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User's Manual for Interactive LINEAR, a FORTRAN Program

1. e 46 FILES e 4 48 USER SUPPLIED SUBROUTINES 49 Aerodynamic Model Subroutines e e e e e e e e e e e e e 00 Control Model Subroutines 4 52 Engine Model Subroutines e e e e e n e e e oe 8 e e n e n n e e e e 53 Mass and Geometry Model Subroutines e e e e e e n 53 111 CONCLUDING REMARKS a APPENDIX A CORRECTIONS TO AERODYNAMIC COEFFICIENTS FOR CENTER OF GRAVITY NOT AT THE AERODYNAMIC REFERENCE POINT APPENDIX B ENGINE TORQUE AND GYROSCOPIC EFFECTS MODEL s e s e APPENDIX C OBSERVATION VARIABLE NAMES RECOGNIZED BY LINEAR APPENDIX D STATE VARIABLE NAMES RECOGNIZED BY LINEAR APPENDIX E ANALYSIS POINT DEFINITION IDENTIFIERS APPENDIX F EXAMPLE APPENDIX G EXAMPLE APPENDIX H EXAMPLE APPENDIX I EXAMPLE APPENDIX J EXAMPLE INPUT FI LE e LINEARIZED STABILITY AND CONTROL OUTPUT ANALYSIS FILE PRINTER OUTPUT FILE USER SUPPLIED SUBROUTINES Aerodynamic Model Subroutines Control Model Subroutine s s e Engine Model Interface Subroutine
2. 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 mome nt coefficient of yawing moment due to sideslip coefficient of yawing moment due coefficient of yawing moment due coefficient of yawing moment due coefficient of yawing moment due coefficient of yawing moment due total coefficient of sideforce coefficient of sideforce due to coefficient of sideforce due to coefficient of sideforce due to coefficient of sideforce due to lateral trim parameter to to to to to alleron deflection rudder deflection differential elevator deflection roll rate yaw rate sideslip aileron deflection rudder deflection differential elevator deflection 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
3. 11 F 12 Y 13 TDOT QDE 1 DF 2 QDOT DF 3 RDOT DF 4 VDOT DF 5 ALPDOT DF 6 BTADOT DF 7 THADOT DF 8 PSIDOT DF 9 PHIDOT DF 10 HDOT 11 DF 12 YDOT DF 13 EQUIVALENCE DA 1 DE DC 5 DT DC 8 DR DC 9 DSB DC 10 COMPUTE TERMS NEEDED WITH ROTATIONAL DERIVATIVES V2 2 0 2 B N2 C2V N2 C ROLLING MOMENT COEFFICIENT C CL CLB BTA CLDA DA CLDR DR CLDT DT B2V CLP p CLR R MOMENT COEFFICIENT CM CMO CMA ALP CMDE DE CMSB DSB C2V CMQ kO CMAD ALPDOT YAWING MOMENT COEFFICIENT 116 CNB BTA CNDA DA CNDR DR CNDT B2V CNP p CNR R COEFFICIENT OF DRAG CDO CDA ALP CDDE DE CDSB DSB C COEFFICIENT OF LIFT C CLFTO CLFTA ALP CLFTDE DE CLFTSB DSB C2V CLFTO 20 CLFTAD ALPDOT C SIDEFORCE COEFFICIENT C CYDA DA CYDR DR CYDT DT C RETURN END Control Model Subroutine The following subroutine UCNTRL models the gearing between the control inputs that is trim parameters and thr and the control surfaces This pro vides LINEAR with a generic interface capable of handling most airborne vehicles Figure 8 shows the control System imple
4. equation 582 95174 P dns 2 5 q 1 0 Vo gt 7 0 is executed until the change from one iteration to the next is less than 0 001 knots Also included in the observation variables are the flightpath related parameters described in app including flightpath angle flightpath acceleration fpa vertical acceleration h flightpath angle rate and for lack of better category in which to place it scaled altitude rate h 57 3 The equations used to determine these quantities are Y sinc B V V E g h sin 0 ay Sin cos 0 a cos cos 0 Vh V Vv2 12 Two energy related terms are included in the observation variables specific energy Eg and specific power Pg defined as lt e ll 2 Es h x 2g yy gt g 20 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 D qSCp N L cos a D sin A L sin a D cos 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 Also included are the time derivativ
5. Mass and Geometry Model s e REFERENCES iv e e L o e 54 56 58 66 73 74 76 80 86 90 109 109 113 115 116 117 interactive FORTRAN program that provides the user with a powerful and flex ible tool for the linearization of aircraft aerodynamic models is documented in this report The program LINEAR numerically determines a linear system model using non linear equations of motion and a user supplied linear or nonlinear aerodynamic model The nonlinear equations of motion used six degree of freedom equations with sta tionary atmosphere and flat nonrotating earth assumptions The system model deter mined by LINEAR consists of matrices for both the state and observation 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 the NASA Ames Research Center to provide a standard docu mented and verified tool to derive linear models for aircraft stability analysis and control law design This development was undertaken to address the need for the aircraft specific linearization programs common in the aerospace industry Also the lack of available documented linea
6. sin sin sin y cos cos y cos sin cos q sin sin y sin cos vy 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 gm sin 0 9 m Yp Y gm cos sin 9 9m az Zp D sin a L cos gm cos cos g m The eguations for body axis accelerometers that are at vehicle center of gravity are an Xp
7. Sin cos F 1 Eo F g sin cos t cos i Py F 0 F Pz s n i COS Es Pz so that Fpy j ES cos COS COS Ej COS 54 cos ny sin sin sin e cos i Eo cos COS COS sin nj sin cos Sin sin cos i Fpz cos COS Sin j sin n cos i The moment arm through which the vectored thrust acts is Ax 7 cos Ej COS Ej as Ar Ay cos g Sin 5 Az sin and the total torque due to thrust vectoring is n to Y Y Ar x AF p 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 9 Ta is the gyroscopic moment produced by the ith engine h the rate of change of the angular momentum of the ith engine le 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 r t If it is assumed that the angular momentum of the engine is constant then and the equation simplifies to Tg 0 x Te Wej Two terms le and Wess remain to be defined in the equation for the gyroscopic moment produced by the ith engine Once again si
8. SPECIFIC POWER PS P SUB S 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 the Analysis Point Definition section of this report these suboptions and the alphanumeric descriptors associated with each Analysis point definition suboptions Straight and level Alpha trim Mach trim Pushover pullup Alpha trim Load factor trim Level turn Alpha trim Load factor trim Thrust stabilized turn Alpha trim Load factor trim Alphanumeric descriptors ALP ALPH ALPHA MACH AMCH ALP ALPH ALPHA LOAD GS G S AN ALPH ALPHA LOAD GS G S AN ALPH ALPHA LOAD GS G S AN The following list defines 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
9. 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 put files from LINEAR is the stability and control derivative file that can be used later to analyze the same case See the Aerodynamic Model section for more infor mation When using a linearized set of stability and control derivatives from a previous case the user must be careful to define a control vector that is consistent with the derivative
10. hi4 Pe Ixe cos Ej Sin cos qe sin j cos j sin Gi fej sin ei 68 Thus the gyroscopic moment induced by the ith engine Tg can be expanded to 1 LS rhj Phi and the total moment induced by gyroscopic interaction of the vehicle dynamics and the rotating engine components is Me r3 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 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 spec ified by the user The input file is formatted as shown in table 1 The alphanumeric data meas urement 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 spec ified
11. The level trim options available in LINEAR require the specification of 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 1 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 defini tions 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 ver tical plane or is calculated using the equation n2 _ cos2 cos Y ttan 1 where the positive sign is used for a right turn and the negative sign is used for left turn From turn rate can be calculated as b 2 tan Using these two definitions the state variables be determined ene sin y cos B q y sin gy Sin y sin 3 sin Y sin 8 LI EOS gt sin2 r tan 4218 cos 8 8 P P cos rs Sin a r pg sin a rg cos a 28 sin 1 p W D 1 The trim surface positions thrust angle of sideslip and either angle of attac
12. e e e s s e e m n s e e 4 6 e e e e e 17 SELECTION OF STATE CONTROL AND OBSERVATION VARIABLES 22 LINEAR 22 ANALYSIS POINT DEFINITION v lt e w a Ge a 04 0808 089 e s e 5 5 5 5 544 4 6 d s I 0 e 2 e 45 289 Straight and Level 27 Pushover Pullup s lt lt lt s e e e e s s s s s e s 27 Level Turi 5 amp E O 28 a a 229 Thrust Stabilized Turn s s 8 e c o e 29 Beta we 5 AR A ww le DO 8 ie neo 109 Specific 29 NONDIMENSIONAL STABILITY AND CONTROL DERIVATIVES n n 31 DATA INPUT 42 5 8 44 24 AA 180 AAA A AA 233 Input Files 33 Title Information 33 Geometry and Mass Data 33 State Control and Observation Variable Definitions e 34 Trim Parameter Specification 7 37 Additional Surface Specification 37 Test Case Specification e e 6 s Bn s n 6 e e e e oo 38 Interactive Data Input 38 AERODYNAMIC MODEL
13. 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 be modified the user must input the entire vector again The choices available for the test case selection are to input the data from the terminal for each case or read the entire case specifica tion from a file 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 cont
14. D cos L sin a g m an Y g m an Zp D sin a L cos a g m an 2 D sin a L cos a g m For orthogonal accelerometers that are aligned with the vehicle body axes but are not at vehicle center of gravity the following equations apply n 2 M 2 o i 7 Any 7 q r xy pq r y pr 4 2 1 4 any pq r xy 2 r2 yy ar 2 1 4 2 25 2 2 An i pr q xz qr p y a2 21 9 Qan i a pr q X tar f plyg gt q2 pola where the subscripts x y and z refer to the x y and z body axes respectively and the symbols x y and z refer to the x y and z body axis locations of the sen sors relative to the vehicle center of gravity The symbol Jo is 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 ambient or free stream pressure total pressure ambient or free stream temperature T and total temperature Tye The selected reference and scaling parameters are Mach number M dynamic pressure speed of sound Reynolds number Re Reynolds number unit length and the
15. Mach meter calibration ratio Ac Pa These quantities are defined as 1 2 a 1 4 Fo Polo 18 eis M a pV 5 Ra u V B d S UM 2 q 2 2 2 5 tan p 1 0 p M gt 1 0 5 6M2 0 8 a 1 0 0 2M2 3 2 1 0 M 1 0 _ 2 5 i 2 1 2 2 2 6M 1 0 mM gt 1 0 5 6 2 0 8 Pa il dc T 1 0 0 2M2 Tc where p is the density of the air is the coefficient of viscosity and the sub Script O refers to sea level standard day conditions Free stream pressure free Stream temperature and the coefficient of viscosity are derived from the U S Stand ard Atmosphere 1962 Also included in the air data calculations are two velocities equivalent air speed Ve and calibrated airspeed Vc both computed in knots calculations assume that internal units are in the English system The equation used for equivalent air Speed is Ve 17 17 Va which is derived from the definition of equivalent airspeed 29 Ve where 0 002378 slug ft3 and Ve 15 converted from feet per second to knots 1 ibrated airspeed is derived from the following definition of impact pressure 19 HQ 2 V 2 5 C 5 76 1 2 22 P 2 E 29 0 8 RT For the case where lt ag the equation for is 2 7 dc Po 1479 116 Calibrated airspeed is found using an iterative process for the case where 2
16. 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 selected the user can specify the direction of the turn by entering TURN RIGHT or TURN LEFT when prompted to initialize any other var iables This is especially 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 5 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 to determine the source of input as explained previously If the user wishes to modify the vehicle geometry and mass data or the stability and control derivatives interactively the program saves all the values from the previous case and allo
