TRIM, and in selecting this

option the user must specify all nonzero state and control variables.

For all of the analysis point definition options, any state or control parameter may be input after the required data is defined when the program prompts as follows:
WHAT OTHER STATE OR CONTROL VARIABLES WOULD YOU LIKE TO INITIALIZE?

TYPE HELP TO LIST VALID VARIABLES NAMES

(ENTER "N" IF NO OTHER CONDITION IS TO BE SPECIFIED)

44

The user should enter the name of the variable to be initialized or type HELP to list the variables that may be initialized.

If a level turn trim is by entering "TURN RIGHT" or iables. This is especially

selected, the user can specify the direction of the turn

"TURN LEFT" when prompted to initialize any other var-

useful for asymmetric aircraft.

Default is "TURN RIGHT".

If data are being read from a file(s), the data can be reviewed by replying "Y" when prompted

DO YOU WISH TO REVIEW THE DATA AS IT

IS READ FROM THE INPUT FILE(S)?

(Y/N)

If any data are entered incorrectly, the user should finish entering all data as prompted by the program until asked if there are any changes to be made:

DO YOU WISH TO CHANGE ANY PARAMETERS

IN YOUR MODEL OR UPDATE THE INPUT

CASE SELECTION FILE?

(Y/N)

At this time, the user selects "Y" and the program prompts

WHAT PARAMETERS DO YOU WISH TO CHANGE?

1) VEHICLE GEOMETRY
2) STATE VECTOR 3) OBSERVATION VECTOR 4) CONTROL VECTOR
5) CASE INPUT FILE
6) STABILITY AND CONTROL DERIVATIVES 7) SURFACES TO BE SET 8) NO FURTHER CHANGES

Any section of data can be updated by reading a new data file or by typing in the

data from the terminal.

The program prompts the user

input as explained previously.

If the user wishes to

to determine the source of modify the vehicle geometry

and

mass all

data or the stability and control the values from the previous case

derivatives interactively, the program saves and allows the user to modify any individual

parameter. ties, a "1"

For example, if the user wishes to modify the vehicle would be entered and the program would continue

geometry

proper-

WHAT IS THE NAME OF THE FILE CONTAINING THE VEHICUE GEOMETRY AND MASS PROPERTIES?

(TYPE 0 TO INPUT THE DATA FROM THE TERMINAL AND STORE ON A FILE)

If the user replies with a zero, and then respond as follows:

the program will prompt

for a new output

file name

45

GEOMETRY AND MASS DATA FOR: USER'S GUIDE
FOR THE PROJECT: DEMONSTRATION CASE

1) WING AREA 2) WING SPAN 3) MEAN CHORD 4) VEHICLE WEIGHT 5) 1x
6) IY 7) 2 8) IXZ 9) Ixy 10) IyYZ

608.000 42.800 15.950
40700.000 28700.000
165100.000 187900.000
~520.000 0.000 0.000

VECTOR DEFINING REFERENCE POINT OF AERODYNAMIC MODEL WITH RESPECT TO THE VEHICLE CENTER OF GRAVITY:

11) DELTA X 12) DELTA Y 13) DELTA Z 14) MINIMUM ANGLE OF ATTACK 15) MAXIMUM ANGLE OF ATTACK

0.000 0.000 0.000 -10.000 40.000

16) FORCE AND MOMENT COEFFICIENTS CORRECTIONS DUE TO THE OFFSET OF THE REFERENCE POINT OF THE AERODYNAMIC MODEL FROM THE VEHICLE CG ARE INTERNALLY CALCULATED IN LINEAR. TYPE IN 16=0 IF CALCULATIONS ARE DONE IN CCALC.

TO MAKE CHANGES, TYPE IN THE DESIRED VALUES.
"N" INDICATES NO CHANGES "R" REFRESHES THE SCREEN
FOR EXAMPLE: TO CHANGE

LINE NUMBER ON THE
ARE DESIRED WITH NEW VALUES. VEHICLE WEIGHT

4=40000.0

If selection "16=0" is chosen, the program will set a logical variable that deter-

mines where center-of-gravity corrections are made to the force and moment coef-

ficients.

The vector defining the reference point of aerodynamic model with respect

to the vehicle center of gravity must be defined in feet.

If the state, control, or observation vectors need to input the entire vector again. The choices available for to input the data from the terminal for each case or read tion from a file.

be modified, the user must the test case selection are the entire case specifica-

46

All surfaces in the control model are initially zero unless specified in the case input file. Surfaces not in the control vector can be set by choosing option "7"
when prompted for parameter changes. The program will prompt for the file name con-
taining the surfaces that can be set for each case. If the user wishes to input the
surface names interactively, the program will prompt for the number of surfaces to be
added, their names, location in common /CONTRL/ and units used in subroutine UCNTRL. If the location in common /CONTRL/ has already been designated for another surface,
the program will save the original name and ignore any other name input for that location, and so inform the user.

The interactive LINEAR program provides the user with two different ways to
calculate the total force and moment coefficients. One method is by using a full aerodynamic model and the subroutines ADATIN and CCALC. The other method uses a set
of linearized stability and control derivatives, and the internal subroutine CLNCAL.
When using the second method, the data can be typed in during execution or can be input from a file.

An example of a linearized stability and control data set is shown in appendix G.
The first line contains the file description. The case number (1-999) at the end of the first line allows the user to correlate the linearized data from a case to other
data from that case. The next line is the title of the run. The following two lines
of text describe the project for which the case was run, and the next two lines list
the reference altitude and Mach number for the case. The next line determines
whether the alpha and beta derivatives are in units of degrees or radians. The
following two lines indicate whether Mach or velocity derivatives will be read in.
This is intended to warn the user against modifying the wrong derivative while
editing outside of LINEAR. If the wrong derivative is changed, LINEAR will ignore
the change. Additionally, if the reference altitude is changed, LINEAR will rely
on the specified derivative and compute the other based on the speed of sound which is a function of altitude. While editing this file in LINEAR, this caution is not
necessary since LINEAR will perform the computation right away. Also, when these derivatives are changed in LINEAR, the last velocity or Mach derivative changed will
determine which one LINEAR will regard as the specified derivative when the file
is output.

The next line determines the number of control surfaces used for control deriva-

tives.

The units of each of the control surfaces, where applicable, are listed next

to the surface name as well as the location of the surface in the common /CONTRL/.

The units defined for each surface correspond to the units defined in user-supplied

subroutine CCALC.

The rest of the file contains the stability and control derivatives. This part

is broken into six sections, one for each of the force and moment coefficients.

In

each section, there is one zero coefficient corresponding to zero angle of attack,

zero sideslip, zero control surface deflection, and constant Mach and altitude. Each

section also contains 10 stability derivatives.

In addition, each section has as

many control derivatives as specified earlier. The six sections must always be in

the same order: rolling moment, pitching moment, yawing moment, drag force, lift

force, and sideforce.

The easiest way to format a derivative file is by running the interactive LINEAR program and inputting the derivatives when prompted by the program. One of the out-

47

pwcmpliuraoattetntheitvrorifnooit.luhl)steeosdedcaraenWfsiahrreevlio,nyamvztaeittuLvihsIeveiNsetnEhgAeuRsoseneasrtai.mstlehmieuntsheIctefaafrsieilab.sezet,easdbucirat(lfrhSsiaaeeettcfetyeulsotfhuiaesrntdofsatAdcaeedcebfreo'iifonsnlidteniryedtoneylafmftaiehacdcanecttdrosniMtvodcradooooetennllsitvrteohvnsleoeetccfttifdoiloehroerarncvieevttsahhftaaocittroavnredrsmicseaonrsmepocfomrobneoenidnmsnitifsunsosagtre-ednt

will not be included in the linear system model calculated. aware that LINEAR expects to have control over all axes to If there are no nonzero control derivatives in a given axis, LINEAR will fail to trim; and it may even generate a fatal

The user should also be determine a trimmed state.
it is highly likely that error in the nonlinear

equation solver (generally a floating-point overflow).

AERODYNAMIC MODEL

As already stated, there are two methods for calculating the total force and

moment coefficients. One method relies on a full nonlinear aerodynamic model to pro-

vide the stability and control derivatives.

These derivatives are stored in common

blocks in the user-supplied routine ADATIN and used in CCALC to calculate the total

force and moment coefficients using the state and control surface values calculated

in LINEAR.

These subroutines and the aerodynamic model are the same ones used in the

manned simulations at Ames-Dryden.

Using a full aerodynamic model with CCALC and ADATIN enables the user to run as

many cases as desired without program generates a structured

changing derivative data sets. linearized derivative data file

From each which can

case, the be used

as

an input file later. This file contains all the stability and control derivatives

for the control surfaces specified in the control vector, and any other controls

specified by the user. The interface required for using a nonlinear aerodynamic

model is described later in the Aerodynamic Model Subroutines section.

The other method for calculating total force and moment coefficients requires the

user to supply linearized stability and internal subroutine CLNCAL to calculate

control derivatives. These are used in the the total force and moment coefficients.

When using this option, care must be taken to ensure that the input data is valid

for the trim point desired. Center-of-gravity shifts will not affect the data that

is output to the derivative file when this method is used.

The method used to calculate the force and moment coefficients is specified

during the input phase for a particular case.

After all the vehicle data, state,

control, observation vectors, and case data have been defined, the program will

prompt

DO YOU WISH TO

1) INPUT A SET OF LINEAR STABILITY AND CONTROL DERIVATIVES
2) OR USE THE NONLINEAR MODEL YOU HAVE INTERFACED WITH THIS PROGRAM
If the internal routines are to be used to calculate the derivatives, the program will prompt

48

DO YOU WISH TO

1) GENERATE A NEW DATA SET
2) OR USE A DERIVATIVE DATA SET YOU HAVE ON FILE 3) OR MODIFY THE OLD ONE

If the derivatives are stored on a file (format for the file is given in app.
program will prompt for a file name. At this point, the user can modify the interactively by choosing one of the following options:

G), the
data

TO CHANGE ANY DERIVATIVES,
APPROPRIATE LINE # TO SEE TYPE IN "N" IF NO CHANGES

INPUT THE
THE DATA. DESIRED

1) TYPE OF ANGLE MEASUREMENT 2) ROLL MOMENT 3) PITCH MOMENT 4) YAW MOMENT 5) DRAG FORCE
6) LIFT FORCE 7) SIDE FORCE 8) ADD SURFACES

If the user selects option one, the program will list the units that are used for

output of the stability derivatives. It will also list the units for each control

surface derivative as used for calculations in the subroutine UCNTRL. There are

three choices for units for the control surfaces: none, radians, and degrees. If the control surface calculations in UCNTRL are nondimensional, the user should spec-

ify none, and no conversion is done. For surface calculations in radians, a conver-

sion to degrees for output only is performed.

