NASA
Technical
Paper 2768
December 1987

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

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

NASA

NASA Technical Paper 2768
1987

User’s Manual for
LINEAR, a FORTRAN Program to Derive
Linear Aircraft Models
Eugene L. Duke, Brian P. Patterson, and Robert F. Antoniewicz
Ames Research Center
Dryden Flight Research Facility
Edwards, California

NASA
National Aeronautics and Space Administration Scientific and Technical Information Division

CONTENTS

SUMMARY . « © © «© © «© e « @

INTRODUCTION

. «6 « « «© «© e

NOMENCLATURE

. + « « «© « «

Variables . . « « Superscripts . . Subscripts . «6 « FORTRAN Variables

« « « «6 « « « «© « ...

PROGRAM OVERVIEW

. 6 « « «6

EQUATIONS OF MOTION . . « «

OBSERVATION EQUATIONS .. -»

SELECTION OF STATE, CONTROL ’ LINEAR MODELS .« « «6 « « « e

ANALYSIS POINT DEFINITION .

Untrimmed . 2. 2 « « « »

Straight-and-Level Trim

Pushover-Pullup ... .

Level Turn

« « « « » »

Thrust-Stabilized Turn

Beta Trim . . 6 » « « «

Specific Power ... -»

NONDIMENSIONAL STABILITY AND CONTROL DERIVATIVES

INPUT FILES « « « « « «© « «

Case Title, File Selection Information, and Project

Geometry and Mass Data State, Control, and Observation
Trim Parameter Specification

Variable

Definitions

Additional Surface Specification

«

Test Case Specification .

°

OUTPUT FILES

.« « « « «© « «

USER~SUPPLIED SUBROUTINES .

Aerodynamic Model... Control Model .« « « « « Engine Model . ...« « Mass and Geometry Model

Page
12 15

@mwyI~ymo

N

CONCLUDING

REMARKS

e ° e e e o ¢ e e ° oe

e

8

o

8

e

e

.

e

.

e

.

e

e

e

e

ote

53

APPENDIX A:

CORRECTION TO AERODYNAMIC COEFFICIENTS FOR A CENTER OF GRAVITY

NOT AT THE AERODYNAMIC REFERENCE POINT . «2 2 0 ee we ee eee

55

APPENDIX B:

ENGINE TORQUE AND GYROSCOPIC EFFECTS MODEL « 2 « « © «© « « « « »

57

APPENDIX C:

STATE VARIABLE NAMES RECOGNIZED BY LINEAR

2. « + «© «© «© © © © @ «

65

APPENDIX D:

OBSERVATION VARIABLE NAMES RECOGNIZED BY LINEAR

. « «© «© « «e « «

66

APPENDIX E:

ANALYSIS POINT DEFINITION IDENTIFIERS

eo © © © © © © © © ew 8 ll

73

APPENDIX F:

EXAMPLE INPUT FILE . 2. « «© © we we ee

o © © © © © © © ew ew ew ltl

lt

75

APPENDIX G:

EXAMPLE OUTPUT ANALYSIS FILE . ..

«6

« eo 0 © © © © © © ew lt ll lt

79

APPENDIX H:

EXAMPLE PRINTER OUTPUT FILES

. . 2s

e¢ « e eo © © © © © © ew ew ee

82

APPENDIX I:

EXAMPLE USER-SUPPLIED SUBROUTINES

Ce

91

Aerodynamic

Model

Subroutine

oe

Engine Model Interface Subroutine

e e e ¢ e¢ # @ © # © © © ©
.......26.ee.e.e.e-s

Control Model Subroutine

.......0.0e8s ec we eee er

© ©
er

© e@ @ @

Mass and Geometry Model Subroutine

......s.e.e.«e © © 6 © ww wt

91
94
95 96

APPENDIX J:

REVISIONS TO MICROFICHE SUPPLEMENT .....

ee © © eo oe ew ew ew

98

REFERENCES

e ° e ee ee @ e e oe oe

@e@ @ eo ee @ eo 6 o @ e ee

oo oe e@ oo e e e@ 6 @ @

1 04

iv

SUMMARY
This report documents a FORTRAN program that provides a powerful and flexible tool for the linearization of aircraft models. The program LINEAR numerically determines a linear system model using nonlinear equations of motion and a usersupplied nonlinear aerodynamic model. The system model determined by LINEAR con sists of matrices for both state and observation equations. The program has been designed to allow easy selection and definition of the state, control, and observa-~ tion variables to be used in a particular model.
INTRODUCTION
The program LINEAR was developed at the Dryden Flight Research Facility of NASA's Ames Research Center to provide a standard, documented, and verified tool to be used in deriving linear models for aircraft stability analysis and control law design. This development was undertaken to eliminate the need for aircraftspecific linearization programs common in the aerospace 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 LINEAR, a FORTRAN program that provides the user with a powerful and flexible tool for the linearization of aircraft models. LINEAR is a program with well-defined and generalized interfaces to aerodynamic and engine models and is designed to address a wide range of problems without requiring program modification.
The system model determined by LINEAR consists of matrices for both state and observation equations. The program has been designed to provide easy selection and definition of the state, control, and observation variables to be used in a particular model. Thus, the order of the system model is completely under user control. Further, the program provides the flexibility of allowing alternative formulations of both 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 flexibility. By generalizing the surface definitions and making no assumptions of symmetric mass distributions, the program can be applied to any aircraft in any phase of flight except hover. The unique trimming capabilities, provided by means of a user-supplied subroutine, allow unlimited possibilities for trimming strategies and surface scheduling, which are particularly important for oblique-wing 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 variables for the linear models, combined with the large number of observation quantities availiable, allows any analysis problem to be solved with ease.

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

used and the methods employed to implement the program.

The trimming capabilities

of LINEAR are discussed from both a theoretical and an implementation perspective.

The input and output files are described in detail. The user-supplied subroutines

required for LINEAR are discussed, and sample subroutines are presented.

NOMENCLATURE

The units associated with the listed variables are expressed in a generalized system (given in parentheses). LINEAR will work equally well with any consistent

set of units with two notable exceptions:

the printed output and the atmospheric

model,

Both the printed output and the atmospheric model assume English units.

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

Variables

A

state matrix of the state equation, x = Ax + Bu; or, axial force (force)

At a an an,i anx anx,i
any any,i
ang anz,i
ay ay ay

state matrix of the state equation, Cx = A'x + Blu speed of sound in air (length/sec) normal acceleration (g) normal acceleration, accelerometer not at center of gravity (g) x body axis accelerometer output, accelerometer at center of gravity (gq) x body axis accelerometer output, accelerometer not at center of
gravity (g) y body axis accelerometer output, accelerometer at center of gravity (g) y body axis accelerometer output, accelerometer not at center of
gravity (q) Zz body axis accelerometer output, accelerometer at center of gravity (g) z body axis accelerometer output, accelerometer not at center of
gravity (g) acceleration along the x body axis (g) acceleration along the y body axis (g) acceleration along the z body axis (gq)

control matrix of the state equation, xe = Ax + Bu

B!

control matrix of the state equation, Cxe = A'x + Btu

wingspan (length)

C matrix of the state equation, Cx = A'x + B'u; or, force or moment coefficient
coefficient of drag coefficient of lift coefficient of rolling moment coefficient of pitching moment coefficient of yawing moment coefficient of sideforce center of mass of ith engine

mean aerodynamic chord (length)

dynamic interaction matrix for state equation, x = Ax + Bu + Dv; or,
drag force (force)
°
dynamic interaction matrix for the state equation, Cx = A'x + B'u + D'v 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 (length) feedforward matrix of the observation equation, y = Hx + Fu total aerodynamic force acting at the aerodynamic center engine thrust vector

F!

feedforward matrix of the observation equation, y = H'x + Gx + F'u

fpa

flightpath acceleration (gq)

G matrix of the observation equation, y = H'x + Gx + Flu acceleration due to gravity (length/sec2) observation matrix of the observation equation, y = Hx + Fu

observation matrix of the observation equation, y altitude (length)

H'x + Gxe + F'u

angular momentum of engine rotor (mass-length2/sec)

aircraft inertia tensor (mass-length2) rotational inertia of the engine (mass-length2)

x body axis moment of inertia (mass-length2)

x-y body axis product of inertia (mass~length2)

x-z body axis product of inertia (mass-length2)

y body axis moment of inertia (mass-length2)

y-z body axis product of inertia (mass-length2)

z body axis moment of inertia (mass~length2) total body axis aerodynamic rolling moment (length-force); or,
total aerodynamic lift (force) generalized length (length) Mach number; or, total body axis aerodynamic pitching moment
(length-force) aircraft total mass (mass) mass of engine

normal force (force); or, total body axis aerodynamic yawing moment (length-force)
load factor specific power (length/sec)

p Pa
Pte q
q de Rey RE
Re Re!
r S T
Te u u Vv Vo Ve Vv Vv W w X Xp
x

roll rate (rad/sec); or, pressure (force/length2) ambient pressure (force/length?2)

total pressure (force/length2) pitch rate (rad/sec)

dynamic pressure (force/length?)

impact pressure (force/length2) axis transformation matrices

Reynolds number Reynolds number per unit length (length7!)
yaw rate (rad/sec)

wing planform area (length?) ambient temperature (K); or,
or thrust (force)

total

angular

momentum

(mass-length2/sec2);

total temperature (K)

velocity in x-axis direction (length/sec)

control vector

total velocity (length/sec)

calibrated airspeed (knots)

equivalent airspeed (knots)

velocity in y-axis direction (length/sec)

dynamic interaction vector

vehicle weight (force) velocity in z-axis direction (length/sec) total force along the x body axis (force) thrust along the x body axis (force)

state vector

sideforce (f£ orce) thrust along the y body axis (force)

observation vector total force along the z body axis (force) thrust along the z body axis (force)

angle of att ack (rad) angle of sid eslip (rad) flightpath a ngle (rad) displacement of aerodynamic reference point from center of gravity displacement from center of gravity along x body axis (length) displacement from center of gravity along y body axis (length)
displacement from center of gravity along z body axis (length) lateral trim parameter

differential stabilator trim parameter

longitudinal trim parameter

Kronecker de lta

directional trim parameter

speed brake trim parameter

thrust trim parameter

incremental rolling moment (length-force)

incremental pitching moment (length-force); or, incremental Mach

incremental yawing moment (length-force)

incremental x body axis force (force)

éY

incremental y body axis force (force)

6Z

incremental z body axis force (force)

angle from t he thrust axis of engine to the x-y body axis plane (rad)

t

angle from the projection of Fp onto the engine x-y plane to the local

x axis (rad)

n

angle from Fp to the engine x-y plane (rad)

)

pitch angle (rad)

u

coefficient of viscosity

E

angle from the projection of Fp onto the x-y body axis plane to the

x body axis (rad)

)

density of air (mass /length3)

EL

total body axis rolling moment (length-force)

=IM

total body axis pitching moment (length-force)

=N

total body axis yawing moment (length-force)

T

torque from engines (length-force)

Tg

gyroscopic torque from engine (length-force)

>

roll angle (rad)

or,

tilt angle of acceleration normal to the flightpath from the vertical

plane (rad)

p

heading angle (rad)

Ww

total rotational velocity of the vehicle

De

engine angular velocity (rad/sec)

Superscripts

a

nondimensional version of variable

.

derivative with respect to time

T

transpose of a vector or matrix

Subscripts

ar

aerodynamic reference point

D

total drag

E

engine

h L g M mM max min n fe) p q r s TP x Y y Zz 6)

altitude total lift rolling moment Mach number pitching moment maximum minimum yawing moment offset from center of gravity roll rate pitch rate yaw rate stability axis thrust point along the x body axis sideforce along the y body axis along the z body axis standard day, sea level conditions; or, along the reference trajectory

FORTRAN Variables

AIX

inertia about the x body axis

AIXE

engine inertia

AIXY AIXZ AIY

inertial coupling between the x and y body axes inertial coupling between the x and z body axes inertia about the y body axis

ATYZ
AIZ

inertial coupling between the y and z body axes
inertia about the z body axis

ALP ALPDOT AMCH AMSENG AMSS B BTA BTADOT CBAR cD CDA CDDE Cho CDSB CL CLB CLDA CLDR CLDT CLFT CLFTA CLFTAD CLFTDE CLFTO CLFTQ CLFTSB
CLP

angle of attack time rate of change of angle of attack 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 angle of sideslip 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

CLR CM CMA CMAD CMDE CMO CMQ CMSB CN CNB CNDA CNDR CNDT CNP CNR CY CYB CYDA CYDR CYDT DAS Dc DELX
DELY
DELZ
DES 10

coefficient of rolling moment due to yaw rate total coefficient of pitching moment 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 longitudinal trim parameter surface deflection and thrust control array displacement of the aerodynamic reference along the x body axis from the center of gravity displacement of the aerodynamic reference along the y body axis from the center of gravity displacement of the aerodynamic reference along the z body axis from the center of gravity lateral trim parameter

DRS DXTHRS EIX ENGOMG
PHIDOT PSI PSIDOT
THADOT THRSTX THRUST TLOCAT

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 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 location of the engine in the x, y, and z axes from the center of gravity

14

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

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 engine axis in x-y body axis plane
orientation of engine axis in x-z body axis plane
position east from an arbitrary reference point time rate of change of east-west position

PROGRAM OVERVIEW

apsacetnboheaqdelipluenaeatbctgTsltihtyaeioeanranndttogsesopcfrwrohoeaaopigfnmeidcraxcthairtmmnirocimatoetncinhfgLottifIonsireNpnocEtslgtiAtasloRani)ntdn,vleeainarnuaarinemavdanaeaudbrrrssliiieyiezacsrnasebat-cldllesllemluuosoypdwrgiiprmnnloobdtggisdyoeesetdlessttrhhaeeemetlninissiegoeniscnpenitlrdsefiioeeynngffgerifeaaaneftrmecfhadlteeniscnatteoeatisasrrnirinoadmlded(yytestssnethuyoiaercespsmhmrttiiemiclnmoiipeannsnoeeimmdanoonstrtddeheeeetlelleio.scntyctthsohueetnrsdaretui.LmsrnIttogNr,blEyamAjonRedovstecanpoltrlre.ioiqiscr-uniyeef,,aaylTrishnoeg

Because the program is designed to satisfy the needs of a broad class of users,

a wide variety of options are those tion variables to

options has been provided. Perhaps the most important of these that allow user specification of the state, control, and observabe included in the linear model derived by LINEAR.

Within the program, the nonlinear equations of motion include

senting earth.

a rigid aircraft flying in a stationary atmosphere Thus, the state vector x is computed internally as

over

a

12 states repreflat nonrotating

x=[p qrva

6b ¢ 8 p hx yjT

12

The nonlinear equations used to determine the derivatives of the quantities are presented in the following section (Equations of Motion). 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,
y = [x7 xT ut yT YoT ¥3OT YqoT Y¥5yf Yeyl Y¥7yt y2y|tl|Tt

—
moa ange
r c
< x TE DEW. <r 3
—_!

where

Tv

Yi

anz Aan 4nx,i 4ny,i 4nz,i 4n,i n]

Yo

Pa dce/Pa Pt T Tt Ve Vel T

. ot

T

¥3

(y fpa Y h_ h/57.3]

Y4

ys

Y6

Y¥7 = [a5 Bi hi h,il T yg = [(T Ps dg Ys] T

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 variables desired in the output linear model (described in the Selection of State, Con-
trol, and Observation Variables section). Figure 1 illustrates the selection of variables in the state vector for a requested linear model. From the internal formulation on the right of the figure, the re-
quested model is constructed, and the linear system matrices are selected in accordance with the user specification of the state, control, and observation variables.

Output model parameters

{>

[

\

Specification of state vector
for linear model
Figure 1. Selection of state ables for linear model.

Internal parameters
-
vari-

13

ancoesLtnrerherIeaubcaNfvitEtateitAiuTroRroshsnsanee..pisrrneyacllilwwiaforinicATwietelonhsehsaengdorsdszei-esttlrheimtieoivnoaro)vsecdnanerelajlaluleoaldycranestbadfdioollesoaruysirptysgitiihvsipatseoow,dn(ihaastni,optctopnphbaiyuoonlsinddtlnhteetLytorhfsIveiicNetmrnEoaramiiAtsrlibtR,bibneineigdwoethniarispdracueoirhlnfpodylitpmeuintotdtpoedieehssndeoetr,tlnamasitiiarlennsiolesetadvalwoaienltnhtloiioatwcotrathnuhbtaeeeutlatrhtrnehdaAssseejn,tpareaneeuidclscauvtyeisdoeorsefrydritri-,ysrcs.azttondeaesrPtafloonetiai-raLnntsilIetmiiyNscodsEoneiAnDtassRdhealleifitipptpinoroaaoitiiocntrnavcmi-tlieao.l-dlny-es mdstohepedosencldrTfiihinbergiessdtl-iutoonniredndeaeerrirtthehusesyrteseLrtriemntsmehcaeornomtfasrttMoraoalind.cdeTelaassrydlTohsarerefecotrusimsedoeurenrlit.eaestrcimaoniTnenhxeepsdaeonflfseobicyrttohmneunlusamtsataetbitraooetuinetcaelqeotfuhqeauptaetithraoietnnouanrl,oybumsataitptsiurotincpeossiyansntdtc,eomrarraees-

or the generalized equation,

x = Ax + Bu

Cx

A'x + Blu

The observation matrices can be selected sponding to the standard equation,

from

either

of

two

formulations

corre-

or the generalized equation,

y = Hx + Fu

y = H'x + Gx + Flu

In addition sional stability are discussed in

to and the

theNcoonnltdirinomelaernsdiesorynisavtleamtiSvtmeaasbtirliaictetsy,theanLdIaNnEaCAloRynstirasollsopoDiecnrotim.vpautteiTsvheesstehesedcentroiinovdnai.tmievne-s

paffriirelloeep,edTrheatefrtieihenesienddpi,sustotcfatuaesn,tfsdhieeldecthoaeniiftnrorrcaorntlaha,LfelItyNsEaIiAnnasRdpndutpoisobsiseFnleiatlernecvstaoAstpSiCtsoIievnIocantrsiivfooeinulc.asetreorpstrhsaoetgldreeacsdmtieerfdei.odnpetsiionTnhsteh.tehedegtWeoaiouittmlhpesiutntryoflthiainntsdehaermianispnmsupotudtel

The output of LINEAR is three files, one

and two documenting the options and analysis

is intended to be used with follow-on design

contain all the information contained in the

of the analysis These files are

point and described

the nondimensional in the Output Files

containing the linear system matrices

points selected by the user.

