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User's Manual for LINEAR , a FORTRAN Program to Derive

1. ADDITIONAL SURFACE 3 LOCCNT 3 CONVR3 5A4 110 A4 l l U q c I m I U gs Test case specification M a n M A ANALYSIS POINT DEFINITION OPTION 20A4 ANALYSIS POINT DEFINITION SUBOPTION A4 VARIABLE 1 VALUE 1 5A4 F15 5 VARIABLE 2 VALUE 2 5 4 15 5 VARIABLE 3 VALUE 3 5 4 15 5 4 ANALYSIS POINT DEFINITION OPTION 20A4 ANALYSIS POINT DEFINITION SUBOPTION A4 VARIABLE 1 VALUE 1 5 4 15 5 VARIABLE 2 VALUE 2 5 4 15 5 VARIABLE 3 VALUE 3 5 4 15 5 END A4 1 There are seven input files for the batch linearizer The first is a file con taining the vehicle title and the names of the six data input files The six data input files are 1 project title 2 mass and geometry properties 3 state control and observation vectors 36 4 trim parameter limits 5 trim definition cases and 6 additional surfaces to be set This is the order in which they must be defined However the sixth file will be read before the fifth file Case Title File Selection Information and Project Title There are two title records that need to be specified for LINEAR a title for the individual test cases case title being analyzed and the name of the specific vehicle project title These records are read with a 20A4 format and are sepa rated by a file selection record Bot
2. ee REN RR FC a DN Input record Format M M n Y M eee d M State control and observation variable definitions NUMSAT STAB 110 4 11 4 Y a M n C STATE 1 DRVINC 1 5A4 F10 0 STATE 2 DRVINC 2 5A4 F10 0 STATE 3 DRVINC 3 5A4 F10 0 NUMSUR 110 CONTROL 1 LOCCNT 1 CONVR 1 CNTINC 1 5A4 110 A4 6X F10 0 CONTROL 2 LOCCNT 2 CONVR 2 CNTINC 2 5A4 110 A4 6X F10 0 CONTROL 3 LOCCNT 3 CONVR 3 CNTINC 3 5 4 110 4 6 10 0 110 4 MEASUREMENT 1 1 1 3 5 4 3 10 0 MEASUREMENT 2 2 1 3 5 4 3 10 0 MEASUREMENT 3 PARAM 3 1 to 3 5 4 3 10 0 A d Trim parameter specification M n T U emin S ense PAR Sud S Pus Penna i Additional surface specification NUMXSR 1 12 ADDITIONAL SURFACE 1 LOCCNT 1 CONVR1 5A4 110 A4 ADDITIONAL SURFACE 2 LOCCNT 2 CONVR2 5A4 110 A4 35 TABLE 1 Concluded A n lt a Input records Format n c ye Pub RN Additional surface specification continued
3. y O FF B ID EE T F h TT Roll attitude rate rad sec PHIDOT PHI DOT Altitude rate length sec h HDOT ALTITUDE RATE Velocity north length sec x XDOT Velocity east length sec y YDOT Accelerations x body axis g ay AX acceleration LONGITUDINAL ACCEL X AXIS ACCELERATION X AXIS ACCELERATION X BODY AXIS ACCEL X BODY AXIS ACCEL y body axis g ay AY acceleration Y AXIS ACCELERATION Y AXIS ACCELERATION AXIS ACCEL Y BODY AXIS ACCEL LATERAL ACCELERATION LAT ACCEL LATERAL ACCEL z body axis g ag AZ acceleration Z BODY AXIS ACCEL Z BODY AXIS ACCEL x body axis acceler g anx ANX ometer at vehicle X AXIS ACCELEROMETER center of gravity X AXIS ACCELEROMETER y body axis acceler g any ANY ometer at vehicle Y AXIS ACCELEROMETER center of gravity Y AXIS ACCELEROMETER i m U U U U T a a U a m o n T 67 eee Observation variable Units Symbol Alphanumeric descriptor _ 5 4_ aama UUU Accelerations continued ee z body axis acceler g anz ANZ ometer at vehicle Z AXIS ACCELEROMETER center of gravity 2 AXIS ACCELEROMETER Normal acceleration g an AN NORMAL ACCELERATION NORMAL ACCEL GS G S x body axis accel g erometer not at ANX I vehicle center of gravity body axis 1 q erometer not at ANY vehicle center of gravity z body axis accel g AZ I erometer not at
4. 92 2 221 40 an 7 pr q Xz qr 42 2 2 1 40 where the subscripts x y and 2 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 sensors relative to the vehicle center of gravity Also included in the set of acceleration equations is 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 Par total pressure py ambient or free stream temperature T and total temperature The selected reference and scaling parameters are Mach number M dynamic pressure q speed of sound a Reynolds number Re Reynolds number per unit length Re and the Mach meter calibration ratio These quantities are defined as 1 ice 0 u pv Re 7 2 V q 1 0 0 242 5 1 0 M 1 0 q 2 5 c 5 76M i 1 ae 1 0 p M gt 1 0 6M2 0 8 1 0 0 2M2 2 2 1 0 M lt 1 0 2 5 522222 0 8 Tt T 1 0 0 2M2 where is length p pressure p the density of the air and y the coefficient of viscosity Free stream pressure free stream temperature and the
5. 744 510111 NOWMWO2 PISNOSN3 35 9 13 H SHHLXG ZXNVAL AXNVAL 5 P TONWZX IDNVAX P ILVODOTL 450 9 0111 NOWHOD S3NIONZ OL gna LN3W3HONI LNS3NOM ONIMVA IVLOL SENIONS OL gna LN3WN3H2NI LNHWONW ONITION IVLOL Toa SANIONS OL LNSWSSONI LN3WOW 5NIHOLId IWLOL SIXV XGO8 Z JHL NI IWLOL 244 SIXW AGOG A AHI NI LSNUYHL IWLOL Ada SIXV AGO X AHL NI LSNYHL IWLOL STOTIIS NOIILI2S3uUIG SIXV X 3NI5N3 SAILV53N HHL NI SAILISOd gaunsv3W LNIOd LSNYHL HHL S3NIS5N3 JO 972 NAJMILIJE SONVLSIG AHL SIXV LSNYHL AINIONGT Z X HHL NO 150 4 HL I IHL INW1Id SIXV ONIONS 1 JHL N33SML38 FIONN AHL 1 ZXNWAL 934 ONIONS X X FHL OLNO HOLOSA LS NHL HL I AHL 40 NOILDILOUd OL SIXV 3NI5N3 X JHL WOUJ INNTA SIXV 3NI5N3 A X SH L NI S3T15NV HHL AXNWAL 236 51 JHL LnOSV N HWN FHL ONISN QATIUNSYAN SI NOILWLOW 3AILISOd 3NIONZ 1 1 JHL JO 1 IYVNOILWLOY FHL 1 SWOONG 59815 S3NI5N3 HL I FHL NI AHS3NIHOWW 5NILVIOH 40 SSYW I ONHSWVX Cs L4 5n T1S S 60 SI 861 320 92 0 0T ST L86T 350 92 ETTO 2110 ITTO OTTO 60TO 8010 LOTO 9010 SOTO FOTO t0TO coto TOTO 0010 6600
6. CNDA CNDR CNDT CDDE 05 CLFTO CLFTA CLFTQ CLFTAD CLFTDE CLF TSB CYB CYDA CYDT C C ROLLING MOMENT DERIVATIVES C C STABILITY DERIVATIVES WITH RESPECT TO C SIDESLIP ROLL RATE YAW RATE C CLB 1 3345 01 2 0000 01 CLR 1 5099 E 01 C C CONTROL DERIVATIVES WITH RESPECT TO AILERON RUDDER DIFFERENTIAL TAIL C CLDA 2 6356 E 02 CLDR 2 3859 03 CLDT 4 0107 E 0 C PITCHING MOMENT DERIVATIVES C STABILITY DERIVATIVES WITH RESPECT TO ANGLE OF C ATTACK PITCH RATE ANGLE OF ATTACK RATE C CMO 4 2204 E 02 C 1 6882 01 CMQ 3 8953 1 1887 01 91 92 C CONTROL DERIVATIVES WITH RESPECT TO C ELEVATOR SPEED BRAKE C CMDE 6 9528 E 01 CMSB 4 1750 E 01 C C YAMING MOMENT DERIVATIVES C C STABILITY DERIVATIVES WITH RESPECT TO C SIDESLIP ROLL RATE YAW RATE C CNB 1 2996 01 CNP 3 3721 E 02 CNR 4 0471 E 01 C C CONTROL DERIVATIVES WITH RESPECT TO C AILERON RUDDER DIFFERENTIAL CNDA 2 1917 03 CNDR 6 9763 E 02 CNDT 3 0531 02 C C COEFFICIENT OF DRAG DERIVATIVES C C STABILITY DERIVATIVES WITH RESPECT TO C ANGLE OF ATTACK C CDO 1 0876 02 C CDA 3 7257 E 01 C C C CONTROL DERIVATIVES WITH RESPECT TO C ELEVATOR SPEED BRAKE C CDDE 4 3831 E 02 CDSB 6 4935 E 02 C C COEFFICIENT OF LIFT DERIVATIVES C C STABIL
7. 2 1 vehicle center of gravity Normal accelerometer g AN I not at vehicle center of qravity Load factor Dimension n N less LOAD FACTOR Tr F mrH T s 0 Alr data parameters n n eee Speed of sound length sec a A SPEED OF SOUND aReynolds number Dimension Re RE less REYNOLDS NUMBER Reynolds number length Re RE PRIME per unit length R LENGTH R FEET R UNIT LENGTH REOR UNE REEL ERRORES et 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 70 Observation variable Units Symbol Alphanumeric descriptor a Ne ees Flightpath related parameters continued Flightpath angle rad sec Y GAMMA DOT rate GAMMADOT Vertical acceler length sec2 h VERTICAL ACCELERATION ation HDOTDOT H DOT DOT HDOT DOT Scaled altitude length sec h 57 3 H DOT 57 3 rate HDOT 57 3 Energy related terms Specific energy length Eg ES E SUB S SPECIFIC ENERGY Specific power length sec Ps PS 50 5 SPECIFIC POWER SPECIFIC THRUST Force parameters O ec ae Lift force force L LIFT Drag force force D DRAG Normal force force N NORMAL FORCE Axial force force A AXIAL
8. CNDA CNDR CNDT CDO CDA 00 CDSB CLFTO CLFTA CLFTQ CLFTAD CLFTDE CLFTSB CYB CYDA CYDR CYDT C C EQUIVALENCE VARIABLE NAMES C EQUIVALENCE T F 2 Q F 3 R F 4 Y F 5 ALP F 6 F 7 THA F 8 51 F 9 F 10 H Ee F 12 Y F 13 PDOT OF 2 9007 DF 3 RDOT DF 4 VDOT DF 5 ALPDOT 6 BTADOT DF 7 THADOT DF EDO 9 PHIDOT DF 10 HDOT DF 11 XDOT DF 12 YDOT DF 13 EQUIVALENCE DA DC 1 DE DC 5 DT DC 8 DR 9 058 DC 10 C e COMPUTE TERMS NEEDED WITH ROTATIONAL DERIVATIVES C V2 2 0 B2V IN2 C2V N C C ROLLING MOMENT COEFFICIENT C CL DA C LDR DR 8 2V CLP CLR R C C PITCHING MOMENT COEFFICIENT C CM CMO CMA ALP CMDE DE 3CMSB DSB 5 2 CMQ Q CMAD ALPDOT C C YAWING MOMENT COEFFICIENT 93 CN CNB CNDA DA C NDR DR NDT DT B 2V CNP CNR gt R OF DRAG C CD 00 CDA ALP CDDE DE 4058 DSB C C COEFFICIENT OF LIFT CLFT LFTO CLFTA ALP C LF TDE DE 3CLF TSB DSB 2 CLFTQ Q CLFTAD ALPDOT C C SIDEFORCE COEFFICIENT C CY C Y8 BTA CYDA DA 3C YDR DR CYDT RETURN END Engine Model Interface Subroutine The followin
9. State variable Units Symbol AT PRA DONETE descriptor Roll rate rad sec p P ROLL RATE Pitch rate rad sec q Q PITCH RATE Yaw rate rad sec r R YAW RATE Velocity ft sec V V VELOCITY VEL VTOT Angle of rad a ALP attack ALPHA ANGLE OF ATTACK Sideslip angle rad B BTA BETA SIDESLIP ANGLE ANGLE OF SIDESLIP Pitch angle rad 9 PITCH ATTITUDE Heading rad PSI HEADING HEADING ANGLE Roll angle rad ROLL ATTITUDE BANK ANGLE Altitude length h H ALTITUDE Displacement length x X north Displacement length y Y east 65 APPENDIX D OBSERVATION VARIABLE NAMES RECOGNIZED BY LINEAR This appendix lists all observation variable names recognized by LINEAR except for state and control variable names If state variables are specified as elements in the observation vector the alphanumeric descriptor must correspond to the names defined in appendix C When control variables are to be included in the observation vector these variables must be identified exactly as they were specified by the user The input file is formatted 5A4 with the alphanumeric data left justified Floating point fields are used to define sensor locations not at the center of grav ity The input name specified by the user for an observation variable serves both to identify the observation variable selected within the program itself and to iden tify observation variables on the printed output of LINEAR An asterisk preceding the variable n
10. Thrust stabilized turn THRUST STABILIZED TURN THRUST LIMITED TURN FIXED THROTTLE TURN FIXED THRUST TURN Beta BETA SIDESLIP Specific power SPECIFIC POWER PS P SUB S Each of these analysis point definitions except the untrimmed beta and spe cific power options has two suboptions associated with it The suboptions are requested using alphanumeric descriptors read using an A4 format These suboptions 73 are defined in the Analysis Point Definition section The following table defines these suboptions and the alphanumeric descriptors associated with each Analysis point definition suboption Straight and level Alpha trim 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 descriptor ALP ALPH ALPHA MACH AMCH ALPH ALPHA LOAD GS G S AN ALP ALPH ALPHA LOAD GS G S AN ALP ALPH ALPHA LOAD GS G S AN 74 APPENDIX F EXAMPLE INPUT FILE The following listing is an example of an input file to LINEAR This file was used with the example subroutines listed in appendix I to generate the analysis and printer output files listed in appendixes G and H respectively BEN V LINEARIZER TEST AND DEMONSTRATION CASES USER S GUIDE 6 080000 02 4 280000 401 1 595000E 1 4 500000E 04 2 870000 404 1 651000E 5 1 879000 05 5 200000
11. Two energy related terms are included in the observation variables specific energy Es and specific power Ps defined as y2 ET 9 g The set of observation variables available in LINEAR also includes four force parameters total aerodynamic lift L total aerodynamic drag D total aerodynamic normal force N and total aerodynamic axial force A These quantities are defined as L N L cos D sina A L sin Q 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 e e a rate w Also included are the time derivatives of these quantities u v and w The equations defining these quantities are u V cos cos V sin V w V sin a cos f 5 gm sin D cos L sin u rv qw m 4 gm cos 0 sin Y v oe ge cr 22 gm cos 0 sin L cos w ee qu pv The final set of observation variables available in LINEAR is a miscellaneous collection of other parameters of interest in analysis and design problems The 22 first group consists of measurements from sensors not located at the vehicle center of gravity These represent angle of attack angle of sideslip 1 altitude hi and altitude
12. axis coefficients The stability axis forces are represented by CD coefficient of drag Cp CLFT coefficient of lift and CY sideforce coefficient Cy Control Model The program LINEAR attempts to trim the given condition using control inputs similar to those of a pilot longitudinal stick lateral stick rudder and throt tle The UCNTRL subroutine utilizes these trim output control values to calculate actual surface deflections and power level angles for the given aircraft fig 2 The location of each surface and power level angle in the CONTRL common block is specified by the user in the input file maximum of 30 surfaces The limits for the control parameters in pitch roll yaw and thrust are user specified see Trim Parameter Specification in the Input Files section and must agree in units with the calculations in CCALC The common block CTPARM contains the four trim parameters that must be related to surface deflections in the subroutine UCNTRL COMMON CTPARM DES DAS DRS THRSTX The output from UCNTRL is through the common block CONTRL described previously in the Aerodynamic Model section The variables DES DAS DRS and THRSTX are used to trim the longitudinal lateral directional and thrust axes respectively For an aircraft using feedback such as a statically unstable vehicle state variables and their derivatives are available in the common block DRVOUT If parameters other than state variables an
13. ALPHA I ALPHA INSTRUMENT AOA INSTRUMENT BETA I BETA INSTRUMENT SIDESLIP INSTRUMENT ALTITUDE INSTRUMENT HDOT I 71 72 Observation variable Units Symbol Alphanumeric descriptor eae ae Ae ee Ino E ee SASER Other miscellaneous parameters M Vehicle total angular mass length2 T ANGULAR MOMENTUM momentum sec ANG MOMENTUM Stability axis roll rad sec Pg STAB AXIS ROLL RATE rate Stability axis pitch rad sec ds STAB AXIS PITCH RATE rate Stability axis yaw rad sec rs STAB AXIS YAW RATE rate a ee APPENDIX E ANALYSIS POINT DEFINITION IDENTIFIERS Analysis point definition options are selected using alphanumeric descriptors The first record read for each analysis case contains these descriptors All these descriptors are read using a 5A4 format The following table associates the analy sis point definition options with their alphanumeric descriptors n a r n lt qEAI Analysis point definition option Alphanumeric descriptor Untrimmed UNTRIMMED NO TRIM NONE NOTRIM Straight and level STRAIGHT AND LEVEL WINGS LEVEL LEVEL FLIGHT Pushover pullup PUSHOVER PULLUP PULLUP PUSHOVER PUSHOVER AND PULLUP PUSHOVER PULLUP PUSHOVER PULL UP PUSHOVER PULLUP PUSH OVER PULL UP Level turn LEVEL TURN WINDUP TURN
14. 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 in 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 C C Bec h C M 5 5 3 8 Xn 2 36 B n Cm Cm CmgB Sh Cg OM Cms L P Cr q Cm Cn Cn Cng Cp Sh Cg DER Enga Cnt 48 n 6 CL Cr a CL Sh 2 oa Cyr Crea 58 32 n Cp Da 2 CpgP T Sh 6M 1 Cn b Cosa Cosh DpP 9 Cp Da C a 1 Cy CyB M 2 Cy q Cy t Cy q p where the stability and control derivatives have the usual meaning Cex ax with Ce being an arbitrary force or moment coefficient subscript 5 denoting rolling moment m pitching moment L total lift D total drag n yawing moment and
15. 02 0 0 0 0 0 CCALC WILL CALCULATE CG CORRECTIONS 1 000000 401 4 000000E 01 4STAN RADI ALPHA Q THETA VEL 3 ELEVATOR 5 THROTTLE 12 SPEED BRAKE 10 2STAN AN AY 2 900 400 5 430 00 4 000 00 4 000E 00 3 250E 00 3 250E 00 1 000E 00 1 000E 00 0 ADDITIONAL SURFACES WINDUP TURN ALPHA H 20000 0 MACH 0 90 AN 3 00 BETA 0 0 NEXT LEVEL FLIGHT ALPHA H 20000 0 MACH 0 9 GAMMA 10 0 END M This input file is for a case called record 1 LINEARIZER TEST AND DEMONSTRA TION CASES and all input data are on logical device unit 1 signified by the second record being blank The project title is USER S GUIDE record 3 Record 4 specifies the mass and geometric properties of the vehicle as S 608 ft2 b 42 8 ft 15 95 ft Q 45 000 lb w Record 5 defines the moments and products of intertia of the vehicle all in units of slug ft2 as 75 Ix 28 700 Iy 165 100 Iz 187 900 Ixz 520 0 Toz 0 Record 6 defines the location of the vehicle center of gravity to be coincident with the aerodynamic reference point of the nonlinear aerodynamic model by setting Ax 0 Ay 0 Az O Record 6 also specifies that LINEAR should not use its internal model to make cor rections for the offset in the vehicle center of gravity from the aerodynamic ref erence point either because the aerodynamic model includes such corrections or because none are to be made Record 7 defines the a