If degrees is specified, no conver-

sion is done. When the user inputs the stability and control derivatives, the pro-

gram does not perform any conversions, therefore, care must be taken to ensure that

the units are consistent with the aerodynamic model and the calculations in UCNTRL.

If option "8" is chosen, the control surfaces defined are listed, and the user can define other surfaces in common /CONTRL/ for which derivatives exist.

If the derivatives are be to calculated from a nonlinear aerodynamic model, the

program will prompt for the names of any other control surfaces not defined in the

control vector for which linearized derivatives will be calculated.

If options "2"

through "7" are chosen, the program will list the current values for the stability

and control derivatives for each respective coefficient.

The user can change these

values interactively, as in the following example:

ROLL MOMENT DERIVATIVES

WITH RESPECT TO:

0) CLO 1) ROLL RATE

(RAD/SEC)

2) PITCH RATE (RAD/SEC)

3) YAW RATE

(RAD/SEC)

-0.000033820 -0.200000003
0.000000000 0.163589999

49

4) VELOCITY

(FT/SEC)

5) MACH NUMBER

6) ALPHA

(DEG/SEC)

7) ‘BETA

(DEG/SEC)

-~0.000000115 ~0.000119522
0.000362335 -0.140983000

TO MAKE CHANGES, TYPE IN THE LINE NUMBER AND THE VALUE OF THE DERIVATIVE.
"N" INDICATES NO CHANGES ARE DESIRED "R" REFRESHES THE SCREEN WITH THE NEW VALUES FOR EXAMPLE: ROLL MOMENT WRT ALPHA
6=0 .003

OUTPUT FILES

There are four output files from LINEAR:

(1) a general-purpose analysis file,

(2) a printer file containing the calculated case conditions and the state-space
matrices for each case, (3) a printer file containing the calculated case conditions

only, and (4) a linearized set of stability and control derivatives.

LINEAR also

outputs any data input by the user during an interactive session that can be used

later either by the batch or interactive programs.

The program prompts the user for

the file names that each of the sections of data will be written to, as described in

the Interactive Data Input section.

The general-purpose analysis file contains the following: the title of the cases
being analyzed; the state, control, and observation variables used to define the
state-space model; and the state and observation matrices calculated in LINEAR. The C and G matrices are printed only if the user has selected an appropriate formulation of the state and observation equations. The output for this file is assigned to FORTRAN device number 15. An example general-purpose analysis file is presented in appendix H corresponding to the format of table 2.

The
file in and the defined

vehicle and case titles are written on the first two records of the analysis

80-character strings and are specified in LINEAR as the title of the title of the cases. The next record contains the number of the case in LINEAR (maximum of 999 cases).

vehicle as

The number of states, controls, and outputs used to define each case is written

on the subsequent record. The formulation of the state and observation equations

is listed next, followed by the names and values of the states, controls, dynamic

interaction variables, and outputs.

These values are followed by the matrices that

describe each case.

The title records only appear at the beginning of the file while all other records are duplicated for each subsequent case calculated in LINEAR. The matrices are written row-wise, five columns at a time as shown in table 3. This table shows a
system containing seven states, three controls, and eleven outputs using the general state equation and standard observation equation.

50

TABLE 2. — ANALYSIS FILE FORMAT

Variable
Title of the case Title of the aircraft Case number Number of states, controls, and outputs State equation formulation Observation equation formulation

State variable names, values and units

Control variable names, values and units

Dynamic interaction variable names and units

Output variable names, values and Matrix name A~Matrix Matrix name B-Matrix Matrix name D-Matrix Matrix name
C-Matrix (if general form chosen) Matrix name H-Matrix Matrix name
F-Matrix
Matrix name E-Matrix Matrix name
G-Matrix (if general form chosen)

units

Format
(4A20) (4A20)
(//,64X,13) (17%,12,22X,12,22x,13) (36X,2A4) (36X,2A4)
(///7/) (1X,5A4,3X,E12.6,2X,A20) (////) (1X,5A4,3X,E12.6,2X,A20) (////) (1X,5A4,17X,A20) (////) (1X,5A4,3X,E12.6,2X,A20) (//,A8,/) (5(E13.6)) (//,A8,/) (5(E13.6)) (//,A8,/) (5(£13.6)) (//,A8,/) (5(£13.6)) (//,B8,/) (5(E13.6)) (//,A8,/) (5(E13.6)) (//,A8,/) (5(E13.6)) (//,A8,/)
(5(E13.6))

51

TABLE 3. — ANALYSIS FILE OUTPUT MATRIX STRUCTURE

Size of matrix
A= [7x 7]
B= {7 x 3]
D= {7x 6]
c=[7x 7]
H = [11 x 7]
F = [11 x 3]
E = [11 x 6]

Output formulation
A= [7x 5] [ 7 x 2]
B= [7 x 3]
D= [7x 5] [7x 1]
c= [7x 5]
[ 7 x 2] H = [11 x 5]
{11 x 2}
F= [11 x 3]
E = {11 x 5}
[11 x 1]

The second output file, which is assigned to FORTRAN device unit number 3, con-

tains the this file

values calculated in LINEAR describing each case. contains a listing of the input data defining the

The first aircraft's

section of geometry and

mass properties, variable names defining the state-space model, and various control

surface limits characteristic of the given aircraft.

Appendix I presents an example

printer output file.

The second section of this file contains the trim conditions of the

the point of interest.

These conditions include the type of trim being

whether trim was achieved, and parameters defining the trim condition.

aircraft at attempted,

The values for the variables of the state-space model at the trim condition are

also printed. equations of the geometry

If trim was not achieved, Py qe xr, v, B, and ra motion) and the force and moment coefficients are and mass properties are also printed.

(calculated from the

printed.

Changes in

The
control aircraft

third section of this output file derivatives for the trim condition
at the given flight condition is

contains the
calculated. also printed.

nondimensional stability and The static Margin of the

The final section of this output file contains the state and observation matrices

for the given flight condition. terms of the matrices chosen by

The formulation of the state equations and the user to define the model are printed.

only

the

The third output file, which is assigned to FORTRAN device unit 2, contains the

trim conditions of the aircraft at the point of interest.

These conditions include

the type of trim being attempted, whether trim was achieved, parameters defining the

trim condition, and the static margin of the aircraft at the given flight condition.

Appendix I also presents an example of this file.

The fourth output file is the set of linearized stability and control derivatives for each case as defined in the Aerodynamic Model section.

USER~SUPPLIED SUBROUTINES

There are five subroutines that must be supplied by the user to interface LINEAR

with a specific aircraft's subsystem models:

ADATIN, CCALC, IFENGN, UNCTRL, and

MASGEO.

The first two subroutines comprise the aerodynamic model.

The subroutine

IFENGN is used to provide an interface between LINEAR and any engine modeling rou-

tines the user may wish to incorporate.

UCNTRL converts the trim parameters used

by LINEAR into control surface deflections for the aerodynamic modeling routines.

MASGEO enables the mass and geometry properties to change as the aircraft configura-

tion does (for example, with wing sweep).

The use of these subroutines is illustra-

ted in figure 3 which shows the program flow and the interaction of LINEAR with the

user~supplied subroutines.

These subroutines are described in detail on the fol-

lowing pages.

Examples of these subroutines can be found in appendix J.

53

_| _ 25.0

DES

"| 5.43

DAS

"»|| 2“04..00

_| 180 “| 8 | 181 0

14.0 L,y 18Tt0

DRS THRSTX

»"}| 330..205 -1,0
af
0,1

»| 18T0

_ -45.0

“18a0

= i bg

>

é a

>~ oo dat

~ _

6,

>_ Sep

.(to ITFHERNGN)

Figure 3. Program flow diagram showing communication with usersupplied subroutines.
Aerodynamic Model Subroutines
It is assumed that the aerodynamic models consist of two main subroutines: ADATIN and CCALC. ADATIN is used to input the basic aerodynamic data from remote storage. ADATIN can also be used to define aerodynamic data. CCALC is the subroutine that uses this aerodynamic data, the state variables, and the surface positions to determine the aerodynamic coefficients. Either routine may call other subroutines to perform related or required functions; however, from the point of view of the interface to LINEAR, only these two subroutines are required for an aerodynamic model.
Externally, ADATIN has no interface to the program LINEAR. The subroutine is called only once when the aerodynamic data is input or defined. The calling program has to provide ADATIN with the input devices it requires, but no other accommodation is necessary. The aerodynamic data is communicated from ADATIN to CCALC through named common blocks that occur only in these two routines.
The interface between CCALC and the calling program is somewhat more complicated. However, the interface is standard and this feature provides a framework about which a general purpose tool can be built. This interface consists of several named common blocks which are used to pass state variables, air data parameters, surface positions, and force and moment coefficients between CCALC and the calling program. CCALC is executed whenever new aerodynamic coefficients are required (for example, once per frame for a real-time simulation).
The main transfer of data into the subroutine CCALC is through five named common blocks. These common blocks contain the state variables, air data parameters, and

54

Externally, ADATIN has no interface to the program LINEAR. The subroutine is called only once when the aerodynamic data is input or defined. The calling program has to provide ADATIN with the input devices it requires, but no other accommodation is necessary. The aerodynamic data is communicated from ADATIN to CCALC through named common blocks that occur only in these two routines.

The interface between CCALC and the calling program is somewhat more complicated.
However, the interface is standard and this feature provides a framework about which a general purpose tool can be built. This interface consists of several named com-
mon blocks which are used to pass state variables, air data parameters, surface posi-~ tions, and force and moment coefficients between CCALC and the calling program. CCALC is executed whenever new aerodynamic coefficients are required (for example, once per frame for a real-time simulation).

The main transfer of data into the subroutine CCALC is through five named common
blocks. These common blocks contain the state variables, air data parameters, and surface positions. The transfer of data from CCALC is through a named common block containing the aerodynamic force and moment coefficients. The details of these common blocks follow.

The common block /DRVOUT/ contains the state variables and their derivatives with respect to time. The structure of this common block is given as follows:

COMMON

/DRVOUT/

T

'

P

1 Q

7R

'

Vv

, ALP

, BIA

'

THA

, PSI

» PHI

’

H

7X

7

TDOT ,

PDoT , QDOT ,

VDOT , ALPDOT,

THADOT, PSIDOT,

HDOT , XDOT ,

x

7

RDOT , BTADOT, PHIDOT, YDOT

The body axis rates p, q, and r appear as P, Q, and R, respectively.