The first

and analysis programs.

The other two

first file and also include the details

stability and control derivatives.

section.

To execute LINEAR, five user-supplied discussed in the User-Supplied Subroutines

subroutines are section, define

required.

These routines,

the nonlinear aerodynamic

14

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 gearing model (fig. 2) defines how the LINEAR trim inputs will be connected to the surface models and allows schedules and nonstandard trimming schemes to be employed. This last feature is particularly important for oblique-wing aircraft.

Inputs from ICTPARM/
DES —————* DAS ———|_ pRS —————+|_ THRSTX —__>

UCNTRL (Gearing)

Outputs to {CONTRL!
|____» DC(1)

[> . .

DC(2)

° /——_» DC(30)

(Pilot stick, pedal, and throttle)

(Surface deflections and power level setting)

Figure 2. Inputs to and outputs of 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 aerodynamic 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 will be detailed in a proposed NASA Reference Publication, "Derivation and Definition of a Linear Aircraft Model," by Eugene L. Duke, Robert F. Antoniewicz, and Keith D. Krambeer (in preparation).
The following equations for rotational acceleration are used for analysis point definition:
b= ((ZL)Iy + (2M)IQ + (EN)I3 - p2(IxgI2 - Ixyl3)
+ pa(Iy,Iy - TyzI2 - Dz1I3) - pr(Ixyyl4 + DyI2 - Iyz13)

+ q?(Iyzly

- Ixyy13)

- gr (DyI4

- Ixyt2 + Iy713)

- r2(Iy2ly - IxzI2)]/det I

15

where

q = [(2L)I5 + (ZM)Iq + (2N)I5 - p2(IyeT4 - IxyI5)
+ palTxgI2 - Iyglq - DgIs5) - pr(IxyI2 + DyI4 - IygIs5) + q2(IygIo - IxyI5) - qr(DyIg - IxyIq4 + IxzI5)
- r2(IygT9 - IypI4)]/det I
t= ((2L)I3 + (EM)I5 + (IN)Ig - p2(1y,I5 - IxyIg)
+ pa(Iyz13 - IygI5 - Dglg) - pr(IxyI3 + DyIs - IygTg) + g@(TygI3 - IxyI6) - qx(DyI3 - IxyI5 + IxzT6) ~ x2(IyzT3 - IxgI5)]/det I

2

2

2

Iq = Iylz - Iy2z

Ig = IxylIzg + TIyzlxz T3 = Ixylyz + IylIxz Iq = Iylg - Ix2gz
I5 = Ixlyz + Ixylxz

Tg = Ixly - Ixy Dy = Ig - ly Dy = Ix - Ig Dz = ly - Ix
vmpTtHloehoiieenhrtmgescied,,ehcneltteseprttrhiaometttcdieacuen,lehnebitonedmgartyono,tdmheeoatnfxhaytinracssdwuhgsartnaayrrgvraaaeiettwateeisy.snyitgnmhemaaenrmmTdtehoosremmuieemdtensnhestttesosit)oagfacrlnneodafeaateflrmelerfogdeimyncerdacinoeeePetsrsrsnciotogdpsqpny,oaianibctanodaetmuunditedctoofrtrZtqo,Lhm,ueotemhtcseeh.oXIne,Mrt,rnseonY,snoa(plnnTadiohcnaenndoneddiaIinNerpn,cqozgiuwdaeabretroeentidproscoyolpendaesryncoanttlxoa-ilfeumvissienectlrddhayu(et.rceome,lod-del

16

are derived in appendix A. The equations defining the engine torque and gyroscopic
contributions to the total moments are derived in appendix B.) The body axis moments
and products of inertia are designated I,, Ty, Iz, Ixy, Ixz, and Iyg.e These moments and products of inertia are elements of the inertia tensor I, defined as

Ty

~Iyy

~Ixz

T=

|-txy

ly

~Iyz

-Ixz

~lyz

Ty J

To derive the state equation matrices for the generalized formulation,

Cx = Ax + Bu

(where A and B are the state and control matrices tional accelerations are cast in a decoupled-axes to derive the linearized matrices are

of the state formulation.

equation), the The equations

rotaused

f 1

“Ixy

-Ixz] 72Pp4

Tx

Ty

~Ixy
Ty
-Ixz
tg

1
~lyz
Iz

-Iyz

q

Ty

1

°

J

p—aTrx- rp Pi=Ix>y +7 PlyzIxoz + org Ty>,ly +* (qe ‘4 2 - v42,) =Tlxy2 - qr =TTxz]

jew ote by tye a oa) ee, =I

7 rmyotragy

~ PAT

+ (r* - p

I

+ pry

Y

Y

Y

Y

Y

Y

T—IzN + ap —TIyx-aqrq

Ixz
Iz

+

Pp pr

Tyz
Iz

+

(pf Pp >

- g. 2, g*) —IIpx—y

- Pg =lTyz |

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

17

< e

W

[-D cos B + Y sin B + Xp cos a cos B + Yp sin B + Zp sin a cos 8B - mg(sin 8 cos a cos B - cos 6 sin $ sin B - cos 6 cos $ sin a cos 8)]/m
a = [-L + 2p cos a - Xp sin a + mg(cos 8 cos $ cos a + sin 9 sin a))/vm cos 8 +q- tan 8 (p cos 4 + r sin a)
B = [D sin B + Y cos B - Xp cos a sin B + Yp cos B - Zp sin a sin B
+ mg(sin 8 cos a sin B + cos 8 sin $ cos B - cos 8 cos $ sin a@ sin 8)]/Vm + p sin a- r cos a where a, B, 9, and $ are angles of attack, sideslip, pitch, and roll, respectively; Xp, Yp, and 2p are thrust along the x, y, and z body axes; and D is drag force, g gravitational acceleration, L total aerodynamic lift, m total aircraft mass, V total velocity, and Y sideforce. The equations defining the vehicle attitude rates are
$ =p+q sin > tan 6 + r cos $ tan 8
§ = q cos ¢ - r sin ¢$
p =q sin 6 sec 8 + r cos ¢ sec 86 where ~ is heading angle.
The equations defining the earth-relative velocities are
vV(cos B cos a sin @ - sin B sin $ cos 6 - cos B sin a cos $ cos 6)
x = V[I[cos B cos a cos 6 cos py + sin 8 (sin $¢ sin 9 cos t - cos ¢$ sin jp) + cos B sina (cos ¢ sin 9 cos » + sin $ sin p)])
Vi{[cos 8 cos a cos 6 sin py + sin B (sin @ sin 8 sin P + cos 6 cos }p) + cos 8 sin a (cos ¢ sin 9 sin yp - sin ¢ cos )])
where h is altitude.

w e

I l

he! °

i

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

18

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 et al., 1967; Etkin, 1972; Gainer

and Hoffman, 1972; Gracy, 1980). Implicit in many of these observation equations is

an atmospheric model. Atmosphere (1962).

The model included in LINEAR is derived from the U.S. Standard

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
and are derived directly from the body axis forces defined in the previous section for translational acceleration. The equations used in LINEAR for the body axis
accelerations ay, ay, and ag are

ay = (Xp - D cos 4 + L sin a - gm sin 8)/gom

ay = (Yp + ¥ + gm cos 8 sin $)/Fgm ag = (2p - D sin a- L cos a + gm cos § cos $)/gpm

where subscript 0 denotes standard day,
outputs of the body axis accelerometers center of gravity are

sea level conditions. (denoted by subscript

The equations for the n) that are at vehicle

anx = (Xp - D cos a + L sin &)/ggm

any = (Yp + Y)/gogm anz = (2p - D sin a - L cos a)/gpm
ay = (-Z2p + D sin a + L cos a)/gpm

For orthogonal not at vehicle
apply:

accelerometers that are aligned with the vehicle center of gravity (denoted by subscript ,i), the

body axes following

but are equations

anx,i = anx - [(q2 + r2)xy - (pq - r)yx - (pr + q)2x]/dg

f o 5
9 )
t i

= any + [(pq + r)xy - (p2 + 2)yy + (ar - plzyl/gy

anz,i = ang + [(pr - q)xz + (qr + p)yz - (q2 + p@)2g]/gy an,i = an - [(pr - q)xz + (ar + p)yg - (q* + p%)z2]/g,

where the subscripts x, y, and z refer to the x, y, and z body axes, respectively, and the symbols x, y, and z refer to the x, y, and z body axis locations of the

19

sensors relative to the vehicle center acceleration equations is load factor, lift and W is the vehicle weight.

of gravity. Also included in n = L/W, where L is the total

the set of aerodynamic

The air data parameters having the greatest application to aircraft dynamics and

control problems are The sensed parameters

the are

sensed parameters impact pressure

and the reference and scaling parameters. da, ambient or free~stream pressure Par

total pressure py, ambient or free-stream temperature T, and total temperature Ty.

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

speed of sound a, Reynolds number Re, Reynolds number
Mach meter calibration ratio Gc/Pae These quantities

per unit length,
are defined as

Re',

and

the

Re = OVE uy

'

pV

Re = va

T= =_ pv">e

q. ¢ =

[t1.0 + 0.2m2)3*° - 1.0]p

1.2M2

5.76M2

2.°5 - 1.0] p

5.6M2

- 0.8

a

(M < 1.0)
(M > 1.0)

q,
_P—. =

(1.0 + 0.2m2)3*> - 1,0

1.2m2

{—22762M“ 2.5 - 1.0 5.6M2 - 0.8

PL =P, + 4,

(M < 1.0) (M > 1.0)

where 2 is length, p pressure, p the density of the air, and yw the coefficient of

viscosity.

Free-stream pressure, free-stream temperature, and the coefficient of

viscosity are derived from the U.S. Standard Atmosphere (1962).

Also included in the air data calculations are two velocities:

equivalent

airspeed Ve and calibrated airspeed Ve, both computed in knots. The calculations

assume that internal units are in the English system. equivalent airspeed is

The equation used for

20

Ve = 17.17 Vg (1b/ft2)

which is derived from the definition of equivalent airspeed,

2q

Ve =

Pg

where Pg = 0.002378 slug/ft3 and V,. is converted from feet per second to knots. Calibrated airspeed is derived from the following definition of impact pressure:

Po 1..0 + 7.0—2p0-, y2© \?r? - 7

(Ve <$ a a.)

° q

=

1.2(—-V)e\? py

\

0

5.76 ——
5.6 - 0.8 (a)/Vq)

2.5 - Py

(Vo > ag)

Por the case where V, < a 0” the equation for V, is

Ve = 1479.116

de

2/7

— Po + 1.0

~ 1.0

(Ve $ ay)

Calibrated airspeed is found using an iterative process for the case where Vo > agi

Ve = 582.95174

/We{s-+ 1.0) 0

|1.0 - ——-*1*.-0 --5 2.5 7., 0(Ve/a, 0 )

(Ve > ag)

is executed until the change in Ve from one iteration to the next is less than 0.001 knots.

Also included in the observation variables are the flightpath-related parameters (described in app. D), 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 h/57.36 these quantities are

The equations used to determine

is)
k -
ton) i
<| a5°
“ee”

I l

~ <

; Pp a=vTo
= a, sin @ - ay sin $ cos 86 - az cos ¢ cos 8

a s

q

21

vh-hVv

v Vv2 - 72

Two energy-related energy Eg and specific

terms power

are Pg,

included in defined as

the

observation

variables:

v2 Bg = h + 55

specific

Pg =h*+ gow

The set parameters: normal force

of observation variables total aerodynamic lift N, and total aerodynamic

available in LINEAR also includes L, total aerodynamic drag D, total
axial force A. ‘These quantities

four force aerodynamic are defined

as

L = qSCy,

N= Lcos a+D sina

A = -L sin a+D cos a where Cp and Cy, are coefficients of drag and lift, respectively.

Six body axis These include the

rates and accelerations x body axis rate u, the

are available as y body axis rate

observation variables. v, and the z body axis

rate w. Also included The equations defining

are the time derivatives these quantities are

of these quantities,

u., ve, and wa,.

u V cos a cos 8

Vv

V sin B

w = V sin a cos B

(=

- gm sin 9 - D cos 4 +L n

sin a ) + rv - qw

U l

S s

q e

(= + gm cos 6 sing +y¥Y

ma

) + pw - ru

W W

° (= + gm cos 9 cos ¢ - D sin a -Lcos a

U l

m

) + qu - pv

=

The final collection of

set of observation variables other parameters of interest

available in LINEAR is in analysis and design

a miscellaneous problems. The

22

first group consists of measurements from sensors not located at the vehicle center of gravity. These represent angle of attack a ;, angle of sideslip B,i, altitude

his and altitude rate hoi measurements displaced from the center of gravity by some

X, Y, and z body axis distances.

The equations used to compute these quantities are

ag - (7 a, =a -(

Vv

)

hee) = )
l i R D
+

(= #4)

U

Vv

hi =h + x sin 6 = y sin 6 cos ®@ - z cos $ cos @

hy = h + 8(x cos ® + y sin $ sin 8 +z cos ¢ sin 9)

- ly cos $¢ cos 8 - z sin $ cos 6)

The remaining miscellaneous parameters are total axis roll rate pg, stability axis pitch rate qs, defined as

angular momentum T, stability and stability axis yaw rate rg,

1 5 (Ixp? - 2IxyP4 - 2Iyzpr + Tyq? - 2Iyzqr + Izr?)

Ps = pcos a+r sina Is= q fg = -p sina +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 input file. Internally, the program uses a

variables are defined by the user in the 12-state model, a 120-variable obser-

vation vector, to specify the

and a 30-parameter control vector. These variables can be formulation most suited for the specific application. The

selected order and

number of parameters in the output model is completely under user control.

Figure 1

illustrates However, it the matrices

the selection of should be noted
in the internal

variables for the state vector of the output model. that no model order reduction is attempted. Elements
formulation are simply selected and reordered in the

of

formulation specified by the user.

The selection of specific state, control, and observation variables for the for-

mulation of the input file. The Files section. tors. Appendix The alphanumeric

output matrices is accomplished by alphanumeric descriptors use of these alphanumeric descriptors is described in the

in the Input

Appendix C lists the state variables and their alphanumeric descrip-

D lists the observation variables descriptors for the selection of

and their alphanumeric descriptors. control parameters to be included

23

in the
input

observation vector
file, as described

are the control variable names
in the Input Files section.

defined

by

the

user

in

the

LINEAR MODELS

eaatSTnimisadavpyslallunooma,Tryprhdeteeedirso1ie9anlv7rsita2iosn;teueiismsaoverbepdit,rdezaoixeipsptndocoatsunhessatdrsstiheyeoessdnsutNielesAt,mSiaAnbmoaimutnaRttrPhtieracibetcyhAsteenisamDluenayk-uncseimao,inlemsyvrpsaiuAirctnPsiaeotdaliolnnnpyttiobeiywnDilitseicLnfIzeiN,a(anErDiAsiRtieaimnusodpdynalosrenetKnersemae,a.tmchpbetpeire1oo9Trnfx7h,.ie8ir;msatvit-naTKiohlwoerianpddkreieetrtrpetynaocarhatanoekttifhriqemousneatn)hpdiaosrf- a

£(x, + Ax) - f(x, ~ Ax)

ox

2 Ax

where f is a general function
may be set by the user but it with the single exception of sound, to obtain a reasonable

of x, an arbitrary independent variable. The Ax

defaults to 0.001 for all state and control velocity V, where Ax is multiplied by a, the

parameters speed of

perturbation size.

From the generalized nonlinear state,

and observation equations,

Tx = f(x, xe, wu)

y = g(x, x, u)

the program
the system:

determines

the

linearized

matrices

for

the generalized

formulation

of

where

Cc 5x = A' éx + B' 6u
Sy= H' 6x + G 6x + F! bu

c=7-2f
ox

A

a

of ax

'-

o2f

B

du

to. 2ag

H

ox

24

F! = 39

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

point

(X50

e
X.0, Uy di the state,

time derivative

of state,

and control

vectors

can be

expressed as small perturbations (6x, 6x, 6u) about the nominal trajectory, so that

X = X, + 6 x

x = x + 6x u = Uy + 6u

In addition to the matrices for this generalized system, the user has the option of requesting linearized matrices for a standard formulation of the system:

&x

A éx + B éu

Sy =H 6x + F du

where

A = j |T -—dTf|-1 xOF

L

OX |

*

B=

r |T - d£ff1-1 >3

ax |

u

*

ax |

)-1

Ox |

*

r

1-1

F - 99,

99

T _ bf

of

du

ax |

axe]

du

with all derivatives evaluated along the nominal trajectory defined by the analysis
poiinnt (xo. Xx o uy) .

LINEAR also provides two nonstandard matrices, D and E, in the equations Ax + Bu + Dv
y = Hx + Fu + Ev

* i

25

or D' and E' in the equations

Cx = A'x + Blu + D'v

Y

H'x + Gx + Flu + E'v

These dynamic interaction matrices include the effect acting on the vehicle. The components of the dynamic mental body axis forces (6x, Sy, 68%) and moments (6L,

of external interaction 6M, 6N):

forces vector

and moments v are incre-

[Sx éY

Thus,

and

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 for-

mulation. 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 specified trajectory (a point at which the rotational and translational accelerations

26

are zero; Perkins and Hage, 1949; Thelander, 1965) or a totally arbitrary state ona
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. 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 the rotational and translational accelerations to zero. The analysis point options are described in detail in the following sections.

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 variables names listed in appendix C. All state variables except velocity
must be specified according to this list. Velocity can also be defined by specifying Mach number (see alternative observation variable names in app. D). Appendix 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) of which the potential user should be aware. 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 vari-

ables must be zero.

(This is not the case, however:

Trim is achieved when the

acceleration-like terms are identically zero; no constraints need to be placed on

the velocity-like terms in x. Thus, for the model used in LINEAR, only P, qe r, v,

a, and & must be zero to satisfy the trim condition.) The trim condition is
achieved for the straight-and-level, pushover-pullup, level turn, thrust-stabilized turn, and beta trim options described in the following sections. In general, the no-trim 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 straightand-level and level turn options force the model to represent sustainable flight conditions. In fact, only in the special case where the flightpath angle is zero does this occur for these options.