16. ZXNWNS I ZXNVSO AX IONN ZX I5NWM 1 1 I 04 LSNYHL 51 4 3Llf dWOO TIYO ANILNOW TAGOW 3NIS5NS 50 TIYO A 2 4 1 6 4 via 9 4 Loax Tt aa Ha LOGQISd 8 44 jag Loaqatw s 44 Loqd 2z 1 x T1 43 ISd 8 4 41 5 34 8662745 LOGH 2 LOdA 1044 1044 d H A WIG U 2 2 3 2 0470 69 0 89 0 L910 9910 910 9 0 910 910 0910 6ST0 8510 1810 94 0 SSTO PSTO ESTO 2410 TSTO 0570 6510 8PTO 9 0 SETO 0 2010 TPTO 6tTO LETO SETO O IO 6c10 8cTO LZTO 9210 210 EZTO 2210 0 6TTO 90 TO LTTO 9110 110 ANIONG 101 3NISNS NOLNY TVGSAXS TLZ L NVHIHNO4 xwa 1 2XNWNSs I ZX NWNS 3XIQ Z VWLYUNIV T WLUNiIwv ILRXNNNS amp 4 I ZXNVSO I ZXNXNS43XIS3 I ZXNVSO I ZXNVSOs I AXNVNS I AXNVNS amp 3XIS Z T WLUNIV 1 AXNVSDe 1 ZXNWNSs I 2XNWSD 9XIQ I AXNVSO I AXNVNS amp I ZXNVSO4 I ZXNYSO 3XIS 1 52 1 52 1 2 52 1 2 52 3 13 7979 14 1 SNIONG HL I JHL 40 HOSN3L WILUANI I LWOo TW C I LV2OJIL T I LV2OIL I XIS I ZX
17. and angle of attack range for the vehicle model being analyzed first record of this set of five records is a vehicle title record in which the vehicle name and Specific configuration can be defined to describe the information presented in the remaining records The second and third records define the vehicle geometry mass and mass dis tribution The second record defines the wing planform area S in units of length squared the wingspan b in units of length the mean aerodynamic chord of the wing units of length and the sea level weight of the vehicle Weight in units of mass length per second squared third record defines the vehicle moments and products of inertia in units of mass length squared fourth record defines the location of the vehicle center of gravity with respect to the aerodynamic reference point This fourth record defines the offset of the aerodynamic reference point with respect to the center of gravity of the vehicle in the normal right handed body axis reference system with the positive x axis forward and DELX DELY and DELZ in table 1 representing the x y and z body axis displacements of the aerodynamic reference in units of length see app A The fourth variable of the fourth record is an alpha numeric variable read using a 12A4 format to specify if corrections due to an off set of the aerodynamic reference with respect to the center of gravity are to be computed in LINEAR or in the
18. cos 0 sin cos cos 0 cos sin sin B Vm p sin r cos a where a B 9 and are angles of attack sideslip pitch and roll respectively and Zp are thrust along the x y and z body axes and D is drag force g gravitational acceleration L total aerodynamic lift m total aircraft mass V total velocity and Y sideforce The equations defining the vehicle attitude rates are p q sin tan r cos tan q cos r sin V q sin sec r cos sec 0 where is heading angle The equations defining the earth relative velocities are h V cos cos sin 8 sin sin cos 0 cos sin cos 9 cos x V cos cos cos 0 cos sin sin sin 9 cos y cos sin jp sin cos sin W sin sin y V cos cos a cos 0 sin W sin sin sin 0 sin W cos cos y sin cos sin sin sin cos y where h is altitude OBSERVATION EQUATIONS The user selectable observation variables computed in LINEAR represent a broad class of parameters useful for vehicle analysis and control design problems These 18 variables include the state time derivatives of state and control variables Also included are air data parameters accelerations flightpath terms and other miscel laneous parameters The equations used to calculate those parameters are derived from a number of sources Clancy 1975 Do
19. 020 2020 020 0020 6610 8610 L6TO 9610 610 PETO 6TO 2610 T6TO 0610 68 0 8810 8 0 9810 810 8 0 ESTO 28 0 810 0810 6110 8110 LLTO 9L10 5 70 tLTO ZLTO INIONG 102 IT1 8qO343 INIONS NOLNW 5 6 TLZ L PA NNuLHOJ XWA SE 60 ST L861 320 9C 0 0 lt 861 320 92 2 z unbgox 2 I SSNI5N3 OL INA LNS SW3SHSONI LNIWOW T I NIONGHs Z I NISNSH s E I NIONSHs I I NIS5NSGH Z I NISNSH I NIS5NSH 2 Ot pd Q4 0 Oi T 61023443 2140 C WLUNIWsZ93WO C NINUNIV amp 4A453WNO I D VLHNIV4XOSWO Nuni3u 2 OUAD Noa OUAD gt TINLOL 3LNdWOD 12222 gt AQNILNOD 9 2 OUAD OUAD 7Z OUAD OUAD T OAD 2 9 gt 070 OUAD 0 0 2 OMA5 0 0 T 3 DSOHAD 104 2 72720 gt JNNILNOD S gt 3nNILNO2 P 2 C I NIONGH 0920 6520 8620 68 0 9520 SSZ0 FSZ0 ESZO 24420 520 0620 6 20 8 20 LEZO 9920 SZO 20 0 2 20 0 20 6620 8520 LEZO 9620 620 TEZO TEZO 0620 6220 ANIONG 103 REFERENCES Chen Robert Kinematic Properties of Rotary Wing and Fixed Wing Aircraft in Steady Coordinated High g Turns AIAA 81 1855 Aug 1981 Chen Robert T N and Jeske James A Kinematic Proper
20. 8600 L600 9600 S600 600 600 600 1600 0600 6800 8800 L800 9800 5800 t800 800 2800 T800 0800 6100 8L 00 1 00 9L00 SL00 FL00 tL00 21 00 TL00 0200 6900 8900 900 9900 5900 8900 900 7900 T900 0900 6S00 8S00 qNISN3 100 1 1 1 4515 TL7Z L PA NNUuLHOd XWA 0 4 L 56 60 lt 4 1 861 320 92 0 0 6 861 320 9 4 1041 Cor 6 L aa 9 4 Loqu c 4a LSNOUHL 51 2 I AXNVSO 1 AXNYNS I AXNVSO I AXNVSO AXMLNS AXNLSO I ZXNVSO I ZXNWNS 1 ZXNWNS 1 XXNV SO I ZXNVSO 1 ZXNWNS I 5 1 2 52 2 2 5 ZXWLSO ZXWLSO 0 0 0 1 1 Z od 0 0 0 244 Add Xdd SIXV X INLOL 3Ll ndNWOO ISUHLIZ ISWUHLX ISUHLZ ISUHLA 158 ISUHLZ ISHHLZ ISHHLX lt x 45 4 x 1 LSOUBL ZX5NWVL NISG 2 1 50024 XXONWL NISA XXONWL SODG u5a I ZXNWVAL wod I AXNWAL AX IONW NISA AX I19NX SODQ ZX I5NX NISQ 2 5024 1 2 1 2 SANILNOD T I LSHHL2 I LSHHLA 1 LSMHLX ISHHLZ ISHHLA ISHHLIX 2 5 ZXWLSO KXVLNS KXVLSO ZXONWL AXONWL I AXNWNS 502 1
21. All aerodynamic effects are modeled at this aerodynamic reference point Thus when this point and the vehicle 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 aerody namic model The total aerodynamic moment M acting at the vehicle center of gravity 1s defined as where Mar Mar Nar is the total aerodynamic moment acting at the aerodynamic reference point denoted by subscript ar of the vehicle Ar Ay Az T is the displacement of the aerodynamic reference point from the vehicle center of gravity and F X Y 717 is the total aerodynamic force acting at the aerodynamic center where X Y and Z are total forces along the x y sideforce and z body axes Thus Lar Ay 2 Az Y M Mar Az X Ax 7 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 45 Cp sin Q Y qSCy 7 45 sin Cy cos a Substituting these equations into the definition of the total aerodynamic moment equation and applying the definitions of the total aerodynamic moments 55 Lar qSbC a
22. ENGOMG I ROTATIONAL VELOCITY RAD SEC COMMON ENGSTF THRUST 4 TLOCAT 4 3 XYANGL 4 XZANGL 4 TVANXY 4 TVANXZ 4 DXTHRS 4 EIX 4 AMSENG 4 ENGOMG A C C EQUIVALENCE VARIABLE NAMES C EQUIVALENCE THR DC 12 C ASSUME THRUST PER ENGINE IS HALF VEHICLE WEIGHT C THRUST 1 24000 0 THR THRUST 2 2 24000 0 C C LET ALL OTHER PARAMETERS DEFAULT TO ZERO C RETURN END 01 F Control Model Subroutine The subroutine UCNTRL provides an interface between the trim parameters and the surface deflections Figure 8 illustrates the gearing model implemented in the example UCNTRL subroutine The thrust demand parameter is also set and passed to the subroutine IFENGN 25 0 180 DES 5 43 n 9 DAS 9 DRS THRSTX THR to IFENGN UCNTRL Figure 8 Gearing model in example UCNTRL subroutine 95 SUBROUTINE UCNTRL eee EXAMPLE TRIM CONTROL SURFACE INTERFACE ROUTINE TO CONVERT TRIM INPUTS INTO CONTROL SURFACE DEFLECTIONS INPUT COMMON BLOCK CONTAINING TRIM PARAMETERS gt 0 COMMON CTPARM DES DAS DRS THRSTX OUTPUT COMMON BLOCK CONTAINING CONTROL SURFACE DEFLECTIONS COMMON CONTRL DC 30 EQUIVALENCE VARIABLE NAMES OOO Cc C EQUIVALENCE DA DC 1 DE UC 5 DT 0 8 08 0 9 DCB DC 10 THR DC 12 DATA DGR 57 29578 CONVERT FROM INCH
23. Fp 229 pz where Fp and Fp are the components of thrust in the x y and z body Xi Yi Zi axes respectively From figures 5 and 6 it can be seen that the following rela tlonships hold F Pes Fp cos cos B Fp sin Fp sin 57 where Fp represents the magnitude of the thrust due to the ith engine the angle from the thrust axis of the engine to the x y body axis plane and the angle from the projection of Fp onto the x y body axis plane to the x body axis Aircraft center of gravity Figure 5 Orientation of the engines Figure 6 Detailed definition of engine in the x y and x z body axis planes location and orientation parameters Denoting the point at which the thrust from the ith engine acts as Arj this offset vector can be defined as Ar Ay Az T where Ax Ayi and Az are the x y and z body axis coordinates respectively of the origin of the ith thrust vector The torque due to offset from the center of gravity of the ith engine ATo is then given by Thus Pz 1 Ay F l The total torque due to engines offset from the center of gravity of the vehicle To is given by 58 a VE n To 2 Sto Ar x where n is the number of engines For the case of vectored thrust the equations for torque produced at the vehicle
24. Hx Fu observation matrix of the observation equation y H x Gx F u altitude length angular momentum of engine rotor mass length2 sec aircraft inertia tensor mass length rotational inertia of the engine mass length2 body axis moment of inertia mass length2 x y body axis product of inertia mass length 2 body axis product of inertia mass length2 y body axis moment of inertia mass length2 2 body axis product of inertia mass length2 z body axis moment of inertia mass length2 total body axis aerodynamic rolling moment length force or total aerodynamic lift force generalized length length Mach number or total body axis aerodynamic pitching moment length force aircraft total mass mass mass of engine normal force force or total body axis aerodynamic yawing moment length force load factor specific power length sec Pa RE Re Tt roll rate rad sec or pressure force length2 ambient pressure force length2 total pressure force length2 pitch rate rad sec dynamic pressure force length impact pressure force length2 axis transformation matrices Reynolds number Reynolds number per unit length length yaw rate rad sec wing planform area length ambient temperature K or total angular momentum mass length2 sec2 or thrust force total temperature K velocity in x axis direction length sec con
25. ROLLING MOMENT YAWING MOMENT 0 0000000 00 POUNDS 0 0000000400 POUNDS 0 0000000 00 POUNDS 0 0000000400 FOOT POUNDS 0 0000000400 FOOT POUNDS 0 000000D 00 FOOT POUNDS OBSERVATION VARIABLES AN AY 0 9852280400 GS 0 000000D 00 GS A MATRIX FOR DX DT A X B U D V 0 1209000 01 0 100000D 01 0 5757300 02 0 701975D 04 0 1491890401 0 2214510401 0 1896400 01 0 2313680 03 0 0000000 00 0 100000D 01 0 0000000 00 0 0000000 00 0 576868002 0 0000000 00 0 3162510 02 0 4604350 02 B MATRIX FOR DX DT A X B U D 0 1419610400 0 448742D 03 0 9289320 02 0 2207780402 0 1478120 02 0 1350740402 0 0000000 00 0 000000D 00 0 000000D 00 0 1051860 02 0 343162D 02 0 1558320 02 D MATRIX FOR DX DT A X B U D 0 9348800 08 0 000000D 00 0 738119D 06 0 3079410 07 0 000000D 00 0 2431290 05 0 0000000 00 0 000000D 00 0 0000000 00 0 714920D 03 0 4958290 13 0 9054970 05 0 000000D 00 0 000000D 00 0 000000D 00 0 0000000 00 H MATRIX FOR Y H X 0 35042 4D 02 0 128333D 06 0 632314 02 0 0000000 00 0 000000D 00 0 000000D 00 FOR Y F 0 4113230401 0 4928450 03 0 2632880 00 0 0000000400 0 000000D 00 0 000000D 00 E MATRIX FOR Y H X F U E 0 1026760 07 0 0000000 00 0 2141160 04 0 0000000 00 0 2222220 04 0 0000000 00 0 000000D 00 0 000000D 00 0 0000000 00 0 000000D 00 0 6056940 05 0 000000D 00 0 00
26. Y force along the y body axis and x an arbitrary nondimensional variable denoting angle of attack B angle of sideslip h altitude M Mach number p roll rate q pitch rate and r yaw rate rotational terms in the equations are nondimensional versions of the corresponding state variable with PUO 4 ov A br r 2V 2V 3 b gt v where b is wingspan and is mean aerodynamic chord The 6 in the summations are the n control variables defined by the user The effects of altitude and velocity Mach are included in the derivatives with respect to those parameters and in the incremental multipliers Sh h h 33 where the subscript zero represents the current analysis point u described in the Analysis Point Definition section All stability derivatives are computed as nondimensional terms except the alti tude and velocity parameters The control derivatives are in whatever units are used in the nonlinear aerodynamic model The derivatives with respect to velocity are multiplied by the speed of sound at the analysis point altitude to convert them to derivatives with respect to Mach number Derivatives with respect to angle of attack and angle of sideslip can be obtained in units of reciprocal degrees These derivatives are simply the corresponding nondimensional derivatives multiplied by 180 0 INPUT FILES The LINEAR input file defined in table 1 is se
27. and control variables determine the analysis point com pletely For all other trim options only certain states are not varied all con trols connected to the aerodynamic and engine model are varied OUTPUT FILES There are three output files from LINEAR a general purpose analysis file a printer file containing the calculated case conditions and the state space matrices for each case and a printer file containing the calculated case conditions only The general purpose analysis file contains the title for the cases being ana lyzed the state control and observation variables used to define the state space model and the state and observation matrices calculated in LINEAR The C and G 42 matrices are printed only if the user has selected an appropriate formulation of the state and observation equations The output for this file is assigned to FORTRAN device unit 15 An example of a general purpose analysis file is pre sented in appendix G corresponding to the format of table 3 TABLE 3 ANALYSIS FILE FORMAT Variable te Title of the case Title of the aircraft Case number Number of states controls and outputs State equation formulation Observation equation formulation State variable names values and units Control variable names values and units Dynamic interaction variable names and units Output variable names values and units Matrix name A matrix Matrix name B matrix Matri
28. cos 6 cos 5 cos cos Bi sin sin cos Piye sin cos 61 PER 0 COS i 1 so that Fp cos Ny cos Gi cos cos 5i cos ny sin Ci sin sin sin cos i F sin sin cos i The moment Ax cos cos Ari Ay Az and the total torque due to thrust vectoring is n To gt 1 1 60 n Ari x i 1 Pp cos ni cos cos sin j cos ni sin Lj cos Ei sin sin ny cos j arm through which the vectored thrust acts is The engine inertia tensor must be defined in an axis system oriented consist ent with the vehicle body axis system This is done in two steps These steps involve rotating the engine inertia tensor into a coordinate system orthogonal to the aircraft body axis system First the ith engine inertia tensor is rotated through an angle about the local y axis so that the new inertia tensor is ori ented with its local x y body axis plane parallel to the X y body axis plane of the vehicle The second step requires a rotation through an angle about the local Z axis so that the local x y and z axes are orthogonal to the x y and z body axes of the vehicle As determined by Gainer and Sherwood 1972
29. 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 The subroutine MASGEO allows the user to vary the center of gravity position and vehicle inertias automatically Nominally this subroutine must exist as one of the user subroutines but it may be nothing more than RETURN and END statements MASGEO is primarily for variable geometry aircraft such as an oblique wing or variable sweep configurations or for modeling aircraft that undergo significant mass or iner tia changes over their operating range The center of gravity position and inertias may be functions of flight condition or any surface defined in the CONTRL common block Changes in center of gravity position are passed in the CGSHFT common block and inertia changes are passed in the DATAIN common block Care must be taken when using the subroutine MASGEO in combination with selecting an observation vector that contains measurements away from the center of gravity If measurements are desired at a fixed location on the vehicle such as a sensor platform or nose boom the moment arm to the new center of gravity location must be recomputed as a result of the center of gravity shift for accurate results This can be accomplished by implementing the moment arm calculations in one of the user subr
30. in the aerodynamic model and a model of the mass and geometry properties of the aircraft The gearing model fig 2 defines how the LINEAR trim inputs will be connected to the surface models and allows schedules and nonstandard trimming schemes to be employed This last feature is particularly important for oblique wing aircraft Inputs from Outputs to ICTPARM ICONTRL DC 1 DC 2 DES DAS UCNTRL DRS Gearing THRSTX DC 30 Pilot stick pedal Surface deflections and throttle and power level setting Figure 2 Inputs to and outputs of the user supplied subroutine UCNTRL EQUATIONS OF MOTION The nonlinear equations of motion used in the linearization program are general six degree of freedom equations representing the flight dynamics of a rigid aircraft flying in a stationary atmosphere over a flat nonrotating earth The assumption of nonzero forward motion also is included in these equations because of this assump tion these equations are invalid for vertical takeoff and landing or hover Th se 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 will be detailed in a proposed NASA Reference Publication Derivation and Definition of a Linear Aircraft Model by