Total velocity

is represented by the variable V and the altitude by H. Angle of attack (ALP), angle

of sideslip (BTA), and their derivatives with respect to time (ALPDOT and BTADOT,

respectively) are also contained within this common block.

The common block /SIMOUT/ contains the main air-data parameters required for the aerodynamic model. The variables in this common block are as follows:

COMMON /SIMOUT/ AMCH, QBAR, GMA, DEL, UB , VB, WB, VEAS, VCAS

Mach number and bolized by AMCH included as UB,

dynamic pressure are the First two entries in the and QBAR, respectively. The body axis velocities VB, and WB, respectively.

common block symu, v, and w are

55

The /CONTRL/ common block contains the surface position information.

The exact

definition of each of the elements used for a particular aerodynamic model is deter-

mined by the implementer of that model.

The structure of the common block /CONTRL/

is as follows:

COMMON /CONTRL/ DC (30)

The common block /DATAIN/ contains geometry and mass information. The first

three variables in the common block, S, B, and CBAR, represent wing area, wing span,

and mean aerodynamic chord, respectively.

The vehicle mass is represented by AMSS.

The structure of common /DATAIN/ is as follows:

COMMON /DATAIN/ S

, B

, CBAR, AMSS,

AIX , AIY , AIZ , AIXZ,

AIXY, AIYZ, AIXE

The information on the displacement of the reference point of the aerodynamic
data with respect to the aircraft center of gravity is contained in the /CGSHFT/
common block:

COMMON /CGSHFT/ DELX, DELY, DELZ

The displacements are defined along the vehicle body axis with DELX, DELY, representing the displacements for the x, y, and z axes, respectively.

and DELZ

The output common block /CLCOUT/ contains the variables representing the aero-
dynamic moment and force coefficients:

COMMON /CLCOUT/ CL, CM, CN, CD, CLFT, CY

The variables CL, CM, and CN are the symbols for the rolling moment (Cg), pitching
moment (Cy), and yawing moment (C,) coefficients, respectively; these terms are body axis coefficients. The stability axis forces are represented by CD, coefficient of
drag (Cp); CLFT, coefficient of lift (Cy); and CY, sideforce coefficient (Cy).

Control Model Subroutines
The LINEAR program attempts to trim the given condition using control inputs similar to that of a pilot: longitudinal stick, lateral stick, rudder, and throttle.
The UCNTRL subroutine uses these trim output control values to calculate actual sur-
face deflections and power lever angles for the given aircraft (fig. 2). The location of each surface and power lever angle in the /CONTRL/ common block is specified by the user in the input file (maximum of 30 surfaces). The limits for the control parameters in pitch, roll, yaw, and thrust are user specified and must agree in units with the calculations in CCALC.
The common block /CTPARM/ contains the four trim parameters that must be related to surface deflections in the subroutine UCNTRL:
COMMON /CTPARM/ DES, DAS, DRS, THRSTX
56

The output from UCNTRL is via Aerodynamic Model Subroutines used to trim the longitudinal,

the common block /CONTRL/ described previously in the

section.

The variables DES, DAS, DRS, and THRSTX are

lateral, directional, and thrust axes, respectively.

For an aircraft using feedback (such as a statically unstable vehicle), state

variables and their derivatives are available in the common block /DRVOUT/.

However,

this control model must contain no states of its own. If parameters other than state

variables and their time derivatives are required for feedback, the user may access

them using the common block /OBSERV/ described later in the Mass and Geometry Model Subroutines section of this report.

Engine Model Subroutines

The subroutine IFENGN computes individual engine parameters used by LINEAR to
calculate force, torque, and gyroscopic effects due to engine offset from the centerline. The parameters for each engine (maximum of four engines) are passed through common /ENGSTF/ as follows:

COMMON /ENGSTF/ THRUST(4), TLOCAT(4,3), XYANGL(4), XZANGL(4),

TVANXY(4), TVANXZ (4), DXTHRS(4),

EIX

(4), AMSENG (4), ENGOMG(4)

The variablesin this common block correspond to thrust (THRUST); x, y, and z body
axis coordinates of the point at which thrust acts (TLOCAT); the orientation of the thrust vector in the x-y body axis plane (XYANGL), in degrees; the orientation of the
thrust vector in the x-z body axis plane (XZANGL), in degrees; the orientation of the thrust vector in the x-y engine axis plane (TVANXY), in degrees; and the orientation of the thrust vector in the x-z engine axis plane (TVANXZ), in degrees. The distance between the center of gravity of the engine and the thrust point (DXTHRS), measured positive in the negative x engine axis direction; the rotational inertia of the engine (EIX); mass (AMSENG); and the rotational velocity of the engine (ENGOMG) are also in this common block. The variables are all dimensioned to accommodate up to four engines. Appendix B provides graphical and physical descriptions of these parameters.

While the common block structure within LINEAR is designed for engines that do not interact with the vehicle aerodynamics, propeller-driven aircraft can be easily modeled by communicating the appropriate parameters from the engine model IFENGN to
the aerodynamic model CCALC.

Mass and Geometry Model Subroutines

The subroutine MASGEO allows the user to vary the center-of-gravity position and

vehicle inertias automatically.

Nominally, this subroutine must exist as one of the

user subroutines, but it may be nothing more than a RETURN and END statement. MASGEO

is primarily for variable-geometry aircraft (such as an oblique-wing or variable-

sweep configuration) or for modeling aircraft that undergo significant mass or iner-

tia changes over their operating range. The center-of-gravity position and inertias

may be functions of flight condition or any surface defined in the /CONTRL/ common

57

block. block,

Changes in and inertia

center-of-gravity position are passed in the /CGSHFT/ changes are passed in the /DATAIN/ common block.

common

brsmfaenoueerbcamrosaoomucbCuprcasotuerroeitemmrneepvendnlaosmtitsussiesahotsanenbdadroabeoevmpebr,yacedttssaeosukstirileihmtnnerpgeltdehmowmfaohtteemhanenetttnhctieonaunensgwtiacfanriegimxnmtnehotsdemetteorhne-mmtlooeotfmhacse-easaungtrubtrmirrnaoeoevnmwuvaietratnimlyoctnuneseencstastehlhaMeicrwAtfau-ShtyloGrvaEfoeOtu-hffiggiorhorrconialmnsevta,ihctctechyoieunmsrbu/aciOoclthnBneoeeSancEttaaRietsorVorife/nosanutolfhcwsteoiemsmtnu.ghmssurotosanrevsriTbetehplbyille.saoctct-ki:cnaIgnf

COMMON /OBSERV/ OBVEC (120), PARAM (120,6)

SpadtcvsmtObsagntiheereeaebiBudeaonsnchilvrVntntksiieemfEte)orcsnaaCe,.poTrlecr(rahbedttei1es-os,rlid2soienCluy0emfroAorom)Ldc~nvdicbaI,toagtandashNnmptsrrtgtieEeoampbtoainitArogaerlavooRinvcen,andnn.iotkamddtetrtnteiehfiyaibnxegimaliloitP(innnso(nATeedxchRmlCs,ake.osAotitsmhoMcnOct(ee(ahfbhyPtw/cn,et1eosAheobOAti2mreeaRlmcBfoe0tmlrAmcSimzsn,aMvoer)Eom.a6(t,anssRmra)anittasVmders.aoie/nennio.btdn=ntlTfttohhoorfaerocoofulAa1emknerVlnbpItemefaocshatpltowersteaoit)/ssiirhsrbetOfaeotvashlrB3obneam)eoeSmblpst,tmermEusheanoitevsReeferuotenaVwraswnottrt/evhfmhDssstiureeaeehbscortteefeenoecierinsobaoacsornfrtsnesucansieneetsbonvtttarrmrooaatieivn-phociidoavrednauusoilenasastttafassrsceiit-biipocPtodonelgrhafeacAentdereesbRriwssdaslAaaoe,igvMemtcnnrMwoiseetuiaAabtatsditmtvSlsovyeosihbiGlneeunrnaetEeccr.slgfrwyOhtvliillttoaoxthehoorwhteefe-coctaisdifaoasspontratotoniehhstublbttsgoecehsshhbsaosneieeeeoiecedsncarruaoiitvvsnpdftbuoaiaedaoooesstnstrottuioseecenininirrhdsptdotsorvheacsnveneadrhuesdtuaabcwioatlinnnmttirofveogteooiaoasenhrturxbrttwsisieteeshhh-m:rdieee~ipttasnrclhhnuevheeedssa-e~ids

CONCLUDING REMARKS
ttCadidlohiloanoxiwsrtcsopeuoormtJTdr-fheherdaeaneseertctipsieegocoAsFdnrrnOc.ti,Ro.mbTmitetRpocodAuro.MTNtlohaTuefihysrsienesptcaorrEhr-opxedrsgar,wedrumopieappgotrlmlrprhiiMetalovsasmuietstLeisIddnaNioilgcsfEsmihAcnosuRuedwuotsasfihrbersefirtweitoattsctsuhsmueaet)osntideitdnehrpoeele-rnvwissnosi.etgulihrponsafappnottmledraetdinhreedafdiaaconitrtrporVdiuMcsvtSrupaearbFforOVoftvoRAipLXiTuleIdRterNseAsiaEt.N1nAta1eRi/bsan7i7g5l7pT0fihralteoressyegoxy(risoDabtuiptamlethgrn,emapiea,tutsltaeyidlsnsepiittosfsewcrEideorqmnmriaucayfbliniuidupnLlndmIb,geeeNadcnEpotApnptaRetohnwnerdri-ottilhesd vaijgcetafeayqrycpruiitaaownoabtshTgrblicihyelooce,enhpstsihpcoefrsthoeaoifgnlseesdretfxacaftmmtterorleicateinticnmLgsoteaIanaiinNrnndEaagiAnanRdntcgsdroylinsinimttnnusaretecmomaaleoultrurpseidtiemziicorvoenad-avndglesra,lluiryapitebaphnailblesngdelidsiseeeenstdde.eedrfioemrirfneineffonecefnbdetcyelsticisntnsigsesp,aaeirnctdliheisefutnatyceeheihearrpnrmrogiodalngysisernnyadaaensnmmateirteaecmnitotatshlhymmeyrsoodrsutddeiseeetstllmeb,.ry mpmuioostcnidiooeLnenrmItlgqpN.uElteAeho,nRetnoen‘lTlacayihisenodnntetrpasaraolpro-iselocn-t

58

LINEAR has several features that make it unique among the linearization programs

common in the aerospace industry. The most significant of these features is flexi-

bility.

