NASA Technical Paper 2835 September 1988 User’s Manual for Interactive LINEAR, a FORTRAN Program To Derive Linear Aircraft Models Robert F. Antoniewicz, Eugene L. Duke, and Brian P. Patterson NASA NASA Technical Paper 2835 1988 User’s Manual for Interactive LINEAR, a FORTRAN Program To Derive Linear Aircraft Models Robert F. Antoniewicz, Eugene L. Duke, and Brian P. Patterson Ames Research Center Dryden Flight Research Facility Edwards, California NASA National Aeronautics and Space Administration Scientific and Technical Information Division CONTENTS SUMMARY . © © © © © © # © © © © 8» #@® #© © © 8 @ © o 6 #© #6 #8 #© # #6 # # #© #6 # # # # & 1 INTRODUCTION . «© © © © © © © © © © © © © © © © © © © © © © © 8 eo ee we ee 1 NOMENCLATURE . 2. « © «© © «© © © © © © «© © © © #© © © © © © © © © © © © © © © © 8 «© 2 Variables « « 6 6 © © © © © © © © oe we we we ee ew we ew ew ew ew ew 2 SuperscriptS « 6 + « «© © © © © © # © © oe ww oe ee we we ee ee ee et 6 Subscripts «© 6 « © © 6 6 6 © © © © oe 8 ww ew ww ee ew we eh ee ww 6 FORTRAN Variables . . «6 6 © © © © © © © © © © © © ee eo ee ee ew wl 7 PROGRAM OVERVIEW . 2. «6 6 0 2 ee ee we ee ee we we ee ww we wt we dd EQUATIONS OF MOTION . 6 6 2 6 6 ee eee ee we ee ww ee we ew we ee ee (4 OBSERVATION EQUATIONS . 2. 2. 2 6 6 6 6 © ew ee ee ee ee ew ew ee eee ee TT SELECTION OF STATE, CONTROL, AND OBSERVATION VARIABLES «. . «© «© «© © © «© © © © «© «© 22 LINEAR MODELS . « 6 « «© © «© © © © © © © © ow ew wh ew ee ww he ew ew lt ew we OR! ANALYSIS POINT DEFINITION . . 2. 6 «© © © «© © © © © © © © © © © © © ee ew hehehehe) 65 Untrimmed . . 2. 2 6 6 te oe ee ew we tw wt wt et tw tt ew le lle wl le 2G Straight-and-Level Trim . 2. 6 2 6 ee ee eo ee ee ew ee ew we ee we ee 27 Pushover-Pullup . 2. 2. 2 2. 6 ee ee © © ee eo ee ee ee ee ee ee ee we 7 Level Turn . .. 6 © © © © © © © © © © © oe we we we ee ww lw lw lw hele tl lhl) lB Thrust-Stabilized Turn . « « 2 © «© © © © © © © © © © © oe ew ee ee ee we we 2D Beta Trim. . «6 «© «© «© © © © © © © © 8 6 wee ee te we et wt tle lw lel wl le RD Specific Power «© 6 6 6 «© © eo ee ee ee ew ew ew ew ee we eee ee 29 NONDIMENSIONAL STABILITY AND CONTROL DERIVATIVES ~. « « «© «© «© © «© © © ©» © © © © © 31 DATA INPUT . 2. 2. 2 2 © © © © © @ 8 © 8 6 eo ew we ee ee we ww et hw we le ew) 8B Input Files 2 6 6 6 6 ee 8 ee we we wwe we we ee ew ee ee eee ew ew 83 Title Information .« . 2. 6 «© «© © © © © © © © oe eo ee ew we we tw ee ee) 83 Geometry and Mass Data . . . + 6 «© © © © © © ee eo © ee ee ee ee ew ew we) 33 State, Control, and Observation Variable Definitions ... +... + +++... 34 Trim Parameter Specification . . « « +» « «© © © © © © © © © © eee ee ew we 87 Additional Surface Specification . « 6 « © « © © © © © © © © © © © © © © we 37 Test Case Specification . 2. 6 6 6 © © © © © © © ew ee we ew ee ew ew ee 8B Interactive Data Input . . 6 6 2» «© © © © © © © © ew we we ee te tw el ee 8B AERODYNAMIC MODEL . . ° * e e e e e e e e e . e o e * e . . e . * e e a e . e e ° 46 OUTPUT FILES «6 6 6 «© «© «© © © © © © we ee we ww ee ww ww ew ee eee «48 USER-SUPPLIED SUBROUTINES . .« 2 2 © 8 © © © © © © © © © © ew ee ww tw ww ew) «4D Aerodynamic Model Subroutines . . + + 6 © 6 6 © © 8 6 ee ew ew ew ew ew ew ew ee (5D Control Model Subroutines . . «6 « «© © © © © © ee ee ee ew ww ww we ew 8B Engine Model Subroutines «© + « 6 © 6 © © © © © we ee ee we ew ew ww ew ee «(5B Mass and Geometry Model Subroutines .« «6 «© 56 6 © © © © © © © © © ee © we ew oe ew (58D iii CONCLUDING REMARKS . ..... es «se ee eo 8 6 APPENDIX A: CORRECTIONS TO AERODYNAMIC COEFFICIENTS FOR AT THE AERODYNAMIC REFERENCE POINT .. A CENTER OF GRAVITY NOT APPENDIX B: APPENDIX C: APPENDIX D: APPENDIX E;: ENGINE TORQUE AND GYROSCOPIC EFFECTS MODEL .. a a OBSERVATION VARIABLE NAMES RECOGNIZED BY LINEAR ........e.6., STATE VARIABLE NAMES RECOGNIZED BY LINEAR .. a re ANALYSIS POINT DEFINITION IDENTIFIERS . APPENDIX F: EXAMPLE INPUT FILE ........s.e ee APPENDIX G: EXAMPLE LINEARIZED STABILITY AND CONTROL APPENDIX H: EXAMPLE OUTPUT ANALYSIS FILE . ..... APPENDIX I: EXAMPLE PRINTER OUTPUT FILE ...... APPENDIX J: EXAMPLE USER-SUPPLIED SUBROUTINES ..., ECAnoegnritonrdeoylnaMmoMidocedlelMoIdnSetulebrrfoSauuctbeirnoeuStuibnre.os.ut.i.n..e4...+.6..4.. .8..8.. 6e. .8. .8-. .e Mass and Geometry Model... .....6e8e ee REFERENCES . . . ° e . . ° e . e . . e . e e e . ° e 54 56 58 66 73 74 76 80 86 90 109 109 113 115 116 117 iv SUMMARY An interactive FORTRAN program that provides the user with a powerful and flex- ible tool report. for The the linearization of aircraft aerodynamic models is program LINEAR numerically determines a linear system documented in model using this non- linear equations of motion and a user-supplied linear or nonlinear aerodynamic model. The nonlinear equations of motion used are six-degree-of-freedom equations with sta- tionary atmosphere and flat, nonrotating mined by LINEAR consists of matrices for earth assumptions. both the state and The system observation model deter- equations. The program has been designed to allow easy selection and definition of the state, control, and observation variables to be used in a particular model. INTRODUCTION The program LINEAR described in this report was developed at the Dryden Flight Research Facility of mented, and verified the NASA Ames Research Center to tool to derive linear models for provide a standard, docuaircraft stability analysis and the control aircraft law design. This development was specific linearization programs undertaken to address common in the aerospace the need for industry. Also, the lack of available documented linearization programs provided a strong motivation for the development of LINEAR; in fact, the only available documented linearization program that was found in an extensive literature search of the field is that of Kalviste (1980). Linear system models of aircraft dynamics and sensors are an essential part of both vehicle stability analysis and control law design. These models define the aircraft system in the neighborhood of an analysis point and are determined by the linearization of the nonlinear equations defining vehicle dynamics and sensors. This report describes a FORTRAN program that provides the user with a powerful and flexible tool for the linearization of aircraft models. with well-defined and generalized interfaces to aerodynamic is designed to address a wide range of problems without the LINEAR is a program and engine models and requirement of program modification. The program LINEAR numerically determines a linear system model using nonlinear equations of motion and a user-supplied linear or nonlinear LINEAR is_also capable of extracting both linearized engine aerodynamic model. effects, such as net thrust, torque, and gyroscopic effects system model. The point at which this and including linear system these model effects in the linear is defined is determined either by completely specifying the state and control variables, or by specifying an analysis point on a trajectory and directing the program to determine the con- trol variables and remaining state variables. The system model determined by LINEAR consists of matrices for both the state pTaofnhrduosgt,rhoaebmstehsertpvaratooetvr,iidodenercsoneotfqtrhuoeatlth,iefolnesasnx.ydisbteiomlTbihsteemyrovdpearoltofigoraniaslmlovcwaohiarmnsipgalbeblteaeeelslntyerdtnoeuasntdibeegerneufdsouesrdemtrouliapnctroaionovtnirpsdoaler.toifecausFblyouatrrhtsheelmtreho,ecdetli.tohne state and observation equations. LINEAR has several features that make it unique among the linearization programs common in the aerospace industry. The most significant of these features is flexi- bility. By generalizing the surface definitions and making no assumptions of sym- metric mass distributions, the program can be applied to any aircraft in any phase of flight except hover. The unique trimming capability, provided by means of a user- supplied subroutine, allows face scheduling, which are unlimited possibilities of trimming strategies particularly important for asymmetrical vehicles and and sur- aircraft having multiple surfaces affecting a single axis. The formulation equations of motion permits the inclusion of thrust-vectoring effects. The of the ability to select variables for models, without program the state, control, and observation vectors of the linear modification, combined with the vast array of observation variables available, allows any analysis problem to be attacked with ease. The interactive LINEAR program is an extension of the batch LINEAR program tasdiornteepasubcctirailisniebpf.teuiydtlesTatiohnnedfoDtruhacekoeenrttohpredraoyonlndgbaramatdoimctechhreitrvhpmsaorrtdooiegulv(rg1eah9sm8.c7aa)nf.otreAbelrlmDtihadnetedaafaltifanldeieodgrschactnrfiinobrpbieomtniawgnotmttoedwtdiahoeyffsi:deaaidtinatrecairrnfeaitssfleettter,sacoatfnosidrivmneioatllneyasdfrtiumlflertcnooasmsnieotonhnc-eaasle linear aerodynamic model as used in simulations at Ames-Dryden. NOMENCLATURE The units associated with the following quantities are system. LINEAR will work equally well with any consistent notable exceptions: the printed output and the atmospheric output and the atmospheric model assume English units. expressed in the English set of units with two model. Both the printed Where applicable, quantities are defined with respect to the body axis system. Variables A A‘ a an,i anx Aanx,i any any,i ang state matrix of the state equation x = Ax + Bu; or, axial force (lb) state matrix of the state equation cx = A'x + Bru speed of sound in air (ft/sec) normal acceleration not at center of gravity (g) x body axis accelerometer at center of gravity (g) x body axis accelerometer not at center of gravity (g) y body axis accelerometer at center of gravity (g) y body axis accelerometer not at center of gravity (g) 2 body axis accelerometer at center of gravity (9g) z body axis accelerometer not at center of gravity (g) acceleration along the x body axis (g) acceleration along the y body axis (9g) acceleration along the z body axis (9) control matrix of the state equation x = Ax + Bu control matrix of the state equation Cx = A'x + B'u wingspan (ft) C matrix of the state equation Cx = A'x + B'u; or, force or moment coefficient mean aerodynamic chord (ft) dynamic interaction matrix for the state equation x = Ax + Bu + Dv; or, drag force (1b) dynamic interaction matrix for the state equation cx = A'x + Blu + D'v determinant dynamic interaction matrix for the observation equation y = Hx + Fu + Ev dynamic interaction matrix for the observation equation y = H'x + Gx + Flu + E'v specific energy (ft) feedforward matrix of the observation equation y = Hx + Fu feedforward matrix of the observation equation y = H'x + Gx + Flu fpa flight path acceleration (g) G matrix of the observation equation y = H'x + acceleration due to gravity (£t/sec2 ) altitude (ft) observation matrix of the observation equation y = Hx + Fu H' observation matrix of the observation equation y = H'x + Gx + Flu angular momentum of engine rotor (slug-ft?/sec) aircraft inertia tensor (slug-ft2) rotational inertia of the engine (slug-ft?) x body axis moment of inertia (slug-ft?) x-y body axis product of inertia (slug-ft2) x~z body axis product of inertia (slug-ft?) y body axis moment of inertia (slug-ft2) y-z body axis product of inertia (slug-ft2) z body axis moment of inertia (slug-ft2) total body axis rolling moment (ft-lb) generalized length (ft) Mach number; or, total body axis pitching moment (ft-lb) aircraft total mass (slug) normal force (lb); or, total body axis yawing moment (ft-lb) load factor specific power (ft/sec) roll rate (rad/sec) ambient pressure (lb/ft?) total pressure (lb/ft?) pitch rate (rad/sec) dynamic pressure (1b/ft2) de impact pressure (1b/ft2) yaw rate (rad/sec) Re Reynolds number Re' Reynolds number per unit length (ft71) wing planform area (£t2) ambient temperature (°R); or, total angular momentum (slug-ft?/sec2) total temperature (°R) velocity in x-axis direction (ft/sec) control vector total velocity (ft/sec) velocity in y-axis direction (ft/sec) dynamic interaction vector calibrated velocity (ft/sec) equivalent velocity (ft/sec) vehicle weight (1b) velocity in z-axis direction (ft/sec) total force along the x body axis (1b) state vector thrust along the x body axis (1b) sideforce (1b) observation vector thrust along the y body axis (1b) total force along the z body axis (1b) thrust along the z body axis (1b) angle of attack (rad) angle of sideslip (rad) flight path angle (rad) displacement of engine from center of gravity along x body axis (ft) Ay displacement of engine from center of gravity along y body axis (ft) Az displacement of engine from center of gravity along z body axis (ft) lateral trim parameter longitudinal trim parameter incremental rolling moment (ft-lb) incremental pitching moment (ft-lb) incremental yawing moment (ft-lb) directional trim parameter thrust trim parameter incremental x body axis force (1b) incremental y body axis force (1b) incremental z body axis force (1b) pitch angle (rad) density of air (slug/ft3) torque from engines (ft-lb) gyroscopic torque from engines (ft-lb) roll angle (rad) tilt angle of acceleration normal to the flightpath from the vertical plane (rad) heading angle (rad) engine angular velocity (rad/sec) Superscripts derivative with respect to time transpose of a vector or matrix Subscripts ar aerodynamic reference point total drag h L L M m max min n p q r s x y Y z co) AIX AIXE AIXY AIXZ AIY AIYZ AIZ ALP ALPDOT altitude total lift rolling moment Mach number pitching moment maximum minimum yawing moment roll rate pitch rate yaw rate stability axis along the x body axis along the y body axis sideforce along the z body axis standard day, sea level conditions; or, along the reference trajectory FORTRAN Variables inertia about the x body axis engine inertia inertia coupling between the x and y body axes inertia coupling between the x and z body axes inertia about the y body axis inertia coupling between the y and z body axes inertia about the z body axis angle of attack time rate of change of angle of attack AMCH AMSENG AMSS B BTA BTADOT CBAR CD CDA CDDE CDO CDSB CL CLB CLDA CLDR CLDT CLFT CLFTA CLFTAD CLFTDE CLFTO CLFTQ CLFTSB CLP CLR CM Mach number total rotor mass of the engine aircraft mass wingspan angle of sideslip time rate of change of angle of sideslip mean aerodynamic chord total coefficient of drag coefficient of drag due to angle of attack coefficient of drag due to symmetric elevator deflection drag coefficient at zero angle of attack coefficient of drag due to speed brake deflection total coefficient of rolling moment coefficient of rolling moment due to beta coefficient of rolling moment due to aileron deflection coefficient of rolling moment due to rudder deflection coefficient of rolling moment due to differential elevator deflection total coefficient of lift coefficient of lift due to angle of attack coefficient of lift due to angle-of-attack rate coefficient of lift due to symmetric elevator deflection lift coefficient at zero angle of attack coefficient of lift due to pitch rate coefficient of lift due to speed brake deflection coefficient of rolling moment due to roll rate coefficient of rolling moment due to yaw rate total coefficient of pitching moment CMA CMAD CMDE CMO CMQ CMSB CN CNB CNDA CNDR CNDT CNP CNR CY CYB CYDA CYDR CYDT DAS DC DELX DELY DELZ DES coefficient of pitching moment due to angle of attack coefficient of pitching moment due to angle-of-attack rate coefficient of pitching moment due to symmetric elevator deflection pitching moment coefficient at zero angle of attack coefficient of pitching moment due to pitch rate coefficient of pitching moment due to speed brake deflection total coefficient of yawing moment coefficient of yawing moment due to sideslip coefficient of yawing moment due to aileron deflection coefficient of yawing moment due to rudder deflection coefficient of yawing moment due to differential elevator deflection coefficient of yawing moment due to roll rate coefficient of yawing moment due to yaw rate total coefficient of sideforce coefficient of sideforce due to sideslip coefficient of sideforce due to aileron deflection coefficient of sideforce due to rudder deflection coefficient of sideforce due to differential elevator deflection lateral trim parameter surface deflection and thrust control array displacement of the aerodynamic reference point along the x body axis from the vehicle center of gravity displacement of the aerodynamic reference point along the y body axis from the vehicle center of gravity displacement of the aerodynamic reference point along the z body axis from the vehicle center of gravity longitudinal trim parameter DRS DXTHRS EIX ENGOMG GMA H HDOT NUMSAT NUMSRF P PpoT PHI PHIDOT PSI PSIDOT Qo QBAR QDOT R RDOT Ss T TDOT THA THADOT THRSTX THRUST 10 directional trim parameter distance between the center of gravity of the engine and the thrust point rotational inertia of the engine rotational velocity of the engine flightpath angle altitude time rate of change of altitude number of states in the state vector number of recognizable control names roll rate time rate of change of roll rate roll angle time rate of change of roll angle heading angle time rate of change of heading angle pitch rate dynamic pressure time rate of change of pitch rate yaw rate time rate of change of yaw rate wing area time time rate of change of time pitch angle time rate of change of pitch angle thrust trim parameter thrust generated by each engine TLOCAT TVANXY TVANXZ UB Vv VB VCAS vboT VEAS WB x XDOT XYANGL XZANGL Y YDOT location of the engine in the x, y, and z axes from the center of gravity orientation of the thrust vector in the x-y engine axis plane orientation of the thrust vector in the x-z engine axis plane velocity along the x body axis velocity velocity along the y body axis calibrated airspeed time rate of change of total vehicle velocity equivalent airspeed velocity along the z body axis position north from an arbitrary reference point time rate of change of north-south position orientation of thrust vector in x-y body axis plane orientation of thrust vector in x-z body axis plane position east from an arbitrary reference point time rate of change of east-west position PROGRAM OVERVIEW The program LINEAR numerically determines a linear system model using nonlinear equations of motion and a user-supplied nonlinear aerodynamic model. LINEAR is also capable of extracting linearized gross engine effects, such as net thrust, torque, and gyroscopic effects and including these effects in the linear system