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

27

tors (such as trim and requirements placed on vided by this program,

sustainable flight conditions) and ultimately by accuracy

the representation.

Thus, in using the linear models pro-

the user should exercise some caution.

Untrimmed
For the untrimmed option, the user specifies all state and control variables that are to be set at some value other than zero. The number of state variables specified is entirely at the user's discretion. If any of the control variables 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 trims 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 alti-
tude, 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 are determined within the following constraints:

p=q=r=0

¢ = 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 9 is determined by iterative solution of the altitude rate equation.

Pushover-Pullup

The pushover-pullup analysis point definition options 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.

Forn< 1, this trim results in a pushover at the maximum alti-

tude with h = O. There are two options

point definition:

(1) the "alpha-trim"

mined from the specified altitude, Mach

available for the pushover-pullup analysis option in which angle of attack is deternumber, and load factor and (2) the "load-

28

factor-trim" option in which angle of attack, altitude, and Mach number are specified and load factor is determined according to the constraint equations.

The analysis point following contraints:

is determined

at the specified

conditions,

subject

to the

p=r=0

a= = vm co1s . 8 [mg mg({(n = cos (6 - a )) -- @ Zp cos a@ + Xp si:n o ] ¢ = 0

The expression for q is derived from the a equation by setting a = 0 and > = 0; 8 is

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

of sideslip, solving the

Level Turn

The level turn analysis point definition options result in non-wings-level,

constant-turn-rate flight at excess thrust to trim at the

n > 1. The vehicle model is assumed to have sufficient 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 to define the desired flight

user can then use condition. These

either angle of two options are

attack or load referred to as

factor “alpha-

trim" and "load-factor-trim," respectively. For either option, the user may also

request a specific flightpath angle definitions may result in ascending

or or

altitude rate. Thus, these analysis point descending spirals, although the default is

for the constant-altitude turn.

The constraint equations for the coordinated level turn analysis point defini-

tions load

are derived factor, the

by tilt

Chen (1981) and Chen and Jeske (1981). angle of the acceleration normal to the

Using the requested flightpath from the

vertical plane, $1, is calculated using the equation

br, _ ttan-1

E

- cos2

wel

cos Y

where the positive sign is used for a right turn and the negative sign is used for a left turn. From $7, turn rate can be calculated as

y) = aev tan $1,

29

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

q = We sin2 $y,

|-sin y sin 8 +

(sin? y sin2 g - sin? Y = cos2 B 1/2 sin2 OL

ro =

g

s tan $7 cos B

Ps =

p sin Y
cos 8

q tan B

P = Pg COS G - rg sin o

Y = Pg sin @ + rg cos a

6 = sin-1(-p/)
¢ = tan~l(q/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 ina constant-

throttle, non-wings-level turn with a nonzero altitude rate.
available in LINEAR are “alpha-trim" and "load-factor-trim."

The two options
These options allow

the user to specify either the angle of attack or the load factor for the analysis
point. The altitude 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 level turn. However, is determined by LINEAR.

for the thrust-stabilized turn are the same as
for this analysis point definition, flightpath

those
angle

Beta Trim

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

flight with heading rate v = 0 at a user-specified Mach number, altitude, and angle

of sideslip.

This trim option is nominally at 1 g, but as 8 varies from zero,

normal acceleration decreases and lateral acceleration increases. For an aero-

dynamically symmetric aircraft, a trim to B = 0 using the beta trim option results

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

However, for an aero-

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

options are not equivalent.

30

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 acceleration.

solving the nonlinear equations for translational Pitch attitude 9 is derived from the equation for

and rotational flightpath angle

y

with Y = 0:

8 = tan-1 (8 8 sin » + cos B sin *) cos B~ cos a

Specific Power

The specific power specified Mach number, the other trim options

analysis
altitude, provided

point definition results thrust trim parameter,
in LINEAR, the specific

ina level and specific power option

turn at a user-

power.

Unlike

does not, in

general, attempt to achieve velocity rate V = 0. In fact, because the altitude rate

h = 0 and specific power is defined by

the resultant velocity rate will be

However, the other requested analysis

acceleration-like terms point is achieved.

e

e

(p, q,

¥e, &, and

8e ) will

be zero

if the

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

dy, = ttan71(n2 - 11/2

(where the positive sign is used for a right turn and the negative sign is used for a left turn),

v = tan OL

q = W sin $7, cos B a
tan $7, cosB

Pg = -q tan B P = Pg cos 4 - rg sin a

31

I

5

= Pg Sin a + rg cos a

8 = sinc (- 5)
y tan-14r

W

o o

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

load factor,
solving the

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

NONDIMENSIONAL STABILITY AND CONTROL DERIVATIVES

daTttnnyhiiodpeorn,enselsseci,.TnthceieoeaonqsrnvuuTnacehalohertnasideeoiraprlnsmoeaysedar,nsyaoosnmbinaielnemoticfineqlfcfaruueosldecr et-mswoitisdnhtneeiafgolsffbteihcelaialtisisostrsnlycugotrmiwnoeatfogafu-intdfdt,wboiuhrlndaodacita:aolndnnadltalyrrpoeaaclrpoffapmoonmlrprodeciumreetctumreaalairttalvsailitanoyeodtndniiavnlmceoostlmonitesnheneeiaactadsoreemll.rpaaceurotdatgeeeeefrrdrfoaeidlxcy-ccbindylleaiaunmsrsLtiseiIccvNteEoeilAfqoeyRuqnauavtaellfithraoiiotnomcesenlrqseau.tlah--e

The nondimensional stability and control derivatives tions for the aerodynamic force and moment coefficients:

assume

the

following

equa-

Cy == Cy + Cy a + Cy 8 + Cy Sh + Coy 6 OM + 2n “ng. 6.A

+C, ape” p+tc vagt qc© Mae” T+* C,Sgae® +C*,,SBys

Cm = Cm)

* Cg

+ Cmg8

+ Cm,

Sh

+ Cry

SM

n
+ 28;

8

+ CmpP + Cnt + Cn F + Cin 0 + Cing 8

Ch =

Cn

+ Cny®

+ CngB

+ Cry

oh

+ Cy,

OM

n
+ 20785

éy

+ CnpP + Cngt + Cnt + Cyjo + CngB

CL= = CL.

+ Cry@

+ Crg8

+ Cry

6h + Cry

n
6 6M + 2%,

83 i

a
+ CLpP + Crd + cyt + Craa + CygB

32

n
Cp = Cb, + Cp + Cog 8 + Cp, oh + Chu 6M + Y 5, oy i=1
+ Cp. DpPb ++ CpC.pan4gd ++ CpCoyt + CoDsga + Cppeg
n Cy = CY) + Cyy a + Cyg8 + Cyn éh + CY éM + 2 Cyg. 85:

A

Aa

+ Cy,Pb + C+ yCydG++ CyCy, t + CYygtta ++ CygChys e

where the stability and control derivatives have the usual meaning,

with Ce being an arbitrary force or moment coefficient (subscript % denoting rolling moment, m pitching moment, L total lift, D total drag, n yawing moment, and Y force along the y body axis) and x an arbitrary nondimensional variable (a denoting angle of attack, B angle of sideslip, h altitude, M Mach number, p roll rate, q pitch rate, and r yaw rate). The rotational terms in the equations are nondimensional versions of the corresponding state variable with
*P* . ®2PV

Ta oy£4

A

br

r=2V

¢ _ Ga ~ 2V

> ob
B = 2V

where b is wingspan and c is mean aerodynamic chord. The 6;
the n control variables defined by the user. The effects of (Mach) are included in the derivatives with respect to those incremental multipliers

in the summations are altitude and velocity parameters and in the

z

i ]

a

i }

Sh

=

I

ic-4

W t

6M

33

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

point

(x) 1 Xe

Uyde

All stability derivatives are computed as nondimensional terms except the altitude and velocity parameters. The control derivatives are in whatever units are used in the nonlinear aerodynamic model. The derivatives with respect to velocity are multiplied by the speed of sound (at the analysis point altitude) to convert them to derivatives with respect to Mach number. Derivatives with respect to angle of attack and angle of sideslip can be obtained in units of reciprocal degrees. These derivatives are simply the corresponding nondimensional derivatives multiplied by 180.0/T.

INPUT FILES

The LINEAR input file (defined in table 1) is separated into seven major sections: case title and file selection information; project title; geometry and mass data for the aircraft; state, control, and observation variable definitions for the state-space model; 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 above 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.
TABLE 1. — INPUT FORMAT FOR LINEAR

Input record

Format

Title and file selection information

Case title

(20A4)

Input file names

(6A10)

Project title

Project title

(20A4)

Geometry and mass data

Ss

b

Cc

Weight

Ix

Iy

Iz

Ixz

Ixy

Tye

DELX DELY DELZ LOGCG

Onin max

(4F13.0) (6F13.0) (3F10.0,12A4) (2F13.0)

34

TABLE 1. — Continued.

Input record

Format

State, control, and observation variable definitions

NUMSAT
STATE 1 STATE 2 STATE 3

EQUAT

STAB

DRVINC 1

DRVINC 2

DRVINC 3

:

(110, a4, 11X84)

-

(5A4,F10.0)

(5A4,F10.0)

(5A4,F10.0)

NUMSUR
CONTROL 74 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) (5A4,1106X,, AF410,.0) (5A4,110,A4,6X,F10,0)

NUMYVC

EQUAT

MEASUREMENT 1

PARAM 1 (1 to 3)

MEASUREMENT 2

PARAM 2 (1 to 3)

MEASUREMENT 3

PARAM 3 (1 to 3)

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

Trim parameter specification

fmin

max

@min

max

‘min ormax Sthnin Sthmax

NUMXSR

Additional surface specification

ADDITIONAL

SURFACE 1

LOCCNT 1

CONVR1

ADDITIONAL

SURFACE 2

LOCCNT 2

CONVR2

(8F10.0)
(1X,12)
(5A4,110,A4) (5A4,110,A4)

ADDITIONAL

TABLE 1. — Concluded.

Input records

Format

Additional surface specification (continued)

SURFACE 3

LOCCNT 3

CONVR3

(5A4,110,A4)

Test case specification ANALYSIS POINT DEFINITION OPTION ANALYSIS POINT DEFINITION SUBOPTION

VARIABLE VARIABLE VARIABLE

VALUE 1 VALUE 2 VALUE 3

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

NEXT ANALYSIS ANALYSIS VARIABLE VARIABLE VARIABLE

DEFINITION OPTION DEFINITION SUBOPTION VALUE 1 VALUE 2 VALUE 3

(A4) (2004) (A4) (5A4,F15.5) (5A4,F15.5)
(5A4,F15.5)

END

(A4)

There are seven taining the vehicle input files are

input title

files for the and the names

batch linearizer,. The first of the six data input files.

is a file conThe six data

1. project title,

2. mass and geometry properties,

3. state, control, and observation vectors,

36

4, trim parameter limits,
5. trim definition cases, and
6. additional surfaces to be set.
This is the order in which they must be defined. read before the fifth file,

However, the sixth file will be

Case Title, File Selection Information, and Project Title
There are two title records that need to be specified for LINEAR: a title for the individual test cases (case title) being analyzed and the name of the specific vehicle (project title). These records are read with a 20A4 format and are separated by a file selection record. Both titles appear on each page of the line printer output file. The file selection record contains the names of the files from which the data are read, The data contained on the files specified by the six fields of the file selection record are shown in table 2. The input file names are written in character strings 10 columns long, and if not specified, the data are assumed to be on the same file as the first title record and the file selection record. The local name of the file containing these first two records must be attached to FORTRAN device unit 1.

TABLE 2. — DEFINITION OF FILES SPECIFIED IN FILE SELECTION RECORD

Fi+ee ld nanobser 1

Data contained on selected file
Project title

2

Geometry and mass data

3

State, control, and

observation variable definitions

4

Trim parameter

specifications

5

Test case specification

6

Additional surface

definitions

Geometry and Mass Data
The geometry and mass data file consists of four records that can either follow the title and file selection records on unit 1 or be stored on a separate file

37

rsdarlthepeeoniemcfcgsocaialrinitedfeni~.sidiooecnntfg-bayocoTtffhorteneatcfhfctoeiihkgregvedeuosfrm.iaraeraerttsnreirtgcooyeodnryodnfasfanocmdraitnhicesmthabeesarisenpfdvvueeeedtrfhhaieitinanccefcllideeelreecmwttoooiinrtdtahdelmdsleeesrcedrdbrseeeeipifcfbnoieiegrncndetitatihnittoenoahnelsywitzhhngeeifedocnoo.hrmcemettanhtrttehTyieeh,orenfivlemofefhaipsirrscsegsltsreeealpnverrticoenettcpadyimoe,eorrdntiinaaennsoddf,the

stqruiabruTethdie,on.sethceonTdhweinagnssdepcaontnhdirbd,recironercdourndidstesfidneoeffsinleetnhegtthhe,winvgethheipcllameneafnogremoameeratorredyay,naSm,miacsisn,

and units
chord

mass disof length
of the wing,

crictsnurrprpmlcnaehoon,uoeeeroatuauimmisfopcsstttpnepedarstesiitureurhte-no.nitecceifslcecettnoeeaLsdoscnununfIUtnestgNhsivTetitEeteowhasilonoAhsniirfrfeRgsst-inaythaSgsehLtiptiobutethrInfehnnltfhpeeoerNoheoeptmEedru,cllAytrLaeieRvs(nitInetnesexewaahoN,tergihdcrmEeetotioeoiArhhdancrmcRyr,iddyla,.neSaenkpcuioetapnofbhruamu.asrernsneniqcddodiifActuetgnutnehhanrApgytseertr)aodzis.teeevserhenrndiiefeoueabo.ctftiffosseyinedecTo1vseyrrohf2etemseer-haAwsaonTe4rsfiehsacgeutecxserxftchlpti-ahoffettpsielvuohovialoteiererlxoornihttlnmlifn.eiedohyasgofrssdifvttwsdp:fs,ewiheesphenvtfwceiliTigaaotrtcahtgrreetselhchoiwrccqeteaaooaustormbrdolfatvsrdledyrcaapeenioneutrnefstttadleicphhsad.mamaeieeneoittebftcdffhchnilyteeitecooanfesneDtTenhEshrsinstveteLfooouLeteXdrObhhi,renGmyritcftftCanoocohohoGlalueeuruotaDemtrfhrrEbe,iieteteLrdncrahvhYcoegeeie,tdfrLprhgWiayOaeioherorrunGvcrdtieCeaenlCailfgy-cCncfstGmteodhoAnhoesyirtLraarrcC,dmmdne,fdeDmidnuiEdatoceceeoLrmdirdeliZesendeedneffabtrtysoieatibpesucnrnonwoosfnereiditeisasilnynanturbaltbacttsbeelnaeee-ndpbdoatchlfxhweoeafiefii--stn h1

NO CG CORRECTIONS BY LINEAR

CCALC WILL CALCULATE CG CORRECTIONS

FORCE AND MOMENT CORRECTIONS CALCULATED IN CCALC daTrehtagentsgraeeecTekhse,pfoatrorfaimnbweaehltiecurhssre,edctohreOfdopriunsofetrra-nitdsmhumisipOnpmglagixeeeodtmheeStpnreoaycniilrficaynrneadafrtthmeasmasoemdrieondlida.mytaunmamTsihecatensdemdoemdfapeixalniremasumm(eCtCteAhvrLeasCl)uaeansrgiesleof-vionaflai-nudagn.tlitetasckofof

State, Control, and Observation Variable Definitions vatstthahreieearoltnieeacupTbtshrolieefeeforvi.nlitseohdtueaerstfedleice,lynofiriindtne.dcieaeodornsnitczrrTbeoiyhsdlbee,etdthnseuyiasmsnsbdteseetedrmsceootbnesooadrffremerivorrnafeedetccediooftorrihnddenbsseydivnaiotnnphrieunitatFbhOnrleuRefemTcisbRloseeArtrNdasttosee,oufnlbtieehtcacsttuouicns1ohteendrioortvlha,diernearirfeaiatbnnhfldieosetltsilooooorubnwetsdsdpeeurfttivhoonaennetdfioatolhranesmbstuyelpfa-io-lfe

The states
defined in the definitions.

to be used in the output first set of records in The first record of this

formulation of the linearized system are the state, control, and observation variable set defines the number of states to be used

38

(NUMSAT), the formulation of the output equation (EQUAT), and whether the sional stability derivatives with respect to angle of attack and angle of are to be output in units of reciprocal radians or degrees. The variable read using an A4 format and is tested against the following list:

nondimensideslip EQUAT is

NONSTANDARD
NON-STANDARD GENERALIZED EXTENDED

If EQUAT matches the first four characters put formulation of the state equation is

of any of the listed words,

the out-

Cx = A'x + Blu If EQUAT is read in as STANDARD or does not match the preceding list, then the default standard bilinear formulation of the state equation is assumed, and the output matrices are consistent with the equation

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 nondimensional stability derivatives with respect to angle of attack and angle of sideslip are printed in units of reciprocal degress 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 taMvWtiavchnhhnoaaanedednedrrcneiirlaaesvbtstbbmehthaDlleeearReentiVstseaIepitbNceinaolCtmrcclveteiiropaofehrniiimaist.bdieeansesednburtslmispieutevsgeisTuraehmncsideteoiictirnfotvgeioswdedeitbeddetarebthteishaoeevfcnufdarsouatiersnvDtdipeyaRhadvtarVveotIiewfsrNaolatsiCbrnaroseltnctxeeiiactvtlnttehhiagyeeepsrn,vttiaaeamronmvdebiuueaDsflatsmroRebprseiV.iluaraInaetgibNrdecCleeaaerplfT(iphoicisenevsrhp(naemeipdtcnucewcuiiklirrnaxevaftiildcetittucasiestrulCoseb.lcnfnaauoawttrlriaiieatooItnthffonvgMiaatnclhatgtrheihhdeteedhesniepsautntemchnuplyecreimriAtit'binbseneaiCertasgdateranoaoridifnnizndoinevstncdeHftt'rhilGeleotremchesmmt.eeihcyaasntesotttyttgrraL..eniitiTsmeiccnhtezeeeasesat))dre, 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 put model. to be used
(CONTROL),

those The

defining the first record

variables to be used in of this set defines the

the control vector of the outnumber of control parameters

(NUMSUR). The their location

remaining records define the (LOCCNT) in the common block

names of /CONTRL/

these variables (see the User-

Supplied (CONVR),

Subroutines section), the units and the increments (CNTINC) to

associated be used with

with these control these variables in

variables determining

the B' and F' matrices.