17. 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 acceler ation Pitch attitude 0 is derived from the equation for flightpath angle with Y 0 and is defined as B sin cos B sin 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 LINEAR the Specific power option does not in general attempt to achieve the zero velocity rate V In fact because the altitude rate h 0 and specific power are defined by the resultant velocity rate will be PJ V Y However the other acceleration like terms T 8 will be zero if the requested analysis point is achieved The constraint equations used with the specific power analysis point definition option can be derived from the load factor tilt angle equation used in the level turn analysis point definition with y 0 or ttan n 1j V where the positive sign is used for a right turn and the negative sign is used for a left turn 9 7 tan q 41 cos B
18. AE tan cos ps tan B P Pg COS rs sin r pg sin a COS 0 9 sin 1 E 1 d tan analysis point surface positions load factor angle of attack angle of sideslip are determined by numerically solving the nonlinear equations for transla tional and rotational acceleration 30 NONDIMENSIONAL STABILITY AND CONTROL DERIVATIVES The nondimensional stability and control derivatives computed by LINEAR from the nonlinear aerodynamic model assume broadly formulated linear aerodynamic equations These equations include effects of what are normally considered exclusively lateral directional parameters in the longitudinal force and moment coefficient equations and conversely effects of longitudinal parameters the lateral directional equa tions The reason for this is two fold application to a larger class of vehicle types such as oblique wing aircraft and computational ease The nondimensional stability and control derivatives assume the following equa tions for the aerodynamic force and moment coefficients n Cg Cp Cp CgoB Co bh Cg 2 Cag i i Cy Bt Cyd Ce E Cg 42 n Cm Cng Cm V gt i l Cmgd CmQf Cmgf n Cn Cho Chaa ChaB Cnm h Chy V i h 121 C p r T npP Cn d Enga Cj Cy ot Cy at
19. AXIS ACCEL AY Y AXIS ACCELERATION AXIS ACCELERATION Y BODY AXIS ACCEL Y BODY AXIS ACCEL LATERAL ACCELERATION LAT ACCEL LATERAL ACCEL AZ BODY AXIS ACCEL 2 BODY AXIS ACCEL ANX 5 ACCELEROMETER X AXIS ACCELEROMETER ANY Y AXIS ACCELEROMETER Y AXIS ACCELEROMETER ANZ Z AXIS ACCELEROMETER Z AXIS ACCELEROMETER 71 72 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 ANX I vehicle center of gravity y body axis g AY I accelerometer not at ANY I vehicle center of gravity z body axis g anz i AZ I accelerometer not at 2 1 vehicle center of gravity normal accelerometer g an i not at vehicle center of gravity load factor Dimension n less LOAD FACTOR Air data parameters Speed of sound ft sec a A SPEED OF SOUND tReynolds number Dimension Re RE less REYNOLDS NUMBER Reynolds number RE PRIME per unit length R LENGTH R FEET R UNIT LENGTH Mach number Dimension M M less MACH AMCH a Observation variable Units Symbol Alphanumeric descriptors 222 22 2 Dynamic pressure Impact pressure Static pressure Impact ambient pressure ratio Total pressure Temperature Total temperature Equivalent airspeed Calibrated ai
20. Division Hawthorne California Nov 1980 Kwakernaak Huibert and Sivan Raphael Linear Optimal Control Systems Wiley Interscience New York 1972 Perkins C D and Hage R E Airplane Performance Stability and Control John Wiley amp Sons New York 1949 Thelander J A Aircraft Motion Analysis Air Force Flight Dynamics Laboratory FDL TDR 64 70 Mar 1965 0 5 Standard Atmosphere 1962 0 5 Government Printing Office 1962 121 UMA Report Documentation Page Space Admwnstration 1 Report No NASA TP 2835 2 Government Accession No Recipient s Catalog No 4 Title and Subtitle Report Date September 1988 User s Manual for Interactive LINEAR a FORTRAN Program To Derive Linear Aircraft Models Performing Organization Code H 1443 7 Author s Performing Organization Report No Robert F Antoniewicz Eugene L Duke RTOP 505 66 11 and Brian P Patterson 10 Work Unit No 9 Performing Organization Name and Address Contract or Grant No NASA Ames Research Center Dryden Flight Research Facility Box 273 Edwards CA 93523 5000 Type of Report and Period Covered 12 5 ing Age Name and Add A ponsoring Agency an ress Technical Paper National Aeronautics and Space Administration Washington DC 20546 Sponsoring Agency Code 15 Supplementary Notes A listing of the progr
21. 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 center of gravity position are passed in the CGSHFT common block and inertia changes are passed in the DATAIN common block Care must be taken when using the subroutine MASGEO in combination with selecting an observation vector that contains measurements away from the center of gravity If measurements are desired at a fixed location on the vehicle such as a sensor plat form or nose boom the moment arm to the new center of gravity location must be recomputed as a result of the center of gravity shift for accurate results This can be accomplished by implementing the moment arm calculations in one of the user subroutines and passing the new moment arm values through the OBSERV common block COMMON OBSERV OBVEC 120 PARAM 120 6 The common block OBSERV allows the user to access all the observation vari ables during trim as well as to pass
22. OUTPUT ANALYSIS FILE The following is an example analysis file This file was produced using the example input file listed in appendix F and the example user supplied subroutines listed in appendix J LINEARIZER TEST AND DEMONSTRATION CASES USER S GUIDE TEST CASE kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk 4 X DIMENSION 4 U DIMENSION Y DIMENSION 2 STATE EQUATION FORMULATION STANDARD OBSERVATION EQUATION FORMULATION STANDARD STATE VARIABLES ALPHA 0 4656950 01 RADIANS Q 0 921683D 01 RADIANS SECOND THETA 0 159885D 01 RADIANS VEL 0 933232D 03 FEET SECOND CONTROL VARIABLES ELEVATOR 0 538044D 01 THROTTLE 0 214105D 00 SPEED BRAKE 0 000000D 00 DYNAMIC INTERACTION VARIABLES X BODY AXIS FORCE 0 134742D 06 POUNDS Y BODY AXIS FORCE 0 102657D 05 POUNDS Z BODY AXIS FORCE 0 938155D 03 POUNDS PITCHING MOMENT 0 680525D 02 FOOT POUNDS ROLLING MOMENT 0 000000D 00 FOOT POUNDS YAWING MOMENT 212007D 02 FOOT POUNDS OBSERVATION VARIABLES 0 300163D 01 GS 0 941435D 00 GS AN AY 90 A MATRIX FOR DX DT B U D y 0 1214360 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 0 141961D 00 0 220778D 02 0 000000D 00 0 105186D 02 D MATRIX FOR 0 343642D 07 0 113192 06 0 000000D 00 0 7
23. POINT DEFINITION SUBOPTION VARIABLE 1 VARIABLE 2 VARIABLE 3 VALUE 1 VALUE 2 VALUE 3 Sthmin 1 CONVR2 CONVR3 INPUT FORMAT FOR LINEAR Format 20A4 4F13 0 6F13 0 3F10 0 12A4 2F 13 0 110 4 11 4 5 4 10 0 5 4 10 0 5 4 10 0 110 5A4 110 A4 6X F10 0 5A4 110 A4 6X F10 0 5A4 110 A4 6X F10 0 110 4 5 4 3 10 0 5 4 3 10 0 5 4 3 10 0 8F10 0 1 12 5 4 110 4 5 4 110 4 5 4 110 4 2044 4 5A4 F15 5 5 4 15 5 5A4 F15 5 TABLE 1 Concluded a A A AA cOT r r r am aa raw awr cr rr E CFF FT EEE VEIA Input record Format NEXT A4 ANALYSIS POINT DEFINITION OPTION 20A4 ANALYSIS POINT DEFINITION SUBOPTION 4 VARIABLE 1 VALUE 1 5A4 F15 5 VARIABLE 2 VALUE 2 5 4 15 5 VARIABLE 3 VALUE 3 5 4 15 5 END A4 35 The first and second records describe the vehicle geometry mass and mass 415 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 r
24. THE SQUARES TRIM PARAMETERS LBS LBS FT FT SEC KTS FT SEC FT 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 TRIM PITCH AXIS PARAMETER TRIM ROLL AXIS PARAMETER TRIM YAW AXIS PARAMETER TRIM THRUST PARAMETER CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE OBSERVATION VARIABLES AN AY 96 DEG I li 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 0 66958 0 01526 0 02125 0 21410 0 05380 0 21410 0 00000 3 00163339 GS 94136286 GS 00 G00000 0 00 400000 0 00 400000 0 00 400000 0 00 400000 0 00 400000 0 LO GOf f OP L 6 00 d00000 0 00 400000 0 00 4 00000 0 00 400000 0 00 400000 0 00 400000 0 0 62 26 6 4015 cOo aocevL e 00 400000 0 10 4 0962 4 00 400000 0 LO GOCECL 00 400000 0 00 400000 0 00 0090L8 P cO GEOS80S 1 S0 GdtEPSU L 0 10440866 51 00 400000 0 10 4 096 6 1 LAJIT 2 0 00566 9 00 400000 0 20 001686 0 00 400000 0 00 d00000 0 00 400000 0 00 d00000 0 L0 dOZSscL 00 400000 0 00 400000 0 00 400000 0 00 d00000 0 00 d00000 0 c0 d09 80 owd ATANL
25. 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 Symbol Alphanumeric descriptors Derivatives of state variables Roll acceleration rad sec PDOT ROLL ACCELERATION Pitch acceleration rad sec q ODOT PITCH ACCELERATION Yaw acceleration rad sec2 r RDOT YAW ACCELERATION Velocity rate ft sec V VDOT VELOCITY RATE ALPDOT ALPHA DOT ALPHADOT Angle of attack rate rad sec We Angle of sideslip rate rad sec BTADOT BETA DOT BETADOT De THADOT THETA DOT Pitch attitude rate rad sec Heading rate rad sec Y PSIDOT PSI DOT PHIDOT PHI DOT Ce Roll attitude rate rad sec 70 Observation variable Units Symbol Derivatives of state variables Altitude rate Velocity north Velocity east x body axis acceleration y body axis acceleration z body axis acceleration x body axis accelerometer at vehicle center of gravity y body axis accelerometer at vehicle center of gravity body axis accelerometer at vehicle center of gravity ft sec h ft sec x ft sec y Accelerations 9 9 a g anx g Any anz Alphanumeric descriptors continued HDOT ALTITUDE RATE XDOT YDOT AX LONGITUDINAL ACCEL 5 ACCELERATION X AXIS ACCELERATION AXIS ACCEL X BODY
26. center of gravity along y body axis ft 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 1b directional trim parameter thrust trim parameter incremental x body axis force lb incremental y body axis force lb 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 aerodynamic reference point total drag AIXE 2 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 engine inertia inertia coupling between inertia coupling between inertia about the y body inertia coupling between inertia about the z body angle of att
27. center of gravity are not coincident the forces acting at the aerodynamic reference point 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 M Mar Ar x F where Mar Lay 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 X Y 71 is the total aerodynamic force acting at the aerodynamic center where x y and z are total forces along the x y and 2 body axes Thus Lar Ay Z Az Y M Mar Az X Z Nar Ax Y Ay X 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 qS Cp cos a Cy sin a Y qSCy Z qS Cp sin a Cy cos 60 Equations obtained by substituting these equations into the definition of the total aerodynamic moment equation and applying the definitions of the total aerodynamic moments result in qScC qSbC il Nar Nar Expressions for total aerodynamic moment coefficients correcte
28. 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 inter active 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 15 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 in
29. 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 instead of reading the data from a formatted file the program will prompt WHAT 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
30. dh SV 2 A n C Cp Cpo t 0 CpgB Sh Cp ov 1 1 L DEP CpyA Cpr Cp A Cp gb 31 n t CygB Cy h t cy oa em 1 1 A Cy Cy d Cy C B where the stability and control derivatives have the usual meaning x with Ce being an arbitrary force or moment coefficient and x an arbitrary nondimen Sional variable The rotational terms in the equations are nondimensional versions of the corresponding state variable with bp 5 A Sq y PE 27 2 y 2 _ bB V 6 in the summations the control variables defined by the user The effects of altitude and velocity are included in the derivatives with respect to those parameters and in the incremental multipliers V V Vo where the subscript zero represents the current analysis point Uo described in the Analysis Point Definition section All stability derivatives are computed as nondimensional terms except the alti tude and velocity parameters The units of the control derivatives correspond to the units used in the nonlinear aerodynamic model LINEAR is capable of handling both Mach and velocity derivatives Using the equation Cey where is the speed of sound at the reference altitude LINEAR will compute the M