31. of two main subroutines ADATIN and CCALC ADATIN is used to input the basic aerodynamic data from remote storage and can also be used to deflne aerodynamic data CCALC is the subroutine that uses these aerodynamic data the state variables and the surface positions to determine the aerodynamic coefficients Either routine may call other subrou tines to perform related or required functions however from the point of view of the interface to LINEAR only these two subroutines are required for an aero dynamic model 46 Externally ADATIN has no interface to the program LINEAR The subroutine is called only once when the aerodynamic data are input or defined The calling program must provide ADATIN with the input devices it requires but no other accom modation is necessary The aerodynamic data are communicated from ADATIN to CCALC through named common blocks that occur in only these two routines The interface between CCALC and the calling program is somewhat more compli cated However the interface is standard and this feature provides a framework about which a general purpose tool can be built This interface consists of several named common blocks that are used to pass state variables air data parameters sur face positions and force and moment coefficients between CCALC and the calling program CCALC is executed whenever new aerodynamic coefficients are required for example once per frame for a real time simulation The main tran
32. option in which angle of attack altitude and Mach number are spec ified and load factor is determined according to the constraint equations The analysis point is determined at the specified conditions subject to the following contraints p r 0 mgl 9 2 in H o aoe es Zp cos Xp sin 0 0 The expression for q is derived from the equation by setting a 0 and 0 9 is 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 non wings level constant turn rate flight at n gt 1 The vehicle model is assumed to have sufficient excess thrust to trim at the condition specified If thrust is not sufficient trim will not result and the analysis point thus defined will have a nonzero in fact negative velocity rate The level trim options available in LINEAR require the specification of an alti tude and a Mach number user can then use either angle of attack or load factor to define the desired flight condition These two options are referred to as alpha trim and load factor trim respectively For either option the user may also request a specific flightpath angle or altitude rate Thus these analysis point definitions may resu
33. output accelerometer at center of gravity 4 anz i z body axis accelerometer output accelerometer not at center of gravity g ay acceleration along the x body axis g ay acceleration along the y body axis 4 acceleration along the z body axis control matrix of the state equation x Ax Bu control matrix of the state equation Cx A x Btu wingspan length C matrix of the state equation Cx A x B u or force or moment coefficient coefficient of drag coefficient of lift coefficient of rolling moment coefficient of pitching moment coefficient of yawing moment coefficient of sideforce center of mass of ith engine mean aerodynamic chord length dynamic interaction matrix for state equation x Ax Bu Dv or drag force force dynamic interaction matrix for the state equation Cx B u D v dynamic interaction matrix for the observation equation y Hx Fu EV dynamic interaction matrix for the observation equation y H x Gx F u E v specific energy length feedforward matrix of the observation equation y Hx Fu total aerodynamic force acting at the aerodynamic center engine thrust vector feedforward matrix of the observation equation y H x Gx F u fpa flightpath acceleration q G matrix of the observation equation y H x Gx F u acceleration due to gravity length sec2 observation matrix of the observation equation y
34. q r V a B q y h yr 12 The nonlinear equations used to determine the derivatives of the quantities are presented in the following section Equations of Motion The internal control vector u can contain up to 30 controls The internal observation vector y contains 120 variables including the state variables the time derivatives of the state variables the control variables and a variety of other parameters of interest Thus within the program _ T yf X Y Y Y Ys Y y7 where Yi ax anx anz T 4nx i 4nz i 4n i n T Y2 a Re q qc dc Pa Pt Ve T ya Es Ps T ys L D N A e T v w u v T Y wi 8 1 hi h il T Y8 T ps 95 rs The equations defining these quantities are presented in the Observation Equations section From the internal formulation of the state control and observation variables the user must select the specific vari ables desired in the output linear model described in the Selection of State Con trol and Observation Variables section Figure 1 illustrates the selection of var iables in the state vector for a requested linear model From the internal formula tion on the right of the figure the re quested model is constructed and the linear system matrices are selected in accordance with the user specification of the state control an
35. rate h measurements displaced from the center of gravity by some x y and z body axis distances The equations used to compute these quantities are qx PY sa o SG rx pz V hi h x sin 9 y sin cos 0 2 cos cos 9 9 li US 2e il h 9 x cos y sin sin 9 z cos sin y cos cos 2 sin cos 9 The remaining miscellaneous parameters are total angular momentum T stability axis roll rate pg stability axis pitch rate qs and stability axis yaw rate rs defined as 1 T gt xP 2IyyPd 21 Iyq2 2Iyzqr Izr Ps p cos Q r sina ds Gd p sin r cos SELECTION OF STATE CONTROL AND OBSERVATION VARIABLES The equations in the two preceding sections define the state and observation variables available in LINEAR The control variables are defined by the user in the input file Internally the program uses a 12 state model a 120 variable obser vation vector and a 30 parameter control vector These variables can be selected to specify the formulation most suited for the specific application The order and number of parameters in the output model is completely under user control Figure 1 illustrates 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 f
36. subroutine IFENGN is used to provide an interface between LINEAR and any engine modeling rou tines the user may wish to incorporate UCNTRL converts the trim parameters used by LINEAR into control surface deflections for the aerodynamic modeling routines The subroutine MASGEO allows the user to define the mass and geometry properties of the vehicle as a function of flight condition or control setting The use of these sub routines is illustrated in figure 3 which shows the program flow and the interaction of LINEAR with the user supplied subroutines These subroutines are described in detail in the following sections Examples of these subroutines are given in appen dix I 45 User supplied subroutines Wasara pw Aeromodel Read aircraft specific input data files Read in data for next analysis point option requested Main program Interface common block names ICONTRL CTPARM Control surface gearing UCNTRL Yes ICONTRL ENGSTF Thrust effector Determine analysis model trim point conditions 1 IFENGN Determine linear model Select matrix elements for output ICGSHFT CLCOUT Aerodynamic model CCALC IDATAIN DRVOUT OBSERV CGSHFT Mass gis IDATAINI CONTRLI MASGEO Run another case 2 No Figure 3 Program flow diagram showing communication with user supplied subroutines Aerodynamic Model It 1s assumed that the aerodynamic models consist
37. user supplied aerodynamic subroutine CCALC described in the User Supplied Subroutines section The variable LOGCG defaults to a state that causes the aerodynamic reference offset calculations to be performed by sub routines 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 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 These parameters Qnin and max Specify the minimum and maximum values of angle of attack to be used for trimming the aircraft model These parameters are in units of degrees State Control and Observation Variable Definitions The state control and observation variables to be used in the output formula tion of the linearized system are defined in records that either follow the last of the previously described sets of records on FORTRAN unit 1 or are stored on a sep arate file defined by the second of the input file selection definitions on the file selection record 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 st
38. wide range of problems without requiring program modification The system model determined by LINEAR consists of matrices for both state and observation equations The program has been designed to provide easy selection and definition of the state control and observation variables to be used in a particular model Thus the order of the system model is completely under user control Further the program provides the flexibility of allowing alternative formulations of both state and observation equations LINEAR has several features that make it unique among the linearization programs common in the aerospace industry The most significant of these features is 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 capabilities provided by means of a user supplied subroutine allow unlimited possibilities for trimming strategies and surface scheduling which are particularly important for oblique wing vehicles and aircraft having multiple surfaces affecting a single axis The formulation of the equations of motion permit the inclusion of thrust vectoring effects The ability to select without program modification the state control and observation vari ables for the linear models combined with the large number of observation quan tities availiable allows any analysis problem to be solved
39. 000000D 00 0 1051860 02 0 34281 7D 02 0 155832D 02 D MATRIX FOR DX DT A X B U D V 0 343642D 07 0 000000D 00 0 7373780 06 0 0000000400 0 000000D 00 0 1131920 06 0 0000000 00 0 2428850 05 0 605694D 05 0 000000D 00 0 0000000 00 0 000000D 00 0 000000D 00 0 000000D 00 0 0000000 00 0 7142030 03 0 3984920 06 0 3328420 04 0 000000D 00 0 0000000 00 79 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 H MATRIX FOR Y F U E y 0 351752D402 0 191245D 05 0 150046D 02 0 640771D 02 0 000000D 00 0 000000D 400 0 150534D 01 0 274248D 09 F MATRIX FOR Y F E 0 4128450401 0 1809780 02 0 2916990 00 0 0000000 00 0 000000D 00 0 0000000 00 E MATRIX FOR H X U E 0 3770370 07 0 000000D 00 0 214132D 04 0 000000D 00 0 000000D 00 0 0000000 00 0 222222D 04 0 000000D400 0 0000000 00 0 0000000 00 0 000000D 00 0 000000D 00 TEST CASE FEISS III I I IOS IIR IOSD A 2 X DIMENSION 4 U DIMENSION 3 Y DIMENSION 2 STATE EQUATION FORMULATION STANDARD OBSERVATION EQUATION FORMULATION STANDARD STATE VARIABLES ALPHA 1266500 01 RADIANS Q 0 000000D 00 RADIANS SECOND THETA 0 161868D 00 RADIANS VEL 0 933232D403 FEET SECOND CONTROL VARIABLES ELEVATOR 0 637734D 01 THROTTLE 0 225092D 00 SPEED BRAKE 0 000000D 00 DYNAMIC INTERACTION VARIABLES X BODY AXIS FORCE Y BODY AXIS FORCE Z BODY AXIS FORCE PITCHING MOMENT
40. 0000D 00 0 000000D 00 0 0000000 00 0 0000000 00 0 203434D 02 0 0000000 00 0 0000000 00 0 000000D 00 0 000000D 00 0 000000D 00 81 APPENDIX H EXAMPLE PRINTER OUTPUT FILES The following listings are the printer output files generated by LINEAR using the example input file in appendix F and the example user supplied subroutines listed in appendix I Example printer output file 1 unit 3 GEOMETRY AND MASS DATA FOR LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE WING AREA 608 000 FT 2 WING SPAN 42 800 FT MEAN CHORD 15 950 FT VEHICLE WEIGHT 45000 000 LB IX 28700 000 SLUG FT 2 IY 165100 000 5106 FT 2 IZ 187900 000 SLUG FT 2 IXZ 520 000 SLUG FT 2 I XY 0 000 SLUG FT 2 IYZ 0 000 5106 FT 2 VECTOR DEFINING REFERENCE POINT OF AERODYNAMIC MODEL WITH RESPECT TO VEHICLE CENTER OF GRAVITY DELTA X 0 000 FT DELTA Y 0 000 FT DELTA Z 0 000 FT FORCE AND MOMENT COEFFICIENT CORRECTIONS DUE TO THE OFFSET OF THE REFERENCE POINT TO THE AERODYNAMIC MODEL FROM THE VEHICLE CG ARE CALCULATED IN CCALC MINIMUM ANGLE OF ATTACK 10 000 DEG MAXIMUM ANGLE OF ATTACK 40 000 DEG PARAMETERS USED IN THE STATE VECTOR FOR LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ALPHA Q THETA VEL THE STANDARD FORMULATION OF THE STATE EQUATION HAS BEEN SELECTED THE FORM OF THE EQUATION IS DX DT A X B U D Y SURFACES TO BE USED FOR THE CONTROL VE
41. 1d 1 NI H T5NV TONWAX d3NI5N3 JHL 40 HLVNIGNOOO SIXV 08 2 AHL OL 54 0453 2 SNISN3 HL I 40 JLWNIGUOOD SIXV JHL OL 54 4534 JNIS5NS HL I 3HL JO S31LWNIGHOOO SIXW AHL OL SGNOdSSHHOD L334 ANIONG S 60 ST1 1861 390 92 0 0 lt 861 3920 92 333 HM 76730 3oN3H343H LiJVHOHIV JHL WOU HL I HHL JO LNS3NS3OV 1dSIG r I LV2O L 541 S3NISNS HL I ZO LSNUHL IWLOL FHL I LSOUHILI 910111 1 SNDOTG NONWOO 5 2 NI 531 IHL YANIT JO 1519 AHL OL 53114405 ANY 10111 NI S318WVIHXVA HHL 40 SINIYA JHL S3LhndWOO ANILNOU SIHL NON34JI ANILNOW d311ddhS UISN AHL NI G3NIJS3G JUV 910111 NI S31NIUVA AHL ANILNOU SIHL 4 5 201 NOWWOO O I SHL 111 910111 SROOTA NOWNOD JHL ZXNWNS bP 2 502 5 09 52 NOISNIWIG 2123ALI NOLSNIWIG 2 NOISN3MWIG E VIMNIV OHNHAD E V NOISN3MWIQ P LSHHLZ P LSHHLA P LSHHLX NOISNIWIA 2JAAWWN p UWALOIVAWHD 2 0 NOISIOS3Hd 31184004 LIOI IdNI H N NOISI23Hd S I8nOGQ LIOI dNIi 5 S3 1HWd S86T 3Nn f NOILINIJSQ S315NW LNS3H3JJIG AOA G3IJIGOM 330n 3371 86 1 NALLIUM S 3NI5N3 JO 51044434 3L dWOD O
42. 522771 GS 0 00000000 GS AN AY VEHICLE STATIC MARGIN IS 3 5 MEAN AERODYNAMIC CHORD STABLE AT THIS FLIGHT CONDITION OI APPENDIX I EXAMPLE USER SUPPLIED SUBROUTINES The following subroutines are examples of user supplied routines that provide the aerodynamic control and engine models to LINEAR These subroutines are based F 15 aircraft simulation and are typical of the routines needed to interface LINEAR to a set of nonlinear simulation models These subroutines are meant to illustrate the use of the named common blocks to communicate between LINEAR and the user s routines These subroutines are used with all examples in this report Included with this report are microfiche listings of these subroutines Aerodynamic Model Subroutines The following two subroutines define a linear aerodynamic model Even though this model is greatly simplified from the typical nonlinear aerodynamic model the example illustrates the functions of the subroutines ADATIN and CCALC SUBROUTINE ADATIN C C EXAMPLE AERODYNAMIC DATA DEFINITION OR INPUT SUBROUTINE C C ROUTINE TO DEFINE STABILITY AND CONTROL DERIVATIVES FOR THE C AERODYNAMIC MODEL C C C COMMON BLOCK TO COMMUNICATE AERODYNAMIC DATA BETWEEN THE C SUBROUTINES ADATIN AND CCALC C COMMON ARODAT CLP CLDR CMQ CMAD CMDE 5
43. 9364 TRIM ROLL AXIS PARAMETER 0 00000 TRIM YAW AXIS PARAMETER 0 00000 TRIM THRUST PARAMETER 0 22509 86 CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE OBSERVATION VARIABLES AN AY 0 06377 0 22509 0 00000 0 98522771 GS 0 00000000 GS NON DIMENSIONAL STABILITY AND CONTROL DERIVATIVES FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ROLLING PITCHING YAWING MOMENT MOMENT MOMENT DRAG ZERO COEFFICIENTS 9 371490 20 4 220400 02 6 115950 19 1 087600 02 ROLL RATE RAD SEC 2 000000 01 0 000000 00 3 372100 02 0 000000 00 PITCH RATE RAD SEC 0 00000D 00 3 895300 00 0 000000 00 0 000000 00 YAW RATE RAD SEC 1 509900 01 0 00000D 00 4 04710D 01 0 000000 00 VELOCITY FT SEC 8 281110 17 1 017770 09 7 42372D 16 0 00000D 00 MACH NUMBER 8 586880 14 1 055350 06 7 69784D 13 0 000000 00 ALPHA RAD 0 00000D 00 1 68820D 01 0 00000D 00 3 72570D 01 BETA RAD 1 334500 01 0 00000D 00 1 299600 01 0 00000D 00 ALTITUDE FT 0 00000D 00 0 00000D 00 0 00000D 00 0 00000D 00 ALPHA DOT RAD SEC 0 00000D 00 1 188700 01 0 00000D 00 0 00000D 00 BETADOT RAD SEC 0 00000D 00 0 000000 00 0 00000D 00 0 00000D 00 ELE VATOR 0 000000 00 6 952800 01 0 00000D 00 4 383100 02 THROTTLE 0 00000D 00 0 00000D 00 0 00000D 00 0 000000 00 SPEED BRAKE 0 00000D 00 4 17500D 01 0 00000D 00 6 49350D 02 VEHICLE STATIC MARGIN IS 3 54 MEAN AERODYNAMIC CHORD ST