model. The point at which this linear system model is defined is determined either by specifying the state and control variables or by selecting an analysis point on a trajectory, selecting a trim option, and allowing the program to determine the control variables and remaining state variables to satisfy the trim option selected. Because the program is designed to satisfy the needs of a broad class of users, a wide variety of options has been provided. Perhaps the most important of these options are those that allow user specification of the state, control, and observation variables to be included in the linear model derived by LINEAR. Within the program, the nonlinear equations senting a rigid aircraft flying in a stationary earth. Thus, internally, the state vector x is of motion include atmosphere over a computed as 12 states repre- flat, nonrotating x=([p qrvVvaBb $6 ph x yi" 11 The nonlinear equations used to determine the derivatives of the states are presented in the following section. The internal control vector u can contain up to 30 controls. The internal observation vector y contains 120 variables including the state variables, the time derivatives of the state variables, the control variables, and a variety of other parameters of interest. Thus, within the program, where y= [= x.T ul yT1° yT2 yT3 yT4 yP5 yP“6 yt“7 "|“8 T y11 = [a x a y a Zz a nx a ny a nz a n a nx,i a ny,i ‘ nz,i a n,i n]* yo =_ [a Re Re ' M q=d Gc Pa Qc/Pa Pt T Tt Ve Vo | T ¥3 = Ly fpa y. h"* ¢h@ /57.3 | T y4 = [Eg Pg] T ys= [Lb Dw a]? ye=[u vwudev e wi*5T Y7 == [o,4 Ba A h° al T and yg = [T Ps dg rg] The equations defining these quantities are presented in the Observation Equations section. From the internal formulation of the state, control, and observation variables, the user must select the specific vari- ables desired in the output linear model (described in the Selection of State, Con- trol, and Observation Variables section). Internally, the program uses a 12-state vector, 120-variable observation vector, and a 30-parameter control vector. These variables can be selected to specify the formulation most suited for the specific application. The order and number of parameters in the output model are com- pletely under user control. Figure 1 shows the selection of the variables in the state vector for a requested linear model. From the internal formulation on Output model Parameters y a le q 8 | Figure 1. ables for f \ [ Specification of ' state vector or linear model Selection of linear model. Internal parameters lp | a q V \ Q B > 6 L x y | 7 state vari- 12 the right, the requested model is constructed, and selected in accordance with the user specification vation variables. the linear system matrices of the state, control, and are obser- aiattsasrhlfreieeliicdtoemteodwistuzsolasTetnilehrhtperolert)yhaoifjwmlaseiiaioactrnnrrthteechbaoauriinarrsavtaysferlettrayvomsrsretoe.iy(rdapasiaelonlllrpyctop.wAoiolindonisdnepatnoirtgrtiitsbTvoi-ihaineolttetnsdnesrocevalweruhbbdlyayedibeaceosnhducaLdsftrlIetliyNafitibsEtighswieeAehnhsdRtei,adcorhnopiinsnotaposiaunatntshtddtiehreeatooittrnvameatdaleimrjrelirlimuenfsenicei,gntanaieorinnsdrdttypoie.uptmohlatoatnetliddruoeyanpl-naTAossnshnp,telataiialisnattoynpelnisarewtsiolovohsnyegicaslrcolabiahenlPsmdlotoiiuatdwtnrhpcLetineocrIosiiNent,nvElhDtueAeese.dRrefo.oarnriutnspiierLazodrtoIeenivNrfssoiEoiptn-dAneoReecs-s The linear system matrices are determined by numerical perturbation and are the first-order terms of a Taylor in the Linear Models section. user control. The user can series expansion about the analysis point as described The formulation of the output system model is under select state equation matrices corresponding to either the standard formulation of the state equation or the generalized equation “ i ] Ax + Bu Cx A'x + B'u The observation matrices can be selected from either of two formulations corresponding to the standard equation y = Hx + Fu or the generalized equation y = H'x + Gx + Flu In sional addition stability to the linear and control system matrices, derivatives at the LINEAR also computes the analysis point. These nondimen- derivatives are discussed in the Aerodynamic Model section. Aircraft geometry and mass properties and analysis point definition are input interactively to specified by the the program or are read from ASCII files of 80-character records user. The state, control, and observation vectors are also de- fined in either manner. The details of data input are discussed in the Data In- put section. The output of LINEAR consists of the following four files: one is intended to batmceianovntaetrlusriysocseleid,ess.xtwripataonhcdiTtnhetefosdoblssleeasotrcewvol-naetodtchniteoefndidalenesabviylegyccnstotiohnsretaanidpunnosaseimarenn,ta.alllyliassnidtTsosfheonpatehnrfedoigrcrsovttanarmtilsmau,oeifsn,sitntwhfoeosasrntedmadaboticftiluhioiemlntee,ysnsttaacttnhoteedhnetacsciopoonannpscttterrioootllnh,sseysdteassenrttmdiaavttaee,-, 13 and observation values, the trim control input positions, and the nondimensional stability and control derivative data. It will also contain the output of the equations of motion if the program fails to attain a trimmed condition. The third file contains all of the information in the second file plus the state space system model matrices. The fourth file contains the stability and control derivatives extracted at the analysis point in a form suitable to be read back into LINEAR as a linear aerodynamic model. To execute LINEAR, five user-supplied subroutines are required. These routines, discussed in the User-Supplied Subroutines section, define the nonlinear aerodynamic model, the gross engine model, the gearing between the LINEAR trim inputs and the surfaces modeled in the aerodynamic model, and a model of the mass and geometry properties of the aircraft. The control model, shown in figure 2, defines how the LINEAR trim inputs will be connected to the surface models and allows schedules and nonstandard trimming schemes to be used. The last feature is particularly important for asymmetrical aircraft. Inputs from ICTPARM/ DES ————_—_> DAS ——————|__ DRS ——————_»|__ THRSTX ———_» UCNTRL (Gearing) Outputs to ICONTRL/ 1» DC(1) -——> DC(2} . ° +—— DC(30) (Pilot stick, pedal, and throttle) (Surface deflections and power level setting) Figure 2. Inputs and outputs to the user-supplied subroutine UCNTRL. EQUATIONS OF MOTION The nonlinear equations of motion used in the linearization program are general six-degree-of-freedom equations representing the flight dynamics of a rigid aircraft flying in a stationary atmosphere over a flat nonrotating earth. The assumption of nonzero forward motion also is included in these equations. Because of this assumption, these equations are invalid for vertical takeoff and landing or hover. These equations contain no assumptions of either symmetric mass distribution or aerody- namic properties and are applicable to asymmetric aircraft (such as oblique-wing aircraft) as well as to conventional symmetric aircraft. These equations of motion were derived by Etkin (1972), and the derivation is detailed in the NASA Reference Publication, “Derivation and Definition of a Linear Aircraft Model," by Eugene L. Duke, Robert F. Antoniewicz, and Keith D. Krambeer. The following equations for rotational acceleration are used for analysis point definition: 14 where p= ((ZL)Iq + (EM)Ip + (EN)I3 - p2(Iyelo - Igyl3) + pq(Ixzly - Iyzt2 - DgI3) - pr(Iyyly + DyIg - TyzI3) + aq? (Iyzly ~ r2(Iy2Tq ~ IxyI3) - ar(Dyly - IxzI2)]/det I - Tyyt2 + Ixz13) q = [(ZL)Ig + (IM)Ig + (IN)I5 - p2(ty,I4 - Tyyl5) + pq(IxzI2 ~ Iyzly ~ DyIg) ~ pr(IyyIg + DyIg - Iyzts) + a? (Iyzl2 ~ Ixyt5) ~ ar(DyIg - IxyTg + Ixzt5) - r@(IygI2 - IxzI4)]/det I K e I [(2L)I3 + (EM)Is + (IN)Ig - p*(IyzI5 - Iyylg) + pq(IxzI3 - TygI5 - Dlg) - pr(IyyI3 + DyI5 - Iyzlg) + q*(IyzT3 - Ixylg) - ar(DyI3 - IRxyt5 + Ixztg) - r2(IyzI3 - IyzI5)]/det 1 det I = IylyIz - 2Ixylyglyz - Ixl*yz - Tyl?xz - Igl@xy I5 = Ixlyz + Ixylxz Ig = Ixly - I?xy Dy = Iz - ly Here, the body axis rates are designated p, q, and r, corresponding to roll rate, pitch rate, and yaw rate. The total moments about the x, y, and z body axes are designated IL, =M, and IN, respectively. These total moments are the sums of all 15 aerodynamic moments and powerplant-induced moments due to thrust asymmetries and gyroscopic torques. The equations used to determine the change in moment coef- ficients due to the noncoincidence of the vehicle center of gravity and the refer- ence point of the nonlinear aerodynamic model are derived in appendix A. The equa- tions defining the engine torque and gyroscopic contributions to the total moments are derived in appendix B. The body axis moments of inertia are designated I,, Ty. and Iz: The products of inertia are designated I xy’ I “x ze and Iyz° These moments and products of inertia are elements of the inertia tensor I, defined as Ix ~Ixy Ixy Ixz Iy -Iyz ~Ixz ~Iyz Iz To derive the state equation matrices for the generalized formulation Cx Ax + Bu the rotational accelerations are cast in a decoupled-axes formulation. tions used to derive the linearized matrices are The equa- 1 Ixy Ix Ixz 3) Ix Ixy ~Iyz . Ty ly q aTxL _y P > I x XI Z Iz, y“I z I, 1 Fae pq —— + rq aI + (q@ - r2) x HI + rq IPL - pg I HE + (2-?52) ly y y y aI=,N + —TILx, = qr ITyxz + pr II,yz + (p 2 - q*2) — The translational acceleration equations used in the program LINEAR for both analysis point definition and perturbation are Vv [-D cos B + ¥ sin 8 + Xp cos a cos B + Ym sin 8 + Zp sin a cos B ~ mg(sin 0 cos acos 8 - cos 0 sin $ sin 8B - cos Q cos ¢ sin a cos £B)]/m 16 a= [“L + Zp cos a - Xp Sin a+ mg(cos 8 cos ¢ cos a + sin 60 sin a)]/Vm cos 8B + q- tan B (p cos a+ r sin aq) B = [D sin 8 + Y cos B - Xp cos a sin 8 + Ym cos B - Zp sin a sin 8B + mg(sin 6 cos a sin 8 + cos 8 sin ¢ cos 8 - cos 8 cos $¢ sin a sin £)]/Vm + p sin a- r cos a The equations used to define the vehicle attitude rates are $ =p t+q sin $? tan 8+ r cos ¢ tan 6 8 q cos ¢- r sin 6 v q sin ¢ sec 6 + r cos » sec 6 The equations used to define the earth-relative velocities are W l r e Vi(cos 8 cos a sin 8 - sin 8 sin $¢ cos 6 - cos B sin a cos ¢ cos 9) K X e V[cos 8 cos a cos 6 cos ~ + sin B (sin > sin 6 cos ~ - cos $ sin jp) + cos 6B sin a (cos » sin 6 cos p+ sin @ sin »)] y = Vi[cos 8B cos a cos 8 sin ~ + sin 8 (sin ¢ sin 6 sin p+ cos ¢ cos jp) + cos 8 sin a (cos $¢ sin 8 sin p - sin $ cos })]) OBSERVATION EQUATIONS The user-selectable observation variables computed in LINEAR represent a broad class of parameters useful for vehicle analysis and control design problems. These variables include the state, time derivatives of state, and control variables. Also included are air data parameters, accelerations, flightpath terms, and other miscel- laneous parameters. The equations used to calculate those parameters are derived from a number of sources (Clancy, 1975; Dommasch and others, 1967; Etkin, 1972; Gainer and Hoffman, 1972; Gracey, 1980). Implicit in many of these observation equations is an atmospheric model. The model included in LINEAR is derived from the U.S. Standard Atmosphere (1962). The vehicle body axis accelerations constitute the set of observation variables that, except for state variables themselves, are most commonly used in the aircraft control analysis and design problem. These accelerations are measured in g units 17 and are derived directly from the body axis forces defined in the previous section for translational acceleration. The body axis acceleration equations used in LINEAR are ay = (Xp - D cos a + L sin a - gm sin 9)/g ym (Yp + Y + gm cos 8 sin $)/gym ag (Zp - D sin a - L cos a+ gm cos 6 cos $)/gm The equations for body axis accelerometers that are at vehicle center of gravity are an, = (Xp - D cos a + L sin a)/gom any = (¥p + ¥)/gym an, = (Zp - D sin a- L cos a)/gG om an = (-2p + D sin a+ L cos a)/gGom For orthogonal accelerometers that are aligned with the vehicle body axes but are not at vehicle center of gravity, the following equations apply: Nyx,i ~ 4nx 7 [(q? + x?) xy - (pq - LY x ~ (pr + D2x1/5, any i = any + [(pq + r)xy - (p2 + r)yy + (qr - p)Zy]/g, angi = an, + [(pr - qx, + (qr + Py, ~ (q2 + p*)z2]/9, an,i = ay - [lpr - q) x, + (qr + P)y, - (q2 + p*)zz]/9, where the subscripts x, y, and z refer to the x, y, and z body and the symbols x, y, and z refer to the x, y, and z body axis sors relative to the vehicle center of gravity. The symbol Jo axes, respectively, locations of the senis the acceleration due to gravity at sea level. Also included in the set of acceleration parameters is the load factor n = L/W, where L is the total aerodynamic lift and W is the vehicle weight. The air data parameters having the greatest application to aircraft dynamics and control problems are the sensed parameters and the reference and scaling parameters. The sensed parameters are impact pressure q,, ambient or free-stream pressure pa, total pressure p;,, ambient or free-stream temperature T, and total temperature T,. The selected reference and scaling parameters are Mach number M, dynamic pressure qe speed of sound a, Reynolds number Re, Reynolds number per unit length Re', and the Mach meter calibration ratio Ic/Pa> These quantities are defined as 1/2 a= (1.4 PPoolo n) 18 rae a R, € = Ve u Rg e' u= p&v - _ pv 4 2 [(1.0 + o.2m2)3°5 - 1.0]p, qd. Cc = 4 ( 1.2m2 (5.—62M2:2e~ 2M0e.8 2.5 - 1.0] p a ae ( (1.0 + 0.2mM2)"°°3?.5 ~ 14.0 Pp a [ t.2m2 {5.—62M2.:76- 2M0*.8 2.5 - 1.0 (Mm < 1.0) (M > 1.0) (M < 1.0) (mM > 1.0) Tt T(1.0 + 0.2M2) where p is the density of the air, yw is the script O refers to sea level, standard day stream temperature, and the coefficient of ard Atmosphere (1962). coefficient of viscosity, and the conditions. Free-stream pressure, viscosity are derived from the U.S. sub- freeStand- Also included in the air data calculations are two velocities: speed Ve and calibrated airspeed V,, both computed in knots. The that internal units are in the English system. speed is The equation used equivalent aircalculations assume for equivalent air- Ve = 17.17Vq which is derived from the definition of equivalent airspeed, Ole where Po = 0.002378 slug/ft3 and Ve is converted from feet per second to knots. ibrated airspeed is derived from the following definition of impact pressure: Cal- 19 3.5 fc =~ Po[\'*°1.0 *+=-To~p2, Yey? 1] (Vy c *§ Fo ag) 1 i t Vo 2 5.76 2.5 ° of “915.6 = 0.8 (ac/Vg)* ° For the case where Vo * ag, the equation for Vg is Vo = 1479.116 (2q + 1.0) 2/7 - 1.0 (Vy < ag) Po Calibrated airspeed The equation is found using an iterative process for the case where Vo ? age Vo = 582.95174 (= + 1-0) Po 1.0 7.0 1:0 (Vo/ao) 7 2.5 (Ve > ag) is executed until 0.001 knots. the change in V, from one iteration to the next is less than Also included in the observation variables are the flightpath-related parameters (described in app. C), including flightpath angle y, flightpath acceleration fpa, vertical acceleration h, flightpath angle rate Y and (for lack of a better category in which to place it) scaled altitude rate f/57.3. these quantities are The equations used to determine Jo h = a sin @ - ay sin ¢ cos 6 ~ a, cos » cos 8 -_ovh - iv Y= v Vv2 - £2 Two energy-related terms are included in the observation variables: energy Eg, and specific power P,, defined as Eg =h+ V42 2g Ww s g Po =ht— specific 20 The set of observation variables available in LINEAR also includes four force parameters: total aerodynamic lift L, total aerodynamic drag D, total aerodynamic normal force N, and total aerodynamic axial force A. These quantities are defined as L = qSCy, D = gSCp N= L cos a+ D sin a A = -L sin a+D cos a where Cp and Cy, are coefficients of drag and lift, respectively. Six body axis rates and accelerations are available as observation variables. These include the x body axis rate u, the y body axis rate v, and the z body axis rate w. Also included are the time derivatives The equations defining these quantities are of these quantities, u, vy, and We u i H V cos a cos £6 v= V sin £ w = V sin acos B . Xm — gm sin 6 - D cos a+ L sin a u = m + rv - qw ° Yp + gm cos 6 sin ¢+ Y¥ v= m + pw - ru ° Zp + gm cos 8 cos 9 - D sin a-~ L cos a The final set of observation variables available in LINEAR is a miscellaneous collection of other parameters of interest in analysis and design problems. The first group consists of measurements from sensors not located at the vehicle center of gravity. These represent angle of attack a,i, angle of sideslip B,i, altitude his and altitude rate hi measurements displaced from the center of gravity by some x, y, and z body axis distances. The equations used to compute these quantities are ai — = o — ( {AX_v—_by ) Bhi = B+ (>) h,i = h+ x sin 6- y sin $¢ cos 6 = 2 cos ¢ cos 6 21 hoi =h+ (x cos 6 + y Sin » sin 6 + z cos q¢ sin 6) - oly cos 9 cos 8 - z sin » cos 6) The remaining roll rate pg, as miscellaneous parameters are total angular momentum T, stability axis stability axis pitch rate q,, and stability axis yaw rate r s’ defined T =-12 (Ixp 2 - 21.xyP4 - 21,,,Pr + Ta 2_-. 