By generalizing the surface definitions and making no assumptions of sym-

metric mass distributions, the program can be applied to any aircraft in any phase

of flight except hover. The unique trimming capability, provided by means of a user-

supplied subroutine, allows unlimited possibilities of trimming strategies and sur-

face scheduling, which are particularly important for asymmetrical vehicles and

aircraft having multiple surfaces affecting a single axis. The formulation of the
equations of motion permit the inclusion of thrust-vectoring effects. The ability

to select, without program modification, the state, control, and observation vari-

ables for the linear models, combined with the large number of observation quantities

available, allows any analysis problem to be attacked with ease.

This report documents the use of the program LINEAR, defining the equations and

the methods used to implement the program.

The trimming capabilities of LINEAR are

discussed from both theoretical and implementation perspectives.

The input and out-

put files are described in detail.

The user-supplied subroutines required for LINEAR

are discussed and sample subroutines are presented.

Ames Research Center Dryden Flight Research Facility National Aeronautics and Space Administration Edwards, California, November 3, 1987

59

APPENDIX A:

CORRECTION TO AERODYNAMIC COEFFICIENTS FOR A CENTER OF GRAVITY NOT AT THE AERODYNAMIC REFERENCE POINT

The point on the vehicle at which the nonlinear force and moment coefficients are

defined is referred to as the aerodynamic reference point.

All aerodynamic effects

are modeled at
vehicle center reference point

this aerodynamic reference point.

Thus, when this point and the

of gravity are not coincident, the forces acting at the aerodynamic

effectively induce moments that act incrementally on the moments

defined at the aerodynamic reference point by the nonlinear aerodynamic model.

The total aerodynamic moment M acting at the vehicle center of gravity is defined as

where

M = Mar + Ar x F

Mar = [Lar Mar Nar]? is the total aerodynamic moment acting at the aerodynamic reference point (denoted by subscript ar) of the vehicle,
Ar = [Ax Ay Az]T
is the displacement of the aerodynamic reference point from the vehicle center of gravity, and
F= {x y g]t is the total aerodynamic force acting at the aerodynamic center where x, y, and z are total forces along the x, y, and z body axes. Thus,

Lar + Ay Z - Azy

M =

Mar + Az X - Ax Z@

Nar + Ax Y - Ay xX

The total aerodynamic moment acting at the vehicle center can be expressed in terms of the force and moment coefficients derived from the user-supplied nonlinear aerodynamic modeling subroutine CCALC by defining the body axis forces in terms of stability axis force coefficients:

X = gS(-Cp cos a+ Cy sin a)
Y = gSCy Z = qS(-Cp sin a - Cy, cos a)

60

Equations obtained aerodynamic moment moments, result in

by substituting these equation and applying

equations into the the definitions of

definition of the total the total aerodynamic

Lar = GSbC 9,

Mar = gScCm,+
Nar = GSbC,

Expressions for total aerodynamic moment coefficients corrected to the vehicle center of gravity can be derived as follows:
Cg == Cet pAy (Cp si:n a = Cy cos a) = DApz Cy
Cm _= Cmay + IZ (“Cp cos a + Cy si.n a) - =Ax (-Cp si.n a ~ Cy, cos a) c
Cy = Char t DAx Cy 7 aAs (-Cp cos a + Cy; si.n a)

These calculations are normally performed within
CGCALC. However, if the user elects, the calculation user-Supplied aerodynamic model CCALC.

LINEAR can be

in the subroutine performed within the

61

APPENDIX B: ENGINE TORQUE AND GYROSCOPIC EFFECTS MODEL

Torque and gyroscopic effects represent (after thrust) the main contributions of
the engines to the aircraft dynamics. The torque effects arise due to thrust vectors not acting at the vehicle center of gravity. The gyroscopic effects are a consequence of the interaction of the rotating mass of the engine and the vehicle dynamics. These effects can be either major or virtually negligible, depending on the vehicle.

The torque effects can be modeled by considering the
where the thrust vector is aligned with the local x axis point Ar from the center of gravity of the vehicle (fig.

thrust
of the 4).

of an engine F,, engine acting at

some

Center of

Figure 4. Definition of location of engine
center of mass (cm;) relative to vehicle center of gravity.

The thrust vector for the ith engine Fo can be defined as

FP, i . = [ Fexs

Foy;

T
Fpz |

where
axes, ships

PP!

Foy! i

and Fp _ ave the components of i

respectively. From figures 5 and 6 it can

hold:

thrust in the x, be seen that the

y, and z body following relation-

PPK, = | Fp, | cos €; cos &

*pys =| Fp, | cos €; sin &
a = - | Fp, | Sin €;

62

where | Fp, | represents the magnitude of the angle from the thrust axis of the engine to plane, and &; the angle from the projection
x-y body axis plane to the x body axis.

thrust due to the ith engine, €, the a plane parallel to the x-y body axis of Fo: onto the plane parallel to the

< =

— —

F

bo

Po

7

Z

i

1 \T

|

aw

U

5

F P,

| |

xX <« Vaan

_

+

Zvz

(-}----56---------

> “

y

“ 7a

UC

é

<*
nd

ns Zz

[

ra

Engine center a,

of gravity —~~]

\

a 7

v

|

a

F

7

an

a, m

Aicrecnratfetr of gravity
Az

a

a

Ay

|

~t—_ Ax

z

'

Figure 5. Orientation of the engines in the x-y and x-z body planes.
Denoting the point at which the thrust vector can be defined as

Figure 6. Detailed definition of engine location and orientation parameters.
from the ith engine acts as Ary, this offset

Ar; = [Axy Ayy Az,]T

where Ax;, Ay;, and Az; are the x, y, and z body axis coordinates, respectively, of
the origin of the ith thrust vector.
The torque due to offset from the center of gravity of the ith engine To; is then given by

Thus,

OF

Pi

At OF =

[ay Fp,
Az Px

= Az Foy. i |
- Ax Fpzy

63

The total torque due to engines offset from the center of gravity of the vehicle To is given by

where n is the number of engines.
For the case of vectored thrust, the equations for torque produced at the vehicle center of gravity from the ith engine ATo, are somewhat more complicated. Figure 7 schematically represents an engine with thrust vectoring whose center of gravity is located at Ar; relative to the vehicle's center of gravity.

T [\"

Thrust point +
y TP

AW

YE

|

|

tp 2

\'
gt
— Engine
cofentgerravity

Xe Xqp

Figure 7, Detailed definition of thrust-vectoring parameters,

The thrust is assumed to act at ~Axm in the local (engine) x axis, with the engine

center of gravity being the origin of this local coordinate system. The thrust is also assumed to be vectored at angles nj and ¢; relative to the local coordinate

axes, with n,; being the angle from the thrust vector to the engine x-y plane and oi

the angle from the projection of the thrust vector onto the engine x-y plane to the

local x axis.

Thus,

letting

Fp, , TAL

Fr.Py,’

and F.Pa;

represent

the x,

y, and z thrust

components in the local engine coordinate system, repectively, these terms can be

defined in terms of the total thrust for the ith engine Foi and the angles n;, and

Si as

Px; =| Fp, | cos Ny cos fj

Foy, = | Fp, | cos ny sin tj

Foe,

- | Fp. | sin ni

64

where

To transform this equation the transformation matrix

FoP.i

=|F[

t
Px

'
“Pyy

from the ith engine

'
pa
axis

| T
system

to the

body

axis

system,

cos €j cos &} cos €; cos &

“sin & cos &

Sin €; cos ey Sin €; cos &;

“Sin €j

0

is used.

The resultant force in body axis coordinates is

lr Px;

— cos €,1 cos €,1

-sin &,;i

COs Ey

—

1 4

Sin €.i cos &;1 FE Pxy

F Pi _ = F Py;

= [cos e€;i Siinn & i&;

cos E€;i

Sin £,i cos E i€.

F!Py;

so that

a F Pe; |

i

“Sini

€;1

9)

cos

€: 1

Fr! | a Pa; |

PPxs =| Fy, | (cos Ny cos f, cos &€}, cos Ei - cos ny sin 4 sin Ey
~ sin nj sin ej; cos &;) FPy; =| Fy, | (cos Nj Cos tj cos €; sin ay + cos n; sin oi cos Ey
~ sin nj sin €; cos &) * pz, =| Fp, | (-cos Nn; cos ; sin €; — sin ny cos €;)

The moment arm through which the vectored thrust acts is i” Ax. cos €; COS E. |

Ar; = Ay; 7 Ax 5 cos €; sin oi

Azi t+ Axp 5 sin €j

|

and the total torque due to thrust vectoring is

n
To = > AT). = > (Ar; x AF, )

65

The gyroscopic effects due to the interaction of the rotating mass of the engines and the vehicle dynamics can be derived from the equation for the rate of change of angular momentum,
Tg, = Be, + © x(Te;%;)
where Tg; is the gyroscopic moment produced by the ith engine, h ei the rate of change of the angular momentum of the ith engine, Te; the inertia tensor for the ith engine, We; the rotational velocity of the ith engine, and w the total rotational velocity of the vehicle given by
w= [p q r]t
If it is assumed that the angular momentum of the engine is constant, then

and the equation simplifies to

Tg, = © X(Tej ei)

Two terms, Te; and Wess remain
moment produced by the ith engine. is assumed that the inertia tensor

to be defined in the equation for the gyroscopic

Once again, simplifying assumptions are made.

It

of the engine contains a single nonzero entry,

xe,

0

0

' te; =

0

0

0

0

0

0

where Tx... i is the inertia of the ith engine at the location of the ith engine, oriented with the local x axis coincident with the rotational velocity of the engine. The rotational velocity of the engine has components in the x, y, and z body axes
(Pe, dejs and Tey! respectively) so that
Mei . = (Pei . dex . Loe.i) T

where

Pe; =| we; | cos €; cos by

de; =| we | cos €; sin &

66

with | we ; | being the total angular velocity of the ith engine.

The engine inertia tensor must be defined in the vehicle body axis system and at the vehicle center of gravity. This is done in three steps. The first two steps involve rotating the engine inertia tensor into a coordinate system orthogonal to the aircraft body axis system, while the third step involves defining the engine inertia tensor about the vehicle center of gravity rather than about the center of gravity of the engine itself. First, the ith engine inertia tensor is rotated through an angle €; about the local y axis so that the new inertia tensor is oriented with its

local x-y body axis plane parallel to the x-y body axis plane of the vehicle.

The

second step requires a rotation through an angle &; about the local z axis so that

the local x, y, and z axes are orthogonal to the x, y, and z body axes of the

vehicle.