31. force lb A AXIAL FORCE Body axis parameters x body axis velocity ft sec u UB X BODY AXIS VELOCITY X BODY AXIS VELOCITY AXIS VEL X BODY AXIS VEL U BODY U BODY y body axis velocity ft sec V VB Y BODY AXIS VELOCITY Y BODY AXIS VELOCITY Y BODY AXIS VEL Y BODY AXIS VEL V BODY V BODY Observation variable Units Symbol Alphanumeric descriptors Body axis parameters z body axis velocity ft sec W WB 2 AXIS VELOCITY Z BODY AXIS VELOCITY Z BODY AXIS VEL 2 BODY AXIS VEL W BODY W BODY Rate of change of t sec u UBDOT velocity in UB DOT x body axis Rate of change of ft sec2 VBDOT velocity in VB DOT y body axis Miscellaneous measurements not at vehicle center of gravity Rate of change of ft sec2 WBDOT velocity in WB DOT z body axis Angle of attack not rad a i ALPHA I at vehicle center ALPHA INSTRUMENT of gravity AOA INSTRUMENT Angle of sideslip not rad Bi BETA I at vehicle center BETA INSTRUMENT of gravity SIDESLIP INSTRUMENT Altitude instrument tt hi H I not at vehicle center ALTITUDE INSTRUMENT of gravity Altitude rate ft sec hi HDOT I instrument not at vehicle center of gravity Other miscellaneous parameters Vehicle total angular slug ft2 T ANGULAR MOMENTUM momentum sec ANG MOMENTUM Stability axis roll rad sec Ps STAB AXIS ROLL RATE rate 75 76 Observation variable Units Symbol Alphanumeric descriptors Other miscellaneous parameters conti
32. name lists and values and the state space system matrices The second file contains all of the trim information the control state 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 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 Outputs to ICONTRL DC 1 DC 2 UCNTRL DRS Gearing THRSTX DC 30 Pilot stick pedal Surface deflections and throttle and power level setting Figure 2 In
33. parameters associated with the observations back to LINEAR common block OBSERV contains two single precision vectors OBVEC 120 and PARAM 120 6 A list of the available observations and parameters is given in appendix C Access to the observation variables allows the users to imple ment trim strategies that are functions of observations such as gain schedules and surface management schemes parameters associated with the observations are used primarily to define the moment arm from the center of gravity to the point at which the observation is to be made the user subroutine MASGEO is used to change the center of gravity location and observations are desired at fixed locations on the vehicle then the moment arm from the new center of gravity location to the fixed sensor location x y Z in feet must be computed in one of the user subroutines and passed back in the first three elements of the PARAM vector associated with the desired observation PARAM n 1 to 3 where n is the number of the desired observa tion Additional information on observations and parameters can be found in the State Control and Observation Variable Definitions section CONCLUDING REMARKS The FORTRAN program LINEAR was developed to provide a flexible powerful and documented tool to derive linear models for aircraft stability analysis and control law design This report discusses the interactive LINEAR program describing the interfaces to
34. reference point along the z body axis from the vehicle center of gravity longitudinal trim parameter DRS DXTHRS GMA H HDOT NUMSAT NUMSRF P PDOT PHI PHIDOT PSI PSIDOT Q QBAR QDOT RDOT 5 T TDOT THA 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 number of number of roll rate time rate of change roll angle time rate of change heading angle time rate of change pitch rate dynamic pressure time rate of change yaw rate time rate of change wing area time time rate of change pitch angle time rate of change of of of of of of of thrust trim parameter of change of altitude states in the state vector recognizable control names roll rate roll angle heading angle pitch rate yaw rate time pitch angle thrust generated by each engine TLOCAT location of the engine in the x y and z axes from the center of gravity TVANXY orientation of the thrust vector in the x y engine axis plane TVANXZ orientation of the thrust vector in the x z engine axis plane UB velocity along the x body axis velocity VB velocity along the y body axis VCAS calibrated airspeed VDOT time rate of change of total vehic
35. that the generalized formulation is desired the observation equation y H x Gx F u is used Otherwise the standard formulation is assumed and the form of the obser vation equation used is y Hx Fu The records defining the observation variables to be used in the output formu lation 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 com pared with the list of observation variables given in appendix D If the parameter name is recognized as a valid observation variable name that observation variable is 38 included 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 vari able In that case PARAM 1 is used to specify the characteristic length When no value is i
36. the state equation will be in the standard form x A X Bu and that the nondimensional stability and control derivatives with respect to angle of attack and angle of sideslip should be scaled in 4 571 The next four records define the output formulation of the state vector to be lt os e 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 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 and THRSTX used to trim the longitudinal lateral and directional axes and thrust respec tively The ranges for these parameters are defined by record 20 to be 2 9 DES 5 43 4 0 lt DAS lt 4 0 3 25 lt DRS lt 3 25 1 0 5 lt 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 app J The thrust trim parameter is specified in this manner because of the imple mentation of UCNTR
37. user for the specific data it needs for this section If the user wishes to input all the data from the terminal selection 5 should be chosen and the program will prompt for a data file name on which to store the input data The program will prompt the user for all the input data required 41 When inputting the title data the program reads character strings of 80 char acters When prompting for vehicle mass and geometry data the program reads a floating 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 user must define the size and formulation of the state equation as well as the units in which the angle of attack and sideslip stability derivatives will be output 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 A X B U D V 1 DX DT 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 ou
38. 14203D 03 DX DT A X B U D V 0 164948D 02 0 928933D 02 0 543324D 02 0 135074D 02 0 000000D 00 0 000000D 00 0 342817D 02 0 155832D 02 DX DT A xX B Uw D V 0 000000D 00 0 737378D 06 0 000000D 00 0 000000D 00 0 000000D 00 0 242885D 05 0 605694 05 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 0 398492D 06 0 332842D 04 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 E V 0 412845 0 1 0 180978 02 0 291699D 00 0 000000 00 0 000000D 00 0 000000D 00 E MATRIX FOR Y H X F U E V 0 377037D 07 0 000000D 00 0 214132D 04 0 000000D 00 0 000000D 00 0 000000D 00 0 222222D 04 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 TEST CASE X DIMENSION 4 STATE EQUATION FORMULATION OBSERVATION EQUATION FORMULATION STATE VARIABLES ALPHA Q THETA VEL CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE DYNAMIC INTERACTION AXIS FORCE Y BODY AXIS FORCE Z BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT YAWING MOMENT 126650D 01 0 000000D 00 0 161868D 00 0 933232D 03 0 637734D 01 0 225092D 00 0 000000D 00 VARIABLES 0 443769D 05 0 300494D 04 0 445795D 09 528481D 11 0 000000D 00 0 114239D 10 OBSERVATION VARIABLES AN AY A M
39. 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT F 15 SIMULATION 2 0000D 04 9 0000D 01 REFERENCE ALTITUDE REFERENCE MACH ANGLES ARE IN RADIANS FOR OUTPUT WARNING DERIVATIVES WITH RESPECT TO MACH WILL BE USED VELOCITY DERIVATIVES WILL BE COMPUTED FROM MACH DERIVATIVES THERE ARE 3 CONTROL SURFACES ROLL MOMENT DERIVATIVES CLO 1 25377 16 ROLL RATE RAD SEC 2 00000D 01 PITCH RATE RAD SEC 00000D 00 YAW RATE RAD SEC 1 50990D 0 1 VELOCITY FT SEC 00000D 00 MACH NUMBER 00000D 00 ALPHA RAD 00000D 00 BETA RAD 1 33450D 01 ALTITUDE FT 00000D 00 ALPHA DOT RAD SEC 00000D 00 BETA DOT RAD SEC 00000D 00 ELEVATOR 5 00000 THROTTLE 12 00000D 00 SPEED BRAKE 10 00000D 00 PITCH MOMENT DERIVATIVES CMO 4 22040D 02 ROLL RATE RAD SEC 00000D 00 PITCH RATE RAD SEC 3 89530D 00 YAW RATE RAD SEC 00000D 00 VELOCITY FT SEC 7 79749D 10 MACH NUMBER 8 08541D 07 ALPHA RAD 1 68819D 01 BETA RAD 00000D 00 ALTITUDE FT 00000 00 ALPHA DOT RAD SEC 1 18870D 01 BETA DOT RAD SEC 00000D 00 ELEVATOR 5 6 95279 01 THROTTLE 12 00000D 00 SPEED BRAKE 10 4 17500 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 CDO ROLL RATE RAD SEC PITCH RA
40. 2 body axis accelerometer not at center of gravity 4 acceleration along the x body axis g acceleration along the y body axis g acceleration along the z body axis g control matrix of the state equation x Ax Bu control matrix of the state equation Cx A x Blu wingspan ft 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 Btu 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 F u E v specific energy ft feedforward matrix of the observation equation y Hx Fu feedforward matrix of the observation equation y H x Gx F u flight path acceleration 4 G matrix of the observation equation y H x Gx F u acceleration due to gravity ft sec altitude ft observation matrix of the observation equation y Hx Fu observation matrix of angular momentum of engine rotor slug t sec the observation equation y aircraft inertia tensor 6104 8692 rotational inertia of x body axis moment of body axis product 2 body axis product y body axis moment of 2 body axis product body axis moment of the engine slug ft2 inertia slu
41. 4 119 EQUIVALENCE VARIABLE NAMES C EQUIVALENCE THR DC 12 C C ASSUME THRUST PER ENGINE IS HALF VEHICLE WEIGHT C THRUST 1 24000 0 THR THRUST 2 24000 0 THR C C LET ALL OTHER PARAMETERS DEFAULT TO ZERO C RETURN END Mass and Geometry Model The subroutine MASGEO enables the user to model mass inertia and center of gravity changes due to vehicle configuration or mass changes This particular ver Sion is simply a RETURN and END statement because the aircraft that it models is a fixed wing vehicle and any mass changes can be done through the input files or even interactively However this subroutine does provide an elegant way of including a model of those changes for other aircraft SUBROUTINE MASGEO SUBROUTINE TO COMPUTE THE MASS AND GEOMETRY PROPERTIES THE AIRCRAFT COMMON CONPOS DAP FLAT DATRIM DEP FLON DETRIM DRP FPRF DRTRIM DSBP DFP PLAPL PLAPR THSL THSR COMMON CONTRL DC 30 COMMON DATAIN S B CBAR AMSS AIX AIY AIZ AIXZ AIXY AIYZ AIXE COMMON SIMOUT AMCH QBAR GMA DEL UB VB WB VEAS VCAS COMMON DRVOUT F 13 DF 13 EQUIVALENCE T F 1 2 0 3 R 4 V F 5 ALP 6 BTA e 7 8 PSI 9 PHI F 10 F 11 X F 12 Y 13 A TDOT DF 1 PDOT DF 2 QDOT DF 3 RDOT DF 4 VDOT DF 5 ALPDOT DF 6 BTADOT DF 7 THADOT DF 8 PS
42. ACH 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 RAD ALTITUDE FT ALPHA DOT RAD SEC BETA DOT RAD SEC ELEVATOR THROTTLE SPEED BRAKE 2 25747 04 3 37217D 02 00000D 00 4 04710D 01 3 21400D 07 3 33268D 04 00000D 00 1 29960D 01 00000D 00 00000D 00 00000D 00 5 00000D 00 12 00000D 00 10 00000D 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 38318D 02 12 00000D 00 10 6 49355D 02 1 57360D 01 00000D 00 1 72315D 01 000000 00 1 452860 05 1 50651D 02 4 87061D 00 00000D 00 00000D 00 1 72315D 01 00000D 00 5 5 72950D 01 12 00000D 00 10 3 74913D 02 85 SIDE FORCE DERIVATIVES 5 32725D 04 ROLL RATE RAD SEC 00000D 00 PITCH RATE RAD SEC 00000D 00 YAW RATE RAD SEC 00000D 00 VELOCITY FT SEC 00000D 00 MACH NUMBER 00000D 00 ALPHA RAD 00000D 00 BETA RAD 9 74030D 01 ALTITUDE FT 00000D 00 ALPHA DOT RAD SEC 00000D 00 BETA DOT RAD SEC 00000D 00 ELEVATOR 5 00000D 00 THROTTLE 12 00000D 00 SPEED BRAKE 10 00000D 00 hitt a s T a e A RY ar i E t Ta laat a TA t 86 STABILITY AND CONTROL DERIVATIVES FOR CASE