2tyzar + I,ré2) Ps = pcos a+r sina ds = q Yg = -p Sin a+r cos a SELECTION OF STATE, CONTROL, AND OBSERVATION VARIABLES The equations in the two preceding sections define the state and observation variables available in LINEAR. The control variables are defined by the user in the input file. Internally, the program uses a 12~state model, a 120-variable obser- vation vector, anda 30-parameter control vector. These variables can be selected to specify the formulation most suited for the specific application. The order and number of parameters in the output model are completely under user control. Figure 1 shows the selection of variables for the state vector of the output model. However, it should be noted that no model order reduction is attempted. Elements of the matrices in the internal formulation are simply selected and reordered in the for- mulation specified by the user. Specific state, control, and observation variables for the formulation of the output matrices are selected by alphanumeric descriptors in the input file. The use of these alphanumeric descriptors is described in the Data Input section. Appendix C lists the observation variables and their alphanumeric descriptors. Appendix D lists the state variables and their alphanumeric descriptors. The alphanumeric descriptors for the selection of control parameters to be included in the observation vector are the control variable names defined by the user in the input file, as described in the Input Files section. LINEAR MODELS The linearized system matrices computed by LINEAR are the first-order terms of a Taylor series expansion about the analysis point (Dieudonne, 1978; Kwakernaak and Sivan, 1972; NASA RP by Duke, Antoniewicz, and Krambeer) and are assumed to result in a time-invariant linear system. The validity of this assumption is discussed in the Analysis Point Definition section. The technique used to obtain these matrices numerically is a simple approximation to the partial derivative, that is, dg £ (xq + AK) - E(x ~ dx) Ox 2 dx where f is a general function of x, an arbitrary independent variable. The Ax may be set by the user, but defaults to 0.001 for all state and control parameters, with the single exception of velocity V, where Ax is multiplied by a, the speed of sound, to obtain a reasonable perturbation size. From the generalized nonlinear state equation and the observation equation Tx £ (x, x, u) Y g(x, x, u) the program determines the linearized matrices for the generalized formulation of the system: Cc éx = A' 6x + B' éu oy H' 6x + G éx + F' éu where Q I B t | At =—_— 23 with all derivatives evaluated along the nominal trajectory defined by point (Xo, Xo, Uo). The state, expressed as small perturbations time derivative of state, and about the nominal trajectory, control so that the analysis vectors can be KX = Xo + 6x Xs = Xeo + 6xe u=Uo+ du of In addition to the matrices for this requesting linearized matrices for a generalized system, standard formulation the user has the of the system: option é&xk = A 6x + B 6u where éy = H 6x + F fu a=|r r-2 7-1 2 a ox | * B= r- pie a ox n= 2842p[ ahr 7)- at a ox| °* r= 2“3,a%x 1-2Lox]13 with all derivatives evaluated along the nominal trajectory defined by the analysis point (Xo, Xor Uo). LINEAR also provides two nonstandard matrices, D and E, in the equations xX = Ax + Bu + Dv or D' and E' in the equations Y Hx + Fu + Ev Cx = A'x + Blu + D'v y = H'x + Gx + F'u + Ey 24 These dynamic interaction matrices include the effect acting on the vehicle. The components of the dynamic mental body axis forces and moments: of external interaction forces vector and moments v are incre- is éY 6Z éL 6M L ON Thus, and E' = 2g I s l e [ an of ov Oxe T - ~a~x noTv These matrices allow the effects of unusual subsystems or control effectors to be easily included in the vehicle dynamics. The default output matrices for LINEAR are those for the standard system formulation. However, the user can select matrices for either generalized or standard state and observation equations in any combination. Internally, the matrices are computed for the generalized system formulation and then combined appropriately to accommodate the system formulation requested by the user. ANALYSIS POINT DEFINITION The point at which the nonlinear system equations are linearized is referred to as the analysis point. This can represent a true steady-state condition on the spec~ ified trajectory (a point at which the rotational and translational accelerations are zero; Perkins and Hage, 1949; Thelander, 1965) or a totally arbitrary state on a tra- jectory. LINEAR allows the user to select from a variety of analysis points. These analysis points are referred to as trim conditions, and several options are available to the user. The arbitrary state and control option is designated NOTRIM, and in selecting this option the user must specify all nonzero state and control variables. For the equilibrium conditions, the user specifies a minimum number of parameters, and the program numerically determines required state and control variables to force 25 the rotational and translational accelerations to zero. are described in detail in the following subsections. The analysis point options For all the analysis point definition options, any state or control parameter may be input by the user. Those state variables not required to define the analysis point are used as initial estimates for the calculation of the state and control conditions that result in zero rotational and translational accelerations. As each state variable is read into LINEAR, the name is compared to the list of alternative state variable names listed in appendix D. All state variables except velocity must be specified according to this list. Velocity can also be defined by specify- ing Mach number (see alternative observation variable names in appendix C). Appen- dix E lists analysis point definition identifiers that are recognized by LINEAR. It should be noted that the option of allowing the user to linearize the system equations about a nonequilibrium condition raises theoretical issues (beyond the scope of this report) the potential user should be aware of. Although all the analysis point definition options provided in LINEAR have been found to be useful in the analysis of vehicle dynamics, not all the linear models derived about these analysis points result in the time-invariant systems assumed in this report. However, the results of the linearization provided by LINEAR do give the appearance of being time invariant. The linearization process as defined in this report is always valid for some time interval beyond the point in the trajectory about which the linearization is done. However, for the resultant system to be truly time invariant, the vehicle must be in a sustainable, steady-state flight condition. This requirement is something more than merely a trim requirement, which is typically represented as x(t) = 0, indicating that for trim, all the time derivatives of the state variables must be zero. (This is not the case, however: Trim is achieved when the acceleration-litekrems are identically zero; no constraints need to be placed on the velocity-like terms in Ke Thus, for the model used in LINEAR, only PB, qe r, V, Oy and é must be zero to satisfy the trim condition.) The trim condition is achieved for the straight-andlevel, pushover-pullup, level turn, thrust-stabilized turn, and beta trim options described in the following sections. In general, the untrimmed and specific power analysis point definition options do not result in a trim condition. Of these analysis point options resulting in a trim condition, only the straight- and-level and level turn options force the model to represent sustainable flight con- ditions. In fact, only in the special case where the flightpath angle is zero does a truly sustainable trim occur. As previously stated, the linearization of a nonlinear model and its represen- tation as a time-invariant system are always valid for some time interval beyond the analysis point on the trajectory. This time interval is determined by several fac- tors (such as trim and sustainable flight conditions) and ultimately by accuracy requirements placed on the representation. Thus, in using the linear models provided by this program, the user should exercise some caution. Untrimmed For the untrimmed analysis point, the user specifies all state and control variables that are to be set at some value other than zero. The number of state var- 26 iables specified is entirely at the user's discretion. If any of the control var- iables are to be nonzero, the user must specify the control parameter and its value. The untrimmed option allows the user to analyze the vehicle dynamics at any flight condition, including transitory conditions. Straight-and-Level Trim The straight-and-level analysis points available in LINEAR are in fact wings- level, constant-flightpath-angle trims. Both options available for straight-and- level trim allow the user to specify either a flightpath angle or an altitude rate. However, since the default value for these terms is zero, the default for both types of straight-and-level trim is wings-level, horizontal flight. The two options available for straight-and-level trim require the user to specify altitude and either an angle of attack or a Mach number. If a specific angle of attack and altitude combination is desired, the user selects the “Mach=-trim" option, which determines the velocity required for the requested trajectory. Likewise, the "alpha-trim" option allows the user to specify Mach number and altitude, and the trim routines in LINEAR determine the angle of attack needed for the requested trajectory. The trim condition for both straight-and-level options is determined within the following constraints: p=q=re=od0 ¢= 0 The trim surface positions, thrust, angle of sideslip, and either velocity or angle of attack are determined by numerically solving the nonlinear equations for the translational and rotational acceleration. Pitch attitude 8 is determined by itera-~ tive solution of the altitude rate equation. Pushover-Pullup The pushover-pullup analysis points result in wings-level flight at n # 1. For n> 1, the analysis point is the minimum altitude point of a pullup when h=0. For n < 1, this trim results in a pushover at the maximum altitude with h = 0. There are two options available for the pushover-pullup analysis point definition: (1) the “alpha-trim" option in which angle of attack is determined from the specified alti- tude, Mach number, and load factor, and (2) the "load-factor-trim" option in which angle of attack, altitude, and Mach number are specified, and load factor is deter- mined according to the constraint equations. The analysis point is determined at the specified lowing contraints: conditions, subject to the fol- p=r=0 q e Yme co1s 8 [mg(n - cos(6 - a)) ~ Zp cos a + Xp sijn a] ¢= 0 27 The expression for q is derived from the a equation by setting a = 0 and ¢ = 0; Ois derived from the h equation. The trim surface and either angle of attack or load factor are nonlinear equations for the translational and positions, determined rotational thrust, angle by numerically accelerations. of sideslip, solving the Level Turn The level turn analysis point definition options result in nonwings-level, constant-turn-rate flight at load factors greater than one. The vehicle model is assumed to have sufficient excess thrust to trim at the condition specified. If thrust is not sufficient, trim will not result, and the analysis point thus defined will have a nonzero (in fact, negative) velocity rate. The level trim options available in LINEAR require the specification of an alti- tude and a Mach number. The user can then use either angle of attack or load factor to define the desired flight condition. These two options are referred to as “"alpha- trim" and "load-factor-trim," respectively. For either option, the user may also request a specific flightpath angle or altitude rate. Thus, these analysis point definitions may result in ascending or descending spirals, although the default is for the constant-altitude turn. The constraint equations for the coordinated level turn analysis point definitions are derived by Chen (1981), and Chen and Jeske (1981). Using the requested load factor, the tilt angle of acceleration normal to the flightpath from the vertical plane 47, is calculated using the equation dy, = ttan™1 E = ccooss? wt) Y where the positive sign is used for a right turn and the negative sign is used for a left turn. From 9$,, turn rate can be calculated as b= deana Using these two definitions, the state variables can be determined: 1 Q t l y sin? g |-sin y sin B + (ss02 y sin? g - sin? Y- cos sin2 or, :) 1//2 Le S = tan @7, cos B = -¥SiIN Y _ tan P. cos Bp 4 8 P= Ps, COS O- Lg sin a r= P, sina + Ys cos a 28 o c I sin71(-p/p) > tan7l(g/r) The trim surface positions, thrust, angle of sideslip, and either angle of attack or load factor are determined by numerically solving the nonlinear equations for the translational and rotational accelerations. Thrust-Stabilized Turn The thrust-stabilized turn analysis point definition results in a constant throt- tle nonwings~level turn with a nonzero altitude rate. The two options available in LINEAR are "alpha-trim" and "load-factor-trim." These options allow the user to spe- cify either the angle of attack or the load factor for the analysis point. The alti- tude and Mach number at the analysis point must be specified for both options. The user also must specify the value of the thrust trim parameter by assigning a value to the variable THRSTX in the input file after the trim has been selected. The constraint equations for the thrust-stabilized turn are the same as those for the level turn. However, for this analysis point definition, flightpath angle is determined by LINEAR. Beta Trim The beta trim analysis point definition results in nonwings-level, horizontal flight with the heading rate d equal to zero at a user-specified Mach number, alti- tude, and angle of sideslip. This trim option is nominally at 1 g, but as 8B and vary from zero, normal acceleration decreases and lateral acceleration increases. For an aerodynamically symmetric aircraft, a trim to £8 = 0 using the beta trim option results in the same trimmed condition as the straight-and-level trim. However, for an aerodynamically asymmetric aircraft, such as an oblique-wing vehicle, the two trim options are not equivalent. The constraint equations used with the beta trim option are The trim surface positions, thrust, angle of attack, and bank angle are determined by numerically solving the nonlinear equations for translational and rotational acceleration. Pitch attitude 8 is derived from the equation for flightpath angle y with y = 0 and is defined as 6 = tan-1 (2 B sin » + cos B sin =) cos 8 cos a Specific Power The specific power analysis point definition results in a level turn at a user- specified Mach number, altitude, thrust trim parameter, and specific power. Unlike 29 the other trim options provided in LINEAR, the Specific power option does not, in general, attempt to achieve the zero velocity rate Vv. In fact, because the altitude rate h = 0 and specific power are defined by the resultant velocity rate will be v= 82 Vv However, the other acceleration-like terms (p, qe r, Oy and B) will be zero if the requested analysis point is achieved. The constraint equations used with the option can be derived from the load factor analysis point definition with y = 0: specific power analysis point definition tilt angle equation used in the level turn OL = ttan71(n2 - 1) 1/7? (where the positive sign is used for a right turn and the negative sign is used for a left turn), yy= 7Fd tan qy, q = p sin dy, cos £ Ys = tan 91S,. cos £ Ps = -q tan B P= Pg coS a- rg sin a Y= Pg sin a + Yg COS o mf5nH.f —_> “4a -1 CLs; rt a n e + Crop + Crd + CLF + Cyga + Cr 38 n Cp = Cpg + Cpga + Cr gb + Ch, dh + Cp, ov + 2 D6; 83 1= + Cp.B + Cpud + Cp f + a Cpta + a Cp g8e 31 Cy = Cyg + Cya t+ CygB + Cy, St cy vt n i=SY1 cy 6; ok aA + Cy ¥pPB++ CCyy aG++ CyCry,t+tcoYtQ ai”t+c?y e 38 where the stability and control derivatives have the usual meaning, Cex FT with C, being an arbitrary force or moment sional variable. The rotational terms in of the corresponding state variable with coefficient the equations and are x an arbitrary nondimensional nondimenversions P- ™. b2Vpv a4 *= £o&vJ p= b2eV 2V * _ ca : _ bB BR 5 The 6; in the summations are the n control variables effects of altitude and velocity are included in the those parameters and in the incremental multipliers 6h =h = ho defined by the user. The derivatives with respect to " < t < fe) 6V where the subscript zero represents the current analysis described in the Analysis Point Definition section. point (xo, Xo, Uo), All stability tude and velocity units used in the Mach and velocity derivatives are computed as nondimensional terms except the parameters. The units of the control nonlinear aerodynamic model. LINEAR derivatives. Using the equation derivatives is capable of correspond handling altito the both Cey = Cey 4h (where a, is the Mach derivatives speed of from the sound at velocity the reference altitude), LINEAR will compute the derivatives or vice versa. Internally, however, 32 LINEAR uses velocity derivatives in the computations of the total force and moment coefficients. Derivatives with respect to angle of attack and angle of sideslip can be obtained in units of reciprocal degrees. These derivatives are simply the cor- responding nondimensional derivatives multiplied by 180/n. DATA INPUT The interactive linearizer allows the user to input data in various manners: 1. All input data can be loaded from a single file (either a file assigned to logical unit 1 or any file named with 10 characters or less); 2. Data can be separated onto different files, each containing different sec- tions of data; 3. Data can be typed in from the terminal; or 4. A combination of (2) and (3). Input Files The interactive LINEAR input file (defined in table 1) is separated into six major sections: vehicle title information; geometry and mass data for the aircraft; state, control, and observation variable definitions for the state-space model; the trim parameter specification; additional control surfaces that may be specified for each case; and various test case specifications. All the input data can be written on one file or various files according to the divisions specified at the beginning of this paragraph and according to the input format defined in table 1. An example input file is listed in appendix F. All the input records to LINEAR are written in ASCII form. Title Information There is one title record name of the specific vehicle. title appears on each page of that needs to be specified for interactive LINEAR: the This record is read with a 20A4 format. The vehicle the line printer output file. Geometry and Mass Data The geometry and mass data consist of four records that either can follow the vehicle title, exist as a file on its own, or be input interactively from the ter- minal. If it is input from the terminal, then it will be stored either on the same file as the vehicle title, or on a separate file. The geometry and mass data records define the geometry, mass properties, location of the aerodynamic reference point with respect to the center of gravity, and angle-of-attack range for the vehicle model being analyzed. 