44. ABLE AT THIS FLIGHT CONDITION MATRIX A USING THE FORMULATION OF THE STATE EQUATION DX DT A X B U D V FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ALPHA Q THETA VEL TIME DERIVATIVES WRT ALPHA 0 120900D 01 0 100000D 401 0 5757300 02 0 7019750 04 0 0 1491890 01 0 2214510401 0 1896400 01 0 231368 03 0 0000000 00 0 1000000 01 0 0000000 00 0 000000D 00 VEL 0 5768680402 0 0000000 00 0 3162510 02 0 4604350 02 MATRIX B USING THE FORMULATION OF THE STATE EQUATION DX DT B U D FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE LIFT 1 573600 01 0 00000D 00 1 723200 01 0 000000 00 7 18527D 09 7 450580 06 4 870600 00 0 000000 00 0 00000D 00 1 723200401 0 000000 00 5 729600 01 0 00000D 00 3 74920D 02 SIDE FORCE 4 314320 19 0 000000 00 0 000000 00 0 00000D 00 0 00000D 00 0 000000 00 0 00000D 00 9 74030D 01 0 000000 00 0 000000 00 0 000000 00 0 000000 00 0 000000 00 0 000000 00 87 TIME DERIVATIVES WRT ALPHA Q THETA VEL ELEVATOR THROTTLE SPEED BRAKE 0 141961D 00 0 4487420 03 0 928932D 02 0 2207780402 0 147812D 02 0 135074D 02 0 000000D 400 0 000000D 00 0 000000D 400 0 1051860 02 0 343162D402 0 155832D402 MATRIX D USING THE FORMULATION OF THE STATE EQUATION DERIVATIVES WRT TIME OF ALPHA Q THETA VEL D
45. ABLES ELEVATOR 0 05380 THROTTLE 0 21410 SPEED BRAKE DEG 0 00000 89 OBSERVATION VARIABLES AN AY 3 00163339 GS 0 94136286 GS Uu 1 VEHICLE STATIC MARGIN IS 3 5 MEAN AERODYNAMIC CHORD STABLE AT THIS FLIGHT CONDITION TRIM CONDITIONS FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE STRAIGHT AND LEVEL TRIM WHILE VARYING ALPHA TRIM ACHIEVED COEFFICIENT OF LIFT 0 13221 COEFFICIENT OF DRAG 0 00895 LIFT LBS 44376 86258 DRAG LBS 3004 93778 ALTITUDE FT 20000 MACH 0 90000 VELOCITY FT SEC 933 23196 EQUIVALENT AIRSPEED KTS 403 42303 SPEED OF SOUND FT SEC 1036 92440 GRAVITATIONAL ACCEL FT SEC 2 32 11294 NORMAL ACCELERATION 6 5 0 98523 LOAD FACTOR 0 98803 DYNAMIC PRESSURE LB FT 2 552 05302 DENSITY SLUG FT 3 0 00126774 WEIGHT ALTITUDE 185 44914 60434 BETA DEG 0 00000 ALPHA DEG 0 72565 PHI DEG 0 00000 THETA DEG 9 27435 ALTITUDE RATE FT SEC 162 05403 GAMMA DEG 10 00000 ROLL RATE DEG SEC 0 00000 1 DEG SEC 0 00000 YAW RATE DEG SEC 0 00000 THRUST LBS 10804 39673 SUM OF THE SQUARES 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 0 98
46. ATION OF THE STATE EQUATION DX DT THROTTLE 0 16494 8D 02 0 5433240 02 0 000000D 00 0 34281 7D 02 LINEARIZER DEMONSTRATION AND TEST CASES F R THE PROJECT USER S GUIDE SPEED BRAKE 0 9289330 02 0 1350740 02 0 000000D 00 0 155832D402 A X B U D V DERIVATIVES X BODY AXIS FORCE Y BODY AXIS FORCE Z BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT WRT TIME OF ALPHA 0 3436420 07 0 000000D 00 0 7373780 06 0 000000D 00 0 000000D 00 Q 0 113192D 06 0 0000000 00 0 2428850 05 0 6056940 05 0 000000D 00 THETA 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 VEL 0 714203D 03 0 3984 92D 06 0 332842D 04 0 000000D 00 0 000000D 00 DERIVATIVES YAWING MOMENT WRT TIME OF ALPHA 0 000000D 00 Q 0 000000D 00 THETA 0 000000D 00 VEL 0 000000D 00 AN AY AN AY MATRIX H USING THE FORMULATION OF THE OBSERVATION EQUATION Y H X FeU E V FOR CASE 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ALPHA Q THETA VEL 0 3517520402 0 191245 05 0 150046D 02 0 6407710 02 0 000000D 00 0 0000000 00 0 150534D 01 0 274248 09 MATRIX F USING THE FORMULATION OF THE OBSERVATION EQUATION Y F U E 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 2916990 00 0 000000D 00 0 000000D 00 0 0000000 00 85 MATRIX E US
47. AXIS FORCE Z BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT AN 0 1026760 07 0 000000D 00 0 2141160 04 0 0000000400 0 0000000 00 0 000000D 00 0 2222220 04 0 000000D 00 0 000000D 00 0 000000D 00 YAWING MOMENT AN 0 000000D 00 AY 0 000000D 00 n n M a U I M M aaa Example printer output file 2 unit 2 TRIM CONDITIONS FOR CASE 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE LEVEL TURN WHILE VARYING ALPHA TRIM ACHIEVED COEFFICIENT OF LIFT 0 40144 COEFFICIENT OF DRAG 0 03058 LIFT LBS 134741 68807 DRAG LBS 10265 70661 ALTITUDE FT 20000 MACH 0 90000 VELOCITY FT SEC 933 23196 EQUIVALENT AIRSPEED KTS 403 42303 SPEED OF SOUND FT SEC 1036 92440 GRAVITATIONAL ACCEL FT SEC 2 32 11294 NORMAL ACCELERATION 6 5 3 00163 LOAD FACTOR 2 99995 DYNAMIC PRESSURE LBS FT 2 552 05302 DENSITY SLUG FT 3 0 00126774 WEIGHT ALTITUDE LBS 44914 60434 BETA DEG 0 03193 ALPHA DEG 2 66824 PHI DEG 70 62122 THETA DEG 0 91607 ALTITUDE RATE FT SEC 0 00000 GAMMA DEG 0 00000 ROLL RATE DEG SEC 0 08951 PITCH RATE DEG SEC 5 28086 YAW RATE DEG SEC 1 85749 THRUST LBS 10277 03515 SUM OF THE SQUARES 0 00000 TRIM PARAMETERS TRIM PITCH AXIS PARAMETER 0 66958 TRIM ROLL AXIS PARAMETER 0 01526 TRIM YAW AXIS PARAMETER 0 02125 TRIM THRUST PARAMETER 0 21410 CONTROL VARI
48. CTOR FOR LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE LOCATION IN CONTROL ELE VATOR 5 THROTTLE 12 SPEED BRAKE 10 82 PARAMETERS USED IN THE OBSERVATION VECTOR FOR LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE AN AY THE STANDARD FORMULATION OF THE OBSERVATION EQUATION HAS BEEN SELECTED THE FORM OF THE EQUATION IS Y H X F U E LIMITS FOR TRIM OUTPUT PARAMETERS MINIMUM MAXIMUM PITCH AXIS PARAMETER 2 900 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 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 0 40144 COEFFICIENT OF DRAG 0 03058 LIFT LBS 134741 68807 DRAG LBS 10265 70661 ALTITUDE FT 20000 MACH 0 90000 VELOCITY FT SEC 933 23196 EQUIVALENT AIRSPEED KTS 403 42303 SPEED OF SOUND FT SEC 1036 92440 GRAVITATIONAL ACCEL FT SEC 2 32 11294 NORMAL ACCELERATION G S 3 00163 LOAD FACTOR 2 99995 DYNAMIC PRESSURE LBS FT 2 552 05302 DENSITY SLUG FT 3 0 00126774 WEIGHT ALTITUDE LBS 44914 60434 BETA DEG 0 03193 ALPHA DEG 2 66824 PHI DEG 70 62122 THETA DEG 0 91607 ALTITUDE RATE FT SEC 0 00000 GAMMA DEG 0 00000 ROLL RATE DEG SEC 0 08951 PITCH RATE DEG
49. DEP FLON DETRIM DRP FPED DRTRIM DSBP DFP PLAPL PLAPR THSL THSR COMMON CONTRL DC 30 COMMON DATAIN S 8 CBAR AMSS AIX AIY AIZ AIXZ AIXY AIYZ AIXE COMMON SIMOUT AMCH QBAR GMA DEL UB VB WB VEAS VCAS COMMON DRVOUT 13 0 13 EQUIVALENCE T 5 TCI P F 2 Q F 3 R F A V F 5 F 6 7 THA F 8 51 F 9 F 10 H F 11 0 F 12 Y F 13 TOOT DF 1 PDOT DF 2 QDOT DF 3 RDOT DF 4 VDOT DF 5 ALPDOT DF 6 BTADOT DF 7 THADOT DF 8 5100 0 9 PHIDOT DF 10 DF 11 XDOT DF 12 YDOT DF 13 RETURN END 97 APPENDIX J REVISIONS TO MICORFICHE SUPPLEMENT The following listing of subroutine ENGINE incorporates revisions not contained in the microfiche supplement included with this report Program LINEAR and Subrou tines Utilized by LINEAR This listing should be used in place of the subroutine ENGINE listing on the microfiche supplement 98 1 gt 5 2 1 4 XWA 3NI5NS3 HL I JHL JO SIXW S3NISNGd X 40 LN3WON JHL 1 13 SIXW ANIONS AGOG 2 AHL NO GG3LlI23roOHd SIXV 3NI5NS3 GNW S3NW Id SIXW AGOG JHL N33A L3dH S3 IDNV JHL I 15NVZX 530 SN Td SHI OLNO SIXV ANIONS HL I JHL 40 NOIL2S3DONd FHL OL 51 1408 HHL SNV
50. ERIVATIVES WRT TIME OF ALPHA Q THETA VEL AN AY AN 88 DX DT A X B U p y FOR CASE 42 LINEARIZER DEMONSTRATION AND TEST CASES FOR THE PROJECT USER S GUIDE X BODY AXIS FORCE Y BODY AXIS FORCE 7 80 AXIS FORCE PITCHING MOMENT ROLLING MOMENT 0 934880D 08 0 000000D 00 0 7381190 06 0 307941D 07 0 000000D 00 0 243129D 05 0 000000D 00 0 000000D 00 0 000000D 00 0 714920D 03 0 495829D 13 0 905497D 05 YAWING MOMENT 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 0 000000D 00 0 605694D 05 0 000000D 400 0 000000D 00 MATRIX H USING THE FORMULATION OF THE OBSERVATION EQUATION Y H X F U E JY FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ALPHA Q THETA 0 3504240402 0 1283330 06 0 632324D 02 0 000000D 00 0 000000D 00 0 000000D 00 VEL 0 203434D 02 0 000000D 00 MATRIX F USING THE FORMULATION OF THE OBSERVATION EQUATION Y H X F U E y FOR CASE 2 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ELEVATOR THROTTLE SPEED BRAKE 0 411323D 01 0 4928450 03 0 2632880 00 0 000000D 00 0 000000D 00 0 0000000 00 0 0000000 00 0 000000D 00 0 000000D 00 0 0000000 00 MATRIX E USING THE FORMULATION OF THE OBSERVATION EQUATION Y H X F U E FOR CASE 2 LINEARIZER DEMONSTRATION AND TEST CASES FOR THE PROJECT USER S GUIDE X BODY AXIS FORCE Y BODY
51. ES OF STICK AND PEDAL TO DEGREES OF SURFACE DEFLECTION c0 DA DAS 20 0 DE DES 25 0 DR DRS 30 0 Wn gt 0 43 25 SET DIFFERENTIAL TAIL BASED ON AILERON COMMAND DT DA 4 0 e CONVERT THRUST TRIM PARAMETER TO PERCENT THROTTLE COMMAND OOO O 0 0 IF THRSTX GE 0 0 THR C USE SPEED BRAKE IF NEEDED C DB 0 0 IF THRSTX LT 0 0 DSB THRSTX 45 0 C CONVERT SURFACE COMMANDS TO RADIANS C DA DA DGR DE DE DGR DR DR DGR DT DT DGR DSB DSB DGR RETURN END Mass and Geometry Model Subroutine The following subroutine MASGEO is an example of the mass and geometry sub routine that the user must provide to LINEAR If the mass and geometry character istics of the aircraft do not change or can be easily specified as a function of flight condition and surface setting vehicle configuration then the mass and geometry characteristics of the vehicle can be defined using the input file and MASGEO would be a dummy subroutine as is the following example However for an aircraft in which these mass and geometry properties are complicated or already defined using FORTRAN subroutines MASGEO provides an interface between LINEAR and those subroutines 96 c 0 c SUBROUTINE MASGEO SUBROUTINE TO COMPUTE THE MASS AND GEOMETRY PROPERTIES OF THE AIRCRAFT COMMON CONPOS DAP FLAT DATRIM
52. Eugene L Duke Robert F Antoniewicz and Keith D Krambeer in preparation The following equations for rotational acceleration are used for analysis point definition 1 211 IygI2 0213 Pr IxyIi DyI2 Iyg13 r2 Iyz11 IxzI2 det I 15 q EL Ip ZM I4 IN Is 2 1 214 IxyI5 Pq IxzI2 IyzI4 0 15 pr IxyI2 DyI4 IyzIs q2 IyzI2 IxyI5 12 IxyI4 1 215 E 125215 _ IxzI4 det I r ZL I3 XM Is XN Ig p2 I xzl5 IxyIg d IyzI3 IxyIg qr DyI3 15 1 216 r2 IyzI3 IxzI5 det I where det I 11 12 T3 14 5 Ig 2 2 IxIyIz 2IxyIxzIyz Ixlyz EZ IyIyz IzIxy 2 Ty IxyIz IyzIxz 2 Iylz Ixz IxIyz Ixylxz 2 Ix Iz Here the body axis rates are designated p q and The total moments about the x y and z body axes rol ling pitching and yawing moments are designated 21 and IN respectively These total moments are the sums of all aerodynamic moments and powerplant induced moments due to thrust asymmetries and gyroscopic torques The equations used to determine the change in moment coefficients due to the noncoincidence of the vehicle center of gravity and the reference point of the nonlinear aerodynamic model pitch rate and yaw rate 16 2 r corresponding to roll rate are derived in appendix A The equations defining the engine torque and gyro
53. FORCE Body axis parameters ae es Se x body axis velocity length sec u UB X BODY AXIS VELOCITY X BODY AXIS VELOCITY AXIS VEL X BODY AXIS VEL U BODY U BODY IT MM M Observation variable y body axis velocity z body axis velocity Rate of change of velocity in x body axis Rate of change of velocity in y body axis Rate of change of velocity in z body axis H length sec length sec length sec length sec2 Length sec2 Body axis parameters continued V ce qe ze Symbol VB Y BODY Y BODY Y BODY Y BODY V BODY V BODY WB Z BODY Z BODY Z BODY Z BODY W BODY W BODY UBDOT UB DOT VBDOT VB DOT WBDOT WB DOT Alphanumeric AXIS AXIS AXIS AXIS AXIS AXIS AXIS AXIS descriptor VELOCITY VELOCITY VEL VEL VELOCITY VELOCITY VEL VEL Miscellaneous measurements not at vehicle center of gravity MM Se a sas Angle of attack not at vehicle center of gravity Angle of not at center Altitude not at center Altitude sideslip vehicle of gravity instrument vehicle of gravity rate instru ment not at vehicle center of gravity rad rad length length sec Bi r 12
54. ING THE FORMULATION OF THE OBSERVATION EQUATION Y H X F U E FOR CASE 1 LINEARIZER DEMONSTRATION AND TEST CASES FOR THE PROJECT USER S GUIDE X BODY AXIS FORCE Y BODY AXIS FORCE Z BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT AN 0 3770370 07 AY 0 000000D400 0 000000D 00 0 222222 04 0 214132D 04 0 000000D 00 0 000000D 00 0 0000000 00 0 000000D 00 0 000000D 00 YAWING MOMENT AN 0 000000D 00 AY 0 000000D 00 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 0 13221 COEFFICIENT OF DRAG 0 00895 LIFT LBS 44376 86258 DRAG LBS 3004 93778 ALTITUDE FT 20000 MACH 0 90000 VELOCITY FT SEC 933 23196 EQUIVALENT AIRSPEED KTS 403 42303 SPEED OF SOUND FT SEC 1036 92440 GRAVITATIONAL ACCEL FT SEC 2 32 11294 NORMAL ACCELERATION 0 5 0 98523 LOAD FACTOR 0 98803 DYNAMIC PRESSURE LB FT 2 552 05302 DENSITY SLUG FT 3 0 00126774 WEIGHT ALTITUDE 185 44914 60434 BETA DEG 0 00000 ALPHA DEG 0 72565 PHI DEG 0 00000 THETA DEG 9 27435 ALTITUDE RATE FT SEC 162 05403 GAMMA DEG 10 00000 ROLL RATE DEG SEC 0 00000 PITCH RATE DEG SEC 0 00000 YAW RATE DEG SEC 0 00000 THRUST LBS 10804 39673 SUM OF THE SQUARES 0 00000 TRIM PARAMETERS TRIM PITCH AXIS PARAMETER 0 7
55. ITY DERIVATIVES WITH RESPECT TO ANGLE OF C ATTACK PITCH RATE ANGLE OF ATTACK C CLFTO 1 5736 E 01 C CLF TA 4 8706 CLFTQ 1 7232 E 01 CLFTAD 1 7232 01 C C CONTROL DERIVATIVES WITH RESPECT TO C ELEVATOR SPEED BRAKE C CLFTDE 5 7296 01 CLFTSB 3 7492 E 0 C C SIDEFORCE COEFFICIENT DERIVATIVES C C STABILITY DERIVATIVES WITH RESPECT TO C SIDESLIP 4 4 C 9 7403 E 01 C C CONTROL DERIVATIVES WITH RESPECT TO C AILERON RUDDER DIFFENTIAL TAIL C CYDA 1 1516 E 03 CYDR 1 5041 E 01 CYDT 7 9315 E 02 C RETURN END C gt SUBROUTINE CCALC EXAMPLE AERODYNAMIC MODEL ROUTINE TO CALCULATE THE AERODYNAMIC FORCE AND MOMENT COEFFICIENTS COMMON BLOCKS CONTAINING STATE CONTROL AND AIR DATA PARAMETERS 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 BAR GMA DEL UB VB WB VEAS VCAS COMMON CGSHFT DELX DELY DELZ C C COMMON BLOCK TO OUTPUT AERODYNAMIC FORCE AND MOMENT C COEFF IC IENTS C COMMON CLCOUT CL CN CD CLFT C C COMMON BLOCK TO COMMUNICATE AERODYNAMIC DATA BETWEEN C THE SUBROUTINES ADATIN AND CCALC C COMMON ARODAT CLP CLDA CLDT CMO CMQ CMAD CMDE CMSB CB
56. L ANILNOU 3NILnOWNGS UUUOUUUUU 600 9500 4600 FS00 S00 2800 1500 06500 6900 8900 1900 9700 stoo 00 00 2900 6 00 8600 1600 9600 58600 00 00 2600 00 0 lt 00 6200 8200 200 9200 5200 200 200 2200 TZ00 0200 6100 8100 100 9100 STOO TOO ETOO 2100 TT00 0100 6000 8000 000 9000 5000 t000 000 2000 1000 ebed 99 T HO4 INIONS NOLNW Tvasas ILC L A XWA l b a t 14 Oy 2 ja d S 02 02 23 5 0 9 OZT SOduva 580416 071 DAAD01 DAAWON 1 213a aiT13a x 3d SYDDA SIAM gA gn T1390 YHO Uvad AD LATS 02 NOD HW5 IS VHLSOO WHLINIS ISdSOD ISdNIS IHdSO2D IHdNIS VLS8SO O WLSNIS d IVSO O d IXNIS aqvotw so WNV WWV WIV HNOJdIS 5WVHGQ L4 lLl 3XIV ZAIWV XXIV ZXIV ZIXV AIX XINV SSWV WHN82 8 S NV ZNV ANV XNWV ZW XN XVX d L SJONAIWAINGT tt0111 OVOTITS STRPOTTIS 100111 L00 111 6 0111 800111 50111 820 111 6TOTTIS 210111 8 111 NOWNOD NOWWOD NOWWOD NOWWO2 NOWNOD NONWO2D NOWWO2D NOWWOSD NOWNOD NONNWO2 NOWWOS NOWWOD 540415 1901 TOT111
57. LE IN THE USER SUPPLIED SUBROUTINES USING THE OBVEC ARRAY Location index in OBVEC n Q i 25 26 27 28 29 30 31 32 33 M n h n p i ss F a EE PR i Variable State Variables Roll rate Pitch rate Yaw rate Velocity Angle of attack Angle of sideslip Pitch attitude Heading Roll attitude Altitude Displacement north Displacement east Derivatives of state variables Y Roll acceleration Pitch acceleration Yaw acceleration Velocity rate Angle of attack rate Angle of sideslip rate Pitch attitude rate Heading rate Roll attitude rate Altitude rate Velocity north Velocity east Accelerations x body axis acceleration y body axis acceleration z body axis acceleration x body axis accelerometer center of gravity y body axis accelerometer center of gravity z body axis accelerometer center of gravity Normal acceleration x body axis accelerometer center of gravity y body axis accelerometer center of gravity M M M M M M Fs P R HM pn Vq at vehicle at vehicle at vehicle not at vehicle not at vehlcle 51 TABLE 4 Continued Location index in OBVEC Variab
58. NASA Technical Paper 2768 December 1987 User s Manual for LINEAR a FORTRAN Program to Derive Linear Aircraft Models Eugene L Duke Brian P Patterson and Robert F Antoniewicz NASA NASA Technical Paper 2 68 1987 User s Manual for LINEAR a FORTRAN Program to Derive Linear Aircraft Models Eugene L Duke Brian P Patterson and Robert F Antoniewicz Ames Research Center Dryden Flight Research Facility Edwards California NASA National Aeronautics and Space Administration Scientific and Technical Information Division SUMMARY e INTRODUCTION gt e CONTENTS NOMENC LATURE Lj L e e Var tables s e Superscripts s lt lt e o o e l Subscripts e e e e e e e e e e ef FORTRAN Variables e e e PROGRAM OVERVIEW EQUATIONS OF MOTION OBSERVATION EQUATIONS SELECTION OF STATE CONTROL LINEAR MODELS e ANALYSIS POINT DEFINITION Untr immed e e s e o Straight and Level Trim Pushover Pullup eee o 9 e Level Turn e e e o Thrust Stabilized Turn s e e gt e o n Beta Trim e o Specific Pow
59. NVNS I 5WO5N3 1 5 x I ZXNVSO I 5WO5NS3 I AXNVSO I ZXNVSO x I 5WO5N3 SINI NI 40 W LNSWONW UYIN I 2UI23AL I LISUHLX I I 23123AL I LSURLA 3nOWuOL I I 2TIOSALs I LSHHIZ I 2T23AL I LSHHLX 4 2 12 1 15 5 42 T anduoL I ZXNVNS x 1 5 14 I LYDOTL 1 5 x I ZXNWSO I SuHLxqa C I LV2OIL I AXNVSO ID ZXNVSO x I SMHHLXG T I Lwoo0 ts ONIWOLIIA LSNYHL ANY 135440 AJNIONG OL ZNA 51223443 IVNOILVIOH 104 2 I LSHUHLZ I LSYHLA I LSHUVHLX 60 6 861 320 9 0 01 ST L 861 320 97 p E E VINNIVM E NLMNNIVXM T E NIUNIVX 2 NIHNIV Z Z NLMNIVX T Z NLHUNIM T WLYNIW Z T WLUNIWV T T WLYUNIV V HHL Lnogy JHI 23 Z53WO ADS3NO P T I od 3nNILNOO t 3n WNHOL 2z 3nO uoL I1 3nO uol I 27123AL C I 27123AL I I 2123AL P I I anduoL Z 9nd oL 1 0 4 1 SNITHG3LN32 Q Q CO JANILNOD 7 244 Add 244 Add 8220 1220 9220 5220 220 20 2220 1220 0220 6120 8 20 1120 9120 58120 120 TZ0 2120 20 orco 6020 8020 LOZO 9020 5020 Poco