43. ATRIX FOR DX 0 985228D 00 0 990655D 14 DT A X k e k k k k k k k k k k k kk k k k k k kk kk k k k k k k k k k k k k kk kk k k kk k k U DIMENSION 3 STANDARD STANDARD RADIANS RADIANS SECOND RADIANS FEET SECOND POUNDS POUNDS POUNDS FOOT POUNDS FOOT POUNDS FOOT POUNDS GS GS HT Ur PV 0 120900D 01 0 100000D 01 0 575730D 02 0 701975D 04 0 149189D 01 0 221451D 01 0 189640D 01 0 2313680 03 0 000000D 00 0 100000D 01 0 000000D 00 0 000000D 00 0 576868D 02 0 000000D 00 0 316251D 02 0 460435D 02 FOR DX DT A X B U D 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 Y DIMENSION D MATRIX FOR DX DP A X B U D V 0 934880 08 0 000000D 00 0 738119D 06 0 000000D 00 0 000000D 00 0 307941D 07 0 000000D 00 0 243129D 05 0 605694D 05 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 0 714920D 03 0 000000D 00 0 905497D 05 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 JU 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 50 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 1026760 07 0 000000D 00 0 214116
44. D 04 0 000000D 00 0 000000D 00 0 000000D 00 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 Q THETA VEL THE STANDARD FORMULATION OF THE STATE EQUATION HAS BEEN SELECTED THE FORM OF THE EQUATION IS DX DT A X B U SURFACES TO USED FOR THE CONTROL VECTOR FOR LINEARIZER 5 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 t F U LIMITS FOR TRIM OUTPUT PARAMETERS MINIMUM MAXIMUM PITCH AXIS PARAMETER 2 900 5 430 ROLL AXIS PARAMETER 4 000 4 000 YAW AXIS PARAMETER 3 250 3 250 THRUST PARAMETER 1 000 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
45. D 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 ALTITUDE 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 FT 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 I 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 109 5 5 0 66958 TRIM ROLL AXIS PARAMETER 0 01526 TRIM YAW AXIS PARAMETER 0 02125 5 0 21410 CONTROL VARIABLES ELEVATOR 0 05380 THROTTLE 0 21410 SPEED BRAKE DEG 0 00000 OBSERVATION VARIABLES AN 3 00163339 GS AY 94136286 GS VEHICLE STATIC MARGIN IS 3 5 MEAN AERODYNAMIC CHORD STABLE AT THIS FLIGHT CONDITION 110 TRIM CONDITIONS FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE
46. ED SPEED OF SOUND GRAVITATIONAL ACCEL NORMAL ACCELERATION LOAD FACTOR DYNAMIC PRESSURE DENSITY WEIGHT ALTITUDE BETA ALPHA PHI THETA ALTITUDE RATE GAMMA ROLL RATE PITCH RATE YAW RATE THRUST SUM OF THE SQUARES 102 LBS LBS FT FT SEC KTS FT SEC FT SEC 2 G S LB FT 2 SLUG FT 3 LBS DEG DEG DEG DEG FT SEC DEG DEG SEC DEG SEC DEG SEC LBS 0 13221 0 00895 44376 86258 3004 93778 20000 00000 0 90000 933 23196 403 42303 1036 92440 32 11294 0 98523 0 98803 552 05302 0 00126774 44914 60434 0 00000 0 72565 0 00000 9 27435 162 05403 10 00000 0 00000 0 00000 0 00000 10804 39673 0 00000 TRIM PARAMETERS TRIM PITCH AXIS PARAMETER TRIM ROLL AXIS PARAMETER TRIM YAW AXIS PARAMETER TRIM THRUST PARAMETER CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE OBSERVATION VARIABLES AN AY 0 79364 0 00000 0 00000 0 22509 0 06377 0 22509 0 00000 0 98522771 GS 0 00000000 GS 103 00 400000 0 00 400000 0 00 400000 0 00 d00000 0 00440000020 00 400000 0 10 060 7 6 00400000 0 00 400000 0 004400000 0 00 400000 0 00 400000 0 00 400000 0 91 Gazte 4Ssi v 415 cOo aocevL e 00 400000 0 10 0096202 00 400000 0 LO GOCTETL i 00 400000 0 00 400000 0 0040090 8 7 LO C6S60L 6 1 01920 6 00 400000 0 10 0d0Z ZL 1 00 d00000 0 10 009624 1 LAIT co ao
47. ERENCE ALTITUDE 2 0000D 04 REFERENCE MACH 9 0000D 01 ANGLES ARE IN RADIANS FOR OUTPUT WARNING DERIVATIVES WITH RESPECT MACH WILL BE USED VELOCITY DERIVATIVES WILL BE COMPUTED FROM MACH DERIVATIVES THERE ARE 3 CONTROL SURFACES ROLL MOMENT DERIVATIVES CLO ROLL RATE RAD SEC PITCH RATE RAD SEC YAW RATE RAD SEC 4 02966D 05 2 00000D 01 00000D 00 1 50990D 01 VELOCITY FT SEC 1 266960 07 1 31375 04 ALPHA RAD 00000D 00 BETA RAD 1 33450D 01 ALTITUDE FT 00000D 00 ALPHA DOT RAD SEC 00000D 00 BETA DOT RAD SEC 00000 00 ELEVATOR 5 00000D 00 THROTTLE 12 00000D 00 SPEED BRAKE 10 00000 PITCH MOMENT DERIVATIVES CMO 4 22040D 02 ROLL RATE RAD SEC 00000D 00 PITCH RATE RAD SEC 3 89527D 00 YAW RATE RAD SEC 00000D 00 VELOCITY FT SEC 3 28490D 06 MACH NUMBER 3 40620D 03 ALPHA RAD 1 688190 01 RAD 00000D 00 ALTITUDE FT 00000D 00 ALPHA DOT RAD SEC 1 18870D 01 BETA DOT RAD SEC 00000D 00 ELEVATOR 5 6 952810 01 THROTTLE 12 00000D 00 SPEED BRAKE 10 4 17500 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 RAD ALTITUDE FT ALPHA DOT RAD SEC BETA DOT RAD SEC ELEVATOR THROTTLE SPEED BRAKE DRAG FORCE DERIVATIVES CDO ROLL RATE RAD SEC PITCH RATE RAD SEC YAW RATE RAD SEC VELOCITY FT SEC M
48. FFICIENT DERIVATIVES C C STABILITY DERIVATIVES WITH RESPECT TO C SIDESLIP CYB 9 7403 D 01 C C CONTROL DERIVATIVES WITH RESPECT TO C AILERON RUDDER DIFFENTIAL TAIL C CYDA 1 1516 D 03 CYDR 1 5041 D 01 CYDT 7 9315 D 02 C RETURN END SUBROUTINE CCALC C C EXAMPLE AERODYNAMIC MODEL C C ROUTINE TO CALCULATE THE AERODYNAMIC FORCE AND MOMENT COEFFICIENTS C C COMMON BLOCKS CONTAINING STATE CONTROL AND AIR C DATA PARAMETERS C C COMMON DRVOUT F 13 DF 13 COMMON CONTRL DC 30 C COMMON DATAIN S B CBAR AMSS AIX AIY AIZ AIXZ AIXY AIYZ AIXE COMMON TRIGFN SINALP COSALP SINBTA COSBTA SINPHI COSPHI SINPSI COSPSI SINTHA COSTHA COMMON SIMOUT AMCH QBAR GMA DEL UB VB WB VEAS VCAS COMMON CGSHFT DELX DELY DELZ 115 COMMON BLOCK OUTPUT AERODYNAMIC FORCE AND MOMENT C COEFFICIENTS COMMON CLCOUT CL CM CN CD C C COMMON BLOCK TO COMMUNICATE AERODYNAMIC DATA BETWEEN THE SUBROUTINES ADATIN AND CCALC COMMON ARODAT CLR 2 CLDA CLDR CLDT A CMO CMQ CMAD 5 CNB CNP CNR CNDA CNDR CNDT CDO CDA CDDE CDSB CLFTO CLFTQ CLFTAD 2 CLFTDE CLFTSB CYDA CYDR CYDT 2 EQUIVALENCE VARIABLE NAMES EQUIVALENCE F 1 FC 2 Q 3 R 4 V F 5 ALP F 6 BTA 7 F 8 PSI F 9 PHI F 10
49. IDOT DF 9 PHIDOT DF 10 DF 11 XDOT DF 12 YDOT DF 13 RETURN END 120 REFERENCES Chen R T N Kinematic Properties of Rotary Wing and Fixed Wing Aircraft in Steady Coordinated High g Turns AIAA 81 1855 Aug 1981 Chen R T N and Jeske J A Kinematic Properties of the Helicopter in Coordinated Turns NASA TP 1773 1981 Clancy L J Aerodynamics John Wiley amp Sons New York 1975 Dieudonne J E Description of a Computer Program and Numerical Techniques for Developing Linear Perturbation Models From Nonlinear Systems Simulations NASA TM 78710 1978 Dommasch D O Sherby S S and Connolly T F Airplane Aerodynamics Fourth edi tion Pitman Publishing Company New York 1967 Duke E L Antoniewicz R F and Krambeer K D Derivation and Definition of a Linear Aircraft Model NASA RP 1207 1988 Duke E L Patterson B P and Antoniewicz R F User s Manual for LINEAR a FORTRAN Program to Derive Linear Aircraft Models NASA TP 2768 1987 Etkin Bernard Dynamics of Atmospheric Flight John Wiley amp Sons New York 1972 Gainer T G and Hoffman Sherwood summary of Transformation Equations and Equa tions of Motion Used in Free Flight and Wind Tunnel Data Reduction and Analysis NASA SP 3070 1972 Gracey William Measurement of Aircraft Speed and Altitude NASA RP 1046 1980 Kalviste Juri Fixed Point Analysis Program NOR 80 165 Northrop Corporation Aircraft
50. JO NOILYINNAOA AHL ONISN XIYLYW 4 HSVO NOA 99 100 MATRIX H USING THE FORMULATION OF THE OBSERVATION EQUATION Y F Ut tE V FOR CASE 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ALPHA Q THETA VEL 0 351752D 02 0 2775560 16 0 150046D 02 0 640771D 02 0 000000D 00 0 000000D 00 0 150534D 01 0 446789D 10 MATRIX F USING THE FORMULATION OF THE OBSERVATION EQUATION Y H X FY UTE FV FOR CASE 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ELEVATOR THROTTLE SPEED BRAKE 0 412845D 01 0 180978D 02 0 291699 400 0 000000D 00 0 000000D 00 0 000000D 00 00 4000000 0 00 4000000 0 LNSNON 00 0000000 0 00 4000000 0 00 4000000 0 vo deeecec 0 00 4000000 0 00 1000000 0 00 0000000 0 VO ACELVLC O 00 4000000 0 LO GLEOLLE 0 LNIWOW LBNAWOW SNIHOLId SIXV AGOH Z SIXV AGOG A 4109404 SIXV AGOG X S NHMS LOXCOSNd HHL NOA SASVO 41544 NOILV LSNONAG JAZTAVANTI ASVO NOA 166440 y d X x H gt NOILWNOA NOILVANHSHO AHL JO NOILVIQWHO4 HHL SNISn H XINLVW KV NV AV NV 101 TRIM CONDITIONS FOR CASE 4 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE STRAIGHT AND LEVEL TRIM WHILE VARYING ALPHA TRIM ACHIEVED COEFFICIENT OF LIFT COEFFICIENT OF DRAG LIFT DRAG ALTITUDE MACH VELOCITY EQUIVALENT AIRSPE
51. L Speed brake is scheduled when THRSTX lt 0 and thrust is com manded when THRSTX gt 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 These records request a level turn trim option at h 20 000 ft M 0 9 an 3 0 g and 0 02 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 final six records records 27 point option at and 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 to 32 define a straight and level analysis 20 000 0 9 10 0 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 LINEARIZED STABILITY AND CONTROL DERIVATIVE FILE STABILITY AND CONTROL DERIVATIVES FOR CASE 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT F 15 SIMULATION REF
52. NASA Technical Paper 2835 September 1988 User s Manual for Interactive LINEAR a FORTRAN Program To Derive Linear Aircraft Models Robert Antoniewicz Eugene L Duke and Brian P Patterson 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 5 SUMMARY gt e e 1 INTRODUCTION e 2 4 1 NOMENCLATURE gt e e s 9 o Variables s e e e 8 8 0 6 oo SuperscriptS 4 4 4 4 2 SubscriptS e lt s rr 9 n9 FORTRAN Variables s 6 9 6 9 6 J OY O to N PROGRAM OVERVIEW s e s e e e e e 0 6 s o c 0 c s c e e e e s 11 EQUATIONS OF MOTION e s e 6 e e e e e e e e e e 6 e 6 6 14 OBSERVATION EQUATIONS lt s
53. NVUINI I c HSVO YO Aa G n xy X x Y NOILYN A ALVIS AHL JO AHL ONISN G XINLVW 106 MATRIX H USING THE FORMULATION THE OBSERVATION EQUATION Y H X E V FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ALPHA Q THETA VEL 0 350424D 02 0 277556 16 0 632324D 02 0 203434D 02 0 000000D 00 0 000000D 00 0 000000D 00 0 212306D 16 MATRIX F USING THE FORMULATION OF THE OBSERVATION EQUATION Y F U E v FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ELEVATOR THROTTLE SPEED BRAKE 0 411323D 01 0 492845D 03 0 263288D 00 0 000000D 00 0 000000D 00 0 000000D 00 107 a a TTT EEE E t E TT EET pr py vine 00 4000000 0 00 4000000 0 LNINON ONIMVA 00 0000000 0 00 4000000 0 00 4000000 0 0 0 00 4000000 0 00 4000000 0 00 4000000 0 0 0911 1270 00 4000000 0 0 49 9201 0 LNYWOW DNITION LNAWOW ONTHOLId SIXV AGOS Z HOUOA SIXV AGOH A HINOA SIXV AGOG xX S AASA LOY OYd FHL SHSVO Sal ANY NOILVALSNOWHG HAZINVANITT C HSVO A x A l x A X xy HH A 2 0 NOILVAHSSHO HHL JO NOILVINNJOA HHL ONISN H XINLVW NW AW NW 108 TRIM CONDITIONS FOR CASE 1 LINEARIZER TEST AN
54. OF THE STATE EQUATION DX DT B 0 FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ALPHA Q THETA VEL TIME DERIVATIVES WRT ALPHA 0 120900D 01 0 100000D 01 0 575730D 02 0 701975 04 Q 0 1491890 01 0 221451 401 0 189640D 01 0 231368D 03 THETA 0 000000D 00 0 100000D 01 0 000000D 00 0 000000D 00 VEL 0 576868D 02 0 000000D 00 0 316251D 02 0 4604350 02 MATRIX USING THE FORMULATION OF THE STATE EQUATION DX DT B U D y FOR CASE 4 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ELEVATOR THROTTLE SPEED BRAKE TIME DERIVATIVES WRT ALPHA 0 141961D 00 0 448742D 03 0 928932D 02 0 220778D 02 0 147812 02 0 135074D 02 THETA 0 000000D 00 0 000000D 00 0 000000D 00 VEL 0 105186D 02 0 343162D 02 0 155832D 02 00 4000000 0 THA 00 4000000 0 VLAHI 00 4000000 0 00 4000000 0 VHd IV JO LIM LNAWOW ONIMYA SHAILVATAJA 00 4000000 0 00 4000000 0 50 4 67506 0 00 4000000 0 0 Gd0cevLZL O THA 00 4000000 0 00 4000000 0 004 d000000 0 00 4000000 0 00 4000000 0 VLAHJL 00 000000 0 0 09769509 0 0 6 1 0 00 4000000 0 LO GLT6LOE O 00 4000000 0 00 4000000 0 90 d6LLBEL O 00 4000000 0 80 d088Tt6 0 4O INTL LAM SYAILYVAIN4A LNINON LNAWOW SNIHOLIA 20404 SIXV AGOH Z SIXV AGOH A SIXV AGOH X SASN LOHCOdd HHL NOA SHSVO 415454 NY NOTLVALSNONO HdZI