39

wpTihahnepernurtteiBfceuoucsbrliayeanu,grslethyetchoLenIwuhNsseEeiunrAsnRttertanihrhcmeeaymseduussineenordandtaheafelfotyratsueuilmssstpeutbsspocfeooqintnutocterononetltsdrteoaflivbidalnreiiinvstatahibirolifneaicbcolanoenttpairtmooiennlosa,nm.evosfatrheiitashbecloeenxcttorrnioetnlmrioetlliyvaalrviaiarmcbipolaonebrdltieatsnni.atom,ness

The CONVR field in the control variable records is

trol variables are given in degrees or radians. and is compared to the following list:

CONVR

used to is read

specify if the conusing an A4 format

DEGREES DGR RADIANS
RAD

If CONVR agrees is assumed that agrees with the assumed that the assumed if CONVR that the control control variable

with the first four characters of either of the first the control variable is specified in units of degrees. first four characters of either of the last two listed

two names, If CONVR
names, it

it is

control variable is specified does not agree with any of the variable is specified in units is converted to degrees before

in units of radians. No units are listed names. When it is assumed of radians, the initial value of the being written to the printer file.

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

control that the sion is

surface when the units of CNTINC

B' and F' matrices are being calculated. agree with those used for the surface, and

attempted on these increments.

If CNTINC is not specified

It no
for

is assumed unit convera particular

surface, a default value of 0.001 is used.

trvlst(heeeiiMetcocnEnteoAsoarSodTrrdUhue,seRtfpEpiemMunroioftEetdnibNseansTliae)mtnlrohtdvihaseeanastltdoneidtnsoeufntttmohhaoreebfneyserpqafsreueoppcacoresoitfosmecfirinuiyodtslfonsoiaib,rotcsntiaehioternnanivonioatdnvntthfaoieforlooirobnoamfcstsbaaehtltterapveietvaasdoearron,tiauiamtatobeptocn(lutoPetenAbstretRhspeAraeMorq)il(aun,vaNmaceUsetlhMsttiuYiohaeoVdcacnnrCeltdi)sdea.tmoe(acbidEtynoesQnUeTtArhbbwteTeevehir)eat.htrufiseioooefrqbntdsushTtiehegrerrivevanadroavrebrtiicsettioaehmytorbeora.ndlivenacoitouvonifmtedgoppcenuutftttoiherniis-

The variable is compared with state equation.

used to specify the formulation of the observation equation (EQUAT) the same list of names used to determine the formulation of the If it is determined that the generalized formulation is desired,

the observation equation

y = H'x + Gxe + F'u

is used.

Otherwise, the standard formulation is assumed, and the form of the obser-

vation equation used is

y = Hx + Fu

The records defining the observation variables to be used in the output for-

mulation of the linear model contain a (MEASUREMENT) and three fields (PARAM)

variable defining,

that includes the when appropriate,

parameter name the location of

40

the sensor relative to the vehicle center of gravity.

The parameter name is com-

pared 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 included in the formulation of the output observation vector. If the param-

eter 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 length.

If the selected observation

variable 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 characteristic 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 4,, Sa, 5;, and

Sth, representing the longitudinal, lateral, directional, and thrust trim parame-

ters, respectively.

The units associated with these parameters are defined by the

implementation of UCNTRL.

Additional Surface Specification

The first record of this 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.

Only the

controls are defined in the additional surface specification records; actual control

4)

positions are defined in the analysis point definition records.

Because there may be

no additional controls, these secondary trim parameter specification records may not

be present. If such controls are defined, the records defining them will have the

format specified in table 1. The control variable name (ADDITIONAL SURFACE), loca-

tion (LOCCNT) in the common block /CONTRL/, and the units associated with this con-

trol variable (CONVR) are specified for each additional control.

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 definition identifiers described in appendix E. The second record of a test case specification set, defining an analysis point definition suboption (ANALYSIS POINT DEPFINI-
TION 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

initial altitude must be specified before Mach number if the correct initial veloc-

ity 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 variable will be accepted. Trim parameters also can be set in these

records.

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

variables has little effect.

However, for some of the trim options defined in the

Analysis 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 aerodynamic and engine model are varied.

OUTPUT FILES

There are three output files from LINEAR: a general-purpose analysis file, a printer file containing the calculated case conditions and the state-space matrices for each case, and a printer file containing the calculated case conditions only.

The general-purpose analysis file contains the title for the cases being ana-

lyzed; 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

42

matrices are printed only if the user has selected an appropriate the state and observation equations. The output for this file is FORTRAN device unit 15. An example of a general-purpose analysis sented in appendix G, corresponding to the format of table 3.

formulation of assigned to file is pre-

TABLE 3. — 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 units Matrix name A matrix Matrix name B matrix
Matrix name D matrix Matrix name
C matrix (if general form chosen)
Matrix name
H matrix

Format (4A20) (4A20) (//,64X,13) (17X,12,22X,12, 22X,13) (36X, 2A4) (36X, 2A4)
(///7)
(1X, 5A4, 3X,B12.6, 2X,A20)
(////)
(1X, 5A4, 3X,E12.6, 2X,A20)
(////)
(1X, 5A4,17X,A20)
(////)
(1X, 5A4,3X,E12.6,2X,A20) (//,A8,/) (5(£13.6)) (//,A8,/) (5(E13.6))
(//,A8,/) (5(B13.6)) (//,A8,/)
(5(E£13.6))
(//,A8,/)
(5(E13.6))

TABLE 3. — Concluded.

Matrix name

Variable

Fo matrix

Matrix name

E matrix

Matrix name

G matrix (if general form chosen)

Format
(//,A8,/) (5(E13.6)) (//,A8,/) (5(E13.6)) (//,A8,/)
(5(B13.6))

The strings

titles are written on the first two records and are specified in LINEAR as the title of

of the file the vehicle

in 80-character and the title for

the cases.
(999 cases

The next
maximum).

record

contains

the

number

of

the

case

as defined

in LINEAR

on

The number of states,
the following record.

controls, and outputs used to define each case are written
The formulation of the state and observation equations

are listed next, followed by the names and
interaction variables, and outputs. These describe each case.

values
values

of the states, controls, dynamic
are followed by the matrices that

The titles records appear only at the beginning of the file; all other records

are duplicated for each subsequent case calculated in LINEAR. The matrices are

written row-wise, five columns at a time, as illustrated
which shows a system containing 7 states, 3 controls, and general state equation and standard observation equation.

in
11

the following
outputs using

tabulation,
the

Size of matrix
A=[{7x7]
B= [7x 3] D= [7x 6]
c= [7x 7]
H = [11 x 7]
Fo = {11 x 3] E = [11 x 6]

Output formulation
A=([{7x 5] { 7 x 2]
B= [7x 3] D= [7x 5]
[7x1] c= [7x 5]
{7x 2] H = [11 x 5]
[11 x 2] F = [11 x 3]
E= [11 x 5] {11 x 1]

44

The second output file, which is assigned to FORTRAN device unit 3, contains the values calculated in LINEAR describing each case. The first section of this file contains a listing of the input data defining the aircraft's geometry and mass properties, variable names defining the state-space model, and various control surface limits characteristic of the given aircraft. Appendix H presents an example of this
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.

e

e

e

e

If trim was not achieved, p, q, r, V, 8, and a (calculated from the

equations of motion) and the force and moment coefficients are printed. in the geometry and mass properties are also printed.

Changes

The third section of this output file contains the
control derivatives for the trim condition calculated. aircraft at the given flight condition is also printed.

nondimensional stability and The static margin of the

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

ces for the given flight condition.

The formulation of the state equations and only

the terms of the matrices chosen by the user to define the model are printed.

A

subset of this output file containing only the trim conditions is assigned to

FORTRAN device unit 2.

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 inelude

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 H presents an example of this file.

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, UCNTRL, and

MASGEO.

The first two subroutines constitute 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. The

subroutine MASGEO allows the user to define the mass vehicle as a function of flight condition or control routines is illustrated in figure 3, which shows the

and geometry properties of the setting. The use of these subprogram flow and the interaction

of LINEAR with the user-supplied subroutines.

These subroutines are described in

detail in the following sections. Examples of these subroutines are given in appen-

dix I.

45

Main program

Inivteieaplizatigt on

| 1

v
Read aircraft-specific input data files

¥ Read in data for next
analysis point

oppetoin on rrenquested

Usseurb-rsouuptpi.nleised

a

A(eArDoAmToIdNe)l

Interface common block names
ICONTRLI,/CTPARM/

cor earing (UCNTRL)

Determine analysis trim point conditions

i ~<«!—
<—,

ICONTRL/,JENGSTF/

>

ICONTRL/,/CGSHFT/

Thrust effector model
(IFENGN)

Determine linear model
Y Select matrix elements
for output
OutputY data

! f ICGSHFCTO/N,T/RCLLCOUT/,
IDATAIN/,/DRVOUT/

|> Aerody(nCaCmAiLcC” ) model

/OBSERV/,/CGSHFT/,

Mass and geometry

IDATAJIEN/N,G/SCTOFN/TRL/, >

(M mAoSGEO)

Run

Yes

another case

2

Figure 3. Program flow diagram showing communication with user-supplied subroutines.

Aerodynamic Model

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 and can also be used to define aerodynamic data. CCALC is the subroutine

that uses these aerodynamic data, the state variables, and the surface positions

to determine the aerodynamic coefficients.

Either routine may call other subrou-~

tines 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 aero-

dynamic model.

46

Externally, ADATIN has is called only once when the program must provide ADATIN modation is necessary. The through named common blocks

no interface to the program LINEAR.

The subroutine

aerodynamic data are input or defined.

The calling

with the input devices it requires, but no other accom-

aerodynamic data are communicated from ADATIN to CCALC

that occur in only these two routines.

acnpfabaartocmoeeuegddtrT.ahpmeow.chsoiimHictmonhoiwtCnoeeCnvrAsaefL,brCal,gcoeecnaiksnetsdrhbeaeeltxtf-wehoipeacrneutcutnreteperaodfrCsaaenCecdAewLhuCesmtenooideosmalvenednsrttotcaatnnhpcnedaeoaswebrsefcdfa,ailebsclurtiiiaoaneltdngtnedy.tnsaptvmrhaiiorTbcsgiheriatasbwfcmleeoeeaesintf,niusftrieCecrCsaifAioearLmpcnCeretwodshvaaaitnctaddoaenrsesmptiohasrerarteaesmqcfeuarctiloaoerlfmmreipesdnlw,sgioe-rvk(efsrouarrlexample, once per frame for a real-time simulation).

bcsluoorncftkaaTscih.eeninmpgaoTishneitstheietornacsano.esmrfmoeodrnyTnhaebomflioctcdrkaastnfasofrcecoirennttoaoaifnndthdeamtthoaemseunbsftrtrooamtucetoieCnfCveAfaLirCciCiaCebAinlLsteCss,.thisroauiTtgrhhheroduadagtehatnaaimpsleiasdxramnocefaotmmeemrtdohsne,sceobmlamoncodconkmmon blocks follow.

The respect

common block /DRVOUT/ contains the state variables and their

to time.

The structure of this common block is as follows:

derivatives

with

COMMON /DRVOUT/ T

,

P

7 ©

, R

'

Vv

, ALP , BTA ,

PDOT

, QDOT

, RDOT

,

VDOT

, ALPDOT, BTADOT,

THADOT, PSIDOT, PHIDOT,

HDOT

, XDOT

, YDOT

The body axis rates p, ity is represented by of sideslip (BTA), and respectively) are also

q, and r appear as P, Q, and R, respectively. the variable V, altitude by H; angle of attack
their derivatives with respect to time (ALPDOT contained within this common block.

Total veloc-
(ALP), angle and BTADOT,

the

The common block /SIMOUT/ contains the main air

function generation subroutine.

The variables

data parameters required in this common block are

for

47

COMMON /SIMOUT/ AMCH, QBAR, GMA, DEL,

UB

, VB, WB '

Mach number and bolized by AMCH
included as UB,

VEAS, VCAS

dynamic pressure are the and QBAR, respectively. VB, and WB, respectively.

first entries The body axis

in the common velocities u,

block, v, and

symw are

definTihteio/nCONoTf RLe/achcomomf onthebloeclkemecnotnstaiunssed mined by the implementer of that model, is as follows:

the surface position information. for a particular aerodynamic model The structure of the common block

The exact is deter/CONTRL/

COMMON /CONTRL/ DC (30)

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

COMMON /DATAIN/ S

, B-

, CBAR, AMSS,

AIX , AIY , AIZ , AIXZ,

AIXY, AIYZ, AIXE

The first wingspan, by AMSS.

three variables in the common block, S, B, and mean aerodynamic chord, respectively.

and The

CBAR, represent vehicle mass is

wing area, represented

The information on aerodynamic data with /CGSHFT/ common block:

the displacement of the respect to the aircraft

aerodynamic reference point of center of gravity is contained

the in the

COMMON /CGSHFT/ DELX, DELY, DELZ

The displacements are defined along the vehicle representing the displacements in the xX, y, and

body axis with DELX, DELY, Zz axes, respectively.

and

DELZ

The common block normal accelerometer

/SIMACC/ output at

contains the the center

accelerations, accelerometer of gravity of the aircraft.

outputs,

and

COMMON /SIMACC/ AX , AY , AZ,

ANX, ANY, ANZ,

AN

The output common block /CLCOUT/ namic moment and force coefficients:

contains

the

variables

representing

the

aerody-

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

48

The variables CL, CM, and CN are the symbols for the coefficients of rolling moment Ce, pitching moment C,, and yawing moment C,, 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

The program LINEAR attempts to trim the given condition using control inputs

similar to those of a pilot:

longitudinal stick, lateral stick, rudder, and throt-

tle. The UCNTRL subroutine utilizes these trim output control values to calculate

actual surface deflections and power level angles for the given aircraft (fig. 2).

The location of each surface and power level 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 (see Trim

Parameter Specification in the Input Files section) 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

The output from UCNTRL is through the common block /CONTRL/ described previously in

the Aerodynamic Model section.

The variables DES, DAS, DRS, and THRSTX are used to

trim the longitudinal, 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/. 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 in the
Mass and Geometry Model section of this report.

Engine Model

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),

TVANXY(4),

ELX

(4),

TLOCAT(4,3), TVANXZ (4), AMSENG (4),

XYANGL(4), DXTHRS(4), ENGOMG(4)

XZANGL(4),

The variables in this common block correspond to thrust (THRUST); the 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; 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; the rotational inertia of

49

the engine (EIX); mass (AMSENG); and the rotational velocity of the engine (ENGOMG). The variables are all dimensioned to accommodate up to four engines.

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 to the aerodynamic model

the appropriate (CCALC).

parameters

from the engine model

(IFENGN)

Mass and Geometry Model

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 RETURN and END statements.

MASGEO

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

sweep configurations) 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

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

block, and inertia changes are passed in the /DATAIN/ common block.

Care must be taken when using the subroutine MASGEO in combination with selecting an observation vector that contains measurements away from the center of gravity. If measurements are desired at a fixed location on the vehicle, such as a sensor platform or nose boom, the moment arm to the new center-of-gravity location must be recomputed as a result of the center-of-gravity shift for accurate results.
This can be accomplished by implementing the moment arm calculations in one of the
user subroutines and passing the new moment arm values through the /OBSERV/ common block:

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

The common block /OBSERV/ allows the user to access all the observation vari-

ables during trim as well as to pass parameters associated with the observations

back to LINEAR. The common block /OBSERV/ contains two single-precision vectors:

OBVEC(120), and PARAM(120,6).

A list of the available observations and parameters

is given in table 4. Access to the observation variables allows the user to imple-

ment trim strategies that are functions of observations, such as gain schedules and

surface management schemes.

The parameters associated with the observations are

used primarily to define the moment arm from the center of gravity to the point at

which the observation is to be made. If the user subroutine MASGEO is used to change the center-of-gravity location and observations are desired at fixed loca-

tions on the vehicle, then the moment arm from the new center-of-gravity location to

the fixed sensor location ((x, y, z), in feet) must be computed in one of the user

subroutines and passed back in the first three elements of the PARAM vector asso-

ciated with the desired observation (PARAM(n = 1 to 3), where n is the number of

the desired observation).

Additional information on observations and parameters

can be found in the State, Control, and Observation Variable Definitions section.

50

TABLE

4. -- OBSERVATION

VARIABLES USING

AVAILABLE THE OBVEC

IN THE USER-SUPPLIED ARRAY

SUBROUTINES

LocaitnioOnBVE(Cindex)
1 2
3 4 5
6 7
8
9 10 11 12
13
14
15 16 17
18 19 20 21 22 23 24
25 26
27
28
29
30
31
32
33

Variable

State Variables

Roll rate Pitch rate
Yaw rate
Velocity
Angle of attack
Angle of sideslip Pitch attitude
Heading
Roll attitude Altitude Displacement north Displacement east

Derivatives of state variables

Roll acceleration
Pitch acceleration
Yaw acceleration
Velocity rate Angle-of-attack rate
Angle-of-sideslip rate Pitch attitude rate Heading rate Roll attitude rate Altitude rate Velocity north Velocity east

Accelerations

x body axis acceleration y body axis acceleration
z body axis acceleration
x body axis accelerometer center of gravity
y body axis accelerometer
center of gravity
z body axis accelerometer
center of gravity Normal acceleration
x body axis accelerometer
center of gravity y body axis accelerometer
center of gravity

at vehicle at vehicle
at vehicle
not at vehicle
not at vehicle

51

LocaitnioOnBVE(Cindex)
34 99 98
91 37 103 35 36 56 55 57 58 59 60 92 93
39 38 40 43
46 47
94 95 96 97
52 53 54

TABLE 4. — Continued.