60. ROTTLE 0 00000D 00 0 00000D 00 0 00000D 00 0 00000D 00 SPEED BRAKE 0 000000 00 4 17500D 01 0 00000D 00 6 49350D 02 VEHICLE STATIC MARGIN IS 3 5 MEAN AERODYNAMIC CHORD STABLE AT THIS FLIGHT CONDITION MATRIX A USING THE FORMULATION OF THE STATE EQUATION DX DT A X B U D V FOR CASE 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ALPHA Q THETA VEL DERIVATIVES WRT TIME OF ALPHA 0 1214360 01 0 100000D 01 0 136756D 02 0 121605D 03 Q 0 147423D 01 0 221451D401 0 450462D 02 0 29401 9D 03 THETA 0 000000D 00 0 331812D 00 0 000000D 00 0 000000D 00 VEL 0 7908530 02 0 0000000 00 0 3208220402 0 1572970 01 MATRIX B USING THE FORMULATION OF THE STATE EQUATION DX DT FOR CASE 1 A X B U D V LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE 84 LIFT 1 573600 01 0 00000D 00 1 723200401 0 00000D 00 1 45433D 05 1 508030 02 4 87060D 400 0 00000D 00 0 000000 00 1 723200401 0 000000 00 5 729600 01 0 00000D 00 3 749200 02 SIDE FORCE 5 428820 04 0 00000D 00 0 00000D 00 0 00000D 00 0 00000D 00 0 00000D 00 0 00000D 00 9 74030D 01 0 00000D 00 0 00000D 00 0 00000D 00 0 00000D 00 0 00000D 00 0 00000D 00 DERIVATIVES WRT TIME OF ALPHA Q THETA VEL FOR CASE 1 ELEVATOR 0 1419610 00 0 2207780402 0 000000D 00 0 1051860 02 MATRIX D USING THE FORMUL
61. S 2 1 3 qe sin 6 cos amp sin i 8112 sin e Thus the gyroscopic moment induced by the ith engine Tg be expanded to i qe rhe 2 Phe m Ge 63 and the total moment induced by gyroscopic interaction of the vehicle dynamics and the rotating engine components is n u 2 tg 1 1 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 64 APPENDIX C STATE VARIABLE NAMES RECOGNIZED BY LINEAR This appendix lists the alphanumeric descriptors specifying state variables that are recognized by LINEAR In the input file the field containing these descriptors uses a 54A format and all characters are left justified The input alphanumeric descriptor specified by the user serves both to identify the state variable selected by the user within the program itself and to identify state variables on the printed output of LINEAR as described in the Output Files section FE SL O R DS a _
62. SB CN CNB CNDA CNDR CNDT CNP CNR CY CYB CYDA CYDR DAS DC DELX DELY DELZ DES 10 coefficient of rolling moment due to yaw rate total coefficient of pitching moment coefficient of pitching moment due to angle of attack coefficient of pitching moment due to angle of attack rate coefficient of pitching moment due to symmetric elevator deflection pitching moment coefficient at zero angle of attack coefficient of pitching moment due to pitch rate coefficient of pitching moment due to speed brake deflection total coefficient of yawing moment coefficient of yawing moment due to sideslip coefficient of yawing moment due to aileron deflection coefficient of yawing moment due to rudder deflection coefficient of yawing moment due to differential elevator deflection coefficient of yawing moment due to roll rate coefficient of yawing moment due to yaw rate total coefficient of sideforce coefficient of sideforce due to sideslip coefficient of sideforce due to aileron deflection coefficient of sideforce due to rudder deflection coefficient of sideforce due to differential elevator deflection longitudinal trim parameter surface deflection and thrust control array displacement of the aerodynamic reference along the x body axis from the center of gravity displacement of the aerodynamic reference along the y body axis from the center of gravity displacement of the aerodynamic referenc
63. SEC 5 28086 YAW RATE DEG SEC 1 85749 THRUST LBS 10277 03515 SUM OF THE SQUARES 0 00000 TRIM PARAMETERS TRIM PITCH AXIS PARAMETER 0 66958 TRIM ROLL AXIS PARAMETER 0 01526 TRIM YAW AXIS PARAMETER 0 02125 TRIM THRUST PARAMETER 0 21410 83 CONTROL VARIABLES ELEVATOR THROTTLE SPEED BRAKE OBSERVATION VARIABLES AN AY teu ou 0 05380 0 21410 0 00000 3 00163 0 94136 339 GS 286 GS NON DIMENSIONAL STABILITY AND CONTROL DERIVATIVES FOR CASE 1 LINEARIZER TEST AND DEMONSTRATION CASES FOR THE PROJECT USER S GUIDE ROLLING PITCHING YAWING MOMENT MOMENT MOMENT DRAG ZERO COEFFICIENTS 4 029660 05 4 220400 02 2 257470 04 1 428820 04 ROLL RATE RAD SEC 2 000000 01 0 00000D 00 3 37210D 02 0 00000D 00 PITCH RATE RAD SEC 0 00000D 00 3 89530D 00 0 000000 00 0 00000D 00 YAW RATE RAD SEC 1 50990D 01 0 00000D 00 4 04710D 01 0 00000D 00 VELOC ITY FT SEC 1 27955D 07 3 287390 06 3 210960 07 0 00000D 00 MACH NUMBER 1 32680D 04 3 408780 03 3 32952D 04 0 00000D 00 ALPHA RAD 0 000000400 1 688200 01 0 00000D 00 3 72570D 01 BETA RAD 1 334500 01 0 00000D 00 1 29960D 01 0 00000D 400 ALTITUDE FT 0 00000D 00 0 00000D 00 0 00000D 00 0 00000D 00 ALPHA DOT RAD SEC 0 000000400 1 188700 01 0 00000D 00 0 00000D 00 BETA DOT RAD SEC 0 000000 00 0 00000D 00 0 00000D 00 0 00000D 00 ELEVATOR 0 00000D400 6 95280D 01 0 00000D 00 4 38310D 02 TH
64. T time rate of change of north south position XYANGL orientation of engine axis in x y body axis plane XZANGL orientation of engine axis in x z 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 vari able and remaining state variables to satisify 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 observa tion 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 rigid aircraft flying in a stationary atmosphere over a flat nonrotating earth Thus the state vector x is computed internally as x p
65. VARIABLE NAMES RECOGNIZED BY LINEAR e e o o o 65 APPENDIX D OBSERVATION VARIABLE NAMES RECOGNIZED BY LINEAR e e 66 APPENDIX E ANALYSIS POINT DEFINITION IDENTIFIERS 4 e o o 73 APPENDIX F EXAMPLE INPUT FILE 4 9 4 9 9 e o 75 APPENDIX EXAMPLE OUTPUT ANALYSIS oo o o 79 APPENDIX H EXAMPLE PRINTER OUTPUT FILES 9 o o 82 APPENDIX I EXAMPLE USER SUPPLIED SUBROUTINES e do e 91 Aerodynamic Model Subroutine ee 4 6 6 6 o o 91 Engine Model Interface Subroutine e e e e 94 Control Model Subroutine s e s esos e 2 95 Mass and Geometry Model Subroutine 4 96 APPENDIX J REVISIONS TO MICROFICHE SUPPLEMENT 0 98 REFERENCES e e e e e 1 04 SUMMARY This report documents a FORTRAN program that provides a powerful and flexible tool for the linearization of aircraft models The program LINEAR numerically determines a linear system model using nonlinear equations of motion and a user supplied nonlinear aerodynamic model The system model determined by LINEAR con sists o
66. ach 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 non wings level horizontal flight with heading rate 7 0 at a user specified Mach number altitude and angle of sideslip This trim option is nominally at 1 g but as varies from zero normal acceleration decreases and lateral acceleration increases For an aero dynamically symmetric aircraft a trim to B O using the beta trim option results in the same trimmed condition as the straight and level trim However for an aero dynamically asymmetric aircraft such as an oblique wing vehicle the two trim options are not equivalent 30 The constraint equations used with the beta trim option are The trim surface positions thrust angle of attack and bank angle are determined by numerically solving the nonlinear equations for translational and rotational acceleration Pitch attitude 0 is derived from the equation for flightpath angle Y with Y O eo sin B sin cos sin Specific Power The specific power analysis point definitio
67. ame indicates 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 location in that order of the sensor These offsets from the vehicle center of gravity are defined in units of length Observation variable Units Symbol Alphanumeric descriptor Derivatives of state variables Roll acceleration rad sec2 p PDOT ROLL ACCELERATION Pitch acceleration rad sec2 q QDOT PITCH ACCELERATION Yaw acceleration rad sec r RDOT YAW ACCELERATION lt je VDOT VELOCITY RATE Velocity rate length sec2 Re ALPDOT ALPHA DOT ALPHADOT Angle of attack rate rad sec Angle of sideslip rate rad sec BTADOT BETA DOT BETADOT Pitch attitude rate rad sec 8 THADOT THETA DOT lt PSIDOT PSI DOT Heading rate rad sec 66 A A ES TS I r mHe Pa M P e Observation variable Units Symbol Alphanumeric descriptor Po SD ER i i tl e e a ite m rO Derivatives of state variables continued M M amp a
68. 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 indicating that for trim all the time derivatives of the state vari ables must be zero This is not the case however Trim is achieved when the acceleration like terms are identically zero no constraints need to be placed on the velocity like terms in x Thus for the model used in LINEAR only 4 8 must be zero to satisfy the trim condition The trim condition is achieved for the straight and level pushover pullup level turn thrust stabilized turn and beta trim options described in the following sections In general the no trim and specific power analysis point definition options do not result in a trim condition Of these analysis point options resulting in a trim condition only the straight and level and level turn options force the model to represe
69. and Thelander 1965 this rotation is a similarity transformation that yields a new inertia tensor lei such that 1 1 where Rr and Re are axis transformation matrices that perform the previously described rotations through and respectively These matrices are given as COS j 0 Re i 0 1 0 Sin i 0 COS i cos sin amp i 0 R 61 0 0 0 0 so that cos cos j sin 51 cos sin j Re Res sin cos j sin sin j sin i 0 COS i Because m T i Rei 1 RE and the matrices are unitary Rei RE Res RE RE Re 62 cos cos sin cos j sin j T as sin 51 cos 0 cos sin i sin i sin COS j Therefore cos j cos cos j amp sin Ei cos sin cos j le cos i Cos 5 sin i sin Ei cos i sin cos sin i cos sin j cos sin cos sin 51 sin i The angular momentum of the ith engine he can now be expressed as i ITe We he he he i ei 1 with 2 d E Pe cos cos 7 cos cos sin si 2 E 1xe COS Sin j COS Pe cos sin i i e 1x sin 6 cos 1 1 re 1 sin cos sin zl i ei 2 he Pej COS sin j CO
70. at the trim condition are also printed If trim was not achieved r V and 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 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 matri ces for the given flight condition The formulation of the state equations and only the terms of the matrices chosen by the user to define the model are printed A subset of this output file containing only the trim conditions is assigned to FORTRAN device unit 2 The third output file which is assigned to FORTRAN device unit 2 contains the trim conditions of the aircraft at the point of interest These conditions inelude the type of trim being attempted whether trim was achieved parameters defining the trim condition and the static margin of the aircraft at the given flight condition Appendix H presents an example of this file USER SUPPLIED SUBROUTINES There are five subroutines that must be supplied by the user to interface LINEAR with a specific aircraft s subsystem models ADATIN CCALC IFENGN UCNTRL and MASGEO first two subroutines constitute the aerodynamic model
71. ate control and observation variable definitions The first record of this set defines the number of states to be used 38 NUMSAT the formulation of the output equation EQUAT and whether the nondimen sional 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 NONS TANDARD NON STANDARD GENERALIZED EXTENDED If EQUAT matches the first four characters of any of the listed words the out put formulation of the state equation is Cx A x Btu If EQUAT is read in as STANDARD or does not match the preceding list then the default standard bilinear formulation of the state equation is assumed and the output matrices are consistent with the equation x Ax Bu The variable STAB is also read using an A4 format and is compared with the following list DEGREES DGR If STAB matches the first four characters of either of these words the nondimen sional stability derivatives with respect to angle of attack and angle of sideslip are printed in units of reciprocal degress on the printer file Otherwise these derivatives are printed in units of reciprocal radians The remaining records of the state variable definition set are used to specify the state variables to be used in the output formulation of the linearized system and the increments to be used for the
72. ause of the implementation of the subroutine UCNTRL discussed in app I thrust trim parameter is used again because of the implementation of UCNTRL to schedule speed brake when THRSTX lt O and to command thrust when THRSTX gt O Record 21 specifies that no additional control surfaces are to be set The next seven records 22 to 28 define an analysis point option These records request a level turn trim option at h 20 000 ft M 0 9 an 3 0 g 0 The second record of this set record 23 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 anal ysis point option definition set contains the key word NEXT to indicate both an end to the current analysis point option definition and that another analysis point option definition follows The final six records records 29 to 34 define a straight and level analysis point option at h 20 000 ft M 0 9 Y 10 0 77 The second record of this set record 30 identifies the Alpha trim suboption in which angle of attack is varied until trim is achieved at the specified condition The final record of this set contains the key word END to indicate the termination of the current analysis point definition as well as the termination of input cases 78 APPENDIX G EXAMPLE OUTPUT ANALYSIS FILE The following listing is a
73. be applied to any aircraft in any phase of flight except hover The unique trimming capability implemented through a user supplied subroutine allows unlimited possibilities of trimming strategies and sur face scheduling which are particularly important for oblique wing vehicles and aircraft having multiple surfaces affecting a single axis The formulation of the equations of motion permit the inclusion of thrust vectoring effects The ability to select without program modification the State control and observation variables for the linear models combined with the large number of observation quantities available allows any analysis problem to be solved with ease This report documents the use of the program LINEAR defining the equations used and the methods employed to implement the program The trimming capabilities of LINEAR are discussed from both theoretical and implementation perspectives The input and output files are described in detail The user supplied subroutines required for LINEAR are discussed and sample subroutines are presented National Aeronautics and Space Administration Ames Research Center Dryden Flight Research Facility Edwards California March 6 1985 54 APPENDIX A CORRECTION TO AERODYNAMIC COEFFICIENTS FOR A CENTER OF GRAVITY NOT AT THE AERODYNAMIC REFERENCE POINT The point on the vehicle at which the nonlinear force and moment coefficients are defined is referred to as the aerodynamic reference point
74. center of gravity from the ith engine ATo are somewhat more complicated Figure 7 schematically represents an engine with thrust vectoring whose center of gravity is located at Ar relative to the vehicle s center of gravity Thrust point Engine y center TP 2Tp ZE of gravity Figure 7 Detailed definition of thrust vectoring parameters The thrust is assummed to act at Axp in the local engine x axis with the engine center of gravity being the origin of this local coordinate system The thrust is also assumed to be vectored at angles n and relative to the local coordinate axes with n being the angle from the thrust vector to the engine x y plane and Oj the angle from the projection of the thrust vector onto the engine x y plane to the local x axis Thus letting Foy Fpy and represent the and 2 thrust 1 1 L components in the local engine coordinate system respectively these terms can be defined in terms of the total thrust for the ith engine Fpi and the angles nj and as Fox cos nj cos Gi Fpy cos n sin Fp Sin n 59 where t T Fez To transform this equation from the transformation matrix COS cos 51 6 Sin i is used The resultant force in body axis sin sin i cos sin i 0 cos coordinates is the ith engine axis system to the body axis system