55. ONSTRATION CASES FOR THE PROJECT USER S GUIDE ALPHA 0 121436D 01 0 147423D 01 0 000000D 00 0 790853D 02 MATRIX B USING THE DX DT FOR CASE 8 1 0 100000D 01 0 221451D 01 0 331812D 00 0 000000D 00 FORMULATION THETA 0 136756D 02 0 450462D 02 0 000000D 00 0 320822D 02 A X B U D V LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ELEVATOR 0 141961D 00 0 220778D 02 0 000000D 00 0 105186D 02 THROTTLE 0 164948D 02 0 543324D 02 0 000000D 00 0 342817D 02 SPEED BRAKE 0 928933D 02 0 135074D 02 0 000000D 00 0 155832D 02 EOUATION VEL 0 121605D 03 0 294019D 03 0 000000D 00 0 157297D 01 OF THE STATE EQUATION 00 4000000 0 00 4000000 0 00 0000000 0 00 4000000 0 LNINON 00 4000000 0 00 0000000 0 0 129826670 90 426 86 70 0 4602 1270 00 4000000 0 00 4000000 0 00 4000000 0 00 0000000 0 00 4000000 0 00 4000000 0 lt 0 0 69509 0 50 46588272 0 00 4000000 0 90 426161170 00 4000000 0 00 4000000 0 90 48 2626270 00 4000000 0 0 0 LNAWOW ONTTIOY LNAWOW ONTHOLId SIXV AGOH Z HOYOA SIXV AGOH A SIXV THA AWIL DUM SHATLVAIAHA TIA VLHHL VHd IV 10 LIM SHAILVATAAA LAIAS S AASA LOHCOdd HHL NOA SHSWO 41535404 ANW NOILVHLSNOWSHG YAZIAVINI I Ag A A x A X x V NOILVN H 15 AHL
56. PROJECT USER S GUIDE STRAIGHT AND LEVEL TRIM WHILE VARYING ALPHA TRIM ACHIEVED COEFFICIENT OF LIFT COEFFICIENT OF DRAG LIFT DRAG ALTITUDE MACH VELOCITY EQUIVALENT AIRSPEED SPEED OF SOUND GRAVITATIONAL ACCEL NORMAL ACCELERATION LOAD FACTOR DYNAMIC PRESSURE DENSITY WEIGHT CALTITUDE 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 FT SEC 2 G S LB FT 2 SLUG FT 3 LBS DEG DEG DEG DEG FT SEC DEG DEG SEC DEG SEC DEG SEC LBS H 0 13221 0 00895 44376 86258 3004 93778 20000 00000 0 90000 933 23196 403 42303 1036 92440 32 11294 0 98523 0 98803 552 05302 0 00126774 44914 60434 0 00000 0 72565 0 00000 9 27435 162 05403 10 00000 0 00000 0 00000 0 00000 10804 39673 0 00000 TRIM PARAMETERS TRIM PITCH AXIS PARAMETER 0 79364 TRIM ROLL AXIS PARAMETER 0 00000 TRIM YAW AXIS PARAMETER 0 00000 TRIM THRUST PARAMETER 0 22509 CONTROL VARIABLES ELEVATOR 0 06377 THROTTLE 0 22509 SPEED BRAKE 0 00000 OBSERVATION VARIABLES AN 0 98522771 GS AY 0 00000000 GS VEHICLE STATIC 15 3 5 MEAN AERODYNAMIC CHORD STABLE AT THIS FLIGHT CONDITION 112 APPENDIX J EXAMPLE USER SUPPLIED SUBROUTINES The following subroutines are examples of the user supplied routines that provide the aerodynamic cont
57. RADI ALPHA Q THETA VEL 3 ELEVATOR 5 THROTTLE 12 10 25 2 900E 00 5 430E 00 4 000E 00 4 000E 00 3 250E 00 3 250E 00 1 000E 00 1 000E 00 _ 0 ADDITIONAL SURFACES WINDUP TURN ALPHA 20000 0 0 90 3 00 0 0 LEVEL FLIGHT ALPHA 20000 0 0 9 10 0 END This input file is for a case called USER S GUIDE record 1 The project title is input from the terminal as described in the Data Input section Record 2 speci fies the mass and geometric properties of the vehicle as S 608 ft b 42 8 ft 80 15 95 ft il 45 000 lbs Record 3 defines the moments and products of intertia of the vehicle as Ix 28 700 slug ft Iy 165 100 slug t2 187 900 slug ft Ixz 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 Ay 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
58. RIVATIVES WITH RESPECT TO AILERON RUDDER DIFFERENTIAL TAIL 114 0000020 O Q O oO 0 O 000000 A CLDA 2 6356 D 02 CLDR 2 3859 03 CLDT 4 0107 D 02 PITCHING MOMENT DERIVATIVES STABILITY DERIVATIVES WITH RESPECT TO ANGLE OF ATTACK PITCH RATE ANGLE OF ATTACK CMO 4 2204 02 CMA 1 6882 D 01 CMQ 3 8953 CMAD 1 1887 D 01 CONTROL DERIVATIVES WITH RESPECT TO ELEVATOR SPEED BRAKE CMDE 6 9528 D 01 CMSB 4 1750 D 01 YAWING MOMENT DERIVATIVES STABILITY DERIVATIVES WITH RESPECT TO SIDESLIP ROLL RATE YAW CNB 1 2996 D 01 CNP 3 3721 02 CNR 4 0471 D 01 CONTROL DERIVATIVES WITH RESPECT TO AILERON RUDDER DIFFERENTIAL TAIL CNDA 2 1917 D 03 CNDR 6 9763 D 02 CNDT 3 0531 D 02 COEFFICIENT OF DRAG DERIVATIVES STABILITY DERIVATIVES WITH RESPECT ANGLE OF ATTACK CDO 1 0876 D 02 CDA 3 7257 D 01 CONTROL DERIVATIVES WITH RESPECT TO ELEVATOR SPEED BRAKE CDDE 4 3831 D 02 CDSB 6 4935 D 02 COEFFICIENT LIFT DERIVATIVES C C STABILITY DERIVATIVES WITH RESPECT TO ANGLE OF C ATTACK PITCH RATE ANGLE OF ATTACK RATE C CLFTO 1 5736 D 01 C CLFTA 4 8706 CLFTQ 1 7232 0401 CLFTAD 1 7232 D 01 C C CONTROL DERIVATIVES WITH RESPECT TO C ELEVATOR SPEED BRAKE C CLFTDE 5 7296 D 01 CLFTSB 3 7492 02 C C SIDEFORCE COE
59. S QHOHO 6 6 00 400000 0 00 d00000 0 00 d00000 0 00 400000 0 00 400000 0 004400000 0 10 4109662 1 00 000000 0 VO ACSE6CEPE LO d9601c 10 4 01 0 00 G00000 0 CO JOlcLtt 0 42 0 LNINON ONIMVA 10 40054 p 00 400000 0 10 1082 lt 6 9 00 400000 0 10440 8817 00 400000 0 00 400000 0 10 002889 1 0 08 80976 90 Gd6 L8C E 004400000 0 00 40 568 00 400000 0 20 00 022 8 LNIWOW SNIHOLId NOLLIGNOO 4 SIHL LV SI DILVIS 00 00000 0 ayva 5 00 400000 0 ATLLOJHL 00 100000 0 YJOLVAI TA 00 Gd00000 0 DIS AVA LOG WLad 00 d00000 0 045 AVA LOG 00 d00000 0 Ld AANLILTY 10 40 lt 671 AVA viag 00 d00000 0 AVA VHd IV 0 408926 1 M ANAN 0 48466 2 1 5 44 1 L0 d0660S L DAS AVA ALVY MWA 00 400000 0 O4S awu HLVAI HOLId L0 4G00000 Z 245 AVA ALVA TION S0 q996eco vr LNINON ONT TIO SLN11901414400 2 5 845 1 4 AHL SdSWO NOILVHILSNONHG ANY 15844 HAZIAVANIT 1 SVO HOJ SAATLVAIYIG IONLNOO ANY ALITIGVIS TWNOISNAWIG NON 97 DERIVATIVES WRT ALPHA Q THETA VEL TIME DERIVATIVES WRT ALPHA Q THETA VEL 98 MATRIX A USING THE FORMULATION OF THE STATE DX DT FOR CASE 1 A X B U D V LINEARIZER TEST AND DEM
60. TE 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 RAD ALTITUDE FT ALPHA DOT RAD SEC BETA DOT RAD SEC ELEVATOR THROTTLE SPEED BRAKE 88 1 22535D 16 3 37217D 02 00000 00 4 04710 01 00000D 00 00000D 00 00000D 00 1 29960D 01 00000D 00 00000D 00 00000D 00 5 00000D 00 12 00000D 00 10 00000D 00 1 08760D 02 00000D 00 00000D 00 00000D 00 00000 00 00000D 00 3 725700 01 00000D 00 00000D 00 00000D 00 00000D 00 5 4 38313D 02 12 00000D 00 10 6 49346D 02 1 57360D 01 00000D 00 1 72320D 01 00000D 00 3 592630 09 3 72529D 06 4 87061D 00 00000D 00 00000D 00 1 72320D 01 00000D 00 5 5 72961D 01 12 00000D 00 10 3 74913 02 SIDE FORCE DERIVATIVES 4 15737 16 ROLL RATE RAD SEC 00000D 00 PITCH RATE RAD SEC 00000D 00 YAW RATE RAD SEC 00000D 00 VELOCITY FT SEC 00000D 00 MACH NUMBER 00000D 00 ALPHA RAD 00000D 00 BETA RAD 9 74030D 01 ALTITUDE FT 00000D 00 ALPHA DOT RAD SEC 00000D 00 BETA DOT RAD SEC 00000D 00 ELEVATOR 5 00000D 00 THROTTLE 12 00000D 00 SPEED BRAKE 10 00000 00 m IM r Jkh T CGU 89 APPENDIX H EXAMPLE
61. ach derivatives from the velocity derivatives or vice versa Internally however 32 LINEAR uses velocity derivatives 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 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 be typed in from the terminal or 4 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
62. ack axis the x and y body axes the x and z body axes axis the y and z body axes axis time rate of change of angle of attack AMSENG AMSS B BTA BTADOT CBAR CD CDA CDDE CDO CDSB CL CLB CLDA CLDR CLDT CLFT CLFTA CLFTAD CLFTDE CLFTO CLFTO 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
63. am Interactive LINEAR is provided in the microfiche Supplement included with this report 4 sheets total 16 Abstract An interactive FORTRAN program that provides the user with a powerful and flex ible tool for the linearization of aircraft aerodynamic models is documented in this report The program LINEAR numerically determines a linear System model using nonlinear 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 stationary atmosphere and flat nonrotating earth assumptions The system model determined by LINEAR consists of matrices for both the state and observation equations The program has been designed to allow easy selection and definition of the state control and observation variables to be used particular model 17 Key Words Suggested by Author s Aircraft model Computer program Control law design Linearization 18 Distribution Statement Unclassified Unlimited Subject category 66 21 No of pages 22 Price 19 Security Classif of this report 20 Security Classif of this page Unclassified 124 Unclassified 6 1626 86 For sale by the National Technical Information Service Springfield Virginia 22161 NASA Langley 1988
64. amic 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 sub routine that uses this aerodynamic data the state variables and the surface posi tions 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 aero dynamic 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 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
65. 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 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 is not specified for a particular surface a default value of 0 001 is used The final set of records in the state control and observation variable defini tions pertain to the specification of parameters associated with the obseryation 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
66. are written in ASCII form Title Information There is one title record that needs to be specified for interactive LINEAR the name of the specific vehicle This record is read with a 20A4 format The vehicle title appears on each page of 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 34 TABLE 1 Input record VEHICLE TITLE S b iy DELX min NUMSAT STATE 1 STATE 2 STATE 3 NUMSUR CONTROL 1 CONTROL 2 CONTROL 3 NUMYVC MEASUREMENT MEASUREMENT MEASUREMENT 6 Semin emax NUMXSR ADDITIONAL ADDITIONAL ADDITIONAL c Weight 1 Ixz Ixy Ivz DELY DELZ LOGCG Omax EQUAT STAB DRVINC 1 DRVINC 2 DRVINC 3 LOCCNT 1 CONVR 1 CNTINC 1 LOCCNT 2 CONVR 2 CNTINC 2 LOCCNT 3 CONVR 3 CNTINC 3 EQUAT 1 PARAM 1 1 3 2 PARAM 2 1 3 3 PARAM 3 1 3 6 Amin max SURFACE 1 SURFACE 2 SURFACE 3 n 6 Ymin Ymax LOCCNT 1 LOCCNT 2 LOCCNT 3 ANALYSIS POINT DEFINITION OPTION ANALYSIS
67. ass information 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 CBAR AMSS AIX ALY 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 CGSHET common block COMMON CGSHFT DELX DELY DELZ The displacements are defined along the vehicle body axis with DELX DELY and DELZ representing the displacements for the x y and z axes respectively 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 Cm and yawing moment Cp coefficients respectively these terms are body axis coefficients The stability axis forces are represented by CD coefficient of drag Cp CLFT coefficient of lift Cr and CY sideforce coefficient 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
68. 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 botn 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 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 g r 0 0 The trim surface positions thrust angle of sideslip and either velocity or angle of attack are determined by numerically solving the nonlinear eguations for the translational and rotational acceleration Pitch attitude 9 is determined by itera tive solution of the altitude rate equation Pushover Pullup pushover pullup analysis points result in wings level flight at n 1 For n gt 1 the analysi
69. d 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 control derivatives These are used in the internal subroutine CLNCAL to calculate 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 G the program will prompt for a file name At this point the user can modify the data interactively by choosing one of the followin
70. d to the vehicle center of gravity can be derived as follows Az Cg Car sin cos a 12 2 Ax Cm Cmar 7 Cp cos a sin sin a cos En Cnar Cy Cp cos a sin a These calculations are normally performed within LINEAR in the subroutine CGCALC However if the user elects the calculation be performed within the user Supplied aerodynamic model CCALC 61 APPENDIX 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 tors not acting at the vehicle center of gravity The gyroscopic effects are a con sequence of the interaction of the rotating mass of the engine and the vehicle 4 mics These effects can be either major or virtually negligible depending on the vehicle The torque effects can be modeled by considering the thrust of an engine where the thrust vector is aligned with the local x axis of the engine acting at some point Ar from the center of gravity of the vehicle fig 4 Center of Figure 4 Definition of location of engine center of mass cm relative to vehicle center of gravity The thrust vector for the ith engine be defined Pi where F are the components o