33 TABLE 1. — INPUT FORMAT FOR LINEAR Input record VEHICLE TITLE Ss b c Ix ly I, DELX Omin NUMSAT STATE 1 STATE 2 STATE 3 DELY Oma x EQUAT DRVINC DRVINC DRVINC Weight Tye Ixy DELZ STAB 1 2 3 LOGCG Format (20A4) (4F13.0) (6F13.0) (3F10.0,12A4) (2F13.0) (110,A4,11X,A4) (5A4,F10.0) (5A4,F10.0) (5A4,F10.0) NUMSUR CONTROL 1 CONTROL 2 CONTROL 3 LOCCNT 1 LOCCNT 2 LOCCNT 3 CONVR 1 CONVR 2 CONVR 3 CNTINC 1 CNTINC 2 CNTINC 3 (110) (5A4,110,A4,6X,F10.0) (5A4,110,A4,6X,F10.0) (5A4,110,A4,6X,F10.0) NUMYVC MEASUREMENT MEASUREMENT MEASUREMENT EQUAT 1 2 3 PARAM 1 (1-3) PARAM 2 (1-3) PARAM 3 (1-3) (I110,A4) (5A4,3F10.0) (5A4,3F10.0) (5A4,3F10.0) Semin Semax NUMXSR ADDITIONAL ADDITIONAL ADDITIONAL Samin Samax SURFACE 1 SURFACE 2 SURFACE 3 Srmin Srmax LOCCNT 1 LOCCNT 2 LOCCNT 3 Sthmin Sthmax (SF 10.0) (1X,12) (5A4,110,A4) (5A4,110,A4) (5A4,110,A4) ANALYSIS ANALYSIS VARIABLE VARIABLE VARIABLE POINT POINT 1 2 3 DEFINITION DEFINITION VALUE 1 VALUE 2 VALUE 3 OPTION SUBOPTION (20A4) (A4) (5A4,F15.5) (5A4,F15.5) (5A4,F15.5) 34 TABLE 1. — Concluded Input record NEXT ANALYSIS ANALYSIS VARIABLE VARIABLE VARIABLE POINT POINT 1 2 3 DEFINITION DEFINITION VALUE 1 VALUE 2 VALUE 3 OPTION SUBOPTION Format (A4) (20A4) (A4) (5A4,F15.5) (5A4,F15.5) (5A4,F15.5) END (A4) 35 The first and second records describe the vehicle geometry, mass, and mass dis- tribution. The first record defines the wing planform area S in units of feet squared; the wingspan b in units of feet; the mean aerodynamic chord of the wing c in units of feet; and the sea level weight of the vehicle Weight, in units of pounds. The second record defines the vehicle moments and products of inertia in units of slug-feet squared. The third record defines the offset of the aerodynamic reference point with respect to the vehicle center of gravity in the normal right-handed body axis reference system with the positive x axis forward. The format of the data can be seen in table 1 with DELX, DELY, and DELZ representing the x, y, and z body axis displacements of the aerodynamic reference point with respect to the center of gra- vity in units of feet (see app. A). The fourth variable of the third record is an alphanumeric variable, read using a 12A4 format, to specify if corrections due to a center-of-gravity offset are to be computed in LINEAR or in the user-supplied aerody- namic subroutine CCALC. The variable LOGCG defaults to a state that causes the aero- dynamic reference point offset calculations to be performed by subroutines within LINEAR. Any of the following statements in the LOGCG field will cause LINEAR not to make these corrections: NO CG CORRECTIONS BY LINEAR CCALC WILL CALCULATE CG CORRECTIONS FORCE AND MOMENT CORRECTIONS CALCULATED IN CCALC However, LINEAR will read only the first four characters of the string. The final record of this geometry and mass data set defines the angle-of-attack range for which the user-supplied nonlinear aerodynamic model (CCALC) is valid. The parameters Omin and max specify the minimum and maximum values of angle of attack to be used for trimming the aircraft model. These parameters are in units of degrees. State, Control, and Observation Variable Definitions The state, control, and observation variables to be used in the output formula- tion of the linearized system are defined in records that either follow the last of the previously described sets of records on the same file, are stored on a separate file, or are input through the terminal. If they are input through the terminal, they will be stored in a file specified by the user. The number of records in the state, control, and observation variable definition set is determined by the number of such variables defined by the user. The states to be used in the output formulation of the linearized system are defined in the first set of records in the state, control, and observation variable definitions. The first record of this set, as shown in table 1, defines the number of states to be used (NUMSAT), the formulation of the state equation (EQUAT), and whether the nondimensional stability derivatives with respect to angle of attack and angle of sideslip are to be output in units of reciprocal radians or degrees. The variable EQUAT is read using an A4 format and is tested against the following list: 36 NONSTANDARD NON~STANDARD GENERALIZED EXTENDED If EQUAT matches the first four characters of any of the listed words, the output formulation of the state equation is Cx = A'x + Btu If EQUAT is read in as STANDARD or does not match the preceding list, fault standard bilinear formulation of the state equation is assumed, put matrices are consistent with the equation then the deand the out- x = Ax + Bu The variable STAB is also read using an A4 format and is compared with the following list: DEGREES DGR If STAB matches the first four characters of either of these words, the nondimen- sional stability derivatives with respect to angle of attack and angle of sideslip are printed in units of reciprocal degrees on the printer file. Otherwise, these derivatives are printed in units of reciprocal radians. The remaining records of the state variable definition set are used to specify the state variables to be used in the output formulation of the linearized system and the increments to be used for the numerical perturbation described in the Linear Models section. The state variable names are checked for validity against the state variable alphanumeric descriptors listed in appendix D. If a name is not recognized, the variable is ignored and a warning message is written to the printer file. The increment to be used with any state variable (in calculating the A' and H' matrices) and the time derivative of that state variable (in calculating the C and G matrices) can be specified using the DRVINC variable. The DRVINC specified for velocity will be multiplied by the speed of sound within LINEAR in order to scale up the pertur- bation size to a reasonable value while keeping DRVINC on the same order of magnitude as for the other states. If DRVINC is not specified by the user, the default value of 0.001 is used, The next set of records in the state, control, and observation variable defini- tions are those defining the variables to be used in the control vector of the output model. The first record of this set defines the number of control parameters to be used (NUMSUR). The remaining records define the names of these variables (CONTROL), their location (LOCCNT) in the common block /CONTRL/ (see the User~Supplied Sub- routines section), the units associated with these control variables (CONVR), and the increments (CNTINC) to be used with these variables in determining the B' and F' matrices. Because LINEAR has no default control variable names, the control variable names input by the user are used for subsequent identification of the control variables. Therefore, consistency in the use of control variable names is extremely important, 37 particularly when the user attempts to establish control variable initial conditions when using the untrimmed analysis point definition option. The CONVR field in the control variable variables are given in degrees or radians. compared to the following list: records is used to specify if the control CONVR is read using an A4 format and is DEGREES DGR RADIANS RAD If CONVR agrees with the first four characters of either of the first two names, it is assumed that the control variable is specified in units of degrees. If CONVR agrees with the first four characters of either of the last two listed names, it is assumed that the control variable is specified in units of radians. No units are assumed if CONVR does not agree with any of the listed names. When it is assumed that the control variable is specified in units of radians, the initial value of the control variable is converted to degrees before being written to the printer file. The variable CNTINC can be used to specify the increments used for a particular control surface when the B' and F' matrices are being calculated. It is assumed that the units of CNTINC agree with those used for the surface, and no unit conversion is attempted on these increments. If a default value of 0.001 is used. CNTINC is not specified for a particular surface, The final set of records in the state, control, and observation variable defini- tions pertain to the specification of parameters associated with the observation vec- tor, observation equation, and observation parameters. The first record of this set defines the number of observation variables (NUMYVC) to be used in the output linear model and the formulation of the output equation (EQUAT). The remaining records in this set specify the variables to be included in the observation vector (MEASUREMENT) and any position information (PARAM) that may be required to compute the output model for a sensor not located at the vehicle center of gravity. The variable used to specify the formulation of the observation equation (EQUAT) is compared with the same list of names used to determine the formulation of the state equation. If it is determined that the generalized formulation is desired, the observation equation y = H'x + Gx + F'u is used. Otherwise, the standard vation equation used is formulation is assumed, and the form of the obser- y = Hx + Fu The records defining the observation variables to be used in the output formulation of the linear model contain a variable that includes the parameter name (MEASUREMENT) and three fields (PARAM) defining, when appropriate, the location of the sensor relative to the vehicle center of gravity. The parameter name is compared with the list of observation variables given in appendix D. If the parameter name is recognized as a valid observation variable name, that observation variable is 38 included in the formulation of the output observation vector. If the parameter name is not recognized, an error message is printed and the parameter named is ignored. The three variables represented by PARAM(1), PARAM(2), and PARAM(3) provide the x- axis, y-axis, and z-axis locations, respectively, of the measurement with respect to the vehicle center of gravity if the selected observation is one of the following: The unit associated with these variables is feet. If the selected observation var- iable is not in the preceding list, the PARAM variables are not used. The sole exception to this occurs when Reynolds number is requested as an observation variable. In that case, PARAM(1) is used to specify the characteristic length. When no value is input for PARAM(1), the mean aerodynamic chord ¢ is used as the character- istic length. Trim Parameter Specification There is one record in the trim parameter specification set that is associated with the subroutine UCNTRL (described in the User-Supplied Subroutines section). This record specifies the limits to be used for the trim parameters 64, Sar Sy, and Sth representing the longitudinal, lateral, directional, and thrust trim parameters, respectively. The units associated with these parameters are defined by the implementation of UCNTRL. Additional Surface Specification The first record of the set of additional surface specifications defines the number of additional controls to be included in the list of recognized control names (NUMXSR). The purpose of defining these additional controls is to allow the user to set such variables as landing gear position, wing sweep, or flap position. The controls are only defined in the additional surface specification records; actual control positions are defined in the analysis point definition records. If such controls are defined, the records defining them will have the format specified in table 1. The control variable name (ADDITIONAL SURFACE), location (LOCCNT) in the common block /CONTRL/, and the units associated with this control variable (CONVR) are specified for each additional control. 39 Test Case Specification The test case specification records allow the user to define the flight condi- tion, or analysis point, at which a linear model is to be derived. Multiple cases can be included in the test case specification records. The final record in each set directs the program to proceed (NEXT) or to stop (END) execution. The first record of a test case specification set determines the analysis point, or trim, option to be used for the current case. The ANALYSIS POINT DEFINITION OPTION parameter is read in and checked against the list of analysis point defini-~ tion identifiers described in appendix E. The second record of a test case specifi- cation set, defining an analysis point definition suboption (ANALYSIS POINT DEFIN~ ITION SUBOPTION), will be read only if the requested analysis point definition option has a suboption associated with it. These suboptions are defined in the Analysis Point Definition section. The valid alphanumeric descriptors for these suboptions are described in appendix E. The remaining records in a test case specification set define test conditions or initial conditions for the trimming subroutines. These records consist of a field defining a parameter name (VARIABLE) and its initial condition (VALUE). ‘These records may be in any order; however, if initial Mach number is to be defined, the altitude must be specified before Mach number if the correct initial velocity is to be determined. The parameter names are checked against all name lists used within LINEAR. Any recognized state, time derivative of state, control, or observation var- iable will be accepted. Trim parameters also can be set in these records. In general, setting observation variables and time derivatives of the state vari- ables has little effect. However, for some of the trim options defined in_the Anal- ysis Point Definition section, Mach number and load factor are used. The thrust trim parameter only affects the specific power trim. For the untrimmed option, the initial values of the state and control variables determine the analysis point com- pletely. For all other trim options, only certain states are not varied; all con- trols connected to the control and engine models are varied. Interactive Data Input Upon starting program execution, LINEAR will ask the user: WHAT IS THE TITLE OF THE CASE BEING RUN? The user should answer with some meaningful title followed by a carriage return. This title, along with the vehicle title discussed later, will identify this interactive session on all of the output data files. The program will then display the following menu: PROGRAM LINEAR READS ALL INPUT DATA FROM 1 UNLESS A DIFFERENT FILE NAME IS SPECIFIED. 1) ALL INPUT DATA IS ON 1 2) DATA IS ON ONE OTHER FILE 40 3) DATA IS ON VARIOUS FILES 4) INPUT DATA FROM A FILE(S) AND FROM THE TERMINAL 5) INPUT ALL DATA FROM THE TERMINAL The user must answer with the appropriate number followed by a carriage return. If the user answers with a "1", the program will read all the input data from the file assigned logical unit one. If the user selects choice "2", the program will ask for the file name that contains all the input data required by LINEAR. If the user selects "3" or "4", the program will prompt the user for each file name pertaining to each section of input data. If option "5" is chosen, the program will prompt the user for the input data values. The data is separated into the six sections described previously: 1. Vehicle title information, 2. Vehicle geometry and mass properties, 3. State, control, and observation variable definitions, 4. Limits for the trim parameters, 5. Additional control surface definitions, and 6. Test case specifications. The program will prompt the user for the name of the file that contains each section of data. An example prompt follows: WHAT IS THE NAME OF THE FILE CONTAINING THE TITLE INFORMATION? (TYPE 0 TO INPUT THE TITLES FROM THE TERMINAL AND STORE ON A FILE) If the user already has created a file containing the appropriate data needed for the section, type the a 10-character (maximum) file name may be entered. data from the terminal and have the program format If the a file, user wishes to a zero may be entered and the program will prompt WHAT WOULD YOU LIKE THE FILE TO BE CALLED? At this time the user must input a file name (10 characters maximum) on which all data will typed from the terminal will then prompt the user for the be stored for specific data that section it needs for of input data. this section. LINEAR If the user wishes to input all the data from the be chosen and the program will prompt for a data file input data. The program will prompt the user for all terminal, selection "5" should name on which to store the the input data required, 41 When acters. floating inputting the title data, the program reads character strings of 80 charWhen prompting for vehicle mass and geometry data, the program reads a point field. An example data prompt is as follows: INPUT THE VEHICLE GEOMETRY AND MASS PROPERTIES WING AREA (FT**2) = The user would input the vehicle wing area in square feet followed by a carriage return. The remainder of the vehicle geometry data is input similarly. When inputting the state vector from the terminal, the and formulation of the state equation as well as the units attack and sideslip stability derivatives will be output. user must define the size in which the angle of The program will prompt HOW MANY STATE VARIABLES ARE IN YOUR MODEL? (MAXIMUM 12) The program will read an integer variable and then prompt WHICH FORM OF THE STATE EQUATION DO YOU WISH TO USE? 1) DX/DT = A*X + B*U + D*V 2) C * DX/DT A*X + B*U + D*V If the standard formulation is being used, the user should input "1" and for the generalized equation, "2". The LINEAR program uses the generalized formulation for internal calculations and then performs any transformations necessary for the output formulation requested. The program will then prompt for the units of the angle of attack and sideslip stability derivatives. The program uses radians for all inter- nal calculations and transforms the stability derivatives to degrees for output if desired. The variables used in the state vector are input next and are read in strings 20 characters in length. Each variable name entered is checked against legal names for state variables. If a variable is incorrect, the program will respond VARIABLE so~and-so IS AN INVALID STATE PARAMETER PLEASE CHOOSE ANOTHER OR TYPE HELP FOR MORE INFO. If the user types HELP, the program will list all the valid names for the state variables. The names are given in appendix D. After the entire state vector has been entered, the program will prompt THE STEP SIZE USED FOR DERIVATIVE EXTRACTION IS INITIALIZED FOR ALL VARIABLES TO 0.001. DO YOU WISH TO INPUT A DIFFERENT VALUE FOR ANY OF THE STATE VARIABLES? (Y/N) 42 To change the perturbation step size for any of the state variables, the user should enter "Y" and, after being prompted, the variable name to be updated and the desired step size. For the interactive input of the control variables, similar prompts are given with one major difference: after each variable name is input, the program prompts for the location of the variable in the common block /CONTRL/ (the named common block /CONTRL/ reserves 30 locations for control surface variable values used as interface to the user-supplied aerodynamic model). The program also prompts for the units to be used in calculations involving the control surface: WHAT UNITS ARE USED IN CALCULATIONS WITH THIS CONTROL SURFACE? 