As determined by Gainer and Sherwood (1972) and Thelander (1965), this

rotation is a similarity transformation that yields a new inertia tensor Ie

that

iv 5°

IeC,i = ReiRRoe.gletj] Re-y1R. ey-1

where Re and Re are axes transformation matrices that perform the previously

described rotations through € and e€, respectively.

These matrices are given as

cos €j4

0

sin €4]

Re; =

0

1

0

-sin €j

0

cos €j |

so that

cos &j

-sin &3

07

Re; =

sin &

cos &3

0

rt)

0

11

ReiRes =

cos €| cos €j Sin &, cos &€

-sin & cos &

cos €; sin €4 Sin €; sin Ey

Because

“sin €j

0

Re, = Rey

cos €j

67

and and the matrices are unitary,

Rr-G.1i

=

T
Re

REeRpyg, = RoeRe = R;rN.ERG,, ey 7

[cos Ei cos

( R Ej R €;) TL=

-sSiin &

sin €; cos €j Cccos €;

~sin €
0

cos §; sin ey

sin Ey sin ey

cos €; |

Therefore,

cos 2 €¢;. cos* 2¢.&j

A

Xe;

cos* 2 €;¢. cos € . si<jn oy.

- cos €;. Sisin €;; cos & ;

cos* 2 e€;oc. cos &;. siofn a . Ssiin¢nd €)&: cos* 2 e}¢.
-sSiin €j; cos €;. sisin & ;

~- cos €j. Siojn €>; cos & -ssiin €;. cos €;. sisjn
sini*n? ¢€;¢:

The angular momentum of the ith engine hj; can now be expressed as
hyi *= “1ei %ei = [pit Mig his] 7

with
iq * Pei[Txe,; cos* ej cos? ei * Joi] xe
+ res[ - i xe; cos €j;. Sigin €, cos 2 g

Nis = Pei[txe; cos? €j cos € sin 4] + de, | Tx

sin@ Ej cos2 e|

Si

+ rei[-Txe, sin €j; cos €; sin ei]

hig = Pei[~Txe, cos €; Sin €; cos ei | + dei[-Tke, sin e€; cos €; sin ei] + Tei[Txes sin? e3

68

Thus, the gyroscopic moment induced by the ith engine Tg, i can be expanded to
tg, =] hi, 7 Phi;
| Phin ~ ahi, —
and the total moment induced by gyroscopic interaction of the vehicle dynamics and the rotating engine components is
Engine torque and gyroscopic effects are modeled within the subroutine ENGINE using information provided by the user from the engine modeling subroutine IFENGN. These effects are calculated as incremental moments and are included directly in
the equations of motion for both analysis point definition and derivation of the
linearized system matrices.
69

APPENDIX C: OBSERVATION VARIABLE NAMES RECOGNIZED BY LINEAR
This appendix lists all observation variable names recognized by LINEAR except for state and control variable names. If state variables are specified as elements in the observation vector, the alphanumeric descriptor must correspond to the names defined in appendix D of this report. When control variables are to be included in the observation vector, these variables must be identified exactly as they were specified by the user.
The input file is formatted as shown in table 1. The alphanumeric data (measurement) is left-justified in a 5A4 format. The floating point fields (PARAM) are used to define sensor locations not at the center of gravity. The input name specified by the user for an observation variable serves both to identify the observation variable selected within the program itself as well as to identify observation variables on the printed output of LINEAR.

Observation

variable

Units

Derivatives of

Symbol

Alphanumeric

state variables

descriptors

Roll acceleration

rad/sec?

p

PDOT ROLL ACCELERATION

Pitch acceleration

rad/sec2

q

QDOT PITCH ACCELERATION

Yaw acceleration

rad/sec2

h e

RDOT YAW ACCELERATION

Velocity rate Angle-of-attack rate

ft/sec? rad/sec

< e

VDOT
VELOCITY RATE

a

ALPDOT

ALPHA DOT

ALPHADOT

Angle-of-sideslip rate

rad/sec

W e

BTADOT
BETA DOT BETADOT

Pitch attitude rate

rad/sec

D e

THADOT THETA DOT

Heading rate

rad/sec

y

PSIDOT

PSI DOT

Roll attitude rate

rad/sec

G e

PHIDOT PHI DOT

70

Observation variable Derivatives

Units

Symbol

of state variables

Alphanumeric (continued)

descriptors

Altitude rate

ft/sec

h

HDOT

ALTITUDE RATE

Velocity north

ft/sec

x

XDOT

Velocity east

ft/sec

y

YDOT

x body axis acceleration
y body axis acceleration
zZ body axis acceleration
x body axis accelerometer at vehicle center of gravity
y body axis accelerometer at vehicle center of gravity
z body axis accelerometer at vehicle center of gravity

Accelerations

g

ay

g

ay

g

ag

g

anx

AX LONGITUDINAL ACCEL
X-AXIS ACCELERATION X AXIS ACCELERATION X“BODY AXIS ACCEL X BODY AXIS ACCEL

AY Y-AXIS ACCELERATION
Y AXIS ACCELERATION Y-BODY AXIS ACCEL Y BODY AXIS ACCEL LATERAL ACCELERATION LAT ACCEL
LATERAL ACCEL

AZ Z~BODY Z BODY

AXIS AXIS

ACCEL ACCEL

ANX A-AXIS X AXIS

ACCELEROMETER ACCELEROMETER

g

any

ANY

Y-AXIS ACCELEROMETER

Y AXIS ACCELEROMETER

g

ang

ANZ

Z-AXIS ACCELEROMETER

Z AXIS ACCELEROMETER

71

Observation variable

Units

Symbol

Alphanumeric descriptors

Accelerations (continued)

normal acceleration

g

an

AN

NORMAL ACCELERATION

NORMAL ACCEL

GS

G's

*x body axis

g

accelerometer not at

vehicle center of

gravity

anx,i

AX, I ANX,I

*y body axis

g

accelerometer not at

vehicle center of

gravity

any,i

AY,1I ANY,1I

*z body axis

g

accelerometer not at

vehicle center of

gravity

4nz,i

AZ,I ANZ,I

*normal accelerometer

g

4n,i

AN, I

not at vehicle

center of gravity

load factor

Dimension-

n

less

N LOAD FACTOR

Speed of sound tReynolds number Reynolds number
per unit length
Mach number

Air data parameters

ft/sec

a

A

SPEED OF SOUND

Dimension-

Re

less

RE REYNOLDS NUMBER

ft71

Re'

RE PRIME

R/LENGTH

R/FEET

R/UNIT LENGTH

Dimension-

M

less

72

Observation variable

Units

Symbol

Alphanumeric descriptors

Air data parameters (continued)

Dynamic pressure Impact pressure
Static pressure
Impact/ambient
pressure ratio

lb/f£t2 lb/ft2
lb/ft2
Dimension~
less

q dec Pa q./p,

QBAR
DYNAMIC PRESSURE
QC
IMPACT PRESSURE
DIFFERENTIAL PRESSURE
PA
STATIC PRESSURE
FREESTREAM PRESSURE
QC/PA
OC/P

Total pressure

lb/ft2

Pt

pT

TOTAL PRESSURE

Temperature
Total temperature

deg
absolute
deg absolute

T

TEMP

TEMPERATURE

FREESTREAM TEMPERATURE

Te

TOTAL TEMPERATURE

Equivalent airspeed

knot

Ve

VEAS

EQUIVALENT KEAS

AIRSPEED

Calibrated airspeed

knot

Ve

VCAS

CALIBRATED
KCAS

AIRSPEED

Flightpath angle
Flightpath acceleration
Flightpath angle
rate

Flightpath-related parameters

rad

Y

GAM

GAMMA

FLIGHT PATH ANGLE

GLIDE PATH ANGLE

GLIDE SLOPE

g

fpa

FPA

FLIGHT PATH ACCEL

rad/sec

Y

GAMMA DOT

GAMMADOT

73

Observation

variable

Units

Flightpath-related

Symbol parameters

Alphanumeric (continued)

descriptors

Vertical acceleration

ft/sec2

h

VERTICAL ACCELERATION

HDOTDOT

H-DOT=-DOT

HDOT=DOT

Scaled altitude rate
Specific energy Specific power
Lift force Drag force Normal force Axial force

ft/sec

h/57.3

H-DOT/57.3 HDOT / 57.3

Energy-related terms

ft

Eg

ES

E-SUB-S

SPECIFIC ENERGY

ft/sec

Py

Force parameters

PS P-SUB-S SPECIFIC SPECIFIC

POWER THRUST

lb

L

LIFT

lb

D

DRAG

lb

N

NORMAL FORCE

lb

A

AXIAL FORCE

x body axis velocity y body axis velocity

Body axis parameters

ft/sec

u

UB

X-BODY X BODY X~BODY X BODY U-BODY U BODY

AXIS AXIS AXIS AXIS

VELOCITY VELOCITY VEL VEL

ft/sec

Vv

VB

Y-BODY Y BODY Y-BODY
Y BODY V-BODY V BODY

AXIS AXIS AXIS AXIS

VELOCITY VELOCITY VEL VEL

Observation variable z body axis velocity

Units

Symbol

Alphanumeric descriptors

Body axis parameters

ft/sec

w

WB

Z-BODY AXIS VELOCITY

Z BODY AXIS VELOCITY Z-BODY AXIS VEL

Z BODY AXIS VEL

W-BODY W-BODY

Rate of change of
velocity in x body axis

ft/sec2

a

UBDOT UB DOT

Rate of change of velocity in y body axis

ft/sec2

v

VBDOT

VB DOT

Miscellaneous measurements not at vehicle center of gravity

Rate of change of
velocity in z body axis

ft/sec2

“

*Angle of attack not
at vehicle center
of gravity

rad

OG

*Angle of sideslip not

rad

Bhi

at vehicle center

of gravity

*Altitude instrument

Et

hii

not at vehicle center

of gravity

WBDOT WB DOT
ALPHA,I
ALPHA INSTRUMENT AOA INSTRUMENT
BETA,1 BETA INSTRUMENT SIDESLIP INSTRUMENT
H,1I ALTITUDE INSTRUMENT

*Altitude rate instrument not at
vehicle center of gravity

ft/sec

hy

HDOT, I

Other miscellaneous parameters

Vehicle total angular

slug-ft2/

T

momentum

secé

ANGULAR MOMENTUM ANG MOMENTUM

Stability axis roll rate

rad/sec

Ps

STAB AXIS ROLL RATE

75

Observation variable

Units

Symbol

Other miscellaneous parameters

Alphanumeric (continued)

descriptors

Stability axis pitch rate

rad/sec

ds

STAB AXIS PITCH RATE

Stability axis yaw rate

rad/sec

ts

STAB AXIS YAW RATE

*All parameters with an asterisk are for measurements at some point

other than the vehicle center of gravity. The program LINEAR uses the quantities defined in the first three floating point fields as

definitions of the location of the sensor with respect to the vehicle

center of gravity. The three parameters define the x body, y body,

and z body locations, in that order, of the sensor.

