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. [ \ 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 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 = 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