71. 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 120 variable obser vation vector and a 30 parameter control vector These variables can be selected to specify the formulation most suited for the specific application The order 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 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 t
72. deflections and power lever angles for the given aircraft fig 2 loca tion 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 the common block CONTRL described previously in the Aerodynamic Model Subroutines section The variables DES DAS DRS and THRSTX are used to trim the longitudinal lateral directional and thrust axes respectively For an aircraft using feedback such as a statically unstable vehicle state variables and their derivatives are available in the common block DRVOUT 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 cen terline The parameters for each engin
73. 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 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 Aerodynamic models are described in the next section of this report 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
74. e 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 4 AMSENG 4 4 The variables in 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
75. e 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 records is used to specify if the control variables are given in degrees or radians CONVR is read using an A4 format and is compared to the following list 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
76. ed 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 0 000033820 1 ROLL RATE RAD SEC 0 200000003 2 PITCH RATE RAD SEC 0 000000000 3 YAW RATE RAD SEC 0 163589999 49 4 VELOCITY FT SEC 0 000000115 5 MACH NUMBER 0 000119522 6 ALPHA DEG SEC 0 000362335 7 DEG SEC 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 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 tit
77. es of these quantities We The equations defining these quantities are u V cos v V sin fg w V sin cos B Xp gm sin 0 D cos a L sin THO rv qw m gm cos 0 sin Y UE a pw ru m Zp gm 9 cos D sin q L cos PA a ee ee oJ final set of observation variables available in LINEAR is 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 ais angle of sideslip B i altitude h i and altitude rate h measurements displaced from the center of gravity by some x y and z body axis distances equations used to compute these quantities are 1 8 1 2 h i h x sin 9 y sin cos 9 z cos 21 h Ox 6 y sin q sin 0 2 cos q sin 0 oly cos 9 z sin q 0 The remaining miscellaneous parameters are total angular momentum T stability axis roll rate pg stability axis pitch rate and stability axis yaw rate r defined as Ss H li 1 5015 21xyPA 21 pr 1 4 215291 1 1 Ps p cos a r sin Gs 4 p sin a r cos a SELECTION OF STATE CONTROL AND OBSERVATION VARIABLES The equations in the two preceding sections
78. f thrust in the and 2 body PZ TT axes respectively From figures 5 and 6 it can be seen that the following relation ships hold P Px COS i COS Li F F COS sin 6 pi 1 i Pz Ei 62 where represents the magnitude of the thrust due to the ith engine the Pi P 1 angle from the thrust axis of the engine to a plane parallel to the x y body axis plane and E the angle from the projection of Po onto the plane parallel to the x y body axis plane to the x body axis center Engine center of gravity Figure 5 Orientation of the engines Figure 6 Detailed definition of engine in the x y and x z body planes location and orientation parameters Denoting the point at which the thrust from the ith engine acts Arj this offset vector can be defined as where Axj Ayj and Az are the 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 At then given by 18 1 Ar Oj 1 Thus Ay F Az F Pzi Pyi 2 82 Fox 7 AK Foz 63 The total torque due to engines offset from the center of gravity of the vehicle To is given by n n To 2 ATO 2 Uri x AF 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
79. 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 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
80. g 5 0 16 pr Iyy13 0 1 1 716 q IyzI3 Ixyl6 ar D 13 Iy yIs 1x216 r2 IyzI3 Iy I5 det I where det I 2IyyIyzIyz IyI2 IyI y 1212 yz Y 11 I I 12 yz 12 Ixylz Iyzlxz I3 5 Ixylyz y XZ 14 1 12 12 2 Is re eres ier Ig IxIy I yy Dx Iy Dy Ix Dz Iy Ix 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 XM and XN respectively These total moments are the sums of all aerodynamic moments and powerplant induced moments due to thrust asymmetries 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 equa tions defining the engine torque and are derived in appendix B The body and I The products of inertia are and products of inertia are elements gyroscopic contributions to the total moments axis moments of inertia are designated I Iy of the inertia tensor I defined as designated I These moments Ix Ixy Ixz I Ixy Iy 71 To derive the state equation matrices for the generalized formulation Cx Ax Bu the rotational accelerations are cast decoupled axes formula
81. g ft2 of inertia slug ft2 of inertia slug ft2 inertia slug ft2 of inertia slug ft2 inertia slug ft2 total body axis rolling moment ft 1b 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 lb ft2 impact pressure lb ft2 yaw rate rad sec H x Gx F u Reynolds number Reynolds number per unit length ft wing planform area ft ambient temperature R or total angular momentum slug ft sec 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 lb 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 Az ar displacement of engine from
82. g options TO CHANGE ANY DERIVATIVES INPUT THE APPROPRIATE LINE TO SEE THE DATA TYPE IN IF NO CHANGES 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 calculat
83. ght 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
84. gid aircraft flying in a stationary atmosphere over a flat nonrotating earth Thus internally the state vector x is computed as x p q r V a B 98 y h x 11 nonlinear equations used to determine the derivatives of the states are pre sented 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 y xT ul yT y T y7 1 72 3 4 5 6 7 78 where Y1 a a a a a a a a a gt n 1 x y 2 nz n nx i ny i nz i n i _ y2 Re Re M q qc 4 Pt T Te Ve Vo e T Y fpa Y h h 57 3 T Ya Es ys 2 p N af Y6 V w u V w T Y7 hi h i and Ya T ps ag Ys The equations defining these quantities are presented in the Observation Equations section From the internal formulation of the Output model Internal state control and observation variables Parameters parameters 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 30 parameter control vect
85. he 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 3g f xg Ax f xg Ax Ox D 2 Ax 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 Tx f x x u and the observation equation the program determines the linearized matrices for the generalized formulation of the system C A x B H 6x G x F where of OX 23 with all derivatives evaluated along the nominal trajectory defined by the analysis point Xo Xo The state time derivative of state and control vectors be expressed as small perturbations about the nominal trajectory so that x Xo o X u Uo In addition to the matrices for this generalized system the user has the option of requesting linearized matrices for a standard formu
86. ight 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 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 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 li
87. ing 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 use 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 and H matrices and the time derivative of that state variable in calculating the C and G matrices be specified using the DRVINC variable 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 variabl
88. k 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 equal to zero at a user specified Mach number alti tude and angle of sideslip This trim option is nominally at 1 but as and vary from zero normal acceleration decreases and lateral acceleration increases For aerodynamically symmetric aircraft a trim to B 0 using the beta trim option results in the same trimmed condition as the straight and level trim However for an aerodynamically asymmetric aircraft
89. lation of the system x A 6x B H 6x F Su where 1 ox ox hee E ox se of eS Sq E ES with all derivatives evaluated along the nominal trajectory defined by the analysis point Xo Xo ug LINEAR also provides two nonstandard matrices D and E in the equations x Ax Bu Dv Hx Fu Ev Y or E the equations Cx A x B u D v y H x Gx F u 24 These dynamic interaction matrices include the effect of external forces and moments acting the vehicle The components of the dynamic interaction vector v are incre mental body axis forces and moments Thus and 29 9g p JE o ov These matrices allow the effects of unusual subsystems or control effectors to be easily included in the vehicle dynamics The default output matrices for LINEAR are those for the standard system for mulation However the user can select matrices for either generalized or standard state and observation equations in any combination Internally the matrices are computed for the generalized system formulation and then combined appropriately to accommodate the system formulation requested by the user ANALYSIS POINT DEFINITION The point at which the nonlinear system equations are linearized is referred to as the analysis point This can represent a true steady state condition on the spec ified trajec
90. le 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 formula tion 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 vehicle and case titles are written on the first two records of the analysis file in 80 character strings and are specified in LINEAR as the title of the vehicle and the title of the cases The next record contains the number of the case as defined in LINEAR maximum of 999 cases 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 ob
91. le velocity VEAS equivalent airspeed WB velocity along the z body axis position north from an arbitrary reference point XDOT time rate of change of north south position XYANGL orientation of thrust vector in x y body axis plane XZANGL orientation of thrust vector in 2 body axis plane Y position east from an arbitrary reference point YDOT 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 obser vation variables to be included in the linear model derived by LINEAR Within the program the nonlinear equations of motion include 12 states repre senting a ri
92. mented in this example subroutine UCNTRL DES e DAS Oat DRS THRSTX sb THR to IFENGN UCNTRL Figure 8 Gearing model in example UCNTRL subroutine 118 SUBROUTINE UCNTRL EXAMPLE TRIM CONTROL SURFACE INTERFACE ROUTINE ROUTINE CONVERT TRIM INPUTS INTO CONTROL SURFACE DEFLECTIONS INPUT COMMON BLOCK CONTAINING TRIM PARAMETERS C COMMON CTPARM DES DAS DRS THRSTX OUTPUT COMMON BLOCK CONTAINING CONTROL SURFACE DEFLECTIONS COMMON CONTRL DC 30 EQUIVALENCE VARIABLE NAMES EQUIVALENCE DA DC 1 DE DC 5 DT DC 8 A DR DC 9 DSB DC 10 THR DC 12 C DATA DGR 57 29578 C Co CONVERT FROM INCHES OF STICK AND PEDAL TO DEGREES OF SURFACE C DA DAS 20 0 4 0 DE DES 25 0 5 43 DR DRS 30 0 3 25 SET DIFFERENTIAL TAIL BASED AILERON COMMAND 4 0 CONVERT THRUST TRIM PARAMETER TO PERCENT THROTTLE COMMAND 0 0 IF THRSTX GE 0 0 THR THRSTX USE SPEED BRAKE IF NEEDED DSB 0 0 IF THRSTX LT 0 0 DSB THRSTX 45 0 Cees CONVERT SURFACE COMMANDS RADIANS DA DA DGR DE DE DGR DR DR DGR DT DGR DSB DSB DGR END Engine Model Interface Subroutine The following subroutine IFENGN both defines an engine model and provides the interface to LINEAR In normal usage this subroutine w