75. coefficient of viscosity are derived from the U S Standard Atmosphere 1962 Also included in the air data calculations are two velocities equivalent airspeed Ve and calibrated airspeed Vc both computed in knots The calculations assume that internal units are in the English system The equation used for equivalent airspeed is 20 Ve 17 17 yq 1b ft2 which is derived from the definition of equivalent airspeed 2a Ve 00 where 0 002378 slug ft3 and V is converted from feet per second to knots Calibrated airspeed is derived from the following definition of impact pressure 1 0 0 v2 lt Po 7202 1 as q Po 2 Po d o9 For the case where lt a the equation for is 0 Calibrated airspeed is found using an iterative process for the case where gt ap Vo 582 95174 is executed until the change in Ve from one iteration to the next is less than 0 001 knots Also included in the observation variables are the flightpath related parameters described in app D including flightpath angle y flightpath acceleration fpa vertical acceleration h flightpath angle rate Y and for lack of a better category in which to place it scaled altitude rate h 57 3 The equations used to determine these quantities are T m He 2 lt gt fpa sin ay sin cos 9 cos cos 9 21 Vh hy v Vv2 n2
76. d by the matrices that describe each case The titles records appear only at the beginning of the file all other records are duplicated for each subsequent case calculated in LINEAR The matrices are Written row wise five columns at a time as illustrated in the following tabulation which shows a system containing 7 states 3 controls and 11 outputs using the general state equation and standard observation equation Size of matrix Output formulation A 7 x 7 A 7 x 5 7 x 2 B 7 x 3 B 7x 3 D 7 x 6 D 7 x 5 7 x 1 C 7 x 7 7 x 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 s Aw s F a a n e Un U U 44 The second output file which 15 assigned to FORTRAN device unit 3 contains the values calculated in LINEAR describing each case The first section of this file contains a listing of the input data defining the aircraft s geometry and mass prop erties variable names defining the state space model and various control surface limits characteristic of the given aircraft Appendix H presents an example of this output file The second section of this file contains the trim conditions of the 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
77. d observation variables Output model Internal parameters parameters S lt Specification of state vector for linear model lt x TE DOOEWk lt x 206 Figure 1 Selection of state vari ables for linear model 13 The linear model derived by LINEAR is determined at a specific analysis point LINEAR allows this analysis point to be defined as a true steady state condition on a specified trajectory a point at which the rotational and translational accel erations are zero or as a totally arbitrary state on a trajectory LINEAR provides the user with several options described in detail in the Analysis Point Definition section These analysis point definition options allow the user to trim the air craft in wings level flight pushovers pullups level turns or zero sideslip ma neuvers 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 perturbation 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 user can select state equation matrices corre sponding to either the standard formulation of the state equation x Ax Bu or the generalized equation Cx A x B u The observation matrices can be selected from either of
78. d their time derivatives are required for feedback the user may access them using the common block OBSERV described in the Mass and Geometry Model section of this report Engine Model The subroutine IFENGN computes individual engine parameters used by LINEAR to calculate force torque and gyroscopic effects due to engine offset from the cen terline The parameters for each engine maximum of four engines are passed through common ENGSTF as follows COMMON ENGSTF THRUST 4 TLOCAT 4 3 XYANGL 4 XZANGL 4 TVANXY 4 TVANXZ 4 DXTHRS 4 EIX 4 AMSENG 4 ENGOMG 4 The variables in this common block correspond to thrust THRUST the x y and z body axis coordinates of the point at which thrust acts TLOCAT the orientation of the thrust vector in the x y body axis plane XYANGL in degrees the orientation of the thrust vector in the x z body axis plane XZANGL in degrees the orienta tion of the thrust vector in the x y engine axis plane TVANXY in degrees the orientation of the thrust vector in the x z engine axis plane TVANXZ in degrees the distance between the center of gravity of the engine and the thrust point DXTHRS measured positive in the negative x engine axis the rotational inertia of 49 the engine EIX mass AMSENG and the rotational velocity of the engine ENGOMG The variables are all dimensioned to accommodate up to four engines While the common block structure within LINEAR is designed
79. designated NOTRIM and in selecting this option the user must specify all nonzero state and control variables For the equilibrium conditions the user specifies a minimum number of parameters and the program numerically determines required state and control variables to force the rotational and translational accelerations to zero The analysis point options are described in detail in the following sections For all the analysis point definition options any state or control parameter may be input by the user Those state variables not required to define the analysis point are used as initial estimates for the calculation of the state and control conditions that result in zero rotational and translational accelerations As each state variable is read into LINEAR the name is compared to the list of alternative state variables names listed in appendix C All state variables except velocity must be specified according to this list Velocity can also be defined by specify ing Mach number see alternative observation variable names in app D Appendix E lists analysis point definition identifiers that are recognized by LINEAR It should be noted that the option of allowing the user to linearize the system equations about a nonequilibrium condition raises theoretical issues beyond the scope of this report of which the potential user should be aware Although all the analysis point definition options provided in LINEAR have been found to be useful in the
80. e along the z body axis from the center of gravity lateral trim parameter DRS DXTHRS EIX ENGOMG GMA H HDOT P PDOT PHI PHIDOT PSI PSIDOT Q QBAR RDOT T TDOT THA THADOT THRSTX THRUST TLOCAT directional trim parameter distance between the center of gravity of thrust point rotational inertia of the engine rotational velocity of the engine flightpath angle altitude time rate of change 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 of thrust trim parameter altitude roll rate roll angle heading angle pitch rate yaw rate time pitch angle thrust generated by each engine location of the engine in the x the engine and the 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 V velocity VB velocity along the y body axis VCAS calibrated airspeed VDOT time rate of change of total vehicle velocity VEAS equivalent airspeed WB velocity along the 2 body axis position north from arbitrary reference point XDO
81. e first record of this set of additional surface specifications defines the number of additional controls to be included in the list of recognized control names NUMXSR purpose of defining these additional controls is to allow the user to set such variables as landing gear position wing sweep or flap position Only the controls are defined in the additional surface specification records actual control 41 positions are defined in the analysis point definition records Because there may be no additional controls these secondary trim parameter specification records may not be present If such controls are defined the records defining them will have the format specified in table 1 control variable name ADDITIONAL SURFACE 1 tion LOCCNT in the common block CONTRL and the units associated with this con trol variable CONVR are specified for each additional control Test Case Specification The test case specification records allow the user to define the flight condi tion or analysis point at which a linear model is to be derived Multiple cases can be included in the test case specification records The final record in each set directs the program to proceed NEXT or to stop END execution The first record of a test case specification set determines the analysis point or trim option to be used for the current case The ANALYSIS POINT DEFINITION OPTION parameter is read in and checked against the list of a
82. ed The final set of records in the state control and observation variable defini tions pertain to the specification of parameters associated with the observation vector observation equation and observation parameters The first record of this set defines the number of observation variables NUMYVC to be used in the output linear model and the formulation of the output equation EQUAT The remaining records in this set specify the variables to be included in the observation vector MEASUREMENT and any position information PARAM that may be required to compute the output model for a sensor not located at the vehicle center of gravity The variable used to specify the formulation of the observation equation EQUAT is compared with the same list of names used to determine the formulation of the state equation If it is determined that the generalized formulation is desired the observation equation e 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 for mulation of the linear model contain a variable that includes the parameter name MEASUREMENT and three fields PARAM defining when appropriate the location of 40 the sensor relative to the vehicle center of gravity The parameter name is com pared with the list of observation variables given in appendix D If the pa
83. ed by a the speed of sound to obtain a reasonable perturbation size From the generalized nonlinear state Tx f x x u and observation equations g x u the program determines the linearized matrices for the generalized formulation of the system A 6x 4 Bt H 6x G 6x F lt lI where of ox 24 p 99 with all derivatives evaluated along the nominal trajectory defined by the analysis point u the state time derivative of state and control vectors can be 0 expressed as small perturbations x 6x about the nominal trajectory so that x x x Ox u u u In a dition to the matrices for this generalized system the user has the option of requesting linearized matrices for a standard formulation of the system x A x H x F where 9 1 Of A T mo pm X 9 1 1 9 B 2 ox d 9 1 29 29 gt g 9 1 96 31 x with all derivatives evaluated along the nominal trajectory defined by the analysis point Xx 7 X yj u 0 LINEAR also provides two nonstandard matrices D and E in the equations x Ax Bu Dv y Hx Fu Ev 25 or D and E in the equations Cx A x B u D v Gx Flu E v y These dynamic interaction matrices include the effect o
84. er e e NONDIMENSIONAL STABILITY AND CONTROL DERIVATIVES INPUT FILES e Case Title File Selection Information and Project Title Geometry and Mass Data e s e e e e State Control and Observation Variable Definitions Trim Parameter Specification e e lt Additional Surface Specification Test Case Specification s e OUTPUT FILES e e USER SUPPLIED SUBROUTINES e e e e Aerodynamic Model e s e e Control Model s e e Engine Model s s s t Mass and Geometry Model s e e e e c iil Page N co I N 12 15 18 23 24 26 28 28 28 29 30 30 31 32 34 37 37 38 41 41 42 42 45 46 49 49 50 CONCLUDING REMARKS e e e e 53 APPENDIX A CORRECTION TO AERODYNAMIC COEFFICIENTS FOR A CENTER OF GRAVITY NOT AT THE AERODYNAMIC REFERENCE POINT lt ss 55 APPENDIX B ENGINE TORQUE AND GYROSCOPIC EFFECTS MODEL e 57 APPENDIX C STATE
85. erating system is included with this report Revisions to the microfiche Supplement are given in app J The program LINEAR numerically determines a linear system model using nonlinear equations of motion and 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 as 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 selecting a trim option and directing the program to deter mine the control variables and remaining state variables The system model determined by LINEAR consists of matrices for both state and observation equations The program has been designed to provide an easy selection and definition of the state control and observation variables to be used in a particular model Thus the order of the system model is completely under user control Further the program provides the flexibility of allowing alternative formulations of both state and observation equations LINEAR has several features that make it unique among the linearization programs common in the aerospace industry The most significant of these features is flexi bility By generalizing the surface definitions and making no assumptions of sym metric mass distributions the program can
86. f external forces and moments acting on the vehicle The components of the dynamic interaction vector v are incre mental body axis forces 6x Sy 62 and moments L M N OX Thus Qf 25 Jd II Of T a 32 ov x 1 CL EMA 3f 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 trajectory a point at which the rotational and translational accelerations 26 are zero Perkins and Hage 1949 Thelander 1965 or a totally arbitrary state ona trajectory LINEAR allows the user to select from a variety of analysis points Within the program these analysis points are referred to as trim conditions and several options are available to the user The arbitrary state and control option is
87. f matrices for both state and observation equations The program has been designed to allow easy selection and definition of the state control and observa tion variables to be used in a particular model INTRODUCTION The program LINEAR was developed at the Dryden Flight Research Facility of NASA s Ames Research Center to provide a standard documented and verified tool to be used in deriving linear models for aircraft stability analysis and control law design This development was undertaken to eliminate the need for aircraft specific linearization programs common in the aerospace industry Also the lack of available documented linearization programs provided a strong motivation for the development of LINEAR in fact the only available documented linearization program that was found in an extensive literature search of the field is that of Kalviste 1980 Linear system models of aircraft dynamics and sensors are an essential part of both vehicle stability analysis and control law design These models define the aircraft system in the neighborhood of an analysis point and are determined by the linearization of the nonlinear equations defining vehicle dynamics and sensors This report describes LINEAR a FORTRAN program that provides the user with a power ful 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
88. formation The exact definition of each of the elements used for a particular aerodynamic model is deter mined by the implementer of that model The structure of the common block CONTRL is as follows COMMON CONTRL DC 30 The common block DATAIN contains geometry and mass information COMMON DATAIN S B CBAR AMSS AIX AIY AIZ AIXZ AIYZ AIXE The first three variables in the common block S B and CBAR represent wing area wingspan and mean aerodynamic chord respectively The vehicle mass is represented by AMSS The information on the displacement of the aerodynamic reference point of the aerodynamic data with respect to the aircraft center of gravity is contained in the CGSHFT common block COMMON CGSHFT DELX DELY DELZ The displacements are defined along the vehicle body axis with DELX DELY and DELZ representing the displacements in the X y and z axes respectively The common block SIMACC contains the accelerations accelerometer outputs and normal accelerometer output at the center of gravity of the aircraft COMMON SIMACC AX AY 2 ANX ANY ANZ AN The output common block CLCOUT contains the variables representing the aerody namic moment and force coefficients COMMON CLCOUT CL CM CN CD CLFT CY 48 The variables CL CM and CN are the symbols for the coefficients of rolling moment Cg pitching moment Cm and yawing moment Cp respectively these terms are body
89. g subroutine IFENGN both defines an engine model and provides the interface to LINEAR provide the detailed thrust engine In normal usage this subroutine would call subroutines that rotation and fuel consumption modeling the information from these subroutines would be transferred into the ENGSTF common block SUBROUTINE IFENGN e EXAMPLE SUBROUTINE TO PROVIDE PROPULSION SYSTEM MODEL e e 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 ON THRUST REQUEST TO ENGINE COMMON CONTRL DC 30 OUTPUT COMMON BLOCK CONTAINING DETAILED INFORMATION ON EACH OF UP TO FOUR SEPARATE ENGINES eee ROUTINE TO COMPUTE ENGINE PARAMETERS THRUST I THRUST CREATED BY EACH ENGINE TLOCAT I J XYANGL 1 XZANGL I TVANXY I TVANXZ I DXTHRS I G gt OO O G G gt O O C EIX I 94 LOCATION OF EACH ENGINE IN THE X Y Z PLANE ANGLE IN X Y BODY AXIS PLANE AT WHICH EACH ENGINE IS MOUNTED ANGLE IN X Z BODY AXIS PLANE AT WHICH EACH ENGINE IS MOUNTED ANGLE IN THE ENGINE AXIS PLANE OF THE THRUST VECTOR ANGLE IN THE X Z ENGINE AXIS PLANE OF THE THRUST VECTOR DISTANCE BETWEEN THE ENGINE C G AND THE THRUST POINT ROTATIONAL INERTIA OF EACH ENGINE C AMSENG I MASS OF EACH ENGINE C
