toby
/
zotero-db
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
zotero-db/storage/48KFSPPF/.zotero-ft-cache

7345 lines
145 KiB
Plaintext
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

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
Reference in New Issue View Git Blame Copy Permalink