Variable

Accelerations (continued)

z body axis accelerometer not at vehicle center of gravity
Normal accelerometer not at vehicle center of gravity
Load factor

Air data parameters
Speed of sound Reynolds number Reynolds number per unit length Mach number Dynamic pressure Impact pressure Ambient pressure Impact/ambient pressure ratio Total pressure Temperature Total temperature Equivalent airspeed Calibrated airspeed

Flightpath-related parameters
Flightpath angle Flightpath acceleration Flightpath angle rate Scaled altitude rate

Energy~related terms

Specific power Specific energy

Force parameters
Lift force Drag force Normal force Axial force

Body axis parameters

x body y body Z body

axis axis axis

velocity velocity velocity

TABLE 4. — Concluded.

LocaitnioOnBVE(Cindex)

Variable

Body axis parameters (continued)

100

Rate of change of velocity

in x body axis

101

Rate of change of velocity

in y body axis

102

Rate of change of velocity

in z body axis

Miscellaneous measurements not at vehicle center of gravity

44
45 41 42
48 49 50 51
61 to 90
104 105 106 107

Angle of attack not at vehicle center
of gravity
Angle of sideslip not at vehicle center of gravity
Altitude instrument not at vehicle center of gravity
Altitude rate instrument not at vehicle
center of gravity

Other miscellaneous parameters

Vehicle total angular momentum Stability axis roll rate Stability axis pitch rate Stability axis yaw rate

Control

surface parameters Control surfaces

DC(1)

to DC(30)

Trim parameters
Longitudinal trim parameter Lateral trim parameter Directional trim parameter Thrust trim parameter

CONCLUDING REMARKS
ddlaoewcfuinmTdiehennesgtiegdFnO.tRheTtRoAoTnNlhoinslptiornoregedarperaormritveeqLIudNailEtsiAicnRouenassrsweassmfordodLmeeIlNvsEewAlhRoifpcoehrdfroatmhiteorctrhplaerifontvepaierdrsestpameboacidtelfliilvteeyxiisbaolfndeae,lrayisvppieoosdwteernaaftnnuiddla,ldcoensautcnsdrreori,lb53

VsLiiMnnIuSgNpEptAlhotReehpmeeerainasptitpnietannelagdsrrioefxasecysgdesie.stvseecmnrtioAbiiesnmdui.sceairprnpo-c.fslEiuuxcdpaJhep.med)lpileewdsliitshtosifunbtgrhtiohseuotf irunesetpheseorr-tsp.aunrpdopglrii(anRempeduvtifsosirufobinrlsaeosuV.tAtiXonestT1hh1ee/7a5rmo0eiuctprwpuoirtftehiscehfnrettohemed

pmTaceaihnolqoednsiumeonapttlpgTiehoyctoteihraoneennopsltsapybcracloooaoepsgfnstirtprcraeawmoomfcjholeiiteccfifeLhtyofxIvonieNtarncErrygttAai,hasRacin)sbtdtlihsenneesuaaglnlmidenesucertaslataineiiridtcrnnnea-ecglaslrslruuyeayidpnsmazidptaenleidtgdminrceeiiotdnmnemgettronhrmdegonoeisipsollennttneeialsotiveinensf,aeefrfaaeifdvrcaeeaatlbcnfrisltdiinasenaeseebarddilrnoie(drsosyreu.itsnccshyhtaesbimytdnieegaclmstisenpreetmnmamcheoorietiddnfeeellypds.tiryhnosrgugtuseresiiLtmantIa,mghnNeEmrAoattnRdonooearnbllqyld.iuyiseesnt,ieesra-r

poafcnaoobdnrrsttmeriurTdoclvheluaeaf.lttiaiinrosoiynntssFitumoeroenmtdqohefuela.mortfo,bidooetnthlhTstehh.eusdstes,attptaerTetrohteemgh,rieaannpmedrcodorogdnpoertrbbraroysmovelir,oLdvfIheaaNstsEatiAnhoRdenbtheeeoscnybeosnqsftsueldeiareemstvxstiaiisotgbmniniosoeld.nodieftlymvtoaaistrofripiarccbooaelvmlseiplsdloewefitotneroaglnybbeoaetaluhstnuydeseresrdntsaaettluiienesvecertaainodn

asfeasfombcofevaqiueiorlcaurtplmeeiarcipmfclittrlolttLacyinaihi,Isb.ofeegNclndthEhesmitAe,nawlRdsiihsBsutnyauoetlhefvbxahihaoierlcadngrunlosegmietgopuo,snawttttemsesiirorrpmenodiaoruvwehneblsoleho,aluigptrinvstyraczaie,pihalcrpiea.moelnrlanegmlnscafioar,ometiwelmoansTstybdhthdusieeiuruptitnrffshhasueeieaterndussccrtlnueyaiiirps.iptqmcfrrwnituiuaoiocohetcltbglanaeehalrtu,fdTerasfhmltmieerytdmochiaepntetkhmfotcmieemoiasoininnssmoglntifipabsgitebrottbigiareatseotltshicneuiao,granssatnliinpupntivqnaipsueeugblmftscdaelbiii-nofeeelcdvnordiartewotanrcifmyamotttox,aolothofinok,btrsgilr.oinfiieonaigqamnbagmtsyumnshpetdieeelT.hnn-rheoeeagwevsmfioileaefrbnaitnecssgfsnitcresoeoefttaruranesdfvrmmra.vtapuitatetluzqtitiarahuhoioeettanrTinsniinghosocoetuinaniglnveithysaeaaoisnfrbaeopsdinfisrpafdalholsbiageystultsrxmhuseye-eiares--mrs-to

This used and of LINEAR The input required

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

the are

methods employed discussed from

to both

implement the theoretical

program. The trimming capabilities and implementation perspectives.

and output for LINEAR

files are described in detail. The are discussed, and sample subroutines

user-supplied subroutines are presented.

National Aeronautics and Space Administration Ames Research Center Dryden Flight Research Facility
Edwards, California, March 6, 1985

54

APPENDIX A:

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

aaaetrnhfeedefreocdTdmttyhehosenefmaiemnnpavietroecdseihnitcmdirlsoeeedffoenierlnreceeefetddnenhcertereaarettvdephoottifihhtcneioltsgeraasaeaevreafitorfttdoheyeydcwnythaniiaamacvreihmeercilocydtnyhorneteriafnemedfnciureooccernienelncnicrecniemeedfoaeemrpnreotpenio,tnnifsctnoetr.tchbteeyhpaotiaTftnnhhotdeur.ascc,etmnsoomnAwielhlannleicctnntreieaanmecrtgerohnioestdfaaayftelnirplacooytmidihieynecn-ttons namic model.
The total aerodynamic moment M acting at the vehicle center of gravity is defined as
M = Mar + Ar x F

where

is the total by subscript

Mar = [Lar aerodynamic moment acting at ar) of the vehicle,

Mar Nar! the aerodynamic

reference

point

(denoted

Ar = [Ax Ay Az]T

is the displacement of the aerodynamic reference point from the vehicle center of gravity, and

r= (x y az]?

igs the total aerodynamic force acting at the are total forces along the x, y (sideforce),

aerodynamic center, where and z body axes, Thus,

X,

Y,

and

Z

Lay + Ay 2 - Az Y

M = May + Az X - Ax Z Nar + Ax Y - dy X
ateerrmosdTyhenoafmtioctthealmofdoaereclreiondgyannadsmuibmcroomeumntotimneenctoeCfCafAcLitCcinigebnytsatdefdtiheenriinvvgeedhitchflereombocdetynhetearxuissecarn-fsourbpce pesleixepdirn esntsoeerndmlsinienoafr stability axis force coefficients:

cos @ + Cy, sin a)

i t

tad

1Qn i=) IQ

MeiQaen)h

W

K

Q n I iw)9l)

sin a - Cy, cos a)

U l

N

Substituting equation and

these equations into the applying the definitions

definition of the total aerodynamic of the total aerodynamic moments,

moment

55

Lar = GSbC p ar
Mar = qScCn

Nar = qSbCy

expressions for total aerodynamic moment of gravity can be derived as follows:

coefficients

corrected

to the vehicle

center

Cy -= Cee + ’57Y ((c-Cpp sisinn aa -- Cy, cos a) _ ba2 CY

Cy

= Cmar

+t = (-Cp c

cos

a + Cy

sin

a)

~ =Ax Cc

(-Cp

si; n

a - Cy

cos

a)

Ch = Char +=A

Cy

-

A —

(-Cp cos a + Cy, sin a)

These calculations are normally performed within LINEAR

CGCALC.

However,

the user-supplied

if the user aerodynamic

selects, the calculation model, CCALC.

can

in be

the subroutine performed within

56

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 some point Ar from the center of gravity of the vehicle,

thrust of an engine, Fp,
of the engine acting at as shown in figure 4.

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, Fp. can be defined as
Fp; = [Fex; Ppy, Foz, | T

where Foy. i ! Foy. i ! and Fo. i are the components of axes, respectively. From figures 5 and 6 it can tionships hold:

thrust in the x, be seen that the

y, and z body following rela-

aon = | Fp, | cos €; cos &j

Fey; =| Fp, | cos €; sin &}

Foo. = - | Fp, | sin €;

57

where | Fp, | represents the magnitude of the thrust due to the ith engine, €; the angle from the thrust axis of the engine to the x-y body axis plane, and E; the angle from the projection of Fo: onto the x-y body axis planeto the x body axis.
y

bo
4 ‘<<
A |
5

F
Po
Z -
x
FD,

TI =
a
f |

; T ' '

(- |. ----74---------

; “

y

“

Ay Aircraft

“oo

'

center

“__[_| «“2z m4

/ é

Engine center

gy —L| y i“
_ —? an i,

of gravity
|
Az

of gravity —~|

m”

l

Figure 5. Orientation of the engines in the x-y and x-z body axis planes.

Figure 6. Detailed definition of engine location and orientation parameters.

Denoting the point at which the thrust from the ith engine acts as Arj, this offset vector can be defined as
Ar; = [Ax; Ay; Az,J]T
where Ax;, Ayj, and Azj 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, ATo., is then given by

Ato. = Ar; x Fp.

Thus,

Ay F Pz; - Ag F Py;

Ato, =

|Az Foy. ~ Ax Fo,

i

i

L

Xe 1

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

58

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.

|
Thrust point + gy y ™?

Ba"

4

g

YE

\

Y

=Tp Ze

= Engine center of gravity

Xp XTp

Figure 7. Detailed definition of thrust-vectoring parameters.

The thrust is assummed to act at -Axp in the local (engine) x axis, with the engine

center of gravity being the also assumed to be vectored

origin of at angles

this local coordinate n; and 04 relative to

system. The thrust is the local coordinate

axes, with n; being the angle from the thrust vector to the engine x-y plane and 0;

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

local x axis. Thus, letting Foy. Poy, ! and Foe. represent the x, y, and z thrust

i

1

L

components in the local engine coordinate system, respectively, these terms can be defined in terms of the total thrust for the ith engine, Fp. and the angles nj

and Cj as

Fox, = | Fp, | cos nj cos i

Foy, =| Fp, | cos nj sin oi

Foe. = - | Fp, | sin ny

59

where
roi" = [Fotx, "y' y *x1 ,] T

To transform this equation the transformation matrix

from

the

ith engine

axis

system

to

the body

axis

system,

is used.

cos €; cos §j cos €; cos &;
-sin €;

“sin &y cos &j
0

sin €j cos & sin €; cos &j
cos €j

The resultant force in body axis coordinates is

"Px;

[cos €; cos €; -sin &

sin €j cos &;]

Px;

Fo; == Foy,

= cos €;. siotn & .

cos €:L

sin €;i cos &;L

FYPy;

so that

PFei}

-sin €;

)

cos €j | | {rpPei]

"Px; = | Fp, | (cos Nj cos 4 cos €j cos &j - cos ny sin fj sin Ey
- sin nj sin €j cos &)

Foy; = | Fp, | {cos nj cos Sj cos €j sin &j + cos nj sin ty cos &j
- sin ny sin €3 cos &;)

Foz; = | Fp, | (-cos nj cos fj sin €j - sin ny cos €;)

The moment arm through which the vectored thrust acts is

Ax; - Axp. cos €; cos Ej

Ar; =|

Ay; - Axp; cos €j sin &j
Agi + Ax; sin €;

and the total torque due to thrust vectoring is

to= i}=n Ato; = iD=nYt (arj x AFp,)

60

ent

The with

engine inertia tensor the vehicle body axis

must be system.

defined This

in an is done

axis in

system oriented two steps. These

consiststeps

involve rotating the engine inertia tensor the aircraft body axis system. First, the

into a coordinate system orthogonal to ith engine inertia tensor is rotated

through an ented with

angle €; about the its local x-y body

local y axis so that the new inertia tensor is axis plane parallel to the X-y body axis plane

oriof the

vehicle.

The second step requires a rotation through an angle &; about the local

z axis so that the local axes of the vehicle. As
(1965), this rotation is tensor Tey such that

x, y, and z axes are orthogonal to the x, y, and z body determined by Gainer and Sherwood (1972) and Thelander
a similarity transformation that yields a new inertia

Tey = REiReibtaxRteeia,RE;

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

described rotations through € and €, respectively.

These matrices are given as

cos €y

0

sin €j

Roent. =

6)

1

0

-sin €j

0

cos €4

so that Because

cos §j

-sin &j

6)

RE; =

sin &j

cos §i

e)

6)

¢)

6)

Re Rey =

cos §; cos ej sin €; cos €j
-sin €j

-sin &{ cos §jy
6)

cos 4 sin ej sin €| sin €j
cos €j

-1

T

ei 7 Rey

Re-;j = RET
and the matrices are unitary,
Re-i1_R-e-y1 = RegTReT; = (ReiRej) T

62

Therefore, I ey = I Xe

(Re,Re;)7 =

cos €; cos €4 -sin §j
cos €; sin €j

sin § cos €j cos §4
sin €| sin ej

cos? €4 cos? &j
cos? €;1 cos &;1 sin &.;1

cos? €4 cos & sin &j
sin? &;1 cos’ é;1

-sin €j 0
cos €j
-cos €; sin €; cos &j
-sin €;1 cos €;1 sin &;1

-cos €j, sin €4 cos Ej

-sin €j cos €j sin &j

sin2 €}

]

The angular momentum of the ith engine, he, can now be expressed as
hee,i == TejI%e:eWe;. = [ [hef,i, NeP:lo hee,i,| VF
with
nei, == Pei | Ixe, cos’ 2e.€; cos 2,ei]
+ de; [Ixe, i cos* €; cos & sin ei] + rei[-Ixe, cos €; sin €4 cos S|

hei,. = Pei . | txe; cos €;€; cos &j; s sin ei§&j

+ Fei | Txe, i sin? &; cos? ei | + Tei|-Txe, i sin €; cos &; sin ei] ei; =7 Pej [-Txe, cos €;. si;n €4. cos 2&55.]

ox[txe, 2m? 6] + de; [-Ixe; sin €; cos &j sin ei]

+ Ye. |/T

sin e,

[ahe;, ~ Rex,

Tg, =|rhe, - P ei,

i phe iy

qhe, id |

63

and and

the total moment induced by gyroscopic the rotating engine components is

interaction

of

the

vehicle

dynamics

n
tg= 21=1 ta

Engine torque using information These effects are the equations of linearized system

mompcatnaraidtolorvcniiugcdlyeearsfdtoo.ersdcboybpoaitscthheinaencfuarfsleeerycmsteisnsftraolamrpeoimtnhomtemoednedetlensegfdiinnaeinwtdiitmohoanirdneelainitndhngeclduessdruueibbdvrraootuudittioiirnnneeectolfyEIFNEGtNIhieGNnNE.

64

APPENDIX C: STATE VARIABLE NAMES RECOGNIZED BY LINEAR

This appendix lists the alphanumeric descriptors specifying state variables that

are recognized by LINEAR.

In the input file, the field containing these descriptors

uses a 54A 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 and 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

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

Angle of

rad

attack

Sideslip angle

rad

Pitch angle
Heading
Roll angle
Altitude Displacement
north Displacement
east

rad rad rad length length length

Symbol Pp q Yr Vv
a B
) wp cu) h x y

Adlepshcrainputmoerric

Pp

ROLL
Q

RATE

PITCH RATE R

YAW RATE Vv

VELOCITY

VEL
VTOT ALP ALPHA

ANGLE OF ATTACK BTA

BETA

SIDESLIP ANGLE OF
THA

ANGLE SIDESLIP

THETA PITCH
PSI

ATTITUDE

HEADING HEADING
PHI

ANGLE

ROLL ATTITUDE
BANK ANGLE
H ALTITUDE X

Y

65

APPENDIX D:

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 defined

observation in appendix

vector, the alphanumeric descriptor C. When control variables are to

must correspond to the names be included in the observation

vector these variables must be identified exactly as they were specified by the user.

The input floating-point ity. The input

file is formatted fields are used to
name specified by

5A4 with the alphanumeric data left-justified. The define sensor locations not at the center of gravthe user for an observation variable serves both

to identify the observation variable selected within the program tify observation variables on the printed output of LINEAR.

itself

and

to

iden-

An asterisk preceding the variable name indicates measurements at some point

other than defined in

the the

vehicle center of gravity. first three floating-point

The program LINEAR uses fields as definitions of

the the

quantities location

of the sensor with respect to the vehicle center of gravity.

The three parameters

define the x body, y body, and z body location, in that order, of the sensor.

These offsets from the vehicle center of gravity are defined in units of length.