93. mplifying assumptions are made It is assumed that the inertia tensor of the engine contains a single nonzero entry Ixe 0 0 le 0 0 0 0 0 0 where Ixe is the inertia of the ith engine at the location of the ith engine i 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 Pei dej and Tej respectively so that T o as de i Pei where Pe we COS cos t qe ue j sin 5 66 sin 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 g 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 b
94. nalysis 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 anal ysis 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 anal ysis 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 indi cating 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 like terms are identically zero no constraints need to be placed on the velocity like terms in Thus for the model used LINEAR only d r v and must be zero to satisfy the trim condition The trim condition is achieved f
95. nearized 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 A x Btu If EQUAT is read in as STANDARD or does not match the preceding list then the de fault standard bilinear formulation of the state equation is assumed and the out put matrices are consistent with the equation x Ax Bu The variable STAB is also read us
96. nput for PARAM 1 the mean aerodynamic chord c 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 Ser and Othe representing the longitudinal lateral directional and thrust trim parameters respectively The units associated with these parameters are defined by the imple mentation 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 fli
97. ns 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 DAS DRS Figure 3 Program flow diagram showing communication with user Supplied subroutines Aerodyn
98. nued Stability axis pitch rad sec ds STAB AXIS PITCH RATE rate Stability axis yaw rad sec STAB AXIS YAW RATE 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 Reynolds 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 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 at
99. ody 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 Iei that So 1 1 I cei lei Rei R a 5 where Rr and Re are axes transformation matrices that perform the previously described rotations through and respectively These matrices are given as COS i 0 sin 0 1 0 sin i 0 COS j i sin 54 0 R sin cos 0 0 0 1 so that Ej cos Ej sin i cos 6 sin RE iRgi sin cos es 6 sin E sin sin i 0 cos Because Re 67 R the matrices are unitary Ro Ry R TR eL Si Ei j cos sin Rei 64 E Therefore cos i cos Ex le Ixe cos Ej COS Gi sin Ej 8112 cos Ej sin i cos Ej sin j The angular momentum of the ith engine h 2 ej 6 6 sin i cos Ej sin COS 0 ES sin i COS E COS j Sin j cos 6 2 Bi 81 gi Ej Sin sin E Sj in es COS Sin i Sin ej can now be expressed as T hi his Mi Tete 81 with hj Pe Txe cos g cos T Fei Txey ej Sin g cos i hi sin j COS sin 27 Pez cos ej cos Ej sin de 7 sin 6i cos E
100. 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 CORRECTION 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 this aerodynamic reference point Thus when this point and the vehicle
101. of these com mon 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 Q R V ALP BTA THA PSI PHI H 7 7 TDOT PDOT RDOT VDOT ALPDOT BTADOT THADOT PSIDOT PHIDOT HDOT XDOT 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 dynamic pressure are the first two entries in the common block sym bolized by AMCH and QBAR respectively The body axis velocities u v and w are included as UB VB and WB respectively 55 CONTRL common block contains the surface position information 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 m
102. or These variables can be selected to specify the formulation most suited for the specific Specification of application The order and number of state vector for linear model parameters in the output model are com pletely under user control Figure 1 Figure 1 Selection of state vari shows the selection of the variables in ables for linear model the state vector for a requested linear model From the internal formulation on oo ee lt x TE De DO lt x D 12 the right the requested model is constructed and the linear system matrices are selected in accordance with the user specification of the state control and obser vation variables The linear model derived by LINEAR is determined at an analysis point LINEAR allows this analysis point to be defined as a true steady state condition spec ified trajectory point at which the rotational translational accelerations are zero or a totally arbitrary state on a trajectory The program LINEAR provides the user with several options described in detail in the Analysis Point Definition section of this report These analysis point definition options allow the user to trim the aircraft wings level flight pushovers pullups level turns or zero sideslip maneuvers Also included is a nontrimming option in which the user defines a totally arbitrary condition about which the linear model is to be derived The linear system matrices are determined by numerical pe
103. or the straight and level 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 fact only in the special case where the flightpath angle is zero does 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 Untr immed For the untrimmed analysis point the user specifies all state and control var iables 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
104. ould call subroutines that do the detailed thrust engine rotation and fuel consumption modeling the information from these subroutines would be transferred into the ENGSTF common block SUBROUTINE IFENGN EXAMPLE SUBROUTINE PROVIDE PROPULSION SYSTEM MODEL ROUTINE TO COMPUTE PROPULSION SYSTEM INFORMATION FOR LINEAR THIS SUBROUTINE IS THE INTERFACE BETWEEN THE DETAILED ENGINE MODELING SUBROUTINES AND LINEAR INPUT COMMON BLOCK CONTAINING INFORMATION THRUST REQUEST ENGINE COMMON CONTRL DC 30 OUTPUT COMMON BLOCK CONTAINING DETAILED INFORMATION ON EACH OF UP TO FOUR SEPARATE ENGINES ROUTINE TO COMPUTE ENGINE PARAMETERS THRUST I THRUST CREATED BY EACH ENGINE TLOCAT I J LOCATION OF EACH ENGINE IN THE 2 PLANE XYANGL I ANGLE IN BODY AXIS PLANE AT WHICH EACH ENGINE IS MOUNTED XZANGL I ANGLE IN 2 BODY AXIS PLANE AT WHICH EACH ENGINE IS MOUNTED TVANXY I ANGLE IN THE X Y ENGINE AXIS PLANE OF THE THRUST VECTOR TVANXZ I ANGLE IN THE X Z ENGINE AXIS PLANE OF THE THRUST VECTOR DXTHRS I DISTANCE BETWEEN THE ENGINE AND THE THRUST POINT ROTATIONAL INERTIA OF EACH ENGINE AMSENG I MASS OF EACH ENGINE ENGOMG I ROTATIONAL VELOCITY RAD SEC 200000 OOO OO OOO COMMON ENGSTF THRUST 4 TLOCAT 4 3 XYANGL 4 XZANGL 4 TVANXY 4 TVANXZ 4 DXTHRS 4 ELX 4 AMSENG 4 ENGOMG
105. program is an extension of the batch LINEAR program described in Duke and others 1987 Data describing the aircraft and test case are input to the program through a terminal or formatted data files similar to the input files for the batch program All data can be modified interactively from case to case The aerodynamic model can be defined in two ways a set of nondimensional stability and control derivatives for the flight point of interest or a full non linear aerodynamic model as used in simulations at Ames Dryden NOMENCLATURE The units associated with the following quantities are expressed in the English system LINEAR will work equally well with any consistent set of units with two notable exceptions the printed output and the atmospheric model Both the printed output and the atmospheric model assume English units Where applicable quantities are defined with respect to the body axis system Variables A State matrix of the state equation X Ax Bu or axial force 1b A state matrix of the state equation CX A x B u a speed of sound in air ft sec an normal acceleration net at center of gravity g x body axis accelerometer at center of gravity g x body axis accelerometer not at center of gravity any y body axis accelerometer at center of gravity g y body axis accelerometer not at center of gravity g ang 2 body axis accelerometer at center of gravity det
106. put 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 O 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 a 10 character maximum file name may be entered If the user wishes to type the data from the terminal and have the program format a file 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 typed from the terminal will be stored for that section of input data LINEAR will then prompt the
107. puts 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 assump tion 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 ZEL I4 FM I2 IN I3 P2 IxzI2 1 13 pa Ixz11 IyzI2 0 13 pr Ixyl4 DyI2 Iyzl3 q2 IyzI1 IxyI3 qr D 1 Ixyl2 1713 r Iyzl1 Ixz17 det I q FL I2 IM I4 IN Ig p2 Iyzl4 1 015 pq IyzI Iygl4 7 D I5 Pr Iy 12 D 14 1 15 a Iyzl2 IxyI5 ar DyIg 1 014 1 215 r2 IyzlI2 Ixzl4 det I r IL I3 IM Ig 18016 2 1 215 IyyI
108. remaining state variables The system model determined by LINEAR consists of matrices for both the state and observation equations The program has been designed to provide easy selection of the state control and observation variables to be used ina particular model Thus the order of the system model is completely under user control Further the program provides the flexibility of allowing alternate formulations of both the 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 11 4 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 permits the inclusion of thrust vectoring effects The ability to select variables for the state control and observation vectors of the linear models without program modification combined with the vast array of observation variables available allows any analysis problem to be attacked with ease The interactive LINEAR
109. rization 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 LINEAR is a program with well defined and generalized interfaces to aerodynamic and engine models and is designed to address a wide range of problems without the 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 aerodynamic model LINEAR is_also capable of extracting both linearized 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 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
110. rol engine and mass and geometry models to LINEAR These sub routines are based on 15 simulation and are typical of the routines needed to interface LINEAR to a set of nonlinear simulation models These subroutines are meant to show the use of the named common blocks to communicate between LINEAR and the user s routines These subroutines are used with all examples in this report Included with this report are microfiche listings of these subroutines Aerodynamic Model Subroutines The following two subroutines define a linear aerodynamic model Even though this model is greatly simplified from the typical nonlinear aerodynamic model the example shows the functions of the subroutines ADATIN and CCALC SUBROUTINE ADATIN EXAMPLE AERODYNAMIC DATA DEFINITION OR INPUT SUBROUTINE ROUTINE DEFINE STABILITY AND CONTROL DERIVATIVES FOR THE AERODYNAMIC MODEL COMMON BLOCK COMMUNICATE AERODYNAMIC DATA BETWEEN THE SUBROUTINES ADATIN AND CCALC COMMON ARODAT CLB CLP CLDA CLDR CLDT CMQ CMDE CMSB CNB CNP CNR CNDA CNDR CNDT CDO CDA CDDE CDSB CLFTO CLFTA CLFTQ CLFTAD A CLFTDE CLFTSB CYB CYDA ROLLING MOMENT DERIVATIVES STABILITY DERIVATIVES WITH RESPECT SIDESLIP ROLL RATE YAW RATE CLB 1 3345 D 01 CLP 2 0000 D 01 CLR 1 5099 p 01 CONTROL DE
111. rol 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