The units of

these offsets from the vehicle center of gravity are defined in units of feet.

tReynolds number is defined in terms of an arbitrary unit of length that is input by the user. This length is input using the first
floating point field; however, if no value is input, c is used as the default value.

76

APPENDIX D: STATE VARIABLE NAMES RECOGNIZED BY LINEAR

The alphanumeric descriptors specifying state variables that are recognized by

LINEAR are listed in this appendix. In the input file, the field containing these

descriptors uses a 5A4 format, and all characters are left justified.

The input

alphanumeric descriptor specified by the user serves both to identify the state

variable selected by the user within the program itself as well as to identify state

variables on the printed output of LINEAR as described in the Output Files section.

State variable Roll rate Pitch rate Yaw rate Velocity
Angle of attack Sideslip angle
Pitch attitude Heading Roll attitude Altitude Displacement north Displacement east

Units rad/sec rad/sec rad/sec ft/sec
rad rad
rad rad rad
ft Et ft

Symbol p q Yr Vv
a B
6 ¥ @ h x y

Alphanumeric descriptors

P
ROLL

RATE

Q PITCH

RATE

R YAW RATE

Vv VELOCITY
VEL VTOT

ALP ALPHA ANGLE OF ATTACK

BTA BETA SIDESLIP
ANGLE OF

ANGLE SIDESLIP

THA THETA PITCH

ATTITUDE

PSI
HEADING HEADING

ANGLE

PHI
ROLL BANK

ATTITUDE ANGLE

H ALTITUDE
Xx

XY

77

APPENDIX E: ANALYSIS POINT DEFINITION IDENTIFIERS
Analysis point definition options are selected using alphanumeric descriptors. These descriptors are the first record read for each analysis case. All these descriptors are read using a 5A4 format. The following list associates the analysis point definition options with their alphanumeric descriptors:

Analysis point definition option Untrimmed
Straight-and-level Pushover~pullup
Level turn Thrust stabilized turn Beta Specific power

Alphanumeric descriptors

UNTRIMMED NO TRIM NONE NOTRIM

STRAIGHT AND
WINGS LEVEL LEVEL FLIGHT

LEVEL

PUSHOVER AND PULLUP
PULLUP PUSH-OVER/PULL=-UP PUSH-OVER / PULL-UP PUSHOVER / PULLUP PUSHOVER PULLUP PUSH OVER PULL UP PUSHOVER PUSHPULL

LEVEL TURN WINDUP TURN
THRUST STABILIZED TURN THRUST LIMITED TURN FIXED THROTTLE TURN FIXED THRUST TURN

BETA SIDESLIP

SPECIFIC
PS P-SUB-S

POWER

Each of these analysis point definitions except the untrimmed, beta, and specific power options has two suboptions associated with it. The suboptions are requested
using alphanumeric descriptors read using an A4 format. These suboptions are defined

78

in the Analysis Point Definition section of this report. The following list defines these suboptions and the alphanumeric descriptors associated with each:

Analysis point definition suboptions
Straight-and-level: Alpha-trim
Mach-trim
Pushover-pullup: Alpha-trim

Alphanumeric descriptors
ALP ALPH ALPHA MACH AMCH
ALP

Load-factor-trim

LOAD

Level turn: Alpha-trim
Load-factor-trim

ALP LOAD

Thrust stabilized turn: Alpha-trim
Load-factor-trim

ALP LOAD

79

APPENDIX F: EXAMPLE INPUT FILE

The following listing is an example of an input file to LINEAR. This file was

used with the example subroutines listed in appendix J to generate the analysis and

line printer files shown in appendixes H and I, respectively.

The formats specified

in table 1 would be the same whether each type of input has its own file or all of

the data is in one file.

USER'S GUIDE

6.080000E+02 4.280000E+01 1.595000E+01 4.500000E+04

2-870000E+04 1.651000E+05 1.879000E+05-5.200000E+02 0.

0.

0.

0.

0.

CCALC WILL CALCULATE CG CORRECTIONS

-1.000000E+01 4.000000E+01

4STAN

RADI

ALPHA

Q
THETA

VEL

3

ELEVATOR

5

THROTTLE

12

SPEED BRAKE

10

2STAN

AN

AY

~2.900E+00 5.430E+00-4.000E+00 4.000E+00-3.250E+00 3.250E+00-1.000E+00

_ 0 ADDITIONAL SURFACES

WINDUP TURN

ALPHA

H

20000.0

MACH

0.90

AN

3.00

BETA

0.0

NEXT

LEVEL FLIGHT

ALPHA

H

20000.0

MACH

0.9

GAMMA

10.0

END

1.000E+00

This
is input fies the

input file is for a case called USER'S GUIDE (record 1).
from the terminal as described in the Data Input section. mass and geometric properties of the vehicle as

The project title Record 2 speci-

t t

S = 608 ft2

b

f l

42.8 ft

80

c = 15.95 ft w = 45,000 lbs

Record 3 defines the moments and products of intertia of the vehicle as

Ix = 28,700

slug-ft?

Ty = 165,100 slug-ft4

Iz, = 187,900 slug-ft?

Ix, = -520

slug-ft?

Ixy = 0.0

slug-ft2

Iyz = 0.0

slug-ft2

Record 4 defines the location of the aerodynamic reference point to be coincident with the vehicle center of gravity of the nonlinear aerodynamic model by setting
Ax = 0.0 ft

fy = 0.0 ft Az = 0.0 ft

Record 4 also specifies that LINEAR should not use its internal model to make corrections for the offset in the vehicle center of gravity from the aerodynamic reference point because the aerodynamic model includes such corrections.

Record 5 defines the angle-of-attack range of the aerodynamic model.

Record 6 specifies that there will be four state variables in the linear model, that the formulation of the state equation will be in the standard form

(x =Ax + Bu), and that the nondimensional stability and control derivatives with

respect to angle of attack and angle of sideslip should be scaled in radians~!.

The

next four records define the output formulation of the state vector to be

o qa 3) Vv
Record 11 specifies that the linear model will have three parameters in the control vector. The following three records (12 to 14) specify that

elevator u = ] throttle
speed brake

81

and that elevator, throttle, and speed brake are located in DC(5), DC(12), and DC(10) of the /CONTROL/ common block.
Record 15 specifies that two observation variables will be used, and that the observation equation will be in the standard form,
y = Hx + Fu

The next two records (16 and 17) define the elements of the output vector to be

=|
y=
ay

Record 18 specifies the ranges for the trim parameters DES, DAS, DRS,
used to trim the longitudinal, lateral, and directional axes, and thrust, tively. The ranges for these parameters are defined by record 20 to be

and THRSTX, respec-

~2.9
-4.0 -3.25
-~1.0

< DES < 5.43
< DAS < 4.0 < DRS < 3.25 < THRSTX < 1.0

The first three parameters essentially represent stick and rudder positions and are
so specified because of the implementation of the subroutine UCNTRL (discussed in app. J). The thrust trim parameter is specified in this manner because of the implementation of UCNTRL. Speed brake is scheduled when THRSTX < 0 and thrust is commanded when THRSTX > 0.

Record 19 specifies that no additional control surfaces are to be set.

The next seven records (20 to 26) define an analysis point option. request a level turn trim option at

These records

h = 20,000 ft

M = 0.9

an = 3.0 g

and

B = 0.0°

The second record of this set (record 21) indicates which level turn suboption is

requested.

The alphanumeric descriptor ALPHA indicates that angle of attack is to be

varied until the specified 3.0-g turn is achieved.

The final record of this analysis

point option definition set contains the keyword NEXT to indicate both an end to the

current analysis point option definition and that another analysis point option defi-

nition follows.

82

The final six records (records 27 to 32) define a straight-and-level analysis point option at
20,000 ft

0.9

and

Y

10.0°

The second record of this set (record
which angle of attack is varied until The final record of this set contains the current analysis point definition

28) identifies the Alpha Trim suboption in trim is achieved at the specified condition. the keyword END to indicate the termination of
as well as the termination of input cases.

83

APPENDIX G: EXAMPLE LINEARIGED STABILITY AND CONTROL DERIVATIVE FILE

STABILITY AND CONTROL DERIVATIVES FOR CASE #1 LINEARIZER TEST AND DEMONSTRATION CASES

FOR THE PROJECT: F-15 SIMULATION

REFERENCE ALTITUDE REFERENCE MACH

2.0000D+04 9.0000D-01

ANGLES ARE IN RADIANS FOR OUTPUT
WARNING!!!! DERIVATIVES WITH RESPECT VELOCITY DERIVATIVES WILL BE COMPUTED

TO MACH WILL BE USED. FROM MACH DERIVATIVES.