Observation variable

Units

Symbol

Alphanumeric descriptor

Derivatives of state variables

Roll acceleration

rad/sec2

p

PDOT

ROLL ACCELERATION

Pitch acceleration

rad/sec2

Q e

QpoT PITCH ACCELERATION

Yaw acceleration Velocity rate

rad/sec2

length/sec2

V

N e

RDOT YAW ACCELERATION
VDOT
VELOCITY RATE

Angle-of-attack rate

rad/sec

R e

ALPDOT
ALPHA DOT ALPHADOT

Angle-of-sideslip rate

rad/sec

B

BTADOT

BETA DOT BETADOT

@WMe

Pitch attitude rate

rad/sec

THADOT THETA DOT

Heading rate

rad/sec

G e

PSIDOT PSI DOT

66

Observation variable

Units

Symbol

Alphanumeric descriptor

Derivatives of state variables (continued)

Roll attitude rate Altitude rate Velocity north

rad/sec

>

length/sec

h

length/sec

x

PHIDOT
PHI DOT

HDOT
ALTITUDE

RATE

XDOT

Velocity east
x body axis acceleration
y body axis acceleration

length/sec

y

Accelerations

g

ay

g

ay

z body axis

g

ag

acceleration

x body axis acceler-

g

anx

ometer at vehicle center of gravity

YDOT

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

AY Y-AXIS
Y AXIS

ACCELERATION 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 X-AXIS
X AXIS

ACCELEROMETER ACCELEROMETER

y body axis acceler-

g

ometer at vehicle

center of gravity

any

ANY

Y-AXIS ACCELEROMETER

Y AXIS ACCELEROMETER

Observation variable

Units

Symbol

Alphanumeric descriptor

Accelerations (continued)

Z body axis acceler-

g

ometer at vehicle

center of gravity

Normal acceleration

g

*x body axis accel-

g

erometer not at

vehicle center of

gravity

anz an
4nx,i

ANZ Z-AXIS 4 AXIS

ACCELEROMETER ACCELEROMETER

AN
NORMAL NORMAL
GS G'S

ACCELERATION ACCEL

AX,I ANX, I

*y body axis accel-

g

erometer not at

vehicle center of

gravity

any,i

AY,I ANY,I

*z body axis accel-

g

erometer not at

vehicle center of

gravity

4nz,i

AZ,1 ANZ,1I

*Normal accelerometer

g

not at vehicle

center of gravity

an,i

AN, I

Load factor

(Dimension-

n

less)

Air data parameters

N LOAD FACTOR

Speed of sound
@Reynolds number
Reynolds number per unit length

length/sec

a

(Dimension-—

Re

less

length-1

Re!

A SPEED OF SOUND

RE REYNOLDS

NUMBER

RE PRIME R/LENGTH R/PEET R/UNIT LENGTH

8Reynolds input by
however,

number is defined in terms of an arbitrary unit of length that

the user.

This length is input using the first floating-point

if no value is input, ¢ is used as the default value.

is field;

68

Observation variable

Units

Symbol

Alphanumeric descriptor

Flightpath-related parameters (continued)

Flightpath angle rate

rad/sec

Y e

GAMMA DOT

GAMMADOT

Vertical acceleration

Scaled rate

alti. tude

Specific energy
Specific power

Lift force
Drag force Normal force
Axial force
x body axis velocity

length/sec2

h

length/sec

e
h/57.3

Energy~related terms

length

Eg

length/sec

Pg

Force parameters

force

L

force

D

force

N

force

A

Body axis parameters

length/sec

u

VERTICAL ACCELERATION HDOTDOT H-DOT-DOT HDOT=DOT
H~DOT/57. 3 HDOT / 57.3

ES
E-SUB=S SPECIFIC

ENERGY

PS
P-SUB=-S
SPECIFIC SPECIFIC

POWER THRUST

LIFT
DRAG NORMAL FORCE
AXIAL FORCE

UB
X-BODY X BODY X-BODY X BODY U-BODY U BODY

AXIS AXIS AXIS AXIS

VELOCITY VELOCITY VEL VEL

Observation variable

Units

Symbol

Alphanumeric descriptor

Body axis parameters (continued)

y body axis velocity

length/sec

Vv

VB Y-BODY Y BODY
Y-BODY Y BODY V~-BODY
V BODY

AXIS AXIS
AXIS AXIS

VELOCITY VELOCITY
VEL VEL

z body axis velocity

length/sec

w

WB Z-BODY Z BODY
Z-BODY Z BODY W-BODY W-BODY

AXIS AXIS
AXIS AXIS

VELOCITY VELOCITY
VEL VEL

Rate of change of

length/sec2

a

velocity in x body

axis

Rate of change of

length/sec2

v

velocity in y body

axis

Rate of change of

length/sec2

w

velocity in z body

axis

UBDOT UB DOT
VBDOT VB DOT
WBDOT WB DOT

Miscellaneous measurements not at vehicle center of gravity

*Angle of attack not at vehicle center
of gravity

*Angle of
not at
center

sideslip vehicle
of gravity

*Altitude
not at
center

instrument
vehicle of gravity

rad
rad
length

ad

ALPHA, I

ALPHA INSTRUMENT

AOA INSTRUMENT

Bi

BETA, I

BETA INSTRUMENT

SIDESLIP INSTRUMENT

hoi

H,1I

ALTITUDE INSTRUMENT

*Altitude rate instru-

length/sec

hoi

HDOT, I

ment not at vehicle

center of gravity

71

Observation variable

Units

Symbol

Alphanumeric descriptor

Other miscellaneous parameters

Vehicle total angular

mass~length2/

T

momentum

sec2

Stability axis roll

rad/sec

Ps

rate

ANGULAR MOMENTUM ANG MOMENTUM
STAB AXIS ROLL RATE

Stability axis pitch rate

rad/sec

ds

STAB AXIS PITCH RATE

Stability axis yaw
rate

rad/sec

Ys

STAB AXIS YAW RATE

APPENDIX E: ANALYSIS POINT DEFINITION IDENTIFIERS

Analysis point definition options are selected using alphanumeric descriptors.
The first record read for each analysis case contains these descriptors. All these descriptors are read using a 5A4 format. The following table 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 descriptor

UNTRIMMED NO TRIM NONE
NOTRIM

STRAIGHT AND WINGS LEVEL LEVEL FLIGHT

LEVEL

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

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 spe-
cific power options has two suboptions associated with it. The suboptions are requested using alphanumeric descriptors read using an A4 format. These suboptions

73

are defined in the Analysis Point Definition section. The following table defines these suboptions and the alphanumeric descriptors associated with each,

Analysis point definition suboption
Straight-and-level
Alpha-trim

Alphanumeric descriptor

Mach-trim Pushover-pullup
Alpha-trim

Load-—factor-trim

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

ALP ALPH ALPHA
LOAD GS G'S AN
ALP ALPH ALPHA
LOAD GS G'S AN

74

APPENDIX F: EXAMPLE INPUT FILE

The following listing is an example of an input file to LINEAR. used with the example subroutines listed in appendix I to generate printer output files listed in appendixes G and H, respectively.

This file was the analysis and

LINEARIZER TEST AND DEMONSTRATION CASES

USER'S GUIDE

6. 080000E+02 4.280000E+01 1.595000E+-1 4.500000+E04

2.870000E+04 1.651000E+-5 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 input file is for a case called (record 1) LINEARIZER TEST AND DEMONSTRA-
TION CASES, and all input data are on logical device unit 1, signified by the second
record being blank. The project title is USER'S GUIDE (record 3). Record 4 specifies the mass and geometric properties of the vehicle as
S = 608 ft2
b = 42.8 ft

c = 15.95 ft

w = 45,000 lb

Record 5 defines the moments
units of slug-ft2) as

and products

of intertia

of the vehicle

(all in

75

Iy = 28,700

Iy = 165,100

Iz = 187,900

Ixz = 520

Ixy = 0

Tyz = 0

Record 6 defines the aerodynamic

the location of reference point

the vehicle center of gravity to of the nonlinear aerodynamic model

be by

coincident setting

with

Ax= 0

Ay= 0

Az =0

Record 6 also
rections for erence point,
because none

atrheseeiptehcoteiofrffisbeeebstecmaaitudnhsea,ettheLthIeNvEeAhaRiecrlosedhyonucalemdnitcenrotmoodfueslegraiivtnisctlyuidnetsferromnsaulcthhemocdoaerelrreocdttyoinoamnmasikce

cor-
refor

Record 7 defines the angle-of-attack range of the aerodynamic model,

Record 8 specifies that that the output formulation (x = Ax + Bu), and that the derivatives with respect to as reciprocal radians.

there will be four state variables in the output, of the state equation will be in the standard form output for the nondimensional stability and control angle of attack and angle of sideslip should be scaled

The next four records (9 to 12) define the output formulation of the state
vector to be

* i l <I @Q Rg

Record 13 trol vector.

specifies that the output model The following three records (14

will have three parameters to 16) specify that

in the

con-

elevator u = } throttle
speed brake

and that elevator, throttle, and of the /CONTROL/ common block.

speed

brake

are

located

in pc(5),

DC(12),

and

DC(10)

76

Record 17 specifies that two observation variables will be output and that the
observation equation will be in the standard form,
y = Hx + Fu

The next two records (18 and 19) define the elements of the output vector to be

y~ [a] ~ f@n y.

Record 20 specifies the ranges for the trim parameters, DES, DAS, DRS, and

THRSTX, used to trim the longitudinal, lateral, and directional axes and thrust,

respectively.

The ranges for these parameters are defined by record 20 to be

-2.9 © DES < 5.43 -4.0 < DAS * 4.0 -3.25 < DRS < 3.25 ~1.0 © THRSTX < 1.0

The first three parameters essentially represent stick and rudder limits and are so specified because of the implementation of the subroutine UCNTRL (discussed in app. I). The thrust trim parameter is used, again because of the implementation of UCNTRL, to schedule speed brake when THRSTX < 0 and to command thrust when
THRSTX > O.-

Record 21 specifies that no additional control surfaces are to be set.

The next seven records (22 to 28) define an analysis point option.
request a level turn trim option at

These records

h = 20,000 ft

M = 0.9

an = 3.0 9g

B = 0

The second requested. be varied ysis point end to the

record of this set (record 23) indicates which level turn suboption is

The until

alphanumeric descriptor ALPHA indicates that angle of attack is the specified 3.0-g turn is achieved. The final record of this

to anal-

option definition set contains the key word NEXT to indicate both an

current analysis point option definition and that another analysis point

option definition follows.

The final six records option at

(records

29 to 34) define a straight-and-level

analysis

point

h = 20,000 ft

M = 0.9

10.0°

~
W

77

oTwTfhhheeicthhfseiencaaonclngudlrerreernceotofcrodraadnttaoalfocyfkstihstihissisposvieasntretitecdod(nertfeuaicninotnirisldtiontt3h0re)imaskiediysewenltwaliocfrhdiiaeessvEeNtdDhtehetaotte"ritAmhnleidpnihacasta-ptiteeorcniimf"tihoeefd

suboption in condition.
termination input cases,

78

APPENDIX G: EXAMPLE OUTPUT ANALYSIS FILE

The following listing is an example analysis was produced using the example input file listed
supplied subroutines listed in appendix I.

file output in appendix

on unit 15. This F and the example

file user-

LINEARIZER TEST AND DEMONSTRATION CASES USER'S GUIDE

TEST CASE § 0 Jee R Sap SDE OSE OSS TAIT IS, 1

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

= 0.538044)01 = 0.214105+)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

0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00

POUNDS POUNDS POUNDS FOOT -POUNDS FOOT -POUNDS FOOT -POUNDS

OBSERVATION VARIABLES

AN

0.300163D+01 GS

AY

0.941435D+00 3=GS

A-MATRIX FOR: DX /DT = A*X + B*¥UtTD*YV -0.1214360+01 0.100000D+01 0.13-06 2 -70.152166 05DD-03 -0. 1474230401 -0. 221451D+01 -0. 450462D 02 0.29490D 103 0.000000D+00 0.331812D+00 0.000000D+00 0.000000+D00 -0.790853D+02 0. 000000+D00 -0. 320822D+02 -0.157297D -01

B-MATRIX FOR: DX /DT = A*X + B*¥U+tD *V -0. 141 961D+00 ~0, 16494 8D -02 -0. 928933D -02 -0.220778D+02 0.5403 2-03 . 2 1354 074D 0 +02 0.0000000+00 0.000000D+00 0.000000D+00 -0.105186D+02 0. 34281 7D+02 -0. 155832+D02

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

-0. 343642007 0.113192D-06 0.000000D+00 0.714203D -03

0.000000D+00 0.737378D-06 0.00+00 0 0 -0.0 2420 885D D-05 0.000000D+00 0.000000D+00 0.398492) -06 0.332842D-04

0.000000D0+00 0.605694D05 0.000000D+00 0.000000D+00

0.000000+D00 0.0000000+00 0.000000D+00 0.000000D+00

719

0. 000000D +00 0. 000000D +00 0. 0000000 +00 0. 000000D +00

H-MATRIX FOR: Y = H*X + FUSE*Y

00..30501007005D2+D0+002-0.00.0109010204D+5-0D005-00..115500503446DD--0012

0.640771-D02 0.274248-D09

F-MATRIX FOR: Y = H*X +FRU+E*Y
0. 412845D+01 -0.180978D -02 0.291699D+00 0.000000D +00 0.000000D+00 0.000000D+00

E-MATRIX FOR: Y=S=H*X+F *U+E *Y

-0.377037D-07 0.000000D+00 0. 000000D +00 0. 000000D +00

0.000000D+00-0.214132D -04 0.222222D-04 0.000000D+00

0.000000D+00 0.000000D+00

0.000000D+00 0. 0000000 +00

TEST CASE

HRIKIIK III IKI IK KRISTI IKI EAI EIS

SOX TBAS-TEERDVIAMETEQINUOSANITOINOENQU=AFTOIR4OMNULAFTOIROMNUU:L-ATDIIOMNE:NSIONSSTTAANN=DDAA3RRDD

INIA AKAIKE

I IK

Y - DIMENSION =

2 2

STATE VARIABLES ALPHA Q THETA VEL

= -.1266500-01 = 0.000000D+00 = 0.161868D+00 = 0.933232D+03

RADIANS RRAADDIIAANNSS/SECOND
FEET/SECOND

CONTROL VARIABLES

ELEVATOR THROTTLE SPEED BRAKE

= 0.637734-D01 = 0. 225092D+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

= 0.000000D+00 = 0.0000000+00 = 0.000000D+00 = 0.000000D+00 = 0.000000D+00 = 0,0000000+00

POUNDS POUNDS POUNDS FOOT -POUNDS FOOT -POUNDS FOOT -POUNDS

OBSERVATION VARIABLES

AN

= 0.985228D+00 GS

AY

= 0.000000D+00 GS

A-MATRIX FOR: DX / DT = A*X + B*UtD*Y

-0.120900D+01 0. 100000D +01 ~0. 1491 89D+01-0.221451D+01 0.0000000+00 0.100000D+01 -0.576868D+02 0.000000D +00

-0.575730D -02 -0. 701 975D -04 0.189640D-01 0.231368-D03 -00..0300106002D+5001D0+.0040620040-305000D+.-0002

80

B-MATRIX FOR: DX /DT = A*X + B*YUF+D*YV 0. 1419610400 0. 448-037-04 . 922 893D 20 02 ~0. 2207780 +02 -0. 147812D -02 -0, 1350740 +02 0..000000+D00 0.000000D+00 0.000000D+00 -0.105186D+02 0. 34316-20, D15+58302D2+02

D-MATRIX FOR: DX /DT+A*X+B*U+tD*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 0.0000000+00 0.000000D+00 0.495829D -13-0. 905497D -05

0.000000D+00 0.605694D-05 0.000000D+00 0.000000D+00

0.000000D+00 0.000000D+00 0.000000D+00 0.0000000+00

H-MATRIX FOR: Y = H*X + F*U 0. 3504240 +02 -0. 128333D -06-0.632314D 02 0.203434-D02 0.000000D+00 0.000000D+00 0.000000D+00 0.0000000+00
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 *UTE *V

0.102676D-07 0,000000D+00 0, 0000000 +00 0,.000000D+00

0.000000D+00-0.214116D-04 0, 2222220 -04 0.000000D+00

0.000000D+00 0.000000D+00

0.000000D+00 0.000000D+00

81

APPENDIX H:

EXAMPLE PRINTER OUTPUT FILES

The following listings are the the example input file in appendix listed in appendix I.

printer output files generated by LINEAR using F and the example user supplied subroutines

Example printer output file 1 (unit 3)

GEOMETRY AND MASS DATA FOR: LINEARIZER TEST AND DEMONSTRATION CASES

FOR THE PROJECT: USER'S GUIDE
VWWM11L111ZYYXXXEIEIZZYANNHNGGICLCEASHRPEAOANRWDEIGHT

1168425758519076200000041000082500..........000000890000000000050000000000

((((((((((LFFSFSSSSSBTTTLLLLL)L)*UUUUUU*GGGGG2G----)-FFFFTTTTTT********###*222*22))2))))

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

DDDEEELLLTTTAAA ZYX

000...000000000 (((FFFTTT)))

MODEL

FORCE AND MOMENT COEFFICIENT CORRECTIONS DUE TO THE OFFSET OF THE REFERENCE POINT TO THE AERODYNAMIC MODEL FROM THE VEHICLE CG ARE CALCULATED IN CCALC.