90. h titles appear on each page of the line printer output file file selection record contains the names of the files from which the data are read The data contained on the files specified by the six fields of the file selection record are shown in table 2 The input file names are written in character strings 10 columns long and if not specified the data are assumed to be on the same file as the first title record and the file selection record The local name of the file containing these first two records must be attached to FORTRAN device unit 1 TABLE 2 DEFINITION OF FILES SPECIFIED IN FILE SELECTION RECORD s EY ir J p Data contained on selected file 3ns o M M P P Aa H Field number 1 Project title 2 Geometry and mass data 3 State control and observation variable definitions 4 Trim parameter specifications 5 Test case specification 6 Additional surface definitions Geometry and Mass Data The geometry and mass data file consists of four records that can either fol low the title and file selection records on unit 1 or be stored on a separate file 37 defined by the first of the input file name definitions on the file selection record The geometry and mass data records define the geometry mass properties location of the aerodynamic reference with respect to the center of gravity
91. he x and z body axes inertia about the y body axis inertial coupling between the y and z body axes inertia about the z body axis ALP ALPDOT AMCH AMSENG AMSS B BTA BTADOT CBAR CD CDA CDDE CDO CDSB CL CLB CLDA CLDR CLDT CLEFT CLFTA CLFTAD CLFTDE CLFTO CLFTQ CLFTSB CLP angle of attack time rate of change of angle of attack Mach number total rotor mass of the engine aircraft mass wingspan angle of sideslip time rate of change of angle of sideslip mean aerodynamic chord total coefficient of drag coefficient of drag due to angle of attack coefficient of drag due to symmetric elevator deflection drag coefficient at zero angle of attack coefficient of drag due to speed brake deflection total coefficient of rolling moment coefficient of rolling moment due to angle of sideslip coefficient of rolling moment due to aileron deflection coefficient of rolling moment due to rudder deflection coefficient of rolling moment due to differential elevator deflection total coefficient of lift coefficient of lift due to angle of attack coefficient of lift due to angle of attack rate coefficient of lift due to symmetric elevator deflection lift coefficient at zero angle of attack coefficient of lift due to pitch rate coefficient of lift due to speed brake deflection coefficient of rolling moment due to roll rate CLR CM CMA CMAD CMDE CMO CMQ CM
92. ight The two options available for straight and level trim require the user to specify altitude and either an angle of attack or a Mach number If a specific angle of attack and altitude combination is desired the user selects the Mach trim option which determines the velocity required for the requested trajectory Likewise the alpha trim option allows the user to specify Mach number and alti tude and the trim routines in LINEAR determine the angle of attack needed for the requested trajectory The trim condition for both straight and level options are determined within the following constraints 0 0 trim surface positions thrust angle of sideslip and either velocity or angle of attack are determined by numerically solving the nonlinear equations for the translational and rotational acceleration Pitch attitude 9 is determined by itera tive solution of the altitude rate equation Pushover Pullup The pushover pullup analysis point definition options result in wings level flight at n 1 For gt 1 the analysis point is the minimum altitude point of a pullup when h 0 For n lt 1 this trim results in a pushover at the maximum alti tude 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 deter mined from the specified altitude Mach number and load factor and 2 the load 28 factor trim
93. le Accelerations continued 34 z body axis accelerometer not at vehicle center of gravity 99 Normal accelerometer not at vehicle center of gravity 98 Load factor Air data parameters 91 Speed of sound 37 Reynolds number 103 Reynolds number per unit length 35 Mach number 36 Dynamic pressure 56 Impact pressure 55 Ambient pressure 57 Impact ambient pressure ratio 58 Total pressure 59 Temperature 60 Total temperature 92 Equivalent airspeed 93 Calibrated airspeed a oe ey ee A Flightpath related parameters Se a a et 39 Flightpath angle 38 Flightpath acceleration 40 Flightpath angle rate 43 Scaled altitude rate Energy related terms 46 Specific power 47 Specific energy Force parameters 94 Lift force 95 Drag force 96 Normal force 97 Axial force Body axis parameters 52 X body axis velocity 53 y body axis velocity 54 z body axis velocity 52 TABLE 4 Concluded h e T a M FH M M HU Location index in OBVEC Variable Body axis parameters continued 100 Rate of change of velocity in x body axis 101 Rate of change of velocity in y body axis 102 Rate of change of velocity in z body axis Miscellaneous measurements not at vehicle center of gravity a M t n 44 Angle of attack not at vehicle center of gravit
94. local x axis rad n angle from to the engine x y plane rad 8 pitch angle rad u coefficient of viscosity angle from the projection of onto the body axis plane to the x body axis rad p density of air mass length LL total body axis rolling moment length force total body axis pitching moment length force LN total body axis yawing moment length force T torque from engines length force Tg gyroscopic torque from engine length force roll angle rad tilt angle of acceleration normal to the flightpath from the vertical plane rad V heading angle rad total rotational velocity of the vehicle engine angular velocity rad sec Superscripts i nondimensional version of variable derivative with respect to time T transpose of a vector or matrix Subscripts ar aerodynamic reference point D total drag E engine max min altitude total lift rolling moment Mach number pitching moment maximum minimum yawing moment offset from center of gravity roll rate pitch rate yaw rate Stability axis thrust point along the x body axis sideforce along the y body axis along the z body axis Standard day sea level conditions or along the reference trajectory FORTRAN Variables AIX AIXE AIXZ AIY AIYZ AIZ inertia about the x body axis engine inertia inertial coupling between the x and y body axes inertial coupling between t
95. lt 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 the acceleration normal to the flightpath from the vertical plane is calculated using the equation bebe cos Y where the positive sign is used for a right turn and the negative sign is used for a left turn From turn rate can be calculated as V tan 91 29 Using these two definitions the state variables can be determined sin2 Y cos B is sin Q II sin sin Y sin sin Y sin S tan cos VU sin Y r ps Q rg cos a 1 tan7l q r The trim surface positions thrust angle of sideslip and either angle of attack or load factor are determined by numerically solving the nonlinear equations for the translational and rotational accelerations Thrust Stabilized Turn The thrust stabilized turn analysis point definition results constant throttle non wings level turn with a nonzero altitude rate two options available in LINEAR are alpha trim and load factor trim These options allow the user to specify either the angle of attack or the load factor for the analysis point The altitude and M
96. mmasch et al 1967 Etkin 1972 Gainer and Hoffman 1972 Gracy 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 and are derived directly from the body axis forces defined in the previous section for translational acceleration The equations used in LINEAR for the body axis accelerations ay ay and ag are ax Xp D cos L gm sin 8 g qm ay Y gm cos 9 sin 9 g om az D sin L cos gm cos 9 cos 9 g om where subscript O denotes standard day sea level conditions The equations for the outputs of the body axis accelerometers denoted by subscript n that are at vehicle center of gravity are anx Xp D cos L sin a gopm any Y ggm anz Zp D sin L cos a ggm an Zp D sin Q L cos a g For orthogonal accelerometers that are aligned with the vehicle body axes but are not at vehicle center of gravity denoted by subscript i the following equations apply anx i anx q2 r2 xx pq r yx pr q zx 90 any i any r xy p2 r2 yy qr p zyl gp anz i anz pr
97. n example analysis file output on unit 15 This file was produced using the example input file listed in appendix F and the example user supplied subroutines listed in appendix I LINEARIZER TEST AND DEMONSTRATION CASES USER S GUIDE TEST CASE kkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk X DIMENSION 4 U DIMENSION 3 Y DIMENSION 2 STATE EQUATION FORMULATION STANDARD OBSERVATION EQUATION FORMULATION STANDARD STATE VARIABLES ALPHA 0 4656950 01 RADIANS Q 0 9216830 01 RADIANS SECOND THETA 0 1598850 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 Y BODY AXIS FORCE Z BODY AXIS FORCE PITCHING MOMENT ROLLING MOMENT YAWING MOMENT 0 0000000 00 POUNDS 0 000000D 00 POUNDS 0 000000D 00 POUNDS 0 0000000 00 FOOT POUNDS 0 0000000 00 FOOT POUNDS 0 000000D 00 FOOT POUNDS OBSERVATION VARIABLES AN AY 0 3001630401 GS 0 9414350 00 65 A MATRIX FOR DX DT A X B U D V 0 1214360 01 0 1000000 01 0 1367560 02 0 1216050 03 0 1474230401 0 221451D 01 0 450462D 02 0 294019 03 0 0000000 00 0 3318120400 0 000000D 00 0 000000D 00 0 7908530402 0 000000D 00 0 3208220 02 0 1572970 01 B MATRIX FOR DX DT A X B U D V 0 1419610 00 0 1649480 02 0 92 8933D 02 0 2207780 02 0 543324D 02 0 1350740 02 0 000000D 00 0 000000D 00 0
98. n results ina level turn at a user specified Mach number altitude thrust trim parameter and specific power Unlike the other trim options provided in LINEAR the specific power option does not in general attempt to achieve velocity rate 0 In fact because the altitude rate h 0 and specific power is defined by the resultant velocity rate will be e e L e However the other acceleration like terms p q and B 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 ttan n pue where the positive sign is used for a right turn and the negative sign is used for a left turn 122223 V tan q sin cos q tan cosB ps tan 8 p ps cos rs Sin 31 r pg Sin rg cos a sin 2 tanc n 1 The analysis point surface positions load factor angle of attack and angle of 1 are determined by numerically solving the nonlinear equations for transla tional and rotational accelerations 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
99. nalysis 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 DEFINI TION SUBOPTION will be read only if the requested analysis point definition option has a suboption associated with it These suboptions are defined in the Analysis Point Definition section The valid alphanumeric descriptors for these suboptions are described in appendix E The remaining records in a test case specification set define test conditions or initial conditions for the trimming subroutines These records consist of a field defining a parameter name VARIABLE and its initial condition VALUE These records may be in any order however if initial Mach number is to be defined the initial altitude must be specified before Mach number if the correct initial veloc ity is to be determined The parameter names are checked against all name lists used within LINEAR Any recognized state time derivative of state control or observation variable will be accepted Trim parameters also can be set in these records In general setting observation variables and time derivatives of the state variables has little effect However for some of the trim options defined in the Analysis Point Definition section Mach number and load factor are used The thrust trim parameter only affects the specific power trim For the untrimmed option the initial values of the state
100. ngle of attack range of the aerodynamic model Record 8 specifies that there will be four state variables in the output that the output formulation of the state equation will be in the standard form x AX Bu and that the output for the nondimensional stability and control derivatives with respect to angle of attack and angle of sideslip should be scaled as reciprocal radians The next four records 9 to 12 define the output formulation of the state vector to be lt DA Record 13 specifies that the output model will have three parameters in the con trol vector The following three records 14 to 16 specify that elevator u throttle speed brake and that elevator throttle and speed brake are located in DC 5 DC 12 and DC 10 of the CONTROL common block 76 Record 17 specifies that two observation variables will be output and that the observation equation will be in the standard form y Hx Fu The next two records 18 and 19 define the elements of the output vector to be a 3 Y Record 20 specifies the ranges for the trim parameters DES DAS DRS and THRSTX used to trim the longitudinal lateral and directional axes and thrust respectively The ranges for these parameters are defined by record 20 to be 2 9 lt DES 8 5 43 4 0 S DAS amp 4 0 3 25 lt DRS amp 3 25 1 0 S THRSTX 4 1 0 The first three parameters essentially represent stick and rudder limits and are so specified bec
101. nt sustainable flight conditions In fact only in the special case where the flightpath angle is zero does this occur for these options As previously stated the linearization of a nonlinear model and its 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 27 tors such as trim and sustainable flight conditions and ultimately by accuracy requirements placed on the representation Thus in using the linear models pro vided by this program the user should exercise some caution Untrimmed For the untrimmed option the user specifies all state and control variables that are to be set at some value other than zero The number of state variables specified is entirely at the user s discretion If any of the control variables are to be nonzero the user must specify the control parameter and its value The untrimmed option allows the user to analyze the vehicle dynamics at any flight con dition including transitory conditions Straight and Level Trim The straight and level trims available in LINEAR are in fact wings level constant flightpath angle trims Both options available for straight and level trim allow the user to specify either a flightpath angle or an altitude rate However since the default value for these terms is zero the default for both types of straight and level trim is wings level horizontal fl
102. 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 C 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 can be specified using the DRVINC variable The units are the units of the state variable or its time derivative except for derivatives with respect to velocity When DRVINC is specified for velocity DRVINC specifies a Mach number increment If DRVINC is not specified by the user the default value of 0 001 is used The next set of records in the state control and observation variable defini tions are those defining the variables to be used in the control vector of the out put 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 Subroutines 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 39 Because LINEAR has no default control variable names the control variable names input by
103. ormulation specified by the user The selection of specific state control and observation variables for the for mulation of the output matrices is accomplished by alphanumeric descriptors in the input file The use of these alphanumeric descriptors is described in the Input Files section Appendix C lists the state variables and their alphanumeric descrip tors Appendix D lists the observation variables and their alphanumeric descriptors The alphanumeric descriptors for the selection of control parameters to be included 23 in the observation vector are the control variable names defined by the user in the input file as described in the Input Files section LINEAR MODELS The linearized system matrices computed by LINEAR are the first order terms of a Taylor series expansion about the analysis point Dieudonne 1978 Kwakernaak and Sivan 1972 proposed NASA RP by Duke Antoniewicz and Krambeer in preparation 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 employed to obtain these matrices numerically is a simple approximation to the par tial derivative that is f x Ax f x Ax 9 2 where f is general function of x an arbitrary independent variable The Ax may be set by the user but it defaults to 0 001 for all state and control parameters with the single exception of velocity V where Ax is multipli