THERE ARE 3 CONTROL SURFACES

ROLL MOMENT DERIVATIVES

CLO

ROLL RATE (RAD/SEC)

PITCH RATE (RAD/SEC)

YAW RATE

(RAD/SEC )

VELOCITY

(FT/SEC)

MACH NUMBER

ALPHA

(RAD)

BETA
ALTITUDE

(RAD }
(FT)

ALPHA DOT
BETA DOT

(RAD/SEC)
(RAD/SEC)

-4.02966D-05 -2.00000D-01
-00000D+00 1.50990D-01 -1.26696D-07 -1.31375D-04
-00000D+00
-1.33450D-01
-00000D+00
-00000D+00
-00000D+00

ELEVATOR THROTTLE SPEED BRAKE

5

-00000D+00

12

-00000D+00

10

-00000D+00

PITCH MOMENT DERIVATIVES

CMO

ROLL RATE (RAD/SEC)

PITCH RATE (RAD/SEC)

YAW RATE

(RAD/SEC)

VELOCITY

(FT/SEC)

MACH NUMBER ALPHA

(RAD)

BETA

(RAD)

ALTITUDE

(FT)

ALPHA DOT (RAD/SEC)

BETA DOT

(RAD/SEC )

4.22040D-02
-00000D+00 3.89527D+00
-00000D+00 -3.28490D-06
~3.40620D-03 -1.68819D-01
-90000D+00 -00000D+00 -1.18870D+01 -00000D+00

ELEVATOR
THROTTLE SPEED BRAKE

5

-6.95281D-01

12

-00000D+00

10

-4.17500D-01

84

YAW MOMENT DERIVATIVES

CNO

ROLL RATE (RAD/SEC)

PITCH RATE (RAD/SEC)

YAW RATE

(RAD/SEC )

VELOCITY

(FT/SEC)

MACH NUMBER

ALPHA

(RAD)

BETA ALTITUDE

(RAD) (FT)

ALPHA DOT (RAD/SEC)

BETA DOT

(RAD/SEC)

ELEVATOR THROTTLE SPEED BRAKE

DRAG FORCE DERIVATIVES

cbD0

ROLL RATE (RAD/SEC)

PITCH RATE (RAD/SEC)

YAW RATE

(RAD/SEC)

VELOCITY

(FT/SEC)

MACH NUMBER

ALPHA

(RAD)

BETA

(RAD)

ALTITUDE

(FT)

ALPHA DOT (RAD/SEC)

BETA DOT

(RAD/SEC)

ELEVATOR
THROTTLE SPEED BRAKE

LIFT FORCE DERIVATIVES

CLFTO

ROLL RATE (RAD/SEC)

PITCH RATE (RAD/SEC)

YAW RATE

(RAD/SEC )

VELOCITY

(FT/SEC)

MACH NUMBER

ALPHA

(RAD)

BETA ALTITUDE

(RAD) (FT)

ALPHA DOT (RAD/SEC)

BETA DOT

(RAD/SEC)

ELEVATOR
THROTTLE SPEED BRAKE

2.25747D-04
-3.37217D-02 -00000D+00
-4.04710D-01 3.21400D-07
3.33268D-04 -00000D+00
1.29960D-01 -00000D+00 »90000D+00 -00000D+00

-00000D+00

12

-00000D+00

10

-00000D+00

1.08760D-02
-00000D+00 -00000D+00 -00000D+00 -90000D+00 -00000D+00 3.72570D-01 -00000D+00 -00000D+00 -Q0000D+00 -00000D+00

5

4.38318D-02

12

-00000D+00

10

6.49355D-~02

1.57360D-01
-00000D+00 -1.72315D+01
-00000D+00
1.45286D-05 1.50651D-02
4.87061D+00 -00000D+00 -00000D+00
1.72315D+0 1 -00000D+00

5

5.72950D-01

12

-00000D+00

10

3-74913D-02

85

SIDE FORCE DERIVATIVES

cy0o

ROLL RATE
PITCH RATE YAW RATE VELOCITY

(RAD/SEC)
(RAD/SEC) (RAD/SEC )
(FT/SEC)

MACH NUMBER

ALPHA

(RAD )

BETA
ALTITUDE

(RAD)
(FT)

ALPHA DOT
BETA DOT

(RAD/SEC)
(RAD/SEC)

ELEVATOR THROTTLE
SPEED BRAKE

5.32725D-04
-Q00000D+00
-00000D+00 -00000D+00 -00000D+00 -00000D+00 -00000D+00
-9.74030D-01
-00000D+00
-00000D+00
-00000D+00

5

-00000D+00

12

-00000D+00

10

-00000D+00

86

STABILITY AND CONTROL DERIVATIVES FOR CASE #2 LINEARIZER TEST AND DEMONSTRATION CASES

FOR THE PROJECT: F-15 SIMULATION

REFERENCE ALTITUDE

REFERENCE MACH

=

2-0000D+04 9.0000D-01

ANGLES ARE IN RADIANS FOR OUTPUT WARNING!!!! DERIVATIVES WITH RESPECT VELOCITY DERIVATIVES WILL BE COMPUTED

TO MACH WILL BE USED. FROM MACH DERIVATIVES.

THERE ARE 3 CONTROL SURFACES

ROLL MOMENT DERIVATIVES

CLO

ROLL RATE PITCH RATE YAW RATE

(RAD/SEC) (RAD/SEC) (RAD/SEC )

VELOCITY

(FI/SEC)

MACH NUMBER ALPHA

(RAD)

BETA

(RAD)

ALTITUDE ALPHA DOT

(FT) (RAD/SEC)

BETA DOT

(RAD/SEC)

1.25377D-16
~2.00000D-01 -00000D+00
1.50990D-01 -00000D+00 -00000D+00 -Q0000D+00
-1.33450D-01
-00000D+00 -00000D+00 -00000D+00

ELEVATOR THROTTLE SPEED BRAKE

5

-00000D+00

12

-00000D+00

10

-00000D+00

PITCH MOMENT DERIVATIVES

CMO

ROLL RATE (RAD/SEC)

PITCH RATE (RAD/SEC)

YAW RATE

(RAD/SEC)

VELOCITY

(FT/SEC)

MACH NUMBER

ALPHA

(RAD)

BETA

(RAD)

ALTITUDE

(FT)

ALPHA DOT (RAD/SEC)

BETA DOT

(RAD/SEC)

4.22040D-02
-00000D+00 3.89530D-00
-00000D+00 -7.79749D-10 -8.08541D-07 -1.68819D-01
-00000D+00 -00000D+00 -1.18870D+01 -00000D+00

ELEVATOR THROTTLE SPEED BRAKE

5

-6.95279D-01

12

-00000D+00

10

-4.17500D-01

YAW MOMENT DERIVATIVES

CNO

ROLL RATE (RAD/SEC)

PITCH RATE (RAD/SEC)

YAW RATE

(RAD/SEC)

VELOCITY

(FT/SEC)

MACH NUMBER

ALPHA

(RAD )

BETA

(RAD)

ALTITUDE

(FT)

ALPHA DOT (RAD/SEC)

BETA DOT

(RAD/SEC)

ELEVATOR
THROTTLE SPEED BRAKE

DRAG FORCE DERIVATIVES

CbDO

ROLL RATE (RAD/SEC)

PITCH RATE (RAD/SEC)

YAW RATE

(RAD/SEC)

VELOCITY

(FT/SEC)

MACH NUMBER

ALPHA

(RAD)

BETA
ALTITUDE

(RAD)
(FT)

ALPHA DOT BETA DOT

(RAD/SEC) (RAD/SEC)

ELEVATOR THROTTLE
SPEED BRAKE

LIFT FORCE DERIVATIVES

CLFTO
ROLL RATE PITCH RATE
YAW RATE VELOCITY MACH NUMBER ALPHA BETA
ALTITUDE ALPHA DOT BETA DOT

(RAD/SEC) (RAD/SEC) (RAD/SEC)
(FT/SEC)
(RAD) (RAD)
(FT) (RAD/SEC) (RAD/SEC)

ELEVATOR
THROTTLE SPEED BRAKE

88

1.22535D-16
~3.37217D-02 »00000D+00
-4.04710D-01
-00000D+00 -00000D+00
-90000D+00 1.29960D-01
-00000D+00 -00000D+00 -00000D+00

5

-00000D+00

12

-00000D+00

10

-900000D+00

1.08760D-02
-00000D+00 -00000D+00 -00000D+00 -00000D+00 -00000D+00
3.72570D-01 -00000D+00 -00000D+00 -00000D+00 -00000D+00

5

4.38313D-02

12

-00000D+00

10

6.49346D-02

1.57360D-01
-90000D+00 -1.72320D+01
-00000D+00 3.59263D-09
3.72529D-06 4.87061D+00
-00000D+00
-90000D+00 1.72320D+01
-00000D+00

5

5.72961D-01

12

-00000D+00

10

3.74913D-02

SIDE FORCE DERIVATIVES

cyo

ROLL RATE (RAD/SEC)

PITCH RATE (RAD/SEC)

YAW RATE

(RAD/SEC )

VELOCITY

(FT/SEC)

MACH NUMBER

ALPHA

(RAD)

BETA

(RAD )

ALTITUDE

(FT)

ALPHA DOT (RAD/SEC)

BETA DOT

(RAD/SEC )

ELEVATOR
THROTTLE SPEED BRAKE

4.15737D-16
-00000D+00 -00000D+00 -00000D+00 -00000D+00 -00000D+00 -00000D+00 -9.74030D-01 -00000D+00 -00000D+00
-00000D+00

5

-00000D+00

12

-00000D+00

10

-00000D+00

89

APPENDIX H: EXAMPLE OUTPUT ANALYSIS FILE

The following is an example analysis file. example input file listed in appendix F and the listed in appendix J.

This file was produced using the example user-supplied subroutines

LINEARIZER TEST AND DEMONSTRATION CASES USER'S GUIDE

TEST CASE #

TTT TTT TT CTT TTT eT TC PCPS SSCS SCC

ee SS SS

te

X - DIMENSION = 4

U - DIMENSION = 3

STATE EQUATION FORMULATION:

STANDARD

OBSERVATION EQUATION FORMULATION:

STANDARD

Y¥ ~ DIMENSION = 2

STATE VARIABLES
ALPHA Q THETA VEL

= 0.465695D-01
= 0.921683D-01 = 0.159885D-01 = 0.933232D+03

RADIANS
RADIANS/SECOND RADIANS FEET/SECOND

CONTROL VARIABLES

ELEVATOR
THROTTLE SPEED BRAKE

= 0.538044D-01
= 0.214105D+00 0.000000D+00

DYNAMIC INTERACTION VARIABLES

X-BODY AXIS FORCE
Y-BODY AXIS FORCE Z-BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT YAWING MOMENT

= 0.134742D+06
= 0.102657D+05 = 0.938155D-03 = 0.680525D+t02 = 0.000000D+00 = -.212007D+02

POUNDS
POUNDS POUNDS FOOT-POUNDS FOOT-POUNDS FOOT-POUNDS

OBSERVATION VARIABLES

AN

0.300163D+01

Gs

AY

i l

0.941435D+00

GS

90

A~MATRIX FOR: DX / DT = A*X

+ B¥*¥U+t+D*YV

-0.121436D+01 0.100000D+01 0.136756D-02-0.121605D-03 -0.147423D+01-0.221451D+01-0.450462D-02 0.294019D-03
0.000000D+00 0.331812D+00 0.000000D+00 0.000000D+00 -0.790853D+02 0.000000D+00-0.320822D+02-0.157297D-01

B-MATRIX FOR: DX / DT = A*xXK + B*U+tD*¥YV

0.14196 1D+00- 0. 164948D-02-0.928933D-02 0.220778D+02 0.543324D-02-0.135074D+02 0.000000D+00 0.000000D+00 0.000000D+00 0.105186D+02 0.342817D+02-0.155832D+02