MINIMUM ANGLE OF ATTACK MAXIMUM ANGLE OF ATTACK

-10.000 (DEG) 40.000 (DEG)

PARAMETERS USED IN THE STATE VECTOR FOR: LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT: USER'S GUIDE ALPHA
Q
THETA VEL

THE STANDARD FORMULATION OF THE STATE EQUATION HAS BEEN SELECTED. THE FORM OF THE EQUATION IS:
DX/T =A*X+B*U+tD *V

SURFACES TQ 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

82

PARAMETERS USED IN THE OBSERVATION VECTOR FOR: LINEARIZER TEST AND DEMONSTRATION CASES
FOR THE PROJECT: USER'S GUIDE
AANY

THHAES BSETEANNDASREDLECFTOERDM.ULATTIHOEN FOORFM THOEF OTHBESEREVQAUTAITOINON EQIUS:ATION YsH*X+F*U+E*V

LIMITS FOR TRIM OUTPUT PARAMETERS:

MINIMUM

PITCH AXIS PARAMETER

2. 900

ROLL AXIS PARAMETER

4.000

YAW AXIS PARAMETER

-3.250

THRUST PARAMETER

-1.000

NO ADDITIONAL SURFACES TO BE SET WERE DEFINED

MAXIMUM .
4.000 3.250 1.000

TRIM CONDITIONS FOR CASE # 1 LINEARIZER TEST AND DEMONSTRATION CASES

FOR THE PROJECT: USER'S GUIDE

LEVEL TURN WHILE VARYING ALPHA

TRIM ACHIEVED

DCAMCWLVNDDBESGATALPGRYTSPRAHILEEAOLHPUOOOYOELEAQRHIIACWFMTALPEETIRNMENTLETURAGHTADLHTEGMIMACSIFOUFIVAADAHAHTMITSCFVFIRLTOIUTTUIIAITFAFCRDYDTCLCATAAOEERYIEIETFTCAANEETEI(TTCHTNNP@OOEECTTRRASRNEAOELATLUSTALSENESIIOQODRUTFFRURAUSAAETDPRCDLEIEECRI)OESEAFNDLGT

(((FLLTBB)SS)) (((((KGFFFT-TTTSS///))SSSEEECCC))**2) (((((((((((((DLDDDDLLSDDFDEBEEEEBTEEBLEGSGGGGS/GGGSU)))))))S////GESSSF/CEEETF)CCC*T)))#*2*)3)

= = = = = = = = = = = = = = = = = = = = = = = = = =

TRIM PARAMETERS

TTTTRRRRIIIIMMMM RTYPAOHIWLTRLCUHSATXAIXASIXPSIASRPAAPMRAPEARATMAREEMARTEMETERETRER

= = = 2

10130.41945134030045307-63732902002002.,022000.300451611............7.1.2..65420410900,96009007028.,242965013038090600108.523217312000911980060007314267600490559769200080094438200.42144038354007636009584610631754 ---0000....06021621591425216850

CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE OBSERVATION VARIABLES AN AY

touou

0.05380 0, 21410 0.00000 3.00163339 GS 0. 94136286 GS

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

FOR THE PROJECT: USER'S GUIDE

ROLLING MOMENT

PITCHING MOMENT

ZERO COEFFICIENTS

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..02966D -05 ~-2,00000D-01 0.000000 +00
1.50990D -01 -1.27955D-07 -1.32680D-04 0.00000D+00
-1.33450-001 0.000000 +00 0.00000D+00 0.000000 +00

4, 220400 -02 0.00000D+00 3. 89530D +00 0.00000D +00 -3.28739-D06 -3.40878D-03 -1,68820D-01 0. 00000D +00 0. 00000D +00 = -1.18870D+01 0.000000 +00

ELEVATOR THROTTLE SPEED BRAKE

0.00000D+00 0. 00000D +00 0.00000D+00

-6.95280D-01 0.000000 +00 -4.17500D-01

VEHICLE STATIC MARGIN IS AT THIS FLIGHT CONDITION.

3.5% MEAN AERODYNAMIC

YAW ING MOMENT 2.257470 -04 -3.37210D -02 0.000000 +00 ~4.04710D -01 3.21096D -07 3.32952D 04 0.00000D +00 1.29960D -01 0. 00000D +00 0. 00000D +00 0.000000 +00 0. 000000 +00 0.000000 +00 0.00000D +00 CHORD STABLE

DRAG 1. 42882-D04 0.00000D+00 0.00000+000 0. 00000D+00 0. 000000+00 0.00000D+00 3.72570-D01 0. 00000+D00 0. 00000+000 0.000000 +00 0.000000 +00 4. 38310D02 0. 00000+D00 6.49350-D02

LIFT 1.573600 01 0.000000 +00 -1.72320D+01 0.000000 +00 1.45433D -05 1.50803D -02 4. 87060D+00 0.00000D+00 0.00000D +00 1. 72320D+01 0. 00000D +00 5. 72960D -01 0. 00000D +00 3. 74920D -02

SIDE FORCE 5. 42882D -04 0. 00000D+00 0. 00000+D00 0.00000+000 0, 00000+000 0. 000000+00 0. 00000+000 -9.74030D-01 0, 00000D+00 0. 00000D+00 0.000000 +00 0. 00000+000 0.00000+000 0. 00000D+00

DERIVATIVES WRT
TIME OF: ALPHA Q THETA VEL

MATRIX A USING THE FORMULATION OF THE STATE EQUATION:

DX /DT = A*X + BeU+D *YV

FOR CASE # 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT: USER'S GUIDE

ALPHA

Q

THETA

VEL

-0.121436D+01 0. 147423D+01 0. 0000000 +00 -0. 7908530 +02

0. 1000000 +01 0. 221451D+01 0. 331812D +00 0. 0000000 +00

0.136756D -02 -0.450462D -02 0. 0000000 +00 -0.320822D+02

-0.121605-D03 0.29409 1-03 0.000000D+00 -0.157297D01

MATRIX B USING THE FORMULATION OF THE STATE EQUATION: DX / DT = A*X + B*YU+D *V
FOR CASE # 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT: USER'S GUIDE

84

DERIVATIVES WRT
TIME OF: ALPHA Q THETA VEL

ELEVATOR

THROTTLE

SPEED BRAKE

-0. 141 961D+00 -0. 2207780 +02 0. 000000D +00 -0.105186D+02

0.164948) -02 0. 543324D 02 0. 0000000 +00 0. 34281 7D +02

0. 928933-002 0. 135074+002 0.000000+D00 -0.155832+D02

MATRIX D USING THE FORMULATION OF THE STATE EQUATION DX /DT =A*X+B*U+D*V

FOR CASE # 1 LINEARIZER DEMONSTRATION AND TEST CASES

FOR THE PROJECT: USER'S GUIDE
DTIEMREIVWAROTTFI: VES AQTVELHLPEHTAA

X-BODY AXIS FORCE Y-BODY AXIS FORCE

-0000....103710413034106290402230D00D+0---0000637

0000,......030009008000400000009002DDD0++0+0-000006

Z-BODY AXIS FORCE -0000.....7203340370220388078405820D00 --+-00006504

PITCHING MOMENT ROLLING MOMENT 0000....006000000050006000900040DDDD++-+00000050 — —00-0,.0.....00.00000000000000000000000D00++++00000000

DTIEMREIVWAROTTFI: VES AQTVELHLPEHTAA

YAWING MOMENT 0000..0.00000000000000000000DD00+++000+0000000

MATRIX H USING THE FORMULATION OF THE OBSERVATION EQUATION: Y=H*X + F*¥UFE*V

FLOIRNEACRAISZEER# TES1T AND DEMONSTRATION CASES

FUOSRER'TSHE GUPIRDOEJECT:

ALPHA

Q

THETA

VEL

AANY

00..305010705020DD++0020 -00..109010204050D0-+0050

-00..115500503446DD--0012

00..26744027478100--0029

MATRIX F USING THE FORMULATION OF THE OBSERVATION EQUATION: Yo =H*X + F*¥U+E*V

FLOIRNEACRAISEZER# TES1T AND DEMONSTRATION CASES

FUOSRER'TSHE GUPIRDOEJECT: ELEVATOR

THROTTLE

SPEED BRAKE

AAYN

00..4010208040500D4+0010

-00..108000907080DD-+0020 + 90, .00200909019DD+60+000

85

MATRIX E USING THE FORMULATION OF THE OBSERVATION EQUATION: Y=H*X+F*U+E*Y

FOR CASE #1 LINEARIZER DEMONSTRATION AND TEST CASES

FOR THE PROJECT: USER'S GUIDE

X-BODY AXIS FORCE Y-BODY AXIS FORCE Z-BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT

AN -0.377037D-07

AY

0.0000000+00

0. 0000000 +00 0, 2222220 -04

0. 214132D -04 0. 0000000 +00

0.000000D +00 0. 0000000 +00

0. 0000000 +00 0. 0000000 +00

YAWING MOMENT

AN

0. 000000D+00

AY

0. 000000D+00

TRIM CONDITIONS LINEARIZER TEST

FOR AND

CASE #¢ 2 DEMONSTRATION

CASES

FOR THE PROJECT: USER'S GUIDE

STRAIGHT AND LEVEL TRIM ACHIEVED

TRIM WHILE

VARYING ALPHA

COEFFICIENT OF LIFT

COEFFICIENT LIFT

OF

DRAG

DRAG

ALTITUDE

MACH

VELOCITY

EQUIVALENT AIRSPEED

SPEED OF SOUND

GRAVITATIONAL ACCEL

NORMAL ACCELERATION

LOAD FACTOR

DYNAMIC PRESSURE

DENSITY

WEIGHT BETA

(@ALTITUDE )

ALPHA

PHI

THETA

ALTITUDE RATE

GAMMA

ROLL RATE

PITCH RATE

YAW RATE

THRUST

SUM OF THE SQUARES

TRIM PARAMETERS

=

=

(LBS)

=

(LBS)

=

(FT)

=

=

(FT/SEC)

=

(KTS)

=

(FT/SEC)

=

(FT/SEC**2) =

(G-S)

=

=

(LB/FT**2) = = (SLUG/FT**3) =

(LBS)

=

(DEG)

=

(DEG)

=

(DEG)

=

(DEG)

=

(FT/SEC)

=

(DEG)

=

(DEG/SEC)

=

(DEG/SEC)

=

(DEG/SEC)

=

(LBS)

=

=

0.13221 0.00895 44376. 86258 3004, 93778
20000. 0.90000 933. 23196 403. 42303 1036. 92440 32.11294 0, 98523 0. 98803 552.05302 0.00126774 44914. 60434 0.00000 -0. 72565 0.00000 9.27435 162.05403 10. 00000 0.00000 0.00000 0.00000 10804. 39673 0.00000

TRIM TRIM TRIM TRIM

PITCH AXIS PARAMETER

ROLL YAW

AXIS AXIS

PARAMETER PARAMETER

THRUST PARAMETER

=

0.79364

=

0.00000

=

0.00000

=

0.22509

86

CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE

=

0.06377

=

0.22509

=

0.00000

OBSERVATION VARIABLES AN AY

0.98522771 GS 0.00000000 GS

NON-DIMENSIONAL STABILITY AND CONTROL DERIVATIVES FOR CASE #2 LINEARIZER TEST AND DEMONSTRATION CASES

FOR THE PROJECT: USER'S GUIDE

ROLLING MOMENT

PITCHING MOMENT

YAWING MOMENT

ZERO COEFFICIENTS

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 )

BETADOT

(RAD/SEC)

-9, 371490 -20 -2.00000D -01
0. 00000D +00 1.509900 -01 8.28111D-17 8. 58688D -14 0. 00000D +00 -1.33450D -01 0. 00000D +00 0, 000000 +00 0.000000 +00

4.220400 -02 0. 000000 +00 3. 89530D+00 0. 00000D +00 -1.01777D-09 -1.05535D -06 -1.68820D -01 0. 000000+00 0. 00000D +00 -1.18870D0+01 0.000000 +00

-6.11595D -19 ~3.37210D 02
0.000000 +00 -4,047100 -01
7.42372D -16 7.697840 -13 0.00000D +00 1.29960D -01 0. 00000D +00 0. 00000D +00 0.00000D +00

ELEVATOR THROTTLE SPEED BRAKE

0. 00000D+00 0.00000D0+00 0. 00000D +00

6.952 80D -01 0.000000 +00 -4.17500D -01

0.00000D +00 0.000000 +00 0.00000D +00

VEHICLE STATIC MARGIN IS AT THIS FLIGHT CONDITION.

3.5% MEAN AERODYNAMIC CHORD STABLE

DRAG
1.08760D -02 0.000000 +00 0. 00000D +00 0.000000 +00 0.00000+D00 0. 00000D +00 3.725700 01 0. 00000D+00 0. 00000D +00 0.000000 +00 0.00000D +00
4. 38310D -02 0.00000D +00 6..49350D -02

TIME DERIVATIVES
WRT: ALPHA Q THETA VEL

MATRIX A USING THE FORMULATION OF THE STATE EQUATION:

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

FOR CASE # 2 LINEARIZER TEST AND DEMONSTRATION CASES

FOR THE PROJECT: USER'S GUIDE

ALPHA

Q

THETA

VEL

-0.120900D+01 -0.1491890401
0. 0000000 +00 -0.576868D +02

0. 100000D+01 -0.221451D+01 0.100000D+01
0.000000D +00

0. 575730D 02 0.189640D -01 0. 0000000 +00 -0..316251D+02

0. 701975D-04 0. 231368D -03 0.000000D +00 0. 460435-D02

MATRIX B USING DX / DT

THE FORMULATION OF THE STATE A*X + B*euU+D *V

EQUATION:

FOR CASE # 2 LINEARIZER TEST AND DEMONSTRATION CASES

FOR THE PROJECT: USER'S GUIDE

LIFT 1.573600 -01 0. 00000D +00 -1. 723200401 0.000000 +00 7.1852 7D -09 7.45058) -06 4, 87060D+00 0.000000 +00 0.000000 +00 1.72320D+01 0. 00000D +00 5. 72960D -01 0. 00000D +00 3. 74920D -02

SIDE FORCE 4.31432D-19 0.00000+000 0. 00000+D00 0.00000+000 0. 00000D+00 0.00000+000 0. 00000+D00 -9, 74030D-01 0.00000D+00 0. 00000D+00 0.00000+D00 0.00000+000 0. 000000+00 0. Q0000+D00

87

TIME DERIVATIVES
WRT:
TVAQ EHLLEPTHAA

ELEVATOR

THROTTLE

SPEED BRAKE

~0.141961D+00 -0.220778D+02 0. 0000000+00 0.1051 86D+02

0, 448742-D03 0.147812)-02 0,000000D+00 0. 343162+002

-0. 928932-D02 -0,135074D+02 0.000000D+00 -0,155832D+02

MATRIX D USING THE FORMULATION OF THE STATE EQUATION:

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

FOR CASE #2 LINEARIZER DEMONSTRATION AND TEST CASES FOR THE PROJECT: USER'S GUIDE

DERIVATIVES WRT
TIME OF;

X-BODY AXIS FORCE Y-BODY AXIS FORCE Z-B0DY AXIS FORCE PITCHING MOMENT ROLLING MOMENT

ALPHA
TVQ EHLETA

0. 9348800 -08 -0. 307941D-07 0. 000000D+00 0. 714920-D03

0.000000+D00 0.000000+D00 0,.000000D+00 0.49598D 2-13

0. 738119006 0.24391D 2-05 0. 000000+D00 -0. 9054970 -05

0, 000000+000 0. 605694-D05 0,000000D+00 0. 000000+000

0. 000000+000 0..000000+000 0.000000D+00 0, 000000+000

DERIVATIVES WRT
TIME OF: ALPHA
Q
THETA VEL

YAWING MOMENT 0. 000000+D00 0.0000000+00 0. 0000000+00 0. 000000D+00

MATRIX

H USING

THE FORMULATION
Y= H * X +

OF THE OBSERVATION
FUE RY

EQUATION:

FOR CASE # 2? LINEARIZER TEST AND FOR THE PROJECT: USER'S GUIDE

DEMONSTRATION

CASES

ALPHA

Q

THETA

VEL

AN AY

0. 350424+002 0. 000000+000

~0. 128333D -06 0. 000000+D00

-0.632324-D02 0,000000D+00

0. 203434D -02 0.0000000+00

MATRIX

F USING

THE FORMULATION
Y = H*X +

OF THE OBSERVATION
FeU+E#Y

EQUATION:

FOR CASE # 2 LINEARIZER TEST AND FOR THE PROJECT: USER'S GUIDE

DEMONSTRATION

CASES

ELEVATOR

THROTTLE

SPEED BRAKE

AN AY

0. 411323D+01 0. 000000+D00

0. 492 845D-03 0..000000+D00

0, 263288D+00 0.000000D+00

88

MATRIX E USING THE FORMULATION OF THE OBSERVATION EQUATION:
V=H*X+F * UTE *Y

FOR CASE # 2 LINEARIZER DEMONSTRATION AND TEST CASES

FOR THE PROJECT: USER'S GUIDE

X-BODY AXIS FORCE Y-BODY AXIS FORCE Z-BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT

AN AY

0. 102676-D07 0, 000000+000

0. 0000000 +00 0, 2222220 -04

-0.214116-D04 0..000000+D00

0. 000000+000 0. 000000+D00

0. 000000+000 0. 000000+000

YAWING MOMENT

AN

0. 000000+000

AY

0. 000000+000

Example printer output file 2 (unit 2)

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
DAMSCLVERAPILEQOACFETLUEGHTEIOIFDTCVFUIAIDTLCOEYEIFNETNTSOUANIODFRSPDEREADG
GRAVITATIONAL ACCEL NORMAL ACCELERATION LOAD FACTOR
AWBARTDDPTGPYHEOALLEHHIAYEIWTLPETTIRMNNALTHCMGUIASAAHAHSTMIRTTIUTARCDYTAREETAE(TP@ERRAAELTSTESIUTRUEDE)
SUM OF THE SQUARES

(((FLLTBB)SS)) (((KFFTTTS//)SSEECC))
(FT/SEC**2) (G-S)
(((((((((((((DDDDLLDSDDDFLEEEEBBELEEETBGGGGSSGU/GGGS)))))))GS////ESSSF/CEEEF)TCCCT))*)***23 ) )

0.40144 0.03058 134741.68807 10265. 70661
20000. 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 PITCH AXIS PARAMETER TRIM ROLL AXIS PARAMETER TRIM YAW AXIS PARAMETER TRIM THRUST PARAMETER

-0.66958 -0.01526 -0,02125 0.21410

CONTROL VARIABLES

ELEVATOR

THROTTLE

SPEED BRAKE

(DEG)

0.05380 0.21410 0.00000

89

OBSERVATION VARIABLES

AN AY

3.00163339 &S 0.94136286 GS

VEHICLE AT THIS

STATIC FLIGHT

MARGIN IS CONDITION

3.5% MEAN

AERODYNAMIC

CHORD

STABLE

TRIM CONDITIONS FOR CASE # 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT: USER'S GUIDE

STRAIGHT AND LEVEL TRIM 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 (@ALTITUDE ) BETA ALPHA PHI THETA ALTITUDE RATE GAMMA ROLL RATE PITCH RATE YAW RATE THRUST SUM OF THE SQUARES TRIM PARAMETERS

=

=

(LBS)

=

(LBS)

=

(FT)

=

=

(FT/SEC)

=

(KTS)

=

(FT/SEC)

=

(FT/SEC**2) =

(G-S)

=

=

(LB/FT**2) = =

(SLUG/FT**3) =

(LBS)

=

(DEG)

=

(DEG)

=

(DEG)

=

(DEG)

=

(FT/SEC)

=

(DEG)

=

(DEG/SEC)

=

(DEG/SEC)

=

(DEG/SEC)

=

(LBS)

=

=

0.13221 0.00895 44376, 86258 3004. 93778
20000. 0. 90000 933. 23196 403. 42303 1036, 92440 32.11294 0.98523 0. 98803 552.05302 0.00126774 44914. 60434 0.00000 -0. 72565 0.00000 9.27435 162.05403 10. 00000 0.00000 0.00000 0.00000 10804. 39673 0.00000

TRIM PITCH AXIS PARAMETER

TTRRIIMM

RYOAWLL

AXIS AXIS

PARAMETER PARAMETER

TRIM THRUST PARAMETER

CONTROL VARIABLES

=

-0. 79364

=

0.00000

=

0.00000

=

0.22509

ELEVATOR THROTTLE SPEED BRAKE

=

0.06377

=

0.22509

=

0.00000

OBSERVATION VARIABLES

AN AY

0.98522771 GS 0,00000000 «Ss

VEHICLE STATIC MARGIN IS 3.5% MEAN AERODYNAMIC CHORD STABLE AT THIS FLIGHT CONDITION

APPENDIX I: EXAMPLE USER-SUPPLIED SUBROUTINES

The following subroutines are examples of user-supplied routines that provide

the aerodynamic, control, and engine models to LINEAR.