112. rspeed lb ft q lb ft qc lb ft Pa Dimension dc Pa less 1b t2 Pt deg T absolute deg T absolute knot Ve knot Air data parameters continued p DYNAMIC PRESSURE QC IMPACT PRESSURE DIFFERENTIAL PRESSURE PA STATIC PRESSURE FREESTREAM PRESSURE OC PA OC P TOTAL PRESSURE TEMP TEMPERATURE FREESTREAM TEMPERATURE TOTAL TEMPERATURE VEAS EQUIVALENT AIRSPEED KEAS VCAS CALIBRATED AIRSPEED KCAS rt 2 2220 Flightpath angle Flightpath acceleration Flightpath angle rate rad Y g fpa rad sec Y Flightpath related parameters A A AA GAM GAMMA FLIGHT PATH ANGLE GLIDE PATH ANGLE GLIDE SLOPE FPA FLIGHT PATH ACCEL GAMMA DOT GAMMADOT ca e TALT Ja TT E E Pre 73 ku yp a 52 Observation variable Units Symbol Alphanumeric descriptors Flightpath related parameters continued s Vertical acceleration ft sec2 h VERTICAL ACCELERATION HDOTDOT H DOT DOT HDOT DOT Scaled altitude rate ft sec h 57 3 H DOT 57 3 HDOT 57 3 M MMM MMMMMM Energy related terms Specific energy ft Es ES E SUB S SPECIFIC ENERGY Specific power ft sec 5 P SUB S SPECIFIC POWER SPECIFIC THRUST Force parameters Lift force lb L LIFT Drag force lb D DRAG Normal force lb N NORMAL FORCE Axial
113. rturbation and are the first order terms of a Taylor series expansion about the analysis point as described in the Linear Models section The formulation of the output system model is under user control The user can select state equation matrices corresponding to either the standard formulation of the state equation Ax Bu or the generalized equation Cx A x Blu 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 F u In addition to the linear system matrices LINEAR also computes the nondimen sional stability and control derivatives at the analysis point These derivatives are discussed in the Aerodynamic Model section Aircraft geometry and mass properties and analysis point definition are input interactively to the program or are read from ASCII files of 80 character records specified by the 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 be used with follow on design and analysis programs two document the options and analysis points selected by the user and one contains stability and control deriva tives extracted at the analysis point The first of these files contains the state control and observation vector
114. s point is the minimum altitude point of a pullup when h 0 For n lt 1 this trim results 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 conditions subject to the fol lowing contraints p r 0 1 8 en COS B mg n cos 9 Xp sin a 0 27 expression for is derived from the equation by setting a 0 and 0 9 15 derived from the h equation 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 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
115. sed 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 F U E V 2 G DX DT F U 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 C If selected to be input from the terminal the program will prompt the user for the limits to be used for the trim parameters dg ar Ors Stn Which represent the longitudinal lateral
116. seev 9 004400000 0 20 00 1 8670 004400000 0 00 400000 0 00 400000 0 00 400000 0 LO dOLSZL E 00 400000 0 00 400000 0 00 400000 0 00 d00000 0 00 d00000 0 20 d09 80 Y NOILIGNOO LHOI 4 SIHL LY ATANLS GUOHO G t 00 400000 0 00 400000 0 00 400000 0 00 400000 0 004400000 0 00 400000 0 10 do9eec 1 00 400000 0 00 400000 0 00 400000 0 10 d01 v0 b 00 400000 0 CO UOLGLET E 91 dsescce t LNIWOW ONIMVA 10 000521 00 400000 0 10 d08266 9 00 400000 0 10 d0 881 i 00 400000 0 00 400000 0 10 400889 1 0 42 9 01 486687 9 00 400000 0 O0 GQOE G68 00 d00000 0 20 0009022 LNAWOW ONTHOLId 00 400000 0 00 400000 0 00 400000 0 00 400000 0 00 400000 0 00 400000 0 10 008 6 1 00 400000 0 00 400000 0 00 400000 0 10 4 066067 1 00 d00000 0 10 000000 2 91 GIZLEGC 1 LNAWOW INT TION SI NISHVW 211 15 TAVA 4 1 HOLVAS IS D45 AVA 10 4 LOG VHd IV LI viag AVA VHd IV 295 14 ALIDOTIA DAS AVA ALVY MWA DAS AVA ALVA DAS AVA ALVY TION SLNJ4IDIJJJ09 OYAZ AGIND S AASA LOHCOdd AHL YO SHSVO NOILVYLSNONIO GNV 41544 JHAZIAVANI 1 SVO HOI SHAILVAI NHIG TOULNOO ALI IIHVLS TWNOISNAWIG NON 104 MATRIX USING THE FORMULATION
117. servation equation 50 TABLE 2 ANALYSIS FILE FORMAT Lo o s ame Variable Format Title of the case 4A20 Title of the aircraft 4A20 Case number 64 13 Number of states controls and outputs 17 12 22 12 22 13 State equation formulation 36 2 4 Observation equation formulation 36X 2A4 State variable names values and units Control variable names values and units Dynamic interaction variable names and units Output variable names values and units Matrix name A Matrix Matrix name B Matrix Matrix name D Matrix Matrix name C Matrix if general form chosen Matrix name H Matrix Matrix name F Matrix Matrix name E Matrix Matrix name G Matrix if general form chosen 1X 5A4 3X E12 6 2X A20 1 5 4 3 12 6 2 20 1X 5A4 17X A20 1X 5A4 3X E12 6 2X A20 8 5 13 6 A8 5 E13 6 A8 5 13 6 8 5 13 6 8 5 13 6 A8 5 13 6 8 5 13 6 8 5 13 6 51 TABLE 3 ANALYSIS FILE OUTPUT MATRIX STRUCTURE Size of matrix Output formulation 7 x 7 A 7 x 5 7 x 2 B 7 x 3 B 7 x 3 D 7 x 6 D 7 x 5 D 27231 C 7 x 7 c 7x 5 7 x 2 H 11 x 7 H 11 x 5 11 x 2 F 11 x 3 F 11 x 3 E 11 x 6 E 11 x 5 11 x 1 second output file which is assigned to FORTRAN de
118. set If a surface is defined that does not have corresponding control derivatives on the file that surface s effects on the forces and moments will not be included in the linear system model calculated The user should also be aware that LINEAR expects to have control over all axes to determine a trimmed state If there are no nonzero control derivatives in a given axis it is highly likely that LINEAR will fail to trim and it may even generate a fatal 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 changing derivative data sets From each case the program generates a structured linearized derivative data file which can 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 an
119. tack 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 ft ft Symbol p Alphanumeric descriptors P ROLL RATE 9 R YAW RATE VELOCITY VEL VTOT ALP ALPHA ANGLE OF ATTACK BTA BETA SIDESLIP ANGLE ANGLE OF SIDESLIP THA THETA PITCH ATTITUDE PSI HEADING HEADING ANGLE PHI ROLL ATTITUDE BANK ANGLE H ALTITUDE Y 77 APPENDIX 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 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 LEVEL WINGS LEVEL LEVEL FLIGHT 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
120. the ith engine 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 Engine center ZTP le of gravity Figure 7 Detailed definition of thrust vectoring parameters The thrust is assumed to act at in the local engine axis with the engine center of gravity being tne origin of this local coordinate system thrust 15 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 Ce the angle from the projection of the thrust vector onto the engine plane to the local x axis Thus letting bu and represent the x and z thrust maa components in the local engine coordinate system repectively these terms can be defined in terms of the total thrust for the ith engine the angles nj and as F ES COS Nj COS Gj Poy Poy cos nj Sin i 229 Sin 64 Ec Pi Pxi Pyi transform this equation from the ith engine axis system to the body axis system the transformation matrix COS i Ej sin 64 Sin cos 24 COS Ej COS 6 6 Sin i COS i Sin i 0 COS i is used The resultant force in body axis coordinates is F COS Ey COS 6 sin
121. tion equa tions used to derive the linearized matrices are Iz 12 LL Ixy Iy Iy Iy N 1 Lxz I Iz 2 Ixz 1 2 rp pq rq 5 q r 22 qr x Ixy Ixz P E Le Iyz 1 T z A I z Ix Ix I I I Y Y y Luz 1 1 pr p q Y pq 2 2 2 translational acceleration equations used the program LINEAR for both analysis point definition and perturbation are V D cos B Y sin B cos Ym sin Zp sin a cos mg sin cos cos cos 0 sin q sin 0 cos sin a cos 16 B m a L Zp cos a Xp sin a mg cos 0 cos cos sin sin a Vm cos q tan cos r sin sin Y cos Xp a sin Yp cos Z4 sin a sin mg sin 0 sin cos 0 sin cos 0 cos sin a sin B Vm p sin a r cos a The equations used to define the vehicle attitude rates are p sin tan 8 r cos tan 6 9 sin q sin sec 6 r cos sec 0 The equations used to define the earth relative velocities are h V cos B cos a sin 0 sin B sin cos 0 cos sin cos cos 0 x V cos 8 cos cos 0 cos Y sin B sin sin y cos sin y sin cos sin cos y sin sin y y V cos B cos a sin y
122. tory 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 translational accelerations to zero The analysis point options are described in detail in the following subsections For all the analysis point definition options any state or control parameter may be input by the user Those state variables not reguired 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 Appen dix E lists a
123. tput 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 u
124. user supplied subroutines and input files The output from LINEAR is also described Examples of the user supplied subroutines are presented in appen dix 0 microfiche listing of the program for a 11 750 Digital Equipment Corporation Maynard Massachusetts with the VMS Operating system is included with this report The program is written in standard FORTRAN 77 50 that it may be ported to other computers without modification 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 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 completely spec ifying the state and control variables or by specifying an analysis point on a tra jectory selecting a trim option and directing the program to determine the control variables and remaining state variables 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
125. vice unit number 3 con tains the values calculated in LINEAR describing each case first section of this file contains a listing of the input data defining the aircraft s geometry and mass properties variable names defining the state space model and various control surface limits characteristic of the given aircraft Appendix I presents an example printer output file The second section of this file 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 and parameters defining the trim condition The values for the variables of the state space model at the trim condition are also printed If trim was not achieved q r V calculated from the equations of motion and the force and moment coefficients are printed Changes in the geometry and mass properties are also printed The third section of this output file contains the nondimensional stability and control derivatives for the trim condition calculated The static margin of the aircraft at the given flight condition is also printed The final section of this output file contains the state and observation matrices for the given flight condition The formulation of the state equations and only the terms of the matrices chosen by the user to define the model are printed The third output file which is assigned to FORTRAN device unit 2 contains the trim conditio
126. ws the user to modify any individual parameter For example if the user wishes to modify the vehicle geometry proper ties a 1 would be entered and the program would continue WHAT IS THE OF THE FILE CONTAINING THE VEHICLE 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 the program will prompt for a new output file name and then respond as follows 45 GEOMETRY AND MASS DATA FOR USER S GUIDE FOR THE PROJECT DEMONSTRATION CASE 1 WING AREA 608 000 2 WING SPAN 42 800 3 MEAN CHORD 15 950 4 VEHICLE WEIGHT 40700 000 5 1 28700 000 6 165100 000 7 12 187900 000 8 IXZ 520 000 9 1 0 000 10 0 000 VECTOR DEFINING REFERENCE POINT AERODYNAMIC MODEL WITH RESPECT TO THE VEHICLE CENTER OF GRAVITY 11 DELTA X 0 000 12 DELTA Y 0 000 13 DELTA 2 0 000 14 MINIMUM ANGLE OF ATTACK 10 000 15 MAXIMUM ANGLE OF ATTACK 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 LINE NUMBER ON THE DESIRED VALUES N INDICATES NO CHANGES ARE DESIRED R REFRESHES THE SCREEN WITH NEW VALUES FOR EXAMPLE TO CHANGE VEHICLE WEIGHT 4 40000 0 If selection 16 0 is chosen the program will set a logical variable that deter

User's Manual for Interactive LINEAR, a FORTRAN Program

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