D-MATRIX FOR: DX

/DT=A*X+BFUtTtD*V

0.343642p-07
0.113192D-06 0.000000D+00 0.714203D-03

0.Q000000D+00 0.737378D-06
0.000000D+00-0.242885D-05 0.000000D+00 0.000000D+00 0.398492D-06 0.332842D-04

0.000000D+00
0.605694D-05 0.000000D+00 0.000000D+00

0.000000D+00
0.000000D+00 0.000000D+00 0.000000D+00

0.000000D+00
0.000000D+00 0.000000D+00 0.000000D+00

H-MATRIX FOR:

Y = H*X

+ F* U+E* V

0.351752D+02 0.277556D-16 0.150046D-02 0.640771D-02 0.000000D+00 0.000000D+00-0.150534D-01 0.446789D-10

F-MATRIX FOR: Y = H* X +F*U+tT+E*YV
0.412845D+01-0.180978D-02 0.291699D+00 0.000000D+00 0.000000D+00 0.000000D+00

E-MATRIX FOR: Y=H* X+F* U+t+E*V

-0.377037D-07 0.000000D+00-0.214132D-04
0.Q000000Dt00 0.222222D-04 0.000000D+00 0.000000D+00 0.000000D+00

0.000000D+00 0.000000D+00

0.000000D+00 0.000000D+00

TEST CASE

KKEKKKKHKKKEKKKHEKKEEEEEEKEKEEKEKEKEEKEKKKEKKKEKKKKKEKKKEKE

X - DIMENSION = 4

U - DIMENSION = 3

¥Y ~ DIMENSION =

STATE EQUATION FORMULATION:

STANDARD

OBSERVATION EQUATION FORMULATION:

STANDARD

STATE VARIABLES
ALPHA
0)
THETA VEL

u l

-.126650D-01
= 0.0Q00000D+00 0.161868D+00
= 0.933232D+03

RADTANS
RADIANS/ SECOND RADIANS FEET/ SECOND

CONTROL VARIABLES

ELEVATOR THROTTLE SPEED BRAKE

= 0.637734D-01 = 0.225092D+00 = 0.Q000000D+00

DYNAMIC INTERACTION VARIABLES

X-BODY AXIS FORCE Y-BODY AXIS FORCE Z-BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT “YAWING MOMENT

i !

= 0.443769D+05 = 0.300494D+04 = 0.445795D-09 = ~.528481D-11 = 0.000000D+00
0.114239D-10

POUNDS
POUNDS POUNDS FOOT-POUNDS FOOT-POUNDS FOOT-POUNDS

OBSERVATION VARIABLES

AN

0.985228D+00 GS

AY

0.990655D-14 GS

A-MATRIX FOR: DX / pT = A* X + B*YUtTt+D*YV
0.120900D+01 0.100000D+01-0.575730D-02-0.701975D-04 0.149189D+01-0.221451D+01 0.189640D-01 0.231368D-03 0.000000D+00 0.100000D+01 0.000000D+00 0.000000D+00 0.576868D+02 0.000000D+00-0.316251Dt02-0.460435D-02

B-MATRIX FOR:

DX / DT = A*X

+ B*YUTtD*V

0.141961D+00 0.448742D-03-0.928932D-02
0.220778D+02-0.147812D-02-0.135074D+02 0.000000D+00 0.000000D+00 0.000000D+00 0.105186D+02 0.343162D+02-0. 155832D+02

92

D-MATRIX FOR: DX f/ DT +A*X+B*U+t+D* V

0.934880D-08 0.307941D-07 0.000000D+00
0.714920D-03 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00

0.000000D+00 0.738119D-06 0.000000D+00-0.243129D-05 0.000000D+00 0.000000D+00
0.000000D+00-0.905497D-05

0.000000D+00 0.605694D-05 0.000000D+00
0.000000D+00

0.000000D+00 0.000000D+00 0.000000D+00
0.000000D+00

H-MATRIX FOR: Y = H* xX + F* U

0.350424D+02 0.277556D-16-0.632314D-02 0.203434D-02 0.000000D+00 0.000000D+00 0.000000D+00 0.212306D-16

F-MATRIX FOR:

Y = H*X

+ F*U

0.411323D+01 0.492845D-03 0.263288D+00 0.000000D+00 0.000000D+00 0.000000D+00

E-MATRIX FOR: Y+H* X+F* U+E* V

0.102676D-07
0.000000D+00 0.000000D+00 0.000000D+00

0.000000D+00-0.214116D-04 0.222222D-04 0.000000D+00

0.000000D+00 0.000000D+00

0.000000D+00 0.000000D+00

FOR THE PROJECT: USER'S GUIDE
ALPHA QTHETA VEL

THE STANDARD FORMULATION OF THE STATE EQUATION

HAS BEEN SELECTED.

THE FORM OF THE EQUATION IS:

DX/DT

A*xX+B* U

SURFACES TO BE USED FOR THE CONTROL VECTOR FOR: LINEARIZER TEST AND DEMONSTRATION CASES

FOR THE PROJECT: USER'S GUIDE

LOCATION IN /CONTROL/

ELEVATOR

5

THROTTLE

12

SPEED BRAKE

10

PARAMETERS USED IN THE OBSERVATION VECTOR FOR: LINEARIZER TEST AND DEMONSTRATION CASES
FOR THE PROJECT: USER'S GUIDE
AN AY

THE STANDARD FORMULATION OF THE OBSERVATION EQUATION

HAS BEEN SELECTED.

THE FORM OF THE EQUATION IS:

Y=H*X+FE*U

LIMITS FOR TRIM OUTPUT PARAMETERS:
PITCH AXIS PARAMETER ROLL AXIS PARAMETER YAW AXIS PARAMETER THRUST PARAMETER

MINIMUM -2.900 -4.000 -3.250 ~1.000

MAXIMUM 5.430 4.000 3.250
1.000

NO ADDITIONAL SURFACES TO BE SET WERE DEFINED

95

TRIM CONDITIONS FOR CASE # 1 LINEARIZER TEST AND DEMONSTRATION CASES

FOR THE PROJECT: USER'S GUIDE

LEVEL TURN WHILE VARYING ALPHA

TRIM ACHIEVED

COEFFICIENT OF LIFT
COEFFICIENT OF DRAG LIFT DRAG
ALTITUDE MACH VELOCITY EQUIVALENT AIRSPEED SPEED OF SOUND GRAVITATIONAL ACCEL NORMAL ACCELERATION LOAD FACTOR DYNAMIC PRESSURE DENSITY WEIGHT (@GALTITUDE) BETA ALPHA PHI THETA
ALTITUDE RATE GAMMA ROLL RATE
PITCH RATE YAW RATE THRUST
SUM OF THE SQUARES

(LBS) (LBS) (FT)
(FT/SEC) (KTS) (FT/SEC) (PT/SEC**2) (G-S)
(LBS/FT**2 ) (SLUG/FT**3 ) (LBS) (DEG) (DEG) (DEG) (DEG) (FT/SEC) (DEG) (DEG/SEC ) (DEG/SEC) (DEG/SEC) (LBS)

0.40144
0.03058 134741.68807
10265.70661
20000.00000 0.90000
933.23196 403.42303 1036.92440
32.11294 3.00163 2.99995
552.05302 0.00126774
44914.60434 0.03193 2.66824
70 .62122 0.91607
0.00000 0.00000 -0.08951
5-28086 1.85749 10277.03515
0.00000

TRIM PARAMETERS

TRIM
TRIM TRIM TRIM

PITCH AXIS PARAMETER
ROLL AXIS PARAMETER YAW AXIS PARAMETER THRUST PARAMETER

i }

-0.66958
-~0.01526 -0.02125
0.21410

CONTROL VARIABLES

ELEVATOR
THROTTLE SPEED BRAKE

(DEG)

0.05380 0.21410 0.00000

OBSERVATION VARIABLES

AN

3.00163339 GS

AY

-94136286 GS

96

00+0d00000°0 00+000000°0 00+d00000°0
00+4000000°0 00+d000000°0 00+4d00000°0 L0-GOeOPrl*600+d00000°0 00+d00000°0 00+d00000°0 00+d00000°0 00+d00000°0 00+000000°0 VO-ASCLEE*S
qOXOT darts

TO-AOZEPL°E 00+d00000°0 LO-dO9672L°S
00+d00000°0 LO+GOCETL* EL 00+d00000°0 00+d00000°0 00+0090L8°P cO-GEO80S"* 1 SO-CEEPSH LE 00+d00000°0 LO+CO0ZECL*100+d000000°0 LO-d09ELG"t
LITT

cCO-GOSE6P*9 00+d00000°0 CO-GOLEBE'D
00+4000000°0 00+000000°0 00+d00000°0O 00+d00000°0 L0-GOLSZL°E 00+4000000°0 00+d00000°0 00+d00000°0 00+d00000°0 00+d00000°0 cO-d09280° t
owad

ATAaVLS GYOHO OIWYWNAGOYRY NVAW %S°E

*NOITLIGNOO SI NISYWW

LHOITA OILWLS

SIHL LY FIDIHAA

00+d00000°0 00+d00000°0 00+d00000°0

LO-d00SL1°700+d00000°0 t0-d08zs6°9-

00+000000°0 00+d00000°0 00+000000°0

aywvdd dagads A TLLONHL YOLWATTA

00+000000°0
00+000000°0 00+d00000°0 L0-d0966c" tL 00+000000°0 VO-ACGECE°E LO-A96017°E LO-d0LLbO*P00+d00000°0 cO-dO1LzcLe°€PO-GLULSZ°?

00+000000°0 LO+COL88L°T-
00+d00000°0 00+d00000°0 L0-GOZ889°*t€O0-d8Ls80r*e9O0-C6ELBC*E00+d00000°0
00+00€S68°E 00+d00000°0 CO-dOvOCT* BD

00+400000°0 00+000000°0 00+d00000°0 LO-dOSPEE*L— 00+000000°0 PO-d089cE*tLO-QSS6LZ°tLO-d0660S° 1 00+d00000°*0 LO-d00000°z-
S$0-099620°F-

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(oNs/awd)

Lod VWHd Iv

(La)

AGNLILIY

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VLA

(aw)

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UAGWOAN HOWW

(O8S/L4)

ALIDOIWMA

(ogS/avu)

ALWY MWA

(ONS/avu) SALVA HOLId

(04aS/awvy)

ALWY TION

SLNATOTdSAdTOO OY8dZ

LNAWOW ONIMVA

LNAWOW ONTHOLId

LNAWOW ONT TIO’

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SdSWO NOTLWAYLSNOWAG GNWY LSHL YaZIYWaNIT

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