These subroutines are based

on an F-15 aircraft simulation and are typical of the routines needed to interface

LINEAR to a set of nonlinear simulation models. These subroutines are meant to

Lllustrate the use of the named common blocks to communicate between LINEAR and

the user's routines. These subroutines are used with all examples in this report.

Included with this report are microfiche listings of these subroutines.

Aerodynamic Model Subroutines

The following two subroutines define a linear aerodynamic model.

Even though

this model is greatly simplified from the typical nonlinear aerodynamic model, the

example illustrates the functions of the subroutines ADATIN and CCALC.

CCCcCCCCCcCCCCCCCCCcCCcCCCCC ..............ssRERPASCCCCCCCCCCC.........OXOILLLLLMMMMOUELUAERLPTDQAOOADMBRTMRLTADCMROIPIOHODNLNNIUYEEGNTNCSGI==========AOUN/MMBAEAMTCAAIS-S----SMORROTEIO1ICT1M212T44231OOOMTLNR.D.O...A..A...NDEUAET3OE6M01320B6B85ADANCTR3RS8E088D213I90IDEMTTKOI4O8NL085500Y59LLAFOB/,N5N2LT6079I4739NIITIDL,EPTATINEOS,DEYNMYLCECCCCcEEICCCCCEDEEEE£EDPKDML-NY-LNM------R+LCELIEROOBBB00D000DD0000F.I.TRFRAUSDRD1E1112AA1232TCVDIDITOTETEOHALDOAVVDARRDLETTAAEBIIARIITT,ITVVC,RNV,,,,,,6,,AIICLAA,AOCCCCCRCCCEVVCILTTTLNMLNMMAYDLSEETFIAIEPPADDSTDMF..ASSYDD,TVRRVBAETUN.EIS,EEADNFFBSSISIF,AWsACNNEINCAY,,,,,,,,DITRGCA,WCCCWCTCHTCLCCWAEIYLNILDEMEILNDTODTDFLQNCCORHTTHTETORRRNRI.QNEA«O,A,ATFSE,RRLRTPCREE,,,O,(DEOESS,OCACLsCRDCPPY.TLTBTMEYEDFT.A.ATCNCTDASDTTAI.ACICENTMDKyLORP,I,UI.CTT,TVOOR.)AA.TADITNASVTGEUEALB.SER.OB.UEFOTFOTIRWNEEETNHE THE
91

OOO

MADQAAINY

OrANINO

MIQOOND

a

QDOaaND

CONTROL DERIVATIVES WITH RESPECT TO ELEVATOR, SPEED BRAKE...

CMDE = -6,9528 E-01 CMSB = -4.1750 E-01

«eos YAWING MOMENT DERIVATIVES.

STABILITY SIDESLIP,

DERIVATIVES WITH RESPECT TO ROLL RATE, YAW RATE...

CNB = 1.2996 E-01 CNP = -3,3721 E-02 CNR = -4,0471 E-01

CONTROL AILERON,

DERIVATIVES WITH RESPECT TO RUDDER, DIFFERENTIAL TAIL...

CNDA = 2,1917 E-03 CNDR = -6.9763 E-02 CNDT = 3.0531 E-02

eos COEFFICIENT OF DRAG DERIVATIVES,

STABILITY DERIVATIVES WITH RESPECT TO ANGLE OF ATTACK...

CDO == 1.0876 E-02

CDA = 3.7257 E-01

CONTROL DERIVATIVES WITH RESPECT TO ELEVATOR, SPEED BRAKE . . . CDDE = 4,3831 £-02 CDSB = 6.4935 E-02

«ee eCOEFFICIENT OF LIFT DERIVATIVES.

CLFTO

STABILITY DERIVATIVES WITH ATTACK, PITCH RATE, ANGLE

RESPECT OF ATTACK

TO ANGLE OF RATE ee 6

= 1.5736 E-01

CLFTA = 4,8706 CLFTQ = -1.7232 E+01 CLFTAD= 1.7232 E+01

CONTROL DERIVATIVES WITH RESPECT TO ELEVATOR, SPEED BRAKE... CLFTDE= 5.7296 E-01 CLFTSB= 3,7492 E-02

«+++ SIDEFORCE COEFFICIENT DERIVATIVES.

STABILITY DERIVATIVES WITH RESPECT TO SIDESLIP ... CYB = ~9,7403 E-01

CONTROL DERIVATIVES WITH RESPECT TO AILERON, RUDDER, DIFFENTIAL TAIL CYDA = -1,1516 E-03 CYDR = -1,5041 E-01 CYDT = -7,9315 E-02

RETURN END

OMOIMAAN

o O

Ora

92

SUBROUTINE CCALC

C

C....EXAMPLE AERODYNAMIC MODEL.

c

C... ROUTINE TO CALCULATE THE AERODYNAMIC FORCE AND MOMENT COEFFICIENTS.

c

C

COMMON BLOCKS CONTAINING STATE, CONTROL, AND AIR

c

DATA PARAMETERS

C

COMMON /DRVOUT/ F(13),DF(13)

COMMON /CONTRL/ DC(30)

c COMMON /DATAIN/ S,B,CBAR,AMSS,AIX,AIY,AIZ,AIXZ,AIXY,ALYZ,AIXE

COMMON /TRIGFN/ SINALP,COSALP, SINBTA,COSBTA, SINPHI,COSPHI,

.

SINPSI,COSPSI,SINTHA,COSTHA

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

COMMON /CGSHFT/ DELX,DELY, DELZ

C c

COMMON BLOCK TO OUTPUT AERODYNAMIC FORCE AND MOMENT

C

COEFFICIENTS

C

COMMON /CLCOUT/ CL

,coM

CN

SSOCD SCL. CY

c c

COMMON BLOCK TO COMMUNICATE AERODYNAMIC DATA BETWEEN

C

THE SUBROUTINES ADATIN AND CCALC

C

COMMON /ARODAT/ CLB ,CLP ,CLR ,

:

CLOA ,CLDR ,CLDT ,

.

CMO ,CMA ,CMQ ,CMAD ,

.

CMDE ,CMSB,

.

CNB CNP) s,CRRS Cy

.

CNDA ,CNDR ,CNDT ,

.

cdO. 3=,CDA. Ss yCDDE «= ,CDSB Cy

.

CLFTO ,CLFTA ,CLFTQ ,CLFTAD,

:

CLFTDE,CLFTSB,

.

CYB ,CYDA ,CYDR ,CYDT

c C

EQUIVALENCE VARIABLE NAMES

C

EQUIVALENCE (T

» F( 1)),

.

(P

» FC 2)),(Q

» F( 3)),(R

» F( 4)),

.

(¥

» F( 5)),(ALP_) sy F( 6)),(BTA , F( 7)),

(THA

» F( 8)),(PSI , F( 9)),(PHI =, F(10)},

‘ . .

(((HTPDDOOTT

,»,D0FFF(((1112)))))),,,((XQD0T

» F(12)),(V ,OF( 3)),(RDOT

» F(13)), ,OF( 4)),

:

(VDOT

,DF( 5)),(ALPDOT,DF( 6)),(BTADOT,DF( 7)),

.

(THADOT ,DF( 8)),(PSIDOT,DF( 9)),(PHIDOT,DF(10)),

.

(HDOT

,DF(11)),(XDOT ,DF(12)),(YDOT ,DF(13))

c

EQUIVALENCE (DA

,DC( 1)),(DE

,0C( 5)),(DT

.

(DR

,0C( 9)),(DSB_ DC (10))

cC.eeeCOMPUTE TERMS NEEDED WITH ROTATIONAL DERIVATIVES.

,0C( 8)),

c

v2 =2.0

*V

B2V =B

/N2

C2v =CBAR

/V2

C C.e.eeROLLING MOMENT COEFFICIENT.

C

CL -=CLB

* BTA

+CLDA *DA

+CLDR

*DR

+CLDT

*OT

.

4B 2V

*(CLP

oP

4CLR *R

)

C

C.e.ePITCHING MOMENT COEFFICIENT.

C

CM = =CMO

+CMA

*ALP 4CMDE *DE

+CMSB *DSB

+0 2V

*(CMQ*Q

+CMAD *ALPDOT)

C C..+eYAWING MOMENT COEFFICIENT.

93

CN =CNB

.

4B 2V

* BTA *(CNP

Co. COEFFICIENT OF DRAG.

°

cD = =CDO

+ CDA

Co. eeCOEFFICIENT OF LIFT.

°

CLFT =<CLFTO + CLFTA

.

+C2V

*(CLFTQ

Ce +. SIDEFORCE COEFFICIENT.

°

CY =CYB

*BTA

°

RETURN

END

+CNDA *P *ALP *ALP *Q +CYDA

= *DA 4+CNR

+C NDR oR

+CDDE *DE

+C LF TDE*DE +CLFTAD*ALPDOT

= *DA

+C YOR

*DR )

+C NDT

*T

+CDSB *DSB

+CLFTSB*DSB )

*DR

+CYDT = *DT

Engine Model Interface Subroutine bpiilrnnootfcveokirr.TdfmheaeacteitfohoneltolodfwerLiotInmaNgiEAltReh.dseusbertohIunrstuuisbnntro,eor,umatelinIngFeiEsunNseGaNgw,eoruoltbtdaohttiishboen,dseutfbriraanonnedusstfiefnruaereneldweoncugoilinndntseoumcpamttlhoiledoenl/sEuNbmaGrnoSdodTeuFltp/iirnnogev;csiodmemttsohhneatthe

ANNAAARNAOD

ANMAIANANAAAARAANNMARMAOAAO

SUBROUTINE IFENGN Cc C....EXAMPLE SUBROUTINE TO PROVIDE PROPULSION SYSTEM MODEL.

++ eROUTINE TO COMPUTE PROPULSION SYSTEM INFORMATION FOR LINEAR

THIS SUBROUTINE IS THE INTERFACE MODELING SUBROUTINES AND LINEAR.

BETWEEN

THE

DETAILED

ENGINE

INPUT COMMON BLOCK CONTAINING INFORMATION ON THRUST REQUEST TO ENGINE

COMMON /CONTRL/ DC(30)

OUTPUT COMMON BLOCK CONTAINING DETAILED INFORMATION ON EACH OF UP TO FOUR SEPARATE ENGINES

«««sROUTINE TO COMPUTE ENGINE PARAMETERS:

THRUST (1) TLOCAT(I,J)
XYANGL (1)

THRUST CREATED BY EACH ENGINE LOCATION OF EACH ENGINE IN THE X-Y-Z PLANE ANGLE IN X-Y¥ BODY AXIS PLANE AT WHICH EACH

XZANG(LI )

ENGINE IS MOUNTED

ANGLE IN X-Z BODY ENGINE IS MOUNTED

AXIS

PLANE

AT WHICH

EACH

TVANXY (I) = ANGLE IN THE X-Y ENGINE AXIS PLANE OF THE

THRUST VECTOR

TVANXZ(I)

= ANGLE IN THE X-Z THRUST VECTOR

ENGINE

AXIS

PLANE

OF

THE

DXTHRS(I) = DISTANCE BETWEEN THE ENGINE C.G. AND THE

mos

THRUST POINT

EIX(I) = ROTATIONAL INERTIA OF EACH ENGINE

94

C C

AMSENG(I) ENGOMG(I)

= MASS OF EACH ENGINE = ROTATIONAL VELOCITY

(RAD/SEC)

C

COMMON . .

/ENGSTF/

TTVHANRXYUS(4T),(T 4)V,ATLNOXC,ZADXT( TH(R4S4) ,(43)(,)4),,XXZYANAGLNG(4L), EIX(4),AMSENG( 4), ENGOMG(4)

C

c

EQUIVALENCE VARIABLE NAMES

Cc EQUIVALENCE (THR

»DC(12))

C C....ASSUME

THRUST

PER ENGINE

IS HALF

VEHICLE WEIGHT.

Cc

THRUST (1)= 24000.0 *THR

THRUST (2)= 24000.0 *THR

Cc C....LET

ALL OTHER

PARAMETERS

DEFAULT

TO ZERO

Cc RETURN

END

Control Model Subroutine

The surface
example

subroutine UCNTRL provides an interface between the trim parameters and the

deflections.

Figure 8 illustrates the gearing model implemented in the

UCNTRL subroutine.

The thrust demand parameter is also set and passed to

the subroutine IFENGN.

DES

| _ 255..403

20.0

DAS

“| 4.0

_"|lo1n80
_} 180
nt

-. 6 Se

~ S

-

a

qo1 Pa180

toate

DRS

33.02.50

-1,0

THRSTX

se

_~|| 18on0

_- 9

— 45.0 }-»| 18Tl0

> Oo sb

0,1

_

— (to ITFHERNGN)

UCNTRL

Figure 8. Gearing model in example
UCNTRL subroutine.

95

SUBROUTINE UCNTRL

Cc

C....EXAMPLE C

TRIM/CONTROL

SURFACE

INTERFACE

ROUTINE.

Coes cROUTINE TO CONVERT TRIM INPUTS INTO CONTROL SURFACE DEFLECTIONS.

C

INPUT COMMON BLOCK CONTAINING TRIM PARAMETERS

° COMMON /CTPARM/ DES ,DAS ,DRS — ,THRSTX

C

OUTPUT COMMON BLOCK CONTAINING CONTROL SURFACE DEFLECTIONS

° COMMON /CONTRL/ DC(30)

C

EQUIVALENCE VARIABLE NAMES,

° c

EQUIVALENCE .

(DA (DR

»DC( 1)),(DE »DC( 9)),(DCB

=

»DC( 5)), ,DC(10)),

(DT (THR

=,

»DC( 8)), DC(12))

DATA DGR = /57.29578/

C....CONVERT FROM

C

DEF LECTION

INCHES

OF

STICK AND

PEDAL

TO DEGREES

C

DA

=DAS *( 20.0 / 4.0 )

DE

=DES = *(-25.0 / 5.43)

Cc

DR

=DRS = *( 30.0 / 3.25)

C....SET Cc

DIFFERENTIAL

TAIL

BASED

ON AILERON

COMMAND.

DT

=DA

/ 4.0

C

C....CONVERT Cc

THRUST

TRIM

PARAMETER

TO PERCENT

THROTTLE

THR

= 0.0

IF(THRSTX.GE. 0.0 ) THR

=THRSTX

OF SURFACE COMMAND.

CCc ....USE SPEED BRAKE IF NEEDED.

DSB = 0.0

Cc

IF(THRSTX.LT. 0.0 ) DSB

=THRSTX*(-45.0 )

CCc ....CONVERT SURFACE COMMANDS TO RADIANS.

DA

=DA

/DGR

DE

=DE

/DGR

DR

=DR

/DGR

DT

=DT

/DGR

DSB =DSB- Ss /DGR

C

RETURN

END

Mass and Geometry Model Subroutine dMgatrifehAiesloofSrtoiusiGigctmenEchireTeOstnathdefrestyuwocfoboutfursnhcioloinadhltdnutihlagtteorwiiwahbntoiiechFeanncetOsihgeR.rarTcauiRrnsdtssAadeuhuNtfrembitsmrcseyoussmuurudtsofbmsitarauoncsofbenesuro,pttortiuhoanstevneecMiditshAndtaS,eveinG,egngEhegeOioM,tcmoAalesSoe(tGrLvirEsIeiyOsNhccEiaaaAnncnpptRlhrr.eeoobbevpeexieacfdrmoodeeItpflenasilflfseeioiistnlwghaeyieunodnrfagarimsteanpuitstseheoseeicxncnriao)gffmm,maaippnaceldlseedtsih.tcehgaeebaatnseoniedtmndHwepoatetuwhgetreeeonfyvrouemnmrfecaiL,actstlIlhseriNra,yoEerfnAaaoaRdrcnasytdnuoedbfaar-nn-d
96

SUBROUTINE MASGEO

OOoan

«ee» SUBROUTINE TO COMPUTE THE MASS AND GEOMETRY PROPERTIES OF THE AIRCRAFT.

. : . . . . . . . .

CCCCECOOOOQOMMMMUMMMMMIMOOOOOVNNNNNALEN/////CSDCDCEIROAOMVNNTOOPTA((((((((((UUORITPVHHTPTVTTTSLNHDDODH/////AOOOOATTTTDODDAFSDTCS(A,,(MB1PBD,,,,,»»=»,P33D,DDF,D,0CFFF)F(FCF)FFFF(((,(((L(HB((DDA11FA5F,PT11R28)12851(,),))),))))))1Q)))D)))))A),3P,),,,A,,,,,BM)(L(((T((((ASAXQXPRQAPPASLDSDILSL,PIORO,MIPATT,DDI,DOOPXETTLG,A,P,APDMD,,»,,»,RIFFDFD,YFA((LFFFFF(,((CO((,VTAN11HBI22,SD,996363ZL))W)))))),)B,E))))))),,A,,D,,,,,L({(I(TE((((YRHYPXVTBRBP,SEDHTDZRTRHAIOAOI,UASITTM,ADDB,OIODTXVT,R,CY,APD,,,»,,»DS,FDDAFFF(FLFFFF(((CP(((Y1E111Z03D03)4,477))),)))))A)))D)))),I,,,,,,R,XTERIM,

RETURN END

APPENDIX J: REVISIONS TO MICORFICHE SUPPLEMENT

The following in the microfiche tines Utilized by ENGINE listing on

listing of subroutine ENGINE incorporates revisions not contained supplement included with this report (Program LINEAR and SubrouLINEAR). This listing should be used in place of the subroutine the microfiche supplement.

98