1) NONE 2) DEGREES 3) RADIANS The units should be consistent with the calculations in UCNTRL. The program performs transformations based on the surface type for the printer file only. No other con- trol surface transformations are performed. Again, the derivative step size can be changed for individual variables specified by the user. For the interactive input of the observation vector, similar prompts are given with a few variations. The forms available for the observation equation are 1) Y¥ H*X + F*U + E*V N K | H*X + G*DX/DT + F*U + EB*V For variables defined in the observation vector as being offset from the vehicle cen- ter of gravity, the program will prompt for the offset in the x, y, and z stability axes. For any variables that are calculated based on some reference value, such as Reynolds number, the program prompts for the correct value when the name is entered. Again, the step size for derivative extraction can be modified for any observation variable. Valid observation variable names are listed in appendix Cc. If selected to be input from the terminal, the program will prompt the user for the limits to be used for the trim parameters d., 53, Sy, Stns Which represent the longitudinal, lateral, directional, and thrust trim parameters, respectively. The units associated with these parameters are defined by the implementation of UCNTRL. (See app. J of this report for details.) An example prompt is as follows: WHAT IS THE MAXIMUM THRUST PARAMETER? Again, the program reads a floating point field. After the trim axis parameter limits are defined, the program prompts HOW MANY ADDITIONAL SURFACES DO YOU WISH TO DEFINE? 43 At this time, the user can define additional controls to be included in the list of recognized control names. The purpose of defining these additional surfaces is to allow the user to set such variables as landing gear position, wing sweep, or flap position. The control surface name, location in common /CONTRL/, and units are defined here for each surface; actual control surface positions are defined during analysis point definition. The program will then prompt the user to indicate what aerodynamic model will be used for this analysis. this report. Aerodynamic models are described in the next section of If a nonlinear aerodynamic model is used, the program will prompt the user to determine if there are additional control surfaces for which derivatives are desired. These controls will not become part of the control vector. This option of inter- active LINEAR does not allow these control variables to be read from or saved to a file, and is not available on the batch version of LINEAR. However, the stability and control derivative output file will contain these surfaces and their derivatives. The point at which the nonlinear system equations are linearized is referred to as the analysis or trim point. This can represent a true steady-state condition on the specific trajectory (a point at which the rotational and translational accelera- tions are zero) or a totally arbitrary state on a trajectory. LINEAR allows the user to select from a variety of analysis points. Within the program, these analysis points are referred to as trim conditions, and several options are available to the user. If the user is defining the trim conditions interactively, data from a formatted file, the program will prompt: instead of reading the WHAT TYPE OF TRIM DO YOU WISH TO RUN? 1) NO TRIM 2) STRAIGHT AND LEVEL FLIGHT 3) LEVEL TURN 4) PUSH~OVER/PULL-UP 5) THRUST STABILIZED TURN 6) BETA 7) SPECIFIC POWER The user must specify the number corresponding to the type of trim desired. The arbitrary state and control option is designated as NO TRIM, and in selecting this option the user must specify all nonzero state and control variables. For all of the analysis point definition options, any state or control parameter may be input after the required data is defined when the program prompts as follows: WHAT OTHER STATE OR CONTROL VARIABLES WOULD YOU LIKE TO INITIALIZE? TYPE HELP TO LIST VALID VARIABLES NAMES (ENTER "N" IF NO OTHER CONDITION IS TO BE SPECIFIED) 44 The user should enter the name of the variable to be initialized or type HELP to list the variables that may be initialized. If a level turn trim is by entering "TURN RIGHT" or iables. This is especially selected, the user can specify the direction of the turn "TURN LEFT" when prompted to initialize any other var- useful for asymmetric aircraft. Default is "TURN RIGHT". If data are being read from a file(s), the data can be reviewed by replying "Y" when prompted DO YOU WISH TO REVIEW THE DATA AS IT IS READ FROM THE INPUT FILE(S)? (Y/N) If any data are entered incorrectly, the user should finish entering all data as prompted by the program until asked if there are any changes to be made: DO YOU WISH TO CHANGE ANY PARAMETERS IN YOUR MODEL OR UPDATE THE INPUT CASE SELECTION FILE? (Y/N) At this time, the user selects "Y" and the program prompts WHAT PARAMETERS DO YOU WISH TO CHANGE? 1) VEHICLE GEOMETRY 2) STATE VECTOR 3) OBSERVATION VECTOR 4) CONTROL VECTOR 5) CASE INPUT FILE 6) STABILITY AND CONTROL DERIVATIVES 7) SURFACES TO BE SET 8) NO FURTHER CHANGES Any section of data can be updated by reading a new data file or by typing in the data from the terminal. The program prompts the user input as explained previously. If the user wishes to to determine the source of modify the vehicle geometry and mass all data or the stability and control the values from the previous case derivatives interactively, the program saves and allows the user to modify any individual parameter. ties, a "1" For example, if the user wishes to modify the vehicle would be entered and the program would continue geometry proper- WHAT IS THE NAME OF THE FILE CONTAINING THE VEHICUE GEOMETRY AND MASS PROPERTIES? (TYPE 0 TO INPUT THE DATA FROM THE TERMINAL AND STORE ON A FILE) If the user replies with a zero, and then respond as follows: the program will prompt for a new output file name 45 GEOMETRY AND MASS DATA FOR: USER'S GUIDE FOR THE PROJECT: DEMONSTRATION CASE 1) WING AREA 2) WING SPAN 3) MEAN CHORD 4) VEHICLE WEIGHT 5) 1x 6) IY 7) 2 8) IXZ 9) Ixy 10) IyYZ 608.000 42.800 15.950 40700.000 28700.000 165100.000 187900.000 ~520.000 0.000 0.000 VECTOR DEFINING REFERENCE POINT OF AERODYNAMIC MODEL WITH RESPECT TO THE VEHICLE CENTER OF GRAVITY: 11) DELTA X 12) DELTA Y 13) DELTA Z 14) MINIMUM ANGLE OF ATTACK 15) MAXIMUM ANGLE OF ATTACK 0.000 0.000 0.000 -10.000 40.000 16) FORCE AND MOMENT COEFFICIENTS CORRECTIONS DUE TO THE OFFSET OF THE REFERENCE POINT OF THE AERODYNAMIC MODEL FROM THE VEHICLE CG ARE INTERNALLY CALCULATED IN LINEAR. TYPE IN 16=0 IF CALCULATIONS ARE DONE IN CCALC. TO MAKE CHANGES, TYPE IN THE DESIRED VALUES. "N" INDICATES NO CHANGES "R" REFRESHES THE SCREEN FOR EXAMPLE: TO CHANGE LINE NUMBER ON THE ARE DESIRED WITH NEW VALUES. VEHICLE WEIGHT 4=40000.0 If selection "16=0" is chosen, the program will set a logical variable that deter- mines where center-of-gravity corrections are made to the force and moment coef- ficients. The vector defining the reference point of aerodynamic model with respect to the vehicle center of gravity must be defined in feet. If the state, control, or observation vectors need to input the entire vector again. The choices available for to input the data from the terminal for each case or read tion from a file. be modified, the user must the test case selection are the entire case specifica- 46 All surfaces in the control model are initially zero unless specified in the case input file. Surfaces not in the control vector can be set by choosing option "7" when prompted for parameter changes. The program will prompt for the file name con- taining the surfaces that can be set for each case. If the user wishes to input the surface names interactively, the program will prompt for the number of surfaces to be added, their names, location in common /CONTRL/ and units used in subroutine UCNTRL. If the location in common /CONTRL/ has already been designated for another surface, the program will save the original name and ignore any other name input for that location, and so inform the user. The interactive LINEAR program provides the user with two different ways to calculate the total force and moment coefficients. One method is by using a full aerodynamic model and the subroutines ADATIN and CCALC. The other method uses a set of linearized stability and control derivatives, and the internal subroutine CLNCAL. When using the second method, the data can be typed in during execution or can be input from a file. An example of a linearized stability and control data set is shown in appendix G. The first line contains the file description. The case number (1-999) at the end of the first line allows the user to correlate the linearized data from a case to other data from that case. The next line is the title of the run. The following two lines of text describe the project for which the case was run, and the next two lines list the reference altitude and Mach number for the case. The next line determines whether the alpha and beta derivatives are in units of degrees or radians. The following two lines indicate whether Mach or velocity derivatives will be read in. This is intended to warn the user against modifying the wrong derivative while editing outside of LINEAR. If the wrong derivative is changed, LINEAR will ignore the change. Additionally, if the reference altitude is changed, LINEAR will rely on the specified derivative and compute the other based on the speed of sound which is a function of altitude. While editing this file in LINEAR, this caution is not necessary since LINEAR will perform the computation right away. Also, when these derivatives are changed in LINEAR, the last velocity or Mach derivative changed will determine which one LINEAR will regard as the specified derivative when the file is output. The next line determines the number of control surfaces used for control deriva- tives. The units of each of the control surfaces, where applicable, are listed next to the surface name as well as the location of the surface in the common /CONTRL/. The units defined for each surface correspond to the units defined in user-supplied subroutine CCALC. The rest of the file contains the stability and control derivatives. This part is broken into six sections, one for each of the force and moment coefficients. In each section, there is one zero coefficient corresponding to zero angle of attack, zero sideslip, zero control surface deflection, and constant Mach and altitude. Each section also contains 10 stability derivatives. In addition, each section has as many control derivatives as specified earlier. The six sections must always be in the same order: rolling moment, pitching moment, yawing moment, drag force, lift force, and sideforce. The easiest way to format a derivative file is by running the interactive LINEAR program and inputting the derivatives when prompted by the program. One of the out- 47 pwcmpliuraoattetntheitvrorifnooit.luhl)steeosdedcaraenWfsiahrreevlio,nyamvztaeittuLvihsIeveiNsetnEhgAeuRsoseneasrtai.mstlehmieuntsheIctefaafrsieilab.sezet,easdbucirat(lfrhSsiaaeeettcfetyeulsotfhuiaesrntdofsatAdcaeedcebfreo'iifonsnlidteniryedtoneylafmftaiehacdcanecttdrosniMtvodcradooooetennllsitvrteohvnsleoeetccfttifdoiloehroerarncvieevttsahhftaaocittroavnredrsmicseaonrsmepocfomrobneoenidnmsnitifsunsosagtre-ednt will not be included in the linear system model calculated. aware that LINEAR expects to have control over all axes to If there are no nonzero control derivatives in a given axis, LINEAR will fail to trim; and it may even generate a fatal The user should also be determine a trimmed state. it is highly likely that error in the nonlinear equation solver (generally a floating-point overflow). AERODYNAMIC MODEL As already stated, there are two methods for calculating the total force and moment coefficients. One method relies on a full nonlinear aerodynamic model to pro- vide the stability and control derivatives. These derivatives are stored in common blocks in the user-supplied routine ADATIN and used in CCALC to calculate the total force and moment coefficients using the state and control surface values calculated in LINEAR. These subroutines and the aerodynamic model are the same ones used in the manned simulations at Ames-Dryden. Using a full aerodynamic model with CCALC and ADATIN enables the user to run as many cases as desired without program generates a structured changing derivative data sets. linearized derivative data file From each which can case, the be used as an input file later. This file contains all the stability and control derivatives for the control surfaces specified in the control vector, and any other controls specified by the user. The interface required for using a nonlinear aerodynamic model is described later in the Aerodynamic Model Subroutines section. The other method for calculating total force and moment coefficients requires the user to supply linearized stability and internal subroutine CLNCAL to calculate control derivatives. These are used in the the total force and moment coefficients. When using this option, care must be taken to ensure that the input data is valid for the trim point desired. Center-of-gravity shifts will not affect the data that is output to the derivative file when this method is used. The method used to calculate the force and moment coefficients is specified during the input phase for a particular case. After all the vehicle data, state, control, observation vectors, and case data have been defined, the program will prompt DO YOU WISH TO 1) INPUT A SET OF LINEAR STABILITY AND CONTROL DERIVATIVES 2) OR USE THE NONLINEAR MODEL YOU HAVE INTERFACED WITH THIS PROGRAM If the internal routines are to be used to calculate the derivatives, the program will prompt 48 DO YOU WISH TO 1) GENERATE A NEW DATA SET 2) OR USE A DERIVATIVE DATA SET YOU HAVE ON FILE 3) OR MODIFY THE OLD ONE If the derivatives are stored on a file (format for the file is given in app. program will prompt for a file name. At this point, the user can modify the interactively by choosing one of the following options: G), the data TO CHANGE ANY DERIVATIVES, APPROPRIATE LINE # TO SEE TYPE IN "N" IF NO CHANGES INPUT THE THE DATA. DESIRED 1) TYPE OF ANGLE MEASUREMENT 2) ROLL MOMENT 3) PITCH MOMENT 4) YAW MOMENT 5) DRAG FORCE 6) LIFT FORCE 7) SIDE FORCE 8) ADD SURFACES If the user selects option one, the program will list the units that are used for output of the stability derivatives. It will also list the units for each control surface derivative as used for calculations in the subroutine UCNTRL. There are three choices for units for the control surfaces: none, radians, and degrees. If the control surface calculations in UCNTRL are nondimensional, the user should spec- ify none, and no conversion is done. For surface calculations in radians, a conver- sion to degrees for output only is performed. If degrees is specified, no conver- sion is done. When the user inputs the stability and control derivatives, the pro- gram does not perform any conversions, therefore, care must be taken to ensure that the units are consistent with the aerodynamic model and the calculations in UCNTRL. If option "8" is chosen, the control surfaces defined are listed, and the user can define other surfaces in common /CONTRL/ for which derivatives exist. If the derivatives are be to calculated from a nonlinear aerodynamic model, the program will prompt for the names of any other control surfaces not defined in the control vector for which linearized derivatives will be calculated. If options "2" through "7" are chosen, the program will list the current values for the stability and control derivatives for each respective coefficient. The user can change these values interactively, as in the following example: ROLL MOMENT DERIVATIVES WITH RESPECT TO: 0) CLO 1) ROLL RATE (RAD/SEC) 2) PITCH RATE (RAD/SEC) 3) YAW RATE (RAD/SEC) -0.000033820 -0.200000003 0.000000000 0.163589999 49 4) VELOCITY (FT/SEC) 5) MACH NUMBER 6) ALPHA (DEG/SEC) 7) ‘BETA (DEG/SEC) -~0.000000115 ~0.000119522 0.000362335 -0.140983000 TO MAKE CHANGES, TYPE IN THE LINE NUMBER AND THE VALUE OF THE DERIVATIVE. "N" INDICATES NO CHANGES ARE DESIRED "R" REFRESHES THE SCREEN WITH THE NEW VALUES FOR EXAMPLE: ROLL MOMENT WRT ALPHA 6=0 .003 OUTPUT FILES There are four output files from LINEAR: (1) a general-purpose analysis file, (2) a printer file containing the calculated case conditions and the state-space matrices for each case, (3) a printer file containing the calculated case conditions only, and (4) a linearized set of stability and control derivatives. LINEAR also outputs any data input by the user during an interactive session that can be used later either by the batch or interactive programs. The program prompts the user for the file names that each of the sections of data will be written to, as described in the Interactive Data Input section. The general-purpose analysis file contains the following: the title of the cases being analyzed; the state, control, and observation variables used to define the state-space model; and the state and observation matrices calculated in LINEAR. The C and G matrices are printed only if the user has selected an appropriate formulation of the state and observation equations. The output for this file is assigned to FORTRAN device number 15. An example general-purpose analysis file is presented in appendix H corresponding to the format of table 2. The file in and the defined vehicle and case titles are written on the first two records of the analysis 80-character strings and are specified in LINEAR as the title of the title of the cases. The next record contains the number of the case in LINEAR (maximum of 999 cases). vehicle as The number of states, controls, and outputs used to define each case is written on the subsequent record. The formulation of the state and observation equations is listed next, followed by the names and values of the states, controls, dynamic interaction variables, and outputs. These values are followed by the matrices that describe each case. The title records only appear at the beginning of the file while all other records are duplicated for each subsequent case calculated in LINEAR. The matrices are written row-wise, five columns at a time as shown in table 3. This table shows a system containing seven states, three controls, and eleven outputs using the general state equation and standard observation equation. 