104. outines and passing the new moment arm values through the OBSERV common block COMMON OBSERV OBVEC 120 PARAM 120 6 The common block OBSERV allows the user to access all the observation vari ables during trim as well as to pass parameters associated with the observations back to LINEAR The common block OBSERV contains two single precision vectors OBVEC 120 and PARAM 120 6 A list of the available observations and parameters is given in table 4 Access to the observation variables allows the user to imple ment trim strategies that are functions of observations such as gain schedules and surface management schemes The parameters associated with the observations are used primarily to define the moment arm from the center of gravity to the point at which the observation is to be made If the user subroutine MASGEO is used to change the center of gravity location and observations are desired at fixed loca tions on the vehicle then the moment arm from the new center of gravity location to the fixed sensor location x y Z in feet must be computed in one of the user subroutines and passed back in the first three elements of the PARAM vector asso ciated with the desired observation PARAM n 1 to 3 where n is the number of the desired observation Additional information on observations and parameters can be found in the State Control and Observation Variable Definitions section 50 TABLE 4 OBSERVATION VARIABLES AVAILAB
105. parated into seven major sec tions case title and file selection information project title geometry and mass data for the aircraft state control and observation variable definitions for the state space model trim parameter specification additional control surfaces that may be specified for each case and various test case specifications All the input data can be written on one file or various files according to the divisions specified above and according to the input format defined in table 1 An example input file is listed in appendix F All the input records to LINEAR are written in ASCII form TABLE 1 INPUT FORMAT FOR LINEAR MM HI P eR lr cin mm M a Input record Format M r H H F Title and flle selection information Irs t M d m II J U M Case title 20A4 Input file names 6A10 Project title N PP F sn n n A J U l U S l im Project title 20A4 as nw n n lt _ a S b C Weight 4 13 0 Ix Iy Ig Ixz Ixy Iyz 6F13 0 DELX DELY DELZ LOGCG 3F10 0 12A4 Amin max 2F13 0 ss hr N a MR NN HR 34 a n LE
106. r Mar expressions for total aerodynamic moment coefficients corrected to the vehicle center of gravity can be derived as follows Ay Az EN sin a cos 5 CY 2 Cm Cmar 2 Cp cos sin Cp sin Cy cos A Ch Cnar p CY Cp cos Q Cy sin These calculations are normally performed within LINEAR in the subroutine CGCALC However if the user selects the calculation can be performed within the user supplied aerodynamic model CCALC 56 APPENDIX B ENGINE TORQUE AND GYROSCOPIC EFFECTS MODEL Torque and gyroscopic effects represent after thrust the main contributions of the engines to the aircraft dynamics The torque effects arise due to thrust vectors not acting at the vehicle center of gravity The gyroscopic effects are a consequence of the interaction of the rotating mass of the engine and the vehicle dynamics These effects can be either major or virtually negligible depending on the vehicle The torque effects can be modeled by considering the 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 as shown in figure 4 Center of Figure 4 Definition of location of engine center of mass relative to vehicle center of gravity The thrust vector for the ith engine Pp can be defined as gt T
107. rameter name is recognized as a valid observation variable name that observation variable is included in the formulation of the output observation vector If the param eter name is not recognized an error message is printed and the parameter named is ignored The three variables represented by PARAM 1 PARAM 2 and PARAM 3 provide the x axis y axis and z axis locations respectively of the measurement with respect to the vehicle center of gravity if the selected observation is one of the following ae Pe OQ i Pi The unit associated with these variables is length If the selected observation variable is not in the preceding list the PARAM variables are not used The sole exception to this occurs when Reynolds number is requested as an observation vari able In that case PARAM 1 is used to specify the characteristic length When no value is input for PARAM 1 the mean aerodynamic chord c is used as the char acteristic 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 e and Sth representing the longitudinal lateral directional and thrust trim parame ters respectively The units associated with these parameters are defined by the implementation of UCNTRL Additional Surface Specification Th
108. scopic contributions to the total moments are derived in appendix body axis moments and products of inertia are designated I Iy Ig and These moments and products of inertia are elements of the inertia tensor I defined as Ixy Iyz I To derlve the state equation matrices for the qeneralized formulation Cx Ax Bu where A and B are the state and control matrices of the state equation the rota tional accelerations are cast in a decoupled axes formulation The equations used to derive the linearized matrices are p Ix Ix I gt I _ XZ yz 1 r Iz Iz LL Ixy Ixz Iy 2 gt Ig rp rq we 26 gr e Ix p Ix pq EM Ix Ixy Iyz 2 2 ixz I clue rp Z t rq x Pq t r Io Rr y Y y Y Y 28 12 02 uu 2 aka EU sa Iz Iz gt I 2 iz Iz The translational acceleration equations used in the program LINEAR for both analysis point definition and perturbation are 17 ce D cos B Y sin Xp cos a cos sin Zp sin a cos mg sin cos cos cos 8 sin sin cos sin a cos 8 m a L 2 cos Xp sin mg cos 0 cos cos a sin sin a Vm cos f q tan p cos r sin Whe il D sin Y cos Xp cos sin cos Zp sin sin mg sin 9 cos sin
109. sfer of data into the subroutine CCALC is through six named common blocks These common blocks contain the state variables air data parameters and surface positions The transfer of data from CCALC is through a named common block containing the aerodynamic force and moment coefficients The details of these 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 as follows COMMON DRVOUT T P Q R V ALP BTA THA PSI PHI H X Y TDOT PDOT QDOT RDOT VDOT ALPDOT BTADOT THADOT PSIDOT PHIDOT HDOT XDOT YDOT The body axis rates p q and r appear as P Q9 and R respectively Total veloc ity is represented by the variable V 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 function generation subroutine The variables in this common block are 47 COMMON SIMOUT AMCH QBAR GMA DEL UB VB WB VEAS VCAS Mach number and dynamic pressure are the first 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 The CONTRL common block contains the surface position in
110. tatement Unclassified Unlimited 17 Key Words Suggested by Author s Aircraft model Computer program Control law design Linearization Subject category 66 22 Price 06 21 of Pages 108 20 Security Classif of this page Unclassified 19 Security Classif of this report Unclassified For sale by the National Technical Information Service Springfield Virginia 22161 NASA Langley 1987
111. the user are used for subsequent identification of the control variables Therefore consistency in the use of control variable names is extremely important 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 con trol 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 are assumed if CONVR does not agree with any of the listed names When it is assumed that the control variable is specified in units of radians the initial value of the control variable is converted to degrees before being written to the printer file The variable CNTINC can be used to specify the increments used for a particular control surface when the B and F matrices are being calculated It is assumed that the units of CNTINC agree with those used for the surface and no unit conver Sion is attempted on these increments If CNTINC is not specified for a particular surface a default value of 0 001 is us
112. ties of the Helicopter in Coordinated Turns NASA TP 1773 1981 Clancy L J Aerodynamics John Wiley amp Sons New York 1975 Dieudonne James E Description of a Computer Program and Numerical Techniques for Developing Linear Perturbation Models From Nonlinear Systems Simulations NASA TM 78710 1978 Dommasch Daniel O Sherby Sydney S and Connolly Thomas F Airplane Aerodynamics Fourth edition Pitman New York 1967 Etkin Bernard Dynamics of Atmospheric Flight John Wiley amp Sons New York 1972 Gainer Thomas G and Hoffman Sherwood Summary of Transformation Equations and Equations 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 Corpora tion Aircraft Division Hawthorne California Nov 1980 Kwakernaak Huibert and Sivan Raphael Linear Optimal Control Systems Wiley Interscience New York 1972 Perkins Courtland D and Hage Robert E Airplane Performance Stability and Control John Wiley amp Sons New York 1949 Thelander J A Aircraft Motion Analysis FDL TDR 64 70 Air Force Flight Dynamics Laboratory Mar 1965 U S Standard Atmosphere 1962 U S Government Printing Office 1962 104 1 Report No 2 Government Accession No 3 Recipient s Catalog No NASA TP 2768 4 Title and Sub
113. title 5 Report Date December 1987 User s Manual for LINEAR a FORTRAN Program 6 Performing Organization Code to Derive Linear Aircraft Models 8 Performing Organization Report No H 1259 Author s Eugene L Duke Brian P Patterson and Robert Antoniewicz 10 Work Unit No RTOP 505 66 11 9 Performing Organization Name and Address NASA Ames Research Center Dryden Flight Research Facility P O Box 273 Edwards CA 93523 5000 11 Contract or Grant No 13 Type of Report and Period Covered Technical Paper Sponsoring Agency Name and Address National Aeronautics and Space Administration Washington DC 20546 14 Sponsoring Agency Code Supplementary Notes A listing of the program LINEAR is provided in the microfiche supplement included with this report 3 sheets total Abstract This report documents a FORTRAN program that provides a powerful and flexible tool for the linearization of aircraft models The program LINEAR numerically determines a linear system model using nonlinear equations of motion and a user supplied nonlinear aerodynamic model The system model determined by LINEAR consists of matrices for both 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 18 Distribution S
114. trol vector total velocity length sec calibrated airspeed knots equivalent airspeed knots velocity in y axis direction length sec dynamic interaction vector vehicle weight force velocity in z axis direction length sec total force along the x body axis force thrust along the x body axis force state vector 61 OM N x 627 sideforce force thrust along the y body axis force observation vector total force along the z body axis force thrust along the z body axis force angle of attack rad angle of sideslip rad flightpath angle rad displacement of aerodynamic reference point from center of gravity displacement from center of gravity along x body axis length displacement from center of gravity along y body axis length displacement from center of gravity along z body axis length lateral trim parameter differential stabilator trim parameter longitudinal trim parameter Kronecker delta directional trim parameter speed brake trim parameter thrust trim parameter incremental rolling moment length force incremental pitching moment length force or incremental Mach incremental yawing moment length force incremental x body axis force force incremental y body axis force force incremental z body axis force force angle from the thrust axis of engine to the x y body axis plane rad C angle from the projection of Fp onto the engine x y plane to the
115. two formulations corre sponding to the standard equation y Hx Fu or the generalized equation H x Gx Flu 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 Nondimensional Stability and Control Derivatives section The input file for LINEAR is an ASCII file that defines the geometry and mass properties of the aircraft and selects various program options Within this input file the state control and observation vectors desired in the output linear model are defined and the analysis point options are selected The details of the input file are discussed in the Input Files section The output of LINEAR is three files one containing the linear System matrices and two documenting the options and analysis points selected by the user The first is intended to be used with follow on design and analysis programs The other two contain all the information contained in the first file and also include the details of the analysis point and the nondimensional stability and control derivatives These files are described in the Output Files section To execute LINEAR five user supplied subroutines are required These routines discussed in the User Supplied Subroutines section define the nonlinear aerodynamic model the gross engine model the gearing between the LINEAR trim inputs and the surfaces modeled
116. with ease This report documents the use of the program LINEAR defining the equations used and the methods employed to implement the program The trimming capabilities of LINEAR are discussed from both a theoretical and an implementation perspective The input and output files are described in detail The user supplied subroutines required for LINEAR are discussed and sample subroutines are presented NOMENCLATURE The units associated with the listed variables are expressed in a generalized system given in parentheses LINEAR will work equally well with any consistent set of units with two notable exceptions the printed output and the atmospheric model Both the printed output and the atmospheric model assume English units Where applicable quantities are defined with respect to the body axis system Variables A state matrix of the state equation x Ax Bu or axial force force state matrix of the state equation Cx A x B u a speed of sound in air length sec an normal acceleration g normal acceleration accelerometer not at center of gravity 4 x axis accelerometer output accelerometer at center of gravity 4 x body axis accelerometer output accelerometer not at center of gravity g any y body axis accelerometer output accelerometer at center of gravity q body axis accelerometer output accelerometer not at center of gravity q Aang z body axis accelerometer
117. x name D matrix Matrix name C matrix if general form chosen Matrix name H matrix s O A Format a A G s gx x n s 4A20 4A20 64 13 17 12 22 12 22 13 36 2 4 36 2 4 1 5 4 3 12 6 2 20 7 77 1 5 4 3 12 6 2 20 1X 5A4 17X A20 1 5 4 3 12 6 2 20 8 5 13 6 A8 5 E13 6 8 5 13 6 8 5 13 6 8 5 13 6 Y 43 TABLE 3 Concluded Variable Format A8 F matrix 5 13 6 Matrix name 8 E matrix 5 13 6 Matrix name A8 G matrix if general form chosen 5 13 6 The titles are written on the first two records of the file in 80 character strings and are specified in LINEAR as the title of the vehicle and the title for the cases The next record contains the number of the case as defined in LINEAR 999 cases maximum The number of states controls and outputs used to define each case are written on the following record The formulation of the state and observation equations are listed next followed by the names and values of the States controls dynamic interaction variables and outputs These values are followe
118. y 45 Angle of sideslip not at vehicle center of gravity 41 Altitude instrument not at vehicle center of gravity 42 Altitude rate instrument not at vehicle center of gravity M M P Other miscellaneous parameters 48 Vehicle total angular momentum 49 Stability axis roll rate 50 Stability axis pitch rate 51 Stability axis yaw rate a Control surface parameters I 61 to 90 Control surfaces DC 1 to DC 30 A Wa T Trim parameters T E D i i P 104 Longitudinal trim parameter 105 Lateral trim parameter 106 Directional trim parameter 107 Thrust trim parameter M i ll L I I II 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 LINEAR from the perspective of a potential user defining the nonlinear equations from which the linear model is derived and describ 53 ing the interfaces to user supplied subroutines and input files The output from LINEAR is also described Examples of the user supplied subroutines are presented in the appendixes A microfiche listing of the program for a VAX 11 750 with the VMS op

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