50 TABLE 2. — ANALYSIS FILE FORMAT Variable Title of the case Title of the aircraft Case number Number of states, controls, and outputs State equation formulation Observation equation formulation State variable names, values and units Control variable names, values and units Dynamic interaction variable names and units Output variable names, values and Matrix name A~Matrix Matrix name B-Matrix Matrix name D-Matrix Matrix name C-Matrix (if general form chosen) Matrix name H-Matrix Matrix name F-Matrix Matrix name E-Matrix Matrix name G-Matrix (if general form chosen) units Format (4A20) (4A20) (//,64X,13) (17%,12,22X,12,22x,13) (36X,2A4) (36X,2A4) (///7/) (1X,5A4,3X,E12.6,2X,A20) (////) (1X,5A4,3X,E12.6,2X,A20) (////) (1X,5A4,17X,A20) (////) (1X,5A4,3X,E12.6,2X,A20) (//,A8,/) (5(E13.6)) (//,A8,/) (5(E13.6)) (//,A8,/) (5(£13.6)) (//,A8,/) (5(£13.6)) (//,B8,/) (5(E13.6)) (//,A8,/) (5(E13.6)) (//,A8,/) (5(E13.6)) (//,A8,/) (5(E13.6)) 51 TABLE 3. — ANALYSIS FILE OUTPUT MATRIX STRUCTURE Size of matrix A= [7x 7] B= {7 x 3] D= {7x 6] c=[7x 7] H = [11 x 7] F = [11 x 3] E = [11 x 6] Output formulation A= [7x 5] [ 7 x 2] B= [7 x 3] D= [7x 5] [7x 1] c= [7x 5] [ 7 x 2] H = [11 x 5] {11 x 2} F= [11 x 3] E = {11 x 5} [11 x 1] The second output file, which is assigned to FORTRAN device unit number 3, con- tains the this file values calculated in LINEAR describing each case. contains a listing of the input data defining the The first aircraft's section of geometry and mass properties, variable names defining the state-space model, and various control surface limits characteristic of the given aircraft. Appendix I presents an example printer output file. The second section of this file contains the trim conditions of the the point of interest. These conditions include the type of trim being whether trim was achieved, and parameters defining the trim condition. aircraft at attempted, The values for the variables of the state-space model at the trim condition are also printed. equations of the geometry If trim was not achieved, Py qe xr, v, B, and ra motion) and the force and moment coefficients are and mass properties are also printed. (calculated from the printed. Changes in The control aircraft third section of this output file derivatives for the trim condition at the given flight condition is contains the calculated. also printed. nondimensional stability and The static Margin of the The final section of this output file contains the state and observation matrices for the given flight condition. terms of the matrices chosen by The formulation of the state equations and the user to define the model are printed. only the The third output file, which is assigned to FORTRAN device unit 2, contains the trim conditions of the aircraft at the point of interest. These conditions include the type of trim being attempted, whether trim was achieved, parameters defining the trim condition, and the static margin of the aircraft at the given flight condition. Appendix I also presents an example of this file. The fourth output file is the set of linearized stability and control derivatives for each case as defined in the Aerodynamic Model section. USER~SUPPLIED SUBROUTINES There are five subroutines that must be supplied by the user to interface LINEAR with a specific aircraft's subsystem models: ADATIN, CCALC, IFENGN, UNCTRL, and MASGEO. The first two subroutines comprise the aerodynamic model. The subroutine IFENGN is used to provide an interface between LINEAR and any engine modeling rou- tines the user may wish to incorporate. UCNTRL converts the trim parameters used by LINEAR into control surface deflections for the aerodynamic modeling routines. MASGEO enables the mass and geometry properties to change as the aircraft configura- tion does (for example, with wing sweep). The use of these subroutines is illustra- ted in figure 3 which shows the program flow and the interaction of LINEAR with the user~supplied subroutines. These subroutines are described in detail on the fol- lowing pages. Examples of these subroutines can be found in appendix J. 53 _| _ 25.0 DES "| 5.43 DAS "»|| 2“04..00 _| 180 “| 8 | 181 0 14.0 L,y 18Tt0 DRS THRSTX »"}| 330..205 -1,0 af 0,1 »| 18T0 _ -45.0 “18a0 = i bg > é a >~ oo dat ~ _ 6, >_ Sep .(to ITFHERNGN) Figure 3. Program flow diagram showing communication with usersupplied subroutines. Aerodynamic Model Subroutines It is assumed that the aerodynamic models consist of two main subroutines: ADATIN and CCALC. ADATIN is used to input the basic aerodynamic data from remote storage. ADATIN can also be used to define aerodynamic data. CCALC is the subroutine that uses this aerodynamic data, the state variables, and the surface positions to determine the aerodynamic coefficients. Either routine may call other subroutines to perform related or required functions; however, from the point of view of the interface to LINEAR, only these two subroutines are required for an aerodynamic model. Externally, ADATIN has no interface to the program LINEAR. The subroutine is called only once when the aerodynamic data is input or defined. The calling program has to provide ADATIN with the input devices it requires, but no other accommodation is necessary. The aerodynamic data is communicated from ADATIN to CCALC through named common blocks that occur only in these two routines. The interface between CCALC and the calling program is somewhat more complicated. However, the interface is standard and this feature provides a framework about which a general purpose tool can be built. This interface consists of several named common blocks which are used to pass state variables, air data parameters, surface positions, and force and moment coefficients between CCALC and the calling program. CCALC is executed whenever new aerodynamic coefficients are required (for example, once per frame for a real-time simulation). The main transfer of data into the subroutine CCALC is through five named common blocks. These common blocks contain the state variables, air data parameters, and 54 Externally, ADATIN has no interface to the program LINEAR. The subroutine is called only once when the aerodynamic data is input or defined. The calling program has to provide ADATIN with the input devices it requires, but no other accommodation is necessary. The aerodynamic data is communicated from ADATIN to CCALC through named common blocks that occur only in these two routines. The interface between CCALC and the calling program is somewhat more complicated. However, the interface is standard and this feature provides a framework about which a general purpose tool can be built. This interface consists of several named com- mon blocks which are used to pass state variables, air data parameters, surface posi-~ tions, and force and moment coefficients between CCALC and the calling program. CCALC is executed whenever new aerodynamic coefficients are required (for example, once per frame for a real-time simulation). The main transfer of data into the subroutine CCALC is through five named common blocks. These common blocks contain the state variables, air data parameters, and surface positions. The transfer of data from CCALC is through a named common block containing the aerodynamic force and moment coefficients. The details of these common blocks follow. The common block /DRVOUT/ contains the state variables and their derivatives with respect to time. The structure of this common block is given as follows: COMMON /DRVOUT/ T ' P 1 Q 7R ' Vv , ALP , BIA ' THA , PSI » PHI ’ H 7X 7 TDOT , PDoT , QDOT , VDOT , ALPDOT, THADOT, PSIDOT, HDOT , XDOT , x 7 RDOT , BTADOT, PHIDOT, YDOT The body axis rates p, q, and r appear as P, Q, and R, respectively. Total velocity is represented by the variable V and the altitude by H. Angle of attack (ALP), angle of sideslip (BTA), and their derivatives with respect to time (ALPDOT and BTADOT, respectively) are also contained within this common block. The common block /SIMOUT/ contains the main air-data parameters required for the aerodynamic model. The variables in this common block are as follows: COMMON /SIMOUT/ AMCH, QBAR, GMA, DEL, UB , VB, WB, VEAS, VCAS Mach number and bolized by AMCH included as UB, dynamic pressure are the First two entries in the and QBAR, respectively. The body axis velocities VB, and WB, respectively. common block symu, v, and w are 55 The /CONTRL/ common block contains the surface position information. The exact definition of each of the elements used for a particular aerodynamic model is deter- mined by the implementer of that model. The structure of the common block /CONTRL/ is as follows: COMMON /CONTRL/ DC (30) The common block /DATAIN/ contains geometry and mass information. The first three variables in the common block, S, B, and CBAR, represent wing area, wing span, and mean aerodynamic chord, respectively. The vehicle mass is represented by AMSS. The structure of common /DATAIN/ is as follows: COMMON /DATAIN/ S , B , CBAR, AMSS, AIX , AIY , AIZ , AIXZ, AIXY, AIYZ, AIXE The information on the displacement of the reference point of the aerodynamic data with respect to the aircraft center of gravity is contained in the /CGSHFT/ common block: COMMON /CGSHFT/ DELX, DELY, DELZ The displacements are defined along the vehicle body axis with DELX, DELY, representing the displacements for the x, y, and z axes, respectively. and DELZ The output common block /CLCOUT/ contains the variables representing the aero- dynamic moment and force coefficients: COMMON /CLCOUT/ CL, CM, CN, CD, CLFT, CY The variables CL, CM, and CN are the symbols for the rolling moment (Cg), pitching moment (Cy), and yawing moment (C,) coefficients, respectively; these terms are body axis coefficients. The stability axis forces are represented by CD, coefficient of drag (Cp); CLFT, coefficient of lift (Cy); and CY, sideforce coefficient (Cy). Control Model Subroutines The LINEAR program attempts to trim the given condition using control inputs similar to that of a pilot: longitudinal stick, lateral stick, rudder, and throttle. The UCNTRL subroutine uses these trim output control values to calculate actual sur- face deflections and power lever angles for the given aircraft (fig. 2). The location of each surface and power lever angle in the /CONTRL/ common block is specified by the user in the input file (maximum of 30 surfaces). The limits for the control parameters in pitch, roll, yaw, and thrust are user specified and must agree in units with the calculations in CCALC. The common block /CTPARM/ contains the four trim parameters that must be related to surface deflections in the subroutine UCNTRL: COMMON /CTPARM/ DES, DAS, DRS, THRSTX 56 The output from UCNTRL is via Aerodynamic Model Subroutines used to trim the longitudinal, the common block /CONTRL/ described previously in the section. The variables DES, DAS, DRS, and THRSTX are lateral, directional, and thrust axes, respectively. For an aircraft using feedback (such as a statically unstable vehicle), state variables and their derivatives are available in the common block /DRVOUT/. However, this control model must contain no states of its own. If parameters other than state variables and their time derivatives are required for feedback, the user may access them using the common block /OBSERV/ described later in the Mass and Geometry Model Subroutines section of this report. Engine Model Subroutines The subroutine IFENGN computes individual engine parameters used by LINEAR to calculate force, torque, and gyroscopic effects due to engine offset from the centerline. The parameters for each engine (maximum of four engines) are passed through common /ENGSTF/ as follows: COMMON /ENGSTF/ THRUST(4), TLOCAT(4,3), XYANGL(4), XZANGL(4), TVANXY(4), TVANXZ (4), DXTHRS(4), EIX (4), AMSENG (4), ENGOMG(4) The variablesin this common block correspond to thrust (THRUST); x, y, and z body axis coordinates of the point at which thrust acts (TLOCAT); the orientation of the thrust vector in the x-y body axis plane (XYANGL), in degrees; the orientation of the thrust vector in the x-z body axis plane (XZANGL), in degrees; the orientation of the thrust vector in the x-y engine axis plane (TVANXY), in degrees; and the orientation of the thrust vector in the x-z engine axis plane (TVANXZ), in degrees. The distance between the center of gravity of the engine and the thrust point (DXTHRS), measured positive in the negative x engine axis direction; the rotational inertia of the engine (EIX); mass (AMSENG); and the rotational velocity of the engine (ENGOMG) are also in this common block. The variables are all dimensioned to accommodate up to four engines. Appendix B provides graphical and physical descriptions of these parameters. While the common block structure within LINEAR is designed for engines that do not interact with the vehicle aerodynamics, propeller-driven aircraft can be easily modeled by communicating the appropriate parameters from the engine model IFENGN to the aerodynamic model CCALC. Mass and Geometry Model Subroutines The subroutine MASGEO allows the user to vary the center-of-gravity position and vehicle inertias automatically. Nominally, this subroutine must exist as one of the user subroutines, but it may be nothing more than a RETURN and END statement. MASGEO is primarily for variable-geometry aircraft (such as an oblique-wing or variable- sweep configuration) or for modeling aircraft that undergo significant mass or iner- tia changes over their operating range. The center-of-gravity position and inertias may be functions of flight condition or any surface defined in the /CONTRL/ common 57 block. block, Changes in and inertia center-of-gravity position are passed in the /CGSHFT/ changes are passed in the /DATAIN/ common block. common brsmfaenoueerbcamrosaoomucbCuprcasotuerroeitemmrneepvendnlaosmtitsussiesahotsanenbdadroabeoevmpebr,yacedttssaeosukstirileihmtnnerpgeltdehmowmfaohtteemhanenetttnhctieonaunensgwtiacfanriegimxnmtnehotsdemetteorhne-mmtlooeotfmhacse-easaungtrubtrmirrnaoeoevnmwuvaietratnimlyoctnuneseencstastehlhaMeicrwAtfau-ShtyloGrvaEfoeOtu-hffiggiorhorrconialmnsevta,ihctctechyoieunmsrbu/aciOoclthnBneoeeSancEttaaRietsorVorife/nosanutolfhcwsteoiemsmtnu.ghmssurotosanrevsriTbetehplbyille.saoctct-ki:cnaIgnf COMMON /OBSERV/ OBVEC (120), PARAM (120,6) SpadtcvsmtObsagntiheereeaebiBudeaonsnchilvrVntntksiieemfEte)orcsnaaCe,.poTrlecr(rahbedttei1es-os,rlid2soienCluy0emfroAorom)Ldc~nvdicbaI,toagtandashNnmptsrrtgtieEeoampbtoainitArogaerlavooRinvcen,andnn.iotkamddtetrtnteiehfiyaibnxegimaliloitP(innnso(nATeedxchRmlCs,ake.osAotitsmhoMcnOct(ee(ahfbhyPtw/cn,et1eosAheobOAti2mreeaRlmcBfoe0tmlrAmcSimzsn,aMvoer)Eom.a6(t,anssRmra)anittasVmders.aoie/nennio.btdn=ntlTfttohhoorfaerocoofulAa1emknerVlnbpItemefaocshatpltowersteaoit)/ssiirhsrbetOfaeotvashlrB3obneam)eoeSmblpst,tmermEusheanoitevsReeferuotenaVwraswnottrt/evhfmhDssstiureeaeehbscortteefeenoecierinsobaoacsornfrtsnesucansieneetsbonvtttarrmrooaatieivn-phociidoavrednauusoilenasastttafassrsceiit-biipocPtodonelgrhafeacAentdereesbRriwssdaslAaaoe,igvMemtcnnrMwoiseetuiaAabtatsditmtvSlsovyeosihbiGlneeunrnaetEeccr.slgfrwyOhtvliillttoaoxthehoorwhteefe-coctaisdifaoasspontratotoniehhstublbttsgoecehsshhbsaosneieeeeoiecedsncarruaoiitvvsnpdftbuoaiaedaoooesstnstrottuioseecenininirrhdsptdotsorvheacsnveneadrhuesdtuaabcwioatlinnnmttirofveogteooiaoasenhrturxbrttwsisieteeshhh-m:rdieee~ipttasnrclhhnuevheeedssa-e~ids CONCLUDING REMARKS ttCadidlohiloanoxiwsrtcsopeuoormtJTdr-fheherdaeaneseertctipsieegocoAsFdnrrnOc.ti,Ro.mbTmitetRpocodAuro.MTNtlohaTuefihysrsienesptcaorrEhr-opxedrsgar,wedrumopieappgotrlmlrprhiiMetalovsasmuietstLeisIddnaNioilgcsfEsmihAcnosuRuedwuotsasfihrbersefirtweitoattsctsuhsmueaet)osntideitdnehrpoeele-rnvwissnosi.etgulihrponsafappnottmledraetdinhreedafdiaaconitrtrporVdiuMcsvtSrupaearbFforOVoftvoRAipLXiTuleIdRterNseAsiaEt.N1nAta1eRi/bsan7i7g5l7pT0fihralteoressyegoxy(risoDabtuiptamlethgrn,emapiea,tutsltaeyidlsnsepiittosfsewcrEideorqmnmriaucayfbliniuidupnLlndmIb,geeeNadcnEpotApnptaRetohnwnerdri-ottilhesd vaijgcetafeayqrycpruiitaaownoabtshTgrblicihyelooce,enhpstsihpcoefrsthoeaoifgnlseesdretfxacaftmmtterorleicateinticnmLgsoteaIanaiinNrnndEaagiAnanRdntcgsdroylinsinimttnnusaretecmomaaleoultrurpseidtiemziicorvoenad-avndglesra,lluiryapitebaphnailblesngdelidsiseeeenstdde.eedrfioemrirfneineffonecefnbdetcyelsticisntnsigsesp,aaeirnctdliheisefutnatyceeheihearrpnrmrogiodalngysisernnyadaaensnmmateirteaecmnitotatshlhymmeyrsoodrsutddeiseeetstllmeb,.ry mpmuioostcnidiooeLnenrmItlgqpN.uElteAeho,nRetnoen‘lTlacayihisenodnntetrpasaraolpro-iselocn-t 58 LINEAR has several features that make it unique among the linearization programs common in the aerospace industry. The most significant of these features is flexi- bility. By generalizing the surface definitions and making no assumptions of sym- metric mass distributions, the program can be applied to any aircraft in any phase of flight except hover. The unique trimming capability, provided by means of a user- supplied subroutine, allows unlimited possibilities of trimming strategies and sur- face scheduling, which are particularly important for asymmetrical vehicles and aircraft having multiple surfaces affecting a single axis. The formulation of the equations of motion permit the inclusion of thrust-vectoring effects. The ability to select, without program modification, the state, control, and observation vari- ables for the linear models, combined with the large number of observation quantities available, allows any analysis problem to be attacked with ease. This report documents the use of the program LINEAR, defining the equations and the methods used to implement the program. The trimming capabilities of LINEAR are discussed from both theoretical and implementation perspectives. The input and out- put files are described in detail. The user-supplied subroutines required for LINEAR are discussed and sample subroutines are presented. Ames Research Center Dryden Flight Research Facility National Aeronautics and Space Administration Edwards, California, November 3, 1987 59 APPENDIX A: CORRECTION TO AERODYNAMIC COEFFICIENTS FOR A CENTER OF GRAVITY NOT AT THE AERODYNAMIC REFERENCE POINT The point on the vehicle at which the nonlinear force and moment coefficients are defined is referred to as the aerodynamic reference point. All aerodynamic effects are modeled at vehicle center reference point this aerodynamic reference point. Thus, when this point and the of gravity are not coincident, the forces acting at the aerodynamic effectively induce moments that act incrementally on the moments defined at the aerodynamic reference point by the nonlinear aerodynamic model. The total aerodynamic moment M acting at the vehicle center of gravity is defined as where M = Mar + Ar x F Mar = [Lar Mar Nar]? is the total aerodynamic moment acting at the aerodynamic reference point (denoted by subscript ar) of the vehicle, Ar = [Ax Ay Az]T is the displacement of the aerodynamic reference point from the vehicle center of gravity, and F= {x y g]t is the total aerodynamic force acting at the aerodynamic center where x, y, and z are total forces along the x, y, and z body axes. Thus, Lar + Ay Z - Azy M = Mar + Az X - Ax Z@ Nar + Ax Y - Ay xX The total aerodynamic moment acting at the vehicle center can be expressed in terms of the force and moment coefficients derived from the user-supplied nonlinear aerodynamic modeling subroutine CCALC by defining the body axis forces in terms of stability axis force coefficients: X = gS(-Cp cos a+ Cy sin a) Y = gSCy Z = qS(-Cp sin a - Cy, cos a) 60 Equations obtained aerodynamic moment moments, result in by substituting these equation and applying equations into the the definitions of definition of the total the total aerodynamic Lar = GSbC 9, Mar = gScCm,+ Nar = GSbC, Expressions for total aerodynamic moment coefficients corrected to the vehicle center of gravity can be derived as follows: Cg == Cet pAy (Cp si:n a = Cy cos a) = DApz Cy Cm _= Cmay + IZ (“Cp cos a + Cy si.n a) - =Ax (-Cp si.n a ~ Cy, cos a) c Cy = Char t DAx Cy 7 aAs (-Cp cos a + Cy; si.n a) These calculations are normally performed within CGCALC. However, if the user elects, the calculation user-Supplied aerodynamic model CCALC. LINEAR can be in the subroutine performed within the 61 APPENDIX B: ENGINE TORQUE AND GYROSCOPIC EFFECTS MODEL Torque and gyroscopic effects represent (after thrust) the main contributions of the engines to the aircraft dynamics. The torque effects arise due to thrust vectors not acting at the vehicle center of gravity. The gyroscopic effects are a consequence of the interaction of the rotating mass of the engine and the vehicle dynamics. These effects can be either major or virtually negligible, depending on the vehicle. The torque effects can be modeled by considering the where the thrust vector is aligned with the local x axis point Ar from the center of gravity of the vehicle (fig. thrust of the 4). of an engine F,, engine acting at some Center of Figure 4. Definition of location of engine center of mass (cm;) relative to vehicle center of gravity. The thrust vector for the ith engine Fo can be defined as FP, i . = [ Fexs Foy; T Fpz | where axes, ships PP! Foy! i and Fp _ ave the components of i respectively. From figures 5 and 6 it can hold: thrust in the x, be seen that the y, and z body following relation- PPK, = | Fp, | cos €; cos & *pys =| Fp, | cos €; sin & a = - | Fp, | Sin €; 62 where | Fp, | represents the magnitude of the angle from the thrust axis of the engine to plane, and &; the angle from the projection x-y body axis plane to the x body axis. thrust due to the ith engine, €, the a plane parallel to the x-y body axis of Fo: onto the plane parallel to the < = — — F bo Po 7 Z i 1 \T | aw U 5 F P, | | xX <« Vaan _ + Zvz (-}----56--------- > “ y “ 7a UC é <* nd ns Zz [ ra Engine center a, of gravity —~~] \ a 7 v | a F 7 an a, m Aicrecnratfetr of gravity Az a a Ay | ~t—_ Ax z ' Figure 5. Orientation of the engines in the x-y and x-z body planes. Denoting the point at which the thrust vector can be defined as Figure 6. Detailed definition of engine location and orientation parameters. from the ith engine acts as Ary, this offset Ar; = [Axy Ayy Az,]T where Ax;, Ay;, and Az; are the x, y, and z body axis coordinates, respectively, of the origin of the ith thrust vector. The torque due to offset from the center of gravity of the ith engine To; is then given by Thus, OF Pi At OF = [ay Fp, Az Px = Az Foy. i | - Ax Fpzy 63 The total torque due to engines offset from the center of gravity of the vehicle To is given by where n is the number of engines. For the case of vectored thrust, the equations for torque produced at the vehicle center of gravity from the ith engine ATo, are somewhat more complicated. Figure 7 schematically represents an engine with thrust vectoring whose center of gravity is located at Ar; relative to the vehicle's center of gravity. T [\" Thrust point + y TP AW YE | | tp 2 \' gt — Engine cofentgerravity Xe Xqp Figure 7, Detailed definition of thrust-vectoring parameters, The thrust is assumed to act at ~Axm in the local (engine) x axis, with the engine center of gravity being the origin of this local coordinate system. The thrust is also assumed to be vectored at angles nj and ¢; relative to the local coordinate axes, with n,; being the angle from the thrust vector to the engine x-y plane and oi the angle from the projection of the thrust vector onto the engine x-y plane to the local x axis. Thus, letting Fp, , TAL Fr.Py,’ and F.Pa; represent the x, y, and z thrust components in the local engine coordinate system, repectively, these terms can be defined in terms of the total thrust for the ith engine Foi and the angles n;, and Si as Px; =| Fp, | cos Ny cos fj Foy, = | Fp, | cos ny sin tj Foe, - | Fp. | sin ni 64 where To transform this equation the transformation matrix FoP.i =|F[ t Px ' “Pyy from the ith engine ' pa axis | T system to the body axis system, cos €j cos &} cos €; cos & “sin & cos & Sin €; cos ey Sin €; cos &; “Sin €j 0 is used. The resultant force in body axis coordinates is lr Px; — cos €,1 cos €,1 -sin &,;i COs Ey — 1 4 Sin €.i cos &;1 FE Pxy F Pi _ = F Py; = [cos e€;i Siinn & i&; cos E€;i Sin £,i cos E i€. F!Py; so that a F Pe; | i “Sini €;1 9) cos €: 1 Fr! | a Pa; | PPxs =| Fy, | (cos Ny cos f, cos &€}, cos Ei - cos ny sin 4 sin Ey ~ sin nj sin ej; cos &;) FPy; =| Fy, | (cos Nj Cos tj cos €; sin ay + cos n; sin oi cos Ey ~ sin nj sin €; cos &) * pz, =| Fp, | (-cos Nn; cos ; sin €; — sin ny cos €;) The moment arm through which the vectored thrust acts is i” Ax. cos €; COS E. | Ar; = Ay; 7 Ax 5 cos €; sin oi Azi t+ Axp 5 sin €j | and the total torque due to thrust vectoring is n To = > AT). = > (Ar; x AF, ) 65 The gyroscopic effects due to the interaction of the rotating mass of the engines and the vehicle dynamics can be derived from the equation for the rate of change of angular momentum, Tg, = Be, + © x(Te;%;) where Tg; is the gyroscopic moment produced by the ith engine, h ei the rate of change of the angular momentum of the ith engine, Te; the inertia tensor for the ith engine, We; the rotational velocity of the ith engine, and w the total rotational velocity of the vehicle given by w= [p q r]t If it is assumed that the angular momentum of the engine is constant, then and the equation simplifies to Tg, = © X(Tej ei) Two terms, Te; and Wess remain moment produced by the ith engine. is assumed that the inertia tensor to be defined in the equation for the gyroscopic Once again, simplifying assumptions are made. It of the engine contains a single nonzero entry, xe, 0 0 ' te; = 0 0 0 0 0 0 where Tx... i is the inertia of the ith engine at the location of the ith engine, oriented with the local x axis coincident with the rotational velocity of the engine. The rotational velocity of the engine has components in the x, y, and z body axes (Pe, dejs and Tey! respectively) so that Mei . = (Pei . dex . Loe.i) T where Pe; =| we; | cos €; cos by de; =| we | cos €; sin & 66 with | we ; | being the total angular velocity of the ith engine. The engine inertia tensor must be defined in the vehicle body axis system and at the vehicle center of gravity. This is done in three steps. The first two steps involve rotating the engine inertia tensor into a coordinate system orthogonal to the aircraft body axis system, while the third step involves defining the engine inertia tensor about the vehicle center of gravity rather than about the center of gravity of the engine itself. First, the ith engine inertia tensor is rotated through an angle €; about the local y axis so that the new inertia tensor is oriented with its local x-y body axis plane parallel to the x-y body axis plane of the vehicle. The second step requires a rotation through an angle &; about the local z axis so that the local x, y, and z axes are orthogonal to the x, y, and z body axes of the vehicle. As determined by Gainer and Sherwood (1972) and Thelander (1965), this rotation is a similarity transformation that yields a new inertia tensor Ie that iv 5° IeC,i = ReiRRoe.gletj] Re-y1R. ey-1 where Re and Re are axes transformation matrices that perform the previously described rotations through € and e€, respectively. These matrices are given as cos €j4 0 sin €4] Re; = 0 1 0 -sin €j 0 cos €j | so that cos &j -sin &3 07 Re; = sin & cos &3 0 rt) 0 11 ReiRes = cos €| cos €j Sin &, cos &€ -sin & cos & cos €; sin €4 Sin €; sin Ey Because “sin €j 0 Re, = Rey cos €j 67 and and the matrices are unitary, Rr-G.1i = T Re REeRpyg, = RoeRe = R;rN.ERG,, ey 7 [cos Ei cos ( R Ej R €;) TL= -sSiin & sin €; cos €j Cccos €; ~sin € 0 cos §; sin ey sin Ey sin ey cos €; | Therefore, cos 2 €¢;. cos* 2¢.&j A Xe; cos* 2 €;¢. cos € . si; cos & -ssiin €;. cos €;. sisjn sini*n? ¢€;¢: The angular momentum of the ith engine hj; can now be expressed as hyi *= “1ei %ei = [pit Mig his] 7 with iq * Pei[Txe,; cos* ej cos? ei * Joi] xe + res[ - i xe; cos €j;. Sigin €, cos 2 g Nis = Pei[txe; cos? €j cos € sin 4] + de, | Tx sin@ Ej cos2 e| Si + rei[-Txe, sin €j; cos €; sin ei] hig = Pei[~Txe, cos €; Sin €; cos ei | + dei[-Tke, sin e€; cos €; sin ei] + Tei[Txes sin? e3 68 Thus, the gyroscopic moment induced by the ith engine Tg, i can be expanded to tg, =] hi, 7 Phi; | Phin ~ ahi, — and the total moment induced by gyroscopic interaction of the vehicle dynamics and the rotating engine components is Engine torque and gyroscopic effects are modeled within the subroutine ENGINE using information provided by the user from the engine modeling subroutine IFENGN. These effects are calculated as incremental moments and are included directly in the equations of motion for both analysis point definition and derivation of the linearized system matrices. 69 APPENDIX C: OBSERVATION VARIABLE NAMES RECOGNIZED BY LINEAR This appendix lists all observation variable names recognized by LINEAR except for state and control variable names. If state variables are specified as elements in the observation vector, the alphanumeric descriptor must correspond to the names defined in appendix D of this report. When control variables are to be included in the observation vector, these variables must be identified exactly as they were specified by the user. The input file is formatted as shown in table 1. The alphanumeric data (measurement) is left-justified in a 5A4 format. The floating point fields (PARAM) are used to define sensor locations not at the center of gravity. The input name specified by the user for an observation variable serves both to identify the observation variable selected within the program itself as well as to identify observation variables on the printed output of LINEAR. Observation variable Units Derivatives of Symbol Alphanumeric state variables descriptors Roll acceleration rad/sec? p PDOT ROLL ACCELERATION Pitch acceleration rad/sec2 q QDOT PITCH ACCELERATION Yaw acceleration rad/sec2 h e RDOT YAW ACCELERATION Velocity rate Angle-of-attack rate ft/sec? rad/sec < e VDOT VELOCITY RATE a ALPDOT ALPHA DOT ALPHADOT Angle-of-sideslip rate rad/sec W e BTADOT BETA DOT BETADOT Pitch attitude rate rad/sec D e THADOT THETA DOT Heading rate rad/sec y PSIDOT PSI DOT Roll attitude rate rad/sec G e PHIDOT PHI DOT 70 Observation variable Derivatives Units Symbol of state variables Alphanumeric (continued) descriptors Altitude rate ft/sec h HDOT ALTITUDE RATE Velocity north ft/sec x XDOT Velocity east ft/sec y YDOT x body axis acceleration y body axis acceleration zZ body axis acceleration x body axis accelerometer at vehicle center of gravity y body axis accelerometer at vehicle center of gravity z body axis accelerometer at vehicle center of gravity Accelerations g ay g ay g ag g anx AX LONGITUDINAL ACCEL X-AXIS ACCELERATION X AXIS ACCELERATION X“BODY AXIS ACCEL X BODY AXIS ACCEL AY Y-AXIS ACCELERATION Y AXIS ACCELERATION Y-BODY AXIS ACCEL Y BODY AXIS ACCEL LATERAL ACCELERATION LAT ACCEL LATERAL ACCEL AZ Z~BODY Z BODY AXIS AXIS ACCEL ACCEL ANX A-AXIS X AXIS ACCELEROMETER ACCELEROMETER g any ANY Y-AXIS ACCELEROMETER Y AXIS ACCELEROMETER g ang ANZ Z-AXIS ACCELEROMETER Z AXIS ACCELEROMETER 71 Observation variable Units Symbol Alphanumeric descriptors Accelerations (continued) normal acceleration g an AN NORMAL ACCELERATION NORMAL ACCEL GS G's *x body axis g accelerometer not at vehicle center of gravity anx,i AX, I ANX,I *y body axis g accelerometer not at vehicle center of gravity any,i AY,1I ANY,1I *z body axis g accelerometer not at vehicle center of gravity 4nz,i AZ,I ANZ,I *normal accelerometer g 4n,i AN, I not at vehicle center of gravity load factor Dimension- n less N LOAD FACTOR Speed of sound tReynolds number Reynolds number per unit length Mach number Air data parameters ft/sec a A SPEED OF SOUND Dimension- Re less RE REYNOLDS NUMBER ft71 Re' RE PRIME R/LENGTH R/FEET R/UNIT LENGTH Dimension- M less 72 Observation variable Units Symbol Alphanumeric descriptors Air data parameters (continued) Dynamic pressure Impact pressure Static pressure Impact/ambient pressure ratio lb/f£t2 lb/ft2 lb/ft2 Dimension~ less q dec Pa q./p, QBAR DYNAMIC PRESSURE QC IMPACT PRESSURE DIFFERENTIAL PRESSURE PA STATIC PRESSURE FREESTREAM PRESSURE QC/PA OC/P Total pressure lb/ft2 Pt pT TOTAL PRESSURE Temperature Total temperature deg absolute deg absolute T TEMP TEMPERATURE FREESTREAM TEMPERATURE Te TOTAL TEMPERATURE Equivalent airspeed knot Ve VEAS EQUIVALENT KEAS AIRSPEED Calibrated airspeed knot Ve VCAS CALIBRATED KCAS AIRSPEED Flightpath angle Flightpath acceleration Flightpath angle rate Flightpath-related parameters rad Y GAM GAMMA FLIGHT PATH ANGLE GLIDE PATH ANGLE GLIDE SLOPE g fpa FPA FLIGHT PATH ACCEL rad/sec Y GAMMA DOT GAMMADOT 73 Observation variable Units Flightpath-related Symbol parameters Alphanumeric (continued) descriptors Vertical acceleration ft/sec2 h VERTICAL ACCELERATION HDOTDOT H-DOT=-DOT HDOT=DOT Scaled altitude rate Specific energy Specific power Lift force Drag force Normal force Axial force ft/sec h/57.3 H-DOT/57.3 HDOT / 57.3 Energy-related terms ft Eg ES E-SUB-S SPECIFIC ENERGY ft/sec Py Force parameters PS P-SUB-S SPECIFIC SPECIFIC POWER THRUST lb L LIFT lb D DRAG lb N NORMAL FORCE lb A AXIAL FORCE x body axis velocity y body axis velocity Body axis parameters ft/sec u UB X-BODY X BODY X~BODY X BODY U-BODY U BODY AXIS AXIS AXIS AXIS VELOCITY VELOCITY VEL VEL ft/sec Vv VB Y-BODY Y BODY Y-BODY Y BODY V-BODY V BODY AXIS AXIS AXIS AXIS VELOCITY VELOCITY VEL VEL Observation variable z body axis velocity Units Symbol Alphanumeric descriptors Body axis parameters ft/sec w WB Z-BODY AXIS VELOCITY Z BODY AXIS VELOCITY Z-BODY AXIS VEL Z BODY AXIS VEL W-BODY W-BODY Rate of change of velocity in x body axis ft/sec2 a UBDOT UB DOT Rate of change of velocity in y body axis ft/sec2 v VBDOT VB DOT Miscellaneous measurements not at vehicle center of gravity Rate of change of velocity in z body axis ft/sec2 “ *Angle of attack not at vehicle center of gravity rad OG *Angle of sideslip not rad Bhi at vehicle center of gravity *Altitude instrument Et hii not at vehicle center of gravity WBDOT WB DOT ALPHA,I ALPHA INSTRUMENT AOA INSTRUMENT BETA,1 BETA INSTRUMENT SIDESLIP INSTRUMENT H,1I ALTITUDE INSTRUMENT *Altitude rate instrument not at vehicle center of gravity ft/sec hy HDOT, I Other miscellaneous parameters Vehicle total angular slug-ft2/ T momentum secé ANGULAR MOMENTUM ANG MOMENTUM Stability axis roll rate rad/sec Ps STAB AXIS ROLL RATE 75 Observation variable Units Symbol Other miscellaneous parameters Alphanumeric (continued) descriptors Stability axis pitch rate rad/sec ds STAB AXIS PITCH RATE Stability axis yaw rate rad/sec ts STAB AXIS YAW RATE *All parameters with an asterisk are for measurements at some point other than the vehicle center of gravity. The program LINEAR uses the quantities defined in the first three floating point fields as definitions of the location of the sensor with respect to the vehicle center of gravity. The three parameters define the x body, y body, and z body locations, in that order, of the sensor. The units of these offsets from the vehicle center of gravity are defined in units of feet. tReynolds number is defined in terms of an arbitrary unit of length that is input by the user. This length is input using the first floating point field; however, if no value is input, c is used as the default value. 76 APPENDIX D: STATE VARIABLE NAMES RECOGNIZED BY LINEAR The alphanumeric descriptors specifying state variables that are recognized by LINEAR are listed in this appendix. In the input file, the field containing these descriptors uses a 5A4 format, and all characters are left justified. The input alphanumeric descriptor specified by the user serves both to identify the state variable selected by the user within the program itself as well as to identify state variables on the printed output of LINEAR as described in the Output Files section. State variable Roll rate Pitch rate Yaw rate Velocity Angle of attack Sideslip angle Pitch attitude Heading Roll attitude Altitude Displacement north Displacement east Units rad/sec rad/sec rad/sec ft/sec rad rad rad rad rad ft Et ft Symbol p q Yr Vv a B 6 ¥ @ h x y Alphanumeric descriptors P ROLL RATE Q PITCH RATE R YAW RATE Vv VELOCITY VEL VTOT ALP ALPHA ANGLE OF ATTACK BTA BETA SIDESLIP ANGLE OF ANGLE SIDESLIP THA THETA PITCH ATTITUDE PSI HEADING HEADING ANGLE PHI ROLL BANK ATTITUDE ANGLE H ALTITUDE Xx XY 77 APPENDIX E: ANALYSIS POINT DEFINITION IDENTIFIERS Analysis point definition options are selected using alphanumeric descriptors. These descriptors are the first record read for each analysis case. All these descriptors are read using a 5A4 format. The following list associates the analysis point definition options with their alphanumeric descriptors: Analysis point definition option Untrimmed Straight-and-level Pushover~pullup Level turn Thrust stabilized turn Beta Specific power Alphanumeric descriptors UNTRIMMED NO TRIM NONE NOTRIM STRAIGHT AND WINGS LEVEL LEVEL FLIGHT LEVEL PUSHOVER AND PULLUP PULLUP PUSH-OVER/PULL=-UP PUSH-OVER / PULL-UP PUSHOVER / PULLUP PUSHOVER PULLUP PUSH OVER PULL UP PUSHOVER PUSHPULL LEVEL TURN WINDUP TURN THRUST STABILIZED TURN THRUST LIMITED TURN FIXED THROTTLE TURN FIXED THRUST TURN BETA SIDESLIP SPECIFIC PS P-SUB-S POWER Each of these analysis point definitions except the untrimmed, beta, and specific power options has two suboptions associated with it. The suboptions are requested using alphanumeric descriptors read using an A4 format. These suboptions are defined 78 in the Analysis Point Definition section of this report. The following list defines these suboptions and the alphanumeric descriptors associated with each: Analysis point definition suboptions Straight-and-level: Alpha-trim Mach-trim Pushover-pullup: Alpha-trim Alphanumeric descriptors ALP ALPH ALPHA MACH AMCH ALP Load-factor-trim LOAD Level turn: Alpha-trim Load-factor-trim ALP LOAD Thrust stabilized turn: Alpha-trim Load-factor-trim ALP LOAD 79 APPENDIX F: EXAMPLE INPUT FILE The following listing is an example of an input file to LINEAR. This file was used with the example subroutines listed in appendix J to generate the analysis and line printer files shown in appendixes H and I, respectively. The formats specified in table 1 would be the same whether each type of input has its own file or all of the data is in one file. USER'S GUIDE 6.080000E+02 4.280000E+01 1.595000E+01 4.500000E+04 2-870000E+04 1.651000E+05 1.879000E+05-5.200000E+02 0. 0. 0. 0. 0. CCALC WILL CALCULATE CG CORRECTIONS -1.000000E+01 4.000000E+01 4STAN RADI ALPHA Q THETA VEL 3 ELEVATOR 5 THROTTLE 12 SPEED BRAKE 10 2STAN AN AY ~2.900E+00 5.430E+00-4.000E+00 4.000E+00-3.250E+00 3.250E+00-1.000E+00 _ 0 ADDITIONAL SURFACES WINDUP TURN ALPHA H 20000.0 MACH 0.90 AN 3.00 BETA 0.0 NEXT LEVEL FLIGHT ALPHA H 20000.0 MACH 0.9 GAMMA 10.0 END 1.000E+00 This is input fies the input file is for a case called USER'S GUIDE (record 1). from the terminal as described in the Data Input section. mass and geometric properties of the vehicle as The project title Record 2 speci- t t S = 608 ft2 b f l 42.8 ft 80 c = 15.95 ft w = 45,000 lbs Record 3 defines the moments and products of intertia of the vehicle as Ix = 28,700 slug-ft? Ty = 165,100 slug-ft4 Iz, = 187,900 slug-ft? Ix, = -520 slug-ft? Ixy = 0.0 slug-ft2 Iyz = 0.0 slug-ft2 Record 4 defines the location of the aerodynamic reference point to be coincident with the vehicle center of gravity of the nonlinear aerodynamic model by setting Ax = 0.0 ft fy = 0.0 ft Az = 0.0 ft Record 4 also specifies that LINEAR should not use its internal model to make corrections for the offset in the vehicle center of gravity from the aerodynamic reference point because the aerodynamic model includes such corrections. Record 5 defines the angle-of-attack range of the aerodynamic model. Record 6 specifies that there will be four state variables in the linear model, that the formulation of the state equation will be in the standard form (x =Ax + Bu), and that the nondimensional stability and control derivatives with respect to angle of attack and angle of sideslip should be scaled in radians~!. The next four records define the output formulation of the state vector to be o qa 3) Vv Record 11 specifies that the linear model will have three parameters in the control vector. The following three records (12 to 14) specify that elevator u = ] throttle speed brake 81 and that elevator, throttle, and speed brake are located in DC(5), DC(12), and DC(10) of the /CONTROL/ common block. Record 15 specifies that two observation variables will be used, and that the observation equation will be in the standard form, y = Hx + Fu The next two records (16 and 17) define the elements of the output vector to be =| y= ay Record 18 specifies the ranges for the trim parameters DES, DAS, DRS, used to trim the longitudinal, lateral, and directional axes, and thrust, tively. The ranges for these parameters are defined by record 20 to be and THRSTX, respec- ~2.9 -4.0 -3.25 -~1.0 < DES < 5.43 < DAS < 4.0 < DRS < 3.25 < THRSTX < 1.0 The first three parameters essentially represent stick and rudder positions and are so specified because of the implementation of the subroutine UCNTRL (discussed in app. J). The thrust trim parameter is specified in this manner because of the implementation of UCNTRL. Speed brake is scheduled when THRSTX < 0 and thrust is commanded when THRSTX > 0. Record 19 specifies that no additional control surfaces are to be set. The next seven records (20 to 26) define an analysis point option. request a level turn trim option at These records h = 20,000 ft M = 0.9 an = 3.0 g and B = 0.0° The second record of this set (record 21) indicates which level turn suboption is requested. The alphanumeric descriptor ALPHA indicates that angle of attack is to be varied until the specified 3.0-g turn is achieved. The final record of this analysis point option definition set contains the keyword NEXT to indicate both an end to the current analysis point option definition and that another analysis point option defi- nition follows. 82 The final six records (records 27 to 32) define a straight-and-level analysis point option at 20,000 ft 0.9 and Y 10.0° The second record of this set (record which angle of attack is varied until The final record of this set contains the current analysis point definition 28) identifies the Alpha Trim suboption in trim is achieved at the specified condition. the keyword END to indicate the termination of as well as the termination of input cases. 83 APPENDIX G: EXAMPLE LINEARIGED STABILITY AND CONTROL DERIVATIVE FILE STABILITY AND CONTROL DERIVATIVES FOR CASE #1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT: F-15 SIMULATION REFERENCE ALTITUDE REFERENCE MACH 2.0000D+04 9.0000D-01 ANGLES ARE IN RADIANS FOR OUTPUT WARNING!!!! DERIVATIVES WITH RESPECT VELOCITY DERIVATIVES WILL BE COMPUTED TO MACH WILL BE USED. FROM MACH DERIVATIVES. THERE ARE 3 CONTROL SURFACES ROLL MOMENT DERIVATIVES CLO ROLL RATE (RAD/SEC) PITCH RATE (RAD/SEC) YAW RATE (RAD/SEC ) VELOCITY (FT/SEC) MACH NUMBER ALPHA (RAD) BETA ALTITUDE (RAD } (FT) ALPHA DOT BETA DOT (RAD/SEC) (RAD/SEC) -4.02966D-05 -2.00000D-01 -00000D+00 1.50990D-01 -1.26696D-07 -1.31375D-04 -00000D+00 -1.33450D-01 -00000D+00 -00000D+00 -00000D+00 ELEVATOR THROTTLE SPEED BRAKE 5 -00000D+00 12 -00000D+00 10 -00000D+00 PITCH MOMENT DERIVATIVES CMO ROLL RATE (RAD/SEC) PITCH RATE (RAD/SEC) YAW RATE (RAD/SEC) VELOCITY (FT/SEC) MACH NUMBER ALPHA (RAD) BETA (RAD) ALTITUDE (FT) ALPHA DOT (RAD/SEC) BETA DOT (RAD/SEC ) 4.22040D-02 -00000D+00 3.89527D+00 -00000D+00 -3.28490D-06 ~3.40620D-03 -1.68819D-01 -90000D+00 -00000D+00 -1.18870D+01 -00000D+00 ELEVATOR THROTTLE SPEED BRAKE 5 -6.95281D-01 12 -00000D+00 10 -4.17500D-01 84 YAW MOMENT DERIVATIVES CNO ROLL RATE (RAD/SEC) PITCH RATE (RAD/SEC) YAW RATE (RAD/SEC ) VELOCITY (FT/SEC) MACH NUMBER ALPHA (RAD) BETA ALTITUDE (RAD) (FT) ALPHA DOT (RAD/SEC) BETA DOT (RAD/SEC) ELEVATOR THROTTLE SPEED BRAKE DRAG FORCE DERIVATIVES cbD0 ROLL RATE (RAD/SEC) PITCH RATE (RAD/SEC) YAW RATE (RAD/SEC) VELOCITY (FT/SEC) MACH NUMBER ALPHA (RAD) BETA (RAD) ALTITUDE (FT) ALPHA DOT (RAD/SEC) BETA DOT (RAD/SEC) ELEVATOR THROTTLE SPEED BRAKE LIFT FORCE DERIVATIVES CLFTO ROLL RATE (RAD/SEC) PITCH RATE (RAD/SEC) YAW RATE (RAD/SEC ) VELOCITY (FT/SEC) MACH NUMBER ALPHA (RAD) BETA ALTITUDE (RAD) (FT) ALPHA DOT (RAD/SEC) BETA DOT (RAD/SEC) ELEVATOR THROTTLE SPEED BRAKE 2.25747D-04 -3.37217D-02 -00000D+00 -4.04710D-01 3.21400D-07 3.33268D-04 -00000D+00 1.29960D-01 -00000D+00 »90000D+00 -00000D+00 -00000D+00 12 -00000D+00 10 -00000D+00 1.08760D-02 -00000D+00 -00000D+00 -00000D+00 -90000D+00 -00000D+00 3.72570D-01 -00000D+00 -00000D+00 -Q0000D+00 -00000D+00 5 4.38318D-02 12 -00000D+00 10 6.49355D-~02 1.57360D-01 -00000D+00 -1.72315D+01 -00000D+00 1.45286D-05 1.50651D-02 4.87061D+00 -00000D+00 -00000D+00 1.72315D+0 1 -00000D+00 5 5.72950D-01 12 -00000D+00 10 3-74913D-02 85 SIDE FORCE DERIVATIVES cy0o ROLL RATE PITCH RATE YAW RATE VELOCITY (RAD/SEC) (RAD/SEC) (RAD/SEC ) (FT/SEC) MACH NUMBER ALPHA (RAD ) BETA ALTITUDE (RAD) (FT) ALPHA DOT BETA DOT (RAD/SEC) (RAD/SEC) ELEVATOR THROTTLE SPEED BRAKE 5.32725D-04 -Q00000D+00 -00000D+00 -00000D+00 -00000D+00 -00000D+00 -00000D+00 -9.74030D-01 -00000D+00 -00000D+00 -00000D+00 5 -00000D+00 12 -00000D+00 10 -00000D+00 86 STABILITY AND CONTROL DERIVATIVES FOR CASE #2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT: F-15 SIMULATION REFERENCE ALTITUDE REFERENCE MACH = 2-0000D+04 9.0000D-01 ANGLES ARE IN RADIANS FOR OUTPUT WARNING!!!! DERIVATIVES WITH RESPECT VELOCITY DERIVATIVES WILL BE COMPUTED TO MACH WILL BE USED. FROM MACH DERIVATIVES. THERE ARE 3 CONTROL SURFACES ROLL MOMENT DERIVATIVES CLO ROLL RATE PITCH RATE YAW RATE (RAD/SEC) (RAD/SEC) (RAD/SEC ) VELOCITY (FI/SEC) MACH NUMBER ALPHA (RAD) BETA (RAD) ALTITUDE ALPHA DOT (FT) (RAD/SEC) BETA DOT (RAD/SEC) 1.25377D-16 ~2.00000D-01 -00000D+00 1.50990D-01 -00000D+00 -00000D+00 -Q0000D+00 -1.33450D-01 -00000D+00 -00000D+00 -00000D+00 ELEVATOR THROTTLE SPEED BRAKE 5 -00000D+00 12 -00000D+00 10 -00000D+00 PITCH MOMENT DERIVATIVES CMO ROLL RATE (RAD/SEC) PITCH RATE (RAD/SEC) YAW RATE (RAD/SEC) VELOCITY (FT/SEC) MACH NUMBER ALPHA (RAD) BETA (RAD) ALTITUDE (FT) ALPHA DOT (RAD/SEC) BETA DOT (RAD/SEC) 4.22040D-02 -00000D+00 3.89530D-00 -00000D+00 -7.79749D-10 -8.08541D-07 -1.68819D-01 -00000D+00 -00000D+00 -1.18870D+01 -00000D+00 ELEVATOR THROTTLE SPEED BRAKE 5 -6.95279D-01 12 -00000D+00 10 -4.17500D-01 YAW MOMENT DERIVATIVES CNO ROLL RATE (RAD/SEC) PITCH RATE (RAD/SEC) YAW RATE (RAD/SEC) VELOCITY (FT/SEC) MACH NUMBER ALPHA (RAD ) BETA (RAD) ALTITUDE (FT) ALPHA DOT (RAD/SEC) BETA DOT (RAD/SEC) ELEVATOR THROTTLE SPEED BRAKE DRAG FORCE DERIVATIVES CbDO ROLL RATE (RAD/SEC) PITCH RATE (RAD/SEC) YAW RATE (RAD/SEC) VELOCITY (FT/SEC) MACH NUMBER ALPHA (RAD) BETA ALTITUDE (RAD) (FT) ALPHA DOT BETA DOT (RAD/SEC) (RAD/SEC) ELEVATOR THROTTLE SPEED BRAKE LIFT FORCE DERIVATIVES CLFTO ROLL RATE PITCH RATE YAW RATE VELOCITY MACH NUMBER ALPHA BETA ALTITUDE ALPHA DOT BETA DOT (RAD/SEC) (RAD/SEC) (RAD/SEC) (FT/SEC) (RAD) (RAD) (FT) (RAD/SEC) (RAD/SEC) ELEVATOR THROTTLE SPEED BRAKE 88 1.22535D-16 ~3.37217D-02 »00000D+00 -4.04710D-01 -00000D+00 -00000D+00 -90000D+00 1.29960D-01 -00000D+00 -00000D+00 -00000D+00 5 -00000D+00 12 -00000D+00 10 -900000D+00 1.08760D-02 -00000D+00 -00000D+00 -00000D+00 -00000D+00 -00000D+00 3.72570D-01 -00000D+00 -00000D+00 -00000D+00 -00000D+00 5 4.38313D-02 12 -00000D+00 10 6.49346D-02 1.57360D-01 -90000D+00 -1.72320D+01 -00000D+00 3.59263D-09 3.72529D-06 4.87061D+00 -00000D+00 -90000D+00 1.72320D+01 -00000D+00 5 5.72961D-01 12 -00000D+00 10 3.74913D-02 SIDE FORCE DERIVATIVES cyo ROLL RATE (RAD/SEC) PITCH RATE (RAD/SEC) YAW RATE (RAD/SEC ) VELOCITY (FT/SEC) MACH NUMBER ALPHA (RAD) BETA (RAD ) ALTITUDE (FT) ALPHA DOT (RAD/SEC) BETA DOT (RAD/SEC ) ELEVATOR THROTTLE SPEED BRAKE 4.15737D-16 -00000D+00 -00000D+00 -00000D+00 -00000D+00 -00000D+00 -00000D+00 -9.74030D-01 -00000D+00 -00000D+00 -00000D+00 5 -00000D+00 12 -00000D+00 10 -00000D+00 89 APPENDIX H: EXAMPLE OUTPUT ANALYSIS FILE The following is an example analysis file. example input file listed in appendix F and the listed in appendix J. This file was produced using the example user-supplied subroutines LINEARIZER TEST AND DEMONSTRATION CASES USER'S GUIDE TEST CASE # TTT TTT TT CTT TTT eT TC PCPS SSCS SCC ee SS SS te X - DIMENSION = 4 U - DIMENSION = 3 STATE EQUATION FORMULATION: STANDARD OBSERVATION EQUATION FORMULATION: STANDARD Y¥ ~ DIMENSION = 2 STATE VARIABLES ALPHA Q THETA VEL = 0.465695D-01 = 0.921683D-01 = 0.159885D-01 = 0.933232D+03 RADIANS RADIANS/SECOND RADIANS FEET/SECOND CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE = 0.538044D-01 = 0.214105D+00 0.000000D+00 DYNAMIC INTERACTION VARIABLES X-BODY AXIS FORCE Y-BODY AXIS FORCE Z-BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT YAWING MOMENT = 0.134742D+06 = 0.102657D+05 = 0.938155D-03 = 0.680525D+t02 = 0.000000D+00 = -.212007D+02 POUNDS POUNDS POUNDS FOOT-POUNDS FOOT-POUNDS FOOT-POUNDS OBSERVATION VARIABLES AN 0.300163D+01 Gs AY i l 0.941435D+00 GS 90 A~MATRIX FOR: DX / DT = A*X + B¥*¥U+t+D*YV -0.121436D+01 0.100000D+01 0.136756D-02-0.121605D-03 -0.147423D+01-0.221451D+01-0.450462D-02 0.294019D-03 0.000000D+00 0.331812D+00 0.000000D+00 0.000000D+00 -0.790853D+02 0.000000D+00-0.320822D+02-0.157297D-01 B-MATRIX FOR: DX / DT = A*xXK + B*U+tD*¥YV 0.14196 1D+00- 0. 164948D-02-0.928933D-02 0.220778D+02 0.543324D-02-0.135074D+02 0.000000D+00 0.000000D+00 0.000000D+00 0.105186D+02 0.342817D+02-0.155832D+02 D-MATRIX FOR: DX /DT=A*X+BFUtTtD*V 0.343642p-07 0.113192D-06 0.000000D+00 0.714203D-03 0.Q000000D+00 0.737378D-06 0.000000D+00-0.242885D-05 0.000000D+00 0.000000D+00 0.398492D-06 0.332842D-04 0.000000D+00 0.605694D-05 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 H-MATRIX FOR: Y = H*X + F* U+E* V 0.351752D+02 0.277556D-16 0.150046D-02 0.640771D-02 0.000000D+00 0.000000D+00-0.150534D-01 0.446789D-10 F-MATRIX FOR: Y = H* X +F*U+tT+E*YV 0.412845D+01-0.180978D-02 0.291699D+00 0.000000D+00 0.000000D+00 0.000000D+00 E-MATRIX FOR: Y=H* X+F* U+t+E*V -0.377037D-07 0.000000D+00-0.214132D-04 0.Q000000Dt00 0.222222D-04 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 TEST CASE KKEKKKKHKKKEKKKHEKKEEEEEEKEKEEKEKEKEEKEKKKEKKKEKKKKKEKKKEKE X - DIMENSION = 4 U - DIMENSION = 3 ¥Y ~ DIMENSION = STATE EQUATION FORMULATION: STANDARD OBSERVATION EQUATION FORMULATION: STANDARD STATE VARIABLES ALPHA 0) THETA VEL u l -.126650D-01 = 0.0Q00000D+00 0.161868D+00 = 0.933232D+03 RADTANS RADIANS/ SECOND RADIANS FEET/ SECOND CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE = 0.637734D-01 = 0.225092D+00 = 0.Q000000D+00 DYNAMIC INTERACTION VARIABLES X-BODY AXIS FORCE Y-BODY AXIS FORCE Z-BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT “YAWING MOMENT i ! = 0.443769D+05 = 0.300494D+04 = 0.445795D-09 = ~.528481D-11 = 0.000000D+00 0.114239D-10 POUNDS POUNDS POUNDS FOOT-POUNDS FOOT-POUNDS FOOT-POUNDS OBSERVATION VARIABLES AN 0.985228D+00 GS AY 0.990655D-14 GS A-MATRIX FOR: DX / pT = A* X + B*YUtTt+D*YV 0.120900D+01 0.100000D+01-0.575730D-02-0.701975D-04 0.149189D+01-0.221451D+01 0.189640D-01 0.231368D-03 0.000000D+00 0.100000D+01 0.000000D+00 0.000000D+00 0.576868D+02 0.000000D+00-0.316251Dt02-0.460435D-02 B-MATRIX FOR: DX / DT = A*X + B*YUTtD*V 0.141961D+00 0.448742D-03-0.928932D-02 0.220778D+02-0.147812D-02-0.135074D+02 0.000000D+00 0.000000D+00 0.000000D+00 0.105186D+02 0.343162D+02-0. 155832D+02 92 D-MATRIX FOR: DX f/ DT +A*X+B*U+t+D* V 0.934880D-08 0.307941D-07 0.000000D+00 0.714920D-03 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.738119D-06 0.000000D+00-0.243129D-05 0.000000D+00 0.000000D+00 0.000000D+00-0.905497D-05 0.000000D+00 0.605694D-05 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 H-MATRIX FOR: Y = H* xX + F* U 0.350424D+02 0.277556D-16-0.632314D-02 0.203434D-02 0.000000D+00 0.000000D+00 0.000000D+00 0.212306D-16 F-MATRIX FOR: Y = H*X + F*U 0.411323D+01 0.492845D-03 0.263288D+00 0.000000D+00 0.000000D+00 0.000000D+00 E-MATRIX FOR: Y+H* X+F* U+E* V 0.102676D-07 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00-0.214116D-04 0.222222D-04 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 0.000000D+00 FOR THE PROJECT: USER'S GUIDE ALPHA QTHETA VEL THE STANDARD FORMULATION OF THE STATE EQUATION HAS BEEN SELECTED. THE FORM OF THE EQUATION IS: DX/DT A*xX+B* U SURFACES TO BE USED FOR THE CONTROL VECTOR FOR: LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT: USER'S GUIDE LOCATION IN /CONTROL/ ELEVATOR 5 THROTTLE 12 SPEED BRAKE 10 PARAMETERS USED IN THE OBSERVATION VECTOR FOR: LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT: USER'S GUIDE AN AY THE STANDARD FORMULATION OF THE OBSERVATION EQUATION HAS BEEN SELECTED. THE FORM OF THE EQUATION IS: Y=H*X+FE*U LIMITS FOR TRIM OUTPUT PARAMETERS: PITCH AXIS PARAMETER ROLL AXIS PARAMETER YAW AXIS PARAMETER THRUST PARAMETER MINIMUM -2.900 -4.000 -3.250 ~1.000 MAXIMUM 5.430 4.000 3.250 1.000 NO ADDITIONAL SURFACES TO BE SET WERE DEFINED 95 TRIM CONDITIONS FOR CASE # 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT: USER'S GUIDE LEVEL TURN WHILE VARYING ALPHA TRIM ACHIEVED COEFFICIENT OF LIFT COEFFICIENT OF DRAG LIFT DRAG ALTITUDE MACH VELOCITY EQUIVALENT AIRSPEED SPEED OF SOUND GRAVITATIONAL ACCEL NORMAL ACCELERATION LOAD FACTOR DYNAMIC PRESSURE DENSITY WEIGHT (@GALTITUDE) BETA ALPHA PHI THETA ALTITUDE RATE GAMMA ROLL RATE PITCH RATE YAW RATE THRUST SUM OF THE SQUARES (LBS) (LBS) (FT) (FT/SEC) (KTS) (FT/SEC) (PT/SEC**2) (G-S) (LBS/FT**2 ) (SLUG/FT**3 ) (LBS) (DEG) (DEG) (DEG) (DEG) (FT/SEC) (DEG) (DEG/SEC ) (DEG/SEC) (DEG/SEC) (LBS) 0.40144 0.03058 134741.68807 10265.70661 20000.00000 0.90000 933.23196 403.42303 1036.92440 32.11294 3.00163 2.99995 552.05302 0.00126774 44914.60434 0.03193 2.66824 70 .62122 0.91607 0.00000 0.00000 -0.08951 5-28086 1.85749 10277.03515 0.00000 TRIM PARAMETERS TRIM TRIM TRIM TRIM PITCH AXIS PARAMETER ROLL AXIS PARAMETER YAW AXIS PARAMETER THRUST PARAMETER i } -0.66958 -~0.01526 -0.02125 0.21410 CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE (DEG) 0.05380 0.21410 0.00000 OBSERVATION VARIABLES AN 3.00163339 GS AY -94136286 GS 96 00+0d00000°0 00+000000°0 00+d00000°0 00+4000000°0 00+d000000°0 00+4d00000°0 L0-GOeOPrl*600+d00000°0 00+d00000°0 00+d00000°0 00+d00000°0 00+d00000°0 00+000000°0 VO-ASCLEE*S qOXOT darts TO-AOZEPL°E 00+d00000°0 LO-dO9672L°S 00+d00000°0 LO+GOCETL* EL 00+d00000°0 00+d00000°0 00+0090L8°P cO-GEO80S"* 1 SO-CEEPSH LE 00+d00000°0 LO+CO0ZECL*100+d000000°0 LO-d09ELG"t LITT cCO-GOSE6P*9 00+d00000°0 CO-GOLEBE'D 00+4000000°0 00+000000°0 00+d00000°0O 00+d00000°0 L0-GOLSZL°E 00+4000000°0 00+d00000°0 00+d00000°0 00+d00000°0 00+d00000°0 cO-d09280° t owad ATAaVLS GYOHO OIWYWNAGOYRY NVAW %S°E *NOITLIGNOO SI NISYWW LHOITA OILWLS SIHL LY FIDIHAA 00+d00000°0 00+d00000°0 00+d00000°0 LO-d00SL1°700+d00000°0 t0-d08zs6°9- 00+000000°0 00+d00000°0 00+000000°0 aywvdd dagads A TLLONHL YOLWATTA 00+000000°0 00+000000°0 00+d00000°0 L0-d0966c" tL 00+000000°0 VO-ACGECE°E LO-A96017°E LO-d0LLbO*P00+d00000°0 cO-dO1LzcLe°€PO-GLULSZ°? 00+000000°0 LO+COL88L°T- 00+d00000°0 00+d00000°0 L0-GOZ889°*t€O0-d8Ls80r*e9O0-C6ELBC*E00+d00000°0 00+00€S68°E 00+d00000°0 CO-dOvOCT* BD 00+400000°0 00+000000°0 00+d00000°0 LO-dOSPEE*L— 00+000000°0 PO-d089cE*tLO-QSS6LZ°tLO-d0660S° 1 00+d00000°*0 LO-d00000°z- S$0-099620°F- (ogS/awa) Lod WLad (oNs/awd) Lod VWHd Iv (La) AGNLILIY (awy) VLA (aw) WHd1W¥ UAGWOAN HOWW (O8S/L4) ALIDOIWMA (ogS/avu) ALWY MWA (ONS/avu) SALVA HOLId (04aS/awvy) ALWY TION SLNATOTdSAdTOO OY8dZ LNAWOW ONIMVA LNAWOW ONTHOLId LNAWOW ONT TIO’ ddqdInS S,4¥asn *LOeLOdd AHL YO SdSWO NOTLWAYLSNOWAG GNWY LSHL YaZIYWaNIT L # YSWO YOd SHAILWATIYSC IOULNOD GNW ALITIGVLS TWNOISNSWIG-NON 97